5 GHZ CHANNEL CHARACTERIZATION FOR AIRPORT SURFACE AREAS AND
VEHICLE-VEHICLE COMMUNICATION SYSTEMS
A dissertation presented to
the faculty of
the Russ College of Engineering and Technology of Ohio University
In partial fulfillment
of the requirements for the degree
Doctor of Philosophy
Indranil Sen
August 2007
This dissertation titled
5 GHZ CHANNEL CHARACTERIZATION FOR AIRPORT SURFACE AREAS AND
VEHICLE-VEHICLE COMMUNICATION SYSTEMS
by
INDRANIL SEN
has been approved for
the School Electrical Engineering Computer Science
and the Russ College of Engineering and Technology by
David W. Matolak
Associate Professor, School of Electrical Engineering and Computer Science
Dennis Irwin
Dean, College of Engineering and Technology
Abstract
SEN, INDRANIL, Ph.D., August 2007, Electrical Engineering
5 GHZ CHANNEL CHARACTERIZATION FOR AIRPORT SURFACE AREAS AND
VEHICLE-VEHICLE COMMUNICATION SYSTEMS (416 pp.)
Director of Dissertation: David W. Matolak
We empirically characterize the 5 GHz channel for airport surface (AS) area and vehicle to vehicle (VTV) communication systems. The characterization consists of stochastic models for the channel impulse response, which focus on small-scale, and
“medium,” or “meso-“scale effects. Motivation is provided by reviewing the growth in civil aviation and VTV communications, and by describing the utility of the 5 GHz band for these new communication systems. Further motivation arose from our literature survey, which revealed a pressing need for wideband stochastic channel models for these new applications in this band. Data measurement campaigns and environment descriptions are provided. For both the AS and VTV settings, classification schemes are developed. These schemes allow grouping AS and VTV environments into classes, and these classes are further divided into propagation regions, for which channel characteristics are statistically similar. A pre-processing framework to extract the most pertinent channel information from the measured data was developed. A propagation path loss model was also developed for the large airport class. Based upon our measured data, we deduced some unique propagation effects: severe, or “worse than Rayleigh,” fading, correlated scattering, and statistical non-stationarity (NS). To explain the severe fading phenomenon, we present two physical models that yield results in agreement with our measured data. For each propagation region of the AS and VTV classes, three
different small scale fading models (denoted M1, M2, and M3) were developed. These
models are applicable to different values of channel bandwidth, allow tradeoffs between the model’s implementation complexity and fidelity, and allow for the model user’s incorporation of statistical non-stationarity. Channel non-stationarity was modeled using
two different random processes: the multipath component persistence process (for AS
and VTV) models the finite lifetime associated with a multipath component, whereas the
region persistence process (only AS) emulates the transition of the receiver from one
propagation region to another. Each of these processes is based on a first-order Markov
chain. The channel models were implemented in software using a new correlated
multivariate Weibull random process generator. The model outputs were compared with the actual data using both time and frequency domain measures. Our NS model yields best agreement with the data, for all cases. We also present channel models for the cases when the AS transmitter is located at an airport field site.
Approved: ______
David W. Matolak
Associate Professor, School of Electrical Engineering and Computer Science
Acknowledgments
There are quite a few people that I would like to thank for my wonderful
experience as graduate student at Ohio University. First of all, I would like to express my
sincere appreciation and gratitude for my advisor Dr. David Matolak. His in-depth
knowledge of wireless communication and excellent suggestions have been an invaluable asset for me throughout my research. I really appreciate the fact that inspite of his busy schedule; he always found time to discuss my trivial questions and provided me insights on solving the issue on hand.
Next, I would like to express my appreciation to my committee members, Dr.
Jeffrey Dill, Dr. Frank Van Graas, Dr. Trent Skidmore, Dr. Dinh Van Huynh and Dr.
Sergio Lopez. All of them have been extremely helpful throughout the course of this dissertation and I am grateful to them for that. I would also like to extend my sincere gratitude towards Dr. Roger Radcliff who believed in me and provided me with financial assistance through out my time as a PhD student. I would also like to thank Tammy
Jordan for all her help during my stay as a graduate student with the electrical engineering department. I would also like to thank Dr. Joseph Essman for being on my written and oral comprehensive exams. Next, I would like to acknowledge Rafael D.
Apaza of the FAA Aviation Research Office, Brian Kachmar of Analex Corporation and the airport facilities staff at MIA, JFK, and CLE for their support while gathering measurements for this project.
During my stay at Athens, I made quite a few friends and they all have contributed in creating a wonderful experience for me over the past few years. I would like to thank Sumit Bhattacharya, Kamal Ganti and Irina Anitei for being wonderful
support structures and also for the numerous “Grilling Sessions” which helped me relax
after a week of intense research. Next, I would like to thank my lab-mates Wenhui
Xiong, Beibei Wang, Nicholas Yaskoff and Jingtao Zhang for the numerous discussions regarding research, job hunt, politics, etc. They definitely provided a very friendly atmosphere in the MMCL lab and I will always miss those times and stimulating discussions. A special note of thanks for Wenhui and Nick who helped me collect measurements at the various airports.
Lastly, I would like to thank my parents, sister, brother-in-law, fiancé and two
wonderful nephews. Their support and love helped me believe in myself and concentrate on my dissertation. I am grateful to them for being a part of my life and I would like them to know that my dissertation is dedicated to all of them because without them it
would have never been possible.
7
Table of Contents
Page
ABSTRACT...... 3
ACKNOWLEDGMENTS ...... 5
LIST OF TABLES ...... 11
LIST OF FIGURES ...... 14
LIST OF ACRONYMS AND OF ABBREVIATIONS...... 19
LIST OF SYMBOLS ...... 25
1 INTRODUCTION...... 30
1.1 INTRODUCTION TO WIRELESS CHANNELS...... 30
1.2 AIRPORT SURFACE COMMUNICATIONS APPLICATIONS IN THE 5 GHZ BAND...... 34
1.3 INTELLIGENT TRANSPORTATION SYSTEMS...... 36
1.4 DISSERTATION OBJECTIVES ...... 38
1.5 DISSERTATION CONTRIBUTIONS ...... 41
2 LITERATURE REVIEW ...... 44
2.1 INTRODUCTION...... 44
2.2 GENERIC CHANNEL MODELING ...... 45
2.3 MEASUREMENT AND DATA PROCESSING TECHNIQUES ...... 48
2.4 AIRPORT SURFACE AREA CHANNEL MODELS ...... 49
2.5 VEHICLE TO VEHICLE CHANNEL MODELS...... 55
2.6 SEVERE FADING ...... 59
2.7 WEIBULL FADING PROCESS...... 62
2.8 NON STATIONARY CHANNEL MODELS...... 64
3 MEASUREMENT CAMPAIGNS...... 73
3.1 EQUIPMENT DESCRIPTION...... 73 3.1.1 Channel Sounder ...... 73 3.1.2 Antennas and Miscellaneous Equipment...... 81 3.2 AIRPORT MEASUREMENTS ...... 84 3.2.1 Measurement Procedure ...... 84 8
3.2.2 Large Airport Description ...... 88 3.2.3 Medium Airport Description ...... 91 3.2.4 Small Airport Description...... 92 3.2.5 Measured Data Summary...... 94 3.3 VTV MEASUREMENTS ...... 96 3.3.1 Measurement Procedure ...... 96 3.3.2 VTV Region Description...... 98
4 EXTRACTION OF PARAMETERS FOR CHANNEL MODEL DEVELOPMENT ...... 101
4.1 WIDE SENSE STATIONARY UNCORRELATED SCATTERING (WSSUS) CHANNELS...... 101
4.2 DATA PREPROCESSING...... 106 4.2.1 Channel Sounder Errors ...... 106 4.2.2 Channel Sounder Calibration...... 111 4.2.3 Data Format Translation ...... 113 4.2.4 Note for 50MHz PDPs ...... 117 4.2.5 Noise Thresholding...... 119 4.2.6 Converting PowerRecords to PDPs for Different Bandwidths ...... 122 4.2.7 Multipath Threshold...... 124 4.3 KEY PARAMETERS AND DEFINITIONS ...... 128 4.3.1 Parameters Obtained Directly from the IREs ...... 129 4.3.2 Parameters Obtained via Fourier Transform of IREs ...... 137 4.3.3 Labeling Conventions...... 141 4.4 PROCESSING CONSIDERATIONS IN MODEL DEVELOPMENT...... 141 4.4.1 Determination of the Number of Taps (L)...... 142 4.4.2 Markov Modeling: Transitions between Regions and Persistence Processes...... 146 4.4.3 Tap Energies ...... 151 4.4.4 Tap Correlation Matrix ...... 155 4.5 DIFFERENT MODELS FOR AIRPORT SURFACE AREAS AND VTV SETTINGS...... 163 4.5.1 Airport Surface Area ...... 163 4.5.2 VTV Setting ...... 166
5. SEVERE FADING...... 168
5.1 INTRODUCTION...... 168
5.2 MULTIPLICATIVE MODEL ...... 171
5.3 SWITCHING MODEL ...... 177
6. AIRPORT SURFACE AREA CHANNEL MODELS...... 185
6.1 COMMON STATISTICAL PARAMETERS ...... 185 6.1.1 Time (Delay) Domain Statistics ...... 185 9
6.1.2 Frequency Domain Statistics ...... 192 6.1.3 Channel Tap Properties...... 195 6.2 LARGE AIRPORT MODELS ...... 201 6.2.1 Path Loss Model for Large Airports ...... 202 6.2.2 M1 50 MHz models for Large Airport...... 204 6.2.3 [M1 M2 M3, Large Airport, 25] for all Regions ...... 212 6.3 MEDIUM AIRPORT MODELS...... 220 6.3.1 M1 50 MHz Models for Medium Airports...... 220 6.3.2 [M1 M2 M3, Medium Airport, 25] for all Regions ...... 227 6.4 SMALL AIRPORT MODELS ...... 237 6.4.1 M1 50 MHz Small Airport Models...... 237 6.4.2 [M1 M2 M3, Small Airport, 25] for all Regions...... 242 6.5 COMPARISON OF NONSTATIONARY AND STATIONARY MODELS FOR DIFFERENT AIRPORTS ...... 251 6.5.1 Simulation Procedure to Develop Nonstationary and Stationary Models...... 251 6.5.2 Comparing Nonstationary and Stationary Large Airport Models ...... 257 6.5.3 Comparing Nonstationary and Stationary Medium Airport Models ...... 261 6.5.4 Comparing Nonstationary and Stationary Small Airport Models ...... 265
7 AIRPORT FIELD SITE AND POINT TO POINT CHANNEL MODELS.... 269
7.1 EXAMPLE AFS AND POINT-TO-POINT MEASUREMENT LOCATIONS ...... 269
7.2 COMPARISON OF MEASUREMENTS FOR AFS AND ATCT TRANSMISSION ...... 277
7.3 AFS AND PARTIAL ATCT CHANNEL MODELS FOR MIA...... 280
7.4 AFS CHANNEL MODELS FOR JFK ...... 288
7.5 POINT-TO-POINT RESULTS FOR CLE AND MIA...... 291
8 VTV CHANNELS...... 298
8.1 COMMON STATISTICAL PARAMETERS ...... 298 8.1.1 Time and Frequency Domain Statistics...... 298 8.1.2 Channel Tap Properties...... 303 8.2 VTV CHANNEL MODELS...... 309 8.2.1 [M1, All regions-VTV, 5 and 10] ...... 309 8.2.2 [M2 and M3, All VTV Regions-VTV, 10]...... 313 8.2.3 [M2 and M3, UOC and OHT, 5]...... 318 8.3 COMPARISON OF NONSTATIONARY AND STATIONARY MODELS FOR UOC AND OHT...... 320 8.3.1 Simulation Procedure to Develop Nonstationary and Stationary Models...... 320 8.3.2 Comparing Nonstationary and Stationary Models for UOC and OHT...... 325
9 SUMMARY, CONCLUSIONS AND FUTURE WORK...... 333
9.1 SUMMARY AND CONCLUSIONS...... 333 10
9.1.1 Airport Surface Area Channel Characterization...... 333 9.1.2 Vehicle to Vehicle Channel Characterization...... 342 9.2 FUTURE WORK...... 345 9.2.1 Applications of the Developed Channel Models ...... 345 9.2.2. Analytical Extensions to Current Work...... 347 9.2.3 New Non-Stationary Channel Models...... 347 9.2.4 Using Developed Framework to Characterize Other Environments ...... 349
REFERENCES...... 352
APPENDIX A: GENERATION OF MULTIPLE CORRELATED WEIBULL RANDOM VARIATES ...... 359
A.1 RELATIONSHIP BETWEEN CORRELATED RAYLEIGH AND CORRELATED WEIBULL RANDOM VARIABLES ...... 359
A.2 NUMERICAL EXAMPLE ...... 364
A.3 NOTE ON IMPLEMENTING AN LPF FOR DOPPLER SPECTRA...... 366
APPENDIX B: ANTENNA RADIATION PATTERNS ...... 370
APPENDIX C: [M1 M2, ALL AIRPORTS, ALL REGIONS, 5 10 20]...... 373
C.1 [M1 M2, LARGE AIRPORT, ALL REGIONS, 5 10 20]...... 373
C.2 [M1 M2, MEDIUM AIRPORT, ALL REGIONS, 5 10 20]...... 386
C.3 [M1 M2, SMALL AIRPORT, ALL REGIONS, 5 10 20] ...... 398
APPENDIX D: AFS MODELS FOR [M1, MIA JFK, ALL REGIONS, 20] ...... 412
D.1 [M1, MIA, ALL REGIONS, 20] ...... 412
D.2 [M1, JFK, ALL REGIONS, 20] ...... 413
APPENDIX E: VTV MODELS FOR [M2, SMALL UIC OLT, 5] ...... 415
11
List of Tables
Tables Page
TABLE 3.1 CHANNEL SOUNDER PARAMETERS ...... 80 TABLE 3.2 SUMMARY OF MEASURED PDPS FOR EACH PROPAGATION REGION, AIRPORT...... 96 TABLE 4.1 RELATIONSHIP BETWEEN CORRELATION FUNCTIONS OF DIFFERENT SYSTEM FUNCTIONS...... 104 TABLE 4.2 SYSTEM CORRELATION FUNCTIONS FOR WSS CHANNELS...... 105 TABLE 4.3 UPPER BOUND ON THE CHANNEL SOUNDER ERRORS [121] ...... 110 TABLE 4.4 CHAMELEON OUTPUT FORMAT FOR THE NTH RECORD [120]...... 116 TABLE 4.5 SUB-RECORDS GENERATED FOR NTH RECORD [120]...... 117 TABLE 4.6 PARAMETERS FOR CFAR ALGORITHM [120]...... 120 TABLE 4.7 NUMBER OF SAMPLES/CHIP FOR DIFFERENT BANDWIDTHS [120]...... 123 TABLE 4.8 EXAMPLES OF LABELING CONVENTION USED FOR RESULTS [120]...... 141 TABLE 5.1 AMPLITUDE STATISTICS FOR EXAMPLE SWITCHING MODELS [143] ...... 181
TABLE 6.1 SUMMARY OF MEASURED RMS-DS AND DELAY WINDOW STATISTICS FOR ALL AIRPORTS [135] ...... 191 TABLE 6.2 SUMMARY OF COMPUTED FCE VALUES FOR ALL AIRPORTS [135]...... 195
TABLE 6.3 LEAST SQUARES FIT PARAMETERS FOR TAP PROBABILITY OF OCCURRENCE FOR ALL AIRPORTS [142], [134] ...... 198 TABLE 6.4 LEAST SQUARES FIT PARAMETERS FOR CUMULATIVE ENERGY FOR ALL AIRPORTS [133] [134]...200 TABLE 6.5 CUMULATIVE ENERGY FOR [M1, LARGE AIRPORT, NLOS-S, 50] [136]...... 206 TABLE 6.6 TAP AMPLITUDE PARAMETERS FOR [M1, LARGE AIRPORT, NLOS-S, 50] [136] ...... 207 TABLE 6.7 CUMULATIVE ENERGY FOR [M1, LARGE AIRPORT, NLOS, 50] [136] ...... 209 TABLE 6.8 TAP AMPLITUDE PARAMETERS FOR [M1, LARGE AIRPORT, NLOS, 50] [136] ...... 210 TABLE 6.9 TAP AMPLITUDE PARAMETERS FOR [M1, LARGE AIRPORT, NLOS-S, 25] ...... 213
TABLE 6.10 TAP AMPLITUDE PARAMETERS FOR [M2, LARGE AIRPORT, NLOS-S, 25]. (ENERGIES AND WEIBULL B FACTORS ALSO APPLY TO M3.)...... 213 TABLE 6.11 TAP AMPLITUDE PARAMETERS FOR [M1, LARGE AIRPORT, NLOS, 25]...... 216
TABLE 6.12 TAP AMPLITUDE PARAMETERS FOR [M2, LARGE AIRPORT, NLOS, 25]. (ENERGIES AND WEIBULL B FACTORS ALSO APPLY TO M3.) ...... 217 TABLE 6.13 TAP AMPLITUDE PARAMETERS FOR [M1, MEDIUM AIRPORT, NLOS-S, 50] [137]...... 222 TABLE 6.14 CUMULATIVE ENERGY FOR [M1, MEDIUM AIRPORT, NLOS-S, 50] [137]...... 223 TABLE 6.15 TAP AMPLITUDE PARAMETERS FOR [M1, MEDIUM AIRPORT, NLOS-S, 50] [137]...... 224 TABLE 6.16 CUMULATIVE ENERGY FOR [M1, MEDIUM AIRPORT, NLOS, 50] [137] ...... 225 TABLE 6.17 TAP AMPLITUDE PARAMETERS FOR [M1, MEDIUM AIRPORT, NLOS-S, 50] [137]...... 226 TABLE 6.18 TAP AMPLITUDE PARAMETERS FOR [M1, MEDIUM AIRPORT, LOS-O, 25] ...... 228 12
TABLE 6.19 TAP AMPLITUDE PARAMETERS FOR [M2, MEDIUM AIRPORT, LOS-O, 25] ...... 229 TABLE 6.20 TAP AMPLITUDE PARAMETERS FOR [M1, MEDIUM AIRPORT, NLOS-S, 25]...... 230 TABLE 6.21 TAP AMPLITUDE PARAMETERS FOR [M2, MEDIUM AIRPORT, NLOS-S, 25]...... 231 TABLE 6.22 TAP AMPLITUDE PARAMETERS FOR [M1, MEDIUM AIRPORT, NLOS, 25]...... 233 TABLE 6.23 TAP AMPLITUDE PARAMETERS FOR [M2, MEDIUM AIRPORT, NLOS, 25]...... 234 TABLE 6.24 TAP AMPLITUDE PARAMETERS FOR [M1, SMALL AIRPORT, LOS-O, 50] [138]...... 239 TABLE 6.25 CUMULATIVE ENERGY FOR [M1, SMALL AIRPORT, NLOS-S, 50] [138]...... 240 TABLE 6.26 TAP AMPLITUDE PARAMETERS FOR [M1, SMALL AIRPORT, NLOS-S, 50] [138] ...... 240 TABLE 6.27 TAP AMPLITUDE PARAMETERS FOR [M1, SMALL AIRPORT, NLOS, 50] [138]...... 241 TABLE 6.28 TAP AMPLITUDE PARAMETERS FOR [M1, SMALL AIRPORT, LOSO, 25] ...... 243 TABLE 6.29 TAP AMPLITUDE PARAMETERS FOR [M2, SMALL AIRPORT, LOSO, 25] ...... 243 TABLE 6.30 TAP AMPLITUDE PARAMETERS FOR [M1, SMALL AIRPORT, NLOS-S, 25]...... 244 TABLE 6.31 TAP AMPLITUDE PARAMETERS FOR [M2, SMALL AIRPORT, NLOS-S, 25]...... 245 TABLE 6.32 TAP AMPLITUDE PARAMETERS FOR [M1, SMALL AIRPORT, NLOS, 25] ...... 247 TABLE 6.33 TAP AMPLITUDE PARAMETERS FOR [M2, SMALL AIRPORT, NLOS, 25] ...... 248 TABLE 6.34 NUMBER OF CIRS GENERATED FOR [M2, LARGE AIRPORT, NLOS, 25]...... 254
TABLE 6.35 COMPARISON OF STOCHASTIC DISTANCE MEASURES OF M2 AND M3 WITH THOSE OF DATA FOR LARGE AIRPORTS FOR VARYING BANDWIDTH...... 260
TABLE 6.36 COMPARISON OF STOCHASTIC DISTANCE MEASURES OF M2 AND M3 WITH THOSE OF DATA FOR MEDIUM AIRPORTS FOR VARYING BANDWIDTH ...... 263
TABLE 6.37 COMPARISON OF STOCHASTIC DISTANCE MEASURES OF M2 AND M3 WITH THOSE OF DATA FOR SMALL AIRPORTS FOR SEVERAL BANDWIDTHS ...... 267 TABLE 7.1 FIXED POINT-POINT MEASUREMENT LOCATIONS [147]...... 274
TABLE 7.2 RMS-DS STATISTICS FOR COMMON RECEIVING AREAS, WITH AFS AND ATCT TRANSMISSION AT MIA ...... 278 TABLE 7.3 JFK AFS TRANSMISSION RMS-DS STATISTICS ...... 279
TABLE 7.4 CUMULATIVE ENERGY FOR AFS AND PARTIAL ATCT FOR [M1, MIA, NLOS-S AND NLOS, 50] [146]...... 281 TABLE 7.5 AMPLITUDE STATISTICS FOR [M1, MIA-AFS, NLOS-S, 50] [146] ...... 281 TABLE 7.6 AMPLITUDE STATISTICS FOR PARTIAL ATCT CHANNEL [M1, MIA, NLOS-S, 50] [146]...... 283 TABLE 7.7 AMPLITUDE STATISTICS FOR AFS CHANNEL [M1, MIA, NLOS, 50] [147] ...... 284 TABLE 7.8 AMPLITUDE STATISTICS FOR PARTIAL ATCT CHANNEL [M1, MIA, NLOS, 50] [147]...... 286 TABLE 7.9 AMPLITUDE STATISTICS FOR [M1, JFK-AFS, NLOS-S,50] ...... 289 TABLE 7.10 AMPLITUDE STATISTICS FOR [M1, JFK-AFS, NLOS, 50] [147] ...... 290 TABLE 7.11 RMS-DS FOR FIXED POINT-TO-POINT LOCATIONS AT CLE...... 294
TABLE 7.12 FIRST TAP CHANNEL RICEAN K-FACTORS AT CLE FOR DIFFERENT LOCATIONS AND BANDWIDTHS FOR BORESIGHT LINKS [147]...... 297 13
TABLE 7.13 FIRST TAP CHANNEL RICIAN K-FACTORS AT MIA (POINT 1, POINT 2) FOR DIFFERENT BANDWIDTHS [147] ...... 297 TABLE 8.1 SUMMARY OF MEASURED RMS-DS VALUES FOR FIVE REGIONS [144]...... 300
TABLE 8.2 LEAST-SQUARES FIT PARAMETERS FOR TAP PROBABILITY OF OCCURRENCE, EQ (8.1), ALL REGIONS, 5 MHZ BANDWIDTH...... 306
TABLE 8.3 LEAST-SQUARES FIT PARAMETERS FOR AVERAGE TAP ENERGY, EQ (8.2), ALL REGIONS, 5 MHZ BANDWIDTH [144] ...... 308 TABLE 8.4 TAP AMPLITUDE PARAMETERS FOR [M1, ALL REGIONS –VTV, 10] [144] ...... 310 TABLE 8.5 AMPLITUDE PARAMETERS FOR [M1, ALL REGIONS –VTV, 5] [144]...... 311
TABLE 8.6 CORRELATION COEFFICIENT MATRICES FOR [M1, UOC, 10 AND 5]; LOWER TRIANGULAR PART FOR 10 MHZ, UPPER TRIANGULAR PART FOR 5 MHZ ...... 312
TABLE 8.7 CORRELATION COEFFICIENT MATRICES FOR [M1, SMALL, 10 AND 5]; LOWER TRIANGULAR PART FOR 10 MHZ, UPPER TRIANGULAR PART FOR 5 MHZ ...... 313
TABLE 8.8 CORRELATION COEFFICIENT MATRICES FOR [M1, OHTR, 10 AND 5]; LOWER TRIANGULAR PART FOR 10 MHZ, UPPER TRIANGULAR PART FOR 5 MHZ ...... 313
TABLE 8.9 CORRELATION COEFFICIENT MATRICES FOR [M1, OLTR, 10 AND 5]; LOWER TRIANGULAR PART FOR 10 MHZ, UPPER TRIANGULAR PART FOR 5 MHZ ...... 313
TABLE 8.10 CORRELATION COEFFICIENT MATRICES FOR [M1, UIC, 10 AND 5]; LOWER TRIANGULAR PART FOR 10 MHZ, UPPER TRIANGULAR PART FOR 5 MHZ ...... 313 TABLE 8.11 AMPLITUDE PARAMETERS FOR [M2, OHT, 10] [144]...... 314 TABLE 8.12 AMPLITUDE PARAMETERS FOR [M2, UOC, 10] [144] ...... 315 TABLE 8.13 AMPLITUDE PARAMETERS FOR [M2, UIC, 10] ...... 315 TABLE 8.14 AMPLITUDE PARAMETERS FOR [M2, SMALL, 10]...... 316 TABLE 8.15 AMPLITUDE PARAMETERS FOR [M2, OLT, 10]...... 316
TABLE 8.16 CORRELATION COEFFICIENT MATRICES FOR 10 MHZ M2 MODELS FOR UOC AND UIC; LOWER TRIANGULAR IS UOC, UPPER TRIANGULAR IS UIC ...... 317
TABLE 8.17 CORRELATION COEFFICIENT MATRICES FOR 10 MHZ M2 MODELS FOR SMALL AND OLT; LOWER TRIANGULAR IS SMALL, UPPER TRIANGULAR IS OLT...... 318 TABLE 8.18 AMPLITUDE PARAMETERS FOR [M2, UOC AND OHT, 5] ...... 319
TABLE 8.19 CORRELATION MATRICES FOR 5 MHZ M2 MODELS FOR OHT AND UOC; LOWER TRIANGULAR IS OHT, UPPER TRIANGULAR IS UOC ...... 319 TABLE 8.20 NUMBER OF CIRS FOR SIMULATED M2-VTV MODELS FOR 10 MHZ CHANNEL BW ...... 322
TABLE 8.21 COMPARISON OF STATISTICS OF M1, M2 AND M3 WITH THOSE OF DATA FOR UOC FOR TWO BANDWIDTHS [144] ...... 327
TABLE 8.22 COMPARISON OF STATISTICS OF M1, M2 AND M3 WITH THOSE OF DATA FOR OHT FOR TWO BANDWIDTHS...... 330
14
List of Figures
Figure Page
FIGURE 3.1 TAPPED DELAY LINE CHANNEL MODEL ...... 76
FIGURE 3.2 (A) RAPTOR CHANNEL SOUNDING SYSTEM (LEFT – RECEIVER, RIGHT – TRANSMITTER).(B)
TRANSMITTED POWER SPECTRUM, 50 MCPS SETTING, FC=5.12 GHZ ...... 80 FIGURE 3.3 VARIOUS ANTENNAS USED AT THE AIRPORT ...... 83
FIGURE 3.4 AZIMUTH RADIATION PATTERN FOR THE DIRECTIONAL HORN USED AT THE RECEIVER FOR FIXED- TO-FIXED MEASUREMENTS AT THE AIRPORT ...... 83
FIGURE 3.5 AERIAL PHOTOGRAPH OF JFK INTERNATIONAL AIRPORT AND NUMBERED MEASUREMENT TEST POINTS ...... 85
FIGURE 3.6 TRANSMITTER LOCATIONS AT DIFFERENT AIRPORTS, (A) CLEVELAND INTERNATIONAL AIRPORT (B) JFK INTERNATIONAL AIRPORT, (C) TAMIAMI AIRPORT ...... 86 FIGURE 3.7 VIEW FROM THE JFK ATCT IN DIFFERENT DIRECTIONS ...... 90 FIGURE 3.8 VIEW FROM THE MIA ATCT...... 91 FIGURE 3.9 VIEW FROM THE ATCT TOWER AT CLEVELAND ...... 92
FIGURE 3.10 PICTURES FROM DIFFERENT GA AIRPORTS: (A) OHIO UNIVERSITY; (B) BURKE LAKEFRONT; (C) TAMIAMI ...... 94 FIGURE 3.11 ANTENNA LOCATION ON THE TRANSMITTING AND RECEIVING VEHICLES...... 97 FIGURE 3.12 EXAMPLE PICTURES OF THE DIFFERENT VTV PROPAGATION REGIONS...... 100 FIGURE 4.1 BLOCK DIAGRAM OF THE CHANNEL MEASUREMENT SYSTEM [121]...... 107
FIGURE 4.2 AUTOCORRELATION CURVE FOR THE CHANNEL SOUNDER IN BACK-TO-BACK MODE, USING A BW OF 50 MHZ [120]...... 113 FIGURE 4.3 SCREEN CAPTURE FOR CHAMELEON (FORMAT CONVERSION SOFTWARE) [120]...... 114 FIGURE 4.4 EXAMPLE POWERRECORD FOR 50 MCPS AFTER USING CHAMELEON [120]...... 118 FIGURE 4.5 EXAMPLE POWERRECORD FOR 50 MHZ CHANNEL BANDWIDTH [120]...... 122 FIGURE 4.6 PDPS FOR DIFFERENT BWS [120]: (A) 50 MHZ, (B) 10 MHZ, (C) 5 MHZ, AND (D) 1 MHZ ...... 124 FIGURE 4.7 EXAMPLE RMS-DS VALUES FOR CLE DATA APPLYING DIFFERENT VALUES OF MT [120]...... 126 FIGURE 4.8 EXAMPLE FCES FOR CLE DATA APPLYING DIFFERENT VALUES OF MT [120]...... 127 FIGURE 4.9 SUMMARY FLOW DIAGRAM OF PRE-PROCESSING STEPS OF SECTION 4.1 [120]...... 128
FIGURE 4.10 EXAMPLE PDPS WITH SAME MULTIPATH BEHAVIOR AND DIFFERENT ΜΤ [120] ...... 130 FIGURE 4.11(A) EXAMPLE MEASUREMENT LOCATION AT MIA [120]...... 131 FIGURE 4.11(B) RMS-DS VS. IRE NUMBER (TIME) FOR LOCATIONS IN MIA OF FIG. 4.11(A)...... 132
FIGURE 4.12 RMS-DS AND DELAY WINDOW VS. IRE NUMBER (TIME) FOR A MEASUREMENT SEGMENT AT MIA [120]...... 133
FIGURE 4.13 EXAMPLE PERSISTENCE PROCESSES FOR TAPS 2 AND 5 FOR SEGMENT OF TRAVEL AT JFK [120] ...... 135 FIGURE 4.14 EXAMPLE PHASE VARIATION VERSUS TIME FOR 1ST TAP FOR JFK-NLOS [120]...... 137 15
FIGURE 4.15 EXAMPLE FCE FOR CLE [120]...... 139 FIGURE 4.16 HISTOGRAM OF RMS-DS FOR [JFK, NLOS, 50] [120]...... 144
FIGURE 4.17 REGION STATE AND CORRESPONDING RMS-DS FOR AN EXAMPLE MEASUREMENT SET AT CLE [120]...... 148
FIGURE 4.18 STEADY STATE PROBABILITIES OF STATE 1 FOR [JFK, NLOS, 50] AND [JFK, NLOS-S, 50] [120] ...... 151
FIGURE 4.19 AVERAGE ENERGY ASSOCIATED WITH TAP FOR [JFK, NLOS, 50] AND [JFK, NLOS-S, 50] [120] ...... 154 FIGURE 4.20 CUMULATIVE ENERGY GATHERED FOR [JFK, NLOS, 50] AND [JFK, NLOS-S, 50] [120]...... 155
NLOS−S FIGURE 4.21 NUMBER OF PDPS USED TO DETERMINE R12 FOR [JFK, NLOS-S, 50] [120] ...... 157
FIGURE 4.22 SUMMARY OF MODEL EXTRACTION STEPS OF SECTION 4.3 [120]...... 162
FIGURE 5.1 EXAMPLE PDP EVOLUTION IN TIME FOR UIC VTV ENVIRONMENT. DELAY (RIGHTWARD AXIS) IN MICROSECONDS, AND TIME (LEFTWARD AXIS) IN SECONDS [143] ...... 169
FIGURE 5.2 EXAMPLE DISTRIBUTION FITS FOR AMPLITUDE DATA OF 2ND TAP FOR DIFFERENT AIRPORTS; (A) MIA, (B) CLE [143] ...... 170 FIGURE 5.3 EXAMPLE SETTINGS FOR PIN-HOLE CHANNEL AT CLE [143]...... 172 FIGURE 5.4 EXAMPLE PDFS FOR DOUBLE WEIBULL MULTIPLICATIVE MODEL [143]...... 174
FIGURE 5.5 SURFACE PLOT SHOWING VARIATION OF BDW GIVEN B1 = 2 AND POWER1 = 1 FOR VARYING VALUES OF B2 AND POWER2 ...... 175
FIGURE 5.6 PERCENTAGE OF WR FADING CONSIDERING DOUBLE WEIBULL MODEL FOR VARYING B1, B2, ENERGY1 AND ENERGY2 ...... 177 FIGURE 5.7 EXAMPLE MEASUREMENT SET FROM MIA COMPARING RSSI AND POWER IN 1ST TAP FOR “GOOD” AND “BAD” STATES...... 179 FIGURE 5.8 EXAMPLE RMS-DS TIME SERIES FROM MIA [143]...... 180 FIGURE 5.9 AMPLITUDE STATISTICS FOR AS3 FROM TABLE 5.1...... 182 FIGURE 5.10 DISTRIBUTION FITS TO MEASURED AND SIMULATED WR USING SWITCHING MODEL [143] ...... 183 FIGURE 6.1 HISTOGRAM OF MEASURED RMS-DS VALUES, MIA [135]...... 186 FIGURE 6.2 RMS-DS DISTRIBUTION FOR CLE [135]...... 187
FIGURE 6.3 CUMULATIVE DISTRIBUTION FUNCTIONS OF RMS-DS FOR THREE AIRPORTS ( CLE, MIA AND JFK) [135]...... 188 FIGURE 6.4 HISTOGRAM OF MEASURED RMS-DS VALUES (A) BL AND (B) TA [135] ...... 189 FIGURE 6.5 CUMULATIVE DISTRIBUTION FUNCTIONS OF RMS-DS FOR THREE AIRPORTS [135] ...... 190
FIGURE 6.6 DISTRIBUTION OF Wτ,90 FOR (A) [MIA, 50] AND (B) [JFK, 50] [135] ...... 191 FIGURE 6.7 FCES FOR LARGE AIRPORTS FOR 50 MHZ [135] ...... 192 FIGURE 6.8 FCES FOR MEDIUM AIRPORT [135] ...... 193 FIGURE 6.9 FCES FOR SMALL AIRPORT [134]...... 194
FIGURE 6.10 STEADY STATE TAP PROBABILITY FOR STATE 1 (TAP “ON”) VS. TAP INDEX, MIA, JFK, AND CLE, NLOS AND NLOS-S ...... 196 16
FIGURE 6.11 STEADY STATE TAP PROBABILITY FOR STATE 1 (TAP “ON”) VS. TAP INDEX, BL, OU AND TA, NLOS, NLOS-S AND LOSO ...... 197
FIGURE 6.12 CUMULATIVE ENERGY VERSUS TAP INDEX FOR LARGE AND MEDIUM AIRPORTS, ALL REGIONS [133]...... 199 FIGURE 6.13 CUMULATIVE ENERGY VERSUS TAP INDEX FOR ALL GA AIRPORTS, REGIONS [134] ...... 200 FIGURE 6.14 PATH LOSS MODELING FOR MIA-NLOS-S; A=103 DB, N=2.23, =5.3 DB [136...... 204 FIGURE 6.15 COMPARING DIFFERENT DISTRIBUTION FITS WITH DATA...... 208
FIGURE 6.16 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1 AND 2 FOR [M1, LARGE AIRPORT, NLOS-S, 50] [136]...... 209 FIGURE 6.17 AMPLITUDE STATISTICS OF TAPS 1 AND 3 FOR [M1, LARGE AIRPORT, NLOS, 50] [136] ...... 212
FIGURE 6.18 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1 AND 2 FOR [M1, MEDIUM AIRPORT, LOS-O, 50] [137]...... 223
FIGURE 6.19 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1 AND 2 FOR [M1, MEDIUM AIRPORT, NLOS-S, 50] [137]...... 225
FIGURE 6.20 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1, 2, 10, AND 11 FOR [M1, MEDIUM AIRPORT, NLOS, 50] [137] ...... 227
FIGURE 6.21 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1 AND 2 FOR [M1, SMALL AIRPORT, LOS- O, 50] [138]...... 239
FIGURE 6.22 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1, 2, 5, AND 10 FOR [M1, SMALL AIRPORT, NLOS-S, 50] [138]...... 241
FIGURE 6.23 COMPARING THE RMS-DS EVALUATED FOR THE CIRS GENERATED USING [M2 M3, LARGE AIRPORT] WITH MEASUREMENT DATA ...... 259
FIGURE 6.24 COMPARING THE Wτ,90 EVALUATED FOR THE CIRS GENERATED USING [M2 M3, LARGE AIRPORT] WITH MEASUREMENT DATA ...... 259 FIGURE 6.25 COMPARING FCES OF [M2 M3, LARGE AIRPORT, NLOS-S NLOS, 20] WITH THAT OF DATA..261
FIGURE 6.26 COMPARING THE RMS-DS EVALUATED FOR THE CIRS GENERATED USING [M2 M3, MEDIUM AIRPORT] WITH MEASUREMENT DATA ...... 262
FIGURE 6.27 COMPARING THE Wτ,90 EVALUATED FOR THE CIRS GENERATED USING [M2 M3, MEDIUM AIRPORT] WITH MEASUREMENT DATA ...... 263 FIGURE 6.28 COMPARING FCE OF [M2 M3, MEDIUM AIRPORT, NLOS-S NLOS, 25] WITH THAT OF DATA 264 FIGURE 6.29 COMPARING FCE OF [M2 M3, MEDIUM AIRPORT, LOSO, 25] WITH THAT OF DATA ...... 265
FIGURE 6.30 COMPARING THE RMS-DS EVALUATED FOR THE CIRS GENERATED USING [M2 M3, SMALL AIRPORT, 25] WITH MEASUREMENT DATA ...... 266
FIGURE 6.31 COMPARING THE Wτ,90 EVALUATED FOR THE CIRS GENERATED USING [M2 M3, SMALL AIRPORT, 25] WITH MEASUREMENT DATA ...... 266
FIGURE 6.32 COMPARING FCES OF [M2 M3, SMALL AIRPORT, NLOS-S LOS-O, 25] WITH THOSE OF DATA ...... 268 FIGURE 6.33 COMPARING FCES OF [M2 M3, SMALL AIRPORT, NLOS, 25] WITH THOSE OF DATA ...... 268 FIGURE 7.1 AFS LOCATION NEAR ILS, MIA [146]...... 271
FIGURE 7.2 AIRPORT SURFACE AT JFK, (A) AFS LOCATION AT ILS, (B) SHADOWED REGION WHILE TRANSMITTING FROM ATCT. [146]...... 272 17
FIGURE 7.3 RADAR MEASUREMENT SITE AT CLE. [147] ...... 275 FIGURE 7.4 ANTENNA ORIENTATIONS AT CLE FIXED POINT-TO-POINT LOCATIONS [147] ...... 276 FIGURE 7.5 ANTENNA ORIENTATIONS AT MIA FIXED POINT-TO-POINT LOCATIONS [147]...... 276
FIGURE 7.6 DISTRIBUTIONS OF RMS-DS FOR TRANSMISSION FROM BOTH AFS AND PARTIAL-ATCT, MIA [146]...... 278 FIGURE 7.7 FCES OBTAINED WITH COMMON RECEIVE AREAS, AFS AND ATCT TRANSMITTERS, MIA ...... 280
FIGURE 7.8 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1, 2, 3, AND 7 FOR AFS CHANNEL [M1, MIA, NLOS-S, 50] [147] ...... 282
FIGURE 7.9 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1, 2, AND 3 FOR PARTIAL ATCT CHANNEL [M1, MIA, NLOS-S, 50] [147]...... 283
FIGURE 7.10 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1, 2, 5, AND 23 FOR AFS CHANNEL [M1, MIA, NLOS, 50] [147]...... 286
FIGURE 7.11 AMPLITUDE HISTOGRAMS AND CURVE FITS FOR TAPS 1, 2, 19 AND 48 FOR PARTIAL ATCT CHANNEL [M1, MIA, NLOS, 50] [147] ...... 288 FIGURE 7.12 CUMULATIVE ENERGY DISTRIBUTION FOR [M1, JFK-AFS, 50] ...... 289
FIGURE 7.13 MEASURED PDPS: POWER VS. DELAY (IN MICROSECONDS, RIGHTWARD AXIS) AND VS. TIME (IN SECONDS, LEFTWARD AXIS) FOR FIELD (ILS) SITE IN MIA AT 105O ORIENTATION WITH RESPECT TO ATCT [132] ...... 292
FIGURE 7.14 TIME VARYING POWER SPECTRUM (POWER VS. FREQUENCY IN MHZ, RIGHTWARD AXIS) AND VS. TIME (IN SECONDS, LEFTWARD AXIS) FOR FIELD SITE (ILS) IN MIA, 15O ORIENTATION WITH RESPECT TO ATCT [132] ...... 293
FIGURE 7.15 POWER DISTRIBUTION VERSUS AZIMUTH ANGLE FOR MIA FIXED POINT-TO-POINT LINKS [147] ...... 294 FIGURE 7.16 RMS-DS FOR MIA FIXED POINT-TO-POINT LINKS VERSUS AZIMUTH ANGLE [147]...... 295 FIGURE 7.17 FCES FOR CLE FIXED POINT-TO-POINT LINKS FOR BORESIGHT [147] ...... 295 FIGURE 7.18 FCES FOR MIA FIXED POINT-TO-POINT LINKS FOR BORESIGHT [147]...... 296 FIGURE 8.1 CUMULATIVE DISTRIBUTION FUNCTIONS OF RMS-DS FOR FIVE REGIONS [144]...... 300
FIGURE 8.2 CUMULATIVE DISTRIBUTION FUNCTIONS OF Wτ,,90 FOR FIVE REGIONS [144] ...... 301 FIGURE 8.3 FREQUENCY CORRELATION ESTIMATES FOR FIVE REGIONS FOR A 5 MHZ BANDWIDTH [144] ...302 FIGURE 8.4 FREQUENCY CORRELATION ESTIMATES FOR FIVE REGIONS FOR A 10 MHZ BANDWIDTH [149] .303
FIGURE 8.5 EXAMPLE PERSISTENCE PROCESSES FOR TAPS 5 AND 8 FOR SEGMENT OF TRAVEL FOR THE UOC CASE, 10 MHZ BW...... 304
FIGURE 8.6 STEADY STATE TAP PROBABILITY FOR STATE 1 (P[TAP “ON”]) VS. TAP INDEX, ALL REGIONS, 5 MHZ BANDWIDTH [144]...... 306 FIGURE 8.7 AVERAGED TAP ENERGIES WITH LS FITS, ALL REGIONS, 5 MHZ BANDWIDTH [144] ...... 307 FIGURE 8.8 CUMULATIVE ENERGY VERSUS TAP INDEX FOR ALL REGIONS FOR 5 MHZ BANDWIDTH [144] ...308
FIGURE 8.9 HISTOGRAMS AND PROBABILITY DENSITY FUNCTION FITS FOR 2ND TAP OF UOC, AND SMALL CITY, FOR 10 MHZ BANDWIDTH [144]...... 312
FIGURE 8.10 COMPARING PDF AND CDF OF RMS-DS FOR MODEL-1, MODEL-2 AND MODEL-3 WITH THAT OF UOC DATA FOR 10MHZ [144]...... 326 18
FIGURE 8.11 COMPARING PDF OF DW-90 FOR MODEL-1, MODEL-2 AND MODEL-3 WITH THAT OF UOC DATA [144]...... 326
FIGURE 8.12 COMPARING PDF OF RMS-DS FOR MODEL-1, MODEL-2 AND MODEL-3 WITH THAT OF DATA FOR OHT-10MHZ ...... 329
FIGURE 8.13 COMPARING PDF OF DW-90 FOR MODEL-1, MODEL-2 AND MODEL-3 WITH THAT OF DATA FOR OHT-10MHZ ...... 330
FIGURE A.1 COMPARISON OF SIMULATED WEIBULL HISTOGRAM AND THEORETICAL FOR THE “RESTRICTED” METHOD...... 365 FIGURE A.2 EXAMPLE ILLUSTRATION FOR INTERPOLATION ALGORITHM ...... 369 FIGURE B.1 AZIMUTH RADIATION PATTERN FOR HIGH GAIN HORN ANTENNA ...... 370 FIGURE B.2 ELEVATION RADIATION PATTERN FOR HIGH GAIN HORN ANTENNA ...... 370 FIGURE B. 3 AZIMUTH RADIATION PATTERN FOR LOW GAIN HORN ANTENNA ...... 371 FIGURE B.4 ELEVATION RADIATION PATTERN FOR LOW GAIN DIRECTIONAL HORN ANTENNA...... 371 FIGURE B.5 ELEVATION PATTERN FOR OMNIDIRECTIONAL (AZIMUTH) MONOPOLE...... 372
19
List of Acronyms and of Abbreviations
2D Two dimensional
3G Third Generation
4G Fourth Generation
5G Fifth Generation
AC Alternating Current
ACAST Advanced Communications, Navigation, and Surveillance (CNS),
Architectures and System Technologies program
AEC Avionics Engineering Center
AFS Airport Field Site
A/G Air to Ground
AoA Angle of Arrival
AR Autoregressive
ARMA Autoregressive Moving Average
ASCII American Standard Code for Information Interchange
AS Air to Satellite
ASDE Airport Surface Detection Equipment
ATCT Air Traffic Control Tower
AWGN Additive White Gaussian Noise
BER Bit Error Ratio
BL Burke Lakefront Airport (Cleveland, OH)
BPSK Binary Phase Shift Keying 20
BVS Berkeley Varitronics Systems, Inc.
BW Bandwidth cdf cumulative distribution function
CDMA Code Division Multiple Access
CFAP Constant False Alarm Probability
CFAR Constant False Alarm Rate
CHI Chi-square test
CIR Channel Impulse Response
CLE Cleveland Hopkins International Airport
CQI Channel Quality Indicator
CW Continuous Wave dB decibel dBm decibels above 1 milliwatt
DS Direct Sequence
DOA Direction of Arrival
FAA Federal Aviation Administration
FCE Frequency Correlation Estimate
FCC Federal Communications Commission
FFT Fast Fourier Transform
FIR Finite Impulse Response
GA General Aviation
G/A Ground to Air 21
GCIR Gaussian Channel Impulse Response
GG Ground to Ground
GHz giga Hertz
GLSF Generalized Local Scattering Function
GPS Global Positioning System
HF High Fidelity
Hz Hertz
HI Histogram Intersection test
ICAO International Civil Aviation Organization
IEEE Institute of Electrical and Electronics Engineers
ILS Instrument Landing System
IRE Impulse Response Estimate
ISM Industrial, Scientific, and Medical
ITU International Telecommunications Union
ITS Intelligent Transport Systems
JFK John F. Kennedy International Airport km kilometer
KL Kullback-Leibler test
LAN Local Area Network
LOS-O Line-of-sight-Open
LPF Low Pass Filter
LSF Local Scattering Function 22
LTI Linear Time Invariant m meter mm millimeter mph miles per hour ms milli second
μs micro second
M1 Model-1
M2 Model-2
M3 Model-3
MAC Medium Access Control (layer of the communications protocol stack)
MANET Mobile Ad Hoc Networks
MC Multi-carrier
Mcps mega Chips per Second
MHz mega Hertz
MIA Miami International Airport
MIMO Multiple Input Multiple Output
ML Maximum Liklihood
MLS Microwave Landing System
MSE Mean Square Error
NASA National Aeronautics and Space Administration
NLOS-S Non-line-of-sight-Specular 23
NS Non-stationary ns nano second
OFDM Orthogonal Frequency Division Multiplexing
OHT Open Area - High Traffic
OLT Open Area- Low Traffic
OU Ohio University, or Ohio University Airport
PDP Power Delay Profile pdf Probability Density Function
PL path loss
PS Public Safety
QoS Quality of Service
RF Radio Frequency
RFI Radio Frequency Interference
RMS-DS Root-mean-square value of (multipath) Delay Spread
RSSI Received Signal Strength Information
Rx receiver
RVC Road-To-Vehicle Communications sec seconds
S Small City
SNR Signal to Noise Ratio
SS Spread Spectrum
TA Tamiami Airport (Kendall, FL) 24
TDL Tapped Delay Line
Tx transmitter
UHF Ultra High Frequency
UNII Unlicensed National Information Infrastructure
UIC Urban-Antenna Inside Car
UMTS Universal Mobile Telecommunications System
UOC Urban –Antenna Outside Car
US Uncorrelated Scattering
USDR User Selected Dynamic Range
UPS Uninterruptible Power Supply
UWB Ultra Wideband
VTFAR Vector Time Frequency Autoregressive
VTV Vehicle to Vehicle
VTV Vehicle to Vehicle
WIMAX Worldwide Interoperability for Microwave Access
WLAN Wireless Local Area Network
WR Worse than Rayleigh
WRC World Radio Conference
WSS Wide Sense Stationary
WSSUS Wide Sense Stationary Uncorrelated Scattering
25
List of Symbols
α Attenuation (dimensionless, or in dB) a Weibull pdf scale factor aref Complex Amplitude at a reference frequency b Weibull pdf shape factor c speed of light (m/sec)
δ Delta function (ideal impulse) d0 reference distance (m) e1 Commutation Error e2 Pulse Compression Error e3 Misinterpretation Error e4 Aliasing Error
ε1 Upper bound on Commutation Error
ε2 Upper bound on Pulse Compression Error
ε3 Upper bound on Misinterpretation Error
ε4 Upper bound on Aliasing Error
φ phase (radians) fd Doppler shift (Hz) fc carrier frequency (Hz) fD Doppler spread (Hz) fD,max Maximum Doppler spread (Hz) 26
f (w ,w ) Joint pdf of the Weibull RVs Wi and Wj. Wi ,W j i j g Impulse response of channel sounder transmit filter h CIR function hI Input Delay Spread Function
hO Output Delay Spread Function
HI Input Doppler Spread Function
HO Output Doppler Spread Function
H Time-varying channel transfer function
K in Ricean distribution, parameter K indicates fading severity
th K0 Zero order modified Bessel function of the second kind l frequency (Hz)
λ Wavelength
L Number of taps in TDL model based on mean(RMS-DS)
~ L Length of IRE
LAS Number of taps in TDL model based on mean(RMS-DS) of airport
LVTV Number of taps in TDL model based on mean(RMS-DS) of VTV
LW Maximum Likelihood estimate for Weibull distribution
LR Maximum Likelihood estimate for Rayleigh distribution
m in Nakagami-m distribution, parameter m indicates fading severity
M Transfer function of the output delay spread function
MT Multipath Threshold (dB)
μτ mean value of energy delay (sec) 27
th Mi Number of PDPs after interpolating Ni PDPs of the i segment
η delay (sec)
Nc Number of symbols used for correlation by channel sounder
th NTj noise threshold of j power record
th Ni Number of PDPs collected in the i segment of travel
Ω Average energy or power (mean-square value of random process) pw pdf of Weibull random variable pDR Analytical pdf for the Double Rayleigh model pDW Analytical pdf for the Double Weibull model
P τ ,η;v Fourier Transform of R I t, s; , U () h ( τ η)
P I f ,l;v H () Fourier Transform of RT (t, s; f ,l)
P m,η;v Fourier Transform of R O t, s; , V () h ( τ η)
P O f ,l;v H () Fourier Transform of RM (t, s; f ,l)
Pf Constant False Alarm Probability
P Power Delay Profile
Pt Transmit power (watts, or dBm)
Pij Transition probability, from state i to state j
R I t, s; , 2D correlation function of the input delay spread system function h ()τ η
RU ()υ, μ;τ ,η 2D correlation function of the delay-doppler spreading function
RU ()υ, μ;τ ,η 2D correlation function of delay Doppler spreading function
R O f ,l;v, 2D correlation function of output delay spread function H ()μ 28
RT ()f ,l;t, s 2D correlation function of transfer function of input delay spread
R Impulse response of channel sounder receiver filter. ri,j Correlation coefficient between amplitude of channel taps i and j
Rα Correlation coefficient matrix of vector α
Rwc Worst case Correlation coefficient matrix
Rmc Maximum Confidence Correlation coefficient matrix
Rmwc Realistic Worst case Correlation coefficient matrix
Rrmc Realistic Maximum Confidence Correlation coefficient matrix
Rc Chip rate (chips/sec)
Rs Symbol rate (symbols/sec)
Region_TS Markov chain transition matrix for a given propagation region
Region_ES Markov chain emission matrix for a given propagation region s time (sec)
SH Scattering function
j th σ n Noise variance of the j PDP
j th σ m Median level of the j PDP
στ standard deviation of delay spread
SS Markov chain steady state matrix t time (sec) tc Coherence (or, correlation) time (sec) tu update time of the sounder(sec)
τ delay variable (sec) 29
1 τ H Mean Excess Delay (sec)
1 τ R Mean duration of the channel sounder receiver filter
T Transfer function of the input delay spread function
Tc Chip duration (sec)
TS Symbol duration (sec)
TD Unambiguous delay range of the channel sounder (sec)
Tcable Propagation delay due to RS232C cable
Tr Length of channel sounder receiver filter
Tg Length of channel sounder transmit filter
TS Markov chain transition matrix
U Delay Doppler Spreading Function
μ Doppler shift (Hz)
1 μH Mean Delay Doppler Product v Doppler shift (Hz)
1 υH Mean Doppler shift (Hz)
V Doppler-Delay Spreading Function
W90 90% energy delay window
W Weibull Random Variate x Input to a linear time variant system y Output of a linear time variant system z persistence random process 30
1 Introduction
In this chapter, we provide a brief introduction to the topic of wireless channels and the necessity and importance of channel modeling. We follow that by providing motivation for introducing airport surface communication applications in the 5 GHz frequency band.
We then discuss the current and anticipated applications for vehicle to vehicle (VTV or
V2V) communications in the 5 GHz band. We conclude this chapter by listing the objectives of this dissertation and its contribution to the literature.
1.1 Introduction to Wireless Channels
Communications channels are either wired or wireless1. The term channel is a generic term that often groups the effects of the actual physical channel, the transmitting and receiving filters, and antennas. In wired systems, we physically connect a transmitter to a receiver with a "wire" which could be a twisted pair, coaxial cable, or light guide fiber, to name a few examples. In the case of wireless channels, transmission is much more “public,” with a transmitter's antenna radiating a signal that can be received by any antenna which is sufficiently near to it. After the signal is transmitted, the signal often encounters several physical objects in its path before reaching the receiver. Each of these physical objects can behave as a reflector, diffractor, scatterer, or create a total obstruction (blockage) in the path of the signal. Yet in any of the aforementioned forms
(diffractor, reflector, scatterer, obstruction), the physical objects alter the phase and the
1 We restrict attention to electromagnetic wave propagation in this work, and do not consider other types of signaling, e.g., acoustic, in which guiding structures other than “wires” are employed. 31 amplitude of the signal. The manner in which the underlying physical environment affects a signal primarily depends on the size (with respect to a wavelength), location and mobility of the encountered physical objects.
On the basis of this, the most significant channel effects upon any signal can be categorized broadly into small scale fading (often termed “fast fading” or “distortion”) and large scale fading (often termed “slow fading,” “shadowing” or “attenuation”). For a complete model of an environment, one would require modeling of both these channel effects. The developed channel models could be deterministic (using for example ray tracing and/or electromagnetic wave theory) or statistical. There have been several books
[1]-[8] that have been dedicated to, or have covered in depth, the physics behind various propagation environments and ways of modeling the physical effects using analysis and simulations. In terms of stochastic models, researchers have done a considerable amount of work in trying to understand the physics of propagation by evaluating the statistics of the channel when it is modeled as a linear time varying filter. Bello’s paper [9] can be considered as one of the most widely cited research papers in this area.
A wireless channel can also be subject to interference, and this can compromise the reliability of the communication system and yield a detrimental effect on system performance. To achieve optimum performance in any physical media, accurate estimation of the underlying channel is required [5]. Precise knowledge of the channel is necessary to evaluate the performance of a communication system before its actual deployment. This channel knowledge is crucial in the system design phase as well, so that costly remedial measures need not be applied after the fact. Evaluation of a system 32 using simulations can provide useful insights regarding the system’s performance under realistic channel conditions. Often, systems encounter spatial coverage holes that severely limit their data rate capability. By evaluating the system’s performance before actual deployment, one can then take appropriate steps to improve the system performance and also provide estimates of achievable data rates. Some examples of these techniques include (employing diversity, performing equalization, using adaptive modulation, using forward error correction coding, etc.
As more advanced communication systems (3G, 4G and 5G?) are designed, developed, and deployed, researchers are looking at the interoperability between different systems (e.g., terrestrial, indoor, and satellite). The definition (and predicted
“requirement”) of an “always on” type of connection implies a seamless transition between systems is needed, with--or possibly without--the knowledge of the end user. In order to accomplish this constant connectivity, it is crucial to emulate the behavior of the channel as precisely as possible. Adjustment of data rate, the number of users that can be supported with a given quality of service, and adaptive receiver processing to mitigate the effects of the channel depend on the knowledge of the channel at that moment. This estimation of the channel is often cast into a measure known as the “channel quality indicator” (CQI). Statistical stationarity is an underlying assumption when using this measure, since the channel is estimated over a set of consecutive symbols to provide the
CQI, and this inherently assumes that the statistics of the channel don’t change over this duration of a few symbols. 33
Channel models, whether deterministic or stochastic, try to depict the propagation characteristics of a given environment (e.g., specular reflections from the ground, reflections due to buildings, vehicles and people, diffraction from roof-tops, scattering due to “rough” physical objects, etc.). Given the dynamics involved with the propagation environment due to mobility of the scatterers, moving transmitter and receiver, differences in the material composition of fixed and mobile scatterers, etc., no two propagation environments are identical. Also, due to the ever increasing demand for higher data rate applications, research on future communication systems focuses on bandwidths that are much higher than those of conventional (current) communication systems. Hence, trying to adapt existing channel models for newer applications for settings different from those in which the model was designed would not be an optimal way of depicting the actual physical channel that the new system would encounter. What is needed is to create channel models for these new applications based on actual measured data. These new channel models would be useful tools for system designers to optimize the performance of systems that are being considered for potential deployment for these new applications. The evaluation of a system using channel models developed using measured data would also provide realistic estimates of achievable link ranges, feasible data rates, packet error rates, latency involved with the data communication, etc.
Predicting the physical layer performance would also help in estimating the performance of higher layers of the protocol stack and hence will be useful in improving the performance from the point of the upper layers of the communications protocol stack as well. These multiple benefits of developing channel models that we have discussed using 34 measured data serve as our primary motivation, and in the next few sections, we briefly discuss the new applications towards which our research is directed.
1.2 Airport Surface Communications Applications in the 5 GHz Band
There are various organizations such as airlines, transportation and security groups, catering agencies, etc., working within the airport surface boundaries. With a consistent increase in the number of commercial and freight activities at airports, an increase in the above mentioned activities has also taken place. The activities of these organizations need to be well coordinated and additional communication services will be needed to ensure efficiency, safety, and security [10]. Due to the near “saturation” (i.e., complete usage of the existing frequency band for current applications) of the dedicated aeronautical VHF band, it is necessary to look at other available spectrum for implementing new applications. In an effort in this direction, the NASA Glenn Research
Center’s Advanced CNS Architectures and Systems Technologies (ACAST) program was conducted to begin investigations to improve airport surface communications [11].
In addition, more efficient use of existing spectrum is being researched; for example, a proposed method to enable new aeronautical communication applications in the VHF band by employing spectrum efficient multicarrier modulation techniques has been proposed in [12].
The deployment of any new service relies on available radio frequency spectrum
(at least if one does not wish to employ the unlicensed bands, which aviation and safety- critical systems will not). The aeronautical frequency band from 5.091-5.15 GHz—the 35
“microwave landing system (MLS) extension” band—is currently underutilized in much of the world. In fact, this MLS extension (MLS-E) band may never realize appreciable utilization levels due to advances in satellite navigation systems. The current 5 GHz
Industrial, Scientific and Medical (ISM) band used by the IEEE 802.11a Wireless Local
Area Network (WLAN) systems is fast approaching exhaustion due to an increase in the number of users and demands for high data rate for these users. If we try to incorporate new applications in this band and hence increase the number of users in this band, the effective performance of any single user will be severely degraded due to multi-user interference. Hence, the MLS-E band of ~60 MHz at 5 GHz is a potential “lifeline” for the members of the WLAN community, if they could be allowed access to it. A spectrum auction of this MLS-E band could also be a source of income for the FCC [13]. Due to this, it is imperative for the aeronautical community to show concrete steps toward utilization of the MLS-E band for future aeronautical communication applications to prevent the community’s loss of this band. The members of the aeronautical community through the International Civil Aviation Organization (ICAO), are trying to emphasize the importance of this band for current and future aviation applications by participating in the International Telecommunications Union’s (ITU’s) World Radio Conference [14]-
15]. One of the goals of NASA and United States Federal Aviation Administration
(FAA), through the ACAST project, was to try to demonstrate the applicability of the E-
MLS band to deploy wideband airport surface communication systems2. As of now, efforts are still in progress at NASA and the FAA to explore the possibility of setting up wideband communication systems on the airport surface.
2 Due to budgetary constraints, the ACAST project did not last its full planned duration. 36
The work proposed in [14]-[15] is geared toward development and deployment of new short-range wireless systems in the MLS-E band. The first step in this effort is the proper characterization of the MLS-E band radio channel. It is necessary to cover multiple settings that are important from the perspective of designing, deploying, and operating an entire airport surface network, which would serve multiple constituencies.
This includes both fixed (transmitter and receiver fixed) or mobile (transmitter fixed receiver mobile) link settings that need to be investigated. The results of this channel characterization for these various settings would help support the proposed use of wideband signaling for airport surface area communications, which in turn would help the aeronautical community retain the MLS extension band.
1.3 Intelligent Transportation Systems
In recent years, research related to intelligent transport systems (ITSs) [16] has increased dramatically. Two example references that cite applications of ITS are [17], which describes a traffic warning system intended to provide warnings to vehicles about road conditions, and [18], which discusses a similar project that helps to provide optimal travel routes and weather information for commuting vehicles. Other benefits of ITS include reductions in the probability of accidents (improving road safety), increases in commuter awareness of traffic and weather conditions in real-time, improvements in highway traffic flow efficiency, easing of “bottlenecks” at toll-booths and thus saving of time and money for commuters and the government, and the prospect of turning long journeys into times for family activities by enabling flow of multimedia between different 37 traveling cars. The information flowing between vehicles will likely be multimedia: data, images, video, and voice. A good summary of possible applications can be found in [19].
For study of ITS communication services, it is helpful to classify them into road- to-vehicle communications (RVCs) and intervehicle communications (VTV). In addition to using standard, commercially available wireless systems for communications, suggestions regarding implementing VTV communications using Bluetooth-enabled vehicles [20] or vehicular ad hoc networks have also been provided. The scope of VTV communications is not limited to a fixed number of a priori specified vehicles, and can hence be easily extrapolated to numerous vehicles via the concept of vehicular mobile ad hoc networks (MANETs) [21]. Vehicular ad hoc networks are of importance since they remove the dependency on conventional cellular networks for communication between vehicles. In densely populated areas, the current cellular architecture can become a
“bottleneck” for basic cellular calls, so transferring the transmissions of VTV communication onto other reliable networks would save VTV communication users both time and money. Public safety (PS) applications may also employ VTV communications
[23]. Reference [23] lists different frequency bands allocated for PS applications in the
US. One of the proposed bands is 4940-4990 MHz.
A recent standard [22] for VTV communication in the 5.9 GHz Unlicensed
National Information Infrastructure (UNII) band has been also developed, aiming to extend the IEEE 802.11a application environment. The most important goal in these communication services in ITSs is to reduce the number of accidents and eliminate all fatal consequences. Hence it is imperative that reliable transfer of information be enabled 38 by the VTV applications. As described before, to accurately predict the VTV system performance, it is essential to test the wireless systems under realistic channels before their actual deployment. Hence providing statistical channel models for VTV applications in the 5 GHz band will greatly benefit researchers in evaluating the performance of their systems under realistic conditions.
1.4 Dissertation Objectives
In this section, we present the objectives of this dissertation.
1. We perform a thorough literature review to properly place our work in the context
of existing work, and identify the major gaps our work will fill (Chapter 2).
2. We identify airports and cities for measurements, and develop plans for the
successful execution of measurement campaigns at those locations to characterize
the 5 GHz band for airport surface areas and VTV communications (Chapter 3).
3. Next, we obtain data in the form of Power Delay Profiles (PDPs) at various
airports (of different size, traffic density, airport infrastructure, and geographical
surroundings) for characterizing the airport surface area channel (Chapter 3).
4. We also obtain PDP data for different possible locations (cities, highways, etc.)
encountered during travel, for characterizing the VTV channel for ITS
applications (Chapter 3).
5. For both airport surface and VTV settings, we develop a framework (algorithm)
to extract the channel parameters (in the time and frequency domains) from the
collected data. Upper bounds for inherent errors in the collected data due to 39
limitations in resolution capabilities of the measurement equipment will also be
presented (Chapter 4).
6. The dynamic nature of the airport surface and the VTV scattering environment
makes it imperative to have statistically non stationarity models to faithfully
represent the underlying physical channel in both environments. We develop
random models based on first order Markov chains to account for these non-
stationarities. We then implement these non-stationary effects in our channel
models on two different time scales (Chapter 4),
a. The finite lifetime associated with each multipath component is modeled
using a two state first order Markov chain. We refer to this Markov model as
the persistence process. We use the persistence process to model the
nonstationary behavior associated with each multipath component for both
airport surface areas and VTV settings. We term this non-stationary behavior
a “mesoscopic” scale effect (Chapter 4).
b. The second Markov process is used to model the transitions when the
receiver (or transmitter) moves from one region to another within a specific
airport (Chapter 4).
7. We propose physical models to explain the occurrence of severe fading (Chapter
5),
a. The first model employs a statistically non-stationary random process that
switches between two distributions, akin to the multi-state models proposed
for land mobile satellite channels. 40
b. The other model we propose is a multiplicative model of two small scale
fading processes. This multiplicative model is due to multiple scattering,
which produces fading representable by more than one multiplicative process.
c. We develop computer simulations to replicate the statistics of these severe
fading processes.
d. We use our measured data to corroborate these models with the results of our
computer simulations.
8. The next step is to develop multiple non-stationary stochastic models (for
different bandwidths) for airport surface area channels while transmitting from the
air traffic control tower (ATCT). Multiple models (Model-1, Model-2 and
Model-3) were proposed for each region for each individual airport. The
proposed models offer flexibility to potential researchers to choose the model that
is most pertinent to their requirements of complexity and fidelity (Chapter 6).
9. Multiple non-stationary stochastic models (for different bandwidths) for airport
surface area channels when transmitting from airport field site (AFS) locations are
also developed. AFS locations were selected primarily to reach airport areas that
were difficult to reach while transmitting from the air traffic control tower
(ATCT). These channel models provided proof of concept for the idea that
transmitting from an AFS can significantly reduce the channel dispersion
compared to when transmitting from the ATCT (Chapter 7).
10. We also propose multiple non-stationary stochastic models (for different
bandwidths) for different propagation scenarios possible in VTV communication 41
systems. Multiple models (Model-1, Model-2 and Model-3) were proposed for
each propagation region, again allowing the communication system designer a
choice (Chapter 8).
11. Finally, we implement the developed stochastic channel models (from items 8 and
10 here) in MATLAB. We then compare the accuracy of the developed models
with the data. Comparisons are also made based on different time domain and
frequency domain statistics (Chapter 6 and 8).
1.5 Dissertation Contributions
In this section, we list the publications generated out of the research presented in this dissertation3
Published Articles
[J1] D. W. Matolak, I. Sen, W. Xiong, R. D. Apaza, “Channel Measurement/Modeling for Airport Surface Communications: Mobile and Fixed Platform Results,” accepted for publication in IEEE Aerospace & Electr. Syst. Mag.
[J2] D. W. Matolak, I. Sen, W. Xiong, “Measurement and Modeling Results for the 5 GHz Airport Surface Area Channel: Part I, Large Airports,” accepted for publication in IEEE Trans. Veh. Tech., May. 2007.
[J3] I. Sen, D. W. Matolak, W. Xiong, “Measurement and Modeling Results for the 5 GHz Airport Surface Area Channel: Part II, Small Airports,” accepted for publication in IEEE Trans. Veh. Tech., May. 2007.
3 We use the notation J: journal paper, C: conference paper. 42
[C1] I. Sen, D. W. Matolak, “Multi-user Spread Spectrum Modulation Performance in a VTV Environment,” to appear in Proc. IEEE International Symposium on Wireless Vehicular Communications, Baltimore, MD, Sep. 30th- Oct. 1st, 2007.
[C2] B. Wang, I. Sen, David W. Matolak, “Performance Evaluation of 802.16e in Vehicle to Vehicle Channels,” to appear in Proc. IEEE Vehicular Technology Conference, Baltimore, MD, Oct. 1-3, 2007.
[C3] I. Sen, Beibei Wang, David W. Matolak, “Performance of the IEEE 802.16 OFDMA Standard System in Airport Surface Area Environments,” in Proc. ICNS 2007, May. 1-3, 2007, Herndon, VA.
[C4] I. Sen, D. W. Matolak, “Wireless Channels that Exhibit “Worse than Rayleigh” Fading: Analytical and Measurement Results,” Proc. MILCOM 2006, Washington D.C., Oct. 23-Oct. 26, 2006.
[C5] I. Sen, D. W. Matolak, “Channel Modeling for Airport Surface Network Communications: Transmission/Reception from Airport Field Sites,” Proc. Digital Avionics Systems Conf. 2006, Portland, OR, Oct. 15-19, 2006. (Best paper in Communication, Navigation and Surveillance Track).
[C6] D. W. Matolak, I. Sen, W. Xiong, “Channel Modeling for VTV Communications,” invited paper at Vehicle-to-Vehicle Communications Workshop (VTVCOM 2006), July 21, 2006, San Jose, CA, USA.
[C7] D. W. Matolak, I. Sen, W. Xiong, “Multicarrier Multiuser Modulation Performance in Severely Fading Channels,” Proc. Symposium on Trends in Communication, Bratislava, Slovakia, 24-27 June 2006. [C8] D. W. Matolak, I. Sen, W. Xiong, R. D. Apaza, L. Foore, “Wireless Channel Characterization: Modeling the 5 GHz Microwave Landing System Extension 43
Band for Future Airport Surface Communications,” Proc ICNS 2006, Baltimore, MD, Apr. 30–May 3, 2006.
[C9] D. W. Matolak, I. Sen, N. T. Yaskoff, W. Xiong, “5 GHz Wireless Channel Characterization for Small Airport Surface Areas,” Proc. Digital Avionics Systems Conf., Washington DC, Oct. 30-Nov. 3, 2005.
[C10] D. W. Matolak, I. Sen, W. Xiong, N. T. Yaskoff, “5 GHz Wireless Channel Characterization for Vehicle to Vehicle Communications,” Proc. MILCOM 2005, Atlantic City, NJ, Oct. 17-19, 2005.
[C11] D. W. Matolak, I. Sen, R. D. Apaza, “MLS Band Sounding Measurements,”NASA Glenn Research Center's 2005 ACAST/SBT Workshop, Cleveland, OH, August 16-17, 2005.
[C12] I. Sen, D. W. Matolak, W. Xiong, N. T. Yaskoff, “5 GHz Wireless Channel Characterization for Airport Surface Areas,” Proc. 15th MPRG Wireless Symposium, Blacksburg, VA, June 8-10, 2005.
Articles in Review
[J4] I. Sen, D. W. Matolak, “Vehicle to Vehicle Channel Models for the 5 GHz band,” in revision with IEEE Trans. Intelligent Transportation Systems, Feb. 2007.
[J5] D. W. Matolak, I. Sen, W. Xiong, “On the Generation of Multivariate Weibull Random Variates,” submitted to IEE Electronic Letters, March 2007.
44
2 Literature Review
2.1 Introduction
In this chapter we discuss some of the research that is pertinent to this dissertation. The primary objectives of our literature review are,
¾ Gather and understand as much of the research already completed in the areas
related to our work.
¾ Use the work gathered from the review to orient and provide guidance for the
new work.
¾ Understand the limitations and gaps in the current body of work, and try to
fill some of these gaps and remove some limitations.
We will begin with a brief discussion on some of the books and classical papers related to deterministic and stochastic channel modeling. Understanding the key ideas presented in these “foundation” papers is imperative to understanding and developing procedures for creating empirical models.
In general, the area of airport surface channel modeling has seen substantially less attention than other channel modeling settings, and most of the references focus upon air to ground links in which a line of sight (LOS) propagation path exists between transmitter and receiver. Here, we provide references that address channel modeling (through empirical and simulated data) for air-to-ground (A-G) and airport surface area (G-G) applications. Followed by the references related to airport surface channel models, we discuss recent analytical and empirical models that have been proposed for vehicle-to- 45 vehicle (VTV) applications. Next, we provide references that we have gathered on the following related topics: channel measurement and data processing techniques, severe fading (“worse than Rayleigh”), and alternative distributions used to model small scale fading. As is well known, any literature review can never be totally complete
(particularly during the present day, when research is expanding daily), but we have tried to be as extensive as possible. In this process, some of these references were gathered after obtaining preliminary results for the channel models.
2.2 Generic Channel Modeling
References [1], [2] are excellent texts for introduction to digital communications and fundamentals of wireless communication systems. They provide an introduction to the basic model of communication, from the generation of the signal at the source through a noisy channel to reception of the signal at the “sink.” For most cases in the literature and also in this dissertation, the term channel model refers to the representation of the physical (wireless) media without considering transmitter and receiver filtering or antennas. This then allows representation of the channel as a linear, time-varying filter.
As such, it is characterized completely by its impulse response. Thus, for our empirical approach, we can estimate the channel by collecting channel impulse response estimate
(IRE) estimates at different instants of time. (Ideally, one would desire the rate of channel time variation to be slow enough that the channel does not change appreciably during the time that the CIR is estimated.) 46
Next, we cite [3], [4] as very thorough, general text references on digital mobile communications for advanced graduate level course-work. Reference [4] can be considered as a general reference book for mobile communications. This reference covers analog and digital systems, propagation physics, coding, modulation, equalization and diversity. The 2nd chapter in [4] provides a fairly complete coverage of propagation modeling, channel impulse response characterization, and statistics. Focus is on the terrestrial environment. Text reference [5] is one of the few books devoted entirely to propagation and the creation of stochastic and deterministic channel models. The treatment provided in Parson’s book on large and small scale fading is thorough, and provided an excellent reference book throughout the course of our research. Reference
[6] provides a systematic framework to develop the various random processes necessary to create channel characterizations. The sum of sinusoids method is one of the most popular methods that exist to generate Rayleigh fading random processes. Some other books that provide good fundamental knowledge for fading channels are [7], [8].
Reference [7] is very popular for the classical “Clarke” Doppler spectrum for two- dimensional isotropic scattering.
Reference [9] is a classic reference that defined the various relationships existing between signals that are transmitted over linear time variant channels. Most of the literature that followed this paper used the standard (simplifying) assumption of wide- sense-stationary uncorrelated scattering (WSSUS) introduced in this paper. It provides relationships and mathematical derivations regarding the duality of correlation and wide sense stationarity in time and frequency domains. Bello’s use of the classification 47
WSSUS to describe the dynamic scattering environment was seen as a reliable way of statistically modeling the channel, as compared to the complicated analysis that must be done using electromagnetic wave theory. Reference [24] can be suggested as a tutorial paper on fading. The paper introduces long and short term fading phenomena, and then provides a statistical characterization of randomly varying channels. Discussion on the physics behind fading is also discussed and a review of simulation and fading mitigation techniques is included. The author also discusses the different channel parameters corresponding to different propagation settings. The environments considered are very different, ranging from ionospheric skywave HF to microwave line of sight communication links. Reference [25] provides basic definitions of the different statistical parameters that can be used to determine the frequency selectivity of the channel. Two of the key parameters are the root mean square delay spread (RMS-DS) and the delay window. The delay window can be interpreted in a similar manner to the RMS-DS. More discussion of relevant channel parameters will be provided in Chapter 4.
References [26], [27] are some “now-classic” papers that discuss how to statistically model “short-term” variations in the channel for terrestrial and indoor scenarios. Reference [26] was unique since the author proposed to model the fading process (small and large) using a single random process. The author proposed to model the small scale fading using a Rayleigh random process, the mean power of which had lognormal distribution. The author also proposed a similar model using Nakagami and lognormal random processes. This approach removed the necessity of modeling the short and long term channel fading components separately. Reference [27] was the first paper 48 of its kind to model the indoor channel as a sum of various clusters of multipath components—each cluster contains multiple rays. The arrival time of the clusters and their rays were modeled using Poisson random variables, and the amplitude of the individual rays was assumed to have a Rayleigh distribution. The authors corroborated their statistical model with collected measurements, and hence provided valuable insight into the propagation conditions encountered by communication systems in an indoor setting. This approach has since been widely used by system designers to develop models for current WLAN and other indoor communication systems.
2.3 Measurement and Data Processing Techniques
In this section, we discuss the some of the key references that we used to develop our data processing algorithms. Reference [28] uses a measurement procedure which is very similar to the approach that we employ to gather channel impulse responses (CIR).
The authors describe the principle of a “sliding correlator” channel sounder to help understand the process of gathering IREs. A spread spectrum channel sounder [29] gathers these IREs, which in turn can be used to develop stochastic tapped delay line channel models [4]. As with any reading obtained from a communication setting, the measured CIRs are affected by the thermal noise. Due to this, it is possible that we might misinterpret some noise “spikes” as low power multipath components. In order to reduce the likelihood of this event, we need to implement “noise-thresholding” on the measured data before actually doing any additional processing. Reference [30] outlines the “noise- 49 thresholding” algorithm that we use to minimize the probability of mistaking a noise spike for a multipath.
Reference [31] is good source to help understand the need for careful organization for statistical data processing of collected RF data (channel measurements). Despite the difference in application and underlying physical channel (indoor), the different statistical parameters (e.g., multipath-delay distribution, temporal correlation functions, etc.) used by the authors of [31] can be determined for RF data collected for any generic application. After obtaining preliminary results from our data, we determined that our
IREs often didn’t meet conditions required for classification under the conventional
WSSUS channels. Hence, we used a technique proposed in [32] to determine the frequency correlation estimates for our collected data.
A missing factor in all these references is the detailed description of the steps (or the “processing framework”) that precedes extraction of the time and frequency domain channel parameters from the collected RF data files. We tried to address this missing factor in our research. In Chapter 4, we explain the algorithms involved in all the pre- processing and post-processing steps that need to be performed on the “raw” RF data to extract the empirical channel models.
2.4 Airport Surface Area Channel Models
The aeronautical channel comprises various kinds of communication links: Air-
Air (A-A), Air-Space (A-S), A-G and G-G. The G-G link can be termed airport surface 50 communications. In this section, we discuss channel models proposed for the different links but focus on the models proposed for airport surface communication systems (G-G).
Some of the earliest references for small scale fading in an aeronautical environment discuss A-S [33] and A-G [34] channels. Reference [33] discusses the channel between an aircraft and a satellite. In [33], Bello uses his system and correlation functions (from [9]) to characterize the channel. The author proposes that an A-S channel can be modeled as consisting of a direct path, a direct specular reflection and a diffuse scatter component. The direct path and direct specular reflection have time varying delays and complex gains due to antenna gains and free space losses associated with them. The diffuse scatter component accounts for multiple reflected signals from the surface of the earth. The authors of [34] proposed a frequency non-selective A-G link that consists of a dominant “specular” component and a “perturbation.” The perturbation is expressed mathematically as a function of the reflection coefficient of the earth surface, link-distance, bandwidth and frequency, and is accounted for by a multiplicative factor on the received signal. The reflection coefficient can be modeled as a Gaussian random variable to account for terrain irregularities. The “specular” component and the perturbation can be considered analogous to the strong specular component and diffused components in a Rician distribution [4]. The authors treat the multipath (reflected) component of the received signal as a Gaussian random variable with time varying statistics and hence emphasize that the “perturbation” is non-stationary and hence the detection procedures need to be adaptive to compensate for this. The authors verified their model by comparing it with the results of a flight experiment that they conducted. 51
Reference [35] characterizes the signal propagation behavior in the MLS band at an airport surface area, analytically. The results were obtained using ray-tracing and electromagnetic field theory. The authors also provide a small amount of measurements of the ratio of received diffuse power to power in the specular (dominant) component.
This ratio is similar to a Rice factor [4]. The measurements used to corroborate the analysis in [35] were performed for a single-tone.
As compared to the carrier frequency of current aeronautical communication systems (118-137 MHz), the MLS band uses a much higher frequency. Also, due to ever increasing demands for high data rate applications and increasing numbers of users, current and future communication systems require greater bandwidths than the conventional 1-1.5MHz [4] used in terrestrial cellular. To evaluate the performance of high bandwidth (BW) systems under realistic conditions, we need to develop wideband channel models of the propagation environment. The scattering environment is highly dynamic in nature due to the mobility of the transmitter, receiver and scatterers. Hence, the fading environment that a signal encounters is quite variable. Theoretically, it is possible to model such a channel deterministically, but, we would need to accurately model the physical composition and structure of the scatterers. We would also need precise mathematical models to emulate the movement of the transmitter, receiver and scatterers in the model, etc. Such a deterministic model would not only be cumbersome to develop but also would be highly location specific. Hence, it is better to approach modeling such a dynamic channel using stochastic processes. A stochastic model of the channel would rely on the data collected at different locations and would help reproduce 52 the dynamic behavior of the channel with much more ease than a deterministic model
All of the work discussed above has either been narrowband, LOS, or deterministic, and hence has severe application limits. Because of these limits, the actual channel is represented with less accuracy than is possible, and less than is required for future transmission schemes; this is further motivation for our work.
As discussed previously, we are mostly interested in characterizing the channel for GG applications. Among recent papers, there has been some research to characterize the airport surface channel. Reference [36] is one of the few such references. This reference relies very heavily on existing land-mobile channel models. The aeronautical channels are proposed for different “phases of flight”, including “parking,” “taxi,”
“arrival,” and “en-route.” For our research, we are interested in the behavior of the channel in airport surface areas, which include the “taxi” and “parking” phases only. The models proposed in [36] for these phases are based on drawing an analogy between these environments and corresponding terrestrial cellular regions (e.g., urban, rural, etc.). The different Doppler spectra proposed in [36] also rely on the scattering environment of the analogous terrestrial regions. The validity of these “analogies” is questionable since the scattering environment encountered on airport surface areas is significantly different from those observed for terrestrial cellular. The terrestrial cellular models mostly assume isotropic scattering, but it is very rare to encounter isotropic scattering on airport surfaces. Some other differences between the assumptions of [36] and actual conditions on the airport surface include the following: the size of the scatterers (large aircraft, refueling vehicles, etc.) are different; the transmitter (Air Traffic Control Tower) is 53 generally much higher than surrounding buildings, but in the case of terrestrial cellular systems the height of the base station is not necessarily much different from that of the surrounding buildings. Based on these observations, creating analogies between the airport surface channel models and terrestrial cellular regions could easily yield an inaccurate representation of the major propagation effects of the airport surface.
In [37], results on channel characteristic for a wideband “aeronautical telemetry” channel are provided. The aeronautical telemetry channel uses a high-gain, tracking antenna at the ground site. The channel for the measured case was found to have an LOS component and a dominant ground reflection along with a secondary reflection that can be modeled as having a Gaussian distributed amplitude. To the best of our knowledge,
[36] is the only work which has been done for wideband channel models in a GG application at airports. The underlying assumptions (basis on terrestrial models) and the lack of empirical data to corroborate the model proposed in [36] casts substantial doubt upon its utility and accuracy. In [37], the directionality of the antenna makes the channel results very application oriented. Hence to get a more realistic estimate of the G-G propagation environment, we need data actually collected at airports and subsequent models for the channels built from this empirical data.
Next we briefly discuss work done on A-G channel models, and cite one reference for signals propagating inside airplanes. Though these are beyond our scope of G-G applications, we provide this discussion to give readers a sense of the channel model in these distinct aeronautical scenarios. Reference [38] provides a rough depiction of an A-
G channel. The analytical model developed is a flat faded channel, which assumes a 54
Rician distribution for the amplitude statistic. The authors mention that single tone measurements conducted in the VHF aeronautical band (118 – 137 MHZ) at Midway
Airport, Chicago and St. Paul Airport, Minneapolis, confirm their Rician model. In [39],
CW and sliding correlator measurements were used to provide small and large scale fading modeling for A-G applications in the VHF band (135 MHz). Though the authors note that for particular situations, for example, when the plane is taking a turn, there exists a strong multipath component at 1.4 μsec, they provide statistics for a non- frequency selective channel. The flat faded small scale fading was found to have a strong
Rician behavior. The range of the Rician K-factor was between 2.6 dB to 19.7 dB and the average was found to be around 16 dB.
In [40], the authors provide a geometric channel model for the A-G channel and present analytical results for joint probability density functions for delay and direction of arrival (DOA) for multipath components. For a given range and altitude of the airborne station, the scatterers are assumed to be uniformly distributed in an ellipsoid. The ellipsoid has an axial symmetry about the LOS and has the slant range between the ground and airborne station as 2f (where f is the foci of the ellipsoid). By calculating the joint pdf for delay and DOA for different elevation angles (defined as the angle between horizon and airborne station), the authors concluded that the DOA distribution becomes uniform as the elevation angle increases. In [41], the same authors provided actual measurement results to evaluate the channel model developed in [40]. Various statistical channel parameters like RMS-delay spread, excess delay, antenna diversity gain, etc., were provided. The authors inferred that the selectivity of the channel increases with a 55 decrease in elevation angle, and this same trend was verified by evaluating the cdf of the multipath fading envelope.
Another paper that talks about propagation characteristics for signals in a different setting is [42], in which the authors quantify the effects of the in-cabin environment
(within the body of airplane, taking into account furniture, passengers, etc.) and of avionics equipment on the received signal of a CDMA handset. From the above discussion, it is evident that researchers are investigating the propagation conditions for multiple aeronautical communication scenarios. It is necessary to understand that the channel for A-G links would be far less dispersive than the channel expected for a G-G link on the airport surface due to the availability of a LOS link for most of the time in the
A-G case. Hence, modeling of A-G links using ray-tracing approaches would likely be sufficient to provide working models for the A-G channel. In the case of G-G links though, a stochastic approach is necessary to account for the much richer scattering and disparate propagation conditions that can be encountered on the airport surface.
2.5 Vehicle to Vehicle Channel Models
A large amount of previous research on the VTV channel topic has pertained to millimeter wave (mmwave) bands, e.g., [43] and [44], which usually use highly directional antennas to circumvent the large propagation path losses encountered in that band. Analytical studies for this type of channel [45] have also been done. Due to increasing interest in personal area networks, there have been efforts to understand the feasibility of using Ultra Wideband (UWB) based systems for communicating in a VTV 56 environment [46]. Current UWB systems are designed to be short-range communication
(due to regulatory policy) and hence may be suitable for platoon (military) vehicles but for consumer applications, it is difficult to provide an “always connected” communication link between vehicles using such point-point links except in some highway situations. Hence it would be beneficial to have a “broader range” network than can be supported by only these directional point-point links. This would enable multiple vehicles to be in contact irrespective of the obstacles and traffic conditions among them, and at larger distances than are achievable with the point-to-point links. Hence, an increase in link range along with the ability to support reasonably high data rate applications makes the 5 GHz band a more suitable candidate than the 60 GHz band for upcoming VTV systems.
In the literature, we find some references on channel models for VTV applications. For example, a recent paper [47] discusses the implementation issues with respect to the medium access and routing layers for potential VTV applications in this band. The authors considered a simple 2-ray model for their channel. In reference [48], the VTV channels considered were AWGN, single-path Rayleigh fading, and 2-tap
Rayleigh fading. The authors assumed equal power in both channel taps, and as usual, the phases were assumed to have a uniform distribution. Similarly, in reference [21], the authors assumed a flat fading Rayleigh channel for simulating the error-rate of mobile ad- hoc networks in VTV scenarios. Additionally, some wideband and narrowband measurements in the 5 GHz band were recently reported in [49] and [50]. The authors of
[49] corroborated some channel measurements with ray-tracing simulation results for 57 urban and motorway channels. For the simulation, the authors assume the transmitter vehicle to be stationary to limit the complexity of the ray-tracing approach. For the simulation, the authors employed an established “built-in” traffic model. The “built in” traffic model was proposed in [51]. The model uses vehicles, road lane models, road surroundings, etc., as stochastic parameters to provide the randomness in the VTV channel model. The signal propagation is then simulated in this traffic model using a
“ray-throwing” approach. Recent wideband measurements were made for the expressway channels in the 2 GHz and 5 GHz bands were reported in [52] and [53]. In [52] the authors report an 8-tap channel model for a bandwidth of 10 MHz, while in [53], the authors propose 6 and 12 tap channel models for a bandwidth of 20 MHz. The authors of
[53] provide 2 different length channel models to provide a tradeoff between implementation complexity and fidelity. Hence, it is also clear from the limited measurements presented that the VTV channel can be quite dispersive.
To the best of our knowledge, there are fewer than ten research papers that characterize the VTV channel using actual measurements. Also, the measurements that have been reported limit themselves to certain typical environments like urban, expressway, etc. On the basis of these arguments, it is apparent that there is a glaring scarcity of data collected for VTV scenarios in the 5 GHz band. Direct dependence on
“assumed” models without the backing of actual data wouldn’t be the best approach for system evaluations. A VTV communication network could operate as a mobile LAN, and transceivers in this kind of VTV communication system will encounter channels that are much different from those of the cited flat fading and 2-ray models. Some reasons for this 58 are that transmitter (Tx) and receiver (Rx) and significant reflectors/scatterers are all mobile, the antennas for both Tx and Rx are at relatively low elevations, and the channel will often be statistically non-stationary due to mobility of Tx, Rx, and scatterers.
There are also some examples of analytical mobile-to-mobile models without regard to carrier frequency, such as [54]-[57]. These models are mainly based on the sum of sinusoids approach and assume isotropic scattering either at the receiver [57] or at both receiver and transmitter [55]. The models are also developed using a WSSUS assumption. As discussed previously, rarely will one find isotropic scattering, and WSS conditions can pertain for only a limited time. Some examples of measurements in environments with non-isotropic scattering were reported in [58] and [59]. Recent analytical studies involving non-isotropic scattering for mobile-to-mobile communication settings are presented in [60]. These studies indicate that while isotropic scattering always produces symmetric Doppler spectra, for non-isotropic scattering, in certain specific scattering conditions (for example, when the scatterers are concentrated in the front or back of the vehicles) one-sided Doppler shifts are likely. These observations are assumes that the location of the scatterers around the transmitter and receiver follow a
Von-Mises pdf. However, as of now, there have been no measurement results to corroborate the analytical studies.
There is a modest amount of literature that addresses time-varying Doppler spectra for mobile-to-mobile channels at different carrier frequencies, for examples, [52],
[53], [61], [62] and [63]. The presence of different Doppler spectra over time is due to changes in scattering and to changes in the relative velocities of the transmitter and 59 receiver with time. The shape of the Doppler spectrum is quite different from the conventional “Clarke spectrum” associated with terrestrial cellular (fixed-to-mobile) links. In the case of mobile-to-mobile links, the Doppler spectrum looks similar to that of a low pass filter (LPF) for most of the cases. In certain scenarios, when the density of scatterers is large, due to a large range of angles of arrival (AoA) of the waves, the
Doppler spectrum can have the “Clark” “horned spectrum.” The “horns” are located at approximately +/- the maximum Doppler frequency. But even for these specific examples, since the maximum Doppler frequency is much less than the 10 MHz bandwidth (or symbol rate) of VTV systems, the “horned” doppler spectra can still be approximated as an LPF.
2.6 Severe Fading
Normally, one of the worst case channels that can be encountered in fading settings is the Rayleigh fading channel [4]. When we encounter conditions that yield statistics worse than Rayleigh (WR), we consider those channels to be severely faded.
Though occurrence of these scenarios is generally rare, as more research into channel modeling proceeds, investigations have found an increasing number of instances in which channel fading is more severe than Rayleigh.
Some of the earliest references that we have found for severe channel fading are
[64]-[66]. In all these references, fading was observed in the measurement of ionospheric scintillation in the HF band. These authors presented empirical fits to fading data using the Nakagami-m distribution, with some m-factors less than unity. This corresponds to 60 fading more severe than Rayleigh, for which the Nakagami-m factor is equal to one.
More recent examples of severe fading include measured results for the “RF backscatter” channel, encountered in two-way RF identification applications [67]. In reference [68], the authors reported measurements from a suburban scenario at 1.5 GHz, and they used different goodness of fit tests—minimum description length and log-likelihood—to compare different statistical distributions. The authors used several distributions to model their empirical data, e.g Weibull [69], Nakagami, etc. The Weibull distribution with shape factor less than 2 and Nakagami distribution with m factor less than 1 were good models for a significant percentage of their collected data. The same authors of [68] also report WR fading in mobile cellular measurements in urban and rural settings in Japan in reference [70]. Reference [71] reports mobile-to-mobile measurements at 1.8 GHz in a forest environment. This author reports data that also shows WR fading in that distinct environment. Examples of WR fading are also seen in indoor UWB applications at 3-10
GHz, such as the ones reported in [72] and [73]. The Weibull, Nakagami, and lognormal distributions were used to model the experimental data.
In addition to measurements, there have also been efforts to establish physical and/or analytical justifications for the WR behavior. Most of the models proposed have been based on the “multiplicative” process in which two (or more) small scale fading random processes are multiplied to create the complete fading amplitude. In the literature we have found different geographical locations and communication systems that encounter such multiplicative models. For examples, the multiplicative model has been used for propagation around street corners [74], “pin-hole” channels encountered in 61
MIMO settings [75], [76], and “amplify and forward” channels in relay networks [77].
The earliest reference that we have found for such multiplicative modeling is [74], which uses a product of two Rayleigh faded processes. Such a WR channel is often referred to as a “double Rayleigh” channel. The authors of [74] also provided some measurement data from Manhattan and Boston at 900 MHz and 2 GHz in support of their WR channel.
Since then, work has been done on other types of multiplicative models using different distributions such as N independent Rayleighs [78] and N independent Nakagamis [79].
Reference [71] suggests a summation of different order multiplicative models to model the WR channel. By order, the authors mean the number of Rayleigh processes involved in the multiplicative model. Interestingly, the main idea behind such a model is to remove the necessity of separately modeling large scale fading using the typical lognormal distribution. The authors corroborate their model with measurements made in a forest environment.
Another plausible explanation for WR behavior in UWB transmission has been proposed in [72]. The authors conclude that WR fading exists due to the presence of a small number of multipath components within the delay resolution possible for UWB signals. In this case, the central limit theorem is inapplicable, thus invalidating the usual
Rayleigh fading consequence. The authors do not though provide any experimental results that corroborate this hypothesis. An analytical explanation for WR fading may be possible from [80], which proposes an analytical probability density function (pdf) consisting of two specular multipath components in the presence of diffuse multipath components. The authors show that in situations where the specular components are 62 nearly equal in power and have opposite phases, the resultant channel will yield WR fading.
On the basis of the above discussions, the following two things are evident: WR fading occurs in a variety of different settings for a non-negligible percentage of time; and, none of the physical explanations for WR fading has been replicated in simulation for comparison against measurements or for corroboration of these explanations. The objectives of our recent research have been to increase the richness of the relatively sparse measurement database for WR fading, to provide additional physical justifications for the occurrence of WR fading, and to develop simulations that enable generation of
WR faded samples using multiple physical models.
2.7 Weibull Fading Process
The Weibull distribution can be used to model small scale fading in indoor as well as outdoor environments. A few references for indoor scenarios are [81]-[83], and for outdoor are [70], [84]-[86]. In addition to finding use of the Weibull distribution in the literature, this distribution was also found to be a better fit to our measured data during the course of preliminary processing. In general, the Weibull distribution [69], [87], can be expressed as follows:
⎡ b ⎤ b b−1 ⎛ r ⎞ pw (r) = b r exp⎢− ⎜ ⎟ ⎥ (2.1) a ⎣⎢ ⎝ a ⎠ ⎦⎥ where (b > 0) is the fading parameter, which indicates the severity of the fading: as b increases, the channel condition gets better. When b=2, the Weibull distribution becomes 63 the Rayleigh distribution. The average fading power E(r2) is given by a 2Γ(2 b +1), where E( ) denotes expectation, and Γ( ) denotes the gamma function. The second order statistics of the Weibull distribution were studied in [88]. In [89], a closed-form expression was derived for the moment-generating function of the Weibull distribution, valid when its fading parameter assumes integer values. There are many papers that are available that evaluate the performance of digital communication systems in the presence of Weibull fading, e.g., [90]. A few of the papers that provide analytical work for the performance of diversity receivers in the presence of Weibull fading channels is as follows: switched diversity receivers [91], dual selection receivers [92], maximal ratio combining and equal gain combining [93]. Thus the performance of communication systems for Weibull distributed fading is clearly of current and future interest.
In our measured data, we encountered correlated scattering and different fading behavior among the taps of the tapped delay line model [4]. To have an accurate simulation that closely approximates the measured channel; we thus need an algorithm to generate a multivariate Weibull distribution with arbitrary correlations, fading parameters and energy distributions. Reference [94] provides a way of generating bi-variate
Weibulls with the above mentioned criteria. Yet for channels with more than two resolvable multipath components, we require an algorithm for generating multivariate distributions. The earliest reference we have found for generating correlated Weibull variables with the same fading parameter is [95]. Reference [96] provides a method for generating complex Weibull variates with arbitrary covariance using complex Gaussian variables. There has been much research on the generation of multivariate Nakagami 64 variables as well, e.g., [97], [98] but comparatively little on the generalized Weibull case.
Similarly, research has addressed generation of correlated Rayleigh variates [99], [100].
To the best of our knowledge, there exists no method or algorithm to generate multivariate correlated Weibull random variates with arbitrary energies and fading parameters. Creating an algorithm for this was one of the objectives of this dissertation.
2.8 Non Stationary Channel Models
Bello’s paper [9] was the first of its kind to provide a systematic and clear representation of the channel by describing it in terms of delay dispersion and Doppler domain characterizations. Bello’s use of the classification WSSUS to describe the dynamic scattering environment successfully emulated the main characteristics of any given propagation environment with stochastic channel models. Yet, since nonstationary effects in the channel have been observed with different time scales, and have been reported for some time, [24] and [28], the WSSUS classification is simply not adequate for accurate modeling. Modeling non-stationary effects of the channel has not to date gathered much momentum due to the “acceptable accuracy” of the stationary channel models to depict the channel in most applications. But, due to the ever increasing bandwidth of wireless transmission schemes and the ever more stringent quality of service (QoS) requirements, it is necessary to understand whether a WSSUS representation of the channel for current and future applications is sufficient. Over the years researchers have encountered instances of non-stationary effects in disparate 65 environments. A few example cases where non-stationary effects have been cited as important are as follows:
¾ Long range (time) adaptive transmission methods like the ones proposed in [101]
and [102]. In these transmission methods; modulation, power control, channel
coding and antenna diversity can be changed adaptively to account for the time
varying channel. The authors in both the references stress the need for non-
stationary models to accurately depict the channel for applications that operate
over longer periods of time.
¾ Applications that lack a centralized base station like ad-hoc networks [61].
Nonstationary modeling is essential in these cases since there exists a significant
probability of the transceiver moving through widely different propagation
environments within a small duration of time without the compensating effects of
power control for large scale fading
¾ Instances of time varying Doppler spectra. These have been reported for several
VTV communication scenarios [53], [61], [62]. As noted, compared to
conventional terrestrial cellular systems, VTV systems have comparatively
smaller heights of at least one transmitter; have a greater probability of high
mobility for transmitter and receiver, etc. For these reasons, the scattering
geometry changes more frequently than in conventional cellular, yielding time
varying Doppler effects. 66
¾ Almost all standards--whether cellular [103] [104], LAN [105], or VTV [106]
[107]--model the distribution of the RMS-DS. This is a tacit way of implying
non-stationarity without actually modeling it.
From this brief survey of related literature, we note again that non-stationary channel effects occur for a non-negligible amount of time and in a variety of scenarios.
Hence it is clear that non stationary models are needed to enable more accurate evaluations of system performance. There have been several efforts in the literature to model these non-stationary effects. The approaches have been both analytical and based on measurements and simulations. Some of the analytical methods depend on electromagnetic wave theory or deterministic techniques, whereas some analytical approaches follow a statistical treatment. We now describe some of the more common non-stationary modeling approaches and applications.
One of the easiest ways of incorporating non-stationarity is to change the multipath component amplitude distribution as a function of time. This approach relies primarily on the accuracy of the measured data from which the parameters of the distribution are determined, over some intervals of time. Examples of this approach can be found in [108] for the time-varying Nakagami m factor, or the time-varying Ricean K factor [109]. Two limitations of this approach are that modeling a frequency selective channel with multiple taps using this approach can be cumbersome, and the accuracy of the varying parameters of the statistical distribution may be limited due to the limited statistical confidence of measured data when available only from small time intervals. 67
Non stationary models have also been developed for mobile satellite channels
[110]. In this setting, the channel amplitude is modeled using two “states,” and each state has its own statistical distribution. By “state,” we mean the channel conditions, so a
“good” state could be LOS and a “bad” state could be NLOS channel conditions. The complete model can then be expressed as a composite pdf which consists of a weighted sum of the individual state pdfs, with weights corresponding to the probability of being in a particular state. Some non-stationary models have been proposed for terrestrial cellular and LAN systems by creating different stochastic tapped delay line models for different delay spreads, [106], [112]. In this approach, frequently occurring delay spreads are chosen and models are created to emulate these delay spreads. The final model switches between these two independent channel models. The obvious limitations of this method are the need to “interpolate,” or otherwise “smooth” the switch, and the inherent limit of using only two models.
Another proposed way of incorporating non-stationarity is to model the finite lifetime associated with individual multipath components, e.g., [113], [114]. The primary motivation behind this approach is to model multipath components in an “ON/OFF” manner due to the changing scattering environment. It is well known in the literature that large scale fading such as “shadowing,” and small scale fading are often referred to as
“macroscopic” and “microscopic” effects, respectively. In [113], the authors refer to the
“arrival/departure” of multipath as a “mesoscopic” effect. The authors have not specifically proposed any approach to model this “mesoscopic” behavior of the propagation environment, but the terminology captures the spatial scale as being between 68 that of small scale (~λ/2) and large scale (many λ). In [114], the authors have proposed multiple ways for modeling, such as via 2D spatial filtering, turning entire clusters of multipath components ON/OFF, etc. None of these approaches have yet been implemented as far as we know, hence it is difficult to assess the utility of the methods.
The authors of [114] do point out that incorporating non-stationarity by turning ON/OFF clusters of multipath components might be easier than utilizing a 2D spatial filter.
Another way of representing the ON/OFF behavior is using a 2 state Markov model,
[115], [116]. The ON state can be modeled as 1 and the OFF state can be modeled as 0.
In this manner, we can account for the arrival and departure of the multipath components.
As discussed previously, it is very likely to encounter time variant Doppler spectra in VTV communications. One can model the time varying Doppler spectra by switching between two (or more) different Doppler spectra [61]. Since in most cases, the shape of the Doppler spectra for VTV resembles that of an LPF, the time varying Doppler spectra can be implemented by “switching” between different LPFs. The LPFs are generally simulated using FIR filters. Still, for realism we would need to interpolate between the coefficients of the FIR filter to smooth the transition between the switch. To prevent this problem, it is simpler to use an “interpolating function” instead of the FIR filter. This is a more convenient approach to simulate the different LPFs and ensure a smooth transition between the tap amplitudes during the switch between different
Doppler spectra.
A deterministic approach to model non-stationary effects so as to evaluate the performance of a channel coefficient prediction algorithm was proposed in [101] and 69
[102]. This deterministic flat fading model was implemented by approximating the resultant of multiple electromagnetic field components. using an autoregressive (AR) process. Due to the required use of electromagnetic field theory, these single-tap models are much more complicated than corresponding stochastic models, e.g., the conventional flat Rayleigh model. Also, modeling a correlated frequency selective channel with multiple taps would be virtually impossible using this approach, due to the tremendous increase in computational complexity required to account for multiple multipath components that are dependent. The performance of the channel prediction algorithms was also found to be worse on measured data than on simulated non-stationary models.
This might be in part due to inaccuracy in modeling the electromagnetic field resultants with a simple AR process. Another point worth making is that there is no physical explanation behind the use of an AR process to model channel amplitude fading.
Recently, there has also been work [59] to provide a framework to characterize the non-WSSUS channel using a framework similar to the one provided by Bello in [9].
By non-WSSUS we mean both non-WSS in time (equivalent to time varying Doppler spectra) and correlated scattering among taps (equivalent to non-WSS in frequency). The author of [43], Matz, characterizes the Doppler and delay domain using four-dimensional functions, and in a sense has been successful in developing a framework so that the traditional WSSUS model is a particular case of this generalized non-WSSUS model.
Matz also notes that the considered non-WSSUS model can be approximated as WSSUS over small intervals of time and frequency. The generalized local scattering function
(GLSF) of [59] is forced to use smoothing to maintain its analogy to the local scattering 70 function (LSF), where the LSF is a generalization of Bello’s scattering function that does not impose WSS in time or US in delay. Matz was in [43] able to relate certain measured quantities to his LS functions. Nonetheless, due to the inherent complexity of this framework, this approach would appear to have minimal utility on measurements or for developing models based on them.
References [117], [118] use a conventional auto-regressive moving average approach to generate time samples for the tapped delay line models. The non-stationarity is incorporated by varying the generalized coefficients in time and frequency. The vector time frequency ARMA (VTFAR) models presented in [118] can be interpreted as vector extensions of the scalar TFAR models proposed in [117]. The authors simulate two example channels and compare the results obtained with corresponding measured data.
The fidelity with which the VTFAR model can emulate collected measurements relies heavily on the selection of the different parameters (for example, the spectral model order, number of peaks in Doppler spectrum per tap, etc.) of the VTFAR model. Hence, to enable the VTFAR model to accurately represent a particular environment, we need a model for the scattering geometry. Also as noted before regarding AR models in general, the required independent “driving” noise (or, “innovations” process) of such models has no basis in propagation principles.
On the basis of the above discussions, we realized that incorporating non- stationary effects in our channel models should be done with the following things in mind: 71
¾ Complexity: To keep the complexity of the developed channel model within
reasonable limits, we decided to develop non-stationary channel models using
stochastic processes instead of relying on complicated and time consuming
deterministic (electromagnetic field theory) approaches. By using this approach,
we can take advantage of the stochastic methods for modeling channels and
allow researchers to continue to use these known methods.
¾ Time scales: It is important to understand that non-stationary effects can happen
on multiple time scales (“mesoscopic” and macroscopic, e.g., region transitions)
and hence it is convenient to model them separately. Fading effects that take
place on different time scales can affect different layers of the protocol stack.
Hence correct modeling of these effects is imperative for cross layer design of
systems [119].
¾ Physical Justification: We prefer to use an approach that has some physical
justification to model the non-stationary (NS) effects. Out of the approaches
discussed above to incorporate NS, only the approach of modeling the finite
multipath component lifetime has some physical significance. Thus at present we
employ this approach.
¾ Filter instability: To model non-stationarity in the Doppler domain, it is necessary
to create time varying Doppler spectra. The conventional approach of modeling
different Doppler spectra using different filters and then switching between them
doesn’t work for our settings (airport surface areas and VTV) due to the
relatively small ratio of our maximum Doppler spread to channel bandwidth. 72
This would require filters of extremely narrow bandwidth, which are difficult to
create efficiently and when, generated by computer (e.g., Matlab®) can be
unstable. To eliminate this issue, we use interpolation to create an effect
equivalent to that of the low pass filter. 73
3 Measurement Campaigns
In this chapter we describe the equipment used to gather impulse response estimates
(IREs). We start with a small description of our channel sounder’s principles of operation, and present its main features. Next, we describe our several measurement campaigns. One section is devoted to our measurement campaigns at multiple airports, and the last section discusses the campaigns for the VTV channel.
3.1 Equipment Description
3.1.1 Channel Sounder
The underlying physical channel can be modeled either in a deterministic or stochastic manner. Ideally, any channel can be represented in a deterministic manner if the scattering geometry, the detailed physical characteristics of the reflectors, the velocity of the reflectors, topographical information, etc., are known accurately. In practice, it is virtually impossible to even estimate such information accurately, let alone having the true information. Also, even if we have access to such detailed information regarding the underlying physical channel, the use of electromagnetic wave theory (e.g., Maxwell’s equations) to characterize the channel would make the task very complicated and computationally complex. For use in communications systems analysis and design, channel models should be in some sense generic, to allow modeling of various dynamic 74 scattering geometries without site-specific (local) constraints; the models should be
“portable” for use in “similar” environments. Because of this, stochastic modeling of the channel is a favored approach.
Due to the presence of scatterers in between the receiver and transmitter, the receiver gets “multiple copies” or “echoes” of the same signal at different instances of time. This phenomenon is often termed as “multipath propagation.” Each “echo” has traveled a different path and might have encountered reflection, refraction or diffraction from different scatterers. Note that depending on the scattering geometry, it is possible that different echoes might also have encountered some of the same scatterers. In an actual communication system, depending on the resolution capability of the receiver, each “individual” echo might actually be an aggregate of several echoes. Similarly, when characterizing a channel, the “echo” resolution capability is limited by the sampling rate of the channel sounder. As discussed previously, the propagation channel can be thought of as a time varying linear filter. As with any such filter, the channel can be characterized completely by its impulse response. By doing so, each “echo” or multipath component can be characterized by an impulse with a certain amplitude, phase, and delay. So, each impulse response is often actually a collection of discrete “echoes,” grouped together in delay “bins,” the width of which represents the delay resolution of the sounder. This delay resolution is inversely proportional to the transmitted sounding signal’s bandwidth.
The channel impulse response is defined as the function h(τ;t), which represents the response of the channel at time t to an impulse input at time t-τ. The CIR is also 75 known as the input delay spread function [9]. Next, we present the complex baseband representation of the CIR. Mathematically, the CIR can then be expressed as,
L−1 jφl ()t h()τ ,t = ∑α l ()t e δ ()t −τ l ()t (3.1) l=0
th where αl(t) represents the l component received amplitude, and the argument of the
th th exponential term (φl) is the l received phase. The l multipath component has a time- varying delay τl(t), and the δ-function is the Dirac delta. The purpose of stochastic modeling is to statistically model all the above parameters using the collected IREs. As discussed, the CIR can be modeled as a collection of discrete “echoes” arriving at different instants of time. Hence, the CIR can be represented by a tapped delay line model (TDL) [4]. The TDL has gained wide acceptance as a way of modeling channels due to its simplicity and capability to model the physics of propagation (varying amplitude “echos” arriving at different instants of time). Figure 3.1 provides an example figure illustrating the tapped delay line model. The symbols used in the figure have already been discussed in the previous paragraph. 76
x xk xk-1 x k- L+1 k+1 τ τ τ0 τ1-τ0 L-1- L-2
− jφ0 − jφ − jφL α ()t e α (t)e 1 α (t)e 0 1 … L−1
y Σ k
Figure 3.1 Tapped delay line channel model
The channel sounder is used to gather the IREs. The channel can be characterized either in the frequency or time domains. The channel sounder transmits a known signal and at the receiver recovers the channel information from the received signal. From the channel information gathered, the user can then understand and characterize several time domain channel effects such as the spread of the original signal in time, the correlation among the multipath components received at different delays, etc. In the most basic sense, the sounder can be described as a transmitter and receiver assembly. If designed from basic principles to measure the CIR, the transmitter must output a signal that is similar to the δ(t) function, and the receiver must then capture the channel output. This output is then processed to extract the desired channel information. There are several issues involved with using an impulse-like transmit signal to measure the channel:
1. By definition, δ(t) has to be of unit energy and zero duration. This kind of signal is
impossible to generate. 77
2. Considering the progress made in signal processing, we are able to generate very
small duration probing pulses. But these pulses need to be amplified to transmit
them over large link distances. Due to very small time duration of these pulses, the
probing pulse has a nearly white spectrum. An amplifier which would have a linear
response over such a large frequency bad would not only be very costly but would
be very difficult to manufacture.
3. As a consequence of the previous issue, non-linearity in the amplifier’s frequency
response would significantly affect the accuracy of the measured IREs.
4. Use of very short probing pulses increases the susceptibility to interference from
impulsive noise spikes.
Due to these issues, short transmit pulses are generally not used by channel modelers for collecting IREs.
The approach that is commonly used is the one where a spread spectrum signal is used in the channel sounder. This spread spectrum method of channel sounding was introduced by [28]. Our sounder uses this approach as well [29], and readers are referred to these references for more detailed descriptions of underlying operation principles. The spread spectrum sounder uses a direct sequence (DS) type of modulation in which the sinusoidal carrier is modulated using a high rate “chip” sequence (c(t)). This c(t) is generally a pseudo-noise sequence and most often, as in our sounder, BPSK modulation is used. Any two multipath components need to be separated by a minimum delay for them to be resolved unambiguously by the receiver. This minimum delay is referred to as the delay resolution capability of the sounder. Each chip in c(t) functions in the same 78 manner as an impulse. Hence, the delay resolution of the multipath components is limited by the chip duration (Tc). At the receiver, our sounder uses an approach referred to as a “stepped correlator.” In this method, the receiver correlates at one value of delay
(τ) for some period, outputs that result, shifts its delay by Tc, correlates again and outputs the result, etc. In this way, the sounder probes over the entire unambiguous delay range of the sounder. This unambiguous delay range (TD) is given by the length of the PN sequence, L=255 chips, multiplied by the chip time Tc. So, when we operate the sounder using a BW of 25MHz, TD is 10.2 μseconds and for BW of 50MHz, it is 5.1 μseconds.
The stepped correlator collects energy over many (Nc) symbol durations. In this case, a symbol is the set of L chips equal to the duration of 5.1 μseconds (Ts). The time taken by the sounder to output a single IRE is called the update time (tu) of the sounder.
This time tu depends further on time T0, the time taken to collect energy for the Nsam samples (each sample requires Nc symbols) and the time taken to transfer this information for the 255 chips to the laptop using the RS232C cable (Ttrans). So, mathematically for a
50 MHz BW,
tu = []To + 255* ()Nc * Ts + Ttrans (3.1)
For our sounder [29], due to manufacturer proprietary information, we don’t know the values of Ttrans and Nc, but, we do know that tu is 0.5 sec and T0 is 11 msec [29].
So, we can then find the value of (Nc*Ts + Ttrans) which turns out to be ~2 msec. Thus, the sounder takes ~2 msec to determine the value of the IRE for any given chip. Note that this 2 msec value is an upper bound, since actually the sounder takes only (Nc*Ts) for a given chip, but since we don’t know Ttrans, we determined this upper bound for the time 79 spent by the sounder at each chip. Then, assuming that the sounder requires 2 msec to gather energy for any given chip, our sounder can collect data unambiguously for a maximum Doppler spread of 500 Hz.
It is crucial for the transmitter carrier and receiver carrier clocks to be aligned for the stepped correlator approach to work. Hence, before we start measuring, we must calibrate the receiver and transmitter by connecting them back-to-back. The receiver uses a quadrature detector, hence in addition to power, we also collect amplitude and phase information. The receiver also collects the total received signal power (“RSSI,” for received signal strength information). The IRE output by the receiver is actually a convolution of the channel’s impulse response and the autocorrelation of the chip signal.
Modern signal processing has enabled us to generate c(t) with autocorrelation very close to that of an impulse (see Figure 4.2) and hence we can safely assume that the IRE output from the sounder approximates the channel impulse response accurately. Table 3.1 lists the main parameters of our channel sounder and Figure 3.2 shows the channel sounder transmitter and receiver along with the transmitted power spectrum. 80
Figure 3.2 (a) Raptor Channel Sounding System (Left – Receiver, Right – Transmitter).(b) Transmitted power spectrum, 50 Mcps setting, fc=5.12 GHz
Table 3.1 Channel Sounder Parameters
Characteristic Value Transmit power level Adjustable between 6 dBm – 33 dBm in steps of 1dB (set at 33 dBm) Center frequency Adjustable between 5.095 GHz – 5.20 GHz in steps of 25 MHz (set at 5.12 GHz) Chip rate 50 Mcps (25 Mcps also available) Unambiguous delay range 5.1 μsec 99% power bandwidth 52.76 MHz Receiver sampling rate 100 MHz Measurement rate 2-60 PDPs/sec
81
3.1.2 Antennas and Miscellaneous Equipment
The other required equipment used during the measurements is listed below:
1. Battery Pack: the receiver is portable, and can even be hand-carried. In order to
have such flexibility, the receiver is provided with a battery pack. The battery
pack enables us to use the receiver without an AC power supply for ~ 3 hours.
2. Uninterruptible Power Supply (UPS): unlike the receiver, the transmitter doesn’t
have a battery pack (one reason is that the transmitter requires much more power).
Hence, to move the transmitter from one place to another without losing AC
power (e.g. during calibration), the transmitter must be connected to an
uninterruptible power supply (UPS).
3. Laptop Computer: to use the sounder, a laptop computer is also required,
connected to the receiver unit via an RS-232 cable. This laptop is used for
receiver configuration and setup, as well as data recording.
4. Antennas: on the airport, we have taken measurements for both fixed-to-mobile
and fixed-to-fixed communication links. For the fixed-to–mobile case, the
transmitter was placed at either the Air Traffic Control Tower (ATCT) or at an
Airport Field Site (AFS), while the receiver was in a van. In this case, for both
receiver and transmitter, we use omnidirectional monopoles with ground planes.
The gain of the monopoles is 1.5 dBi. Figure 3.3a shows the omnidirectional
antenna atop the transmitter platform (small white hemisphere). For the fixed-to-
mobile cases, the transmitter was placed at the ATCT while the receiver was held
stationary at an AFS. We provide an example anttena pattern in the following 82
page, other antenna patterns are included in the Appendix. In this case, at both
the transmitter and receiver, we used directional high gain antennas:
a) The Tx antenna specifications are gain of 8.5 dB, 3 dB azimuth
beamwidth of 60o and 3 dB elevation beamwidth of 60o.
b) The Rx antenna specifications are gain of 17 dB, 3dB azimuth beamwidth
of 30o and 3 dB elevation beamwidth of 15o.
5. Additional equipment used in the course of the measurements included a
transmitter platform on which the transmitter was mounted (see Fig. 3.3a), RF
adaptors, cables, and attenuators, etc.
Figure 3.3 shows the different antennas that were used along with the transmitter platform. Figure 3.3 (a) shows the monopole, transmitting horn, UPS and the sounder- transmitter at an AFS site in Miami. Figure 3.3 (b) shows the receiving horn on its platform. The receiver platform was made so that we could orient the horn antenna at increments of 15º in azimuth to estimate directional effects of the scattering geometry surrounding the AFS. Figure 3.4 shows the azimuth radiation pattern of the directional horn antenna used at the receiver.4
4 Both the receiver and transmitting platforms were built by Nicholas Yaskoff, an Ohio University graduate student who helped us in collecting the measurements. 83
Monopole Rx Horn
Tx Horn
(a) Transmitter
UPS
(b)
Figure 3.3 Various antennas used at the airport
° 0 15 -15 ° ° 30 -30 ° ° 45 -45 ° ° 60 -60 ° ° 75 -75 ° ° 90 -90 16 7234dB ° ° -60 105 -105 ° ° -40 120 -120 ° ° -20 135 -135 ° ° ° ° 150 ° -150 0 165 180 -165
±
Figure 3.4 Azimuth radiation pattern for the directional horn used at the receiver for fixed-to-fixed measurements at the airport
84
3.2 Airport Measurements
3.2.1 Measurement Procedure
One of our primary objectives is to develop accurate stochastic channel models based on actual measured data. Hence, care needs to be taken at each step of the measurement process so that the collected data provides an accurate description of the measured physical channel. In this section, we outline the measurement procedure and provide some details regarding the actual measurement runs taken at the different airports. No two airports are alike: even airports that are of comparable size, traffic density, geographical surroundings, etc., have their individual unique features. Hence to characterize the airport surface channel,—defined here as all outdoor area on airport property, it is necessary that we sample different scattering environments, so that we can develop models for the several physical environments where it is most likely that future communication systems will be used. We focus on areas where future systems are likely to be used by aircraft, ground vehicles, and pedestrians. We also collect measurements at locations that are prospective locations that can be used for fixed point-to-point communications. Keeping these factors in mind, a convenient way of classification was to divide the airport surface channels into three different classes of airports; large, medium and general aviation (GA). Although this classification is not perfectly precise, it represents a good initial attempt to isolate some of the main propagation conditions common to each airport class. We then selected several airports in each class as representative for model development. The choice of the airports was also affected by 85 other non-technical considerations such as the availability of the airport and airport personnel during our preferred times, the availability of necessary FAA personnel, clearance for the university team to measure at the airports, etc.
Prior to arriving at each airport, we obtained an aerial plan view map or photograph of the airport property. Initial measurement routes were proposed, and later adjusted with input from FAA coordinators and airport staff. As with any science, theory is always different in some ways from practice. Hence, the final measurement route was always selected after observing the measurement site personally. Figure 3.5 shows such a measurement route for the JFK airport.
Figure 3.5 Aerial photograph of JFK International Airport and numbered measurement test points 86
The measurement campaigns were made over a period of one to three days. On the first day, we determined the best location for the transmitter. For all transmissions, we downtilted the antenna at the transmitter so that we could reach maximize range, and also increase signal strength in the areas very close to the ATCT. Figure .3.5 shows example transmitter locations at different airports. Note that for the GA airport at Tamiami (Figure
3.5 (b)), we put our transmitter on the roof of an electronics equipment shed (ILS station) and not on the ATCT due to logistical constraints (more explanation on this difference appears in Chapter 6).
Figure 3.6 Transmitter locations at different airports, (a) Cleveland International Airport (b) JFK International Airport, (c) TaMiami airport
After sitting the transmitter, the next step in the setup process was the channel sounder calibration. We connected the sounder receiver and transmitter for calibration at a place close to the actual transmitter site location. Calibration is needed to ensure the synchronization of the precise Rubidium oscillators used to frequency-lock the Tx-Rx pair (more detail on the calibration appears in Chapter 4). The calibration process must be 87 done before each measurement run. Since no data can be collected during calibration periods, we tried to schedule these periods at convenient times (e.g., overnight, or during airport staff lunch times, or during “slow” periods when there is minimum aircraft traffic).
Prior to actual measurements (often during sounder calibration periods), we took a
“reconnaissance” tour of the airport surface are to confirm that our selected route covered all areas of interest from a propagation and communication system operation perspective.
For example, during these tours (and measurements), we endeavored to follow routes followed by actual ground vehicles and aircraft, but we had ensure that we didn’t interfere with the day to day working on the airport surface. During these reconnaissance tours, we also surveyed possible candidate locations for conducting our Airport Filed
Sites (AFS) measurements and fixed-to-fixed link measurements. Often, some of the initial AFS locations were added or removed from consideration after the first day of measurements and sampling of the collected data. This was done so that we could focus on AFS sites that were both benign and challenging in terms of propagation. Either just before or during the airport surface tours, we also tested the ambient surroundings for possible interference in our measurement band. Although the MLS extension band shouldn’t be in use for any other application, due to its closeness in carrier frequency to the 5 GHz ISM band, some “leaked” energy from the ISM band is possible. We used a spectrum analyzer to try to detect this interference, and by doing so, were convinced that we were the only ones transmitting in the band, and that our measurements were not affected by any spurious emissions. 88
Following this “airport surface tour”, during the afternoon, we did our first measurement run on the airport surface. This measurement run was followed by several runs on the next day. Most of the third day was reserved for conducting the fixed-to– mobile measurements while transmitting from the AFS, and for performing fixed-to-fixed measurements using the high gain directional horn antennas. The testing personnel consisted of teams at both the transmitter and receiver, with one person at each end observing the local environment, and relaying observations to those operating the test equipment. Use of these “observers” was also done to allow “flagging” of the associated
PDPs that corresponded to measurements taken during significant scattering environment changes (e.g., an aircraft moving quickly nearby). We also had a test team member in the receiver van who kept notes of the physical environment. Keeping track of such information helped us to corroborate observation of physical obstructions with observed multipath.
3.2.2 Large Airport Description
With respect to the airport classes, by large airports we mean the airports which not only have a large airport surface area, but also have a high volume of traffic, large airplanes for transcontinental flights, etc. Large airports are often hubs for international traffic, and hence, there are multiple airlines that have facilities on these airports. So, in addition to the usual FAA buildings on the airport surface, we also encountered large airline hangars on the airport surface. Large airports also correspond to a higher volume of customers using these airports; hence, there are multiple hotels and office buildings 89 within the vicinity of the airport, often directly adjacent to airport property. The airports that we measured in this category were Miami International Airport (MIA) and the John
F. Kennedy International Airport (JFK). These airports were measured in June 2005 and
August 2005 respectively. For both airports, we tried to cover as much of the airport surface possible during our measurement runs. The planned routes generally covered concourses, taxiways, access roads, near terminal buildings, near ILS locations, near radar locations, and parked jets, etc. Figure 3.7 shows an example picture taken from the
JFK ATCT. The pictures shown in the figure provides the reader with nearly a 360o view around the ATCT. We can see multiple jets, concourses, access roads, sky-cab (shuttle train) etc.
90
Figure 3.7 View from the JFK ATCT in different directions
A similar picture taken from the MIA ATCT is shown in Figure 3.8. Our measurement vans can also be seen in the picture. The picture also shows aircraft lined up outside the concourse and the several hangars, workshops, etc., present on the airport surface. We can also see the faint outline of Miami downtown in the picture. Also, worth noting is the size of the aircraft compared to our measurement van.
91
Figure 3.8 View from the MIA ATCT
3.2.3 Medium Airport Description
Our second class of airports is the medium airport class. The scattering geometry and the nature of the scatterers for the medium airport is largely similar to those present at the large airports. The differences are that the size of the scatterers are often smaller, the relative traffic of the aircraft and ground vehicles on the airport surface is lower and there are fewer buildings and hangars present on the airport property. Also, medium airports don’t have as many large buildings around the airport, and hence the chance of getting some very high delay multipath is significantly less than that at the large airports.
Also, due to the reduced surface area of the medium airports, the height of the ATCT of a medium airport would be lower than that of the ATCT at a large airport. A medium airport at which we measured is Cleveland Hopkins International Airport (CLE, March 92
2005). Figure 3.9 shows a picture taken from the ATCT of Cleveland. Some of the physical descriptions of the medium airport described above can be seen in this picture.
Figure 3.9 View from the ATCT tower at Cleveland
3.2.4 Small Airport Description
The General Aviation (GA) or small airports are generally used and frequented by aircraft much smaller than those encountered at larger airports. Aircraft commonly seen at GA airports are single and twin propeller aircraft. The example GA airports at which measurements were made are the Ohio University Airport, in Albany, OH, the Burke
Lakefront Airport in Cleveland, and the Tamiami Airport in Kendall, FL. Figure 3.10 shows pictures taken at the different GA airports. The OU airport has only a few buildings on the airport property, including buildings belonging to the OU School of
Aviation, the Avionics Engineering Center, and the airport terminal itself. In addition to these buildings are hangars and equipment sheds. Measurements on the OU airport were 93 made during February and March 2005. Another feature for the OU airport is the proximity of US Route 33 on one side of the airport perimeter. The presence of vehicular traffic on this route sometimes caused long delay multipath components. The second set of measurements was made at Burke Lakefront (BL) airport in Cleveland, Ohio. The proximity of BL to downtown Cleveland, with Lake Erie on the other side, created an interesting geographic location. Some long delay multipath reflections from the large downtown buildings were observed. In fact, the majority of the scatterers at BL were concentrated on the downtown side of the air traffic control tower (ATCT). As with other
GA airports, BL has few buildings on the airport property and is used by smaller airplanes. The measurements at BL were made at the end of March 2005. The measurement vans, Lake Erie, and some of the parked airplanes can be seen in the picture in Figure 3.10 (b), with our receiving measurement van seen moving on the runway.
The final set of GA airport channel measurements was made at Tamiami (TA) airport, Kendall, FL. The TA airport is among the biggest GA airports in the US. At TA, we were unable to place our transmitter on the ATCT, and instead used the roof of an
Instrument Landing System (ILS) shed as the transmitter location. Due to the relatively lower height of the ILS shed (approximately 4 m), larger airport surface area, and larger number of airplanes on the surface, our TA channel has scattering geometry significantly different from the other GA airports. Figure 3.10 (c) is a photo of the airport surface area taken from the ILS shed, in which can be seen the ATCT, some hangars, and quite a few aircraft parked on the surface. The TA airport also has a service road that runs on the outskirts of the airport property. 94
Figure 3.10 Pictures from different GA airports: (a) Ohio University; (b) Burke Lakefront; (c) Tamiami
3.2.5 Measured Data Summary
Within a given airport class, there exists different propagation regions. So, similar to the classification used in terrestrial cellular (urban, suburban, rural, etc.), we need to provide a region classification for each airport class as well. To keep the classification simple and to allow comparison of the same region for different airports, we have segregated each airport class into three different regions. The three regions are: LOS-
Open (LOS-O), NLOS-Specular (NLOS-S), and NLOS. The LOS-O areas are those clearly visible from the ATCT, with no significant scattering objects nearby, e.g., runways and some taxiways. The NLOS-S regions represent the regions in between the 95 other two, and exhibit mostly NLOS conditions, but with a noticeable, often dominant, specular, or first-arriving component in the PDP, in addition to lower energy multipath components. The NLOS regions represent areas of the airport that have a completely obstructed LOS to the ATCT. We separated our collected measurements into different regions on the basis of the RMS-DS of the individual PDPs. The RMS-DS has long been used as an effective channel parameter for categorizing channel dispersiveness. As with the development of any model based on measurements, there is always a possibility of mistaking the PDP of one region as belonging to another. But, as long as these occurrences are infrequent and we collect a sufficient number of PDPs, the overall statistic of the developed channel model are not significantly affected. In Table 3.2, we provide the number of PDPs collected as a function of region and airport. We didn’t collect any LOS-O PDPs in JFK and MIA since the propagation conditions (runways) for the LOS-O region were similar irrespective of the airport class. So, to save time and focus on gathering PDPs in the dispersive regions of the airport, we limited our attention to NLOS and NLOS-S sites at JFK and MIA. We also provide the number of PDPs that we collected while transmitting from the various AFSs. (Note: Collection of PDPs while transmitting from the AFSs was limited to the large airports.) 96
Table 3.2 Summary of measured PDPs for each propagation region, airport
Airport Total Number of PDPs Mobile Field Site Transmit NLOS NLOS-S LOS-O NLOS NLOS-S JFK 6,693 7,103 — 7,272 2,240
MIA 6,299 5,950 — 909 1,408 CLE 1,332 852 443 — —
OU — 1,108 — — — BL — 652 256 — — TA 2,248 2,955 — — —
3.3 VTV Measurements
3.3.1 Measurement Procedure
The measurement procedure associated with the VTV measurements is far less complicated than that of the airport surface measurements. In the following paragraphs, we outline the key steps in the VTV measurement campaigns. The first step is to identify potential environments where it is possible to conduct VTV measurements. In selecting locations, the main points for consideration are accessibility (not too far away from
Athens) so that the measurements can be taken in a day; diversity of environments, so that we sample as many realistic VTV communication settings as possible; and time of day, so that measurements at the same place sample varyious levels of traffic. On the basis of these criteria, the cities were chosen and measurements were conducted accordingly in both cities and on interstate or US highways connecting these cities. 97
The antennas used for the VTV measurements were always omnidirectional monopoles, the same as those used for the airport surface measurements. Since the field of ITS is in its infancy, it is necessary to be as creative as possible regarding possible
VTV applications and conditions scenarios. Hence, in addition to the measurements taken with the antennas outside the car, we also took some measurements with the antennas placed inside the car on the dashboard.
Figure 3.11 Antenna location on the transmitting and receiving vehicles
As in the case of the airport surface measurements, it was necessary to calibrate the sounder transmitter and receiver before taking any measurements. In the case of VTV measurements, it is difficult to calibrate the sounder once we have reached the measurement location, away from AC power. Hence, to optimize time utilization and to avoid searching for an AC power source at the site (city) for calibration, we calibrated the sounders on the way to the measurement sites. We used an inverter, present in the OU 98
Avionics van, to supply AC power for the channel sounder. During the actual measurement run, the procedure followed was similar to that described for the airport surface measurements. Two people (a driver and one additional person) were present in the transmitting and receiving van each. The additional person in the receiver car took care of the laptop computer and recorded the data. The additional person in the transmitter car took notes regarding the ambient surroundings and noted specific scattering geometries. The non-driving team members in both cars also communicated via walkie-talkies for discussions throughout the measurements. One thing worth noting is that the measurement route was not planned before-hand, and the selection of routes in the cities was impromptu. Hence, it was necessary to parse the measured data into appropriate files to help later identify them with the route taken.
3.3.2 VTV Region Description
Measurements were conducted in five cities in Ohio: Cincinnati, Cleveland,
Columbus, Dayton, and Athens. We denote the first four as large cities, and Athens is termed a small city. As noted, VTV communications can be used for so-called “comfort” applications, to make long journeys enjoyable for vehicle passengers. Thus we also collected data on several interstate highways. This included interstate routes I-71, I-75, I-
70, and US routes 33 and 50.
The vehicles employed were two vans. The transmitter was in an Ohio University full-sized van and the receiver was in a Honda Odyssey minivan. In the cities, vehicle velocities were limited to less than approximately 10 m/sec, and inter-vehicle distances 99 were from approximately a few meters to 100 meters. We encountered both heavy and light vehicle traffic, with occasional blockage of the line of sight (LOS) signal by large vehicles such as trucks and buses, and by buildings when the leading vehicle turned a corner, ensuring non-LOS (NLOS) conditions. The city areas traversed were primarily those with tall buildings (usually 4-5 stories for the small city, more than 10 stories for the large cities) on both sides of the street. On the highways, velocities were approximately 26 m/sec, with relative velocities between the vehicles substantially less.
The inter-vehicle distance on the highways was up to approximately 1 km, with most data collected with an inter-vehicle distance on the order of a few tens to a few hundred meters. In all environments, measurements were taken with the receiver vehicle both in front of, and behind, the transmitter vehicle. We have classified the measurement environments into the following types: Urban-Antenna Outside Car (UOC), Urban-
Antenna Inside Car (UIC), Small City (S), Open Area–Low Traffic Density (OLT), and
Open Area High Traffic Density (OHT). The “open” areas are the highways. Figure 3.12 shows several pictures taken at Columbus (UOC) and on I-71 (OHT). The Ohio
University van is visible in both pictures. For the UOC case, moderately tall buildings and mild traffic are visible. In the OHT case, vehicles on both sides of the transmitting van traffic are present. 100
Figure 3.12 Example pictures of the different VTV propagation regions 101
4 Extraction of parameters for Channel Model
Development
In this chapter we initially provide a brief discussion regarding wide sense stationarity and define the conditions necessary for the existence of WSSUS channels [9]. Next, we describe the steps necessary for data pre-processing, and also define the key parameters that we use to describe our channel models. This pre-processing is needed to take into account the limitations of the sounder (sounder errors, calibration errors and sounder autocorrelation) in the measurements and then extract the pertinent features of the IRE.
The next step is to discard effects such as those due to “noise” so that the measured IRE accurately models the physical propagation channel. Discussions regarding the trade-offs between model implementation complexity and the accuracy of the channel description are provided. Note: The discussions presented in this chapter were initially presented in [120].
4.1 Wide Sense Stationary Uncorrelated Scattering (WSSUS) Channels
Bello’s paper [9] provides a statistical characterization of random time varying linear channels. The duality between WSS and US channels in time-frequency is expressed by evaluating 2-D correlation functions of the various system functions of the channel like the CIR, transfer function, etc. As discussed before, the linear time varying channel can be characterized completely using its impulse response. The two main components defining the impulse response of a channel are the scattering gain (due to 102 reflection, diffraction, etc) and the delay associated with each scattering element (τ).
Depending on the order in which these two physical events are expressed while defining the impulse response Bello defined an input delay spread function (delay followed by multiplication) and an output delay spread function (multiplication followed by delay).
So, considering a linear time varying channel with x(t) as input and y(t) as output, the input and output delay spread functions can be defined as follows,
y()t = ∫ x (t −τ )h I (t,τ )dτ (4.1a)
y()t = ∫ x (t −τ )hO (t −τ ,τ )dτ (4.1b)
As discussed, the system functions defining the channel in the frequency domain are duals of the system functions characterizing the channel in the time domain. So, a dual function of the input delay spread function would express the output spectrum as an aggregate of filtered copies of the input spectrum which are shifted in frequency by a
Doppler shift (v). So, the input Doppler spread function (HI(f,v)) is a dual of equation
(4.1a) and can be represented as,
Y()f = ∫ X (f − v )H I (f ,v )df (4.2a)
We can define a similar dual expression for eq (4.1b) and that is defined as the output
Doppler spread function and can be represented as,
Y()f = ∫ X (f − v )H O (f ,v )df (4.2b)
In order to understand the 2-D correlation functions of the system functions, which will in turn be used to characterize the stochastic channel, we need to evaluate the
Fourier transforms of the system functions discussed above. Multiple Fourier transform 103 expressions can be evaluated for the input delay spread function depending on the transformation variable (τ or v) The following equations define T(f,t) and U(τ,v);
T()f ,t = ∫ h I ()t,τ e − j2πfτ dτ (4.3a) h I ()t,τ = ∫U (τ ,v )e j2πvt dv (4.3b)
T(f,t) is the transfer function of the input delay spread function with respect to delay and
U(τ,v) is the delay-doppler spreading function which provides an idea regarding the doppler spectra at each delay. The output can then be represented as a sum of delayed and then doppler shifted elements with a scattering amplitude proportional to U(τ,v).
Analogous to (4.3), fourier transform expressions can be evaluated for the output delay spread function depending on the transformation variable (τ or v) The following equations define M(f,t) and V(v, τ) ;
M ()t, f = ∫ hO ()t,τ e − j2πfτ dτ (4.4a) hO ()t,τ = ∫V (v,τ )e j2πvt dv (4.4b)
M(t,f) can be thought as frequency dependant representation of the channel or a transfer function of the output delay spread function with respect to delay. V(v, τ) is the doppler- delay spreading function. The output can then be represented as a sum of doppler shifted and then delayed elements with a scattering amplitude proportional to V(v, τ). Following the definitions of the system functions in equations (4.1-4.4), we see that the correlation functions of these system functions are related to each other using double or quadrapule
Fourier transforms Table 4.1 summarizes these relationships. In Table 4.1, s, η, l and
μ represent the second variable for time, delay, frequency and Doppler shift respectively. 104
Table 4.1 Relationship between correlation functions of different system functions.
Correlation of Related Using Related Using system function Double Fourier Transform Quadrapule Fourier Transform
R I t, s; , R O f ,l;v, h ()τ η RU (υ, μ;τ ,η) H ()μ
RT ( f ,l;t, s)
R O f ,l;v, R I t, s; , H ()μ RU (υ, μ;τ ,η) h ()τ η
RT ( f ,l;t, s)
R O t, s;τ ,η R I f ,l;v, μ h () RV (υ, μ;τ ,η) H ()
RM ( f ,l;t, s)
R I f ,l;v, μ R O t, s;τ ,η H () RV (υ, μ;τ ,η) h ()
RM ( f ,l;t, s)
For sake of interpretation, we shall discuss the 2D correlation functions presented in the
first row. As can be understood from the expression, R I t, s;τ ,η is the 2D correlation h ( ) function of the input delay spread system function. A double Fourier transform of the same can then obtain the 2D correlation function either of the delay-doppler spreading
function, RU ()υ, μ;τ ,η or the transfer function of the input delay spread function,
RT ()f ,l;t, s depending on the transformation variable. RU (υ, μ;τ ,η) provides statistical
characterization of the channel in the delay-doppler domain and RT ()f ,l;t, s does the
same in the time-frequency domain. R O f ,l;v, is the 2D correlation function of the H ( μ) output-delay spread function and this correlation function helps understand the channel in the frequency-doppler domain. Now, for the existence of a WSS channel; the correlation functions depend on the variables t and s only through the difference m = t - s. Using this 105 restriction, Bello further modified the system correlation functions to present the following results in Table 4.2.
Table 4.2 System Correlation Functions for WSS Channels.
Original system System Correlation function with correlation WSS condition function
R I t, s;τ ,η R I m;τ ,η h () h ( )
R O t, s;τ ,η R O m;τ ,η h () h ( )
RT ()f ,l;t, s RT ( f ,l;m)
RM ()f ,l;t, s RM ( f ,l;m)
RU ()υ, μ;τ ,η PU (τ ,η;v)δ (v − μ) − j2πvm P ()τ ,η;v = R I (m;τ ,η )e dm U ∫ h
R I f ,l;v, P I f ,l;v δ v − μ H ()μ H ( ) ( ) − j2πvm P I ()f ,l;v = R (m; f ,l )e dm H ∫ T
RV ()υ, μ;τ ,η PV (m,η;v)δ (v − μ) − j2πvτ P ()m,η;v = R O ()m;τ ,η e dτ V ∫ h
R O f ,l;v, μ P O f ,l;v δ v − μ H ()H ( ) ( ) − j2πvm P O ()f ,l;v = R (m; f ,l )e dm H ∫ M
From Table 4.2, it is clear that WSS in time translates to uncorrelatedness at different Doppler shifts. This means that transfer functions at different Doppler shifts are not correlated with each other. In a similar manner, the system correlation functions were 106 also derived for uncorrelated scattering (US) in the delay domain and Bello proved that
US in the delay domain translates to WSS in the frequency domain.
4.2 Data Preprocessing
In this section we describe the pre-processing steps necessary to convert the “raw” data files collected during the measurement campaigns (described in Chapter 3) into data files that are used for the development of the channel models. The pre-processing consists of format translation (to translate sounder data into data readable by our primary software package, MATLAB®), noise thresholding to eliminate the effects of thermal noise “spikes,” and multipath thresholding, to effect a suitable compromise between model fidelity and implementation complexity.
4.2.1 Channel Sounder Errors
The sounder records Impulse Response Estimates (IREs), which are our estimates of the IRE, as discussed in Chapter 3, at each measurement point. As with any measurement system, there are inherent errors in the channel sounder. We need to determine these errors before we start developing stochastic models based on the collected IREs. The motive is to make sure that these inherent errors don’t affect the accuracy of the developed models. Reference [121] provides an excellent description of the errors associated with a channel sounder that works on the same principle as ours.
Since we generally don’t see much discussion of these errors in the open literature, we 107 feel that it is a good idea to summarize and provide a general overview of these errors.
This overview is based on the description of these errors in [121]. Figure 4.1 shows the basic block-diagram of a generic measurement system that follows the same processing steps as our sounder. The essential components of Figure 4.1 are described as follows,
Δ(t) = ∑δ ()t − mT (4.5) m where T is defined as the sounding period. The impulse response of the Linear Time
Invariant (LTI) transmit filter (G) is denoted as g(t). In the case of our sounder;
N g(t) = A∑bic()t − iTc , (4.6) i=1
Where bi ∈{-1,1} is a PN sequence and c(t) is a chip of length Tc. The length of the filter is Tg = NTc. The impulse response of our channel (H) is same as (3.1). The impulse response of the LTI receiver filter (R) is given as r(t). The length of the receiver filter is
Tr = Tg = NTc, and it is specified as a filter matched to g(t), r(t) = g(Tg – t), (4.7)
Δ(t) x(t) y(t) hˆO ()t,τ G H R
Figure 4.1 Block Diagram of the channel measurement system [121]
The errors described in [121] are as follows: 108
¾ Commutation Error (e1 ): As the name suggests, this error arises due to the
wrongful assumption of the commutative property for the product of the Impulse
response of the channel (H) and that of transmit filter (G).
¾ Pulse Compression Error (e2 ): While recording the IRE at the sounder receiver,
there is a tacit assumption that the convolution of the input with “G” results in a
δ-function. But, in practical cases, this is not true.
¾ Misinterpretation Error (e3 ): We are interested in modeling the input delay spread
function (see equation 4.1). But the measured function hˆO ()t,τ is actually an
estimate of the output delay spread function, which can be interpreted as the
channel at time (t+τ) to an impulse input at time t . Hence, it is an estimate of the
output CIR or hO(τ;t). Since the channel is time variant, hI(τ;t) and hO(τ;t) are not
exactly the same, and this leads to misinterpretation error.
¾ Aliasing Error ()e4 : Aliasing error can arise due to two conditions: first, if the
absolute delay associated with the maximum delay multipath exceeds the
sounding period (T); and second, if the update rate (1/Trep) of the IRE is less than
twice the maximum Doppler frequency of the measured channel.
To give a quantitative meaning to these sounder errors, the authors in [121]
propose mathematical representations and upper bounds (ε n ) to the above mentioned errors. In order to present these bounds, it is necessary to introduce some additional parameters of the physical channel [121]. The Delay – Doppler Spreading Function is given as, 109
S τ ,υ = h t,τ e− j2πυt dt (4.8) H ()∫ () t
The Mean Excess Delay can be expressed using the delay-doppler spreading function,
1 τ 1 = τ −τ S ()τ ,υ dτ dυ (4.9) H S ∫∫ 0 H H 1 τυ
In (4.9), τ is defined as a suitable reference delay. The mathematical operator . is 0 ( ) p defined is 4.12. The mean Doppler shift is,
1 υ 1 = υ S ()τ ,υ dτ dυ (4.10) H S ∫∫ H H 1 τυ
Lastly, the mean delay-Doppler product can be represented as,
1 μ 1 = τ −τ υ S ()τ ,υ dτ dυ (4.11) H S ∫∫ 0 H H 1 τυ
The Lp-norm of a function f(x), with x = [x1, x2, … xn] is defined as follows,
1/ p L − norm[ f (x)] = f = ... f ()x ,...x p dx ...dx (4.12) p p []∫∫ 1 n 1 n
Note that for our calculations, we will use a discrete version of (4.12) and replace all the integrals by summations. On the basis of (4.8), the infinity-norm of a function f(x) is
defined as sup f (x1, x2 ,...xn ) . The upper bounds for these errors are listed in Table 4.3. 110
Table 4.3 Upper Bound on the Channel Sounder Errors [121]
Error Upper Bound Remark
Commutation Error 1 e (mT ,τ ) τ R is the mean duration of 1 rep ≤ ε g S r 1 the receiver filter and is ∞ H 1 1 given as, = 2 1 1 ε1 πτ Rυ H 1 τ 1 = τ r()τ dτ R r ∫ 1 τ Pulse Compression Error [-B, B] is the measurement e2 (mTrep ,τ ) ≤ ε band S 2 H 1 1 ⎛ k ⎞ ⎛ k ⎞ ε 2 = ∑ R⎜ ⎟G⎜ ⎟ −1 T k ≤BT ⎝ T ⎠ ⎝ T ⎠ Misinterpretation Error e mT ,τ dτ ∫ 1 ()rep τ ≤ ε S 3 H 1 1 ε 3 = 2πμ H
Aliasing Error K is defined as Trep/ T e mT ,τ dτ ∫ 4 ()rep τ ≤ ε S 4 H 1 1 ⎡ τ H 1 ⎤ ε 4 = 2⎢ + 2KTυ H ⎥ ⎣T −τ 0 ⎦
From Table 4.3, we see that to evaluate the upper bound on the different errors, we require actual knowledge of the impulse responses or the spreading function! In the paper [121], the authors have provided example error calculations for two cases, using a
2-ray synthetic channel model and for the second case using estimates of the spreading function from their measured data. Since the determination of the error values is dependent on the knowledge of the spreading function, their usefulness is highly channel 111 specific. Ideally, we would require a bound on the possible channel sounder errors independent of the channel. We computed the upper bounds ε S of the errors ( n H 1 ) described in Table 4.3 using our measured data ( for MIA-NLOS). For the computations; scattering functions were evaluated from the measured IREs, doppler spread considered was 0-250Hz, delay range from 0-1.54μsec (mean (RMS-DS) from MIA-NLOS data) and receiver and transmitter filter was assumed to be raised cosine FIR filter with 0.22 roll- off factor. Our computed upperbounds for commutation, pulse compression, and misinterpretation errors were all very small—relative to the minimum IRE sample. They were below -30 dB for all values of possible Doppler spread (0-250 Hz), and hence are negligible. Also as in [121], only our aliasing error was non-negligible. This translates to an inability to track all the channel time variations and compute Doppler spreads—known to us at the outset of our campaign, and something we do not include in our measured data in any case.
4.2.2 Channel Sounder Calibration
As noted in section 3.1, it is crucial to remove the effect of the sounder’s auto- correlation curve so that the measured data depicts the propagation physics independent of the sounder’s inherent characteristics. This procedure is commonly referred as
“calibration” in the literature; we have also followed the same nomenclature for this procedure in our research. The usual method of doing this “calibration” is to connect the sounder transmitter and receiver in a “back-to-back” manner and then let the sounder take 112
CIR measurements. The Rx and Tx are connected using a high grade RF cable and an attenuator is used to make sure that the Rx is not exposed to inappropriate power levels.
On performing this experiment multiple times, we have concluded that the autocorrelation curve of our sounder is very close to that of an impulse function. Figure
4.2 shows the autocorrelation curve obtained from a back-to-back measurement with the sounder; we also provide an exponential curve fit for the autocorrelation values, in dB units. The sounder was calibrated for a BW of 50 MHz. The autocorrelation curve drops to -14dB at delay equal to that of a single chip duration (which in the case of the 50MHz
BW is 20 nanoseconds). After that delay, the autocorrelation continues to drop sharply and reaches -31dB for the 4th chip. We have taken note of this fact and hence in our measurements, any measured multipath component which is equal to or less than the autocorrelation value is attributed to the sounder’s autocorrelation and is not included as a valid multipath.
As discussed previously, the Tx and Rx also need to be calibrated for a certain amount of time so that the rubidium oscillators of the transmitter and receiver get aligned.
We would like to stress the fact that it is crucial that the oscillators are aligned and that they don’t drift (or don’t drift significantly) to affect the frequency lock. This drift in the center frequency of the Rx and Tx oscillator can be mathematically expressed in terms of the duration of time possible for “reliable measurement time,” denoted by Tmeas. Again, after performing repeated tests, we determined that the Tx and Rx are stable and record accurate data under varying conditions (from severe cold weather on top of CLE ATCT 113 to hot and humid afternoons at MIA) well beyond the allocated Tmeas, up to three times
Tmeas. Based upon our designed experiments, we are confident that all our data is valid.
0 Sounder Connected back-to-back -5 Exponential fit: 35exp(-8.8τ) -38
-10
-15
-20
-25
Autocorrelation (dB) -30
-35
-40
-45 0 0.5 1 1.5 2 2.5 3 Delay ( sec) τ μ
Figure 4.2 Autocorrelation curve for the channel sounder in back-to-back mode, using a BW of 50 MHz [120].
4.2.3 Data Format Translation
The measured data (or the “raw” data, as we like to call it) is recorded on the laptop by connecting it to the receiver using a RS-232C cable. Though the RS-232C limits the maximum data rate at which we can store the data in real time, the constrained update rate of the sounder prevented that from being a problem. One of the key things while recording any data is to have a time stamp associated with every measured profile.
The “time stamp” has a two-fold use: first, it helps us to determine if we lost data at any stage; and second, it also helped us to associate the post-processed data with the location where it was measured. This raw data is in a proprietary format—a .rap format (a 114 patented format used by BVS ([29])). In order to use these files in MATLAB, we need to convert these files into ASCII format. The manufacturer BVS has provided us software
(called “Chameleon”), which converts the data files from their .rap format to a .out format. The .out files are in ASCII format. Chameleon allows us to input the fields we want to see in the .out files. Figure 4.3 provides a screen shot of the front-end of the
Chameleon software. We made sure that while taking measurements, we parsed the measured data into files depending on the location of the measurements. Since this was the first endeavor of its kind to stochastically model the airport surface channel, any unique or peculiar propagation phenomenon affecting the signal at some specific locations or areas of the airport needed to be properly documented. Hence, every file was actually a collection of several records- which are sets of PDPs taken over a certain segment of travel. The output of Chameleon can be imported into MATLAB routines using the MATLAB “csvread” command.
Figure 4.3 Screen capture for Chameleon (format conversion software) [120].
115
Referring to Figure 4.3, the fields that we have used in Chameleon are as follows:
1. Magnitude (dBm): PDP sample power values in dBm. The sample power is
expressed using 2 bytes, which gives us a dynamic range of ~48 dB;
2. Phase (radians): phase associated with each sample value in radians;
3. I channel value (dB): PDP sample value on the in-phase channel;
4. Q channel value (dB): PDP sample value on the quadrature channel;
5. RSSI (dBm): Received Signal Strength Indicator for each row (PDP);
6. GPS Latitude: Latitude of the location where the data was collected;
7. GPS Longitude: Longitude of the location where the data was collected.
“Sample” refers to the correlated output value taken at every half-chip duration since in our case the sampling rate is twice the chip rate of 50 Mcps. Table 4.4 shows an example data recording after using Chameleon. The PDP data is recorded every 0.5 seconds. Each column provides the field value from the above description. The sample values from each column are used to determine the Power Delay Profile (PDP) at that time instant. 116
Table 4.4 Chameleon output format for the nth record [120]. Time Stamp Magnitude (dBm) Phase (radians) Magnitude Phase Magnitude Phase GPS GPS RSSI for for nth record for 1st sample for 1st sample (dBm) for (radians) (dBm) for (radians) Latitude Longitude nth of nth record of nth record 2nd sample for 2nd … 1020th for 1020th for nth for nth record of nth record sample sample sample record record of nth of nth record of nth record record
117
The nth record provided in Table 4.4 is further divided into 3 sub-records:
1. PowerRecord lists all the power values, in dBm, for 1020 samples. The minimum
value recorded is -130 dBm. The PowerRecords are the “un-pre-processed”
versions of the PDPs.
2. PhaseRecord lists all the phase values in radians for 1020 samples.
3. GPS_RSSI lists the GPS latitude, longitude, and RSSI for the record.
An example of these sub-records is provided in Table 4.5.
Table 4.5 Sub-records generated for nth record [120].
Magnitude Magnitude Magnitude (dBm) for (dBm) for … (dBm) for st nd th 1 sample 2 sample 1020 sample
Phase Phase Phase (radians) (radians) … (radians) st nd th for 1 for 2 for 1020 sample sample sample
GPS Latitude GPS Longitude RSSI for nth th th for n record for n record record
4.2.4 Note for 50MHz PDPs
The PN code used in the spread spectrum sounder has a length of 255 (i.e., a processing gain of 255). The number of chips and the sampling rate of 100 MHz remains the same irrespective of the BW used. So, if we conduct measurements using a chip-rate of 25 MHz, then the receiver over samples by a factor of 4, and similarly for a chip-rate of 50 MHz, the over sampling factor is 2. For the 25 MHz mode, Chameleon outputs a
PowerRecord with 1020 samples, since we have 255 chips (1020 samples/4), in which case it provides us a maximum possible unambiguous delay range of 10.2 µsec. 118
Similarly for the 50 MHz chip rate, we have a maximum possible unambiguous delay range of 5.1 µsec, as noted in Chapter 3. Due to hardware and buffering limitations, the receiver records 1020 samples even when the chip-rate is 50 MHz. So, in practice, we are recording 510 additional samples. In this case, samples 511-1020 are a copy of the first
510 samples, and can be discarded. Figure 4.4 shows an example PowerRecord for an actual collected data record taken in the laboratory. Note that the values for samples 134 and 644 are the same.
-80
-85 X: 134 X: 644 Y: -82.22 Y: -82.22 -90
-95
-100
-105
-110 Power in dBm dBm in Power
-115
-120
-125
-130 0 200 400 600 800 1000 1200 Sample Index
Figure 4.4 Example PowerRecord for 50 Mcps after using Chameleon [120].
Since we used only the 50 Mcps mode during all measurements, for the remaining data processing, we discard values of samples (511-1020) for both PowerRecord and
PhaseRecord. 119
4.2.5 Noise Thresholding
This algorithm was initially described by us in [120]. The PowerRecords provide information regarding the dispersion of the propagation environment: they provide an estimate of the power associated with each multipath component. As with all communication systems, the PowerRecords are also affected by thermal noise. In order to minimize the effects of this thermal noise, it is necessary to separate valid multipath components from noise. We have used the method outlined in [30], which determines a
th noise threshold (NTj) for the j PowerRecord by enforcing a constant false alarm rate
(CFAR) for each PowerRecord. The CFAR algorithm is widely used in radar related applications for determination of noise thresholds [28]. The value of NTj is determined
j 2 th using the noise variance (σ n ) of the j PowerRecord. True thermal noise can be assumed to be Rayleigh distributed in amplitude (two Gaussians, one each on “I” and
“Q”) [30]. Given this model, the probability that the noise amplitude will exceed some level z0 is given by
⎛ − z 2 ⎞ P( z ) = exp⎜ 0 ⎟ (4.13) 0 ⎜ j 2 ⎟ ⎝ 2()σ n ⎠
j th The estimated median level σ m (for the j PowerRecord) can be found by setting
j z0 = σ m , and by equating (4.9) to ½, we get,
j j σ n ≅ 0.8493σ m (4.14)
After using noise thresholding, we guarantee that a maximum of one noise sample may be mistaken as a valid multipath in each PowerRecord. This gives us a CFA probability 120
(CFAP) (Pf) as listed in Table 4.6. In this table, η is a constant that is obtained from the following equation:
⎛ −η 2 ⎞ ⎜ ⎟ Pf (η) = exp⎜ ⎟ . (4.15) ⎝ 2 ⎠
Table 4.6 Parameters for CFAR algorithm [120].
Mode Number of Samples in Each CFAP for the Mode η for the Mode PowerRecord 25 MHz 1020 9.8×10-4 3.72 50 MHz 510 2×10-3 3.52
Continuing with the CFAR algorithm, from (4.15) and Table 4.6, we get,
j NT j = ησ n . (4.16)
j We estimate σ m for each PowerRecord and then use (4.14) to find its respective noise
j j standard deviation σ n . In order to estimate σ m , we need to first determine the likely noise samples from each PowerRecord. For each PowerRecord, we must select a threshold, below which the samples can be considered to be from noise (only). We define this threshold as our user-selected dynamic range (USDR). For context, in reference [122], for each PDP the authors discard all sample values that are below 20 dB of the maximum value of that PDP. The reasons for applying this USDR are first, low energy multipath is difficult to track for even advanced receiver processing, and second, these low-energy components do not significantly contribute to the aggregate energy collected at the receiver or the resulting CIR statistics. For all of our processing requirements, we have employed a USDR of 25 dB, since we don’t anticipate receiver techniques will be able to effectively gather and track any multipath that is below 25 dB 121 of the maximum (unless SNR is extraordinarily large). Even if receivers can track these components, inclusion of them often greatly increases the channel model complexity.
j Using this USDR, we estimate σ m and then obtain NTj for each PowerRecord using
(4.10), Table 4.6, and (4.16).
Next, we list the steps that are used to complete the noise thresholding on the
PowerRecords:
1. Using Table 4.6, determine the values of CFAR (CFAP) and η depending on the
mode of the channel sounder;
2. Separate out the samples in the jth PowerRecord that are below the USDR;
j 3. Determine the median value σ m for the samples separated in step 2;
4. Use (4.14) and (4.16) to determine the noise threshold NTj;
th 5. In the j PowerRecord, set all samples below NTj to their minimum value, -130
dBm;
6. Repeat steps 2-5 for each PowerRecord.
Figure 4.5 provides an example set of PowerRecords before and after applying our noise thresholding algorithm. As can be seen, the noise thresholding does nothing to those multipath components that are within approximately 25 dB of the main (largest) component.
122
-80
-90
-100
-110
Power (dBm) -120
-130 0 100 200 300 400 500 600 Sample Number
-80
-90
-100
-110 Power (dBm) -120
-130 0 100 200 300 400 500 600 Sample Number
Figure 4.5 Example PowerRecord for 50 MHz channel bandwidth [120].
4.2.6 Converting PowerRecords to PDPs for Different Bandwidths
After making sure that we don’t mistake a noise sample for a valid multipath component, the next step is to extract the PDPs from the PowerRecords. Depending on the BW of the intended channel model, the procedure will be slightly adjusted for different BWs. As discussed in section 4.1.4, for our 50 Mcps rate, we have 510 samples for each PowerRecord and each PhaseRecord. Looking from the propagation physics point of view, we can assume that each individual sample of the PowerRecord is actually an aggregate of all the rays which arrive at the receiver within the minimum possible delay resolution. This minimum delay resolution for our sounder is 20 nsec since our maximum bandwidth is 50 MHz. This enables us to resolve multipath components which have at least a 6 m difference between their respective path distances. Depending on the intended BW, we must vectorially combine samples to obtain the corresponding PDP. It 123 is feasible to do this since we have both a phase value and power value associated with each sample and using complex addition, we can create chip samples of the PDPs for virtually any BW. Table 4.7 lists the number of samples to be combined to determine the
PDP for a given bandwidth. Note that for a BW of 1MHz, we can have a maximum number of 5 taps in our channel model.
Table 4.7 Number of samples/chip for different bandwidths [120].
Channel Bandwidth 50 MHz 10 MHz 5 MHz 1 MHz Number of Samples to be Combined 2 10 20 100 Effective Number of Chips in Each 255 51 25 5 PDP
Now recall eq. (3.1), the equation for our CIR: L−1 jφl ()t h(τ ,t )= ∑ zl ()t α l ()t e δ ()t −τ l ()t (4.17) l=0 We add an additional term zl(t) to the conventional description of the CIR provided in
(3.1). We refer to this process as the “persistence process” for the lth tap. This process models the finite lifetime associated with the multipath component. Due to the dynamic nature of the scattering environment, each multipath has an ON/OFF process associated with it [24]. Our “persistence process” tries to capture this propagation effect. After the vector addition of the samples according to the desired channel bandwidth, we obtain a set of (amplitudes, delays, phases)=(α’s, τ’s, φ’s) for each Impulse Response Estimate
(IRE), where the IRE consists of h(τ;t) constructed from the set (α, τ, φ). The α’s and τ’s are used to determine the corresponding PDP for each CIR, as follows:
L−1 2 P(τ;t) = ∑ zl ()t α l (t)δ[τ −τ k (t)] (4.18) l=0 124
Since the α2’s are proportional to power, the PDP itself contains no phase information.
In Figure 4.6 we show the same PDP for different BWs. As can be seen, the delay resolution decreases as the bandwidth decreases, as expected.
-60 -60
-80 -80
-100 -100 -120 Power in dBm Power indBm -120 -140 0 100 200 300 Chip Index 0 20 40 60 (a) Chip Index (b) -60 -70
-80 -80
-100 -90 Power in dBm Power indBm
-120 -100
-140 -110 0 5 10 15 20 25 1 2 3 4 5 Chip Index Chip Index (d) (c)
Figure 4.6 PDPs for different BWs [120]: (a) 50 MHz, (b) 10 MHz, (c) 5 MHz, and (d) 1 MHz
4.2.7 Multipath Threshold
As the name suggests, the PDP provides us with a snapshot in time of the distribution of energy with respect to delay. We need to understand that this distribution changes at every time instant. The PDP can be thought of as a collection of rays assumes 125 plane wave propagation—this is a good assumption given our choice of frequency band and our intended measuring distances.
As noted above, at different time instants, multipath components associated with the same delays will have varying signal strengths due to the dynamic scattering environment. Depending on the characteristics of the dispersive medium, we will receive components with a range of strengths. As with any practical system, the utility of the tapped delay line model is also dependent on the allowable complexity of the model. To keep the implementation complexity reasonable and to still retain accuracy in modeling the propagation effects, the system designer needs to remove from consideration certain weak multipath components. The thinking behind this is to neglect those components that don’t have energy substantial enough to affect the fidelity of the developed channel model. Given this, designers need to have a threshold, which we refer to as the
“Multipath Threshold” (MT). Any multipath detected below this MT is not considered for developing the tapped delay line model. It is something of an engineering “judgement call” on the part of the designer as to what the value of MT should be. There are various factors that need to be considered when selecting this value, but it essentially comes down to the intended level of accuracy and acceptable complexity. We now cite a few references that employ a MT, though not using this acronym per se. Reference [122] assumes a MT of 20 dB, i.e., in any given PDP, any multipath that is below 20 dB of the maximum value of that PDP is discarded. Discarding anything below 20 dB may yield optimistic channel models, given the fact that we have contemporary channel models
[123] that have provision for channel taps that are within 25 dB of the main peak. What is important to determine is how selection of a certain MT affects the depiction of the 126 actual physical channel. Using the RMS-DS to quantify the dispersive nature or frequency selectivity of a channel has been a long-used approach. To understand the effect of choosing different values of MT on the computed RMS-DS, we evaluated RMS-
DS by using MT values of 20, 25 and 30 dB and then collected the RMS-DS statistics. As an example illustration, the results for all data from CLE are presented in Figure 4.7.
This plot shows the relative frequency of RMS-DS versus the value of RMS-DS. It is interesting to note that the statistics of RMS-DS using a MT of 25 dB or 30 dB are very close to each other. A similar trend has been observed for some of our other data as well.
Hence, for all our data, we have used an MT of 25 dB, which appears to provide a near optimal tradeoff in terms of a precise representation of the channel and its implementation complexity.
0.25 MT :20dB MT :25dB MT :30dB 0.2
0.15
0.1 Relative Frequency
0.05
0 0 500 1000 1500 2000 2500 RMS-DS in nsec
Figure 4.7 Example RMS-DS values for CLE data applying different values of MT [120]
127
As noted, there is some difference in results for MTs of 25 dB and 30 dB at lower values of the RMS-DS statistic. To further support our use of 25 dB as an appropriate
MT, we compare the FCE for CLE-LOSO for a BW of 50 MHz, using three different
MTs. The results are provided in Figure 4.8. The figure further supports our use of an
MT of 25 dB, due to the negligible differences in the FCEs for MTs of 25 dB and 30 dB.
1 MT-25 0.9 MT-20 MT-30 0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 -25 -20 -15 -10 -5 0 5 10 15 20 25
Figure 4.8 Example FCEs for CLE data applying different values of MT [120]
Finally here, we list the steps used to apply the multipath threshold on the PDPs:
th 1. For the j PDP, find the maximum strength chip sample (maxj);
th 2. In the j PDP, set all chip samples that are 25 dB or more below maxj to the
minimum value, -130 dBm;
3. Repeat steps 1 and 2 for all PDPs. 128
Figure 4.9 summarizes all the steps in our pre-processing discussed thus far.
Input: data recording from sounder in .rap format
Convert data recording from .rap format to .out format. Each data record is segregated into PowerRecord, PhaseRecord, and GPS_RSSI (Table 4.3)
Apply noise thresholding algorithm to remove potential noise samples from PowerRecord.
Use eq. (4.14) to obtain PDPs for different BWs using the PowerRecords.
Apply the multipath threshold to separate the useful multipath from very-low-energy multipath.
Output: Preprocessed PDPs ready for analysis
Figure 4.9 Summary flow diagram of pre-processing steps of Section 4.1 [120]
4.3 Key Parameters and Definitions
This section will discuss the parameters that can be obtained after performing post-processing on the measured PDPs. Most of these parameters are statistics and will be used in the development of the stochastic channel models. Most of these statistical 129 parameters are well established within the research community; hence, the discussion will be brief.
4.3.1 Parameters Obtained Directly from the IREs
Here we describe the parameters that are determined from the PDPs (4.18) or
IREs (4.17) directly; that is, we describe those channel statistics computed from the pre- processed PDP and IRE data files.
1. Mean energy delay (μτ): The mean energy delay is simply the mean value of the energy (or power) delay. The value of μτ for a PDP can be obtained as,
L−1 2 ∑τ kα k k =0 μτ = L−1 (4.19) 2 ∑α k k =0
The mean delay can be associated with the average delay of all the multipath components in an IRE. This information, though useful, is not critical in terms of small scale fading channel models. In fact, this parameter can not provide any information regarding the frequency selectivity of the channel. For example, consider Figure 4.10: part (a) and part
(b) are identical in terms of the number of multipath components and their relative power, but part (b) is displaced in time by 3 seconds. So, even though both PDPs might have been obtained by passing through similar channels, μτ for part (a) is 2.1 seconds, and that for part (b) is 5.1 seconds. 130
8
6 μ = 2.1sec 4 τ PDP 2
0 1 2 3 4 5 6 7 8 9 10 Time in sec (a)
8
6 μ = 5.1sec 4 τ PDP 2
0 1 2 3 4 5 6 7 8 9 10 (b) Time in sec
Figure 4.10 Example PDPs with same multipath behavior and different μτ [120]
2. Root Mean Square (RMS) Delay Spread (στ): The root mean square delay spread provides the rms value of the multipath delay spread. The spread of a PDP in time is with respect to the mean energy delay associated with that PDP. In this dissertation, we use the abbreviation RMS-DS for στ. The value of στ for any PDP can be obtained as
L−1 2 2 ∑τ k α k k=0 2 σ τ = L−1 − μτ (4.20) 2 ∑α k k=0
The RMS-DS provides a metric to quantify the spread of the signal in time. As can be observed from (4.16), the effect of the mean delay is incorporated into the RMS-DS.
Hence, the information regarding the frequency selectivity of the channel due to the delays associated with the multipaths can be understood without the influence of the group delay of these components. For example, consider again Figure 4.10, whose parts
(a) and (b) have an identical number of multipath components and the same relative power: they have the exact same value of στ=1.15 seconds. It is a well established fact in 131
the industry that knowledge regarding στ is beneficial in designing techniques (e.g, equalizers) to alleviate the effect of dispersive channels.
In Figure 4.11, we provide a sequence of RMS-DS values for a certain segment of a measurement run at MIA. Part (a) of this figure is the photograph of Figure 3.9, showing the measurement locations. Part (b) shows the corresponding RMS-DS values recorded in these locations, versus IRE number (time). We see multiple “transitions” within the recorded RMS-DS time series. This is because the mobile Rx was getting a strong specular signal when the LOS signal from the ATCT was not blocked by the parked aircraft, yielding low values of RMS-DS. For the PDP recordings taken when the mobile van was in the shadow of an aircraft, the value of στ was considerably larger.
Figure 4.11(a) Example measurement location at MIA [120]
132
RMS Delay spread in nanosecond 2500 Transitions RMS Delay Spread
2000 NLOS NLOS
1500
1000 In Nano-Seconds In
500
0 LOS 0 20 40 60 80 100 120 IRE number Figure 4.11(b) RMS-DS vs. IRE number (time) for locations in MIA of Fig. 4.11(a)
3. Delay Window (Wτ,x): The delay window is another metric that is used to quantify the spread of a signal in time. The delay window can be defined as the length of the middle portion of an IRE which contributes x% to the total energy of the IRE. [25]. For nearly all cases, Wτ,x can be interpreted in the same manner as στ; the larger the value of
Wτ,x, the more dispersive the channel. In Figure 4.12, we compare values στ and Wτ,90 for another segment of a measurement run at MIA. We can observe from the figure that even though there is a difference in the absolute values of the two metrics, there exists a strong correlation between them. This agrees with our understanding that both the parameters essentially provide us with a quantitative measure of the “temporal spread” of the physical media.
133
3500
RMS-DS in nsec 3000 Delay Window for 90% energy
2500
2000
In nsec 1500
1000
500
0 0 5 10 15 20 25 30 35 40 45 50 Profile Number
Figure 4.12 RMS-DS and Delay Window vs. IRE number (time) for a measurement segment at MIA [120]
4. Persistence Process Parameters: As is extensively reported in the literature on channel modeling, all multipath components can realistically be associated with a “birth and death” (i.e., on/off) process. Chapter 9 in reference [24] describes this nicely. The reason for the on/off behavior is simply the dynamic nature of the propagation physics: the orientation and position of reflectors and scatterers changes with platform mobility, often in such a way that these reflections (giving rise to received multipath components)
“come and go.” Recall from eq. (4.13) that we account for this on/off behavior in our
th channel model using the persistence process zl(t), in that formulation applied to the l multipath amplitude. A widely employed way of modeling such an on/off process (for numerous applications) is by using a Markov chain model. A Markov chain produces a sequence of random variables in which the future variable at time n+1 depends on the present variable at time n, but is independent of how the present variable arose from its predecessors. In signal processing terms, a Markov chain has memory of one time unit, 134 and this pertains strictly to a 1st-order Markov chain. A Markov chain model is typically specified using two matrices, the transition (TS) matrix and the steady state probability
(SS) matrix. An example TS matrix for a 3-state Markov chain is given in (4.21). Each element Pij in the matrix TS is defined as the probability of going from state i to state j.
In our IRE application, the states for the persistence process are two: either on (zl(n)=1) or off (zl(n)=0). Thus the persistence process TS matrix for any channel tap is a two by two matrix.
⎡P00 P01 P02 ⎤ TS = ⎢P P P ⎥ (4.21) ⎢ 10 11 12 ⎥ ⎣⎢P20 P21 P22 ⎦⎥
An example three-state SS matrix is provided in (4.22), where each element Pj gives the
“steady state probability” associated with the jth state. For IRE modeling, we obtain these
(two) elements for the tap persistence as “fractions of time” the multipath components are present (zl(n)=1) or not (zl(n)=0), directly from the PDP data.
⎡P0 ⎤ SS = ⎢P ⎥ (4.22) ⎢ 1 ⎥ ⎣⎢P2 ⎦⎥
In Figure 4.13, parts (a) and (b) show example persistence processes associated with the 2nd and 5th taps, respectively (for the definition of tap, refer to Figure 3.1) for a segment of measurement data from JFK. We also show the TS and SS matrices for both the taps in the figure. As we can infer from the matrices, tap 2 has a higher probability of being on than tap 5, and this tendency is visible from the figure as well.
135
Figure 4.13 Example persistence processes for taps 2 and 5 for segment of travel at JFK [120]
5. Tap Correlations: Each PDP is a collection of multipath components in different delay bins. From the physics of propagation, it is obvious that a given scatterer can contribute to several delay bins. Hence, there will often be a correlation among the multipath components in different delays bins. The amount of correlation will of course depend on several things, like the richness of the scattering environment, the delay resolution capability of the sounder, etc. When we use the term “correlation,” we mean that the multipath components associated with different delay bins will follow a similar trend over time. It is necessary to account for this correlation in the developed channel models since this information is crucial during the design and evaluation of advanced signal processing algorithms (diversity, etc.) at the receiver. We will use the correlation 136 coefficient instead of the correlation matrices so that the effect of energy of the individual taps is removed. The correlation coefficient matrix Rα can be defined as follows
⎡r11 .. r1n ⎤ R = ⎢ : : : ⎥ (4.23a) α ⎢ ⎥ ⎣⎢rn1 .. rnn ⎦⎥
cov(αiα j ) ri, j = (4.23b) var(αi ) var(α j )
th where in (4.19b), αi stands for the amplitude of the i tap, ri,j is the correlation coefficient between the ith and the jth tap, and cov and var stand for covariance and variance, respectively.
6. Phase (φk): Each multipath component is actually complex in nature and hence it has an amplitude and phase associated with it. This is clear from our description of the
CIR equations (4.17). The value of the phase φk is obtained directly from the recorded data (Table 4.3). Figure 4.14 shows an example of the variation of phase with time for the first tap for NLOS measurements at JFK. The range of phase values is from –π to π radians because as distance changes, our (non-phase-locked) correlator receiver outputs phase corresponding to the change in distance.
137
4 Phase in radian 3
2
1 ) k φ 0
Phase( -1
-2
-3
-4 0 5 10 15 20 25 30 35 40 45 50 Profile Index
Figure 4.14 Example phase variation versus time for 1st tap for JFK-NLOS [120]
4.3.2 Parameters Obtained via Fourier Transform of IREs
1. Channel transfer function: The time-varying channel transfer function can be expressed as follows:
∞ H (f ;t) = F{h(τ;t)} = ∫ h(τ;t)e− j2πfτ dτ (4.24) −∞
The function H(f,t) quantifies the time variation of the “complex amplitudes” of different
“spectral lines,” where by “spectral lines” we mean the values of H(f,t) at specific values of frequency f (the nomenclature arises from the finite dimensionality of the discrete
Fourier transform, done numerically via the Fast Fourier Transform (FFT) algorithm).
2. Frequency correlation estimate (FCE): The FCE can be interpreted in a fashion similar to the coherence bandwidth. The FCE provides us with a quantitative measure 138 regarding correlated scattering, but in the frequency domain. This information is of vital importance to researchers who intend to develop channel models in the frequency domain. This is also of significant interest to researchers dealing with narrowband modulation schemes such as OFDM. According to the initial processing of our data, we deduced that we often encounter correlated scattering in our collected measurements.
Hence, we cannot always assume the classical Wide Sense Stationary-Uncorrelated
Scattering (WSSUS) environment widely used in other terrestrial channel models [9].
We thus use a formula—for computing the FCE—that doesn’t rely upon WSSUS to determine correlation in the frequency domain; the formula in [32] satisfies this condition. In this method, the time variations of the complex amplitudes of different spectral lines are directly crosscorrelated with the time variations of a reference spectral line. The crosscorrelation is γH(aref,ai), where the term ai is the amplitude of the spectral lines at frequency index i, and aref is the amplitude at the reference frequency, i.e., a = H ( f ,t) | . The FCE is computed as follows, where index j orders the sequential ref f = fref time estimates:
γ (a ,a ) FCE = H ref i ; γ H (aref ,aref )γ H (ai ,ai ) (4.25) N 1 * γ H (aref ,ai ) = ∑ aref , j a i, j N j=1
Figure 4.15 shows an example FCE plot for CLE NLOS-S (BW = 50 MHz). The abscissa is the frequency in MHz with respect to midband, and the ordinate is the estimate of the channel’s correlation at the given frequency separation. Note that we use the “two-sided” measure of frequency separation here, since this FCE applies to bandpass systems. For an example, the FCE falls to 0.6 at a bandwidth of approximately 10 MHz. 139
1 NLOS-S 0.9
0.8
0.7
0.6
0.5
0.4 Correlation Coefficient Correlation 0.3
0.2
0.1
0 -25 -20 -15 -10 -5 0 5 10 15 20 25 Frequency in MHz
Figure 4.15 Example FCE for CLE [120]
2. Doppler Spread (fD): The Doppler spread is defined as the maximum value of
Doppler shift (fd) incurred by the signal as it transits the channel. For a single plane wave, fd is given as,
v f = cos()θ (4.26) d λ
In our case, since the measurements were conducted at a carrier frequency of 5.12 GHz, we have a wavelength (λ) of approximately 6 cm. Due to our channel sounder’s low update rate, we can not estimate the Doppler spectra from the measured PDPs we collect.
Hence, for the Doppler spectra of our taps, we rely on existing literature and analytical reasoning. We have referred to multiple papers that include experimental and analytical studies on the Doppler spectrum for mobile-to-mobile communications and airport surface communications. Next, we provide a small description regarding each setting:
¾ Airport Surface: The maximum speed (v) at which the measurement van traveled
was approximately 30 miles/hour, which is 14 m/s. This yields a maximum
Doppler shift fD of 234 Hz. This is approximately the value of Doppler spread 140
considered in [36] for “parking” and “taxi” scenarios. Note that in [36], the
author has also assumed these values without any actual measurements. The
Doppler spread provides a measure of the rate of change of the fading samples
across time for a given tap (multipath component). Depending on the relationship
between the maximum communication system symbol rate and the coherence
time, tc, the fading can be assumed to be fast or slow. Since fD in our case is 234
Hz, its reciprocal tc is approximately 4.3 millisec. Hence even for data rates as
low as 10 kbps, our channel can be assumed to be slowly fading. In order to
emulate this effect of the channel in our models, researchers often employ a low-
pass filter (LPF) for each tap’s fading process in the tapped-delay line model.
Similar approximations can also be seen in [36]. The cut-off frequency of the
LPF will depend on fD, and the filter’s shape depends upon the spatial distribution
of scattering. Generally when the value of fD is small relative to a symbol rate,
the actual shape of the Doppler spectrum is immaterial.
¾ VTV channel: The maximum Doppler (fD,max) for UOC would be ~400 Hz (with
a maximum relative vehicle speed of ~24 m/s or ~50 mph) and that for OHT
would be ~880 Hz (maximum relative vehicle speed ~52 m/s or ~120 mph).
These maximum Doppler frequencies are much smaller than our bandwidth of 50
MHz, so it is realistic to assume that the tap Doppler spectra are low pass, as in
[36]. Hence for simulation, after generating the tap amplitude processes, we use a
lowpass filter of bandwidth approximately fD,max to emulate this slow time
variation. 141
4.3.3 Labeling Conventions
In this chapter and in subsequent ones, we present results for different models,
different airports, different airport propagation regions, and different bandwidths (BWs).
To make this representation concise and interpretation easy, we define a convention for
representing these cases. The ordered quadruple of the form [Model Type, Airport
Name, Region, BW] will be used to identify the model type, name of the airport, the
propagation region of the airport (LOS-O, NLOS-S, NLOS) (see Chapter 3), and the BW
in MHz, respectively. Values we use for the BW are {50, 10, 5, 1} MHz. In some of the
labels, if any of the four fields is not pertinent, we then don’t mention it. In Table 4.8, we
list some example cases.
Table 4.8 Examples of labeling convention used for results [120]
Case Representation Model-1 for NLOS-S category at JFK for 10 MHz BW [M1, JFK, NLOS-S, 10] Model-3 for NLOS category at Burke Lakefront for 50 MHz BW [M3, BL, NLOS, 50]
4.4 Processing Considerations in Model Development
This section will discuss the different considerations that are encountered when
determining the number of taps for the channel model, and the different parameters
associated with the taps, such as the amplitude distribution, energy, phase, time
correlation, and persistence. As with most empirical modeling, there are multiple ways of
extracting the same information from the collected data files. We will present a
qualitative analysis using various examples to discuss the advantages and disadvantages
associated with the different ways of deriving our parameters. We believe that such a 142 discussion would be of value to channel modeling researchers since it would enable them to have a better understanding of the entire process of extraction of channel parameters.
Before proceeding, we first introduce another notation. This allows us to express (4.14) in a slightly different manner, which in some cases makes it easier to discuss the PDPs and CIRs: specifically, we let p(j,k) represent the power in the kth tap of the jth PDP.
4.4.1 Determination of the Number of Taps (L)
In this section, we discuss the different ways by which we can define the number of taps required to provide an accurate description of the underlying physical channel.
This directly affects the model complexity.
~ 4.4.1.1 Number of Taps from the Length of IRE ( L )
One of the ways of determining the number of taps is to consider the total length of the IRE [124]-[126]. This method may significantly increase the complexity of the system due to the large number of taps that are required for large BWs. For example, in the case of airport surface areas, where we have a total IRE length5 of 5 μsec, given a
BW of 50 MHz, we would need to consider ~250 taps. Similarly, for a BW of 10 MHz, we would have 50 taps. In general, the number of taps is always reduced as bandwidth decreases, even if the entire IRE is considered. There exists several different ways of reducing the number of taps:
5 We actually have 5.1 usec, but to prevent any possibility of “fold-over” we don’t consider the last 0.1 μsec of the IRE. 143
¾ Combining neighboring taps that have significant correlation (~0.5) among them
[124]. This method hence artificially creates the condition of uncorrelated
scattering among the taps.
¾ Reducing the BW of the channel model [126].
¾ Multiplying the steady state probability (SS) of each tap with its respective energy
and then discarding the taps that are 20 dB or more below the maximum valued
multipath component in the IRE. [125]
When the entire length of the IRE is used, it is often assumed that the taps exhibit
Rayleigh behavior in order to reduce the difficulty of stochastically modeling each tap.
Also, uncorrelated scattering is assumed to simplify the implementation of the model, i.e., it is easier to generate uncorrelated random variables than sets of correlated random variables. This approach of using the entire IRE length also has a distinct advantage of been able to faithfully replicate the worst case channel propagation conditions (e.g., IREs with a very large delay spread).
4.4.1.2 Number of Taps from the RMS-DS (L)
As discussed in Section 1.2, the RMS-DS is often used as a valuable measure of the channel dispersion. The use of RMS-DS to determine the number of taps is also popular among researchers [127], [104]-[107],[112]. This method has been popular to determine the number of taps for different propagation conditions such as 2G terrestrial cellular [127], 3G terrestrial cellular like UMTS [112], local area networks [105], metropolitan area networks like WiMax [104], upcoming standards for high velocity 144 platforms like 802.20 [106] and [107], etc. To proceed, Figure 4.16 provides the distribution of RMS-DS for [JFK, NLOS, 50].
600
500
400
300
Number of Profiles 200
100
0 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 RMS-DS in nsec
Figure 4.16 Histogram of RMS-DS for [JFK, NLOS, 50] [120]
We can of course compute different statistics (e.g., minimum, mean, median and maximum) for RMS-DS. In general any one of these statistics could be used to determine the number of taps (L). It is also possible to use the mode of the distribution to determine
L if it is a single-mode distribution. In our case, depending on the application (either airport or VTV), we have used different statistics to determine the number of taps:
¾ For airports, we have used the mean RMS-DS to determine the number of taps.
We prefer using the mean rather than the median since most of the pdfs of the
RMS-DS for different airports have a “deep tail.” i.e, there is a non-negligible
percentage of PDPs with RMS-DS significantly higher than the mode of the
RMS-DS histogram. The histogram of RMS-DS is asymmetric about the mode 145
and has a “deep tail”. For the airports, we intended to develop 50 MHz models,
so we used the mean and not the maximum delay spread to reduce the number of
taps and hence limit the complexity of the developed channel model.
¾ For VTV, we have used the maximum RMS-DS since the intended BWs of the
DSRC system are 10 MHz and 5 MHz. Also, for VTV measurements, to increase
the update rate of the IREs, we used a reduced maximum delay span of 3 μsec.
The update rate was increased since for most of the scenarios, we don’t expect
long delay multipath components beyond a delay of 3 μsec. This is because we
were rarely in open areas with sparse scattering environment other than the time
when we measured in OLT. Also, the maximum doppler frequency that can be
encountered in VTV scenario is higher than those encountered on the airport
surface. Hence, a slightly higher update rate would be beneficial for VTV
measurements. Due to this and also due to the different scattering geometry, the
maximum RMS-DS of the VTV channel is significantly smaller than that
measured on the airport surface. Hence, for the VTV channel using the maximum
RMS-DS to determine L doesn’t increase the complexity of the tapped delay
model significantly.
On the basis of these considerations, the number of taps in the models are found by
⎡mean()RMS − DS ⎤ LAS = ⎢ ⎥ +1 ⎢ Tc ⎥ ⎡max()RMS − DS ⎤ LV 2V = ⎢ ⎥ +1 (4.27) ⎢ Tc ⎥ 146
In (4.27), Tc is the chip time. For example, in the case of [JFK, NLOS, 50], LAS is then 75 and for [UOC, 10], LVTV is 15. [Will put the reference table number once I complete the other chapters]
4.4.2 Markov Modeling: Transitions between Regions and Persistence Processes
As discussed previously, we use the “persistence process” to model the non- stationarity associated with the ON/OFF behavior of the multipath (VTV and airport surface). We also employ transitions between different regions on the airport surface to model the non-stationarity of a longer time scale than the persistence process. In this section, we discuss extraction of the parameters necessary for modeling the transitions between different regions on the airport. We also describe how we use a Markov chain to model the persistence process associated with the taps (our zl(t) processes of (4.14)).
The former application is new, and could be used to model vehicles traversing all three regions over a period of time. The latter application has been cited in some research papers, ([115], [116]), but is not widely used.
4.4.2.1 Modeling Transitions between Regions
As discussed, while moving on the airport, the receiver may pass through different regions—LOS-O, NLOS-S, and NLOS. A realistic channel model should be able to emulate such conditions. The different regions can be modeled as different states of a Markov chain model. Notationally, the different regions can be assigned state numbers: the LOS-O, NLOS-S, and NLOS regions are denoted states 1, 2, and 3, 147 respectively. The algorithm used to determine the Region_TS and Region_SS matrices for a given airport from the collected data is as follows:
1. Consider an airport (here for example, say CLE)
th 2. Associate a state index (Reg_Statusj) with the j PDP of CLE depending on its
RMS-DS. For example, for CLE, if the RMS-DS for the jth PDP is 234 nsec, then
Reg_Statusj=2, denoting the NLOS-S region. (Refer to Table 3.2 for other
demarcations for RMS-DS).
3. Repeat Step 2 for all the PDPs (for CLE).
4. Obtain Region_TS and Region_SS matrices for CLE using the Reg_Status vectors
developed from steps 2 and 3. The SS matrices are the steady-state probabilities
computed as fractions of time, and the TS matrices are transition probabilities
computed by counting transitions and dividing by the total number of PDPs minus
one.
Figure 4.17 provides an example measurement set of RMS-DS vs. time obtained from CLE. The figure illustrates that the receiver inhabits different regions during the course of travel. Depending on the region, each profile has a region state associated with it. The top part of the figure provides the corresponding RMS-DS values of the profiles.
It can be observed that as the RMS-DS changes, the profile moves from one region to another. The thresholds for each region in term of RMS-DS are provided in Table 3.2.
148
2,000 1,800 1,600 1,400 1,200 1,000 800 600
RMS-DS (nsec) RMS-DS 400 200 0 0 5 10 15 20 25 30 35 40 Profile Index
3
2
Region State Region 1
0 0 5 10 15 20 25 30 35 40 Profile Index
Figure 4.17 Region state and corresponding RMS-DS for an example measurement set at CLE [120]
The corresponding matrices Region_TS and Region_SS for CLE obtained using our measurements and the above algorithm is as follows:
⎡0.4308 0.4099 0.1593⎤ ⎡0.1582⎤ ⎢ ⎥ ⎢ ⎥ Region_TS = ⎢0.2193 0.5244 0.2563⎥ Region_SS = ⎢0.3127⎥ (4.28) ⎣⎢0.0406 0.1586 0.8008⎦⎥ ⎣⎢0.5291⎦⎥
Thus for CLE, from the state probability matrix Region_SS, we see that the receiver spent approximately 16% of the time in the LOS-O region, 31% of the time in the NLOS-S region, and 53% of the time in the NLOS region.
4.4.2.2 Modeling the Tap Persistence Processes 149
The changing fading conditions due to the movement of scatterers, blockage of signal due to obstruction, etc., presents a very dynamic channel for communication systems on the airport surface and VTV applications. Hence, some of the multipath echoes might “persist” only for certain percentages of time and then “disappear” (or take amplitudes which are below our threshold). As noted previously, our “persistence process” is designed to capture this effect by taking values of one or zero depending on whether the multipath component (tap amplitude) is above the multipath threshold or not.
The steady state probabilities are the probabilities of being in either the one or zero state, overall. The transition probabilities are the probabilities of going from one state to itself, or to the other state. All these probabilities were determined empirically from our data.
As discussed in Section 1.2, the persistence process are modeled using a Markov chain.
During the course of model development, because of the prevalence of threshold crossings of the tap processes, we determined that the steady state probability associated with the taps is a key factor for providing realistic and accurate channel models. The steady state probability of a tap provides an estimate of the percentage of time that this tap can be considered as a valid multipath component. So essentially, we need to extract
TS and SS matrices from our collected data for each of the taps of each region for a particular airport. After the “quantization” of RMS-DS into state values as in Figure
5.18, these tap transition and emission matrices are empirically determined by relative
U U frequency. The algorithm used in order to determine the TSi and SSi matrices for the ith tap of a Region U (U is NLOS-S, NLOS, or LOS-O for airports or is UOC, Small,
UIC, OHT or OLT for VTV) is as follows:
150
1. Consider an airport and a specific region associated with it (here, say JFK and
NLOS).
2. Consider the ith tap (for NLOS at JFK).
i th th 3. Associate a state index ( Multipath _ Status j ) with the j PDP of the i tap
depending on whether the tap’s power is within 25 dB of the maximum tap value
for the jth PDP. For example, if the maximum power in the jth PDP is 0 dB, and
the power associated with the ith tap of the jth PDP is -23 dB, then
i Multipath _ Status j is 1.
4. Repeat step 3 for all the PDPs (of JFK NLOS) for the ith tap and obtain a binary
(0,1) vector Multipath _ Status i for the ith tap.
NLOS NLOS th 5. Obtain TSi and SSi matrices for the i tap (of JFK NLOS) using the
Multipath _ Status i vector. (The TS elements are determined by counting
transitions among the various states.)
6. Repeat Steps 2-5 for all taps (L = 75 for JFK NLOS).
Figure 4.18 provides the steady state probability of having a multipath component
(state z=1) for [JFK, NLOS, 50] and [JFK, NLOS-S, 50] versus tap index. We also
NLOS NLOS NLOS −S NLOS −S rd provide example values of (TS3 , SS3 ) and (TS3 , SS3 ) for the 3 tap. As expected, the steady state probability for State 1 (tap “on”) decreases as the tap index increases. We also notice that the steady state probability for State 1 is higher in NLOS than in NLOS-S for higher-index taps. 151
Figure 4.18 Steady state probabilities of State 1 for [JFK, NLOS, 50] and [JFK, NLOS-S, 50] [120]
4.4.3 Tap Energies
In this section, we discuss the algorithm used to determine the energy associated with each tap. We follow with a discussion on cumulative energy gathered with each succeeding tap in the model.
4.4.3.1 Energy Determination for Each Tap
Although the sounder outputs a single amplitude corresponding to each tap, this amplitude is actually almost always an aggregate of multiple rays which couldn’t be resolved due to the limited delay resolution capability of the sounder. For example, in the 152 case of a 50 MHz BW, each tap is associated with a delay bin of 20 nanoseconds.
Depending on the scattering environment, the energy associated with each tap will be different. In all contemporary channel models, the energy associated with each tap is an essential element in defining the model. The presence of high energy multipath components at higher-indexed taps yields a highly dispersive channel. Conversely, in the case of a mildly dispersive channel, a large percentage of the aggregate energy is within the lower-indexed taps. We determine the energy associated with each tap from our measured data.
As discussed in Section 1.3.2, the multipath components (taps) each have a persistence process associated with them. This complicates somewhat the computation of average energy. Hence we need to be careful in defining the average energy for each tap.
Clearly, for averaging we should consider only those PDPs in which the tap is recognized as a valid multipath component (i.e., the energy of the tap is within the MT of 25 dB of the maximum energy tap in that PDP). We also account for the tap persistence. The
U th algorithm used in order to determine Energyi for the i tap of Region U is as follows:
1. Consider an airport and a specific region (here, say JFK and U=NLOS).
2. Consider the jth PDP (for NLOS at JFK)
3. Normalize the energy in the jth PDP so that the total energy in that PDP is unity.
th th i 4. Assign the energy of the i tap in the j PDP toTemp _ Energy j .
5. Repeat step 4 for all taps (in JFK NLOS, L = 66)
i i i th 6. Create Valid _ Energy j = Temp _ Energy j × Multipath _ Status j for the j PDP.
i th Note: Multipath _ Status j =z is either 1 or 0 depending on whether or not the i tap
is a valid multipath component. 153
7. Repeat steps 3-6 for all PDPs collected (for JFK NLOS).
U th 8. Find Energyi for the i tap using the formula
i i ∑Valid _ Energy j ∑ Multipath _ Status j . j j
9. Repeat step 8 for all taps.
Figure 4.19 provides the average energy associated with each tap for [JFK,
NLOS, 50] and [JFK, NLOS-S, 50]. As expected, the average energy decreases as the tap index increases. We also notice that the rate at which the tap energy decreases with tap index is much faster for NLOS-S than for NLOS. We also provide an exponential curve fit for both the plots (τ is the tap index) . Note that the exponential fit is in dB units, in contrast to many academic papers, which use linear (e.g., voltage) units; a direct correspondence between these two forms can be made, if desired.
0 NLOS NLOS fit: 10.8 exp(-14τ) - 18.7 -5 NLOSS NLOSS fit: 34.2 exp(-0.35τ)- 25.3
-10
-15
-20 Average Energy in dB in Energy Average
-25
-30 0 10 20 30 40 50 60 70 80 Tap Index
154
Figure 4.19 Average energy associated with tap for [JFK, NLOS, 50] and [JFK, NLOS-S, 50] [120]
4.4.3.2 Cumulative Energy Gathered
Determining the number of taps to fairly trade off implementation complexity and model fidelity is one of the most crucial steps when designing the channel model. There are several techniques available in the open literature that researchers have used to obtain the best balance between complexity and accuracy. The criterion most often used is the energy associated with the tap, or more precisely, its contribution towards the cumulative energy of the PDP. As noted, we maintain that the tap steady state probability, in addition to the energy contribution, should determine whether or not the tap should be
~ considered in the model. In essence, we determine L or L as an upper bound to the number of taps we plan to include in the model. In order to account for both tap energy and steady state probability, we define the cumulative energy of tap i as
U U U Cumu _Energyi = Energyi × SSi (4.29)
U Note that the aggregate Energy i for a given model is not one. This will be taken care of after truncating the number of taps depending on the cumulative energy. We provide an example plot of this for [JFK, NLOS, 50] and [JFK, NLOS-S, 50] in Figure 4.20. This figure provides the cumulative energy for both these cases as a function of tap index, where we have used L taps for each region, as derived previously. As expected, the increase in cumulative energy is a much steeper function of tap index for NLOS-S than it is for NLOS.
155
1
0.9
0.8 NLOS NLOS-S 0.7 Power
0.6
0.5
0.4 0 10 20 30 40 50 60 70 80 Tap-Index
Figure 4.20 Cumulative energy gathered for [JFK, NLOS, 50] and [JFK, NLOS-S, 50] [120]
4.4.4 Tap Correlation Matrix
Our measurements have shown that the airport surface channel and the VTV channel can often have correlation among taps. The correlation coefficient matrix is calculated using (4.19). There are several things to note before determining these correlation matrices.
1. As described in Chapter 3, for a particular airport we have numbered the route of
travel. During our measurement campaign, we moved from one point to another.
Due to the different behavior of the scatterers and the different physical structures
at different locations on the airport, the correlation matrix needs to be evaluated
for each segment of travel separately. For example, JFK airport was mapped into
U 26 points (25 segments of travel). The notation Rmn will mean that the correlation
coefficient matrix was determined using the PDPs collected while traveling 156
between points m and n in Region U. It was also mentioned in Chapter 3 that
there were no formal routes planned for the VTV measurements. Yet even then,
the data recorded during any specific measurement campaign was parsed into
separate files. Hence, an approach similar to the one mentioned for JFK was
followed for the different regions of VTV as well.
U 2. For a particular Rmn , we compute its individual elements ri,j (4.19). Recall that
each tap has a persistence process associated with it. So it is often not the case
that both taps i and j exist for the same PDPs. Hence ri,j should be calculated
using the PDPs only when both these taps are valid multipath components (zi and
zj both equal 1). Because of this phenomenon, different numbers of PDPs may be
U used to compute the different ri,j entries for a particular Rmn . For example, in the
NLOS−S case of R12 for JFK, r1,2 might have been computed using 45 PDPs, whereas
r3,8 was computed using only 10 PDPs. We illustrate this using Figure 4.21. This
NLOS−S figure is plotted for R12 for JFK, the third (“z”) axis is the number of PDPs
used, and the x and y axes are the tap indices. Note that it is possible that for
higher-indexed taps, there might not be any PDPs in a given segment of travel
where both the taps exist.
157
45
40
50 35 45
40 30 35
30 25
25
20 20 Number of PDPs 15 15 10
5 10 0 0 0 2 2 4 4 6 5 6 8 8 10 10 12 12 14 14 Tap Index Tap Index 16 16
NLOS−S Figure 4.21 Number of PDPs used to determine R12 for [JFK, NLOS-S, 50] [120]
As we have mentioned several times, there are multiple ways of determining the various channel parameters. Similarly for the case of determining the appropriate correlation matrix, there are several choices depending on our choice of statistical confidence. As is evident from the previous discussion, we have some amount of
U flexibility in determining matrices Rmn for each region and travel segment. One method
U could be averaging all the Rmn ’s, but this would defeat the purpose of maintaining the integrity of the real fading behavior in each segment. There are though a number of ways of providing a representative correlation matrix. Here we address several ways: the first provides the worst-case correlations encountered, and the second presents the “most statistically confident” correlation matrix. Two other methods are also proposed.
158
4.4.4.1 Worse Case Correlation Matrix
U The worst case correlation matrix Rwc has the individual worst case values for each matrix element ri,j. We define the worst-case value as the ri,j value with the largest magnitude. For example, for an airport like JFK with 25 segments of travel, we have 25 possible values for each ri,j. In actuality, since we made multiple measurement runs over
U some segments, we had a total of 42 measured correlation matrices. The entries of Rwc are the maximum |ri,j| from these 42 possible values, for each (i,j) pair. (Note that via symmetry, only the lower or upper triangular part of each matrix must be computed, so
2 NLOS for an L-tap channel, there are (L -L)/2 unique elements.) The Rwc for [JFK, NLOS, 5
MHz] is provided in (4.25). The number of taps LAS=9 was determined using (4.22).
⎡ 1 0.4859 0.614 0.4503 0.7092 0.7456 0.4238 0.8633 0.7087⎤ ⎢ ⎥ ⎢0.4859 1 0.5448 0.8929 0.9453 0.5427 0.7310 0.7946 0.7046⎥ ⎢0.6410 0.5448 1 0.6603 0.5474 0.7943 0.5107 0.3491 0.6610⎥ ⎢ ⎥ 0.4503 0.8929 0.6603 1 0.7849 0.8056 0.5386 0.4589 0.6839 ⎢ ⎥ (4.30) ⎢0.7092 0.9453 0.5474 0.7849 1 0.7006 0.4666 0.6691 0.6903⎥ ⎢ ⎥ ⎢0.7456 0.5427 0.7943 0.8056 0.7006 1 0.8208 0.5572 0.4562⎥ ⎢0.4238 0.7310 0.5107 0.5386 0.4666 0.8208 1 0.7626 0.6254⎥ ⎢ ⎥ ⎢0.8633 0.7946 0.3491 0.4589 0.6691 0.5572 0.7626 1 0.8456⎥ ⎢ ⎥ ⎣0.7087 0.7046 0.6610 0.6839 0.6903 0.4562 0.6254 0.8456 1 ⎦
In the next equation (4.31), we present the matrix containing the number of PDPs
NLOS that were used in determining the elements in Rwc of (4.30): 159
⎡ 1 196 8 8 8 5 13 11 7 ⎤ ⎢ ⎥ ⎢196 1 78 5 5 8 8 10 12⎥ ⎢ 8 78 1 17 37 7 25 28 27⎥ ⎢ ⎥ ⎢ 8 5 17 1 6 11 21 9 5 ⎥ ⎢ 8 5 37 6 1 34 6 8 9 ⎥ (4.31) ⎢ ⎥ ⎢ 5 8 7 11 34 1 6 8 13⎥ ⎢ 13 8 25 21 6 6 1 8 21⎥ ⎢ ⎥ ⎢ 11 10 28 9 8 5 8 1 9 ⎥ ⎢ ⎥ ⎣ 7 12 27 5 9 13 21 9 1 ⎦
In (4.31), for the diagonal elements, the number of PDPs is 1. This is because we don’t actually estimate the correlation value of the tap with itself from our data. From (4.31), we can observe that some of the correlations were determined using very few PDPs.
Hence, we can’t be statistically confident of those values. Yet we can not deny the occurrence of the small number of PDPs that yielded these worst case correlations on the airport surface. To ensure that we are accounting for the worst case, and to err on the side of pessimism (high correlation) we thus need to include such “low-confidence” correlations. Note that since there can’t be any single PDP which has all the worst case
U correlations for all taps simultaneously, Rwc is a (pessimistic) upper bound for the correlation values.
4.4.4.2 “Maximum Confidence” Correlation Matrix
We consider the same example of JFK with 25 segments of travel, and 42 possible values for each ri,j, with each value ri,j computed using some number of PDPs.
Out of these 42 values, we select the ri,j that is computed using the maximum number of
PDPs. We denote the resulting correlation matrix the “maximum confidence” correlation
U NLOS−S matrix, Rmc . The Rmc matrix for [JFK, NLOS, 5 MHz] is provided in (4.32). 160
⎡ 1 0.4043 0.3085 0.2245 0.1435 0.1329 0.1112 0.1146 0.0663 ⎤ ⎢ ⎥ ⎢0.4043 1 0.1923 0.0677 0.0707 0.0256 − 0.0038 0.1052 − 0.0239⎥ ⎢0.3085 0.1923 1 0.1399 0.1374 0.0990 0.0836 0.0698 0.0039 ⎥ ⎢ ⎥ 0.2245 0.0677 0.1399 1 0.1165 0.1023 0.0862 0.0633 0.0761 ⎢ ⎥ (4.32) ⎢0.1435 0.0707 0.1374 0.1165 1 0.1348 0.0702 0.0335 0.0712 ⎥ ⎢ ⎥ ⎢0.1329 0.0256 0.0990 0.1023 0.1348 1 0.1249 0.0302 0.0287 ⎥ ⎢0.1112 − 0.0038 0.0836 0.0862 0.0702 0.1249 1 0.0737 0.1076 ⎥ ⎢ ⎥ ⎢0.1146 0.1052 0.0698 0.0633 0.0335 0.0302 0.0737 1 0.0369 ⎥ ⎢ ⎥ ⎣0.0663 − 0.0239 0.0039 0.0761 0.0712 0.0287 0.1076 0.0369 1 ⎦
NLOS Equation (4.33) displays the number of PDPs used to compute the elements in Rmc .
⎡ 1 722 693 661 635 615 573 582 558⎤ ⎢ ⎥ ⎢722 1 657 623 601 581 542 547 525⎥ ⎢693 657 1 616 586 565 528 534 516⎥ ⎢ ⎥ ⎢661 623 616 1 575 547 514 517 497⎥ ⎢635 601 586 575 1 538 504 503 490⎥ (4.33) ⎢ ⎥ ⎢615 581 565 547 538 1 506 505 490⎥ ⎢573 542 528 514 504 506 1 489 475⎥ ⎢ ⎥ ⎢582 547 534 517 503 505 489 1 483⎥ ⎢ ⎥ ⎣558 525 516 497 490 490 475 483 1 ⎦
From (4.33), we can observe that number of PDPs used to determine the correlation matrix is large compared to the entries in (4.31).
4.4.4.3 Realistic “Worse Case” Correlation Matrix
NLOS −S As discussed previously, Rwc is not a practical correlation matrix that we
U actually measured. So to provide a more realistic correlated channel, we introduce Rrwc .
We consider the same example of JFK with 25 segments of travel and 42 measured correlation matrices. Out of the 42 matrices, we select the single correlation matrix that is the worst. We determine this “worst” rating by assigning a “severity value” to each matrix. The “severity value” of the matrix is the sum of the absolute values of the ri,j for 161
NLOS −S that matrix in the lower triangular portion. The Rrwc matrix for [JFK, NLOS, 5 MHz] is provided in (4.34).
⎡ 1 − 0.0879 0.5921 0.1238 −1 − 0.2259 0.2411 0.8121 1⎤ ⎢ ⎥ ⎢− 0.0879 1 − 0.0714 − 0.7769 −1 − 0.1029 − 0.8518 0.3173 1⎥ ⎢ 0.5921 − 0.0714 1 − 0.0201 −1 −1 − 0.1952 0.8708 0⎥ ⎢ ⎥ ⎢ 0.1238 − 0.7769 − 0.0201 1 −1 −1 0.9403 − 0.3097 1⎥ (4.34) ⎢ −1 −1 −1 −1 1 1 1 −1 0⎥ ⎢ ⎥ ⎢− 0.2259 − 0.1029 −1 −1 1 1 −1 1 1⎥ ⎢ 0.2411 − 0.8518 − 0.1952 0.9403 1 −1 1 − 0.3782 1⎥ ⎢ ⎥ ⎢ 0.8121 0.3173 0.8708 − 0.3097 −1 1 − 0.3782 1 1⎥ ⎢ ⎥ ⎣ 1 1 0 1 0 1 1 1 1⎦
As can be noticed from (4.34), some of the elements are 1, -1 or 0, and these actually result from the small number of PDPs (fewer than 5) used to determine the corresponding ri,j values. Thus, these correlation values are “low confidence” ones. The zeros denote the complete absence of PDPs in common (e.g. r9,3=0 means tap 9 was never present above threshold when tap 3 was above threshold). The values +1 and -1 mean that for the very few PDPs in common, the correlation coefficient was near +1 or -1.
4.4.4.4 Realistic “Maximum Confidence” Correlation Matrix
U Similar to Rrmc , to obtain a realistic “high confidence” correlated channel, we
U introduce Rrmc . We consider the same example of JFK with 25 segments of travel (in total 42 sets), yielding 42 possible values for the correlation matrix. Out of these 42 matrices, we select the correlation matrix that is calculated using the maximum number of PDPs. We determine this by assigning a “confidence value” to each matrix. The
“confidence value” of element ri,j is the number of PDPs used to determine ri,j divided by the maximum number of PDPs ever used to determine any ri,j, over the entire airport in 162 this region. Then the “confidence value” of a segment is the sum of all “confidence
NLOS −S values” of the elements in its matrix. The Rrwc matrix for [JFK, NLOS, 5 MHz] turns out to be the same as (4.29). This might not be the case for other airports, of course,
U U which may have distinct matrices Rrmc and Rmc .
Figure 4.22 summarizes all the steps involved in extraction of channel related parameters from the pre-processed PDPs. This figure can be viewed as a continuation of
Figure 4.9.
Select the criteria for determining the number of taps (see section 1.3.1)
Determine the number of taps
Determine the Markov chain models for transitions between regions and the persistence processes for taps
Determine the average energy associated with each tap
Determine the channel tap correlation matrix
Output: Channel parameters to begin definition of empirical model (inputs to Chapter 6, 7, 8 for modeling)
Figure 4.22 Summary of model extraction steps of Section 4.3 [120]
163
4.5 Different Models for Airport Surface Areas and VTV Settings
As observed from the discussions presented in the previous sections, there are multiple ways of creating the stochastic channel models for any setting. In this section, we will present three different kinds of channel models for each propagation region of the airport surface area and VTV environment. The parameters (number of taps, cumulative energy, persistence processes, and correlation matrix) of these models are obtained using the different algorithms presented in the previous sections. By using different combinations of these parameters, we present channel models with varying levels of complexity and fidelity.
4.5.1 Airport Surface Area
For the airport surface area, for every region (LOS-O, NLOS-S and NLOS), we propose 3 different models.
¾ Model-1 (M1) : The main focus of M1 is reasonable fidelity with minimum
implementation complexity.
• The upper bound on the number of taps is determined using the mean RMS-
DS of the respective region (Section 1.3.1.2). Thus the upper bound is LAS,
as in (4.22).
• The number of taps considered for the actual models are reduced depending
on the cumulative energy of the taps (Section 1.3.3.2). Note: For different
regions, we use different values of cumulative energy to limit the number of 164
taps. So, for LOS-O, NLOS-S and NLOS, we use 99%, 99% and 95% of the
aggregate energy, respectively.
• Once the number of taps have been decided, the channel tap energies have
been re-normalized to account for the truncation (95% energy in NLOS,
99% energy in NLOS-S) so that the sum of all tap energies multiplied by
their steady-state probabilities) equals unity.
• We also employ the persistence process for each individual tap. The
parameters for the persistence processes are obtained using the algorithms
presented in Section 1.3.2.
• The last choice is regarding the correlation matrices. Broadly, we can use
any of the correlation matrices presented in Section 1.3.4. To account for
U worst case conditions, we use Rwc .
¾ Model-2 (M2): The main focus of M2 is to have maximum fidelity and reasonable
implementation complexity.
• The upper bound on the number of taps is total length of the IRE (Section
1.3.1.1). So for a given BW, the upper bound on the number of taps will
remain the same irrespective of the region. For example, in the case of a 25
MHz BW, the number of taps will be limited to 100.
• The focus of M2 is to be as close as possible to the actual data. Hence, we
don’t reduce the number of taps depending on the cumulative energy. So to
implement the tapped delay line model, we use the upper bound on the
number of taps, corresponding to 100% of the energy.
• A persistence process is used for each individual tap. 165
• With respect to the correlation matrices, in the actual channel model
simulations, we change the correlation matrix after every n realizations of
the channel. This would realistically simulate moving from one point to
another on the airport surface. The value of n would generally be random,
and the changing of correlation matrices could for example be based upon
actual measured matrices and travel times and regions being simulated.
¾ Model-3 (M3): The main focus of M3 is to provide a WSSUS [16] version of
Model-2. This in essence helps evaluate whether or not non-stationary modeling
provides a more precise representation of the channel than does WSSUS
modeling.
• The upper bound on the number of taps is determined in the same manner as
in M2
• As in M2, the number of taps is not reduced. So the tapped delay line model
uses the upper bound on the number of taps, and 100% energy. Note that in
M2, for any given PDP, the aggregate energy gathered using all the taps will
be less than 1 since the persistence process might turn OFF some of the taps.
While in the case of M3, each PDP will have an aggregate energy of 1. We
have evaluated this energy difference between PDPs for M2 and M3 for
various BW channels and found that on average M3 gathers ~0.1-0.3dB
more energy than M2 for any given PDP.
• The persistence processes are turned off. This is done to maintain WSS in
the time domain. 166
• No correlation matrix is used (or, a diagonal (identity) matrix is used) so we
assume uncorrelated scattering.
4.5.2 VTV Setting
Similar to the airport surface area, in the case of VTV communication links, for every region (UOC, UIC, OHT, OLT and Small), we propose 3 different models.
¾ Model-1 (M1): The main focus of M1 is reasonable fidelity with minimum
implementation complexity.
• The upper bound on the number of taps is determined using the maximum
RMS-DS of the respective region (Section 1.3.1.2). So, the upper bound is
LVTV, as in (4.22).
• The number of taps considered for the actual models are reduced depending
on the cumulative energy of the taps (Section 1.3.3.2). We limit the number
of taps to accumulate 99% of the energy, for all cases. Note: Once the
number of taps have been decided, the channel tap energies have been re-
normalized to account for the truncation (99% energy) so that the sum of all
tap energies multiplied by their steady-state probabilities) equals unity.
• We also employ the persistence process for each individual tap.
U • To account for worse case scenarios, we use Rwc (Section 1.3.4.1).
¾ Model-2 (M2): The main focus of M2 is to have maximum fidelity and reasonable
implementation complexity. 167
• The upper bound on the number of taps is still determined using the
maximum RMS-DS of the respective region (Section 1.3.1.2).
• We don’t reduce the number of taps based on the cumulative energy. Thus
to implement the tapped delay line model, we use the upper bound on the
number of taps and gather 100% energy.
• A persistence process is used for each individual tap.
• As in the airport surface area, in the actual VTV channel model simulations,
we change the correlation matrix after every n realizations of the channel.
The actual value of the correlation matrix and the number of realizations n
depend on the actual data files collected during individual measurement
campaigns.
¾ Model-3 (M3): As in the case of airport surface areas, M3 is used to provide a
WSSUS [9] version of Model-2.
• The upper bound on the number of taps is determined in the same manner as
in M2-VTV using the maximum-RMS-DS for individual regions.
• As in M2, the number of taps is not reduced. The tapped delay line model
uses the upper bound on the number of taps, for 100% energy.
• All persistence processes are turned off. This is done to maintain WSS in the
time domain.
• No (or an identity) correlation matrix is used, to yield uncorrelated
scattering. 168
5. Severe Fading
In this chapter, we present several example cases where we have seen evidence of severe, or “worse than Rayleigh,” (WR) fading. To explain this behavior, we present two physical models. The first model employs a statistically non-stationary random process that switches between two distributions, akin to the multi-state models proposed for land mobile satellite channels. The other model we propose is a multiplicative model of two small scale fading processes. We also describe the analytical pdf for such a multiplicative model using two Weibull random variables as the underlying small scale fading distributions. Measured data is used to corroborate these models, and our computer simulations confirm the ability to replicate the statistics of these severe fading processes.
Note: The text, tables and figures in this chapter is based on the matter presented by the author in [143].
5.1 Introduction
For many years, Rayleigh fading has been used to represent the most severe form of fading that can be encountered [4]. Yet as more research into channel modeling proceeds, investigations have found an increasing number of instances in which channel fading is more severe than Rayleigh. We have already discussed several instances of WR fading in Chapter 2. The amount of measured data available to characterize WR fading is limited. Naturally, WR fading can have detrimental effects on system performance, such as reduced link capacity, increased probability of deep fades, and poor diversity receiver
BER performance. The motive behind this part of the research is to associate the 169 occurrence of severe fading with the physical environment. By doing so, we can predict the regions where we might encounter such severe fading. In this chapter, we present some physical justification for the observance of WR fading at airports and in VTV communication settings and connect the observed severe fading with the existing and on- going research on WR fading.
As discussed, the channel can be assumed to be a time varying linear filter which can be characterized completely using its impulse response (refer eq.(4.13)). As an example, Figure. 5.1 shows a surface plot that depicts the time evolution of the PDP for a
VTV (UIC) case for a 10 MHz bandwidth. Fading in time is clearly evident, with several of the multipath components varying by 10 dB or more over this 200 millisec interval.
Figure 5.1 Example PDP evolution in time for UIC VTV environment. Delay (rightward axis) in microseconds, and time (leftward axis) in seconds [143] 170
Fading severity is determined by the tap amplitude statistics. For example, for the
Weibull distribution, values of b less than 2 equate to severe fading [69] by our definition. Similarly, Nakagami distributed taps with values of m less than 1 are considered severely faded [4]. Examples of WR fading for some channel taps were found in the 50 MHz channel models developed for the airport surface area channels.
Figure 5.2 shows example distribution fits for the 2nd tap amplitude for different airports for a 50 MHz BW. WR fading is apparent from the distribution fits. Similar WR behavior was also found for some tap amplitude statistics of channel models with smaller values of bandwidth for airport surface models (Chapter 6, [111], [133], [134]) and for VTV channel models (Chapter 8, [144]). As discussed, the Weibull distribution has proved to be a better fit for most of our collected data. This can be observed from Figure 5.2
Figure 5.2 Example distribution fits for amplitude data of 2nd tap for different airports; (a) MIA, (b) CLE [143] 171
5.2 Multiplicative Model
Multiplicative models or “cascaded” channels for small scale fading processes have been proposed for different communication conditions, such as the so-called “pin- hole” channels for MIMO applications [76], “amplify and forward” channels for relays
[77], etc. We justify the presence of multiplicative scattering on airport surface areas. The existence of multiplicative scattering is primarily due to certain configurations of scatterers in the propagation environment. Explanations of propagation conditions based on the features of specific scattering environments have been used by system researchers for a long time. Some examples of the same are, the “urban canyon effect” for terrestrial cellular [4], “ducting” in indoor halls [127], “clustering of multipath components” for indoor communications [27], etc. From the previous literature on multiplicative scattering [74]-[76], [145], there are several features that enable the existence of these pin-hole channels. One of the physical settings requires two conditions to be present in the structure of the physical scatterers in the surroundings. First, the size of the “scatter ring” around the transmitter and receiver should be small compared to the distance between the transmitter and receiver; second, there must be negligible scatterers in the medium between the scatter rings of both transmitter and receiver. The second condition can also be described as a rank-1 propagation path [144]. Figure 5.3 shows example places on the airport surface areas where the above two conditions appear to be satisfied.
Figure 5.3(a) shows the transmitter at the CLE ATCT, and parts (b), (c), and (d) are example locations on the CLE airport surface area. We have also highlighted the scatter rings at the receiver end for these example locations. Strictly speaking, the Tx and the Rx are not surrounded on all sides by scatterers but as long as there are sufficient number of 172 scatterers, the conditions for existence of “pin-hole channels” will hold. Fig. 5.3(d) also shows the receiver in the mobile van. Another reason for the existence of “pin- hole” channels in airport surface areas is the large difference between the heights of the transmitter (ATCT) and the other buildings (e.g. hangars, concourses, etc.) which leads to multiple roof-top diffraction scenarios. Roof-top scenarios have been described as potential candidate scenarios for “pin-hole” channels in [75] and [76].
Figure 5.3 Example settings for pin-hole channel at CLE [143]
Evidence of “pin-hole” channels resulting in worse than a Rayleigh distribution for the amplitude envelope in indoor settings has also been recently reported in [145]. The authors in [145] claim their work to be the first of its kind to actually document specific 173 measurements of an indoor “pin-hole” channel. It has been pointed out in [145] that it is critical to identify such “pin-hole” channels in real settings so that proper transmission and reception countermeasures can be adopted to prevent the loss in capacity that is encountered due to severe fading. Considering the critical nature of communications reliability on the airport surface, this strongly motivates us to recognize potential “pin- hole” channels.
Analytical pdfs have been derived for Double-Rayleigh [74], N-Rayleigh [78] and
N-Nakagami [79] multiplicative models. The analytical pdf for the Double Rayleigh
2 2 model, where σ 1 and σ 2 represent the energy in the individual Rayleigh processes is given in [77]:
4r ⎛ r 2 ⎞ p ( r ) = K ⎜2 ⎟ (5.1) DR σ 2σ 2 0 ⎜ σ 2σ 2 ⎟ 1 2 ⎝ 1 2 ⎠
th with K0(x) the zero order modified Bessel function of the second kind. One interesting thing that has not been explored is the presence of a Ricean channel or any channel better than Rayleigh as one of the cascaded channels. Of course, in the case of the N-Nakagami models, any one of the Nakagamis can be approximated as a channel with an equivalent
Ricean factor. Also, we have not found any reference for multiplicative models using
Weibull distributions. As discussed in Section 2.7, the Weibull has been used as an effective distribution to model WR fading in several references: [68], [54], [133] and
[134]. In our research, we are interested in the analytical pdf of Double-Weibull multiplicative process. Recall the pdf for a Weibull distribution from equation (2.1).
This univariate Weibull pdf is
b ⎡ r b ⎤ b−1 ⎛ ⎞ (5.2) pw (r) = b r exp⎢− ⎜ ⎟ ⎥ a ⎣⎢ ⎝ a ⎠ ⎦⎥ 174 where (b > 0) is the fading parameter, which indicates the severity of the fading: as b increases, the channel condition gets better. When b=2, the Weibull distribution becomes the Rayleigh distribution. The average fading power E(r2) is given by a 2 Γ[(2 / b) + 1] , where E(⋅) denotes expectation, and Γ(⋅) denotes the gamma function. Using (5.2) and
[69], the analytical pdf of a Double Weibull in integral form is
b b b −1 a −1 ⎛ z ⎞ 1 ⎛ w ⎞ 2 ∞ 1 2 ⎜ ⎟ ⎜ ⎟ 1 ⎛ b b ⎞⎛ z ⎞ ⎛ w ⎞ −⎜ ⎟ −⎜ ⎟ p (w) = ⎜ 1 2 ⎟⎜ ⎟ ⎜ ⎟ e ⎝ a1 ⎠ e ⎝ za2 ⎠ dz DW ∫ ⎜ ⎟⎜ ⎟ ⎜ ⎟ (5.3) 0 z ⎝ a1a2 ⎠⎝ a1 ⎠ ⎝ za2 ⎠
In Figure 5.4, we show various examples of Double Weibull pdfs. The notation Wi(b,
Power) describes the b factor and energy of the ith Weibull process in the Double Weibull model.
Figure 5.4 Example pdfs for Double Weibull multiplicative model [143]
From Figure 5.5, it is also clear that the Double Rayleigh is in fact a special case of the
Double Weibull model. We also observe that having one of the Weibull processes with 175 parameter b better than Rayleigh still results in the overall process being WR. This point is further illustrated in Figure 5.6. In this figure, we assume that one of the Weibull processes in the Double-Weibull is a Rayleigh (b =2) with energy 1. We observe that even when one of the processes has a high Weibull b parameter, the Double-Weibull still has WR behavior. Secondly, the severity of the Double-Weibull process (decreasing bDW) increases as the power in the second Weibull process (b2) decreases. Also, it is worthwhile to note that, for any given value of Power2, as b2 decreases, bDW decreases too.
Figure 5.5 Surface plot showing variation of bDW given b1 = 2 and Power1 = 1 for varying values of b2 and Power2
Reference [74] suggests that multiplicative scattering can also occur in cases where the transmitter and receiver are both in motion. There are certain scattering geometries, such as scattering around a corner, and diffracting from a vehicle top, which can contribute toward creating a multiplicative process for the Double Rayleigh model. 176
Conditions like this can easily occur for the VTV case when there is non-isotropic scattering. One of the recently proposed analytical models for mobile-mobile channels
[55] assumes scatter rings around the receiver and transmitter. For this model to justify
WR behavior, we need sufficient distance between the two vehicles and the absence of significant scatterers in between the two platforms. Such conditions are easily possible in the OLT case where groups of vehicles are traveling on the highway with large distance separating them. As compared Figure 5.5, where we have considered one of the processes in the Double Weibull to be Rayleigh; in Figure 5.6, we vary b1 and b2 from 2 to 6 in increments of 0.1 and Power1 and Power2 are varied from 0 to 1 in increments of
0.1. Thus for a given b1 and Power1, we have 400 possible combinations of (b2 and
Power2). The color bar denotes the percentage of cases for which bDW is less than 2 (1 denotes 100% while 0 denotes 0%). From Figure 5.6, we can observe that even for processes with high Weibull b1, there is a non-negligible percentage of having WR behavior in the resultant amplitude. Also, as b1 approaches 2, the percentage of cases with WR behavior approaches 100%. 177
Figure 5.6 Percentage of WR fading considering Double Weibull model for varying b1, b2, Energy1 and Energy2
5.3 Switching Model
The second model we propose to explain severe fading is similar in concept to the 2- state land mobile satellite channel model, the Lutz model [110]. Generally, when a receiver moves between different fading environments, e.g., from LOS to obstructed
LOS, the envelope power changes. This change of received power is often termed shadowing or large scale fading. In many cases (e.g., cellular) it is assumed that large scale fading can be compensated using power control at the transmitter. The changes in fading conditions, or the transitions between “good” and “bad” channel conditions not 178 only affect the total received power, but also how this power is distributed among the individual taps of the channel model.
For example, in the case of LOS models, the first tap will have a higher average energy than the first tap of NLOS models—inherently, the NLOS channel disperses the energy more. To illustrate this, in Figure 5.7, we provide an example measurement set taken at MIA. Figure 5.7 (a) shows the received signal strength indicator (RSSI) for each
PDP in the measurement set. Figure 5.7 (b) shows the power in the first tap after removing the RSSI from the tap power for each PDP. Thus, Figure 5.7 (a) provides a measure of the large scale fading and Figure 5.7 (b) shows the small scale fading for the first tap. As described previously, due to changes in scattering environment, removing the large scale fading (similar to what power control would do) doesn’t remove the change in power of the first tap between the “good” and “bad” states. In this particular example, we observe a single transition from “good” to “bad” state, but it is possible to have multiple transitions between these states within a measurement set. So, for a given tap, if we consider the samples of the “good” and “bad” state together, then the tap amplitude statistic of the cumulative will display WR behavior. The main reason behind this is the difference in energy between the samples of the “good” and “bad” state. As noted, the quality of the channel is often characterized in terms of the signal delay spread, and this spread of signal energy in time is often characterized using the root mean square delay spread (RMS-DS) [4]. Less dispersive channels have lower values of RMS-DS than highly dispersive channels. Thus, transitions between “good” and “bad” states can be inferred by observing the RMS-DS transitions. 179
-80
-82
-84
-86
-88
RSSI dBm in -90
-92
-94
-96 0 50 100 150 200 250 300
PDP index 0 Good State -1
-2
-3
-4 Bad State
-5
Normalized Tap Power -6
-7
-8 0 50 100 150 200 250 300 PDP Index
Figure 5.7 Example measurement set from MIA comparing RSSI and power in 1st Tap for “good” and “bad” states
Figure 5.8 shows an example time series of RMS-DSs, illustrating multiple such transitions between “good” and “bad” states. Note that even if power control were 180 compensating for average power variations, any rapid and frequent state transitions would still be present, potentially yielding WR tap amplitude statistics.
Figure 5.8 Example RMS-DS time series from MIA [143]
We have observed WR fading in situations where we encounter transitions between these two states. We have been able to identify these conditions in our measured data both on the airport surface (AS) and in vehicle-vehicle (VTV) settings. Table 5.1 shows the amplitude statistics of some example measurements for both scenarios. We have also provided the statistics on the fraction of time Ts (s=g or b) that the receiver spends in each state. The VTV measurements are for 10MHz bandwidth and AS measurements are for 50MHz. Also, the example measurements presented in Table 5.1 are for different taps. 181
Table 5.1 Amplitude statistics for example switching models [143]
Loc Tg, Tb, Total T Weibull Weibull Weibull (a, b) (a, b) (a, b) VTV1 0.46, 0.54, (0.64, 1.77) (0.89, 2.7) (0.43, 1.87) VTV2 0.52, 0.48 , (0.56, 1.64) (0.78, 2.6) (0.32, 1.87) AS1 0.1, 0.9, (1.12,1.91) (2.02, 5.89) (1.03, 1.99) AS2 0.13, 0.87, (1.03, 1.67) (2.17, 4.74) (0.9, 1.89) AS3 0.11, 0.89, (1.09, 1.78) (2.02, 4.72) (0.96, 1.89) AS4 0.26, 0.74, (1.56, 1.73) (2.57, 6.19) (1.1, 1.89) AS5 0.2, 0.8, (1.41, 1.59) (2.35, 5.12) (0.89,1.9) AS6 0.22, 0.78, (1.43, 1.6) (2.37, 5) (0.92, 1.76)
The “bad” state is generally very close to Rayleigh (b factor close to 2) and the “good” states have a higher b factors. Also note the difference in the a parameters of the “good” and “bad” states: as discussed previously, the a parameter is a function of the received energy hence the good states have a larger values of a than the bad states. If we consider both states together and create a single density, WR behavior is evident from the resulting b factors (less than 2).
Figure 5.9 shows the amplitude distribution fits for AS3 from Table 5.1. Note that it is clear that the amplitude distribution of the total data is worse than the individual distributions of the “good” and “bad” states; this is due to the different energies of the distributions for the “good” and “bad” states. Hence, it is necessary to create a non- stationary model which accounts for both these states to accurately model the channel.
182
Figure 5.9 Amplitude Statistics for AS3 from Table 5.1
In order to evaluate the accuracy of this physical justification, it is helpful to reproduce the above conditions in which the receiver encounters such good and bad states using MATLAB simulations. Figure 5.10 shows example distributions fits to some measured data and simulated data. For this figure, we used the parameters from VTV1.
To the best of our knowledge, this is the first effort in the literature to generate WR samples in this manner to model measured amplitude data. From Figure 5.10, it is clear that our simulated results are in very good agreement with the measured data.
183
Figure 5.10 Distribution fits to measured and simulated WR using switching model [143]
Severe fading is a very big factor in determining the performance of a system and hence knowledge of such conditions is critical to assess system performance. It would also be very helpful to determine the propagation conditions which lead to existence of severe fading. So, in this chapter, we have presented two models that can be used to explain the severe levels of (small scale) fading we have observed in our measurements. The physics of propagation which lead to severe fading have been illustrated with various examples obtained from our measurements taken at different locations.
The first explanation is a multiplicative model, which produces fading representable by more than one multiplicative process. In this case, several small scale fading processes are multiplied together, resulting in a product process with worse than Rayleigh statistics. The second mechanism for severe fading is based upon channel transitions. In this model, the channel’s multipath components can be thought of moving from “good” to “bad” state or vice-versa. A “good” state might be line-of-sight (LOS) condition while a “bad” state might be a non-line-of-sight (NLOS) condition. When these transitions from one state to another occur rapidly, it is then difficult to separate the fading into two completely 184 separate states, with the result that the amplitude fading is severe. We have also confirmed the two models using our measured data and computer simulations.
By using our proposed models, we can hence potentially predict (with the help of more measurements and ray tracing software like Wireless Insite) the conditions which lead to WR fading. Using the apriori knowledge regarding the scattering environment which causes severe fading, researchers can then develop countermeasures to alleviate the damage caused by severe fading. 185
6. Airport Surface Area Channel Models
In this chapter, we present our post-processing results for the measurement data collected at different airports while transmitting from the Air Traffic Control Tower
(ATCT). The first section will discuss the various statistical parameters of the data at different airports; the next sections will discuss the various channel models developed for the different airports, regions and bandwidths. We will end the chapter with a discussion on the comparison of the different models and the data. Note: The text, tables and figures in this chapter have been presented in parts at different venues [128]-[138],
[140].
6.1 Common Statistical Parameters
6.1.1 Time (Delay) Domain Statistics
The first set of results that we present are the RMS-DS distributions for the different airports. As discussed several times in this dissertation, the development of stochastic channel models depends heavily on the statistics of the RMS-DS. Figure 6.1 presents a histogram of the RMS-DS of the PDPs collected at MIA. One thing that is immediately apparent is the bi-modal nature of the distribution. We can conclude two things from this observation: first, there exists multiple propagation regions on the airport surface; and second, to accurately model the propagation physics on the airport surface it is imperative that we consider non-stationary channel modeling. Figure 6.2 shows a 186 histogram for the PDPs collected at CLE. Similar observations to those made for MIA can also be made for this figure. One difference in the case of CLE is that we have a tri- modal distribution. These observations justify our classifying the propagation environment at the different airports into different regions (recall Table 3.2). As discussed previously for Table 3.2, we have used RMS-DS as the criteria for classifying
PDPs into different regions. The use of RMS-DS is not the only means for making this division into regions, but is a useful and common one. For example, in the case of MIA, the threshold between NLOS and NLOS-S is selected to be 1000 nsec, which can be justified by a visual inspection of Figure 6.1. Changing this RMS-DS threshold by modest amounts would not have any substantial effect upon the statistics or models developed. Similar observations regarding the choice of threshold for different regions in the case of CLE can be made by observing Figure 6.2.
1400
1200
1000
800
600 Number of Profiles of Number 400
200
0 0 500 1000 1500 2000 2500 RMS-DS in nsec
Figure 6.1 Histogram of measured RMS-DS values, MIA [135]
187
400
350
300
250
200
Number of Profiles 150
100
50
0 0 500 1000 1500 2000 2500 RMS-DS in nsec
Figure 6.2 RMS-DS distribution for CLE [135]
Figure 6.3 shows the cumulative distribution function (cdf) of RMS-DS for the three largest airports. The 90th percentile values are very close to 1.7 μs for all three airports, whereas median (50th percentile) values differ appreciably (500-1000 nanosec).
Percentile values above 70 also illustrate the smaller dispersion of the medium airport
(CLE). We notice similar slopes of the large airport cdfs and a comparatively steeper slope for CLE. This observation indicates differences in the scattering environments for the large and medium airports. 188
1 CLE 0.9 MIA JFK 0.8
0.7 )
τ 0.6 σ
0.5
0.4 Pr(RMS-DS<
0.3
0.2
0.1
0 0 500 1000 1500 2000 σ (ns) τ
Figure 6.3 Cumulative distribution functions of RMS-DS for three airports ( CLE, MIA and JFK) [135]
Figure 6.4 shows the histogram for the RMS-DS of the PDPs collected at BL and
TA. Similar to the histograms presented for the large and medium airports, these GA
RMS-DS histograms are also multi-modal. Figure 6.5 shows the cdfs for the RMS-DS collected at different small airports. As noted in Chapter 3, at TA, we were transmitting from the top of ILS equipment shed and the propagation conditions at TA were significantly different from those at other GAs. In fact, the TA channel can be viewed as a good model for the case when transmitting from an airport field site node. The TA channel data is thus more dispersive than that for the other small airports, showing a larger number of PDPs with large values of RMS-DS. Figure 6.5 indicates that the BL and OU airports have similar cdfs. 189
90 900
80 800
70 700
60 600
50 500
40 400 Number Profiles of Number Profiles of 30 300
20 200
10 100
0 0 0 100 200 300 400 500 0 500 1000 1500 2000 2500 RMS-DS in nsec RMS-DS in nsec (a) (b)
Figure 6.4 Histogram of measured RMS-DS values (a) BL and (b) TA [135]
For BL and OU, the 50th percentile RMS-DS values are very close to 200 nsec, whereas for TA, with its lower transmitter height, the 50th percentile value is closer to
400 nsec. These values are much smaller than the 50th percentile values seen at medium and large airports (approximately 500-1000 nsec), as expected. Only the low-Tx- elevation TA measurements indicate larger 90th percentile RMS-DS values. The steep rise of the RMS-DS cdf for OU and BL suggests that the channel is less frequency selective (than at TA), yet due to their proximity to US-33 and downtown Cleveland, respectively, there were a small percentage of higher-valued RMS-DS PDPs for those airports.
190
1 BL 0.9 TA OU 0.8
0.7 ) τ
σ 0.6
0.5
( RMS-DS < 0.4 n P
0.3
0.2
0.1
0 0 500 1000 1500 2000 σ (nsec) τ Figure 6.5 Cumulative distribution functions of RMS-DS for three airports [135]
We have discussed in Chapter 3 the applicability of the delay window as a statistic to measure the delay dispersion of the channel. Due to its similarity to RMS-DS, we present only one example result of delay window statistics for the large airports in
Figure 6.6. The results provide conclusions similar to those obtained from the RMS-DS: multiple propagation regions exist for both airports, and channel dispersion is largely similar in both airports. Table 6.1 summarizes the statistics of RMS-DS and delay window for the different airports and different regions. We have tabulated the mean, maximum, and minimum values of these statistics for all airports. The increasing delay dispersion (implying increased frequency selectivity) of the channel with the increase in airport size is obvious from the RMS-DS and delay window numbers.
191
10000
8000
6000
4000
2000 Number of Profiles of Number 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 DW in nsec (a)
3000
2000
1000 Number of Profiles of Number 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 DW in nsec (b)
Figure 6.6 Distribution of Wτ,90 for (a) [MIA, 50] and (b) [JFK, 50] [135]
Table 6.1 Summary of measured RMS-DS and Delay Window statistics for all airports [135]
Airport RMS-DS (ns) [min; mean; max] DW-90 (ns) [min; mean; max] NLOS NLOS-S LOS-O NLOS NLOS-S LOS-O JFK [800; [21.4; — [40; [20; — 1,469; 311; 4,200; 338.2; 2,456] 798.7] 4,980] 3,380] MIA [1,000; [23.1; — [1,080; [20; — 1,513; 459; 4,371; 746.3; 2,415] 999.9] 4,980] 3,960] CLE [500; [125; [14; [20; [20; [20; 1,206; 295; 65; 3,237.9 ; 246.8; 44.8; 2,472] 499] 124] 4,980] 2,060] 420] OU — [14; — — [20; — 293; 372.7; 2,416] 4,940] BL — [126; [5; — [20; [20; 429; 44; 747.4; 53.9; 2,427] 124] 4,980] 320] TA [502; [15; — [20; [20; — 1,390; 256; 3,851; 221.1; 2,404] 499] 4980] 1,860]
192
6.1.2 Frequency Domain Statistics
The temporal spreading of the signal by the channel is quantified by RMS-DS.
We employ a frequency correlation estimate (FCE), akin to a correlation, or coherence, bandwidth, to characterize frequency selectivity. The algorithm used to determine the
FCE has been adapted from [32] and has been described in Section 4.2.2. Figure 6.7 shows the FCE computed for MIA and JFK for a BW of 50 MHz. The similarity of the
FCEs for these airports further strengthens our claim that JFK and MIA have statistically similar propagation conditions. The difference in the width of the NLOS and NLOS-S
FCE main lobes quantifies the different frequency selectivity for these propagation regions. From the figure below, we observe that the FCE for NLOSS is equal to 0.5 for
BW of ~13MHz. This means that fading amplitudes of spectral lines separated by
6.5MHz experience fading with correlation of 0.5.
1 MIA, NLOS-S 0.9 MIA, NLOS JFK, NLOS-S 0.8 JFK, NLOS
0.7
0.6
0.5
0.4 Correlation Coefficient 0.3
0.2
0.1
0 -25 -20 -15 -10 -5 0 5 10 15 20 25 Frequency in MHz
Figure 6.7 FCEs for large airports for 50 MHz [135] 193
Figure 6.8 shows the FCEs of the different regions of CLE for a 50 MHz BW.
The figure highlights the difference in frequency correlation for the different regions.
The increase in the FCE main lobe width in the order of NLOS, NLOS-S and LOS-O is equivalent to the trend of increasing RMS-DS for these regions. Another observation can be made regarding the asymmetry of the FCEs: this is due to the existence of correlation among the neighboring multipath components [32]. Also, if we compare the widths of the main-lobes of the FCEs for Figures 6.7 and 6.8, we can observe that Figure 6.7 (large airports) shows more frequency selectivity than Figure 6.8 (medium airport).
1 NLOS-S 0.9 NLOS LOSO 0.8
0.7
0.6
0.5
0.4 Correlation Coefficient 0.3
0.2
0.1
0 -25 -20 -15 -10 -5 0 5 10 15 20 25 Frequency in MHz Figure 6.8 FCEs for Medium Airport [135]
Figure 6.9 shows the FCEs for the different small airports. The FCEs of NLOS-S and LOS-O are based on the data obtained from BL. The NLOS-FCE is calculated from 194 the measurements obtained from TA. Similar to the asymmetry of the FCEs observed in
Figure 6.8, the FCEs in Figure 6.9 also exhibit asymmetry due to inter-component correlation. The asymmetry is much more obvious in the LOS-O case than in the NLOS.
The reason for this is that for NLOS, the scattering environment is richer and much more dynamic than that for the LOSO region, and along with “fixed scattering” comes a higher probability of encountering correlated scattering.
1 LOSO 0.9 NLOS-S NLOS 0.8
0.7
0.6
0.5
0.4 Correlation Coefficient 0.3
0.2
0.1
0 -25 -20 -15 -10 -5 0 5 10 15 20 25 Frequency in MHz
Figure 6.9 FCEs for Small Airport [134]
Table 6.2 lists bandwidths for which the frequency correlation takes values of 0.9,
0.5, and 0.2. The bandwidth values we cite are associated with the smallest frequency separation for which the FCE attains the correlation value. For the NLOS case, due to our frequency resolution limit (255 points in 50 MHz for approximately 196 kHz per frequency bin), we can’t easily determine the frequency separation for which the FCE 195 reaches 0.9. As expected, the width of the main lobe for LOS-O is largest, followed by
NLOS-S, then NLOS.
Table 6.2 Summary of computed FCE values for all airports [135]
Airport FCE (MHz) for Correlation of [0.9; 0.5; 0.2] NLOS NLOS-S LOS-O JFK [NA; [2.734; — 0.7812; 12.89; 11.33] 20.32] MIA [NA; [1.95; — 0.39; 12.7; 10.54] 20.51] CLE [NA; [3.9; [5.07; 10.93; 10.94; 13.86; 20.12] 19.64] 21.68] BL — [4.28; [6.44; 13.08; 15.6; 20.71] 21.68] OU — [3.31; — 11.32; 17.97] TA [NA; [3.9; — 2.54; 13.47; 14.64] 20.31]
6.1.3 Channel Tap Properties
As discussed in Chapter 4, two key parameters of the taps for a stochastic channel model are the tap probability distribution (?) (Section 4.3.2.2) and the individual energy contributed the taps (Section 4.3.3.2). We have also discussed that to reduce the complexity of the tapped delay line model, it is crucial to know the aggregate energy contributed as the number of taps increases. In this section, we will present results for the 196 tap probability and the cumulative energy contributed by different taps. Please note that all the results in Section 1.1.3 pertain to M1 models.
Figure 6.10 compares the steady state probability of state 1 for MIA, CLE and
JFK, versus tap index (recall the steady state probability is approximately the fraction of time a given tap exists above threshold) for NLOS and NLOS-S. Very similar trends for the NLOS regions of the large airports in the reduction of tap existence probability with increasing tap index can be observed from the figure. Also, there is a strong similarity in the measured tap probability of occurrence (Pr[“on”]) versus tap index (20 ns), for the
NLOS-S statistics for all three airports. Due to for the richer scattering in the NLOS regions, the probability of large delay taps is higher than in the NLOS-S case.
1 NLOS, JFK 0.9 NLOS-S, JFK NLOS, MIA 0.8 NLOS-S, MIA NLOS, MIA, LS fit 0.7 NLOS, CLE NLOS-S, CLE 0.6
0.5
0.4
0.3 Tap Probability of Occurrence
0.2
0.1
0 0 10 20 30 40 50 60 70 80 Tap Index
Figure 6.10 Steady state tap probability for state 1 (tap “on”) vs. tap index, MIA, JFK, and CLE, NLOS and NLOS-S
197
Figure 6.11 shows the steady state probability of state 1 versus tap index for the different GA airports for all regions. As observed in the previous figure, the probability of having a multipath at high delay increases with the severity of fading observed in the different regions.
1 LOS0,BL 0.9 NLOS-S,BL NLOS-S, OU 0.8 NLOS, TA NLOS-S, TA 0.7 NLOSS, LS Fit, TA
0.6
0.5
0.4 Probability of having tap having of Probability 0.3
0.2
0.1
0 0 10 20 30 40 50 60 70 80 Tap-Index
Figure 6.11 Steady state tap probability for state 1 (tap “on”) vs. tap index, BL, OU and TA, NLOS, NLOS-S and LOSO
Least-squares curve fits for these probability of occurrence curves fit the general form,
P( k ) = c0 exp( −c1k )+ c2 (6.1) where k is the tap index, and the c’s are curve fit constants. Table 6.3 lists these constants for the airports and regions, and also the maximum number of taps for each case. For clarity, only one curve fit appears on Figure 6.11 and Figure 6.10. Often in the literature one finds representation of channel models in the form of exponential power 198 decay curves. It is interesting to note that the tap probability of occurrence also follows this form.
Table 6.3 Least Squares fit parameters for tap probability of occurrence for all airports [142], [134]
Airport Eq. (6.1) LS Fit Parameters [c0; c1; c2;kmax] NLOS NLOS-S LOS-O JFK [0.84; [1.294; — 0.202; 0.177; 0.28; 0.006; 75] 17] MIA [0.827; [1.304; — 0.252; 0.263; 0.344; 0.115; 77] 24] CLE [0.982; [1.301; [8.99; 0.249; 0.211; 0.017; 0.207 0.22; -7.82; 62] 16] 5] BL -- [1.2697; [7.1159; 0.2374; 0.0209; 0.1013] -5.9409] OU -- [1.1799; -- 0.1761; 0.1015] TA [0.6185; [1.1966; -- 0.1007; 0.2598; 0.2729] 0.1368]
Figure 6.12 compares the cumulative energy as a function of the number of taps for MIA, CLE and JFK for all regions. The rate of increase in cumulative energy is similar for the large two airports in both regions. As expected, the cumulative energy gathers more slowly for regions with rich scattering and a large number of multipath components. Also, the aggregate energy rises slower in the case of large airports than in the medium airport (CLE). The corresponding figure comparing the cumulative energy for small airports is provided in Figure 6.13. Arguments similar to those made for Figure 199
6.12 can also be provided for Figure 6.13. The flatness of the cumulative energy curve at higher index taps further confirms our argument that we can reduce the number of taps in our developed channel models without much loss in fidelity by discarding some of the higher index, low energy taps.
1.05
1
0.95
0.9
0.85
0.8 NLOS, JFK NLOS-S, JFK Cumulative Energy Cumulative 0.75 NLOS, MIA NLOS-S, MIA 0.7 NLOS, CLE NLOS-S, CLE 0.65 LOS-O, CLE
0 1 2 10 10 10 Tap Index Figure 6.12 Cumulative energy versus tap index for large and medium airports, all regions [133] 200
1
0.95
0.9
LOS0, BL 0.85 NLOS-S, BL NLOS-S, OU NLOS-S, TA Power 0.8 NLOS , TA NLOS, LS-Fit, TA
0.75
0.7
0.65 0 1 2 10 10 10 Tap-Index
Figure 6.13 Cumulative energy versus tap index for all GA airports, regions [134]
As with tap probability of occurrence, we also curve-fit tap cumulative energy:
CE( k ) = 1− c3 exp( −c4k )+ c5 (6.2) where again, k is tap index, with range given in Figures 6.12 and 6.13, and the fitting coefficients are given in Table 6.4. The form of (6.2) arises from the PDP functional form, which is exponentially decaying (in dB) with delay.
Table 6.4 Least Squares fit parameters for cumulative energy for all airports [133] [134]
Airport Eq. (5.2) LS Fit Parameters [c3; c4; c5] NLOS NLOS-S LOS-O JFK [0.249; [0.3715; — 0.041; 0.9415; 0.001] 0.003] MIA [0.334; [0.419; — 0.023; 1.045; 201
0.046] -0.005] CLE [0.169; [0.5; [0.483; 0.229; 1.05; 1.4; -0.013] -0.003] -0.001] BL -- [0.7084; [0.8624; 1.3647; 2.1636; 0.0057] 0.0010] OU -- [0.3712; -- 0.6836; 0.0053] TA [0.2779; [0.3728; -- 0.0555; 1.0326; 0.0053] 0.0044]
6.2 Large Airport Models
We performed measurement campaigns at two large airports, JFK and MIA. The airports are similar with respect to airport layout and size, traffic (planes and ground vehicles), height of ATCTs, and these airports also exhibited similar channel characteristics, including RMS-DS, delay window, energy distribution among taps (see
Figures 6.3, 6.6, 6.10 and 6.12), etc. Figure 6.7 further illustrates the similarity in the two airports in terms of FCEs. On the basis of this and the multiple parameter comparisons, we substantiate our approach of developing and using a single model for a given region for any large airport. Since we have two regions (NLOS and NLOS-S), we have two models, one for each region. Since MIA and JFK are among the biggest and busiest airports in the USA, the models presented here can be used for any other large airport as long as its physical characteristics are similar. Note: The discussions presented in this section are based on the results presented by the author at [136], [137], [138] and
[133]. 202
6.2.1 Path Loss Model for Large Airports
A common approach for path loss modeling is to use the “10nlog10(distance)” formula, where the parameter n is the path loss exponent. The path loss equation can then be expressed in this manner [4]:
PLn,σ (d) = A +10nlog(d / d0 ) + X (6.3) where the path loss between transmitter and receiver at distance d is PLn,σ(d), and quantities A and X are in decibels (dB), and distances d and d0 are typically in meters.
From eq (6.3), we can observe that the path loss will be a linear function of the (d/d0) quantity. The parameter A is the intercept point for the equation. A can be obtained directly from the data or can be estimated using the known transmit power, antenna gains, and RF line losses (e.g., using the Friis transmission equation for free-space propagation
[4]). The parameter X is a normal (Gaussian) random variable with zero mean and variance σ2. The variance can be estimated from the data using a least squares curve fit.
For the airport surface measurements, distances were calculated using the latitude, longitude and altitude data recorded using GPS. Generally, the reference distance d0 is some convenient, short-range value, e.g., on the order of 1 m for indoor settings or around 1 km for cellular systems [139]. For the airport surface communication system, the link ranges are likely to be on the order of a few kilometers, hence a reference distance value of 10-50 m or so would be convenient. For the Tx antenna mounted on the
ATCT, it was generally not possible to obtain measurements at 10 m, so a larger value— roughly the minimum attainable with the Tx antenna atop the tower and the receiver at the tower base—was employed. Before discussing further, we point out some restrictions encountered during these measurements. 203
• During the initial measurement campaigns, due to a software error, our sounder
was unable to record GPS data. This prevented us from doing any path loss
modeling for the CLE, OU, and BL airports.
• After the above-mentioned software problem was corrected, we were still unable
to record GPS altitude data while logging our channel measurement data. Due to
cost constraints, we were not able to fix this problem. This meant that we could
record GPS data (latitude, longitude, and altitude) only when we were stationary
(non mobile). Hence, for path loss, our data was gathered for several seconds
near each numbered stopping point on the routes of travel at MIA (see Chapter 3).
• Our sounder is not capable of recording its Received Signal Strength Indication
(RSSI) below a power level of -111 dBm. This limited our modeling only to the
NLOS-S region for the MIA airport. For the remaining regions of JFK, MIA, and
Tamiami, we were always below the threshold of -111 dBm.
Despite these limitations, we do have some data of value, and from this we have developed an initial path loss model for the NLOS-S regions. For MIA, we have used a reference distance d0 of 462.2 meters. Using this reference distance, our curve fit parameter A (eq. (6.3)) is 103 dB (this is close to the free-space loss at d0, approximately
100 dB). We used a linear least squares regression fit to obtain the path loss exponent n≅2.23. Given the large ATCT transmitter height and the airport geometry, one would expect a path loss exponent close to free space, especially in the open areas. (In fact, the free-space model was found to hold quite well for open areas at distances up to our limit of approximately 4 km.) The large difference in the relative heights of the transmitting and receiving antennas, and the imperfect reflection property of the ground contribute to 204 our path loss exponent result. The standard deviation of the Gaussian random variable X of eq. (6.3) was determined to be 5.3 dB. Figure 6.14 shows the path loss model for
MIA-NLOS-S along with the actual data. The path loss plotted here in dB is that with the intercept A subtracted, so for any given distance, path loss is computed as that given by the figure, plus the 103 dB intercept value.
14
12
10
8
6
4 - A (dB) - A σ n 2 PL
0
-2
-4
-6 2 3 4 10 10 10 Distance (m) Figure 6.14 Path loss modeling for MIA-NLOS-S; A=103 dB, n=2.23, =5.3 dB [136]
6.2.2 M1 50 MHz models for Large Airport
6.2.2.1 Markov Model for Large Airport Region Transitions
As mentioned in connection with Figure 4.22, we need to determine the region transition and steady state probability matrices to model the Markov process that is used to emulate the region transitions. Note that these matrices will remain the same 205 irrespective of the model type or BW of the developed channel model. These matrices depend only on the airport type. An example set of matrices (determined from our measured data) is given in (6.4). Refer to Section 4.3.2.1 for more details on the markov models to emulate the region transitions. Since, we don’t model the LOSO region for large airports, the markov model presented below model the transitions between NLOSS
(state 2) and NLOS (state 3). Pi,j is the transition probability of going from state i to state j and Pi is the steady state probability associated with state i. Also note that the user of our model can choose to utilize different transition and steady state matrices depending on the intended application ( e.g. to model the path taken by an aircraft after it lands, emulating the path taken by a ground vehicle on the airport surface, etc).
⎡P22 P23 ⎤ ⎡ 08840 0.1160⎤ Re gion _TS = ⎢ ⎥ = ⎢ ⎥ ⎣P32 P33 ⎦ ⎣0.1094 0.8906⎦ (6.4) ⎡P2 ⎤ ⎡0.4858⎤ Re gion _ SS = ⎢ ⎥ = ⎢ ⎥ ⎣P3 ⎦ ⎣0.5142⎦
6.2.2.2 [M1, Large Airport, NLOS-S, 50]
Recall from Figure 4.22 that the first step in developing the channel model is to determine the number of taps. We use the mean RMS-DS to determine the number of taps (L) for M1. Thus using Table 6.1, we determine that the number of taps for the
NLOS-S model is 24. As discussed in Chapter 4, to alleviate the complexity of realizing this channel model, we can further reduce the number of taps without much loss in fidelity. Table 6.5 shows the cumulative energy versus tap index for the NLOS-S case.
Clearly, the “law of diminishing returns” is in effect as the number of taps increases. As a suitable threshold, we consider the number of taps L to be 8, which accounts for 206
~99.3% of the energy. We have thus substantially reduced the model complexity by using only L=8 out of the 24 taps.
Table 6.5 Cumulative energy for [M1, Large Airport, NLOS-S, 50] [136]
Tap Index Cumulative Energy (%) 1 85 2 95 3 97.5 4 98.5 6 99 8 99.3 12 99.5
NLOS −S For the channel tap correlation matrix, we use Rwc to account for the worst case conditions. This correlation matrix is given in (6.5).
⎡ 1 0.7881 0.2940 0.3485 0.4782 0.4581 0.8969 0.5644⎤ ⎢ ⎥ ⎢0.7881 1 0.3134 0.6588 0.4255 0.8239 0.7768 0.6160⎥ ⎢0.2940 0.3134 1 0.5758 0.8606 0.6958 0.4222 0.9695⎥ ⎢ ⎥ ⎢0.3485 0.6588 0.5758 1 0.6939 0.6605 0.9513 0.7965⎥ (6.5) ⎢0.4782 0.4255 0.8606 0.6939 1 0.9181 0.4653 0.8869⎥ ⎢ ⎥ ⎢0.4581 0.8239 0.6958 0.6605 0.9181 1 0.6528 0.6052⎥ ⎢0.8969 0.7768 0.4222 0.9513 0.4653 0.6528 1 0.7502⎥ ⎢ ⎥ ⎣⎢0.5644 0.6160 0.9695 0.7965 0.8869 0.6052 0.7502 1 ⎦⎥
Table 6.6 provides all of the necessary channel model parameters for the NLOS-S region of a large airport. The symbols used for Weibull parameters in the table were introduced in Section 2.7 and more discussion appears in the Appendix A. This table contains the fading amplitude parameter (b) and tap energy, which together can be used to specify the Weibull density to model amplitude fading. The TS and SS matrix parameters (see Section 4.2.1) are also provided to specify the Markov models for each tap. All the following tables will follow the same guidelines. 207
Table 6.6 Tap Amplitude parameters for [M1, Large Airport, NLOS-S, 50] [136]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.854 4.83 1.0000 na 1.0000 2 0.102 1.7 0.9304 0.2319 0.9425 3 0.027 1.86 0.8768 0.2224 0.8907 4 0.011 1.91 0.6723 0.4600 0.7367 5 0.003 1.97 0.4166 0.6674 0.5341 6 0.002 1.86 0.3230 0.7403 0.4553 7 0.001 1.88 0.2588 0.7895 0.3968 8 0.001 1.89 0.2069 0.8536 0.4382
A remark regarding the curve fits for the measured data is warranted here. The parameters of the different fitted distributions are obtained by using the dfittool in
MATLAB. This tool uses the maximum likelihood (ML) estimator to determine the distribution’s statistical parameters. We then select the distribution type (e.g., Weibull,
Rayleigh, Nakagami, lognormal, or Rician) depending on the maximum likelihood estimate. For example, the likelihood functions for the Rayleigh and Weibull densities are shown below, where N is the number of samples
x ⎛ x2 ⎞ Rayleigh : f ( x;b ) = exp⎜− ⎟ , b is parameter R 2 ⎜ 2 ⎟ b ⎝ 2b ⎠ N ⎧ ⎛ x ⎞ ⎛ x2 ⎞⎫ L = ln⎜ i ⎟ − ⎜ i ⎟ R ∑ ⎨ 2 ⎜ 2 ⎟⎬ (6.6) i=1 ⎩ ⎝ b ⎠ ⎝ 2b ⎠⎭ n ˆ 1 2 ⎛ d ⎞ b = ∑ xi ⎜Obtained by equating ()LR = 0⎟ 2n i=1 ⎝ db ⎠
208
b−1 b ⎛ b ⎞⎛ x ⎞ ⎡ ⎛ x ⎞ ⎤ Weibull : fW ( x;a;b ) = ⎜ ⎟⎜ ⎟ exp⎢− ⎜ ⎟ ⎥; a and b are parameters ⎝ a ⎠⎝ a ⎠ ⎣⎢ ⎝ a ⎠ ⎦⎥ N ⎧ b ⎫ ⎪ ⎛ b ⎞ ⎡ ⎛ xi ⎞⎤ ⎛ xi ⎞ ⎪ LW = ∑ ⎨ln⎜ ⎟ + ⎢()b − 1 ln⎜ ⎟⎥ − ⎜ ⎟ ⎬ i=1 ⎩⎪ ⎝ a ⎠ ⎣ ⎝ a ⎠⎦ ⎝ a ⎠ ⎭⎪ b−1 d N ⎪⎧⎛ b2 ⎞ ⎡()b − 1 x2 ⎤ b ⎛ x ⎞ ⎪⎫ L = ⎜− ⎟ − i + i (6.7) ()W ∑ ⎨⎜ 3 ⎟ ⎢ 3 ⎥ 2 ⎜ ⎟ ⎬ da i=1 ⎩⎪⎝ a ⎠ ⎣ a ⎦ a ⎝ a ⎠ ⎭⎪ N ⎧ b ⎫ d ⎪⎛ b ⎞ ⎡ ⎛ xi ⎞⎤ ⎛ xi ⎞ ⎪ ()LW = ∑ ⎨⎜ 2 ⎟ − ⎢ln⎜ ⎟⎥ − ln()b ⎜ ⎟ ⎬ db i=1 ⎩⎪⎝ a ⎠ ⎣ ⎝ a ⎠⎦ ⎝ a ⎠ ⎭⎪ ⎛ d ⎞ ⎛ d ⎞ bˆ and aˆ are obtained by solving ⎜ ()L = 0⎟ and ⎜ ()L = 0⎟ ⎝ da W ⎠ ⎝ db W ⎠
The Weibull distribution was found to be a better fit to most of our data using this analysis. In Figure 6.15, we see that the Weibull distribution appears to be a better fit to the data visually, we also verify this by computing the ML estimates: for this data set, LW was -136 and LR was -165 (the larger the likelihood value, the better the fit).
1.4 Data Weibull : b = 2.44, a = 0.73 1.2 Rayleigh : b = 0.5
1
0.8
Density 0.6
0.4
0.2
0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Data
Figure 6.15 Comparing different distribution fits with data 209
Figure 6.16 provides curve fits to the measured data for the amplitudes of the 1st and 2nd taps for [M1, Large Airport, NLOS-S, 50]. Fits for other taps are similarly good; we have plotted only the first two tap fits—which contain about 88% of the channel energy—for brevity.
1.2 Tap 1 Amplitude Tap 2 Amplitude Rician: K = 9.3dB 1.8 Nakagami: m = 0.8 Weibull : b = 4.83 Weibull : b = 1.7 1 1.6
1.4 0.8 1.2
1 0.6 Density Density 0.8
0.4 0.6
0.4 0.2 0.2
0 0 0.5 1 1.5 2 0 0.5 1 1.5 Data Data Figure 6.16 Amplitude histograms and curve fits for taps 1 and 2 for [M1, Large Airport, NLOS-S, 50] [136]
6.2.2.3 [M1, Large Airport, NLOS, 50]
Using Table 6.1, we determine that the number of taps for the NLOS model is 77!
As with the NLOS-S case, we reduce this number of taps based upon cumulative energy.
Table 6.7 shows the cumulative energy with increasing tap index for the NLOS case.
Here we select the number of taps L as 57, which accounts for ~95% of the energy. Note that we would require 72 taps to collect 99% of the energy.
Table 6.7 Cumulative energy for [M1, Large Airport, NLOS, 50] [136]
Tap Index Cumulative Energy (%) 5 76.4 10 80 20 84 30 87.6 210
35 89 40 90.7 45 92.1 50 93.5 57 95.2 72 99
NLOS For the correlation matrix, we again use Rwc to account for the worst case
NLOS conditions. Since we have 57 taps for our channel model, Rwc would be a (57×57) matrix. The full correlation matrix appears in the the MATLAB files accompanying this dissertation. The channel model parameters for each tap are provided in Table 6.8.
Table 6.8 Tap Amplitude parameters for [M1, Large Airport, NLOS, 50] [136]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.653 1.9 1.0000 na 1.0000 2 0.090 1.50 0.8408 0.3569 0.8782 3 0.034 1.7 0.7449 0.4259 0.8035 4 0.018 1.8 0.6422 0.4942 0.7181 5 0.010 1.81 0.5383 0.5710 0.6323 6 0.009 1.74 0.4980 0.5951 0.5916 7 0.007 1.78 0.4664 0.6217 0.5671 8 0.007 1.8 0.4490 0.6435 0.5622 9 0.006 1.73 0.4255 0.6642 0.5466 10 0.006 1.75 0.4301 0.6646 0.5554 11 0.005 1.69 0.4002 0.6889 0.5335 12 0.005 1.77 0.4010 0.6827 0.5263 13 0.004 1.91 0.3951 0.6963 0.5354 14 0.004 1.92 0.3953 0.7001 0.5410 15 0.005 1.62 0.3799 0.7268 0.5541 16 0.005 1.76 0.3972 0.6942 0.5360 17 0.004 1.85 0.3897 0.7063 0.5399 18 0.004 1.91 0.3932 0.7011 0.5386 19 0.004 1.85 0.3763 0.7078 0.5154 20 0.004 1.85 0.3894 0.7163 0.5555 21 0.004 1.86 0.3642 0.7184 0.5085 22 0.004 1.79 0.3675 0.7128 0.5058 211
23 0.004 1.9 0.3734 0.7063 0.5072 24 0.004 1.86 0.3748 0.7209 0.5347 25 0.004 1.81 0.3653 0.7195 0.5128 26 0.003 1.82 0.3586 0.7193 0.4982 27 0.004 1.78 0.3699 0.7170 0.5180 28 0.004 1.72 0.3604 0.7295 0.5201 29 0.004 1.82 0.3648 0.7321 0.5333 30 0.004 1.69 0.3739 0.7220 0.5346 31 0.004 1.87 0.3612 0.7320 0.5262 32 0.003 1.83 0.3539 0.7365 0.5191 33 0.004 1.81 0.3566 0.7300 0.5129 34 0.003 1.74 0.3420 0.7522 0.5235 35 0.003 1.8 0.3437 0.7540 0.5305 36 0.004 1.57 0.3455 0.7453 0.5177 37 0.004 1.8 0.3556 0.7410 0.5308 38 0.003 1.93 0.3494 0.7454 0.5261 39 0.003 2.07 0.3435 0.7460 0.5148 40 0.003 1.97 0.3386 0.7549 0.5213 41 0.003 1.86 0.3512 0.7484 0.5353 42 0.003 1.97 0.3331 0.7576 0.5148 43 0.003 1.85 0.3443 0.7418 0.5085 44 0.003 2 0.3348 0.7501 0.5036 45 0.003 1.85 0.3383 0.7488 0.5089 46 0.003 1.98 0.3366 0.7516 0.5104 47 0.003 1.96 0.3308 0.7591 0.5130 48 0.003 1.81 0.3443 0.7462 0.5168 49 0.003 1.84 0.3412 0.7479 0.5133 50 0.003 1.87 0.3385 0.7518 0.5150 51 0.003 1.92 0.3407 0.7534 0.5228 52 0.003 1.91 0.3275 0.7615 0.5104 53 0.003 2.02 0.3404 0.7520 0.5196 54 0.003 1.92 0.3312 0.7550 0.5053 55 0.003 2.05 0.3332 0.7471 0.4940 56 0.003 1.97 0.3291 0.7621 0.5152 57 0.003 1.96 0.3364 0.7492 0.5054
Figure 6.17 provides example curve fits for the amplitudes of the 1st and 3rd taps for the
[M1, Large Airport, NLOS, 50] channel.
212
3 Amplitude for Tap 1 Amplitude for Tap 3 Lognormal :μ = 0.55, σ2 = 0.11 6 Weibull : b = 1.7 Weibull: b = 1.9 Nakagami : m = 0.83 2.5
5
2 4
1.5 Density Density 3
1 2
0.5 1
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Data Data
Figure 6.17 Amplitude statistics of taps 1 and 3 for [M1, Large Airport, NLOS, 50] [136]
6.2.3 [M1 M2 M3, Large Airport, 25] for all Regions
6.2.3.1 [M1 M2 M3, NLOS-S, Large Airport, 25]
In the previous section, we presented the 50 MHz M1 models for the different regions of the large airport surface area. Recall from Section 4.4.1, using the definition for the M2 models, we would require 250 taps for the 50 MHz BW model. In addition to the 50 MHz models, we also present the M1, M2 and M3 Large Airport models for a 25
MHz bandwidth for NLOS-S and NLOS. Table 6.9 presents the tap amplitude statistics necessary to create the [M1, Large Airport, NLOS-S, 25] model. Note that this model has only L=5 taps. 213
Table 6.9 Tap Amplitude parameters for [M1, Large Airport, NLOS-S, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.929 3.6159 1.0000 na 1.0000 2 0.059 1.7158 0.8350 0.2213 0.8966 3 0.008 1.7735 0.7091 0.5603 0.6345 4 0.003 1.6856 0.6278 0.7249 0.5108 5 0.002 1.5983 0.5783 0.8171 0.4895
Table 6.10 presents the tap amplitude statistics necessary to create the [M2, Large
Airport, NLOS-S, 25] model. Note that since we are creating the model for a 25 MHz
BW, we have 125 taps to account for a total IRE length of 5 μsec. Recall from Section
4.4.1 that this means that the correlation matrices will be of size (125 × 125) and will be changed every n realizations. Hence, we don’t include them here. These correlation matrices are provided in the MATLAB files accompanying this dissertation. Another thing to note is that the tap amplitude statistics will remain the same for the [M3, Large
Airport, NLOS-S, 25] model. The only difference is that in order to implement the M3 models, we don’t use the correlation matrices and the persistence process modeling parameters.
Table 6.10 Tap Amplitude parameters for [M2, Large Airport, NLOS-S, 25]. (Energies and Weibull b factors also apply to M3.)
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0019 1.5164 0.0061 0.9973 0.5556 2 0.2689 0.8543 0.9998 0 0.9998 3 0.3707 1.1814 0.9522 0.1373 0.9567 4 0.1633 1.1949 0.8120 0.2945 0.8366 5 0.0474 1.2727 0.5662 0.5311 0.6406 6 0.0157 1.4797 0.3890 0.7003 0.5290 214
7 0.0088 1.5698 0.3016 0.7787 0.4877 8 0.0076 1.5132 0.2611 0.8164 0.4797 9 0.0058 1.6219 0.2299 0.8452 0.4810 10 0.0052 1.6717 0.2131 0.8513 0.4510 11 0.0051 1.6328 0.1996 0.8639 0.4536 12 0.0046 1.6732 0.1941 0.8619 0.4258 13 0.0044 1.6017 0.1867 0.8721 0.4427 14 0.0052 1.5108 0.1877 0.8748 0.4583 15 0.0043 1.7391 0.1887 0.8724 0.4514 16 0.0048 1.5557 0.1764 0.8784 0.4323 17 0.0045 1.5071 0.1634 0.8875 0.4243 18 0.0037 1.6132 0.1469 0.9023 0.4330 19 0.0041 1.6278 0.1443 0.9018 0.4177 20 0.0031 1.7341 0.1338 0.9116 0.4277 21 0.0028 1.7553 0.1256 0.9184 0.4316 22 0.0021 1.8806 0.1148 0.9156 0.3490 23 0.0018 1.7964 0.1042 0.9188 0.3021 24 0.0022 1.8125 0.1156 0.9124 0.3304 25 0.0024 1.6369 0.0998 0.9274 0.3457 26 0.0015 1.9963 0.0882 0.9297 0.2710 27 0.0016 2.0221 0.0954 0.9233 0.2734 28 0.0016 2.0535 0.0980 0.9270 0.3282 29 0.0017 1.8633 0.0840 0.9340 0.2806 30 0.0013 1.8031 0.0806 0.9361 0.2714 31 0.0017 1.7171 0.0769 0.9398 0.2779 32 0.0024 1.3000 0.0720 0.9416 0.2477 33 0.0020 1.3722 0.0786 0.9370 0.2612 34 0.0008 2.4119 0.0668 0.9479 0.2727 35 0.0010 1.9331 0.0710 0.9445 0.2749 36 0.0008 2.3025 0.0727 0.9385 0.2153 37 0.0012 1.8715 0.0714 0.9431 0.2594 38 0.0014 1.7533 0.0773 0.9402 0.2854 39 0.0016 1.7089 0.0714 0.9442 0.2736 40 0.0011 1.9901 0.0672 0.9446 0.2306 41 0.0009 2.2017 0.0621 0.9485 0.2222 42 0.0008 2.1352 0.0640 0.9471 0.2263 43 0.0007 2.4085 0.0606 0.9477 0.1889 44 0.0008 2.0516 0.0534 0.9564 0.2271 45 0.0008 2.2629 0.0577 0.9527 0.2245 46 0.0007 2.3040 0.0527 0.9565 0.2173 47 0.0006 2.2237 0.0465 0.9590 0.1594 48 0.0007 2.0913 0.0475 0.9601 0.1986 49 0.0006 1.8083 0.0424 0.9634 0.1746 50 0.0004 2.6536 0.0355 0.9688 0.1517 51 0.0004 2.8025 0.0402 0.9639 0.1381 215
52 0.0004 2.7951 0.0417 0.9626 0.1411 53 0.0005 2.3916 0.0397 0.9642 0.1356 54 0.0003 3.1147 0.0374 0.9654 0.1081 55 0.0004 2.7281 0.0365 0.9678 0.1521 56 0.0005 2.3367 0.0387 0.9657 0.1478 57 0.0003 3.2159 0.0345 0.9693 0.1415 58 0.0003 3.2354 0.0345 0.9686 0.1220 59 0.0004 2.7946 0.0338 0.9692 0.1194 60 0.0002 3.0606 0.0306 0.9713 0.0934 61 0.0003 2.9499 0.0340 0.9688 0.1139 62 0.0003 2.9333 0.0291 0.9721 0.0694 63 0.0002 2.7148 0.0293 0.9724 0.0862 64 0.0002 2.7012 0.0296 0.9724 0.0966 65 0.0002 3.0200 0.0335 0.9692 0.1106 66 0.0004 3.3160 0.0486 0.9533 0.0865 67 0.0004 3.0190 0.0431 0.9601 0.1133 68 0.0004 3.2864 0.0552 0.9478 0.1067 69 0.0004 2.9926 0.0522 0.9480 0.0548 70 0.0003 2.9420 0.0406 0.9626 0.1162 71 0.0003 3.1189 0.0321 0.9720 0.1571 72 0.0002 3.1135 0.0264 0.9742 0.0510 73 0.0002 3.3857 0.0283 0.9730 0.0714 74 0.0002 3.3864 0.0284 0.9735 0.0947 75 0.0009 3.0934 0.1224 0.8951 0.2479 76 0.0008 3.2350 0.1084 0.9056 0.2236 77 0.0006 3.3608 0.0808 0.9216 0.1083 78 0.0009 3.2188 0.1092 0.9032 0.2111 79 0.0008 2.9606 0.0779 0.9288 0.1577 80 0.0005 2.7730 0.0412 0.9605 0.0776 81 0.0003 2.9665 0.0337 0.9686 0.0950 82 0.0003 3.0439 0.0310 0.9710 0.0924 83 0.0002 2.9563 0.0290 0.9731 0.0988 84 0.0002 2.9432 0.0274 0.9740 0.0798 85 0.0002 3.0728 0.0258 0.9758 0.0850 86 0.0002 2.9044 0.0268 0.9760 0.1258 87 0.0002 2.9623 0.0232 0.9781 0.0797 88 0.0001 3.3507 0.0234 0.9781 0.0863 89 0.0002 3.2390 0.0273 0.9747 0.0988 90 0.0002 3.4258 0.0251 0.9751 0.0336 91 0.0002 3.3708 0.0254 0.9767 0.1060 92 0.0002 3.2088 0.0254 0.9756 0.0662 93 0.0002 3.3176 0.0264 0.9748 0.0701 94 0.0002 3.4234 0.0271 0.9758 0.1304 95 0.0002 3.1761 0.0259 0.9756 0.0844 96 0.0002 3.6571 0.0271 0.9749 0.0994 216
97 0.0002 3.2384 0.0247 0.9762 0.0612 98 0.0002 3.5419 0.0246 0.9764 0.0616 99 0.0001 3.2986 0.0210 0.9797 0.0560 100 0.0001 3.8418 0.0229 0.9781 0.0662 101 0.0001 3.1632 0.0168 0.9841 0.0700 102 0.0001 3.6155 0.0214 0.9792 0.0472 103 0.0001 3.1911 0.0194 0.9811 0.0435 104 0.0002 3.3932 0.0241 0.9767 0.0559 105 0.0001 3.3473 0.0200 0.9806 0.0504 106 0.0002 3.4522 0.0219 0.9788 0.0538 107 0.0001 3.7998 0.0182 0.9825 0.0556 108 0.0001 3.5107 0.0215 0.9790 0.0469 109 0.0001 3.3721 0.0207 0.9799 0.0488 110 0.0001 3.5350 0.0236 0.9766 0.0286 111 0.0002 3.3452 0.0298 0.9712 0.0621 112 0.0002 3.3204 0.0278 0.9739 0.0848 113 0.0002 3.0608 0.0241 0.9781 0.1119 114 0.0002 2.9382 0.0274 0.9751 0.1166 115 0.0002 3.1233 0.0227 0.9786 0.0815 116 0.0002 2.9664 0.0232 0.9791 0.1232 117 0.0001 3.4901 0.0204 0.9804 0.0579 118 0.0002 3.6265 0.0234 0.9771 0.0432 119 0.0002 3.5286 0.0249 0.9763 0.0743 120 0.0002 3.5772 0.0252 0.9760 0.0733 121 0.0002 3.4876 0.0290 0.9712 0.0349 122 0.0002 2.9529 0.0342 0.9676 0.0837 123 0.0002 3.1747 0.0365 0.9659 0.1014 124 0.0003 3.2220 0.0500 0.9522 0.0909 125 0.0006 3.3278 0.1030 0.9000 0.1291
6.2.3.2 [M1 M2 M3, NLOS, Large Airport, 25]
Similar to the M2 models provided in the earlier part for NLOS-S, the following
Tables 6.11 and 6.12 provides the tap amplitude statistical parameters necessary to simulate models M1, M2 and M3 for Large Airports for a BW of 25 MHz. Note again that the tap amplitude parameters for M2 and M3 are the same.
Table 6.11 Tap Amplitude parameters for [M1, Large Airport, NLOS, 25]
Tap Energy Weibull P1,k P00,k P11,k 217
Index Shape k Factor (bk) 1 0.67 1.9390 1.0000 na 1.0000 2 0.06 1.5422 0.8350 0.3084 0.8633 3 0.02 1.6476 0.7091 0.4005 0.7540 4 0.02 1.5695 0.6278 0.4683 0.6846 5 0.01 1.6314 0.5783 0.5224 0.6516 6 0.01 1.5943 0.5655 0.5359 0.6436 7 0.01 1.6976 0.5472 0.5676 0.6421 8 0.01 1.5773 0.5363 0.5787 0.6355 9 0.01 1.6615 0.5404 0.5712 0.6353 10 0.01 1.6622 0.5285 0.5787 0.6241 11 0.01 1.7240 0.5170 0.5825 0.6098 12 0.01 1.6996 0.5187 0.6033 0.6320 13 0.01 1.6316 0.5203 0.5985 0.6300 14 0.01 1.6199 0.5127 0.6005 0.6205 15 0.01 1.6119 0.5168 0.6020 0.6281 16 0.01 1.6232 0.5040 0.6020 0.6084 17 0.01 1.6351 0.5002 0.6145 0.6149 18 0.01 1.6408 0.4935 0.6275 0.6179 19 0.01 1.6761 0.4952 0.6206 0.6132 20 0.01 1.7922 0.4835 0.6236 0.5980 21 0.01 1.6652 0.4794 0.6360 0.6048 22 0.01 1.7330 0.4819 0.6284 0.6007 23 0.01 1.7476 0.4854 0.6297 0.6076 24 0.01 1.7506 0.4775 0.6628 0.6311 25 0.01 1.7470 0.4851 0.6409 0.6190 26 0.01 1.7352 0.4780 0.6291 0.5950 27 0.01 1.7574 0.4821 0.6307 0.6031 28 0.01 1.7413 0.4746 0.6348 0.5959 29 0.01 1.5782 0.4792 0.6328 0.6011 30 0.01 1.7389 0.4745 0.6325 0.5931 31 0.01 1.8115 0.4805 0.6405 0.6112
Table 6.12 Tap Amplitude parameters for [M2, Large Airport, NLOS, 25]. (Energies and Weibull b factors also apply to M3.)
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0032 1.3513 0.0507 0.9678 0.3969 2 0.0640 1.0676 0.9990 0 0.9990 3 0.1019 1.0657 0.9513 0.1466 0.9563 4 0.0781 1.1042 0.8806 0.2470 0.8979 218
5 0.0449 1.1784 0.7903 0.3336 0.8231 6 0.0268 1.2491 0.7113 0.4214 0.7651 7 0.0197 1.2895 0.6539 0.4732 0.7211 8 0.0144 1.3602 0.6070 0.5240 0.6917 9 0.0131 1.3899 0.6008 0.5256 0.6850 10 0.0100 1.5634 0.5767 0.5521 0.6712 11 0.0095 1.5664 0.5614 0.5663 0.6610 12 0.0087 1.5837 0.5510 0.5590 0.6406 13 0.0081 1.6618 0.5380 0.5630 0.6246 14 0.0076 1.6583 0.5352 0.5839 0.6385 15 0.0071 1.7150 0.5219 0.5918 0.6263 16 0.0072 1.6500 0.5209 0.5907 0.6237 17 0.0071 1.6870 0.5282 0.5835 0.6282 18 0.0071 1.6625 0.5238 0.5967 0.6335 19 0.0066 1.7303 0.5092 0.6076 0.6219 20 0.0067 1.7144 0.5011 0.6165 0.6184 21 0.0064 1.7165 0.4913 0.6219 0.6086 22 0.0064 1.7240 0.4956 0.6136 0.6065 23 0.0062 1.7123 0.4949 0.6235 0.6160 24 0.0059 1.8075 0.4846 0.6303 0.6070 25 0.0057 1.8137 0.4783 0.6337 0.6006 26 0.0056 1.8688 0.4824 0.6431 0.6172 27 0.0058 1.8335 0.4876 0.6227 0.6037 28 0.0057 1.8216 0.4851 0.6329 0.6105 29 0.0057 1.8153 0.4867 0.6250 0.6046 30 0.0057 1.8088 0.4835 0.6242 0.5984 31 0.0059 1.7573 0.4784 0.6415 0.6093 32 0.0055 1.8011 0.4795 0.6323 0.6012 33 0.0056 1.8031 0.4784 0.6389 0.6066 34 0.0054 1.8489 0.4762 0.6382 0.6019 35 0.0052 1.8977 0.4754 0.6369 0.5992 36 0.0054 1.8722 0.4723 0.6575 0.6173 37 0.0051 1.8913 0.4772 0.6392 0.6049 38 0.0052 1.8626 0.4612 0.6611 0.6042 39 0.0054 1.9246 0.4759 0.6440 0.6081 40 0.0049 1.9455 0.4670 0.6572 0.6090 41 0.0052 1.8901 0.4697 0.6496 0.6047 42 0.0053 1.8802 0.4605 0.6599 0.6017 43 0.0050 1.9056 0.4605 0.6493 0.5893 44 0.0049 1.9094 0.4529 0.6594 0.5887 45 0.0051 1.8706 0.4555 0.6573 0.5903 46 0.0051 1.8631 0.4650 0.6645 0.6140 47 0.0054 1.8830 0.4662 0.6478 0.5969 48 0.0052 1.8814 0.4648 0.6549 0.6029 49 0.0051 1.9020 0.4534 0.6533 0.5822 219
50 0.0050 1.9555 0.4505 0.6647 0.5913 51 0.0051 1.8565 0.4586 0.6504 0.5875 52 0.0049 1.8514 0.4528 0.6674 0.5982 53 0.0050 1.8681 0.4513 0.6645 0.5922 54 0.0050 1.8913 0.4480 0.6745 0.5991 55 0.0050 1.8875 0.4453 0.6747 0.5949 56 0.0047 1.9730 0.4458 0.6589 0.5761 57 0.0050 1.9703 0.4477 0.6744 0.5984 58 0.0048 1.9384 0.4436 0.6666 0.5818 59 0.0048 1.8518 0.4488 0.6614 0.5842 60 0.0046 1.9342 0.4453 0.6668 0.5851 61 0.0046 1.9539 0.4418 0.6838 0.6006 62 0.0048 1.9986 0.4536 0.6640 0.5952 63 0.0045 1.9636 0.4493 0.6598 0.5831 64 0.0047 2.0075 0.4463 0.6642 0.5835 65 0.0047 1.9465 0.4491 0.6676 0.5925 66 0.0048 1.9729 0.4534 0.6632 0.5941 67 0.0047 1.9550 0.4491 0.6621 0.5856 68 0.0046 1.9738 0.4491 0.6629 0.5867 69 0.0046 1.9569 0.4490 0.6624 0.5858 70 0.0045 1.9782 0.4461 0.6769 0.5990 71 0.0045 1.9609 0.4385 0.6713 0.5792 72 0.0042 1.9554 0.4336 0.6834 0.5865 73 0.0045 1.9514 0.4451 0.6688 0.5873 74 0.0044 1.8863 0.4401 0.6808 0.5940 75 0.0050 1.9473 0.4643 0.6449 0.5905 76 0.0049 1.9230 0.4607 0.6392 0.5778 77 0.0049 1.8657 0.4597 0.6460 0.5838 78 0.0049 1.9710 0.4667 0.6298 0.5771 79 0.0049 1.9512 0.4556 0.6595 0.5932 80 0.0049 1.9236 0.4482 0.6624 0.5844 81 0.0048 1.9145 0.4471 0.6602 0.5794 82 0.0047 1.9216 0.4478 0.6669 0.5890 83 0.0050 1.9372 0.4558 0.6618 0.5964 84 0.0050 1.8869 0.4486 0.6621 0.5848 85 0.0049 1.8534 0.4593 0.6446 0.5816 86 0.0051 1.8202 0.4505 0.6554 0.5799 87 0.0046 1.9212 0.4413 0.6705 0.5830 88 0.0049 1.8796 0.4421 0.6729 0.5875 89 0.0048 1.9175 0.4448 0.6716 0.5904 90 0.0053 1.7782 0.4491 0.6665 0.5911 91 0.0047 1.8895 0.4426 0.6708 0.5856 92 0.0050 1.8617 0.4472 0.6602 0.5801 93 0.0049 1.9020 0.4412 0.6763 0.5902 94 0.0046 1.9280 0.4439 0.6662 0.5820 220
95 0.0048 1.8860 0.4385 0.6708 0.5787 96 0.0047 1.9078 0.4390 0.6700 0.5784 97 0.0047 1.8885 0.4455 0.6589 0.5754 98 0.0047 1.9115 0.4342 0.6881 0.5936 99 0.0052 1.8296 0.4534 0.6559 0.5853 100 0.0049 1.8831 0.4388 0.6712 0.5797 101 0.0051 1.8860 0.4467 0.6561 0.5742 102 0.0050 1.8774 0.4360 0.6835 0.5908 103 0.0050 1.8873 0.4440 0.6747 0.5929 104 0.0050 1.8579 0.4464 0.6623 0.5813 105 0.0053 1.8175 0.4529 0.6555 0.5840 106 0.0053 1.8137 0.4453 0.6605 0.5773 107 0.0052 1.8723 0.4406 0.6582 0.5660 108 0.0048 1.8287 0.4432 0.6685 0.5837 109 0.0049 1.8970 0.4539 0.6591 0.5899 110 0.0051 1.8742 0.4528 0.6621 0.5917 111 0.0051 1.7993 0.4547 0.6561 0.5877 112 0.0059 1.6430 0.4629 0.6351 0.5767 113 0.0060 1.5350 0.4540 0.6613 0.5929 114 0.0062 1.5490 0.4705 0.6353 0.5898 115 0.0063 1.5917 0.4799 0.6198 0.5880 116 0.0057 1.7006 0.4620 0.6466 0.5885 117 0.0061 1.6700 0.4708 0.6375 0.5928 118 0.0066 1.6019 0.4751 0.6391 0.6015 119 0.0059 1.6424 0.4767 0.6188 0.5813 120 0.0064 1.6247 0.4803 0.6205 0.5896 121 0.0065 1.5901 0.4795 0.6310 0.5997 122 0.0071 1.4976 0.4959 0.6014 0.5949 123 0.0065 1.6162 0.4916 0.6051 0.5917 124 0.0066 1.6110 0.5060 0.5910 0.6009 125 0.0066 1.6445 0.5097 0.5686 0.5851
6.3 Medium Airport Models
6.3.1 M1 50 MHz Models for Medium Airports
6.3.1.1 Markov Model for Medium Airport Region Transitions
221
As mentioned in regard to Figure 4.22, we need to determine the region transition and steady state matrices to model the Markov processes that emulate the region transitions. Note again that these matrices will remain the same irrespective of the model type or BW of the developed channel model. These matrices depend only on the airport type (refer Section 4.3.2.1 for in-detail discussion regarding the extraction of the markov model parameters). The matrices presented in (6.8) will model the transitions between
LOSO (state 1), NLOSS (state 2) and NLOS (state 3).
⎡P11 P12 P13 ⎤ ⎡0.4308 0.4334 0.1358⎤ Re gion _TS = ⎢P P P ⎥ = ⎢0.2074 0.5244 0.2682⎥ ⎢ 21 22 23 ⎥ ⎢ ⎥ ⎣⎢P31 P32 P33 ⎦⎥ ⎣⎢0.0477 0.1516 0.8008⎦⎥ (6.8) ⎡P1 ⎤ ⎡0.1582⎤ Re gion _ SS = ⎢P ⎥ = ⎢0.3127⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎣⎢P3 ⎦⎥ ⎣⎢0.5291⎦⎥
6.3.1.2 [M1, Medium Airport, LOS-O, 50]
We performed the medium airport measurement campaign at CLE, which is among the 50 busiest airports in the USA. As discussed in Chapter 3, CLE is smaller than MIA and JFK in terms of the size of aircraft, the density of traffic (planes and ground vehicles) and the building structures on and near the airport property. In CLE, we also covered the LOS-O region. The characteristics of the LOS-O region are the presence of a dominant LOS component and the absence of large reflectors/scatterers in the local vicinity of the receiver. Hence the channel model for the LOS-O region should remain the same irrespective of the airport size: that is, the channel model presented here for LOS-O can be used for the large airports as well. Since we use the mean RMS-DS to determine the number of taps, using Table 6.1, we determine the number of taps for the 222
LOS-O model as L=5. For the LOS-O region, we use the first 3 taps, which contain 98%
LOS−O of the energy. For the correlation matrix, we use Rwc .
⎡ 1 0.9734 0.9992⎤ ⎢ ⎥ ⎢0.9734 1 0.9538⎥ (6.9) ⎣⎢0.9992 0.9538 1 ⎦⎥
LOS−O As can be observed from (6.9), the elements in Rwc are very close to 1. As described before, for LOS-O regions, there are not many mobile reflectors nor many large scatterers nearby, and hence all the taps emanate from stable reflections, and are hence highly correlated. The number of profiles used to determine the values of the correlation matrix in (6.9) are as follows:
⎡1 6 8⎤ ⎢ ⎥ ⎢6 1 8⎥ (6.10) ⎣⎢8 8 1⎦⎥
Table 6.13 provides the channel model parameters for the LOS-O region.
Table 6.13 Tap Amplitude parameters for [M1, Medium Airport, NLOS-S, 50] [137]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.890 7.53 1.0000 na 1.0000 2 0.093 1.65 0.8851 0.1818 0.8935 3 0.018 1.91 0.7023 0.3772 0.7351
Figure 6.18 provides curve fits for the amplitudes of the 1st and 2nd taps for the [M1,
Medium Airport, LOS-O, 50] model.
223
1.5 Tap 1 Amplitude Data Tap2 Amplitude Data K = 13dB Nakagami : m = 0.75 1.2 Weibull : 7.53 Weibull : b = 1.65
1
1
0.8 Density Density 0.6
0.5 0.4
0.2
0 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Data Data Figure 6.18 Amplitude histograms and curve fits for taps 1 and 2 for [M1, Medium Airport, LOS-O, 50] [137]
6.3.1.3 [M1, Medium Airport, NLOS-S, 50]
Using Table 6.1, the number of taps for the NLOS-S model is 16. Table 6.14 shows the cumulative energy versus tap index. As before, we reduce this number, and employ a value of L=5 for the number of taps, which accounts for ~99.1% of the energy.
Table 6.14 Cumulative energy for [M1, Medium Airport, NLOS-S, 50] [137]
Tap Index Cumulative Energy (%) 2 94.4 4 98.4 5 99.1 12 99.9
NLOS−S For the correlation matrix, Rwc is as follows:
⎡ 1 0.8976 0.9924 0.9931 0.9725⎤ ⎢ ⎥ ⎢0.8976 1 0.9379 0.9518 0.6381⎥ ⎢0.9924 0.9379 1 0.9945 0.9763⎥ (6.11) ⎢ ⎥ ⎢0.9931 0.9518 0.9945 1 0.7920⎥ ⎣⎢0.9725 0.6381 0.9763 0.7920 1 ⎦⎥
224
The numbers of profiles used to determine the correlation matrix of (6.11) are given in
(6.12).
⎡ 1 18 27 14 56⎤ ⎢ ⎥ ⎢18 1 32 74 8 ⎥ ⎢27 32 1 17 47⎥ (6.12) ⎢ ⎥ ⎢14 74 17 1 44⎥ ⎣⎢56 8 47 44 1 ⎦⎥
Table 6.15 provides the amplitude statistics for the taps in this channel model, and Figure
6.19 provides example curve fits for the amplitudes of the 1st and 2nd taps for [M1,
Medium Airport, NLOS-S, 50].
Table 6.15 Tap amplitude parameters for [M1, Medium Airport, NLOS-S, 50] [137]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.818 4.77 1.0000 na 1.0000 2 0.127 0.82 0.9115 0.1940 0.9216 3 0.011 0.94 0.7794 0.2635 0.7912 4 0.025 0.96 0.6539 0.4351 0.7024 5 0.014 0.91 0.5020 0.6037 0.6079 225
Tap1 Amplitude data 1.5 1.2 K = 9.5dB Tap2 Amplitude data Weibull : b = 4.77 Nakagami : m = 0.82 Weibull : b = 1.73 1
1 0.8
0.6 Density Density
0.5 0.4
0.2
0 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Dt Figure 6.19 Amplitude histograms and curve fits for taps 1 and 2 for [M1, Medium Airport, NLOS-S, 50] [137]
6.3.1.4 [M1, Medium Airport, NLOS, 50]
Using Table 6.1, for the NLOS model we obtain 62 taps! Reducing this number via the cumulative energy criterion, Table 6.16 shows the cumulative energy for this
NLOS case. Here we select a threshold of 95% of the energy and obtain L=15 taps.
Table 6.16 Cumulative energy for [M1, Medium Airport, NLOS, 50] [137]
Tap Index Cumulative Energy (%) 3 89.1 5 92.1 10 94 15 95 20 96 30 97.5 40 98.5 47 99
NLOS For the correlation matrix, Rwc would be a (15×15) matrix for our 15 tap channel. Because of this size, we show the correlation matrix in the MATLAB files 226 accompanying this dissertation. Table 6.17 shows the parameters necessary to develop the tapped delay line model for [M1, Medium Airport, NLOS-S, 50].
Table 6.17 Tap Amplitude parameters for [M1, Medium Airport, NLOS-S, 50] [137]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.808 1.64 1.0000 na 1.0000 2 0.102 1.45 0.7916 0.4195 0.8470 3 0.030 1.44 0.6565 0.5818 0.7821 4 0.021 1.48 0.5816 0.5933 0.7083 5 0.010 1.49 0.4793 0.6306 0.5977 6 0.005 1.52 0.3833 0.6831 0.4888 7 0.004 1.51 0.3427 0.7194 0.4601 8 0.004 1.52 0.3232 0.7575 0.4928 9 0.004 1.43 0.3263 0.7541 0.4904 10 0.003 1.45 0.3044 0.7393 0.4051 11 0.002 1.49 0.2693 0.7959 0.4477 12 0.002 1.52 0.2639 0.7890 0.4095 13 0.002 1.57 0.2717 0.7985 0.4582 14 0.002 1.53 0.2662 0.7830 0.4029 15 0.002 1.46 0.2475 0.8060 0.4082
Figure 6.20 provides example curve fits for amplitudes of the 1st, 2nd, 10th, and 11th taps for the [M1, Medium Airport, NLOS, 50] channel model. Note that all of these taps exhibit “worse than Rayleigh” fading, with the Nakagami-m parameter m<1, or the
Weibull “b” parameter less than two. 227
Tap1 Amplitude data Tap2 Amplitude data 1.2 Nakagami : m = 0.72 2 Nakagami : m = 0.61 Weibull : b = 1.64 1 Weibull : b = 1.45
0.8 1.5
0.6 Density Density 1 0.4 0.5 0.2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 Data Data 6 7 Tap 11 Amplitude Data
6 Nakagami : m = 0.63 5 Tap 10 Amplitude Data Weibull : b = 1.49 Nakagami : m = 0.6 5 4 Weibull : b = 1.45 4 3 Density Density 3 2 2 1 1
0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 0.1 0.2 0.3 0.4 0.5 Data Data
Figure 6.20 Amplitude histograms and curve fits for taps 1, 2, 10, and 11 for [M1, Medium Airport, NLOS, 50] [137]
6.3.2 [M1 M2 M3, Medium Airport, 25] for all Regions
6.3.2.2 [M1 M2 M3, LOSO, Medium Airport, 25]
In the previous section, we presented the 50 MHz M1 models for the different regions of the medium airport. We present here the M1, M2 and M3 Medium Airport models for 25 MHz for all the regions. Table 6.18 presents the tap amplitude statistics necessary to create the [M1, Medium Airport, LOS-O, 25] model. 228
Table 6.18 Tap amplitude parameters for [M1, Medium Airport, LOS-O, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.963 5.3 1 na 1.0000 2 0.037 1.57 0.7128 0.4615 0.7761
Table 6.19 presents the tap amplitude statistics necessary to create the [M2, Medium
Airport, LOS-O, 25] model. Even though there should be 125 taps in the M2 25 MHz model for all regions, due to the sparse scattering environment present in the LOS-O regions, most of the taps are non-existent (i.e., well below threshold). This is either due to the fact that they have negligible energy in them or the taps qualify as valid multipath for fewer than 10 PDPs and hence we don’t have statistical confidence in computing the parameters for those taps. Also, note that the first tap (index 1) doesn’t exist in Table
6.19 due to the following two reasons:
¾ To determine the tap amplitude parameters of M1, for any given PDP, for
simplicity we discard the initial few taps of the PDP which account for the rise
time of the filters in the receiver. Hence, for M1, the first tap is always the one
with the maximum energy.
¾ As discussed in Chapter 3, M2 is modeled using the entire length of the PDP.
Hence contrary to M1, for any given PDP, we don’t consider the maximum
amplitude component as the first tap. So, in the developed M2 model, often, the
first tap doesn’t have enough energy to be considered as a valid multipath (i.e the
energy of the first tap is below 25dB of the energy of the maximum amplitude
component). To present M2 in a standardized format, in cases where the first tap 229
has negligible energy, we consider the maximum energy component as the first
tap
As mentioned in Chapter 3, the entire measurement route was divided into multiple segments of travel. In Section 4.4.1, we noted that the correlation matrix for M2 will be changed after every n realizations, where the n realizations depend on the number of
PDPs we actually measured for a given segment of travel. So, in the case of M2, for every segment of travel, we have a unique correlation matrix. Hence we don’t include these matrices here and provide them in the MATLAB files accompanying this dissertation. Another thing to note again is that the tap amplitude statistics will remain the same for the [M3, Medium Airport, LOS-O, 25] model. The only difference is that in order to implement the M3 models, we do not use the correlation matrices or the persistence processes.
Table 6.19 Tap amplitude parameters for [M2, Medium Airport, LOS-O, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.4265 0.8829 0.9986 0 0.9986 2 0.4116 0.9054 0.9385 0.1364 0.9433 3 0.1022 0.7570 0.7972 0.2483 0.8102 4 0.0289 0.8338 0.5692 0.5455 0.6552 5 0.0068 1.2799 0.3580 0.7059 0.4706 6 0.0033 1.6695 0.2643 0.8004 0.4468 7 0.0022 1.7150 0.2238 0.8396 0.4465 8 0.0010 2.2694 0.1790 0.8637 0.3780 9 0.0004 1.2270 0.1413 0.8825 0.2871 10 0.0005 2.3777 0.1594 0.8667 0.2982 11 0.0005 1.6100 0.1259 0.8960 0.2809 12 0.0002 3.2350 0.1315 0.9032 0.3617
6.3.2.3 [M1 M2 M3, NLOS-S, Medium Airport, 25] 230
In this section, we present the M1, M2 and M3 Medium Airport models for a 25
MHz bandwidth for the NLOS-S region. Table 6.20 presents the tap amplitude statistics necessary to create the [M1, Medium Airport, NLOS-S, 25] model.
Table 6.20 Tap amplitude parameters for [M1, Medium Airport, NLOS-S, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.911 3.8416 1.0000 NA 1.0000 2 0.075 1.7161 0.8336 0.2857 0.8571 3 0.014 1.8389 0.5608 0.5414 0.6425
Table 6.21 presents the tap amplitude statistics necessary to create the [M2, Medium
Airport, NLOS-S, 25] model. As with the other models, the tap amplitude statistics for
[M3, Medium Airport, NLOS-S, 25] will be the same as the ones provided in Table 6.21.
There are several things that are worth stating regarding the amplitude statistics of the
[M2, Medium Airport, NLOS-S, 25] model.
¾ For the same reasons provided in the previous section on the [M2, Medium
Airport, LOS-O, 25] model, tap 1 is absent for the [M2, Medium Airport, NLOS-
S, 25] model as well.
¾ The high delay taps (from 3240 nsec (tap 81) to 4920 nsec (tap 123)) are not
included in the mode since the taps qualify as valid multipath for fewer than 10
PDPs.
¾ Note that some of the higher delay taps (from 1600 nsec (tap 41 and higher)) have
large Weibull shape parameters. In general, these taps have negligible energy, but 231
for a very small percentage of PDPs, there might be high delay reflections with
significant energy contributing to those taps.
Table 6.21 Tap amplitude parameters for [M2, Medium Airport, NLOS-S, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (βk) 1 0.2812 0.8634 0.9986 0 0.9986 2 0.4039 1.1061 0.9385 0.1364 0.9433 3 0.1743 0.9338 0.7972 0.2483 0.8102 4 0.0452 1.287 0.5692 0.5455 0.6552 5 0.0149 1.27 0.3580 0.7059 0.4706 6 0.0085 1.3578 0.2643 0.8004 0.4468 7 0.0068 1.3431 0.2238 0.8396 0.4465 8 0.0056 1.4261 0.1790 0.8637 0.3780 9 0.0038 1.3548 0.1413 0.8825 0.2871 10 0.0027 1.9597 0.1594 0.8667 0.2982 11 0.0024 1.7434 0.1259 0.8960 0.2809 12 0.0043 1.5751 0.1315 0.9032 0.3617 13 0.0040 1.4048 0.1147 0.9177 0.3659 14 0.0018 1.7172 0.0853 0.9326 0.2787 15 0.0015 1.6743 0.0881 0.9340 0.3065 16 0.0016 1.7003 0.0797 0.9315 0.2105 17 0.0019 1.7726 0.1007 0.9159 0.2500 18 0.0015 1.7598 0.0783 0.9331 0.2143 19 0.0027 1.485 0.0839 0.9327 0.2667 20 0.0011 1.8854 0.0755 0.9424 0.2963 21 0.0008 1.447 0.0545 0.9541 0.2051 22 0.0008 1.4524 0.0559 0.9540 0.2250 23 0.0008 2.5772 0.0629 0.9507 0.2667 24 0.0012 1.7068 0.0699 0.9307 0.0800 25 0.0011 1.4842 0.0657 0.9370 0.1064 26 0.0007 2.0938 0.0573 0.9465 0.1220 27 0.0004 2.3826 0.0448 0.9619 0.1875 28 0.0004 2.5935 0.0392 0.9606 0.0357 29 0.0004 1.5932 0.0378 0.9651 0.1111 30 0.0006 1.3585 0.0545 0.9511 0.1538 31 0.0004 1.9999 0.0364 0.9651 0.0769 32 0.0007 1.6652 0.0420 0.9649 0.2000 33 0.0006 1.8808 0.0448 0.9589 0.1250 34 0.0009 1.5761 0.0462 0.9604 0.1818 232
35 0.0012 1.6554 0.0545 0.9570 0.2564 36 0.0011 1.9713 0.0587 0.9613 0.3810 37 0.0004 3.2086 0.0490 0.9543 0.1143 38 0.0005 1.7831 0.0406 0.9635 0.1379 39 0.0002 2.7185 0.0336 0.9710 0.1667 40 0.0003 2.6927 0.0266 0.9827 0.3684 41 0.0003 2.4441 0.0336 0.9710 0.1667 42 0.0005 2.2158 0.0350 0.9695 0.1600 43 0.0003 3.2156 0.0322 0.9696 0.0870 44 0.0002 1.0947 0.0266 0.9741 0.0526 45 0.0003 2.4636 0.0308 0.9697 0.0455 46 0.0004 2.1762 0.0336 0.9681 0.0833 47 0.0003 2.166 0.0280 0.9755 0.1500 48 0.0002 3.276 0.0266 0.9799 0.2632 49 0.0003 3.8732 0.0266 0.9784 0.2105 50 0.0003 0.9905 0.0252 0.9756 0.0556 51 0.0001 3.4617 0.0182 0.9829 0 53 0.0001 3.0709 0.0126 0.9887 0.1111 54 0.0003 1.7874 0.0364 0.9753 0.3462 56 0.0002 3.1365 0.0252 0.9770 0.1111 58 0.0001 2.8641 0.0140 0.9872 0.1000 61 0.0001 2.5131 0.0196 0.9800 0 62 0.0001 0.8643 0.0112 0.9901 0.1250 63 0.0002 1.8046 0.0252 0.9799 0.2222 64 0.0001 1.8433 0.0154 0.9886 0.2727 65 0.0001 1.5002 0.0140 0.9872 0.1000 66 0.0001 1.2333 0.0210 0.9785 0 67 0.0002 1.9974 0.0252 0.9784 0.1667 68 0.0001 2.0179 0.0196 0.9815 0.0769 69 0.0002 1.0819 0.0182 0.9815 0 70 0.0003 3.3286 0.0448 0.9560 0.0625 74 0.0002 3.6335 0.0783 0.9331 0.2143 75 0.0001 3.3395 0.0378 0.9636 0.0741 76 0.0002 3.6495 0.0322 0.9667 0 77 0.0001 5.1451 0.0462 0.9589 0.1212 78 0.0002 3.5651 0.0364 0.9666 0.1154 79 0.0004 4.2057 0.0182 0.9829 0.0769 123 0.0002 6.9926 0.0322 0.9682 0.0435 124 0.0002 2.2074 0.0741 0.9274 0.0943
233
6.3.2.4 [M1 M2 M3, NLOS, Medium Airport, 25]
Similar to the M2 models provided earlier for NLOS-S, the following Tables 6.22 and
6.23 provide the tap amplitude statistical parameters necessary to simulate models M1,
M2 and M3 for the NLOS region of Medium Airports for a BW of 25 MHz. (Note again that the tap amplitude parameters for M2 and M3 are the same.) An interesting point regarding the [M2, Medium Airport, NLOS, 25] model is that due to the richer scattering environment of NLOS as compared to NLOS-S and LOS-O, in this model, we have all the 125 taps.
Table 6.22 Tap amplitude parameters for [M1, Medium Airport, NLOS, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.849 1.6054 1 na 1.0000 2 0.073 1.4198 0.7466 0.4490 0.8139 3 0.022 1.5557 0.5908 0.5227 0.6689 4 0.009 1.3772 0.4762 0.6133 0.5739 5 0.011 1.3175 0.4487 0.6296 0.5441 6 0.007 1.3302 0.4019 0.6662 0.5040 7 0.005 1.3831 0.3737 0.7123 0.5162 8 0.005 1.3835 0.3713 0.7073 0.5033 9 0.005 1.4141 0.3761 0.7154 0.5269 10 0.007 1.3071 0.3866 0.7132 0.5460 11 0.009 1.3386 0.41 0.7141 0.5897
234
Table 6.23 Tap amplitude parameters for [M2, Medium Airport, NLOS, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0088 0.8006 0.0686 0.9601 0.4588 2 0.1596 0.9035 0.9952 0.1667 0.9959 3 0.1875 1.0092 0.8830 0.3103 0.9085 4 0.1146 1.0567 0.7676 0.4653 0.8379 5 0.0515 1.1622 0.6336 0.5132 0.7181 6 0.0255 1.2572 0.5456 0.5755 0.6474 7 0.0187 1.2853 0.4762 0.5871 0.5450 8 0.0119 1.3744 0.4197 0.6579 0.5260 9 0.0093 1.4737 0.3826 0.6889 0.4968 10 0.0078 1.5107 0.3906 0.6976 0.5289 11 0.0077 1.4875 0.3914 0.6906 0.5175 12 0.0090 1.5124 0.3947 0.6960 0.5328 13 0.0093 1.4566 0.3931 0.7061 0.5453 14 0.0094 1.4207 0.3834 0.7212 0.5527 15 0.0072 1.4821 0.3398 0.7384 0.4929 16 0.0055 1.5317 0.3366 0.7445 0.4976 17 0.0058 1.4764 0.3406 0.7528 0.5226 18 0.0059 1.4684 0.3285 0.7500 0.4901 19 0.0053 1.4980 0.3099 0.7427 0.4282 20 0.0050 1.5074 0.3035 0.7587 0.4468 21 0.0049 1.5371 0.3002 0.7702 0.4651 22 0.0051 1.5621 0.2962 0.7661 0.4454 23 0.0034 1.4986 0.2744 0.7784 0.4147 24 0.0040 1.5283 0.2986 0.7768 0.4770 25 0.0046 1.5219 0.2857 0.7749 0.4379 26 0.0039 1.5973 0.2906 0.7861 0.4791 27 0.0032 1.5061 0.2583 0.8205 0.4859 28 0.0037 1.5792 0.2534 0.8076 0.4345 29 0.0032 1.4985 0.2494 0.8073 0.4207 30 0.0034 1.6171 0.2583 0.8041 0.4389 31 0.0038 1.4869 0.2502 0.7987 0.3981 32 0.0038 1.5169 0.2720 0.7849 0.4256 33 0.0039 1.4638 0.2631 0.7952 0.4277 34 0.0037 1.5314 0.2647 0.7914 0.4220 35 0.0040 1.4748 0.2583 0.8039 0.4375 36 0.0036 1.4927 0.2526 0.8099 0.4391 37 0.0030 1.4446 0.2446 0.8235 0.4554 38 0.0032 1.4012 0.2470 0.8135 0.4295 39 0.0034 1.4963 0.2559 0.8165 0.4669 40 0.0035 1.4893 0.2316 0.8318 0.4390 235
41 0.0035 1.4614 0.2405 0.8193 0.4310 42 0.0035 1.5460 0.2470 0.8208 0.4542 43 0.0031 1.4497 0.2316 0.8225 0.4126 44 0.0028 1.4809 0.2405 0.8287 0.4597 45 0.0030 1.4425 0.2276 0.8318 0.4306 46 0.0039 1.4135 0.2357 0.8268 0.4399 47 0.0029 1.4439 0.2268 0.8288 0.4179 48 0.0024 1.5174 0.2203 0.8383 0.4286 49 0.0030 1.5374 0.2292 0.8386 0.4577 50 0.0025 1.6018 0.2220 0.8276 0.3964 51 0.0024 1.4540 0.2179 0.8431 0.4349 52 0.0029 1.4648 0.2357 0.8205 0.4158 53 0.0026 1.6484 0.2203 0.8468 0.4559 54 0.0025 1.4608 0.2187 0.8428 0.4354 55 0.0027 1.4938 0.2211 0.8351 0.4161 56 0.0023 1.4427 0.2155 0.8383 0.4120 57 0.0028 1.4621 0.2284 0.8366 0.4488 58 0.0036 1.5397 0.2163 0.8299 0.3843 59 0.0026 1.5025 0.2220 0.8402 0.4416 60 0.0026 1.5469 0.2211 0.8342 0.4176 61 0.0027 1.4069 0.2260 0.8425 0.4624 62 0.0027 1.4051 0.2179 0.8421 0.4349 63 0.0024 1.6563 0.2203 0.8373 0.4249 64 0.0032 1.4318 0.2308 0.8277 0.4266 65 0.0034 1.4439 0.2478 0.8174 0.4463 66 0.0033 1.5362 0.2316 0.8256 0.4231 67 0.0037 1.4235 0.2324 0.8318 0.4460 68 0.0037 1.5039 0.2502 0.8116 0.4369 69 0.0032 1.5477 0.2195 0.8323 0.4044 70 0.0031 1.4080 0.2397 0.8185 0.4257 71 0.0027 1.4992 0.2252 0.8269 0.4050 72 0.0028 1.4729 0.2268 0.8307 0.4235 73 0.0030 1.5823 0.2292 0.8241 0.4099 74 0.0029 1.4938 0.2244 0.8229 0.3885 75 0.0028 1.4048 0.2260 0.8246 0.4000 76 0.0028 1.5116 0.2260 0.8319 0.4214 77 0.0024 1.5269 0.2163 0.8455 0.4419 78 0.0024 1.5796 0.2236 0.8285 0.4058 79 0.0032 1.3690 0.2324 0.8368 0.4618 80 0.0025 1.5721 0.2228 0.8399 0.4384 81 0.0027 1.4356 0.2147 0.8488 0.4474 82 0.0028 1.4756 0.2115 0.8596 0.4771 83 0.0024 1.5107 0.2139 0.8284 0.3698 84 0.0023 1.5113 0.2090 0.8490 0.4302 85 0.0028 1.7122 0.2163 0.8620 0.5019 236
86 0.0025 1.5279 0.2203 0.8342 0.4139 87 0.0020 1.5656 0.2074 0.8573 0.4553 88 0.0028 1.4724 0.2163 0.8373 0.4120 89 0.0029 1.5114 0.2268 0.8339 0.4342 90 0.0025 1.4845 0.2066 0.8483 0.4180 91 0.0023 1.4761 0.2147 0.8489 0.4491 92 0.0027 1.4456 0.2002 0.8566 0.4274 93 0.0030 1.4823 0.2139 0.8407 0.4151 94 0.0027 1.4236 0.1937 0.8539 0.3933 95 0.0034 1.5395 0.2252 0.8384 0.4444 96 0.0038 1.3915 0.2155 0.8445 0.4345 97 0.0033 1.5910 0.2195 0.8387 0.4280 98 0.0023 1.5045 0.2155 0.8486 0.4494 99 0.0024 1.4153 0.2179 0.8450 0.4407 100 0.0032 1.4661 0.2308 0.8309 0.4336 101 0.0024 1.4735 0.1994 0.8517 0.4049 102 0.0023 1.6386 0.2171 0.8392 0.4216 103 0.0033 1.4475 0.2042 0.8538 0.4308 104 0.0034 1.4743 0.2252 0.8427 0.4604 105 0.0033 1.4750 0.2090 0.8488 0.4286 106 0.0033 1.3769 0.2058 0.8505 0.4235 107 0.0028 1.3685 0.2082 0.8592 0.4651 108 0.0035 1.4412 0.2300 0.8218 0.4049 109 0.0026 1.4087 0.2163 0.8495 0.4552 110 0.0032 1.4706 0.2220 0.8454 0.4599 111 0.0032 1.5468 0.2349 0.8152 0.3986 112 0.0028 1.4557 0.2155 0.8354 0.4023 113 0.0033 1.3847 0.2252 0.8240 0.3957 114 0.0028 1.4834 0.2220 0.8297 0.4000 115 0.0033 1.4163 0.2308 0.8267 0.4231 116 0.0033 1.5018 0.2526 0.8076 0.4281 117 0.0032 1.4128 0.2316 0.8370 0.4599 118 0.0043 1.4537 0.2446 0.8021 0.3894 119 0.0048 1.3983 0.2365 0.8212 0.4232 120 0.0039 1.3434 0.2284 0.8188 0.3887 121 0.0033 1.3303 0.2324 0.8284 0.4306 122 0.0045 1.4171 0.2405 0.8193 0.4310 123 0.0045 1.5232 0.2615 0.7847 0.3932 124 0.0047 1.4572 0.2760 0.7879 0.4444 125 0.0060 1.4013 0.2994 0.7523 0.4216
237
6.4 Small Airport Models
6.4.1 M1 50 MHz Small Airport Models
We performed measurement campaigns at BL, TA, and OU GA airports to obtain data for developing models for small (GA) airports. Tamiami is actually one of the largest GAs in the USA, in terms of area. At Tamiami, due to staffing limitations and some other restrictions, we could not use the small ATCT as our transmission site.
Instead we mounted our transmitter platform on the level roof of a small shed near an ILS site. The transmit antenna height was thus only about 5-6 meters. This limited transmitting antenna height prevented us from receiving a sufficiently strong signal at certain locations on the airport surface. The models that we developed for TA can be considered as worst case channel models for GAs (or as example AFS models for small airports), where we might not have an antenna mounted at the ATCT or are for some reason unable to use it for transmission. The BL and OU airports are similar with respect to airport layout, traffic density (planes and ground vehicles), heights of ATCTs, and channel statistics. Our small airport channel models are based primarily on the data collected at BL. As noted in Chapter 3, in the GA airports, we usually don’t have a large percentage of NLOS regions, and most of the airport can be classified either as LOS-O or
NLOS-S. Refer to Table 3.2 for statistics on the number of profiles collected in each region at the different GA airports.
6.4.1.1 Markov Model for Small Airport Region Transitions
238
As mentioned in Figure 4.22, we need to determine the region transition and steady state matrices to model the Markov process to emulate the region transitions. Note that these matrices will remain the same irrespective of the model type or BW of the developed channel model. These matrices depend only on the airport type. As discussed previously, we generally don’t encounter NLOS region on the small airport surface.
Hence, while defining the region transitions and steady state probabilities, we limit ourselves to LOSO (state 1) and NLOSS (state 2).
⎡P11 P12 ⎤ ⎡0.5412 0.4588⎤ Re gion _TS = ⎢ ⎥ = ⎢ ⎥ ⎣P21 P22 ⎦ ⎣0.1810 0.8190⎦ (6.13) ⎡P1 ⎤ ⎡0.2819⎤ Re gion _ SS = ⎢ ⎥ = ⎢ ⎥ ⎣P2 ⎦ ⎣0.7181⎦
6.4.1.2 [M1, Small Airport, LOS-O, 50]
Using the mean RMS-DS to determine the number of taps, Table 6.1 enables us to specify L=4 for the LOS-O model. For our small airport LOS-O case, we consider only 2 taps for our channel model, which account for 98% of the cumulative energy. For the
LOS−O correlation matrix, Rwc is as follows:
⎡ 1 0.0312⎤ ⎢ ⎥ (6.14) ⎣0.0312 1 ⎦
Thus, the second tap can be considered essentially uncorrelated with the first (specular) component. Table 6.24 provides the channel model data for the GA LOS-O region. 239
Table 6.24 Tap amplitude parameters for [M1, Small Airport, LOS-O, 50] [138]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.911 10.1 1.0000 na 1.0000 2 0.089 1.87 0.9453 0.1429 0.9502
Figure 6.21 provides example curve fits for the tap amplitudes of the 1st and 2nd taps for data obtained for the [M1, Small Airport, LOS-O, 50] channel.
Tap 1 Amplitude data 1.6 Tap2 Amplitude data 2 K = 14.5dB Weibull : b =1.87 Weibull : b = 10.1 1.4 Nakagami : m = 0.93
1.2 1.5
1
0.8 Density
1 Density
0.6
0.5 0.4
0.2
0 0 1.2 1.4 1.6 1.8 2 2.2 2.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Data Data
Figure 6.21 Amplitude histograms and curve fits for taps 1 and 2 for [M1, Small Airport, LOS-O, 50] [138]
6.4.1.3 [M1, Small Airport, NLOS-S, 50]
For the NLOS-S region at the small airports, via use of Table 6.1, we determine the number of taps for the NLOS-S model as 23. Table 6.25 shows the cumulative energy accretion with tap index for this case. We use here L=10 taps, and thus account for ~99% of the energy. 240
Table 6.25 Cumulative energy for [M1, Small Airport, NLOS-S, 50] [138]
Tap Index Cumulative Energy (%) 2 95.3 4 98.4 10 99 15 99.8 23 1
NLOSS The Rwc tap correlation matrix for this case is shown next, followed by Table
6.26, which provides the amplitude statistics for the [M1, Small Airport, NLOS-S, 50] model.
⎡ 1 −0.0419 0.0249 −0.0710 0.3393 0.5048 0.8583 0.8270 0.0168 0.6043⎤ ⎢ ⎥ ⎢−0.0419 1 0.1904 0.4993 0.0989 0.0218 0.0665 −0.0565 −0.1870 0.1977⎥ ⎢ 0.0249 0.1904 1 0.4205 0.3469 −0.0490 −0.0341 −0.0301 −0.1726 −0.0714⎥ ⎢ ⎥ ⎢−0.0710 0.4993 0.4025 1 0.0956 0.1393 0.0016 −0.0554 −0.0329 0.2003⎥ (6.15) ⎢ 0.3393 0.0989 0.3469 0.0956 1 0.6944 0.1765 0.8924 0.1197 0.2007⎥ ⎢ ⎥ ⎢ 0.5048 0.0218 −0.0490 0.1393 0.6944 1 0.8334 0.8873 0.6678 0.4322⎥ ⎢ 0.8583 0.0665 −0.0341 0.0016 0.1765 0.8334 1 0.9282 0.8527 0.7094⎥ ⎢ ⎥ ⎢ 0.8270 −0.0565 −0.0301 0.0554 0.8924 0.8873 0.9282 1 0.8619 0.4099⎥ ⎢ 0.0168 −0.1870 −0.1726 −0.0329 0.1197 0.6678 0.8527 0.8619 1 0.5558⎥ ⎢ ⎥ ⎣⎢ 0.6043 0.1977 −0.0714 0.2003 0.2007 0.4322 0.7094 0.4099 0.5558 1 ⎦⎥
Table 6.26 Tap amplitude parameters for [M1, Small Airport, NLOS-S, 50] [138]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.821 5.67 1.0000 na 1.0000 2 0.142 1.72 0.9709 0 0.9699 3 0.025 1.81 0.8436 0.2647 0.8634 4 0.007 4.6 0.6917 0.3930 0.7289 5 0.001 2.76 0.4371 0.6475 0.5474 6 0.001 2.78 0.2991 0.7768 0.4794 7 0.001 2.38 0.2285 0.8426 0.4698 8 0.001 1.95 0.2255 0.8614 0.5274 9 0.001 2.24 0.1994 0.8985 0.5969 10 0.001 2.91 0.2393 0.8609 0.5613 241
Figure 6.22 provides curve fits for the tap amplitudes of the 1st, 2nd, 5th, and 10th taps of this [Small Airport, NLOS-S, 50] channel.
1.2 Tap 1 Amplitude Data Tap2 Amplitude data Weibull : b = 7.12 1 Weibull : b = 1.75 K = 12.5dB Nakagami : m = 0.83 1 0.8
0.6 Density Density 0.5 0.4
0.2
0 0 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Data Data 12 Tap 5 Amplitude Data 10 Tap 10 Amplitude Data 10 Nakagami : m = 2.35 Nakagami : m = 2.56 Weibull : b = 2.84 8 Weibull : b = 3.14 8 6 6 Density Density 4 4
2 2
0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.1 0.15 0.2 0.25 0.3 Data Data
Figure 6.22 Amplitude histograms and curve fits for taps 1, 2, 5, and 10 for [M1, Small Airport, NLOS-S, 50] [138]
6.4.1.4 [M1, Small Airport, NLOS, 50]
As discussed previously, it is somewhat unlikely that one will encounter many
NLOS regions at normal GA airports. An NLOS model (using Tamiami data) is provided for generating such a worst case scenario. The tap amplitude statistics for the tapped delay line model for a BW of 50 MHz are provided in Table 6.27.
Table 6.27 Tap amplitude parameters for [M1, Small Airport, NLOS, 50] [138]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.696 1.83 1.0000 na 1.0000 2 0.087 1.4 0.8403 0.2674 0.8607 3 0.024 1.66 0.7064 0.3848 0.7442 4 0.017 1.66 0.6263 0.4560 0.6752 242
5 0.014 1.6 0.5583 0.5504 0.6446 6 0.011 1.64 0.5245 0.5571 0.5980 7 0.010 1.66 0.5102 0.6000 0.6164 8 0.010 1.56 0.4898 0.6213 0.6049 9 0.009 1.64 0.4858 0.6156 0.5934 10 0.008 1.65 0.4591 0.6354 0.5707 11 0.006 1.6 0.4404 0.6333 0.5343 12 0.005 1.75 0.4141 0.6679 0.5306 13 0.005 1.71 0.4124 0.6773 0.5405 14 0.007 1.65 0.4408 0.6513 0.5580 15 0.008 1.51 0.4288 0.6867 0.5830 16 0.008 1.61 0.4328 0.6743 0.5735 17 0.006 1.63 0.4155 0.6657 0.5300 18 0.006 1.76 0.4097 0.6652 0.5179 19 0.006 1.75 0.4177 0.6835 0.5591 20 0.005 1.74 0.4093 0.6684 0.5217 21 0.006 1.59 0.4079 0.6850 0.5431 22 0.005 1.77 0.3848 0.6997 0.5202 23 0.004 1.81 0.3763 0.7009 0.5047 24 0.004 1.83 0.3857 0.6891 0.5052 25 0.004 1.76 0.3652 0.7251 0.5225 26 0.003 1.96 0.3563 0.6943 0.4482 27 0.003 1.84 0.3359 0.7245 0.4556 28 0.003 1.87 0.3341 0.7172 0.4368 29 0.003 1.86 0.3230 0.7449 0.4656 30 0.002 1.94 0.2994 0.7522 0.4205 31 0.002 1.8 0.3163 0.7357 0.4290 32 0.003 1.74 0.3141 0.7515 0.4575 33 0.003 1.7 0.3136 0.7510 0.4553 34 0.003 1.81 0.3221 0.7328 0.4378 35 0.002 1.86 0.3105 0.7508 0.4470 36 0.002 2 0.3101 0.7529 0.4505
6.4.2 [M1 M2 M3, Small Airport, 25] for all Regions
6.4.2.1 [M1 M2 M3, LOSO, Small Airport, 25]
243
In this section, we present the 25 MHz models for the different regions of the small airport. Table 6.28 presents the tap amplitude statistics necessary to create the [M1,
Small Airport, LOS-O, 25] model.
Table 6.28 Tap amplitude parameters for [M1, Small Airport, LOSO, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.976 5.9476 1 na 1.0000 2 0.024 1.8988 0.7105 0.4688 0.7778
Table 6.29 presents the tap amplitude statistics necessary to create the [M2, Small
Airport, LOS-O, 25] model. As described previously in Section 1.3.2.1, due to the presence of very few scatterers in the LOS-O region (LOS-O regions on the airport are primarily on the runways), most of the taps are non-existent because they have nearly negligible energy in them. (For some more detailed reasoning, please refer to the discussion provided in Section 1.3.2.1.) As with the other M3 models, the tap amplitude statistics are the same as for the [M2, Small Airport, LOS-O, 25] model.
Table 6.29 Tap amplitude parameters for [M2, Small Airport, LOSO, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.4673 1.1072 0.9955 0 0.9954 2 0.4722 1.2856 0.9698 0.1500 0.9735 3 0.0228 1.4607 0.6998 0.4848 0.7780 4 0.0116 1.3668 0.3922 0.6965 0.5308 5 0.0105 1.2944 0.3228 0.8129 0.6103 6 0.0041 1.3840 0.2851 0.8586 0.6489 7 0.0049 1.2823 0.2790 0.8910 0.7189 8 0.0017 1.7451 0.2866 0.9004 0.7526 9 0.0019 1.8401 0.2293 0.8902 0.6316 244
10 0.0006 2.0402 0.1795 0.8877 0.4874
6.4.2.2 [M1 M2 M3, NLOS-S, Small Airport, 25]
In this section, we present the M1, M2 and M3 Small Airport models for a 25
MHz bandwidth for the NLOS-S region. Table 6.30 presents the tap amplitude statistics necessary to create the [M1, Small Airport, NLOS-S, 25] model.
Table 6.30 Tap amplitude parameters for [M1, Small Airport, NLOS-S, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.909 3.6082 1 na 1.0000 2 0.081 1.6471 0.8824 0.2949 0.9058 3 0.004 2.0192 0.457 0.6797 0.6205 4 0.002 1.4935 0.2926 0.8034 0.5258 5 0.001 1.6916 0.267 0.8683 0.6420 6 0.909 1.3392 0.2579 0.8862 0.6765
Table 6.31 presents the tap amplitude statistics necessary to create the [M2, Small
Airport, NLOS-S, 25] model. (As before, for the M3 models, the tap amplitude statistics for [M3, Small Airport, NLOS-S, 25] are the same as those provided in Table 6.31.)
Similar to the observations made in Section 6.3.2.2, we can observe that we don’t have valid multipath (within 25dB of the maximum energy component) between tap 87 (3480 nsec) and tap 120 (4800 nsec). The possible reasons are similar to the ones already presented in the discussion in Section 6.3.2.2. 245
Table 6.31 Tap amplitude parameters for [M2, Small Airport, NLOS-S, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0088 1.5459 0.0317 0.9891 0.6667 2 0.3498 1.0423 0.9955 0 0.9954 3 0.4032 1.3987 0.9698 0.1500 0.9735 4 0.0789 1.2256 0.6998 0.4848 0.7780 5 0.0230 1.2773 0.3922 0.6965 0.5308 6 0.0102 1.3409 0.3228 0.8129 0.6103 7 0.0073 1.3669 0.2851 0.8586 0.6489 8 0.0146 1.2397 0.2790 0.8910 0.7189 9 0.0185 1.3121 0.2866 0.9004 0.7526 10 0.0080 1.3838 0.2293 0.8902 0.6316 11 0.0029 1.54 0.1795 0.8877 0.4874 12 0.0021 1.5074 0.1463 0.9115 0.4845 13 0.0012 1.5393 0.1026 0.9259 0.3529 14 0.0009 1.6552 0.0950 0.9349 0.3810 15 0.0007 2.0421 0.0769 0.9411 0.2941 16 0.0007 1.7565 0.0694 0.9432 0.2391 17 0.0011 1.5147 0.0664 0.9450 0.2273 18 0.0003 2.0544 0.0437 0.9716 0.3793 19 0.0005 1.3075 0.0452 0.9620 0.2000 20 0.0003 1.5687 0.0287 0.9736 0.1053 21 0.0007 1.4703 0.0392 0.9780 0.4615 22 0.0008 1.515 0.0498 0.9618 0.2727 23 0.0012 1.3788 0.0528 0.9617 0.3143 24 0.0008 1.8743 0.0664 0.9660 0.5227 25 0.0019 1.6313 0.0784 0.9623 0.5577 26 0.0016 1.6273 0.0739 0.9657 0.5714 27 0.0016 1.7715 0.0694 0.9578 0.4348 28 0.0013 1.8309 0.0588 0.9615 0.3846 29 0.0013 1.7895 0.0649 0.9628 0.4651 30 0.0009 1.9388 0.0618 0.9646 0.4634 31 0.0012 1.3796 0.0483 0.9683 0.3750 32 0.0003 2.0496 0.0271 0.9783 0.2222 33 0.0005 1.511 0.0287 0.9782 0.2632 34 0.0013 1.4665 0.0528 0.9681 0.4286 35 0.0006 1.7295 0.0483 0.9635 0.2500 36 0.0012 1.351 0.0679 0.9514 0.3333 37 0.0008 1.2246 0.0407 0.9669 0.2222 38 0.0010 1.3925 0.0407 0.9732 0.3704 39 0.0005 1.8196 0.0392 0.9654 0.1538 40 0.0011 1.9353 0.0588 0.9727 0.5641 246
41 0.0009 1.8246 0.0573 0.9631 0.3947 42 0.0007 1.7971 0.0513 0.9586 0.2353 43 0.0005 1.8919 0.0618 0.9549 0.3171 44 0.0012 1.7602 0.0814 0.9572 0.5185 45 0.0022 1.4363 0.0890 0.9502 0.4915 46 0.0022 1.4926 0.0739 0.9543 0.4286 47 0.0012 1.686 0.0664 0.9612 0.4545 48 0.0011 1.7504 0.0664 0.9515 0.3182 49 0.0015 1.7564 0.0724 0.9609 0.5000 50 0.0008 1.8907 0.0618 0.9678 0.5122 51 0.0005 2.4129 0.0679 0.9692 0.5778 52 0.0006 1.4479 0.0287 0.9798 0.3158 53 0.0008 1.7751 0.0347 0.9765 0.3478 54 0.0003 2.3531 0.0241 0.9799 0.1875 55 0.0002 2.2872 0.0241 0.9799 0.1875 66 0.0002 4.3841 0.0181 0.9846 0.1667 68 0.0001 3.1281 0.0075 0.9924 0 69 0.0000 3.6934 0.0060 0.9954 0.2500 70 0.0001 2.0234 0.0075 0.9924 0 72 0.0001 3.4132 0.0136 0.9877 0.1111 73 0.0001 3.2286 0.0166 0.9862 0.1818 74 0.0001 2.1904 0.0136 0.9877 0.1111 75 0.0002 2.9619 0.0166 0.9908 0.4545 76 0.0000 2.0182 0.0075 0.9924 0 77 0.0001 1.6957 0.0090 0.9924 0.1667 78 0.0002 1.7728 0.0347 0.9750 0.2609 79 0.0003 1.5468 0.0106 0.9893 0 80 0.0004 2.0093 0.0679 0.9562 0.4000 81 0.0003 2.1105 0.0498 0.9618 0.2727 82 0.0003 1.7645 0.0317 0.9766 0.2857 83 0.0001 2.3325 0.0136 0.9877 0.1111 84 0.0002 2.9713 0.0196 0.9831 0.1538 85 0.0002 2.3611 0.0317 0.9750 0.2381 86 0.0001 2.2615 0.0226 0.9830 0.2667 120 0.0005 1.1149 0.0860 0.9455 0.4211 121 0.0006 1.2826 0.0437 0.9652 0.2414 122 0.0013 0.9712 0.0483 0.9651 0.3125 123 0.0016 1.5111 0.1388 0.9298 0.5652 124 0.0009 0.9308 0.0513 0.9666 0.3824 125 0.0004 1.0355 0.0317 0.9750 0.2381
247
6.4.2.3 [M1 M2 M3, NLOS, Small Airport, 25]
As noted in Section 6.4.1.4, an NLOS region for the small airport is a worst-case kind of channel. Nonetheless, we still present these small airport NLOS 25 MHz bandwidth models here. Tables 6.32 and 6.33 provide the tap amplitude statistical parameters necessary to simulate models M1, M2 and M3 for NLOS region of Small
Airports for a BW of 25 MHz. (Note again that the tap amplitude parameters for M2 and
M3 are the same.)
Table 6.32 Tap amplitude parameters for [M1, Small Airport, NLOS, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.696 1.83 1.0000 na 1.0000 2 0.087 1.4 0.8403 0.2674 0.8607 3 0.024 1.66 0.7064 0.3848 0.7442 4 0.017 1.66 0.6263 0.4560 0.6752 5 0.014 1.6 0.5583 0.5504 0.6446 6 0.011 1.64 0.5245 0.5571 0.5980 7 0.010 1.66 0.5102 0.6000 0.6164 8 0.010 1.56 0.4898 0.6213 0.6049 9 0.009 1.64 0.4858 0.6156 0.5934 10 0.008 1.65 0.4591 0.6354 0.5707 11 0.006 1.6 0.4404 0.6333 0.5343 12 0.005 1.75 0.4141 0.6679 0.5306 13 0.005 1.71 0.4124 0.6773 0.5405 14 0.007 1.65 0.4408 0.6513 0.5580 15 0.008 1.51 0.4288 0.6867 0.5830 16 0.008 1.61 0.4328 0.6743 0.5735 17 0.006 1.63 0.4155 0.6657 0.5300 18 0.006 1.76 0.4097 0.6652 0.5179 19 0.006 1.75 0.4177 0.6835 0.5591 20 0.005 1.74 0.4093 0.6684 0.5217 21 0.006 1.59 0.4079 0.6850 0.5431 22 0.005 1.77 0.3848 0.6997 0.5202 23 0.004 1.81 0.3763 0.7009 0.5047 24 0.004 1.83 0.3857 0.6891 0.5052 248
25 0.004 1.76 0.3652 0.7251 0.5225 26 0.003 1.96 0.3563 0.6943 0.4482 27 0.003 1.84 0.3359 0.7245 0.4556 28 0.003 1.87 0.3341 0.7172 0.4368 29 0.003 1.86 0.3230 0.7449 0.4656 30 0.002 1.94 0.2994 0.7522 0.4205 31 0.002 1.8 0.3163 0.7357 0.4290 32 0.003 1.74 0.3141 0.7515 0.4575 33 0.003 1.7 0.3136 0.7510 0.4553 34 0.003 1.81 0.3221 0.7328 0.4378 35 0.002 1.86 0.3105 0.7508 0.4470 36 0.002 2 0.3101 0.7529 0.4505
Table 6.33 Tap amplitude parameters for [M2, Small Airport, NLOS, 25]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0046 1.1015 0.0358 0.9787 0.4250 2 0.0971 0.9528 0.9978 0.4000 0.9987 3 0.1545 1.0554 0.9593 0.0989 0.9618 4 0.0946 1.1139 0.8708 0.2630 0.8905 5 0.0462 1.2282 0.7742 0.3433 0.8088 6 0.0297 1.2818 0.7066 0.4473 0.7709 7 0.0220 1.3542 0.6512 0.4981 0.7315 8 0.0190 1.3969 0.6163 0.5181 0.7003 9 0.0164 1.4268 0.5970 0.5378 0.6884 10 0.0173 1.3877 0.6015 0.5303 0.6892 11 0.0158 1.4146 0.5733 0.5635 0.6755 12 0.0132 1.5248 0.5675 0.5590 0.6643 13 0.0113 1.5372 0.5586 0.5629 0.6549 14 0.0102 1.5302 0.5376 0.5847 0.6431 15 0.0087 1.5546 0.5174 0.6132 0.6396 16 0.0082 1.6416 0.4946 0.6094 0.6013 17 0.0071 1.6271 0.4830 0.6069 0.5796 18 0.0070 1.6290 0.4602 0.6310 0.5675 19 0.0064 1.6562 0.4620 0.6265 0.5653 20 0.0062 1.6513 0.4597 0.6404 0.5778 21 0.0062 1.7141 0.4468 0.6456 0.5616 22 0.0058 1.7227 0.4396 0.6422 0.5443 23 0.0063 1.6421 0.4338 0.6490 0.5423 24 0.0054 1.8391 0.4383 0.6653 0.5714 25 0.0052 1.8821 0.4302 0.6622 0.5530 26 0.0054 1.7997 0.4414 0.6490 0.5562 27 0.0052 1.8487 0.4021 0.6759 0.5184 249
28 0.0046 1.8925 0.3994 0.6818 0.5218 29 0.0048 1.8328 0.4061 0.6903 0.5474 30 0.0051 1.6496 0.3940 0.6920 0.5267 31 0.0045 1.7326 0.4038 0.6749 0.5205 32 0.0042 1.8815 0.3980 0.6967 0.5416 33 0.0043 1.9058 0.3877 0.6798 0.4948 34 0.0043 1.8151 0.3573 0.7194 0.4956 35 0.0037 1.8371 0.3815 0.7192 0.5440 36 0.0042 1.9450 0.3761 0.7125 0.5226 37 0.0038 1.8864 0.3484 0.7356 0.5058 38 0.0043 1.7400 0.3739 0.7305 0.5490 39 0.0042 1.7471 0.3569 0.7314 0.5163 40 0.0038 1.8941 0.3506 0.7367 0.5128 41 0.0039 1.8272 0.3645 0.7282 0.5264 42 0.0039 1.8171 0.3551 0.7314 0.5126 43 0.0040 1.8666 0.3600 0.7198 0.5025 44 0.0037 1.8401 0.3394 0.7603 0.5343 45 0.0035 1.9791 0.3551 0.7372 0.5233 46 0.0037 1.9173 0.3435 0.7405 0.5033 47 0.0036 1.8080 0.3399 0.7446 0.5046 48 0.0037 1.9049 0.3439 0.7394 0.5033 49 0.0034 1.9095 0.3269 0.7480 0.4815 50 0.0041 1.7857 0.3475 0.7359 0.5045 51 0.0035 1.8285 0.3323 0.7513 0.5007 52 0.0038 1.8514 0.3488 0.7203 0.4782 53 0.0033 2.0264 0.3390 0.7427 0.4987 54 0.0033 1.9350 0.3538 0.7500 0.5424 55 0.0032 1.8574 0.3296 0.7657 0.5237 56 0.0037 1.7753 0.3336 0.7609 0.5228 57 0.0035 1.9189 0.3296 0.7543 0.5007 58 0.0035 1.9067 0.3354 0.7589 0.5227 59 0.0035 1.7932 0.3283 0.7448 0.4782 60 0.0034 1.9249 0.3216 0.7507 0.4743 61 0.0031 1.8598 0.3207 0.7404 0.4505 62 0.0033 1.9524 0.3148 0.7675 0.4943 63 0.0033 1.9243 0.3211 0.7785 0.5320 64 0.0029 1.8754 0.3153 0.7693 0.4993 65 0.0028 1.9406 0.3099 0.7737 0.4964 66 0.0032 1.9817 0.3309 0.7612 0.5176 67 0.0032 2.0420 0.3359 0.7534 0.5126 68 0.0030 1.9292 0.3220 0.7553 0.4840 69 0.0030 1.9285 0.3166 0.7557 0.4732 70 0.0029 1.8835 0.3166 0.7597 0.4816 71 0.0031 1.9571 0.3175 0.7705 0.5070 72 0.0037 1.9185 0.3229 0.7667 0.5111 250
73 0.0039 1.8378 0.3323 0.7554 0.5087 74 0.0037 1.8738 0.3211 0.7706 0.5153 75 0.0035 1.9394 0.3533 0.7377 0.5203 76 0.0034 2.0046 0.3511 0.7172 0.4777 77 0.0032 1.9814 0.3426 0.7413 0.5039 78 0.0035 1.8975 0.3421 0.7437 0.5065 79 0.0034 1.9238 0.3162 0.7651 0.4922 80 0.0030 1.8897 0.3095 0.7790 0.5072 81 0.0030 1.8257 0.3162 0.7670 0.4965 82 0.0028 1.9660 0.2956 0.7973 0.5174 83 0.0030 2.0007 0.3198 0.7612 0.4923 84 0.0029 1.8934 0.3126 0.7715 0.4979 85 0.0029 1.9899 0.2867 0.7873 0.4711 86 0.0029 1.9498 0.3140 0.7795 0.5185 87 0.0027 1.9874 0.3081 0.7872 0.5225 88 0.0034 1.8156 0.3010 0.7855 0.5022 89 0.0031 1.7913 0.3081 0.7807 0.5080 90 0.0030 1.9103 0.3072 0.7784 0.5007 91 0.0033 1.9435 0.3247 0.7628 0.5069 92 0.0034 1.9007 0.3148 0.7668 0.4929 93 0.0032 1.8445 0.3171 0.7733 0.5120 94 0.0032 1.7881 0.3046 0.7574 0.4464 95 0.0031 1.7904 0.3059 0.7782 0.4971 96 0.0033 1.9315 0.3238 0.7730 0.5262 97 0.0032 1.9345 0.3050 0.7688 0.4736 98 0.0032 1.9593 0.3023 0.7729 0.4763 99 0.0029 1.8759 0.3148 0.7753 0.5114 100 0.0032 1.8341 0.3211 0.7666 0.5070 101 0.0031 1.8651 0.3095 0.7816 0.5130 102 0.0030 1.8501 0.3068 0.7818 0.5073 103 0.0031 1.9287 0.3184 0.7498 0.4649 104 0.0031 1.9085 0.3108 0.7656 0.4806 105 0.0030 1.8516 0.3171 0.7739 0.5134 106 0.0032 1.8942 0.2987 0.7869 0.5000 107 0.0027 1.9150 0.3099 0.7789 0.5079 108 0.0030 1.9240 0.3010 0.7753 0.4785 109 0.0032 1.9496 0.3072 0.7745 0.4920 110 0.0032 1.9025 0.3225 0.7682 0.5132 111 0.0038 1.8497 0.3256 0.7605 0.5041 112 0.0040 1.8033 0.3229 0.7634 0.5042 113 0.0035 1.8154 0.3180 0.7572 0.4796 114 0.0041 1.7266 0.3269 0.7513 0.4884 115 0.0043 1.7116 0.3220 0.7736 0.5236 116 0.0035 1.8308 0.3211 0.7594 0.4916 117 0.0039 1.7671 0.3345 0.7566 0.5160 251
118 0.0051 1.4876 0.3439 0.7497 0.5228 119 0.0056 1.4720 0.3618 0.7230 0.5117 120 0.0064 1.4493 0.3712 0.7117 0.5120 121 0.0071 1.4553 0.4003 0.6791 0.5196 122 0.0069 1.4700 0.4253 0.6511 0.5289 123 0.0070 1.5148 0.4150 0.6710 0.5366 124 0.0081 1.4923 0.4217 0.6641 0.5387 125 0.0065 1.5114 0.4222 0.6563 0.5302
6.5 Comparison of Nonstationary and Stationary Models for Different
Airports
In this section, we compare the PDPs generated using nonstationary and stationary channel models with those of the actual data collected at different airports. The models are evaluated on the basis of their precision in emulating the actual channel in the time and frequency domains.
6.5.1 Simulation Procedure to Develop Nonstationary and Stationary Models
In this section, we outline the steps we have used in simulating the nonstationary
(M1 & M2) and stationary (M3) channel models.
6.5.1.1 Steps for Simulating the M1 Model
Assumptions:
As noted in Section 4.2.2, the maximum vehicle speed on the airport surface can be assumed to be ~30 miles/hr; this yields a maximum Doppler frequency (fD,max) of 234
Hz for a carrier frequency of 5.12 GHz. For the case when planes are landing and taking 252 off, the relative velocity between these scatterers could as high as 150 m/s (~ 540 km/hr)
[36], yielding a maximum Doppler frequency of 2500 Hz. Note that this is the relative velocity and it is obtained by assuming a velocity of 270 km/hr (~170miles/hr) for the planes during landing/takeoff. Given this high velocity, the scattering effects due to takeoff/landing at any point on the airport surface would be transient. In addition, only vehicles nearest these high-velocity aircraft would see appreciable effects. Now, as discussed previously in Section 4.2.2, the effect of Doppler spreading can be mimicked using a LPF or by using an equivalent interpolating function. The key factor to consider when implementing the LPF or interpolating function is the ratio of fD,max to the transmission symbol rate (Rs). Thus for example in the case of a 10 MHz BW channel,
-4 the ratio of fD,max/Rs≅2.5×10 . It is difficult to generate a filter with a relative bandwidth of this value using conventional filter design techniques because of stability issues. In our case, we use the approach followed by [103] and [140]. We assume that the PDPs we generate are samples taken at fD,max (2500 Hz), and we then use interpolation to obtain the intervening samples. Note that we need to take care of the fact that after interpolation, the tap amplitude statistics don’t change. More details on the interpolation algorithm are in
Appendix A. Our interpolation algorithm can be used for multivariate Weibull and
Rayleigh random variables.
Algorithm for M1 Model
We describe the simulation algorithm for [M1, Large Airport, NLOS, 25]. The algorithm can be generalized to any airport size, region and BW channel model.
1. From Table 6.11, we determine the number of taps as 31. 253
2. Obtain the fading amplitude Weibull parameters b and a for each of the 31 taps.
NLOS Also, construct the correlation matrix, Rwc for the 31 taps.
3. Using the correlated Weibull simulation program (refer to [140] and Appendix A)
to generate the correlated Weibull tap amplitudes for 6299 CIRs ( the number of
PDPs recorded for MIA-NLOS, refer to Table 3.2).
4. Interpolate the CIRs using cubic interpolation to account for the desired Doppler
spread. For example, if the BW of the channel is 10 MHz (Rs/fD,max = 4000) and
the number of generated CIRs is 6299, these “original” CIRs will be interpolated
to obtain 4000×6299=2.5196×107 CIRs. The matrix generated is of size
2.5196×106 time samples per tap by 31 taps.
5. Generate the persistence process, z(t), for each tap. Each persistence process has
6299 samples. Increase the number of samples to 2.5196×107 by repeating each
sample 4000 times. The matrix generated is again of size 2.5196×107 time
samples per tap by 31 taps. Note that for the persistence process we can not use
interpolation to obtain the intervening states since z(t) is a two state Markov
process that takes only values one and zero. The persistence process generates
states (0 or 1) at a rate equal to the maximum Doppler frequency. Hence, using
the persistence process, we have modeled the physical effects which occur much
slower than the effects contributing to small scale fading.
6. Multiply the matrices obtained in steps 4 and 5, per tap, in element by element
fashion, to obtain the CIRs. In total, we have 2.5196×107 CIRs.
7. Generate uniformly distributed random variables between (-π ,π ) to model the
phase for each of the taps of the 2.5196×107 CIRs. In our measured data, within 254
a delay resolution of 20nsec (chip time), there are multiple plane waves (phasors)
contributing in the sample. Hence, it is okay to assume that the phases are
uncorrelated in time and hence we use uncorrelated phase samples in our
simulation.
6.5.1.2 Steps for Simulating the M2 Model
Assumptions:
Similar assumptions regarding Doppler frequency are made for M2 and M1. We describe the simulation algorithm for [M2, Large Airport, NLOS, 25]. For example, in
Miami, we collected data using 41 segments of travel. Table 6.34 shows the number of
CIRs that need to be generated for each segment of travel (Ni). Ni is the number of NLOS
PDPs collected at MIA in the ith segment of travel. We have also included the number of
CIRs obtained after interpolating the original CIRs. For the M2 model, we use an algorithm similar in principle to applying the M1 model 41 consecutive times. The algorithm can be generalized to any airport size, region and BW channel model. Here note that the number of taps is 125 (refer to Table 6.12). For each iteration of the M2
NLOS model, we have to use a different Rwc . The Weibull a and b factors for the taps remain the same in all iterations (although this could be changed at the discretion of the model user for future high-accuracy models).
Table 6.34 Number of CIRs generated for [M2, Large Airport, NLOS, 25]
Iteration Number of CIRs Number of CIRs after Index (i) Generated (N,i) Interpolation (M,i) 1 147 588000 2 63 252000 255
3 34 136000 4 103 412000 5 64 256000 6 256 1024000 7 226 904000 8 48 192000 9 161 644000 10 67 268000 11 46 184000 12 39 156000 13 27 108000 14 168 672000 15 94 376000 16 169 676000 17 110 440000 18 214 856000 19 58 232000 20 36 144000 21 63 252000 22 7 28000 23 23 92000 24 33 132000 25 34 136000 26 52 208000 27 382 1528000 28 439 1756000 29 456 1824000 30 30 120000 31 488 1952000 32 343 1372000 33 367 1468000 34 245 980000 35 574 2296000 36 121 484000 37 106 424000 38 96 384000 39 16 64000 40 39 156000 41 255 1020000
Algorithm for M2 Model
1. Determine the number of taps as 125 (refer to Table 6.12). 256
2. Obtain Weibull parameters b and a for each of the 125 taps.
th NLOS 3. Consider the i iteration. Load the corresponding correlation matrix, Rwc .
4. Generate N,i CIRs (refer Table 6.34), each having 125 taps, using the correlated
Weibull simulation program.
5. Interpolate the N,i CIRs to obtain M,i CIRs. We use cubic interpolation for this
simulation. The matrix generated is of size (M,i × 125).
6. Generate the persistence process, z(t), for each tap. Each persistence process has
N,i time samples. Increase the number of samples to M,i by repeating each sample
4000 times. The matrix generated is of size (M,i × 125).
7. Multiply the matrices obtained in steps 5 and 6 for each tap, in element by
th element fashion, to obtain the CIRs. In total, we have M,i CIRs for the i
segment.
8. Generate uniformly distributed random variables between (-π ,π ) to model the
phase for each of the taps of the M,i CIRs.
9. Repeat steps 3-8 for all iterations.
10. In total, this yields 41 data sets with different numbers of PDPs in each set.
6.5.1.3 Steps for Simulating M3 Model
Assumptions:
The same assumptions regarding Doppler frequency made for M2 are made for M3.
We will describe the simulation algorithm for [M3, Large Airport, NLOS, 25]. As discussed in Section 4.4, the number of taps considered for M3 is the same as its corresponding M2 model. So in this case we have 125 taps. Also, since M3 is a 257 statistically stationary version of the channel model, we don’t consider the persistence process and correlation matrices.
1. Determine the number of taps as 125 (refer to Table 6.12).
2. Obtain Weibull parameters b and a for each of the 125 taps.
3. Generate 5950 CIRs (refer to Table 4.2), each having 125 taps, using the
correlated Weibull simulation program. Note: Use an identity matrix for the
correlation matrix.
4. Interpolate the CIRs using cubic interpolation to account for Doppler spread. Thus
in this case when the number of generated CIRs is 5950, the original CIRs will be
interpolated to 2.38×106 CIRs. The matrix generated is of size 2.38×106 time
samples per tap by 125 taps.
6.5.2 Comparing Nonstationary and Stationary Large Airport Models
In this section, we compare the CIRs generated using the M2 and M3 models for the large airport for several values of BW. The first statistic that we compare is the RMS-
DS distribution of both models with those of the data. Figure 6.23 compares the histogram of the RMS-DS for the simulated PDPs for M2 and M3 with those of the data.
As is clear from the figure, M2 provides a more accurate representation of the measured data than M3. M3 appears to be much more pessimistic in terms of delay spread as than the data. M2 on the other hand models the data well in terms of shape and modes of the distribution. A similar figure comparing the histograms of Wτ,90 for M2 and M3 is provided in Figure 6.24. We have used a log scale for the y-axis only to make the plot 258 clearer. Similar to the conclusions drawn from Figure 6.23, we can infer the same from this figure as well—M2 better models the data than M3. Both figures thus illustrate the necessity of using nonstationary models to depict the physical propagation conditions accurately. To compare the shapes of the pdfs for Wτ,,90 and στ, we also provide statistics that are used to measure the “distance” of a "true" distribution (in our case the measured data denoted D) to that of the simulated model (denoted S). The different distance measures that we use are the Kullback-Leibler (KL) [141], Histogram Intersection (HI)
2 [142] and χ statistic (CHI). Consider Si and Di as the set of probability density values for the two pdfs, and then the following are the definitions for the various distances:
M ⎛ D ⎞ ⎜ i ⎟ KL = ∑ Di log2 ⎜ ⎟ i=1 ⎝ Si ⎠ M (6.16) HI = ∑ min()Di , Si i=1 M ()S − D 2 CHI = ∑ i i i=1 Di where M is the number of points in the pdf domain. From (6.16), we can state that if the pdfs are identical then KL and CHI should be zero, and HI should have value one. Table
6.35 compares these distance measures for the different statistics for various large airport models and various BWs. The models that produce statistics closest to those of the measured data are highlighted. For all BWs, M2 creates models that are closest to the data. 259
0.1 Data 0.09 M2 M3 0.08
0.07 τ σ 0.06
0.05
0.04 Probability of 0.03
0.02
0.01
0 0 500 1000 1500 2000 2500 σ in nsec τ Figure 6.23 Comparing the RMS-DS evaluated for the CIRs generated using [M2 M3, Large Airport] with measurement data
0 10 Data M3 -1 M2 10 ,90
τ -2 10
-3 10 Probability of W of Probability
-4 10
-5 10 0 1000 2000 3000 4000 5000 W in nsec τ,90 Figure 6.24 Comparing the Wτ,90 evaluated for the CIRs generated using [M2 M3, Large Airport] with measurement data 260
Table 6.35 Comparison of stochastic distance measures of M2 and M3 with those of data for large airports for varying bandwidth
25 MHz BW Model RMS-DS DW-90 KL CHI HI KL CHI HI M2 0.2641 0.3132 0.7893 0.2815 0.2560 0.7954 M3 1.0212 0.9589 0.6025 0.8733 0.7022 0.6373 20 MHz BW M2 0.2434 0.2667 0.7966 0.2816 0.2386 0.8119 M3 0.4359 0.4705 0.7622 0.523 0.4433 0.7179 10 MHz BW M2 0.2820 0.2459 0.7953 0.2509 0.2129 0.8133 M3 0.8025 0.5620 0.6695 0.6357 0.4901 0.6981
Next, we compare the accuracy of the developed M2 and M3 models in the frequency domain. In order to do so we evaluate the FCEs for the CIRs generated using the M2 and M3 models. We provide an example figure comparing the FCEs for a BW of
20 MHz in Figure 6.25. The closeness of both the models to that of the data is evident from this plot. We should of course note that in the case of the [M2 M3, Large Airport,
NLOS-S, 20] models, the model CIRs are more dispersive than in the case of the measured data, as seen from the RMS-DS histograms of Figure 6.23. This is due to the inherent characteristics of M2 and M3 (see Section 4.4) that aim to capture some of the worst case conditions by considering all the possible taps for the model. 261
1 M2-NLOS-S 0.9 M2-NLOS M3-NLOS-S 0.8 M3-NLOS 0.7 Data-NLOS-S Data - NLOS 0.6
0.5
0.4
Correlation Coefficient Correlation 0.3
0.2
0.1
0 -10 -5 0 5 10 Frequency in MHz
Figure 6.25 Comparing FCEs of [M2 M3, Large Airport, NLOS-S NLOS, 20] with that of data
6.5.3 Comparing Nonstationary and Stationary Medium Airport Models
In this section, we compare the CIRs generated using the M2 and M3 models for the medium airport for several values of BW. Figure 6.26 compares the histogram of the
RMS-DS for the simulated PDPs for M2 and M3 with those of the data. As in the case of the large airports, M2 provides a more accurate representation of the measured data than
M3. Due to the absence of the persistence process, M3 has all the taps ON all the time, and hence it provides a much more pessimistic representation in terms of delay spread than the data. M2 on the other hand, though slightly more dispersive than the data, models the distribution quite well in terms of shape and modes. A similar figure 262
comparing the histograms of Wτ,90 for M2 and M3 is provided in Figure 6.27. Use of the log scale again allows an easier visual discrimination of the plots. Similar conclusions regarding the comparison between the models can be made from this figure. As noted in previous sections and in Chapter 5, we need statistically non-stationary models to accurately capture the physical effects and provide a realistic depiction of the channel.
Table 6.36 compares the stochastic distance measures for the different statistics for various medium airport models and various BWs. The models that produce statistics closest to those of the measured data are highlighted. For all BWs, M2 creates models that are closest to the data.
0.16 Data 0.14 M2 M3 0.12 τ
σ 0.1
0.08
0.06 Probability of of Probability
0.04
0.02
0 0 500 1000 1500 2000 2500 σ in nsec τ Figure 6.26 Comparing the RMS-DS evaluated for the CIRs generated using [M2 M3, Medium Airport] with measurement data
263
0 10 Data M3 M2 -1 10 ,90 τ
-2 10 Probability of W
-3 10
-4 10 0 1000 2000 3000 4000 5000 W in nsec τ,90 Figure 6.27 Comparing the Wτ,90 evaluated for the CIRs generated using [M2 M3, Medium Airport] with measurement data
Table 6.36 Comparison of stochastic distance measures of M2 and M3 with those of data for medium airports for varying bandwidth
25 MHz BW RMS-DS DW-90 KL HI KL HI M2 0.3606 0.7321 0.4411 0.6965 M3 0.5968 0.6860 0.7262 0.6151 20 MHz BW M2 0.3633 0.7487 0.287 0.7637 M3 0.512 0.7081 0.6 0.6467 10 MHz BW M2 0.1363 0.8223 0.2251 0.7715 M3 0.3541 0.7233 0.5011 0.6398
A comparison of M2 and M3 for medium airports in the frequency domain is provided in Figures 6.28 and 6.29. We provide an example figure comparing the FCEs for a BW of 25 MHz in both figures. Again, the closeness of both the models to that of 264 the data is clear. As in the case of the large airports, we can see from Figure 6.28, the case of the [M2 M3, Medium Airport, NLOS-S NLOS, 25] models, the models are again slightly more dispersive than the measured data. For the LOS-O case of Figure 6.29, note that the number of taps used for the [M2 M3, Medium Airport, LOS-O, 25] models is 13
(see Table 6.19), thus the M2 and M3 LOS-O models do not yield much worst case behavior as found for the NLOS and NLOS-S regions.
1 M3-NLOS-S 0.9 M3-NLOS M2-NLOS 0.8 M2-NLOS-S Data-NLOS 0.7 Data -NLOS-S
0.6
0.5
0.4 Correlation Coefficient 0.3
0.2
0.1
0 -10 -5 0 5 10 Frequency in MHz
Figure 6.28 Comparing FCE of [M2 M3, Medium Airport, NLOS-S NLOS, 25] with that of data
265
1 M3-LOSO 0.9 M2-LOSO LOSO 0.8
0.7
0.6
0.5
0.4 Correlation Coefficient 0.3
0.2
0.1
0 -10 -5 0 5 10 Frequency in MHz
Figure 6.29 Comparing FCE of [M2 M3, Medium Airport, LOSO, 25] with that of data
6.5.4 Comparing Nonstationary and Stationary Small Airport Models
In this section, we compare the CIRs generated using the M2 and M3 models for the small airport for various BWs. Figure 6.30 compares the histograms of the RMS-DS for the simulated PDPs for M2 and M3 with those of the data. M2 and M3 both provide a realistic depiction of the measured data, but M2 is again closer to the data than is M3.
Observations similar to those reported for large and medium airports can again be made for small airports. A figure comparing the histograms of Wτ,90 for M2 and M3 is provided in Figure 6.31. Table 6.37 compares the stochastic distance measures for the different statistics for various small airport models and various BWs. The models that produce statistics closest to those of the measured data are highlighted. Once again, for all BWs,
M2 creates models that are closest to the data. 266
0.12 Data M2 0.1 M3
0.08 τ σ
0.06
Probability of Probability 0.04
0.02
0 0 500 1000 1500 2000 2500 σ in nsec τ Figure 6.30 Comparing the RMS-DS evaluated for the CIRs generated using [M2 M3, Small Airport, 25] with measurement data
0 10 Data M3 M2 -1 10 ,90 τ
-2 10 Probability of W
-3 10
-4 10 0 1000 2000 3000 4000 5000 W in nsec τ,90 Figure 6.31 Comparing the Wτ,90 evaluated for the CIRs generated using [M2 M3, Small Airport, 25] with measurement data
267
Table 6.37 Comparison of stochastic distance measures of M2 and M3 with those of data for small airports for several bandwidths
25 MHz BW RMS-DS DW-90 KL HI KL HI M2 0.3059 0.7346 0.4170 0.7263 M3 0.9003 0.6273 0.5580 0.6664 20 MHz BW M2 0.2779 0.7489 0.3747 0.7524 M3 0.7369 0.6603 0.6371 0.6611 10 MHz BW M2 0.2741 0.745 0.2131 0.7804 M3 0.6082 0.6736 0.3583 0.7063
Finally, a comparison of M2 and M3 for small airports in the frequency domain is provided in Figures 6.32 and 6.33. We provide an example figure comparing the FCE for a BW of 25 MHz in both figures. Again, the closeness of both the models to that of the data is clear. One thing to note from Figure 6.32 is that [M3, Small Airport, LOS-O, 25] creates a much more dispersive model than the LOS-O data, whereas [M2, Small Airport,
LOS-O, 25] produces simulated data that has frequency selectivity similar to that of the actual LOS-O data. Figure 6.33 compares the FCEs of the measured data from Tamiami with the FCEs computed using M2 and M3. As noted previously, the existence of NLOS regions on small airport is rare and it has been included here for completeness. 268
1
0.9
0.8
0.7
0.6 M2-NLOS-S M2-LOSO 0.5 M3-NLOS-S M3-LOSO 0.4 LOSO NLOS-S
Correlation Coefficient Correlation 0.3
0.2
0.1
0 -10 -5 0 5 10 Frequency in MHz
Figure 6.32 Comparing FCEs of [M2 M3, Small Airport, NLOS-S LOS-O, 25] with those of data
1 M2-NLOS 0.9 M3-NLOS NLOS 0.8
0.7
0.6
0.5
0.4
Correlation Coefficient Correlation 0.3
0.2
0.1
0 -10 -5 0 5 10 Frequency in MHz
Figure 6.33 Comparing FCEs of [M2 M3, Small Airport, NLOS, 25] with those of data 269
7 Airport Field Site and Point to Point Channel Models
In this chapter, we present our post-processing results for the measurement data collected for two different cases. For the first case, the transmitter is fixed at an Airport Field Site
(AFS) and the receiver traverses the airport surface; for the second case the transmitter is fixed on the ATCT and the receiver is fixed at the AFS. We term the measurements taken in the first case as AFS measurements while the second case is referred to as Point- to-Point measurements. The first section will provide a brief description of the measurement locations for the AFS and Point-to-Point measurements; the next section will compare the measurements collected while transmitting from an AFS or an ATCT.
The following two sections will then present the M1 models developed for specific scenarios at different airports while transmitting from the AFS. The final section will provide the results for the Point-to-Point measurements. All the M1 models provided in this Chapter are for a BW of 50 MHz, M1 models for other BWs are provided in [111].
Note: Some of the text, tables and figures in this chapter have been presented at
[132], [146] and [147].
7.1 Example AFS and Point-to-Point Measurement Locations
The first part of this section will describe the AFS measurements. By AFS we refer to some location on the airport surface where other electronic systems are located (e.g., landing systems near runway ends), and where antenna heights can not be large (e.g., at most ten meters). The AFS can serve different purposes, such as being a relay for signals
“obstructed” from the ATCT, a diversity node, a sensor data source node, etc. 270
Usually a main airport network transmitter is located at the ATCT. This serves as the ideal transmission location to reach most of the areas on the airport surface. There are though several places on airport surface areas that are difficult to reach when transmitting from the ATCT. Some of the reasons for this are blockage of signal due to the presence of a large obstruction (e.g., building), shadowing due to the ATCT itself, etc. To reach such “inaccessible” sites on the airport, future networks will transmit from AFSs on the airport surface. These field sites will be chosen so that they are within proximity of the
“inaccessible” sites. We can also predict that the channel encountered between the AFS and some “inaccessible” region will be less dispersive than the channel between this same region and the ATCT. This is due to the reduced link distance and less obstacles or potential scatterers in the link between the Tx and Rx. One thing to note is that even though we can find similar scattering geometries for airport of comparable sizes, the specific locations of AFS transmitters on the airport surface would be different for every airport. Hence, it is difficult to create truly generic AFS models for a specific airport class. The primary motivation behind this study was to gather some example channel data for the case when transmitting from different areas of the airport, and to also observe the effects of reducing the height of the transmitting antenna. In this dissertation, we will present the measurements taken at example locations at MIA and JFK. The specifi purpose for the AFS measurements is different for MIA and AFS. In the following paragraph, we will describe these purposes in detail.
The first location for our AFS transmitter was an ILS site at Miami International
Airport (MIA), Miami, Florida. Figure 7.1 shows a photograph taken from the AFS pointing towards the “inaccessible” regions. (Note that in the picture, a horn antenna 271 appears. For these AFS measurements an omnidirectional monopole was used.) The signal from the ATCT is obstructed due to buildings, aircraft and possibly other infrastructure (e.g., construction cranes). Any multiple reflections also significantly reduce the signal amplitude before it reaches the receiver. Hence, the physical media that the signal encounters from the ATCT is highly dispersive. In comparison, the link from the AFS is mostly a LOS channel, and hence the channel that the signal encounters will be far less dispersive. We also notice that the AFS can act as a very good relay location since it has a clear LOS to the ATCT. This kind of scattering geometry is common at airports which have large infrastructures like hangars, maintenance shelters for planes, etc. Hence, it is likely that similar environments will be present at medium airports as well. Also, locations of ILS electronics are generally near runway ends, and hence they can potentially serve as excellent relay points due to their LOS with the ATCT.
Figure 7.1 AFS location near ILS, MIA [146]
272
Figure 7.2 shows the example AFS at John F. Kennedy International Airport
(JFK), New York. Fig. 7.2(b) shows the JFK ATCT and the location of the transmitter used for the comparison measurements (ATCT to airport surface). In Figure 7.2(b), the
“shadowed regions” are the areas behind, and hence blocked, by the tower itself. The arrow above the label “shadowed region” indicates those areas. It is evident that there will be substantial signal attenuation in the region toward the back of the transmitter, due to this ATCT shadowing. Figure 7.2(a) shows the AFS transmitter at the ILS site. In
Figure. 7.2(a), the arrow indicates that the shadowed region (not visible in the photo of
Figure 7.2(a)) is in the LOS of the AFS transmitter. Similar situations are possible at other airports since the ATCT itself acts as the main blockage to the signals behind the transmitter. This problem can be resolved either by transmitting from the AFS or by having multiple transmitters on the ATCT to reach different regions of the airport.
Figure 7.2 Airport surface at JFK, (a) AFS location at ILS, (b) Shadowed Region while transmitting from ATCT. [146]
273
In addition to the communication taking place on the airport surface between vehicles and the ATCT, there are also large amounts of sensor data from runway sensors, airport surface detection equipment (ASDE), and other airport management devices which must be transferred in a very timely and extremely reliable manner. This time sensitive information is communicated from the AFS to the ATCT. At present, the bulk of this data is conveyed using wired media. Although highly reliable, wired media is prone to outages due to disruption by construction activities on the airport surface. One application for fixed point-to-point links is to transfer sensor data on the airport surface.
Wireless transport of the sensor data, and potentially some of the information currently carried by VHF links to/from mobiles, is an attractive option, and this has also provided motivation for study of the use of the MLS extension band for these purposes. The point- to-point links are from the ATCT to an airport field site, using directional antennas.
These antennas were either 10 dB or 20 dB standard gain horns. For these, the gains and beamwidths are given in Section 3.1.2. For measurements at Cleveland, two 10 dB horns were used. For the Miami measurements, one 10 dB (Tx) and one 20 dB horn (Rx) was used. No point to point measurements were made at JFK. The purpose of these measurements was to collect RSSI and PDP data for several fixed locations, both “on bore sight,” and as a function of azimuth angle, for eventual use in evaluating communication schemes for such fixed links. Table 7.1 provides some information regarding the measurement locations. 274
Table 7.1 Fixed point-point measurement locations [147]
Airport Location Salient Features Name CLE Radar Site • Good LOS condition, with two small buildings behind (Refer Figure Rx van 7.3) • 0 degrees: no observable multipath on Rx display • 90 degrees: strong multipath, possibly from a building reflection • 180 degrees: Rx main beam pointing directly away from, Tx so possible back lobe reception. Second multipath peak (larger delay) of greater strength than first peak • 270 degrees: Two nearly equal-strength peaks on Rx display. Likely reflection from the IX Center CLE Sensor • Good LOS location, one medium sized building NW of Location Rx van • 0 degrees: no observable multipath on Rx display • 90 degrees: 2 distinct peaks • 180 degrees: very low signal level, no significant reflections discernible • 270 degrees: small amount of observable multipath CLE RTR Site • RTR site north of airport across Brookpark Road, adjacent to some NASA Glenn buildings • Clear LOS, with small buildings ~ 10 m behind Rx van • 0 degrees: no observable multipath on Rx display • 90 degrees: very small multipath • 180 degrees: several peaks observable, likely some from the small buildings in the main Rx lobe • 270 degrees: 2 strong multipath peaks, possibly due to Rx main lobe pointing toward large NASA hangar to SW MIA P1 • “GEM” site, very clear LOS, with no buildings within 100 m • Measurements at 24 azimuth angles, separated by 15°
MIA ILS (P2) • ILS Site near American Airlines hangar (see Figure 7.6.) • Measurements at 24 azimuth angles, separated by 15° • “Blast fence” at approximately 150o azimuth from bore sight
275
A picture showing the Radar Site location at CLE is shown in Figure 7.3. The location of the Rx of the fixed point-to-point link is highlighted in the figure. The orientation of the antennas at one of the other point-to-point sites at CLE is shown in
Figure 7.4. The antenna patterns of the horn antennas can also be seen in Figure 7.4. As described previously, we rotate the horn-antenna at the receiver in increments of 30o to obtain some data on the surrounding scattering geometry. A similar figure for MIA fixed point-to-point links is presented in Figure 7.6. Note that in this figure, the beamwidth of the receiver horn antenna is narrower than that of the receiver horn-antenna at CLE.
Hence at MIA, we were able to spatially sample the scattering geometry with a finer resolution of 15o.
Figure 7.3 Radar measurement site at CLE. [147]
276
Figure 7.4 Antenna orientations at CLE fixed point-to-point locations [147]
Figure 7.5 Antenna orientations at MIA fixed point-to-point locations [147]
277
7.2 Comparison of Measurements for AFS and ATCT Transmission
In this section, we compare the results of the measurements taken on the airport surface while transmitting either from an AFS or from the ATCT. All the measurements reported in this section were collected at MIA. As described in the previous section, the
AFS measurements taken at MIA were to reduce the channel frequency selectivity over a certain area on the airport surface. For the purpose of comparison, the measurements taken while transmitting from the ATCT are denoted “Partial-ATCT,” since they pertain only to a small spatial region on the airport surface. The measurements taken while transmitting from the AFS are simply denoted AFS (and of course refer to the same receiving spatial region). When we compare the selectivity of the two channels (AFS to mobile receiver or ATCT to mobile receiver), we use the data obtained from identical travel segments (same receiver locations). In this section, we don’t provide a similar comparison for JFK since the section of the airport surface covered by the AFS in JFK was unreachable while transmitting from ATCT.
Figure 7.6 compares the RMS-DS histograms for these locations in MIA when transmitting from either the AFS or the ATCT. From Figure 7.6, we observe that most of the data collected using the AFS belongs to the NLOS-S category, whereas that for the
ATCT in these same portions of the airport surface area belongs to the NLOS category. It is obvious that we are able to reduce the channel dispersion in the “difficult-to-reach” locations on the airport surface by transmitting from an AFS. 278
Figure 7.6 Distributions of RMS-DS for transmission from both AFS and partial-ATCT, MIA [146]
Table 7.2 compares the RMS-DS statistics for both settings. As a single metric for comparison of these two settings—but not for use in actual channel modeling—we compute an “average number of channel taps,” Lavg, calculated using the following formula:
Lavg = PNLOS-S × LNLOS-S +PNLOS × LNLOS (7.1) where PX denotes the fraction of time in region X, and LX denotes the number of channel taps in region X. The number of taps L for each region was determined using
⎡⎤μστ Δτ + 1, where μστ is the mean value of RMS-DS, and ⎡y⎤ denotes the smallest integer greater than or equal to y.
Table 7.2 RMS-DS statistics for common receiving areas, with AFS and ATCT transmission at MIA 279
Region % RMS-DS % RMS-DS Average Profiles (nsec), Profiles (nsec), NLOS Number in NLOS-S in [Min, Mean, of Taps NLOS-S [Min, Mean, NLOS Max, L] Lavg Max, L] AFS 61 [8, 443, 39 [1000, 1625, 47 997, 24 ] 2451,83] ATCT 14 [60, 512, 86 [1002, 1524, 71 996, 27] 2228, 78]
From Table 7.2, we see that by virtue of the lower delay spread, use of an AFS can significantly reduce the complexity of the channel model. The reader may notice that the AFS results have a higher mean (and max) value of RMS-DS for the NLOS case; this is attributable to the low height of the AFS antenna as compared to that of the ATCT. The average number of taps, Lavg, required to model the channel is significantly smaller in the case of the AFS channel than in the ATCT channel. Similarly, we provide the RMS-DS statistics for AFS at JFK in Table 7.3. The number of taps is calculated using the mean-
RMS-DS. Note that for JFK only the AFS data is provided in Table 7.3 due to lack of
ATCT data over the specific part of the airport surface.
Table 7.3 JFK AFS transmission RMS-DS statistics
Region RMS-DS (nsec) Number of [Min, Mean, Taps (L) Max] NLOS-S [5.8, 317.3, 799.5] 17 NLOS [802, 1475, 2433] 75
The reduction of the channel dispersion can also be demonstrated by comparing the frequency correlation estimates (FCEs) of the AFS and partial ATCT channels. As 280 noted in the previous chapters, the FCE is analogous to a coherence bandwidth. The less dispersive the channel, the wider the FCE will be. Intuitively then, the FCE for the AFS should be wider than the FCE for the partial ATCT channel. The FCEs for the AFS and partial ATCT channel for MIA are provided in Figure 7.7. As expected, the FCE for the
AFS is much wider than that of the partial ATCT channel.
1 AFS 0.9 ATC
0.8
0.7
0.6
0.5
0.4
0.3 Correlation Coefficient Correlation
0.2
0.1
0 -25 -20 -15 -10 -5 0 5 10 15 20 25 Frequency in MHz
Figure 7.7 FCEs obtained with common receive areas, AFS and ATCT transmitters, MIA
7.3 AFS and Partial ATCT Channel Models for MIA
In this section, we present the M1 channel models for the AFS and partial-ATCT channels using data collected at MIA. For both models, the bandwidth is 50 MHz. From
Table 7.2, we observe that the number of taps required to model the channel is quite large. To reduce the complexity of the tapped delay line model, we reduce the number of 281 taps from this value based upon mean RMS-DS by taking into account the cumulative energy. This approach is similar to the algorithm presented in Section 4.4.1. Table 7.3 shows the cumulative energy with increasing numbers of taps for the NLOS and NLOS-S cases for AFS and ATCT transmissions at MIA. The average number of taps metric Lavg is again determined using the largest percentage energy values for each region, e.g., for
NLOS-S using the AFS we use 9 taps and 61% time in NLOS-S, and for NLOS using the
AFS we use 56 taps and 39% time in NLOS.
Table 7.4 Cumulative energy for AFS and partial ATCT for [M1, MIA, NLOS-S and NLOS, 50] [146]
Transmitti Taps Required for Taps required for Given % Energy, Average ng Site Given % Energy, NLOS Number of NLOS-S Taps for 85% 96% 99% 60% 70% 80% 85% 95% Site, Lavg AFS 3 6 9 1 3 9 20 56 27 ATCT 1 2 5 3 19 39 48 67 58
Table 7.5 provides the tap amplitude statistics for a 50 MHz AFS channel model for MIA for the NLOS-S region. Equation (7.2) provides the tap correlation matrix
AFS−NLOS −S ( Rwc ) for MIA.
Table 7.5 Amplitude statistics for [M1, MIA-AFS, NLOS-S, 50] [146]
Tap Weibull Tap P1 P00 P10 Index Shape Energy Factor (b) 1 4.47 0.748 1.0000 na 0 2 1.63 0.084 0.9396 0.1765 0.0530 3 1.7 0.020 0.8260 0.3714 0.1325 4 1.63 0.038 0.8651 0.2421 0.1183 5 1.66 0.054 0.7706 0.5542 0.1328 6 1.41 0.044 0.6982 0.5976 0.1741 7 1.84 0.007 0.5739 0.6450 0.2627 282
8 1.67 0.003 0.4169 0.7012 0.4174 9 1.81 0.002 0.3203 0.7952 0.4356
⎡ 1 0.1006 0.3386 0.5372 0.3214 0.5746 0.3520 0.3802 0.4561⎤ ⎢ ⎥ ⎢0.1006 1 0.0988 0.5860 0.0916 0.7418 0.5554 0.0880 0.0849⎥ ⎢0.3386 0.0988 1 0.7839 0.5875 0.0981 0.5803 0.2346 0.4681⎥ ⎢ ⎥ 0.5371 0.5860 0.7839 1 0.5971 0.6651 0.2568 − 0.0032 0.4402 ⎢ ⎥ (7.2) ⎢0.3214 0.0916 0.5875 0.5971 1 0.3441 0.3695 0.5799 0.5065⎥ ⎢ ⎥ ⎢0.5746 0.7418 0.0981 0.6651 0.3441 1 0.6098 0.2222 0.6555⎥ ⎢0.3520 0.5554 0.5803 0.2568 0.3695 0.6098 1 0.4238 0.5224⎥ ⎢ ⎥ ⎢0.3802 0.0880 0.2346 − 0.0032 0.5799 0.2222 0.4238 1 0.7985⎥ ⎢ ⎥ ⎣0.4561 0.0849 0.4681 0.4402 0.5065 0.6555 0.5224 0.7985 1 ⎦
Figure 7.8 provides curve fits for tap amplitude distributions for the 1st, 2nd, 3rd, and 7th taps for the AFS channel for [M1, MIA, NLOS-S, 50].
2 Tap2 Amplitude 1.2 Tap1 Amplitude Weibull: b = 1.63 Weibull : b = 4.47 1 1.5 m = 0.76 K = 9dB 0.8 1 0.6 Density Density 0.4 0.5 0.2
0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 Data Data Tap3 Amplitude 4 5 Tap 7 Amplitude Nakagami : m = 0.85 Nakagami : m =1.01 Weibull : b = 1.7 4 3 Weibull : b = 1.84 3 2 Density Density 2 1 1
0 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 Data Data
Figure 7.8 Amplitude histograms and curve fits for taps 1, 2, 3, and 7 for AFS channel [M1, MIA, NLOS-S, 50] [147]
Amplitude statistics for the partial ATCT channel model for a 50 MHz BW for this subset of areas on the airport surface are listed in Table 7.6. Equation (7.3) provides
PartialATCT −NLOS −S the tap correlation matrix ( Rwc ) for MIA. 283
Table 7.6 Amplitude statistics for partial ATCT channel [M1, MIA, NLOS-S, 50] [146]
Tap Weibull Tap P1 P00 P10 Index Shape Energy Factor (b) 1 5.56 0.859 1.0000 na 0 2 1.92 0.107 0.9725 0 0.0284 3 2.04 0.023 0.8791 0.2727 0.0943 4 2.2 0.009 0.6978 0.3889 0.2598 5 2.5 0.002 0.3956 0.6182 0.5775
⎡ 1 0.1498 − 0.115 0.0460 − 0.0367⎤ ⎢ ⎥ ⎢ 0.1498 1 0.2686 0.1391 − 0.2005⎥ ⎢− 0.1105 0.2686 1 0.4321 0.0712 ⎥ (7.3) ⎢ ⎥ ⎢ 0.0460 0.1391 0.4321 1 0.2281 ⎥ ⎣⎢− 0.0367 − 0.2005 0.0712 0.2281 1 ⎦⎥
Figure 7.9 provides example curve fits for the tap amplitude distributions of the 1st, 2nd, and 3rd taps for this partial ATCT channel for [M1, MIA, NLOS-S, 50].
Tap1 Amplitude data 1.2 Tap2 Amplitude data K = 10.4dB 1.5 Nakagami : m =0.96 Weibull : b = 5.56 1 Weibull : b= 1.92
0.8 1 0.6 Density Density 0.4 0.5 0.2
0 0 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Data Data 4 Tap 3 Amplitude data Nakagami : m =1.18 3 Weibull : b = 2.04
2 Density
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Data Figure 7.9 Amplitude histograms and curve fits for taps 1, 2, and 3 for partial ATCT channel [M1, MIA, NLOS-S, 50] [147]
284
Next we present the [M1, MIA, NLOS, 50] models for the AFS and Partial-ATCT channels. Due to the high number of taps, we can’t present the correlation matrices here and hence include them in the Appendix. Table 7.7 provides the tap amplitude statistics for a 50 MHz AFS channel model for MIA for the NLOS region.
Table 7.7 Amplitude statistics for AFS channel [M1, MIA, NLOS, 50] [147]
Tap Weibull Tap P1 P00 P10 Index Shape Energy Factor (b) 1 2.12 0.651 1.0000 na 0 2 1.49 0.064 0.8636 0.2581 0.1173 3 1.87 0.020 0.7613 0.2995 0.2200 4 1.76 0.026 0.7327 0.3416 0.2406 5 1.47 0.041 0.6810 0.4690 0.2492 6 1.5 0.018 0.6161 0.4900 0.3184 7 1.85 0.009 0.5237 0.5612 0.4000 8 1.8 0.006 0.4972 0.6039 0.4013 9 1.75 0.007 0.4631 0.6078 0.4537 10 1.69 0.006 0.4521 0.6767 0.3927 11 1.75 0.005 0.4268 0.6846 0.4253 12 1.88 0.005 0.4466 0.6793 0.3990 13 1.85 0.004 0.4048 0.6852 0.4647 14 2 0.004 0.4136 0.6811 0.4533 15 1.8 0.004 0.4026 0.7109 0.4301 16 1.77 0.005 0.4433 0.6818 0.4005 17 1.42 0.006 0.4345 0.7043 0.3858 18 1.67 0.005 0.4213 0.6882 0.4293 19 1.51 0.005 0.3916 0.6975 0.4719 20 1.85 0.004 0.4268 0.6865 0.4227 21 1.98 0.004 0.4136 0.6842 0.4495 22 1.78 0.004 0.4114 0.7191 0.4037 23 1.59 0.005 0.4169 0.6673 0.4670 24 1.74 0.005 0.4070 0.6877 0.4568 25 1.67 0.004 0.3872 0.6984 0.4786 26 1.82 0.004 0.3927 0.6987 0.4678 27 1.9 0.004 0.3949 0.7195 0.4318 28 1.88 0.005 0.4180 0.7216 0.3895 29 1.89 0.004 0.4037 0.7190 0.4169 30 1.93 0.004 0.4004 0.7132 0.4313 31 1.92 0.004 0.4081 0.7300 0.3935 285
32 1.94 0.005 0.4026 0.7072 0.4356 33 1.98 0.003 0.3762 0.7266 0.4516 34 1.74 0.003 0.3784 0.7465 0.4157 35 1.86 0.003 0.3509 0.7419 0.4796 36 2.35 0.002 0.3333 0.7388 0.5248 37 1.86 0.002 0.3421 0.7605 0.4630 38 2.01 0.002 0.3355 0.7297 0.5377 39 2.04 0.002 0.3234 0.7427 0.5408 40 1.95 0.002 0.3201 0.7553 0.5223 41 2.15 0.002 0.3223 0.7740 0.4778 42 2.11 0.002 0.3069 0.8076 0.4373 43 2.02 0.002 0.3234 0.7752 0.4728 44 2.13 0.002 0.3377 0.7671 0.4593 45 1.74 0.002 0.3322 0.7607 0.4801 46 1.74 0.003 0.3674 0.7509 0.4281 47 1.77 0.002 0.3245 0.7765 0.4678 48 1.7 0.002 0.3102 0.7875 0.4752 49 1.97 0.002 0.3113 0.7904 0.4629 50 1.92 0.002 0.3421 0.7521 0.4791 51 1.38 0.003 0.3465 0.7656 0.4444 52 1.54 0.002 0.3190 0.7654 0.5034 53 1.62 0.002 0.2915 0.7900 0.5132 54 1.75 0.002 0.3014 0.7918 0.4854 55 1.7 0.003 0.3102 0.7843 0.4823 56 1.48 0.003 0.3157 0.7874 0.4599 Figure 7.10 provides curve fits for tap amplitude distributions for the 1st, 2nd, 5th, and 23rd taps for the AFS channel model for [M1, MIA, NLOS, 50].
286
1.5 Tap1 Amplitude Data Tap2 Amplitude data 4 Nakagami: m = 1.19 Nakagami : m = 0.67 Weibull: b = 2.12 Weibull : b = 1.49
1 3
Densi t y 2 Densi t y 0.5
1
0 0.5 1 1.5 2 0 Data 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Data 7 7 Tap 5 Amplitude Data Tap 23 Amplitude Data 6 Nakagami : m = 0.65 6 Nakagami : m = 0.75 Weibull : b = 1.47 Weibull : b = 1.59 5 5
4 4 Densi t y 3 Density 3
2 2
1 1
0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 Data Data
Figure 7.10 Amplitude histograms and curve fits for taps 1, 2, 5, and 23 for AFS channel [M1, MIA, NLOS, 50] [147]
Amplitude statistics for the partial ATCT channel model for a 50 MHz BW NLOS region for this subset of areas on the airport surface are listed in Table 7.8.
Table 7.8 Amplitude statistics for partial ATCT channel [M1, MIA, NLOS, 50] [147]
Tap Weibull Tap P1 P00 P10 Index Shape Energy Factor (b) 1 1.95 0.537 1.0000 na 0 2 1.54 0.067 0.8386 0.3222 0.1306 3 1.94 0.021 0.7139 0.4357 0.2252 4 2.15 0.013 0.6395 0.5149 0.2739 5 1.95 0.008 0.5525 0.5562 0.3588 6 2.23 0.007 0.5193 0.5439 0.4231 7 1.99 0.006 0.5103 0.5982 0.3866 8 2.13 0.007 0.4987 0.5832 0.4198 9 2.1 0.006 0.5013 0.5730 0.4258 10 2.18 0.007 0.4888 0.6116 0.4055 11 2.17 0.006 0.4753 0.5959 0.4472 12 1.91 0.006 0.4753 0.6045 0.4377 13 2 0.006 0.4906 0.5785 0.4369 14 2 0.006 0.4709 0.6017 0.4466 15 1.93 0.006 0.4664 0.6178 0.4365 287
16 1.86 0.006 0.4861 0.6066 0.4170 17 1.97 0.007 0.4915 0.6343 0.3777 18 1.92 0.007 0.5094 0.5923 0.3915 19 1.75 0.007 0.4682 0.5963 0.4579 20 1.86 0.006 0.4789 0.6103 0.4232 21 2.12 0.005 0.4691 0.6091 0.4417 22 1.95 0.006 0.4834 0.6094 0.4164 23 2.18 0.005 0.4556 0.6106 0.4646 24 2.15 0.005 0.4350 0.6407 0.4660 25 2.27 0.005 0.4547 0.6293 0.4438 26 2.2 0.004 0.4332 0.6260 0.4907 27 2.32 0.004 0.4484 0.6319 0.4520 28 2.21 0.005 0.4493 0.6281 0.4551 29 2.23 0.005 0.4753 0.6045 0.4358 30 2.03 0.005 0.4565 0.6518 0.4134 31 1.94 0.005 0.4422 0.6634 0.4239 32 2.25 0.005 0.4547 0.6161 0.4596 33 1.95 0.005 0.4439 0.6220 0.4727 34 2.09 0.005 0.4368 0.6576 0.4403 35 2.05 0.005 0.4278 0.6771 0.4307 36 2.18 0.005 0.4430 0.6280 0.4665 37 2.16 0.005 0.4691 0.6351 0.4138 38 1.84 0.006 0.4646 0.6633 0.3868 39 2.36 0.005 0.4628 0.6137 0.4477 40 2.16 0.006 0.4789 0.6282 0.4034 41 1.77 0.007 0.4753 0.6513 0.3837 42 2.0 0.005 0.4439 0.6365 0.4566 43 1.93 0.007 0.4960 0.6275 0.3779 44 2.16 0.005 0.4601 0.6329 0.4297 45 2.14 0.006 0.4753 0.5908 0.4509 46 2.06 0.006 0.4816 0.6031 0.4264 47 2.12 0.005 0.4493 0.6401 0.4400 48 1.58 0.007 0.4646 0.6449 0.4081 49 1.58 0.007 0.4691 0.6132 0.4368 50 1.66 0.007 0.4601 0.6312 0.4316 51 1.74 0.005 0.4610 0.6356 0.4269 52 1.75 0.007 0.4798 0.6190 0.4139 53 2.01 0.006 0.4646 0.6007 0.4595 54 1.89 0.006 0.4726 0.6116 0.4326 55 1.99 0.005 0.4493 0.6450 0.4340 56 1.94 0.005 0.4511 0.6209 0.4602 57 1.99 0.005 0.4395 0.6256 0.4765 58 1.93 0.005 0.4413 0.6356 0.4603 59 2.2 0.005 0.4422 0.6479 0.4431 60 2.19 0.005 0.4664 0.6118 0.4432 288
61 2.06 0.005 0.4655 0.6252 0.4297 62 1.95 0.005 0.4511 0.6514 0.4235 63 2.1 0.005 0.4475 0.6211 0.4669 64 2.07 0.005 0.4610 0.6240 0.4386 65 1.94 0.005 0.4574 0.6109 0.4608 66 1.98 0.005 0.4359 0.6439 0.4598 67 1.93 0.005 0.4502 0.6324 0.4482
Figure 7.11 provides curve fits for the amplitudes of the 1st, 2nd, 19th, and 48th taps for the partial ATCT channel for the [M1, MIA, NLOS, 50] setting.
3 Tap 1 Amplitude Data 4 Tap2 Amplitude data Weibull: b = 1.95 Nakagami : m = 0.7 Lognormal: mean = 0.47, variance = 0.06 3 Weibull : b = 1.54 2 2 Density Density 1 1
0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 Data Data 10 10 Tap 48 Amplitude Data Tap19 Amplitude data 8 Nakagami : m = 0.77 Nakagami : m = 0.89 8 Weibull : b = 1.58 Weibull : b = 1.75 6 6 Density 4 Density 4
2 2
0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Data Data Figure 7.11 Amplitude histograms and curve fits for taps 1, 2, 19 and 48 for partial ATCT channel [M1, MIA, NLOS, 50] [147]
7.4 AFS Channel Models for JFK
In this section, we provide the channel model for [M1, JFK-AFS, NLOS and
NLOS-S, 50]. Figure 7.12 shows the cumulative energy for [M1, JFK-AFS, 50]. We use the information from this figure to reduce the number of taps to develop the M1 models.
Table 7.9 provides the tap amplitude statistics for a 50 MHz AFS channel model for JFK
AFS−NLOS −S for the NLOS-S region. Equation (7.4) provides the tap correlation matrix ( Rwc ) for JFK. 289
Total Energy 1.1
1
0.9
0.8 NLOS NLOS-S 0.7 Power
0.6
0.5
0.4
0 10 20 30 40 50 60 70 80 Tap-Index
Figure 7.12 Cumulative energy distribution for [M1, JFK-AFS, 50]
Table 7.9 Amplitude statistics for [M1, JFK-AFS, NLOS-S,50]
Tap Weibull Shape Tap P1 P00 P10 Index Factor (b) Energy 1 5.6 0.853 1.0000na 0 2 1.72 0.110 0.90950.1498 0.0841 3 1.71 0.021 0.76180.3835 0.1923 4 1.7 0.010 0.59970.5371 0.3085 5 1.82 0.003 0.38640.7308 0.4270 6 1.79 0.002 0.33650.7510 0.4902
⎡ 1 − 0.1023 0.0193 0.2135 0.2740 0.0653 ⎤ ⎢ ⎥ ⎢− 0.1023 1 0.3218 0.0933 0.1496 − 0.0929⎥ ⎢ 0.0193 0.3218 1 0.3040 0.1357 0.3645 ⎥ ⎢ ⎥ (7.3) ⎢ 0.2135 0.0933 0.3040 1 0.38 0.0805 ⎥ ⎢ 0.274 0.1496 0.1357 0.38 1 0.5107 ⎥ ⎢ ⎥ ⎣⎢ 0.0653 − 0.0929 0.3645 0.0805 0.5107 1 ⎦⎥ 290
Table 7.10 provides the tap amplitude statistics for a 50 MHz AFS channel model for
JFK for the NLOS region.
Table 7.10 Amplitude Statistics for [M1, JFK-AFS, NLOS, 50] [147]
Tap Weibull Tap P1 P00 P10 Index Shape Energy Factor (b) 1 1.61 0.398 1.0000 na 0 2 1.22 0.053 0.83850.4000 0.1156 3 1.57 0.018 0.74820.4667 0.1795 4 1.7 0.012 0.70520.4998 0.2091 5 1.7 0.011 0.68640.5138 0.2222 6 1.81 0.010 0.67710.5287 0.2249 7 1.91 0.010 0.67390.5485 0.2186 8 1.91 0.009 0.66360.5460 0.2303 9 1.89 0.010 0.66610.5342 0.2336 10 1.89 0.009 0.66440.5354 0.2347 11 1.93 0.009 0.66190.5624 0.2236 12 1.81 0.009 0.66360.5490 0.2287 13 1.85 0.009 0.65850.5612 0.2277 14 1.92 0.009 0.65260.5534 0.2378 15 1.81 0.008 0.63940.5819 0.2358 16 1.89 0.009 0.65400.5630 0.2314 17 1.92 0.009 0.65180.5590 0.2357 18 1.91 0.009 0.65040.5560 0.2387 19 1.93 0.009 0.65620.5600 0.2306 20 1.91 0.008 0.64750.5737 0.2322 21 1.94 0.008 0.64530.5773 0.2322 22 1.96 0.008 0.64370.5802 0.2325 23 1.82 0.009 0.64670.5735 0.2331 24 1.87 0.009 0.64930.5795 0.2271 25 1.92 0.008 0.64550.5820 0.2297 26 1.96 0.008 0.65230.5773 0.2254 27 2 0.008 0.64170.5700 0.2401 28 1.94 0.008 0.64120.5795 0.2354 29 2 0.008 0.64040.5827 0.2344 30 1.94 0.008 0.64520.5843 0.2288 31 2.02 0.008 0.63900.5807 0.2370 32 1.94 0.008 0.64010.5775 0.2377 33 1.9 0.008 0.64440.5953 0.2235 291
34 1.95 0.008 0.63500.5947 0.2331 35 1.95 0.008 0.63820.5910 0.2320 36 1.98 0.008 0.63720.5894 0.2338 37 1.98 0.008 0.64390.5710 0.2373 38 1.95 0.008 0.63390.6016 0.2302 39 1.96 0.008 0.64560.5985 0.2205 40 1.91 0.008 0.63910.5844 0.2348 41 1.96 0.008 0.63320.5960 0.2339 42 2.05 0.008 0.64550.5886 0.2259 43 2.02 0.008 0.63670.5806 0.2394 44 1.98 0.008 0.63600.5961 0.2313 45 1.98 0.008 0.63970.5963 0.2273 46 1.99 0.008 0.63230.6015 0.2319 47 1.96 0.008 0.63970.5963 0.2275 48 1.95 0.008 0.63740.5993 0.2281 49 2.02 0.007 0.63500.6004 0.2298 50 1.89 0.007 0.63270.6034 0.2303 51 1.98 0.008 0.63420.6011 0.2300 52 1.96 0.008 0.63830.5969 0.2285 53 2.02 0.008 0.64460.6156 0.2119 54 1.89 0.008 0.64000.6015 0.2242 55 1.94 0.007 0.63240.6071 0.2284 56 1.87 0.008 0.64340.5846 0.2303 57 1.96 0.008 0.63900.5874 0.2332 58 1.92 0.008 0.63170.5909 0.2386 59 1.92 0.007 0.62480.6022 0.2387 60 1.93 0.007 0.63760.6011 0.2268 61 1.95 0.008 0.63120.6023 0.2325 62 1.98 0.007 0.63030.6140 0.2265 63 1.96 0.007 0.62370.6131 0.2335 64 1.95 0.007 0.62200.6241 0.2285 65 1.91 0.007 0.62790.5956 0.2397 66 1.93 0.008 0.62860.5882 0.2434 67 1.94 0.007 0.63450.5957 0.2331 68 1.95 0.007 0.63280.5964 0.2343
7.5 Point-to-Point Results for CLE and MIA
Figure 7.13 is an example 3-D representation of a sequence of PDPs collected at a field location (an instrument landing system—ILS) in MIA. The significant difference as 292 compared to the PDPs measured for fixed-mobile links is the presence of stable (i.e., non-fading) multipath at various delays (0.3, 2.5, and 4.1 microseconds). Because of the antenna directionality, multipath was significantly attenuated, so these channels are far less dispersive, and far less time varying, than the mobile channels.
-80
-90
-100
-110
Power in dBm -120
-130 1.5 6 1 5 4 0.5 3 Time in sec 2 1 0 0 Delay in μsec
Figure 7.13 Measured PDPs: power vs. delay (in microseconds, rightward axis) and vs. time (in seconds, leftward axis) for field (ILS) site in MIA at 105o orientation with respect to ATCT [132]
Figure 7.14 shows the time varying power spectrum for the point-to-point channel with an orientation of 15° off “boresight.” There is some slow time variation of the frequency- domain characteristics, but the channel amplitude is relatively flat over several MHz. 293
Figure 7.14 Time varying power spectrum (power vs. frequency in MHz, rightward axis) and vs. time (in seconds, leftward axis) for field site (ILS) in MIA, 15o orientation with respect to ATCT [132]
As explained before, for point-to-point measurements, in addition to measuring the channel when both antennas were aimed at each other (boresight), we also took measurements when the receiver antenna, located at the field site, was rotated through the full 360° in azimuth. This provides some information on potential multipath sources, and the variation of received power versus azimuth angle. This information is critical if diversity links are planned at the field site to provide back-up to the boresight link.
Figure 7.15 shows the power distribution versus azimuth angle for the two locations (P1 and P2) at MIA. Notice that for P2, the RSSI collected at angle 150o is greater than that at boresight, by approximately 2 dB. As mentioned in Table 7.1, there is a large blast 294 fence at approximately that angle. Reflection of energy from that fence increases the amount of scattered energy collected at the receiver.
0 0 Point 1 Point 2 -5 -5
-10
-10
-15
-15
Relative Power (dB) -20 Relative Power (dB)
-20 -25
-30 -25 -200 -150 -100 -50 0 50 100 150 200 -200 -150 -100 -50 0 50 100 150 200 Angle (degree) Angle (degree)
Figure 7.15 Power distribution versus azimuth angle for MIA fixed point-to-point links [147]
Table 7.11 shows the RMS-DS for the fixed point-to-point locations at CLE, for four azimuth angles. With directional antennas, even the maximum RMS-DS is not very large (312 nanosec).
Table 7.11 RMS-DS for fixed point-to-point locations at CLE.
Location RMS Delay Spread (nanoseconds) for Four Azimuth Angle (link distance) Orientations of Receive Antenna (0° is boresight) 0° 90° 180° 270° 1 (1.4 km) 31.8 101 70 273 2 (3.3 km) 48.2 101 170 312 3 (1.3 km) 40 294 239 146
Figure 7.16 shows the RMS-DS distribution as a function of angle for the two
MIA fixed point-to-point locations. Points 1 and 2 were referred to as “GEM” and
“MFA,” respectively, by the airport authorities. 295
1800
GEM Site 1600 MFA Site
1400
1200 P1
1000
800 RMS in nsec 600 400 P2 200
0 0 50 100 150 200 250 300 350 Angle in degrees
Figure 7.16 RMS-DS for MIA fixed point-to-point links versus azimuth angle [147]
Figure 7.17 provides the FCEs for all three locations in CLE, and also the aggregate (average) FCE. The broad main lobe of the FCE is an indication of small channel dispersion for the fixed point-to-point links.
1 Point 3 0.9 Point 2 Point 1 0.8 CLE Aggregate
0.7
0.6
0.5
0.4
0.3 Correlation Coefficient Correlation
0.2
0.1
0 -25 -20 -15 -10 -5 0 5 10 15 20 25 Frequency in MHz
Figure 7.17 FCEs for CLE fixed point-to-point links for boresight [147] 296
Figure 7.18 shows the FCEs for the fixed point-to-point measurements at locations P1 and P2 in MIA. We also provide the aggregate FCE in the figure. Since we used a higher gain antenna for the receiver at MIA than at CLE (see Table 4.1), the main lobes of these FCEs are even wider than those of CLE.
1 Point-2 Point-1 Aggregate FCE 0.9
0.8
0.7
0.6 Correlation Coefficient Correlation
0.5
0.4 -25 -20 -15 -10 -5 0 5 10 15 20 25 Frequency in MHz
Figure 7.18 FCEs for MIA fixed point-to-point links for boresight [147].
The channel model for a fixed point-to-point link generally consists of a strong specular component which has a very large Ricean “K factor.” The reasons for this are the lack of mobility and the use of high gain directional antennas for our measurements.
Table 7.12 shows the estimated Ricean K factors for the first taps of the channel models, for the boresight direction, at the CLE sites, for different channel bandwidths. All the
CLE point-to-point channels have L=1 tap.
297
Table 7.12 First tap channel Ricean K-factors at CLE for different locations and bandwidths for boresight links [147]
Location K factors (dB) for Different Channel Bandwidths (MHz) 50 10 P1 15 14 P2 14.1 12 P3 15 14 Table 7.13 shows the Ricean K factors for the first taps of these channel models, for various azimuth angles, at the MIA sites, for different channel bandwidths. (Note that for
P1 here, a second tap with K=19 dB exists, with energy 5.6 dB down from the first tap, but only for the 50 MHz bandwidth; L=1 for all smaller values of bandwidth.)
Table 7.13 First tap channel Rician K-factors at MIA (point 1, point 2) for different bandwidths [147]
Ricean K-factors (dB) for given Channel Bandwidth (MHz) Azimuth 5 10 50 Angle (°) 0 (24.4, 23) (25.0, 23.1) (24.3, 24.5) 90 (17.3, 13.1) (17.2, 15.9) (20.4, 18.6) 150 (18.5, 25.9) (16.8, 21.3) (19.8, 24.9) 180 (15, 20.2) (17, 22) (22, 23.6) 270 (11, 18.8) (11.6, 21.3) (15.6, 17.2)
298
8 VTV Channels
In this chapter, we present our post-processing results for the VTV measurement data collected at various locations. We have already provided a detailed discussion regarding the measurement locations in Chapter 3, so in this chapter we shall mainly focus on the processing results. Also, the results will be presented in a similar order as used in
Chapter 6. Several statistical channel models are presented, and using simulation results we elucidate tradeoffs between model implementation complexity and fidelity. The channel models presented should be useful for system designers in future VTV communication systems. Note: Some of the text, tables and figures in this chapter have been presented at [144], [148], [149] and [150].
8.1 Common Statistical Parameters
8.1.1 Time and Frequency Domain Statistics
As discussed previously, we have parsed our VTV data into several regions;
UOC, UIC, Small, OHT and OLT. In the three large cities, over 6,000 power delay profiles (PDPs) were gathered. We obtained over 500 PDPs in the small city, and over
5,000 PDPs on the highways. After observing PDPs from all these categories, it was determined that the channel for the UIC case was the most severe in terms of dispersion.
This agrees with intuition, since in addition to the reflections and scattering from buildings and other traffic present between the vehicles, the reduced antenna elevation 299
(dashboard instead of vehicle roof) further reduces the likelihood of a specular or line of sight (LOS) signal between transmitter and receiver. When the antenna was mounted outside the car, the amount of LOS blockage was reduced, but not completely eliminated
(e.g., during turns around corners, or when large vehicles came between transmitter and receiver). For the small city, the nature of the physical obstacles is similar to that of large cities, but the dimensions of some obstacles are smaller. Channel conditions on the highway can vary widely depending on the time of the day, the traffic conditions (density of vehicles), and also depending on the road size. For example state roads (like US-33) exhibit different characteristics than interstate highways. Due to this, we have found it best to differentiate the highway channel models into low traffic and high traffic densities. As one would expect intuitively, the channel for the OLT case is characterized by the presence of a very strong specular component and is the least dispersive.
The first set of results that we present are the RMS-DS distributions for the different
VTV regions. Again, the RMS-DS provides an idea regarding the frequency selectivity of the underlying physical channel. Figure 8.1 shows the cumulative distribution function of RMS-DS for the five regions. As expected, στ is largest for UIC and smallest for OLT. The 90th percentile value for UIC is close to 600 nsec whereas it is approximately 125 nsec for OLT. The UOC and OHT environments provide delay spreads comparable to those for the UIC case. The RMS-DS values for the small city are between those of OLT and UOC. The presence of larger inter-vehicle distances on the open (highway) environments allows larger delay spreads. In the highway measurements, the transmitted signals could be reflected from large trucks and buses at large distances without much loss in strength. Table 8.1 shows the minimum, mean and maximum 300
values of στ for all regions. As noted previously, UIC is the worst channel in terms of the mean RMS-DS and OLT is the least dispersive channel. Another interesting thing to note is that even though the maximum RMS-DS is the largest for OHT, the mean RMS-DS for
OHT is smaller than that of UIC. This is because even though there are instances when long delay multipaths affect the channel for OHT, the percentage of time that this phenomenon occurs is lower than in the UIC case.
1 UOC 0.9 UIC OHT 0.8 OLT Small ) τ
σ 0.7
0.6
0.5
0.4
0.3 Probability ( abcissa < 0.2
0.1
0 0 200 400 600 800 1000 1200 1400 1600 1800 σ in nsec τ Figure 8.1 Cumulative distribution functions of RMS-DS for five regions [144]
Table 8.1 Summary of measured RMS-DS values for five regions [144]
Region RMS-DS (ns) [min; mean; max] UIC [2.7; 236; 1210] UOC [3; 125.8; 1328.6] OLT [0.3; 53.2; 1113.4] OHT [0.5; 126.8; 1773.4] Small [0.7; 160; 1276]
301
The next statistic that we present is the delay widow (Wτ,x ) for 90% of the total
CIR energy. Figure 8.2 shows the cumulative distribution function of Wτ,,90 for the five regions. As expected, the value of Wτ,,90 for the UIC case is largest.
1 UOC UIC 0.9 OHT OLT
) 0.8 Small ,90 τ
0.7
0.6
0.5
0.4 Probability(abcissa < W
0.3
0.2 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 W in nsec τ,90 Figure 8.2 Cumulative distribution functions of Wτ,,90 for five regions [144]
Next we provide a frequency domain characterization of the frequency selectivity of the channel. We use the same definition for our FCE as described in the previous chapters. Figures 8.3 and 8.4 present the FCEs for all the regions for a bandwidth of 5 and 10 MHz respectively. As with our interpretations regarding the frequency selectivity of the channel on the basis of RMS-DS, the FCE also provides similar conclusions. For both bandwidths, OLT is the least dispersive channel. (Note that for the case of UIC, due to limited data, we are not statistically confident in its FCE. For the 10MHz case, UIC appears to be the most dispersive which agrees with our intuition but for the 5MHz case, 302
UIC appears to be less dispersive than UOC and OHT). The frequency selectivity for
OHT and UOC are comparable and are more dispersive than Small region. As an example of interpretation, for a 5 MHz BW, for the OHT case, the correlation is approximately 0.7 at frequency separation of 1.3 MHz; thus the coherence bandwidth for this case is approximately 2.6 MHz.
1 OHT UOC 0.9 UIC Small OLT 0.8
0.7
0.6
Frequency Correlation Coefficient Correlation Frequency 0.5
0.4 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Frequency in MHz
Figure 8.3 Frequency Correlation Estimates for five regions for a 5 MHz bandwidth [144]
303
1 OLTR 0.9 OHTR Small 0.8 UIC UOC 0.7
0.6
0.5 FCE
0.4
0.3
0.2
0.1
0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Frequency in MHz
Figure 8.4 Frequency Correlation Estimates for five regions for a 10 MHz bandwidth [149]
8.1.2 Channel Tap Properties
As discussed in Chapter 4, three key parameters (Section 4.3.2.2) of the taps for a stochastic channel model are the steady state tap probability, tap transition probability and the individual energy contributed by the tap (the remaining key parameter is the
Weibull fading (b) factor). The former two are critical to model the non-stationary behavior of the channel model which we capture using the persistence matrix. We have also discussed that to reduce the complexity of the tapped delay line model, it is crucial to know the aggregate energy contributed as the number of taps increases. In this section, we present results for the tap probability and the cumulative energy contributed by different taps. Note that all the results in Section 8.1.3 pertain to both M1 and M2 304 models. Recall from Section 4.4.2 that in the VTV case, for both M1 and M2, the number of taps is determined using the maximum RMS-DS; but for M1, we reduce the number of taps to account for 99% of the aggregate energy, whereas for M2, we consider all the taps.
As discussed in Chapter 4, Figure 8.5 shows example persistence processes associated with the 5th and 8th taps, for a segment of measurement data from the UOC setting. We also show the TS and SS matrices for both the taps in the figure. These matrices have been defined in Chapter 4. As expected, tap 5 is ON for a longer percentage of time than is tap 8. For the TS matrix, we observe that P00 and P10 increase for higher-index taps. This is also in agreement with intuition that the higher-index taps
(associated with longer delays) will remain “off” for longer periods of time. The Markov modeling parameters are extracted directly from all the PDP data of a given region.
2 ⎡0.5187 ⎤ ⎡0.6955 0.3045 ⎤ 1.5 SS 5 = ⎢ ⎥ TS 5 = ⎢ ⎥ ⎣0.4813 ⎦ ⎣0.3284 0.6716 ⎦
1
ON/OFF 0.5
0 1 5 10 15 20 25 30 35 40 45 PDP Index
2 ⎡0.8447 ⎤ ⎡0.9136 0.0864 ⎤ SS 8 = TS 8 = 1.5 ⎢0.1553 ⎥ ⎢0.4696 0.5304 ⎥ ⎣ ⎦ ⎣ ⎦ 1
ON/OFF 0.5
0 1 5 10 15 20 25 30 35 40 45 PDP Index
Figure 8.5 Example persistence processes for taps 5 and 8 for segment of travel for the UOC case, 10 MHz BW
305
Figure 8.6 shows the measured tap probability of occurrence versus tap index
(100 ns) for a bandwidth of 5 MHz. It is obvious that the UIC case is the most dispersive and has larger delay multipath for the largest percentage of time. It is clear that the steady state probabilities depend on the type of fading environment. The trend in the steady state probabilities agrees with our intuition: UIC, UOC, Small City, OHT, and
OLT appear in descending order of dispersion, where dispersion is roughly the number of taps and their probabilities. Least-squares curve fits for these probabilities of occurrence curves fit the general form,
P( k ) = c0 exp( −c1k )+ c2 , (8.1) where k is the tap index, and the c’s are curve fit constants. Table 8.2 lists these constants for the different regions, and also the maximum number of taps in each case. As with
Figure 8.6, from Table 8.3, we observe that coefficient c1 increases as the channel dispersion of the regions decrease in the order UIC, UOC, OHT and OLT. We also plot the fits for OHT and UIC in Figure 8.6. Similar curves for tap probability of occurrence as a function of tap index for the 10 MHz case are shown in [149]. 306
OLT Data 1 OHT Data OHT- LS Fit UIC Data 0.8 UIC- LS Fit UOC Data Small Data
0.6
0.4
0.2 TapProbability of occurance
0
-0.2 1 2 3 4 5 6 7 8 9 10 Tap Index Figure 8.6 Steady state tap probability for state 1 (P[tap “on”]) vs. tap index, all regions, 5 MHz bandwidth [144]
Table 8.2 Least-squares fit parameters for tap probability of occurrence, eq (8.1), all regions, 5 MHz bandwidth
Region Eq. (8.1) LS Fit Parameters [c0; c1; c2; kmax] UIC [1.9047; 0.0980; -0.6699; 8] UOC [1.6706; 0.2347; -0.2561; 8] OHT [1.5916; 0.4167; -0.0097; 10] OLT [1.7750; 0.4400; -0.0974; 7] Small [1.8160; 0.3890; -0.1477; 8]
Figure 8.7 shows the averaged measured tap energy versus tap index (100 ns) for a bandwidth of 5 MHz. This can be considered similar to the averaged power delay profile (APDP) which is often used to describe the spread of energy in delay for a given channel [124]. From Fig. 8.7, it is evident again that the energy contribution of the taps depends on how dispersive the setting is. The trend is the same as in Fig. 8.6. The small increase in energy for higher-index taps for the small city for indices 6 and 7 is 307 attributable to the small number of PDPs with which the average energies were computed for these taps. Similar to the tap probability of occurrence, the least-squares curve fits for these tap energies fit the general form,
E(k) = c3 exp(−c4k) + c5 . (8.2)
Table 8.3 lists these constants for the different regions, and a 5 MHz bandwidth. A trend similar to that observed in Table 8.2 is also applicable for Table 8.3. A similar figure for the 10 MHz case appears in [149].
0
OLT- LS Fit -5 OLT Data OHT Data OHT- LS Fit -10 UIC- LS Fit UIC Data UOC- LS Fit -15 UOC Data Small Data -20 Small- LS Fit
-25
-30 Tap EnergyTap in dB
-35
-40
-45
-50 1 2 3 4 5 6 7 8 9 10 Tap Index
Figure 8.7 Averaged tap energies with LS fits, all regions, 5 MHz bandwidth [144]
308
Table 8.3 Least-squares fit parameters for average tap energy, eq (8.2), all regions, 5 MHz bandwidth [144]
Region Eq. (9) LS Fit Parameters [c3; c4; c5] UIC [32.5; 0.383; -22.47] UOC [42.05; 0.4357; -27.58] OHT [45.71; 0.4419; -29.9] OLT [54.13; 0.58; -32.1] Small [71.22; 0.28; -53.17]
Figure 8.8 shows the cumulative energy as a function of tap index for all the regions for a 5 MHz bandwidth.
1 OHT 0.98 UOC UIC 0.96 Small OLT 0.94
0.92
0.9
0.88 Cumulative Energy 0.86
0.84
0.82
0.8 1 2 3 4 5 6 7 8 9 10 Tap-Index
Figure 8.8 Cumulative energy versus tap index for all regions for 5 MHz bandwidth [144]
309
For all the regions, the cumulative energy curve flattens as it approaches unity.
Hence, to reduce implementation complexity, we only consider the taps that contribute to
99% of the energy for M1. A similar figure for the 10 MHz bandwidth is presented in
[149]. For each channel category, the tap energies are normalized such that the total energy is 1. From Fig.8.8, there are two points worth noting. First, the rate at which the energy is gathered with increasing tap index is inversely proportional to the channel dispersion (~RMS-DS), and second, as the tap index increases, its corresponding energy contribution decreases.
8.2 VTV Channel Models
In this section, we present the channel models (M1, M2 and M3) for the two different BW (5 and 10MHz) for all the VTV regions. These bandwidths were selected based upon bandwidths of likely wireless technologies that may be used for VTV applications [132], [23].
8.2.1 [M1, All regions-VTV, 5 and 10]
Recall from Figure 4.22 that the first step in developing the channel model is to determine the number of taps. For this model, we use the maximum RMS-DS to determine the number of taps (L). Thus using Table 8.2, we have the number of taps for all the regions. As discussed in Chapter 4, to alleviate the complexity of realizing this channel model, we can further reduce the number of taps without much loss in fidelity.
Figure 8.8 shows the cumulative energy versus tap index for all the regions for a 5 MHz 310
BW. Clearly, the “law of diminishing returns” is in effect as the number of taps increases. As a suitable threshold, we consider the number of taps LVTV that accounts for
~99% of the energy. By doing this we can substantially reduce the model complexity. A similar reduction in complexity can also be performed for the 10 MHz BW models.
Figures illustrating that case are provided in [144]. Tables 8.4 and 8.5 provide the parameters that can be used to simulate the tap amplitude fading processes for all the regions for a 10 and 5 MHz BW, respectively. These tables contain the fading amplitude parameter (b) and tap energy, which together can be used to specify the Weibull density to model amplitude fading. The TS and SS matrix parameters (Recall Section 4.2.1) are also provided to enable generation of the Markov models for each tap.
Table 8.4 Tap Amplitude parameters for [M1, All regions –VTV, 10] [144]
Tap Energy Weibull P00,k P11,k P1 Index Shape k Factor (bk) UOC 1 0.88 3.19 na 1.0000 1.0000 2 0.08 1.61 0.2717 0.9150 0.8956 3 0.03 1.63 0.4401 0.8171 0.7538 4 0.01 1.73 0.5571 0.7488 0.6382 Small City 1 0.90 3.95 na 1.0000 1.0000 2 0.08 1.91 0.4839 0.9446 0.9034 3 0.02 2.02 0.3452 0.7712 0.7383 OHT 1 0.95 4.3 na 1.0000 1.0000 2 0.04 1.64 0.3625 0.8366 0.7960 3 0.01 1.89 0.5999 0.6973 0.5696 OLT 1 0.96 5.15 na 1.0000 1.0000 2 0.04 1.63 0.3836 0.8525 0.8073 UIC 1 0.756 2.49 na 1.0000 1 2 0.120 1.75 0.0769 0.9640 0.9625 311
3 0.051 1.68 0.3103 0.8993 0.8732 4 0.034 1.72 0.3280 0.8521 0.8199 5 0.019 1.65 0.5217 0.7963 0.7017 6 0.012 1.6 0.6429 0.7393 0.5764 7 0.006 1.69 0.6734 0.6686 0.4971
Table 8.5 Amplitude parameters for [M1, All regions –VTV, 5] [144]
Tap Energy Weibull P00,k P11,k P1 Index Shape k Factor (βk) UOC 1 0.92 3.21 na 1.0000 1.0000 2 0.07 1.57 0.2952 0.9003 0.8761 3 0.02 1.74 0.5415 0.7550 0.6518 Small City 1 0.94 3.91 na 1.0000 1.0000 2 0.06 1.77 0.3400 0.8815 0.8442 OHT 1 0.96 4.51 na 1.0000 1.0000 2 0.04 1.63 0.4455 0.8324 0.7680 OLT 1 0.97 4.71 na 1.0000 1.0000 2 0.03 1.64 0.3803 0.8084 0.7643 UIC 1 0.824 2.42 na 1.0000 1 2 0.116 1.65 0.1026 0.9465 0.9438 3 0.045 1.69 0.3333 0.8646 0.8314 4 0.016 1.63 0.5880 0.7418 0.6153
By comparing the number of taps that we have used to model the different regions
(Tables 8.4 and 8.5) and by comparing them to the number of taps that would be required to model using the maximum RMS-DS (Table 8.2), we can conclude that we have significantly reduced the model complexities. What is important to understand is the tradeoff between this reduction in complexity and loss of fidelity. We address this issue in the following section. Finally in Figure 8.9, we provide example distribution fits for two tap amplitude statistics. 312
2 UOC Tap2 Data Small Tap 2 Data Weibull: beta = 1.61 1.8 Weibull : beta = 1.91 2 Nakagami : m = 0.73 Nakagami: m = 0.97 1.6
1.4 1.5 1.2
1 Density 1 Density 0.8
0.6
0.5 0.4
0.2
0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 Data Data (a) (b) Figure 8.9 Histograms and probability density function fits for 2nd tap of UOC, and Small City, for 10 MHz bandwidth [144]
V 2V −Re gion The following Tables 8.6 –8.10 contain the correlation coefficients ( Rwc ) for the different regions. Since the correlation coefficient matrix is symmetric about the diagonal, we need only specify the upper or lower triangular part; for brevity, in these tables, the lower triangular matrix corresponds to correlation between taps for a 10 MHz bandwidth, whereas the upper triangular part of the matrix corresponds to the correlations between taps for a 5 MHz bandwidth.
Table 8.6 Correlation coefficient matrices for [M1, UOC, 10 and 5]; lower triangular part for 10 MHz, upper triangular part for 5 MHz
i , j 1 2 3 4 1 1 0.7243 0.5528 na 2 0.6898 1 0.5260 na 3 0.6518 0.4922 1 na 4 0.5772 0.5142 0.8479 1
313
Table 8.7 Correlation coefficient matrices for [M1, Small, 10 and 5]; lower triangular part for 10 MHz, upper triangular part for 5 MHz
i , j 1 2 3 1 1 0.0293 na 2 0.0338 1 na 3 0.6813 0.0684 1
Table 8.8 Correlation coefficient matrices for [M1, OHTR, 10 and 5]; lower triangular part for 10 MHz, upper triangular part for 5 MHz
i , j 1 2 3 1 1 0.3953 na 2 0.5441 1 na 3 0.4157 0.1707 1
Table 8.9 Correlation coefficient matrices for [M1, OLTR, 10 and 5]; lower triangular part for 10 MHz, upper triangular part for 5 MHz
i , j 1 2 1 1 0.0754 2 0.1977 1
Table 8.10 Correlation coefficient matrices for [M1, UIC, 10 and 5]; lower triangular part for 10 MHz, upper triangular part for 5 MHz
i , j 1 2 3 4 5 6 7 1 1 0.2353 0.1798 0.2590 na na na 2 0.1989 1 0.2447 0.2671 na na na 3 0.0555 0.1477 1 0.2436 na na na 4 0.0481 0.1495 0.2298 1 na na na 5 0.0977 0.0974 0.0106 0.2189 1 na na 6 0.1074 0.2329 0.1368 0.2088 0.1554 1 na 7 0.3504 0.1999 0.1496 0.1143 0 0.2591 1
8.2.2 [M2 and M3, All VTV Regions-VTV, 10]
Recall from Section 4.4.2, using the definition for the M2 models and Table 8.1, we would require 14, 15, 13, 19 and 13 taps for UIC, UOC, OLT, OHT and Small 314 regions, respectively. Tables 8.11- 8.15 present the tap amplitude statistics necessary to create the [M2, All VTV Regions, 10] models. Note that for the M2 models, we still use
V 2V −Re gion the correlation coefficient matrix ( Rwc ) for different regions, but the difference with respect to the M1 models is that the number of taps is larger.
Table 8.11 Amplitude parameters for [M2, OHT, 10] [144]
Tap Energy Weibull P1 P00,k P11,k Index Shape k Factor (bk)
1 0.8982 4.30 1.0000 na 1.0000 2 0.0527 1.64 0.7960 0.3625 0.8366 3 0.0159 1.89 0.5696 0.5999 0.6973 4 0.0084 1.85 0.3739 0.7325 0.5514 5 0.0064 1.72 0.2764 0.8121 0.5084 6 0.0042 1.67 0.1976 0.8641 0.4486 7 0.0027 1.70 0.1444 0.9037 0.4297 8 0.0026 1.84 0.1178 0.9222 0.4172 9 0.0018 1.79 0.0856 0.9414 0.3750 10 0.0010 2.01 0.0576 0.9592 0.3333 11 0.0012 1.99 0.0598 0.9544 0.2844 12 0.0010 1.92 0.0467 0.9640 0.2647 13 0.0006 2.16 0.0404 0.9694 0.2721 14 0.0008 2.06 0.0296 0.9759 0.2130 15 0.0006 4.30 0.0362 0.9729 0.2803 16 0.0003 1.64 0.0285 0.9794 0.2981 17 0.0006 1.89 0.0264 0.9808 0.2917 18 0.0005 1.85 0.0206 0.9829 0.1867 19 0.0005 1.72 0.0178 0.9869 0.2769
315
Table 8.12 Amplitude parameters for [M2, UOC, 10] [144]
Tap Energy Weibull P00,k P11,k P1 Index Shape k Factor (βk)
1 0.8319 3.19 na 1.0000 1.0000 2 0.0817 1.61 0.2717 0.9150 0.8956 3 0.0322 1.63 0.4401 0.8171 0.7538 4 0.0186 1.73 0.5571 0.7488 0.6382 5 0.0109 1.81 0.6955 0.6716 0.4813 6 0.0059 1.95 0.7843 0.5737 0.3362 7 0.0038 1.85 0.8693 0.5417 0.2218 8 0.0026 1.70 0.9136 0.5304 0.1553 9 0.0024 1.59 0.9322 0.4796 0.1156 10 0.0019 1.55 0.9436 0.4941 0.1003 11 0.0019 1.35 0.9549 0.5105 0.0843 12 0.0024 1.34 0.9655 0.5510 0.0714 13 0.0022 1.34 0.9707 0.5364 0.0594 14 0.0008 1.33 0.9772 0.5635 0.0495 15 0.0007 1.29 0.9778 0.5179 0.0440
Table 8.13 Amplitude parameters for [M2, UIC, 10]
Tap Energy Weibull P00,k P11,k P1 Index Shape k Factor (βk)
1 0.0033 2.51 0.9460 0.6098 0.1196 2 0.3764 1.76 na 1.0000 1 3 0.3274 1.69 0.1818 0.9853 0.9827 4 0.1337 1.73 0.1379 0.9213 0.9164 5 0.0614 1.65 0.2500 0.8735 0.8559 6 0.0367 1.59 0.4294 0.8245 0.7651 7 0.0255 1.71 0.5837 0.7723 0.647 8 0.0163 1.92 0.6644 0.7494 0.5706 9 0.0080 2.04 0.7201 0.6800 0.4683 10 0.0049 1.84 0.7850 0.6528 0.3833 11 0.0024 2.10 0.8460 0.6359 0.2968 12 0.0020 2.00 0.8205 0.6267 0.3242 13 0.0014 1.60 0.8708 0.6632 0.2752 14 0.0010 2.45 0.8640 0.5848 0.2464
316
Table 8.14 Amplitude parameters for [M2, Small, 10]
Tap Energy Weibull P00,k P11,k P1 Index Shape k Factor (βk)
1 0.893 3.95 na 1.0000 1.0000 2 0.0807 1.91 0.4839 0.9446 0.9034 3 0.019 2.02 0.3452 0.7712 0.7383 4 0.00631 1.72 0.5985 0.6995 0.5701 5 0.00128 2.11 0.7217 0.4537 0.3364
Table 8.15 Amplitude parameters for [M2, OLT, 10]
Tap Energy Weibull P00,k P11,k P1 Index Shape k Factor (βk)
1 0.949 5.15 na 1.0000 1.0000 2 0.040 1.63 0.3836 0.8525 0.8073 3 0.007 1.7070 0.5771 0.6469 0.5443 4 0.003 1.6996 0.7411 0.5212 0.3507 5 0.001 1.8066 0.8603 0.4618 0.2060
It is interesting to note that for the M2 models for the Small and OLT regions, the of number of taps is limited to 5. This is directly attributable to the relatively sparse scattering environment. Next, we present the correlation coefficient matrix for the M2 models of the different regions. We follow a similar framework as used previously for the M1 models of the VTV regions. We don’t present the correlation matrix for OHT due to size constraints. Table 8.16 and Table 8.17 present the correlation coefficient matrices for the remaining regions. 317
Table 8.16 Correlation coefficient matrices for 10 MHz M2 models for UOC and UIC; lower triangular is UOC, upper triangular is UIC
i , j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 1.0000 0.3949 0.3916 0.3566 0.6763 0.6783 0.6497 0.5708 0.8626 0.5314 0.7084 0.8371 0.9574 0.3273 2 0.6898 1 0.3994 0.32160.4566 0.5849 0.6498 0.7201 0.4509 0.3726 0.6117 0.5486 0.6491 0.0621 3 0.6518 0.4922 1 0.55650.5030 0.6744 0.5008 0.7197 0.2093 0.5330 0.4617 0.7632 0.7071 0.1480 4 0.5772 0.5142 0.8479 1 0.5818 0.6428 0.4832 0.7651 0.7539 0.7450 0.5646 0.9118 0.7698 0.1127 5 0.5134 0.5262 0.4608 0.7101 1 0.7720 0.8672 0.8403 0.6569 0.4201 0.7257 0.9726 0.9462 0.1964 6 0.7501 0.5541 0.3986 0.5363 0.7270 1 0.8070 0.7088 0.8271 0.3278 0.8502 0.9844 0.9768 0.0038 7 0.9522 0.7808 0.4838 0.7313 0.7839 0.8413 1 0.7755 0.6597 0.5900 0.5607 0.9906 0.9113 0.3609 8 0.9890 0.4247 0.7468 0.6595 0.5451 0.7973 0.7380 1 0.3360 0.4905 0.5584 0.8052 0.6397 0.1383 9 0.9348 0.7381 0.3924 0.7877 0.8442 0.7301 0.8339 0.9855 1 0.4961 0.8640 0.8203 0.2806 0.1506 10 0.9412 0.6023 0.5346 0.5491 0.8658 0.8113 0.6887 0.9668 0.9655 1 0.5780 0.8185 0.9397 0.2159 11 0.9731 0.8094 0.8106 0.8117 0.5222 0.5793 0.5320 0.8102 0.7460 0.9733 1 0.8663 0.0379 0.0338 12 0.6051 0.5705 0.4102 0.5430 0.5172 0.4769 0.4025 0.5804 0.6115 0.7123 0.4480 1 0.3147 0.4452 13 0.7346 0.6595 0.6172 0.7608 0.6300 0.6317 0.5764 0.5776 0.6177 0.4677 0.7211 0.5206 1 0.2167 14 0.6530 0.4105 0.5672 0.4196 0.6398 0.5645 0.5247 0.6089 0.4530 0.5200 0.5569 0.6018 0.5452 1 15 0.8183 0.7555 0.8850 0.5653 0.9420 0.7032 0.6053 0.6263 0.5558 0.5973 0.6088 0.7300 0.6150 0.6425
318
Table 8.17 Correlation coefficient matrices for 10 MHz M2 models for Small and OLT; lower triangular is Small, upper triangular is OLT
i , j 1 2 3 4 5 1 1 -0.1977 0.0775 0.1403 0.3361 2 0.0338 1 0.1535 -0.0610 0.1189 3 0.6813 0.0684 1 0.1995 0.4203 4 0.8157 0.1024 0.7745 1 0.3360 5 0.9229 -0.0025 0.7234 0.8819 1
As noted in Chapter 6 when discussing the M3 models for the airports, in the
VTV region models, the tap amplitude statistics for [M3, All VTV Regions, 10] and [M2,
All VTV Regions, 10] are the same. The only difference being that in order to implement the M3 models, we do not require the correlation matrices and the persistence process modeling parameters.
8.2.3 [M2 and M3, UOC and OHT, 5]
Similar to the M2 models presented in the previous section for [M2, All VTV
Regions, 10], we can also present the M2 models for a 5 MHz bandwidth. These 5 MHZ models are presented in the Appendix. In this section, we do present [M2, UOC and
OHT, 5] since we will be using these models to compare the accuracy of our simulated models with the data. Table 8.18 presents the tap amplitude statistics for [M2, UOC and
V 2V −Re gion OHT, 5]. Table 8.19 provides the correlation coefficient matrices Rwc for both regions. As noted, for the [M3, UOC and OHT, 5] models, the tap amplitude statistics stay same as those for [M2, UOC and OHT, 10]. 319
Table 8.18 Amplitude parameters for [M2, UOC and OHT, 5]
Tap Index Energy bk P1 P00,k P11,k k [UOC M-2 5 MHz] 1 0.8843 3.21 1.0000 na 1.0000 2 0.0716 1.56 0.8761 0.2952 0.9003 3 0.0228 1.74 0.6518 0.5415 0.7550 4 0.0084 1.83 0.3510 0.7821 0.5972 5 0.0049 1.61 0.1736 0.9050 0.5470 6 0.0031 1.33 0.1160 0.9397 0.5407 7 0.0036 0.96 0.0751 0.9630 0.5445 8 0.0014 1.28 0.0574 0.9679 0.4726 [OHT M-2 5 MHz] 1 0.9228 4.51 1.0000 na 1.0000 2 0.0466 1.63 0.7680 0.4455 0.8324 3 0.0130 1.71 0.4505 0.7021 0.6368 4 0.0062 1.63 0.2495 0.8456 0.5358 5 0.0038 1.76 0.1540 0.9020 0.4617 6 0.0027 1.71 0.1051 0.9325 0.4256 7 0.0019 1.74 0.0725 0.9506 0.3674 8 0.0012 2.05 0.0607 0.9605 0.3891 9 0.0011 4.51 0.0453 0.9684 0.3333 10 0.0008 1.63 0.0288 0.9782 0.2667
Table 8.19 Correlation matrices for 5 MHz M2 models for OHT and UOC; lower triangular is OHT, upper triangular is UOC i , j 1 2 3 4 5 6 7 8 9 10 1 1 0.7243 0.5528 0.9468 0.9347 0.8401 0.8442 0.7527 na na 2 0.3953 1 0.5260 0.7410 0.7027 0.5741 0.6376 0.6427 na na 3 0.6516 0.3335 1 0.8384 0.8656 0.8562 0.5448 0.8277 na na 4 0.5725 0.2771 0.2771 1 0.9453 0.7520 0.6826 0.5571 na na 5 0.8020 0.4435 0.4435 0.4435 1 0.9680 0.4729 0.5494 na na 6 0.6811 0.4871 0.4871 0.4871 0.4871 1 0.7222 0.7667 na na 7 0.8682 0.9034 0.9034 0.9034 0.9034 0.9034 1 0.7114 na na 8 0.7029 0.1184 0.1184 0.1184 0.1184 0.1184 0.1184 1 na na 9 0.7052 0.2899 0.2899 0.2899 0.2899 0.2899 0.2899 0.2899 1 na 10 0.9558 0.6725 0.6725 0.6725 0.6725 0.6725 0.6725 0.6725 0.6725 1 320
8.3 Comparison of Nonstationary and Stationary Models for UOC and
OHT
In this section, we compare the PDPs generated using nonstationary and stationary channel models with the data collected in different regions. The models will be evaluated on the basis of their accuracy in emulating the actual channel. We focus our comparison on the UOC and OHT channels since they are the most popular regions anticipated for
VTV communications.
8.3.1 Simulation Procedure to Develop Nonstationary and Stationary Models
In this section, we outline the steps we have used in simulating the nonstationary
(M1 & M2) and stationary (M3) channels.
8.3.1.1 Steps for Simulating M1 Model
Assumptions:
As noted in Section 4.2.2, we assume that the maximum relative vehicle speed while traveling in UOC is ~50 miles/hr which translates to a maximum Doppler frequency (fDmax,UOC) of 386 Hz for a carrier frequency of 5.12 GHz. Using a similar analysis for OHT, the maximum relative vehicle speed of ~120 miles/hr would translate to a fDmax,OHT of 925 Hz. As discussed in Section 6.5.1.1, the effect of Doppler can be mimicked using a LPF or using an interpolating function. The key factor used to implement this LPF or interpolating function is the ratio of fD,max to the symbol rate (Rs). 321
For VTV regions as well, we use the approach followed in Section 6.5.1.1 and generate the PDPs of the respective region at fD,max, and then use interpolation to obtain the
“intervening” samples.
Algorithm for M1 Model
We describe the simulation algorithm for [M1, UOC, 10]. The algorithm can be generalized to any region and BW channel model.
1. From Table 8.4, we determine the number of taps as L=4.
2. Obtain fading amplitude Weibull parameters b and a for the 4 taps. Also, provide
UOC the correlation coefficient matrix, Rwc for the 4 taps (see Table 8.6).
3. Using the correlated Weibull simulation program (refer to [121]), generate the
correlated Weibull amplitude samples for 6000 CIRs. The number of CIRs is
assumed to be 6000 since that is the number of PDPs that we gathered for UOC
during our measurements. Also, since we are comparing the statistics of the
simulated channel model with that of the measured data, we considered the same
number of PDPs in both the cases.
4. Interpolate the CIRs using cubic interpolation to account for Doppler spread. For
example, if the BW of the channel is assumed to be 10 MHz (Rs/fDmax,UOC =
10,081) and the number of generated CIRs is 6000, so the original CIRs will be
interpolated to 64,865,000 CIRs. The matrix generated is of size 64,865,000 time
samples per tap by 4 taps.
5. Generate the persistence process, z(t), for each tap. Each persistence process has
6000 samples. Increase the number of samples to 64,865,000 by repeating each 322
sample 10,081 times. Note that for the persistence process we can not use
interpolation to obtain the intervening states since z(t) is a two state Markov
process that takes only values one and zero. The matrix generated is of size
64,865,000 time samples per tap by 4 taps.
6. Multiply the matrices obtained in steps 4 and 5, per tap, in element by element
fashion, to obtain the CIRs. In total, we have 64,865,000 CIRs.
7. Generate uniformly distributed random variables between (-π ,π ) to model the
phase for each of the taps of the 64,865,000 CIRs. A similar argument as
presented in Section 6.5.1.1 regarding the uncorrelated phases (due to multiple
plane waves contributing to the tap’s complex gain) can be used for this case too.
8.3.1.2 Steps for Simulating the M2 Model
Assumptions: Similar to the assumptions presented in Section 6.5.1.2 for creating the M2 models for airports, we generate CIRs for different segments of travel. In the case of
VTV, we assume three segments of travel with 2000 CIRs in each segment. We assume the same correlation matrix but different fDmax in each segment. Table 8.20 provides details regarding these issues.
Table 8.20 Number of CIRs for simulated M2-VTV models for 10 MHz channel BW
UOC Number of CIRs [Maximum Relative Velocity, after Interpolation Region fDmax] BW/fDmax (Mi) Segment 1 [30 mph, 232] 43,103 86,207,000 Segment 2 [40 mph, 309] 32,362 64,725,000 Segment 3 [50 mph, 386] 25,907 51,813,000 323
OHT Segment 1 [100 mph, 771] 12,970 25,940,000 Segment 2 [110 mph, 848] 11,792 23,585,000 Segment 3 [120 mph, 925] 10,811 21,622,000
We describe the simulation algorithm for [M2, UOC, 10]. For the M2 model, we use an algorithm similar in principle to applying the M1 model 3 consecutive times. The algorithm can be generalized to any region and BW. For each iteration of the M2 model, the Weibull a and b factors for the taps and the correlation matrix remain the same.
Algorithm for M2 Model
1. Determine the number of taps as L=15 (refer to Table 8.12).
2. Obtain Weibull parameters b and a for each of the 15 taps.
th 3. Consider the i iteration. Determine the ratio BW/fDmax.
4. Generate 2000 CIRs (refer to Table 8.20), each having 15 taps, using the
correlated Weibull simulation program.
5. Interpolate the 2000 CIRs to obtain (Mi) CIRs. We use cubic interpolation for this
simulation. The matrix generated is of size (Mi × 15).
6. Generate the persistence process, z(t), for each tap. Each persistence process has
2000 samples Increase the number of samples to Mi by repeating each sample
(BW/fDmax) times. The matrix generated is of size (Mi× 15).
7. Multiply the matrices obtained in steps 5 and 6 for each tap, in element by
element fashion, to obtain the CIRs. In total, we have Mi CIRs.
8. Generate uniformly distributed random variables between (-π ,π ) to model the
phase for each of the taps of the Mi CIRs. 324
9. Repeat steps 3-8 for 3 iterations.
10. In total, this yields 3 data sets with varying number of CIRs in each set.
8.3.1.3 Steps for Simulating the M3 Model
Assumptions: Assumptions regarding Doppler frequency for M1 can also be made for
M3. We describe the simulation algorithm for [M3, UOC, 10]. As discussed in Section
8.2, the number of taps considered for M3 is the same as its corresponding M2 model. So in this case we have 15 taps. Also, since M3 is a stationary version of the channel model, we don’t consider the persistence process and correlation matrices.
Algorithm for M3 Model
1. Determine the number of taps as L=15 (refer to Table 8.12).
2. Obtain Weibull parameters b and a for each of the 15 taps.
3. Generate 6000 CIRs, each having 15 taps, using the correlated Weibull
simulation program. Note: Use an identity matrix for the correlation
matrix.
4. Interpolate the CIRs using cubic interpolation to account for Doppler spread.
The matrix generated is of size 64,865,000 time samples per tap by 15 taps.
5. Generate uniformly distributed random variables between (-π ,π ) to model
the phase for each of the taps of the 64,865,000 CIRs. 325
8.3.2 Comparing Nonstationary and Stationary Models for UOC and OHT
In this section, we compare the CIRs generated using the M1, M2 and M3 models for the different VTV regions for two values of BW.
8.3.2.1 Comparing Models for UOC
As discussed previously in Section 6.5.2, the accuracy of the simulated channel models with respect to the data is compared on the basis of how well they agree in terms of
RMS-DS, the shape of the simulated PDPs, Wτ,,90, etc. In Figure 8.10, we compare the pdf and cdf of the RMS-DS for the simulated channel models with those of the collected data for a BW of 10 MHz. Figure 8.11 compares the pdf of Wτ,,90 for the simulated channel models with that of the data. Table 8.21 compares the shapes of the pdfs for
Wτ,,90 and στ, using the stochastic distance measures. The distance measures have been discussed in Section 6.5.2.
326
0.7 Data 1 Model - 2 Data 0.6 0.9 Model - 1 Model - 2 Model -3 0.8 Model - 3 0.5 Model - 1 ) τ σ
τ 0.7 σ 0.4 0.6
0.5 0.3 0.4 Probability of
0.2 0.3 Probability ( abcissa < 0.2 0.1 0.1 0 2 3 0 10 10 0 200 400 600 800 1000 1200 σ in nsec σ in nsec τ τ
Figure 8.10 Comparing pdf and cdf of RMS-DS for Model-1, Model-2 and Model-3 with that of UOC data for 10MHz [144]
0.8 Data 0.7 Model - 2 Model -1 Model - 3 0.6 ,90 τ 0.5
0.4
0.3 Probability of W
0.2
0.1
0 0 500 1000 1500 2000 2500 3000 W in nsec τ,90
Figure 8.11 Comparing pdf of DW-90 for Model-1, Model-2 and Model-3 with that of UOC data [144] 327
Table 8.21 Comparison of statistics of M1, M2 and M3 with those of data for UOC for two bandwidths [144]
10 MHz BW RMS-DS DW-90 KL HI KL HI M1 0.29 0.7234 0.1784 0.7792 M2 0.1167 0.7377 0.0517 0.78 M3 1.0061 0.6268 0.1399 0.6439 5 MHz BW M1 0.1053 0.8072 0.1031 0.8922 M2 0.0543 0.8562 0.0216 0.9150 M3 0.1366 0.6359 0.0696 0.6439 On the basis of the results provided in Table 8.21 and Figures 8.10 and 8.11, we make the following observations:
• From the plots in Figures 8.10 and 8.11, it is clear that Model-1 produces power
th delay profiles that yield values of στ and W ,,90 up to the those of the 80
percentile of the measured data. From the distance measures, we observe that the
M1 simulated pdfs for Wτ,,90 and στ match well with those of the data for lower
values of Wτ,,90 and στ . Since the bulk of the probability mass is concentrated at
these low values, the values of KL and HI for Model-1 are close to the desired
values of zero and one respectively. This indicates a fairly good match between
the pdfs of Wτ,,90 and στ for Model-1 and the collected data. One disadvantage of
Model-1 is that the number of taps is limited to 4, hence the maximum delay
spread possible is 400 nsec (Tc = 100 nsec for 10 MHz). But overall Model-1
emulates the measured UOC channel reasonably well. 328
• From the plots in Figure 8.10 and 8.11, we see that Model-2 produces power
nd th delay profiles that yield values of Wτ,,90 and στ up to the 92 and 99 percentile of
the measured data, respectively. From the distance measures, we observe that the
simulated pdfs for Wτ,,90 and στ match very well with the data. The distance
measures are better for Model-2 than for the other models, which is expected
since we do no truncation of large-delay taps.
• Using the same approach as described in the previous paragraphs, we see that
Model–3 produces power delay profiles that yield values of Wτ,,90 and στ up to the
92nd and 99th percentile of the measured data, respectively. Yet from Figure 8.11
and the distance measures, we observe that the simulated pdfs for Wτ,,90 and στ of
Model–3 are broader than the ones provided by Model-2, indicating higher
channel dispersion. This higher dispersion of Model-3 is due to the absence of tap
persistence.
On the basis of the above discussion, we conclude that Model-1 provides a fair approximation of the channel with minimum implementation complexity. For a more accurate representation, we recommend use of Model-2. Model-3 is more dispersive than
Model-2, and hence Model-2 better represents the statistics of the data than does Model-
3.
8.3.2.2 Comparing Models for OHT
Following the same framework as for the UOC models, in this section, we compare the accuracy of the simulated channel models with respect to data for the OHT 329 channel. In Figure 8.12, we compare the pdf of the RMS-DS for the simulated channel models with those of the collected data for a BW of 10 MHz. Figure 8.13 compares the pdf of Wτ,,90 for the simulated channel models with those of the data. Table 8.22 compares the shapes of the pdfs for Wτ,,90 and στ, using the stochastic distance measures.
0.7 Data 0.6 Model-2 Model-3 Model-1 0.5 τ σ 0.4
0.3 Probability of 0.2
0.1
0 0 200 400 600 800 1000 1200 1400 1600 1800 σ in nsec τ Figure 8.12 Comparing pdf of RMS-DS for Model-1, Model-2 and Model-3 with that of data for OHT-10MHz
330
1 Data 0.9 Model-2 Model-3 0.8 Model-1 0.7 ,90 τ 0.6
0.5
0.4
Probability of W 0.3
0.2
0.1
0 0 500 1000 1500 2000 2500 W in nsec τ,90 Figure 8.13 Comparing pdf of DW-90 for Model-1, Model-2 and Model-3 with that of data for OHT-10MHz
Table 8.22 Comparison of statistics of M1, M2 and M3 with those of data for OHT for two bandwidths
10 MHz BW RMS-DS DW-90 KL HI KL HI M1 0.7429 0.5325 0.1690 0.7774 M2 0.0276 0.8579 0.02 0.8673 M3 0.3664 0.5729 1.8889 0.2854 5 MHz BW M1 0.8898 0.5674 0.2038 0.8305 M2 0.0135 0.9555 0.0008 0.9655 M3 0.0757 0.7786 0.0160 0.9563
On the basis of the results provided in Table 8.22 and Figures 8.12 and 8.13, we make the following observations: 331
• From the plots in Figures 8.12 and 8.13, it is clear that Model-1 produces power
th th delay profiles that yield values of στ and Wτ,,90 up to the those of the 65 and 90
percentile of the measured data respectively. From the distance measures in
Table 8.22, we observe that the simulated pdfs for Wτ,,90 and στ do not match that
well with those of the data.
• From the plots in Figure 8.12 and 8.13, we see that Model-2 produces power
th delay profiles that yield values of στ and Wτ,,90 up to the 99 percentile of the
measured data. From the distance measures, we observe that the simulated pdf
for Wτ,,90 and στ matches very well with the data. The distance measures are
better for Model-2 than for the other models, which is expected since we do no
truncation of large-delay taps.
• Model–3 produces power delay profiles that yield values of στ and Wτ,,90 up to the
99th percentile of the measured data, respectively. Yet from Figure 8.11 and the
distance measures, we observe that the simulated pdfs for Wτ,,90 and στ of Model–
3 are broader than the ones provided by Model-2. Hence, the distribution of Wτ,,90
and στ for Model-3 are less similar than the data than is Model-2. This
observation is verified by looking at the distance measures in Table 8.22. This
higher dispersion of Model-3 is due to the absence of tap persistence.
On the basis of the above discussion, we conclude that Model-1 provides a fair approximation of the channel with minimum implementation complexity. Though Model-
2 and Model-3 can both emulate the worst case conditions well, Model-3 in general is 332 more pessimistic than Model-2 in modeling the data due to the absence of the persistence process. 333
9 Summary, Conclusions and Future Work
In this dissertation, we have addressed the issue of channel characterization for two upcoming areas of communication system design in the 5 GHz band. In this chapter, we summarize the contributions of this dissertation and discuss avenues for future research for academia and industry.
9.1 Summary and Conclusions
9.1.1 Airport Surface Area Channel Characterization
We initially discussed the growth in civil aviation and how that is driving the necessity to devlop new communication systems and services for multiple applications on the airport surface (AS). We then discussed the utility of the 5 GHz E-MLS band as a prospective candidate for hosting these new communication systems. As we noted throughout this dissertation, optimum system performance cannot be guaranteed without accurate knowledge of the underlying physical channel. This implies the need for channel models in the 5 GHz E-MLS band around the airport surface areas. We then cited related references, with the conclusion that wideband stochastic channel models for this band and environment did not previously exist and this further motivated our work on statistical characterization of the E-MLS band channel for airport surface communications. In our literature review, we also discussed certain issues needing further investigation, discovered after the initial data processing of our data. Some of 334 these issues were the existence of severe (worse than Rayleigh) fading, modeling of statistically non-stationary channels, and the need for development of multiple correlated random variables for accurate modeling.
Several striking differences between the scattering environment of the AS and contemporary terrestrial cellular settings were presented, and we hence concluded that creating channel models for AS areas based on existing terrestrial cellular channel models would not yield accurate results for the AS channel. To base our channel models on actual measured data, we conducted several measurement campaigns6 at different airports to gather power delay profiles, which characterize the underlying physical channel in the delay domain. Measurements were made using the popular pseudo-noise (PN) direct- sequence spread spectrum (DS-SS) correlator signaling approach. The transmitter- receiver pair, denoted the “sounder,” has an adjustable center frequency, transmit power, and bandwidth. Different airports were selected depending on convenience and also to account for a range of conditions of traffic density on the airport, size of scatterers
(General Aviation (GA) aircraft, 737s, 747s, etc.) and infrastructure (airline buidlings, hangars, workshops, etc). Due to the dynamic scattering environment of the AS channel, no two airports are exactly alike. One of our main objectives was to create different channel models that could emulate most of the propagation conditions encountered on the
AS. To separate the different propagation environments, we created 3 classes of airports: large, medium and small (GA). Each class of airport was also divided into 3 propagation regions; LOS-O (Line of Sight-Open), NLOS-S (Non Line of Sight-Specular) and NLOS
6 “Wireless Channel Characterization in the 5 GHz Microwave Landing System Extension Band for Airport Surface Areas," NASA ACAST, Grant no. NNC04GB45G. 335
(Non Line of Sight) depending on the different scattering environment. We collected data at Ohio University airport (Small), Albany, OH, Cleveland Hopkins International
Airport (Medium) and GA airport Burke Lakefront, Cleveland, OH, Miami International
Airport (Large) in Miami, FL, GA airport Tamiami (TA), Kendall, FL, and John F.
Kennedy International Airport (Large),New York, NY. A basic overview of the equipment (channel sounder, antennas, measurement principles, sounder calibration, etc.), measurement campaigns (measurement plan, details on transmitter and receiver locations, measurement routes, etc.), and descriptions of the salient features of each airport and propagation environment were provided. Overall, most of the areas where future communication systems for fixed or mobile links could be deployed were covered during the measurement campaign. In total, over 35,000 power delay profiles (PDPs) were taken for the mobile setting (transmitter fixed on ATCT and receiver mobile), just under 5,000 PDPs for the point-to-point setting (transmitter fixed on ATCT and receiver fixed at airport field sites), and over 11,800 PDPs for the airport field site measurements
(transmitter fixed at airport field site and receiver mobile). In the case of the TA airport, for the mobile setting, the transmitter was mounted on top of an ILS equipment station instead of the ATCT.
To create stochastic models from our empirical data, we created a pre-processing framework to extract usable information from the data. Under the pre-processing steps, we took into account the autocorrelation function of the sounder, calibration errors associated with the sounder, potential errors associated with a stepped correlator spread spectrum sounder (pulse compression, misinterpretation and commutation), and also 336 removed the effects due to “noise” so that the processed data accurately models the physical propagation channel. It was determined that all the above mentioned factors had a negligible effect on our channel statistics. The pre-processing framework description was followed by a discussion of the essential channel modeling parameters (in time and frequency domains) which were obtained by processing the measured PDPs. Parameters discussed were the mean energy delay, RMS-delay spread, delay window, tap energy, tap probability, inter-tap correlations, tap phase, channel transfer function, frequency correlation estimate and Doppler spread. For each region of a given airport class, we then presented three different models (M1, M2 and M3) based on the model user’s tolerance of implementation complexity, fidelity requirements with respect to the actual data, and need for consideration of stationarity.
We next documented some unique propagation effects associated with our collected measurements. These propagation effects are not limited to the AS channel but for the scope of this dissertation, we have explained the effects using the AS and VTV measurements. The first effect is the existence of several cases of severe, or “worse than
Rayleigh,” fading. To explain this behavior, we presented two physical models. The first model employs a statistically non-stationary random process that switches between two distributions, akin to the multi-state models proposed for land mobile satellite channels.
The other model we proposed is a multiplicative model of two small scale fading processes. We also described the analytical pdf for such a multiplicative model using two
Weibull random variables as the underlying small scale fading distributions. Measured data was used to corroborate these models, and our computer simulations confirm the 337 ability to replicate the statistics of these severe fading processes. These proposed models can be used by researchers investigating communication system designs in severe fading environments. It is well known in the literature that large scale fading like shadowing and small scale fading are often referred to as “macroscopic” and “microscopic” effects, respectively. In [113], the authors refer to the “arrival/departure” of multipath components as a “mesoscopic” effect since the ON/OFF behavior of the multipath components occurs on a larger spatial scale than small scale fading but on a smaller spatial scale than large scale fading. The other physical phenomena that we documented was the existence of non-stationarity at different time scales (from the order of a few seconds to several minutes, depending on velocities), which includes propagation region transitions. We have modeled this non-stationarity using two different processes: the tap persistence process models the finite lifetime associated with a multipath component using a 2 state first order Markov chain for each tap, whereas the region persistence process was implemented using a 3 state first order Markov chain to emulate the transition of the receiver from one region to another while moving on the AS.
Next, we presented the statistics of the different time- and frequency-domain channel parameters for different airports. Our stochastic channel models are based on the
RMS-delay spread, which has been widely used in the literature as a reliable indicator of channel frequency selectivity. Values of mean RMS delay spread ranged from 44-1500 nanoseconds for these airports. The smallest values apply to the Small airport LOS-O region, and the largest apply to the large airport NLOS region. A similar trend in the values can also be observed for the delay window statistic. An interesting thing to note is 338 that the RMS-delay spread values for TA were much larger than those for other GA airports due to the lower transmitter antenna height. Corresponding frequency correlation bandwidths (correlation value ½) range from approximately 0.39 MHz in non-line of sight (NLOS) settings to 16 MHz in LOS-O settings. We also presented results for steady state tap probabilities and tap energies for all the airport classes and regions. As expected, the steady state probability associated with the taps decreases as delay increases. For a specific delay, the steady state probability of the multipath component increases with an increase in frequency selectivity (the order being as follows: lowest is
LOS-O and highest is NLOS). This increase in frequency selectivity is also expressible as an increase in the richness of scattering. Similar observations were also made with respect to tap energy and tap cummulative energy.
We also provided exponential curve fits for the steady state probabilities and the cumulative energies for the different airport models. Tap probability of occurrence follows a decaying exponential function of tap index (delay), and tap cumulative energy follows a “1 minus decaying exponential” function of tap index. The tap phase followed a uniform distribution between (-π,π). The last parameter to estimate was the Doppler spread, which account for the channel variation in time. The Doppler spread depends mostly on the scattering geometry and the velocity of the scatterers. The scattering encountered by any vehicle on the airport surface will be non-isotropic in azimuth for all but the rarest of cases (e.g., when a vehicle is completely surrounded by other large vehicles in all azimuth directions). Thus for assessing the rate of fading and Doppler spreading, asymmetric Doppler spectra will pertain. As noted previously, due to the very 339 small velocities expected, this will not generally pose any difficulty for most communication signaling schemes. Due to limitations in our hardware update rate, we could not measure PDPs fast enough to estimate the Doppler frequency. Hence, we used analysis to determine the theoretical maximum value for the Doppler spread expected on the AS. The computed value was ~250 Hz using a maximum relative velocity of ~35 mph. Hence, even for low data rates (< 10 kbps), our channel variations are very slow.
Following the discussion regarding the channel parameters, we presented the large scale fading model. Due to hardware limitations, we were able to create a path loss model only for large airports. The propagation path loss fit well to a log-distance model, with path loss exponents near 2-2.3 for the LOS and NLOS-S areas. We then discussed tapped-delay line channel models (M1, M2 and M3) for two values of channel bandwidth
(50 and 25 MHz), for all propagation regions and airport classes. The channel tap amplitudes were modeled using the Weibull distribution. For a given region of a certain airport class, a complete description of the tap amplitude statistics, tap energies, tap transition and steady state probabilities needed to create Markov models for persistence, and tap correlation coefficient matrices were provided. Using analysis, we concluded that the maximum Doppler spread encountered on the AS is very small compared to our channel bandwidth, and hence it was determined that the effect of Doppler could be easily emulated with the use of a low pass filter (LPF). For the extremely slow fading that will be encountered in the AS environment, this approach poses difficulties in implementation due to the extremely small filter bandwidth required. To circumvent this problem, we use an interpolation method of filtering, which has also been used in other 340 channel modeling approaches such as cellular radio. To simulate the region transitions, we also provided the parameters needed to simulate the 3 state first order Markov model.
Due to the huge amount of details involved in simulating the stochastic model, as a reference, we provided an example illustration using the NLOS region of the large airport.
The next step was to verify the fidelity of our developed channel models. For this, we simulated CIRs using our M2 and M3 models and compared them to the measured IREs. The models were simulated using a correlated multivariate Weibull generator we created. The models were compared with the data using both time domain
(RMS-delay spread and delay window) and frequency domain (frequency correlation estimates) measures. We also used stochastic distance metrics to compare the shapes of the pdfs of the simulated and measured CIRs for RMS-delay spread and delay window.
After comparing these channel parameters, it was determined that the non-stationary M2 model provides a much more accurate representation of the channel than the stationary
M3 model. Due to the absence of the persistence process, the M3 model provides a pessimistic representation of the channel. This conclusion was valid for all the models developed, irrespective of the channel bandwidth, airport class, and region.
Also as mentioned, we tried to model the channel for different possible communication links. The next part of the research focused on developing models for links where the transmitter location was at an airport field site (AFS). The AFS locations were selected primarily to reach airport areas that were difficult to reach while transmitting from the air traffic control tower (ATCT). For some cases, transmitting 341 from an AFS can also significantly reduce the channel dispersion compared to when transmitting from the ATCT. As an example proof of concept argument, measurements were made in JFK for the former reason and at MIA for the latter reason. In the case of the MIA measurements, after comparing the channel parameter statistics (RMS-delay spread and frequency correlation estimate) of the IREs obtained while transmitting from the AFS or the ATCT, it was concluded that transmitting from the AFS drastically reduced the frequency selectivity of the channel. The main reasons for this were the reduction of link distance and the reduced occurrence of obstruction/blockage of the LOS signal. We also presented example 50 MHz-M1 channel models for the AFS data collected at MIA and JFK. AFS models are location specific and cannot necessarily be generalized. These links could potentially be used as a diversity nodes to bolster mobile links that have the destination/source transmitter/receiver located on the ATCT.
The final kind of AS channel that we modeled was for point-to-point links. For these links, the receiver was fixed at an AFS and the transmitter was fixed on the ATCT. The measurements were taken at various location at CLE and MIA. Different locations such as an ILS site, radar site, etc., were used as AFS locations. We presented descriptions of the different AFS locations and discussed the surrounding scattering geometry. Due to the fixed nature of these links, high gain directional horn antennas were used at both transmitter and receiver. Due to the high gain and narrow beamwidth of the transmitting and receiving antennas, the observed IREs did not show any substantial fading in the specular or LOS signal. The LOS component was modeled using a Rician distribution with a very large K factor. Due to the narrow beamwidth of the horn antennas, we were 342 also able to sample the power (RSSI) obtained from different directions at the receiver.
This helped us to further characterize the scattering geometry, and these results could also be helpful in selecting directions in which diversity links can be employed. We also provided results on the variation in the RMS-delay spread as a function of angle with respect to the boresight link. In general, it can be expected that the channel observed for fixed point-to-point links will remain statistically the same irrespective of the airport due to use of high gain and narrow beamwidth antennas, and the sparse scattering environment around the AFS locations.
9.1.2 Vehicle to Vehicle Channel Characterization
We initially provided a brief list of some possible applications of intelligent transportation systems (ITS) and also described the substantial increase in interest in this area by the research community. We next discussed how the use of vehicle-to-vehicle
(VTV) communication will be an integral part of ITSs. We then described the growth in
VTV communications and cited various current and future applications of VTV communications. The growth in potential applications implies the need for new channel models in bands and environments not previously characterized, specifically the 5 GHz band. A literature review discussing the different analytical, simulation-based, and empirical channel models developed for various VTV application was presented. New areas and areas requiring additional work were highlighted We concluded that there is a pressing need for developing wideband stochastic channel models in several VTV environments in the 5 GHz band. 343
Channel measurements were made with the transmitter and receiver in two vans, and data was collected in various cities and on highways. Our measurement campaigns were planned with the purpose of collecting data in different environments where VTV communication may be used. The equipment used was essentially the same as that used for the AS measurements. Within time and equipment limitations, we aimed to measure a range of potential VTV conditions by varying antenna locations and collecting data at different times of the day and in different traffic conditions. Measurements were conducted in five cities in Ohio: Cincinnati, Cleveland, Columbus, Dayton, and Athens.
We denote the first four as large cities, and Athens is termed a small city. We also collected data on several highways. This included interstate routes I-71, I-75, I-70, and
US routes 33 and 50. We have classified the measurement environments into the following types: Urban-Antenna Outside Car (UOC), Urban-Antenna Inside Car (UIC),
Small City (S), Open Area–Low Traffic Density (OLT), and Open Area High Traffic
Density (OHT).
Similar to the AS measurements, the same framework for data pre-processing and computation of the channel parameters was followed for the VTV measurements. RMS- delay spread statistics and frequency correlation estimates were provided, and it was observed that frequency selectivity was worst for UIC and minimal for the OLT setting.
The mean-RMS delay spreads ranged from 54-236 nsec. We also presented figures comparing the tap energies, steady state tap probabilities and cumulative energies for the different regions. Example exponential curve fits were also provided for steady state 344 probability and cumulative energy for all the regions. The trend in dispersiveness was similar to that observed for RMS-delay spread.
The non-stationary channel features for the VTV channel were then described, which included modeling multipath component persistence via Markov chains similar to those developed for the AS models. Fading tap amplitude statistics were again modeled using the flexible Weibull distribution, and severe (worse than Rayleigh) fading was found for several conditions. Tapped-delay line channel models (Model-1) for two values of channel bandwidth (5 and 10 MHz) were presented, for all propagation regions, including specification of correlations between taps. Using the multivariate correlated
Weibull algorithm, we generated fading amplitude time-series for Model-1. In order to evaluate the accuracy of the simulated data, several statistics for the different channel parameters obtained from the simulated channel models were compared with those of the measured data. A second model, Model-2, with higher implementation complexity but which modeled data with more fidelity than Model-1, was also discussed. The last model presented, Model-3, does not have correlated scattering and excludes our persistence process. Due to the absence of the persistence process, the PDPs generated by Model-3 provide a pessimistic realization of the channel.
Due to the inclusion of the persistence processes and correlation between multipath components in Model-1 and Model-2, both these models can not be classified as either Wide Sense Stationary or Uncorrelated Scattering. Hence Model-1 and Model -2 can be referred to as non-(WSSUS) models. Model-3, which is a WSSUS approximation of Model-2, was also discussed. It was concluded that inclusion of non-stationarity 345
(persistence processes) provides a more faithful representation of the channel than the
WSSUS model.
9.2 Future Work
In this section, we describe the various avenues for future research based on the results presented in this dissertation.
9.2.1 Applications of the Developed Channel Models
For a given airport, we could compare the PDPs simulated using our stochastic models with PDPs obtained by using a commercial propagation software package, such as “Wireless InSite” from RemCom [154]. To use “Wireless InSite,” we would require as much information about the propagation environment as possible. This includes the size and location of scatterers, information regarding mobility of receiver and transmitter, and the electrical properties of building materials and scatterers. Hence for convenience, in the beginning, it would be advisable to do the comparison for a very simple case, e.g., a certain section of the OU airport. The goal of this research would be to determine agreement between results of the ray tracing approach and our stochastic models, including how either approach may be augmented by some aspects of the other.
The purpose of developing stochastic channel models for AS and VTV applications is to evaluate the system performance under realistic environments. So one of the key areas of research following this dissertation would be to evaluate the performance of various modulation waveforms (e.g., spread spectrum, OFDM, etc.) under these 346 stochastic channel models using simulation and analysis. Initial efforts towards this have already presented by the author along with fellow OU researchers in [150], [151] and
[153]. The results presented in these papers were an initial performance evaluation of the different modulation waveforms for the worst case channels encountered at the different airports (i.e., [NLOS Large Airport and Medium Airport] and [NLOS-S Small Airport]).
A thorough evaluation of all the channel models using different modulation waveforms would also be of potential interest to members of academia and industry, particularly for other potential candidate systems that could meet application requirements for the AS area.
In the past few years, the new IEEE 802.16e wireless metropolitan area network
(WMAN) standard has gathered a lot of momentum, and is being considered a promising potential candidate for the AS area application due to its high data rate, large coverage, scalable bandwidth, quality of service (QoS) support, and mobility support. Under [155], we plan to evaluate the performance of 802.16e on airport surface areas for aviation applications. The results presented in [152] were part of the work done for [155]. In
[152], we have provided a description of three proposed channel estimation methods, and then provided simulation results for the BER performance of the 802.16e system over the proposed non-stationary channel models for different airports. The next part of this project would be to configure, develop, and deploy a wireless network testbed at the OU airport. This part of the project has been delayed due to inavailability of equipment vendors who could provide the necessary equipment in the required MLS-extension band.
Thus the project is still “work in progress” and results are expected in the near future. 347
9.2.2. Analytical Extensions to Current Work
In Chapter 5, we presented the analytical pdf of the random variate obtained by multiplying two Weibull random variates. A potential extension to that work would be to develop new probability density functions for several multiplied random processes. We can then generalize our multiplicative Weibull fading model to higher orders (i.e., N >2
Weibulls). The procedure could be complex, but the developed pdf would be more flexible than the pdf of N-Rayleigh random variates.
In Appendix A, we have presented a new algorithm to generate correlated multivariate Weibull random variates. We generate the Weibull variates using an exponentiation on Rayleigh variables, and this exponent depends on the shape factors of the desired Weibull variates. As an extension of this work, we could analyze operations on other random variables that can yield Weibull variates. The motivation for this would be two-fold: first, it would provide another method of generating such variates that could be more convenient or less complex, and second, it could provide insight useful to physically explain Weibull fading. We could then attempt to associate physical environment features with constituent distributions.
9.2.3 New Non-Stationary Channel Models
The first topic for investigation under this area would be to improve on the implementation of the persistence process. For the purpose of this dissertation, we implemented the persistence process using first order two state Markov chains and found 348 that the simulated PDPs were statistically very close to the measured PDPs (Chapters 6 and 8). As stated, the persistence process was used to model the finite lifetime associated with the multipath components. In practical cases, for a given tap, the persistence process will likely have some memory associated with it, and the first-order Markov chains may not capture this adequately. We could try to implement the persistence processes using 2nd (or higher) order 2 state Markov models, and hence by doing so incorporate additional memory in the persistence processes. Another approach of potential interest would be to create persistence processes that are correlated among taps
(in delay). It would also be of interest to evaluate the accuracy of the simulated Markov models as a function of the number of samples that are simulated.
By incorporating the above two modifications to the persistence process, the developed stochastic channel models could be made more realistic. We would evaluate the efficacy of these new models with the modified persistence processes by comparing these model outputs with those of the non-stationary models developed in this dissertation. For comparison, we could use the same framework used in Chapters 6 and
8. Intuitively, it would seem that the gain in fidelity (in modeling the measured data) might not be significant when using the modifed persistence processes, but for shorter simulations this could yield an improvement. In addition, this endeavor could be of academic interest in furthering understanding of the non-stationarity of the physical channel.
The next area of work would be to investigate ways of modeling the non- stationarity in the tap amplitudes in ways other than by multiplying the persistence 349 process and the tap amplitude process. One idea is to model the “complete” fading amplitude more compactly as a single non-stationary process, instead of separating multipath persistence and amplitude fading as two multiplicative processes. One possible approach is the use of a fading process (e.g., Weibull) whose very statistics vary, at time- varying, random, rates.
As described in Chapter 5, one of the physical models proposed to model severe fading is region transitions. An area of future research would be to obtain measured data to model these transitions. For example, we can actually emulate a region transition by moving a receiver on a pre-determined path such that the receiver moves from a LOS region to an obstructed region followed by a NLOS region and so on. Such experiments would help us to track the change in the amplitude statistic (from high Rician K factor to low Rician K factor to Rayleigh, etc.) of the first and other taps, and provide more understanding regarding how to model severe fading due to region transitions.
9.2.4 Using Developed Framework to Characterize Other Environments
The framework of extracting usable information from the collected data and then performing data processing on the usable data can be used to create stochastic channel models for data gathered in other environments. There are several environments and conditions where we could conduct measurements using our channel sounder. Below, we list a few of these environments and conditions: 350
• Develop a Markov model to model region transitions for VTV communication.
For the purpose of this dissertation, VTV measurements were collected for
individual regions during different measurement campaigns. We can conduct a
long duration VTV measurement campaign and collect VTV data for multiple
regions and then using the collected data develop the region transition matrices
for VTV.
• Inside-vehicle communication systems: For this kind of setting, the transmitter
and receiver will be inside the car and the receiver will be moved to various
locations within the car. The gathered measurements could be compared with
PDPs obtained by simulating this environment using Wireless Insite.
• Long-duration, Multiple Time Scale Modeling: Collecting data and developing
models for longer durations and multiple time scales will aid current and future
“cross-layer” system designs, since the different layers of the protocol stack
operate on different time scales.
• “Ubiquitous” Links: Future communication system designs are planning for the
existence of ubiquitous communication links. As an effort towards this, it would
be interesting to statistically model the channel transitions encountered when the
receiver moves from indoor to outdoor or vice-versa. Such a measurement
campaign could be easily conducted on university grounds and would provide
useful data regarding region transitions between indoor and outdoor. 351
• Near-ground Channels: Statistically modeling channels encountered when
transmitter and receiver antennas are at very low heights (e.g., sensor networks)
would also be of interest to the academic and industrial communities.
• Forest Channels: Characterizing the forest environment where the transmitter and
receiver are both on back-packs is of considerable interest for military
applications. 352
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359
Appendix A: Generation of Multiple Correlated Weibull
Random Variates
The utility of having an algorithm that can generate multiple correlated Weibull random variables with arbitrary shape and scale factor has been presented in Section 2.7.
In this appendix, we describe a general method7 that maps correlated Rayleigh RVs to correlated Weibull RVs, to generate multivariate Weibull RVs with arbitrary correlation matrix, arbitrary fading parameters, and arbitrary average power for each Weibull RV.
This algorithm is unique, and generation of such variables has not been reported in the literature. The appendix is organized as follows: Section A.1 outlines our method and describes the mapping between correlated Rayleigh RVs and correlated Weibull RVs,
Section A.2 provides a numerical example, and conclusions are given in Section A.3.
A.1 Relationship Between Correlated Rayleigh and Correlated Weibull
Random Variables
The pdf of a Weibull RV W is given by b ⎛ − wb ⎞ b−1 ⎜ ⎟ fW (w) = w exp⎜ ⎟ , (A.1) Ω ⎝ Ω ⎠ where b>0 is the fading parameter that indicates the severity of fading: as b increases, the channel condition gets better. When b=2, the Weibull distribution becomes the Rayleigh
7 We would like to extend our appreciation towards Wenhui Xiong who helped us in developing this algorithm.
360 distribution. The average power is given by E[w2 ] = Ω 2 / bΓ[1+ (2 / b )] where E( ) denotes expectation and Γ the gamma function. The idea behind our method for generating multivariate Weibull RVs is to first generate correlated Rayleigh random variates, then utilize the relationship between Rayleigh and Weibull variables to obtain the desired
Weibull RVs. The relationship is given as W = R 2 / b , where R is a Rayleigh RV. To generate N correlated Weibull RVs with correlation coefficient matrix ρ(W), fading parameter vector b=[b1 , b2, ...bN], and parameter vector Ω=[Ω1 , Ω2, ... ΩN] using
W = R 2 / b , one must first find the correlation coefficient matrix of the multivariate
Rayleigh RVs, ρ(R), such that when W = R 2 / b is applied to each individual Rayleigh RV
Ri, the result is a corresponding Weibull RV Wi, and the collection of Weibull variates has the desired correlation coefficient matrix ρ(W). The element in the ith row and jth column of ρ(R) is
()R cov[RRij , ] ρij, = , (A.2) var[Rij ]var[R ] where Ri, Rj are two Rayleigh RVs, and cov(X,Y) is the covariance of RVs X and Y. Using
W = R 2 / b in (A.2), we have
bi / 2 b j / 2 bi / 2 b j / 2 (R) E[Wi W j ] − E[Wi ]E[W j ] ρi, j = (A.3) bi / 2 b j / 2 var[Wi ]var[W j ] The expectation of the ith Rayleigh RV, expressed in terms of a Weibull RV, can be computed as
∞ 1 E[]W bi / 2 = wbi / 2 f ()w dw = πΩ (A.4) i ∫ i wi i i i 0 2 361
Using this and the definition of Ω given above, the variance of the Rayleigh RV is given by,
bi / 2 bi 2 bi / 2 var[Wi ]= E[Wi ]− E [Wi ] (A.5) The first term of the numerator in (A.3) is given by
∞∞ b / 2 b / 2 E W bi / 2W j = wbi / 2 w j f ()w , w dw dw (A.6) []i j ∫∫ i j WiW j i j i j 00 where f (w ,w ) is the joint pdf of the Weibull RVs Wi and Wj. In working to solve Wi ,W j i j
(A.6), we found the joint pdf given in two forms. One of these forms is given in [156],
b −1 b −1 b (W ) b / 2 b j / 2 i j ⎡ bi j ⎤ ⎡ i ⎤ b b w w ⎛ w w ⎞ 2 ρi, j wi w j i j i j 1 ⎜ i j ⎟ ⎢ ⎥ fW W ()wi w j = exp⎢− + ⎥I 0 (A.7) i j Ω Ω ()1− ρ (W ) ⎢ ()1− ρ (W ) ⎜ Ω Ω ⎟⎥ ⎢ 1− ρ (W ) Ω Ω ⎥ i j i, j ⎣ i, j ⎝ i j ⎠⎦ ⎣ ()i, j i j ⎦ where all terms have been defined previously, except I0(x), which is the modified Bessel function of the first kind, order zero. This joint pdf assumes that the correlation coefficients of the Rayleigh and Weibull RVs are identical. Thus, when bi=bj=2, the joint pdf given by (A.7) reduces to joint pdf for correlated Rayleighs. The Weibull distribution can also be viewed as coming from another family of distributions, which cannot reduce to the correlated Rayleigh distribution when bi=bj=2 [94]. The joint pdf of two correlated
Weibull RVs in this case is,
δ δ b b j b b j ⎧ b b j ⎫ ⎧ b b j ⎫ i −1 −1 ⎡ i ⎤ ⎡ i ⎤ ⎡ i ⎤ b b ⎛ w ⎞ δ ⎛ w ⎞ δ ⎛ w ⎞ δ ⎛ w ⎞ δ ⎪ ⎛ w ⎞ δ ⎛ w ⎞ δ 1 ⎪ ⎪ ⎛ w ⎞ δ ⎛ w ⎞ δ ⎪ (A.8) i j i ⎜ j ⎟ ⎢ i ⎜ j ⎟ ⎥ ⎪⎢ i ⎜ j ⎟ ⎥ ⎪ ⎪ ⎢ i ⎜ j ⎟ ⎥ ⎪ fW W ()wi w j = ⎜ ⎟ ⎜ ⎟ + × ⎨ ⎜ ⎟ + + −1⎬× exp⎨− ⎜ ⎟ + ⎬ i j a a ⎜ a ⎟ ⎜ a ⎟ ⎢⎜ a ⎟ ⎜ a ⎟ ⎥ ⎢⎜ a ⎟ ⎜ a ⎟ ⎥ δ ⎢⎜ a ⎟ ⎜ a ⎟ ⎥ i j ⎝ i ⎠ ⎝ j ⎠ ⎢⎝ i ⎠ ⎝ j ⎠ ⎥ ⎪⎢⎝ i ⎠ ⎝ j ⎠ ⎥ ⎪ ⎪ ⎢⎝ i ⎠ ⎝ j ⎠ ⎥ ⎪ ⎣ ⎦ ⎩⎪⎣ ⎦ ⎭⎪ ⎩⎪ ⎣ ⎦ ⎭⎪
1/ bi Where the parameter a relates to the average power Ω by ai = Ω , and δ is implicitly given by
Γ()δ b +1 Γ(δ b +1)Γ(1 b +1 b +1)− Γ(1 b +1)Γ(1 b +1)Γ(δ b + δ b +1) ρ ()w = i j i j i j i j (A.9) 2 2 Γ()δ bi + δ b j +1 ()Γ()2 bi +1 − Γ ()2 bi +1 ()Γ()2 b j +1 − Γ ()2 b j +1 362
For reference, we denote (A.8) the “general” form, and (A.7) the “restricted” form. Our aim is to find the correlations among the Rayleigh variates that will yield the desired correlated Weibulls after applying the transformation W = R 2 / b . Depending on which pdf is used to determine (A.3), we term the algorithm the “restricted” algorithm (when (A.7) is used) and the “general” algorithm (when (A.8) is used). By expressing the Bessel
∞ 2 n 2 function in power series form as I 0 (x) = ∑ ()x / 4 /(n!) , after some algebra, using n=0
(A.7), (A.6) is given by,
(W ) 2 π Ωi Ω j ()1− ρi, j bi / 2 b j / 2 (W ) E[]Wi W j = H ()[]3/ 2,3/ 2 ,1, ρi, j (A.10) 4 where H(a, d, z) is a generalized Hypergeometric function as defined in [156, eq. (13)].
The correlation coefficient of any pair of Rayleigh random variables in terms of the desired Weibull coefficient is thus
π ρρ()RW=−[(1 ( )2 )H ([3/2,3/2],1, ρ ( W ) ) − 1] (A.11) ij,,4 −π ij ij , It is interesting to note that (A.11) is not a function of fading parameters (b's), nor is it a function of the average power parameters (Ω's). A simpler approximate relationship
(R) (W ) between ρi, j and ρi, j was obtained via a least-squares polynomial fit to (A.11), expressed as,
()RW ( )2 ( W ) ρij,,≅+0.10(ρρ ij ) 0.89 ij ,. (A.12)
The correlation mapping can also be calculated using the general pdf. Using this general pdf, we have to solve (A.6), using (A.8) and (A.9), numerically. With the correlation mapping between the correlated Weibull RVs and correlated Rayleigh RVs, 363 we can generate correlated Weibull RVs using correlated Rayleigh RVs. These correlated
Rayleighs are often generated from correlated Gaussian variates [99]. As with the
Rayleigh-Weibull correlation coefficient mapping, the mapping between Rayleigh and
Gaussian coefficients is given by [99]
⎛⎞2|ρ ()G | ()G ⎜⎟ij, π (1+− |ρij, |)E i ()G ⎜⎟1|+ ρij, | 2 ρ ()R = ⎝⎠ (A.13) ij, π 2 − 2
nd (G) where Ei( ) is complete elliptic integral of the 2 kind, and ρi, j is the correlation coefficient of the Gaussian RVs. The algorithm to generate correlated Weibull RVs can then be written as follows:
• Find ρ(R), from the desired ρ(W). This can be done by using either the restricted or
general methods.
a. Restricted Method: find ρ(R) from ρ(W) using (A.11) or (A.12).
b. General Method: determine δ from the desired ρ(W) and b factors using
(A.9), for every pair of Weibull RVs; substitute (A.8) into (A.6) and
numerically find the value of the integral of (A.6), and then evaluate
(A.3);
• Generate N correlated Rayleigh random variables with parameter vector Ω and
correlation coefficient matrix ρ(R) using the method in [99] or the tabular mapping of
[100];
• Use the transformation W = R 2 / b to generate the Weibull RVs from the correlated
Rayleigh RVs with desired fading parameter vector β. 364
A.2 Numerical Example
We illustrate our approaches using an example. The desired Weibull correlation coefficient matrix is,
⎡ 1 0.795 0.604 0.372⎤ ⎢0.795 1 0.795 0.604⎥ ρ ( W ) = ⎢ ⎥ ,(A.14) ⎢0.604 0.795 1 0.795⎥ ⎢ ⎥ ⎣0.372 0.604 0.795 1 ⎦ with fading parameter vector β=[4, 3.5, 2, 1.3], and normalized average power parameter
( R ) vector given by Ω=[4, 3, 2, 1]. Using the restricted algorithm, ρ Re s is,
⎡ 1 0.775 0.578 0.349⎤ ⎢0.775 1 0.775 0.578⎥ ρ ( R ) = ⎢ ⎥ .(A.15) Re s ⎢0.578 0.775 1 0.775⎥ ⎢ ⎥ ⎣0.349 0.578 0.775 1 ⎦ Then, four correlated Rayleigh RVs with correlation coefficients given by (A.15) and parameters Ω8 are generated using the method given by [100]. Finally, each fading parameter, bi, is used with the Rayleigh RVs to obtain the Weibull RVs. The Weibull
(W ) correlation coefficient matrix ρ Res for a single realization using 5000 samples is
⎡ 1 0.743 0.6 0.342⎤ ⎢0.743 1 0.764 0.614⎥ ρ (W ) = ⎢ ⎥ . (A.16) Re s ⎢ 0.6 0.764 1 0.781⎥ ⎢ ⎥ ⎣0.342 0.614 0.781 1 ⎦ We compute the mean square error (MSE) between the matrices of (A.16) and
(A.14) using 1000 trials of 5000 samples each. The MSE in vector form can be expressed
( W ) as E=[e1,2, e1,3, e1,4, e2,3, e2,4, e3,4] where ei,j is the (normalized) MSE between the ρ Re s
8 Note that via W=R(2/β), Ω=E(R2). 365
( W ) and ρ correlation coefficients of the Weibull RVs. In our experiment, ERes=[3.2, 2.9,
1.5, 1.8, 2.2, 0.4]×10-3. To illustrate this example, we compare the theoretical and simulated pdf (histogram) for the first Weibull RV with b=4 and Ω=4. Figure. A.1 shows that the simulated and theoretical pdfs are an excellent match.
Simulated Theoretical 1.2
1
0.8 PDF 0.6
0.4
0.2
0 0 0.5 1 1.5 2 2.5 3 RV value
Figure A.1 Comparison of simulated Weibull histogram and theoretical for the “restricted” method
( R ) When we use the general method, ρGen is given by
⎡ 1 0.76 0.578 0.388⎤ ⎢ 0.76 1 0.789 0.634⎥ ρ ( R ) = ⎢ ⎥ . (A.17) Gen ⎢0.578 0.789 1 0.822⎥ ⎢ ⎥ ⎣0.388 0.634 0.822 1 ⎦
( R ) ( W ) We then use ρGen to generate correlated Rayleigh RVs. The matrix ρGen for a single
(W ) ( W ) realization using 5000 samples is given in (18). The EGen computed for ρGen and ρ using 1000 trials of 5000 samples each is [3.6, 0.4, 1,1,2,9,0.2]×10-3. 366
⎡ 1 0.749 0.619 0.404⎤ ⎢0.749 1 0.775 0.671⎥ ρ (W ) = ⎢ ⎥ (A.18) Gen ⎢0.619 0.775 1 0.798⎥ ⎢ ⎥ ⎣0.404 0.671 0.798 1 ⎦ The MSE values for both the general and restricted methods are very small and are comparable, indicating that the restricted pdf is a very good approximation to the general pdf; both methods generate multivariate Weibull random variables with good accuracy.
The correlation mapping between the correlated Weibull and Rayleigh RVs given by
(A.12) is applicable to arbitrary fading parameters, arbitrary correlations, and arbitrary average fading powers. (A.12) is also computationally more efficient than evaluating ρ(R) using the general method. Yet, (A.12) assumes equal Rayleigh and Weibull correlations through the pdf of (A.7). Hence, the restricted method is inaccurate when bi=bj=2. In this special case, the two Weibull RVs obtained by applying the transformation W = R 2 / b are identical to the input two Rayleigh RVs. Hence, ρ(R) should be equal to ρ(W). But upon applying (A.11) in this case, we find, ρ ()RW≠ ρ ( ) (even though they are close). In constrast, the correlation mapping using the general pdf yields the correct mapping when bi=bj=2 for all the values of δ , (δ ranges from 0.1 to 0.9 [156]). For all other values of fading parameters, correlations, and average powers, the restricted algorithm works very well.
A.3 Note on Implementing an LPF for Doppler Spectra
367
As noted in Chapters 4, 6 and 8; we implement the effect of the Doppler spread by using an LPF to filter. Due to implementation complexities (discussed in Chapter 4), we use an interpolating function to emulate a LPF. Now, it is necessary to make sure that the tap amplitude statistics are not modified after the interpolation operation. Hence, we need to do a small modification to the interpolated samples so that the amplitude distribution of each tap doesn’t change after interpolation. As described in the previous sections, we generate the correlated Weibull RVs from correlated Rayleigh variates which in turn are generated from correlated Gaussian variables. Next, we present the interpolation algorithm9 using the same example used in Section 6.5.1.1. The BW of the intended channel model is 10 MHz, and the fD,max is 2500 Hz. So the ratio of BW/fD,max =
4000. We also know that we have 31 taps in each simulated CIR. We need to interpolate
6299 CIRs by 4000 to obtain 2.5196×107 CIRs. The number 6299 is equal to the number of NLOS PDPs that we collected at MIA. Note that the interpolation algorithm is performed independently for the real and imaginary parts of the Gaussian CIRs (GCIR)
Algorithm for Interpolation
We describe the simulation algorithm for [M1, Large Airport, NLOS, 25]. The algorithm can be generalized to any airport size, region and BW channel model.
8. Initialize NumCIR = 1.
9. This step is performed only for k=1. Generate two CIRs using the fading
amplitude Weibull parameters b and a for the 31 taps and the correlation matrix,
NLOS Rwc for the 31 taps. Denote the 2 CIRs as CIR1 and CIR4000. Note that CIR1
9 The author would like to ackonwledge Beibei Wang for her help in developing the interpolating algorithm. 368
and are CIR4000 are both row matrices with 31 columns. Go to Step 4 unless
iteration number k=1.
10. Generate one CIR using the fading amplitude Weibull parameters b and a for the
NLOS 31 taps and the correlation matrix, Rwc for the 31 taps. Denote this CIR as
CIR4000.
CIR − CIR 11. Set Delta = 1 4000 , so Delta is a row matrix with 31 taps. 3999
12. Initialize S = 0 and count =1.
2 2 ⎛ S ⎞ ⎛ 3999 − S ⎞ 13. Power _ adj = ⎜ ⎟ + ⎜ ⎟ ⎝ 3999 ⎠ ⎝ 3999 ⎠
14. CIRcount+1= CIR1 + count*Delta
CIRcount+1 = CIRcount+1 / Power _ adj
15. S = S+1 and count = count +1. If S > 3999, go to Step 9, else go to Step 6. As an
example, we show in Figure A.2 an illustration of how the samples look after
interpolation for 2 taps.
369
Figure A.2 Example illustration for interpolation algorithm
16. NumCIR = NumCIR + 1 and k = k +1.
17. If NumCIR <= 6299 then CIR1 = CIR4000 and go to Step 3. The reason we
select the last CIR of this iteration as the initial CIR of the next iteration is to have
a smooth transition between the “block edge” CIR samples. If NumCIR > 6299,
go to Step 11.
18. We obtain 2.5196×107 CIRs with 31 taps in each. 370
Appendix B: Antenna Radiation Patterns
In this appendix, we present the radiation patterns (azimuth and elevation) of the different antennas used in our measurements. First, we present the elevation and azimuth patterns for our high gain directional horn antenna. This antenna has a gain of 17 dB and a 3 dB beamwidth of 30o in the azimuth and 15o in the elevation planes. Figure B.1 and
B.2 illustrate these patterns.
Figure B.1 Azimuth radiation pattern for high gain horn antenna
Figure B.2 Elevation radiation pattern for high gain horn antenna
371
The lower gain horn antenna patterns are presented next. This antenna has a gain of 8.5 dB and a 3 dB beamwidth of 60o in both the azimuth and elevation planes. Figure
B.3 and B.4 show these patterns.
Figure B. 3 Azimuth radiation pattern for low gain horn antenna
Figure B.4 Elevation radiation pattern for low gain directional horn antenna
The last figure presented is the elevation pattern for the monopole antennas used in the project. The pattern has a standard “donut shape” as expected. 372
Figure B.5 Elevation pattern for omnidirectional (azimuth) monopole
373
Appendix C: [M1 M2, All Airports, All regions, 5 10 20]
In this appendix, we present the different models for BWs of 5, 10 and 20 MHz for all the airports. This appendix is a follow-up on the models presented in Chapter 6.
C.1 [M1 M2, Large Airport, All regions, 5 10 20]
This section in the appendix presents the 5, 10 and 20 MHz models for different regions of the large airport. M1 for 50 MHz and [M1 M2 and M3] for 25 MHz have already been presented in Chapter 6.
Table C.1 Channel parameters for the [M1, Large Airport, NLOS, 5] model
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.6496 2.28 1.0000 na 1.0000 2 0.0789 1.65 0.9172 0.1748 0.9255 3 0.0579 2.0 0.8699 0.2003 0.8804 4 0.0516 1.63 0.8445 0.2543 0.8626 5 0.0440 2.0 0.8213 0.2678 0.8407 6 0.0437 1.67 0.8093 0.2571 0.8249 7 0.0390 2.0 0.7926 0.2737 0.8098 8 0.0352 2.0 0.7657 0.2957 0.7847
Table C.2 [M2, Large Airport, NLOS, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0294 1.4011 0.4328 0.7413 0.6612 2 0.0977 1.1241 0.9952 0.0800 0.9955 3 0.0901 1.2072 0.9774 0.1111 0.9795 4 0.0698 1.2837 0.9489 0.1925 0.9565 5 0.0549 1.3755 0.9301 0.2320 0.9423 374
6 0.0499 1.3951 0.9054 0.2857 0.9254 7 0.0416 1.5005 0.8958 0.2815 0.9164 8 0.0394 1.5050 0.8861 0.2831 0.9079 9 0.0367 1.5768 0.8805 0.3215 0.9082 10 0.0372 1.4625 0.8734 0.3130 0.9006 11 0.0318 1.6711 0.8651 0.3262 0.8951 12 0.0318 1.6322 0.8603 0.3108 0.8883 13 0.0310 1.6635 0.8570 0.2996 0.8833 14 0.0307 1.6581 0.8551 0.2889 0.8797 15 0.0299 1.6966 0.8672 0.2849 0.8907 16 0.0295 1.6857 0.8524 0.3268 0.8836 17 0.0297 1.6562 0.8622 0.2899 0.8865 18 0.0283 1.6792 0.8566 0.2840 0.8801 19 0.0301 1.6499 0.8551 0.2703 0.8765 20 0.0305 1.6410 0.8614 0.2925 0.8864 21 0.0295 1.6618 0.8537 0.2929 0.8790 22 0.0293 1.6400 0.8526 0.2736 0.8746 23 0.0298 1.6072 0.8655 0.2683 0.8865 24 0.0309 1.5697 0.8711 0.2440 0.8883 25 0.0305 1.5639 0.8680 0.2427 0.8848
Table C.3 [M1, Large Airport, NLOS-S, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (βk) 1 0.9503 3.5 1.0000 na 1.0000 2 0.0356 1.47 0.6941 0.5312 0.7935 3 0.0142 1.43 0.5196 0.6942 0.7173
Table C.4 [M2, Large Airport, NLOS-S, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0248 0.9558 0.0798 0.9752 0.7145 2 0.5043 1.1231 0.9922 0.5455 0.9964 3 0.2286 1.0443 0.8107 0.4305 0.8671 4 0.0617 1.1752 0.5812 0.6463 0.7453 5 0.0373 1.2473 0.4920 0.7236 0.7147 6 0.0264 1.2888 0.4119 0.7663 0.6661 375
7 0.0207 1.3371 0.3789 0.7806 0.6403 8 0.0152 1.3609 0.3368 0.7903 0.5872 9 0.0140 1.3760 0.3113 0.8134 0.5875 10 0.0114 1.4483 0.2925 0.8210 0.5673 11 0.0088 1.4627 0.2540 0.8350 0.5156 12 0.0075 1.4889 0.2292 0.8524 0.5037 13 0.0060 1.5889 0.2176 0.8557 0.4815 14 0.0052 1.6777 0.2045 0.8623 0.4644 15 0.0042 1.9348 0.2369 0.8195 0.4188 16 0.0037 2.1124 0.2348 0.8229 0.4228 17 0.0046 2.3884 0.3666 0.6912 0.4668 18 0.0031 2.1646 0.2244 0.8354 0.4306 19 0.0025 2.0518 0.1827 0.8715 0.4253 20 0.0021 2.2262 0.1804 0.8753 0.4341 21 0.0019 2.2387 0.1671 0.8880 0.4424 22 0.0016 2.3778 0.1555 0.8861 0.3816 23 0.0015 2.4985 0.1640 0.8837 0.4076 24 0.0016 2.4443 0.1707 0.8782 0.4083 25 0.0015 2.4058 0.1606 0.8852 0.4000
Table C.5 [M1, Large Airport, NLOS, 10]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.6350 2.10 1.0000 na 1.0000 2 0.0641 1.58 0.8794 0.1975 0.8899 3 0.0363 1.56 0.7890 0.3258 0.8197 4 0.0323 1.68 0.7747 0.3301 0.8051 5 0.0285 1.71 0.7519 0.3363 0.7809 6 0.0278 1.64 0.7437 0.3599 0.7794 7 0.0265 1.67 0.7288 0.3789 0.7690 8 0.0236 1.75 0.7102 0.4013 0.7556 9 0.0226 1.75 0.7060 0.4063 0.7529 10 0.0207 2.0 0.6930 0.4324 0.7488 11 0.0223 1.73 0.7065 0.4052 0.7528 12 0.0219 1.71 0.7000 0.3868 0.7374 13 0.0192 2.0 0.6798 0.4453 0.7386 14 0.0194 2.01 0.6992 0.4067 0.7449
376
Table C.6 [M2, Large Airport, NLOS, 10]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.0111 1.3908 0.1747 0.9008 0.5316 2 0.0855 1.3400 0.9969 0 0.9969 3 0.1055 1.2700 1 NA NA 4 0.0678 1.3800 0.9102 0.1970 0.9207 5 0.0457 1.4900 0.8642 0.2506 0.8822 6 0.0384 1.5900 0.8307 0.3246 0.8623 7 0.0296 1.3971 0.8036 0.3440 0.8397 8 0.0254 1.4382 0.7871 0.3437 0.8227 9 0.0227 1.4711 0.7626 0.3785 0.8065 10 0.0199 1.5432 0.7495 0.3961 0.7984 11 0.0188 1.5092 0.7388 0.4137 0.7929 12 0.0176 1.5794 0.7414 0.4007 0.7909 13 0.0165 1.6293 0.7184 0.4383 0.7800 14 0.0163 1.6234 0.7194 0.4378 0.7807 15 0.0152 1.6252 0.7150 0.4478 0.7801 16 0.0144 1.6905 0.7082 0.4533 0.7749 17 0.0148 1.6966 0.7158 0.4281 0.7732 18 0.0139 1.7364 0.7043 0.4304 0.7610 19 0.0145 1.6180 0.7014 0.4524 0.7670 20 0.0137 1.7258 0.7032 0.4392 0.7636 21 0.0138 1.7800 0.6976 0.4654 0.7685 22 0.0132 1.7217 0.6954 0.4469 0.7580 23 0.0135 1.7212 0.6951 0.4442 0.7564 24 0.0127 1.7690 0.6920 0.4547 0.7575 25 0.0124 1.7380 0.6854 0.4613 0.7529 26 0.0123 1.7383 0.6810 0.4600 0.7471 27 0.0127 1.8041 0.6917 0.4691 0.7635 28 0.0128 1.7438 0.6850 0.4665 0.7548 29 0.0127 1.7778 0.6925 0.4400 0.7515 30 0.0127 1.7662 0.6895 0.4526 0.7536 31 0.0127 1.7794 0.6956 0.4402 0.7549 32 0.0125 1.7533 0.6961 0.4423 0.7567 33 0.0126 1.7548 0.7051 0.4212 0.7579 34 0.0122 1.7416 0.6915 0.4674 0.7625 35 0.0128 1.7467 0.6803 0.4588 0.7458 36 0.0129 1.7158 0.6907 0.4631 0.7597 37 0.0129 1.7144 0.6827 0.4558 0.7472 377
38 0.0134 1.7142 0.6874 0.4491 0.7497 39 0.0131 1.7238 0.6915 0.4793 0.7679 40 0.0126 1.7846 0.6840 0.4550 0.7484 41 0.0124 1.7431 0.6827 0.4456 0.7425 42 0.0127 1.6913 0.6774 0.4657 0.7457 43 0.0128 1.6938 0.6878 0.4627 0.7562 44 0.0130 1.7356 0.6861 0.4499 0.7484 45 0.0129 1.7044 0.6937 0.4406 0.7532 46 0.0142 1.5515 0.7082 0.4230 0.7624 47 0.0145 1.5250 0.6939 0.4425 0.7542 48 0.0142 1.5565 0.6946 0.4264 0.7477 49 0.0145 1.5858 0.7141 0.4187 0.7675 50 0.0147 1.5480 0.7004 0.4128 0.7490
Table C.7 [M1, Large Airport, NLOS-S, 10]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.9472 3.62 1.0000 na 1.0000 2 0.0405 1.51 0.7527 0.4600 0.8227 3 0.0068 1.53 0.4215 0.7205 0.6162 4 0.0055 1.49 0.3632 0.7809 0.6158
Table C.8 [M2, Large Airport, NLOS-S, 10]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.0077 0.9026 0.0272 0.9885 0.5896 2 0.4108 1.0202 0.9991 0.5000 0.9995 3 0.3616 1.1888 0.8997 0.2535 0.9168 4 0.0616 1.2129 0.5739 0.5904 0.6961 5 0.0293 1.2987 0.4255 0.7323 0.6384 6 0.0191 1.3715 0.3655 0.7680 0.5975 7 0.0151 1.4010 0.3540 0.7672 0.5756 8 0.0136 1.4218 0.3172 0.8069 0.5844 9 0.0100 1.4638 0.2706 0.8365 0.5592 10 0.0069 1.5469 0.2290 0.8483 0.4894 11 0.0058 1.5939 0.2202 0.8543 0.4833 12 0.0057 1.5568 0.2035 0.8603 0.4526 378
13 0.0051 1.5582 0.1937 0.8624 0.4274 14 0.0045 1.4186 0.1731 0.8696 0.3775 15 0.0046 1.3895 0.1668 0.8813 0.4077 16 0.0039 1.5998 0.1656 0.8804 0.3975 17 0.0031 1.6714 0.1530 0.8857 0.3672 18 0.0026 1.8337 0.1518 0.8899 0.3847 19 0.0023 1.7839 0.1309 0.8974 0.3193 20 0.0020 1.8855 0.1255 0.8999 0.3029 21 0.0018 1.6713 0.1036 0.9179 0.2894 22 0.0014 1.8594 0.1039 0.9178 0.2915 23 0.0013 2.0904 0.0986 0.9215 0.2818 24 0.0011 2.0793 0.0945 0.9255 0.2862 25 0.0010 2.3074 0.0957 0.9209 0.2529 26 0.0008 2.5389 0.0833 0.9296 0.2260 27 0.0008 2.5639 0.0895 0.9240 0.2263 28 0.0010 2.6663 0.1166 0.8971 0.2207 29 0.0011 2.4930 0.1117 0.9010 0.2135 30 0.0007 2.5165 0.0767 0.9318 0.1803 31 0.0012 2.9665 0.1499 0.8719 0.2733 32 0.0020 2.8820 0.2285 0.7939 0.3043 33 0.0017 2.6577 0.1684 0.8532 0.2749 34 0.0008 2.3734 0.0871 0.9225 0.1856 35 0.0006 2.6767 0.0738 0.9363 0.2000 36 0.0007 2.4939 0.0782 0.9339 0.2209 37 0.0006 2.4335 0.0807 0.9315 0.2198 38 0.0006 2.7335 0.0749 0.9354 0.2013 39 0.0005 2.7079 0.0711 0.9373 0.1814 40 0.0005 2.6419 0.0717 0.9378 0.1952 41 0.0004 2.9897 0.0573 0.9474 0.1346 42 0.0004 2.6639 0.0590 0.9470 0.1547 43 0.0005 2.8143 0.0665 0.9393 0.1486 44 0.0004 2.8535 0.0623 0.9446 0.1662 45 0.0004 2.8068 0.0664 0.9403 0.1611 46 0.0005 2.9235 0.0722 0.9364 0.1826 47 0.0005 2.5063 0.0723 0.9384 0.2109 48 0.0005 2.6866 0.0662 0.9434 0.2014 49 0.0004 2.8176 0.0700 0.9381 0.1771 50 0.0005 2.6752 0.0829 0.9269 0.1913
379
Table C.9 [M1, Large Airport, NLOS, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.6500 2.0081 1.0000 Na 1.0000 2 0.0658 1.5691 0.8540 0.2409 0.8702 3 0.0246 1.6665 0.7168 0.3772 0.7538 4 0.0192 1.6556 0.6563 0.4535 0.7137 5 0.0158 1.6616 0.6321 0.4875 0.7019 6 0.0154 1.5953 0.6065 0.5119 0.6832 7 0.0137 1.7345 0.6000 0.5075 0.6715 8 0.0129 1.7410 0.5973 0.5086 0.6686 9 0.0119 1.7367 0.5837 0.5254 0.6614 10 0.0120 1.7104 0.5774 0.5328 0.6580 11 0.0127 1.6093 0.5742 0.5428 0.6611 12 0.0124 1.6512 0.5773 0.5444 0.6663 13 0.0114 1.6726 0.5637 0.5394 0.6437 14 0.0109 1.6763 0.5516 0.5647 0.6464 15 0.0102 1.7093 0.5471 0.5637 0.6390 16 0.0100 1.7910 0.5419 0.5801 0.6449 17 0.0100 1.7192 0.5455 0.5662 0.6387 18 0.0095 1.7540 0.5465 0.5713 0.6443 19 0.0091 1.7838 0.5398 0.5860 0.6473 20 0.0092 1.8190 0.5413 0.5837 0.6474 21 0.0097 1.7933 0.5447 0.5741 0.6441 22 0.0092 1.7990 0.5339 0.5886 0.6407 23 0.0094 1.7907 0.5350 0.5815 0.6364 24 0.0088 1.8581 0.5350 0.5756 0.6313 25 0.0085 1.8594 0.5247 0.5952 0.6331 26 0.0081 1.8851 0.5297 0.5847 0.6311
Table C.10 [M2, Large Airport, NLOS, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0047 1.3570 0.0673 0.9585 0.4245 2 0.0681 1.0911 0.9989 0 0.9989 3 0.1095 1.0773 0.9621 0.1277 0.9656 4 0.0730 1.1350 0.8856 0.2161 0.8987 380
5 0.0443 1.1876 0.8024 0.3170 0.8318 6 0.0287 1.2375 0.7273 0.3968 0.7737 7 0.0209 1.3132 0.6834 0.4549 0.7474 8 0.0174 1.3721 0.6631 0.4655 0.7285 9 0.0138 1.4586 0.6411 0.4966 0.7182 10 0.0122 1.5142 0.6319 0.4991 0.7082 11 0.0110 1.5759 0.6166 0.5128 0.6970 12 0.0100 1.6293 0.6015 0.5265 0.6862 13 0.0095 1.5847 0.5866 0.5150 0.6581 14 0.0093 1.6220 0.5892 0.5261 0.6698 15 0.0093 1.5823 0.5735 0.5480 0.6639 16 0.0087 1.6360 0.5640 0.5553 0.6565 17 0.0084 1.6433 0.5647 0.5615 0.6621 18 0.0080 1.6704 0.5602 0.5561 0.6516 19 0.0077 1.7050 0.5487 0.5828 0.6567 20 0.0075 1.7081 0.5532 0.5722 0.6547 21 0.0069 1.7604 0.5406 0.5850 0.6476 22 0.0073 1.7572 0.5487 0.5749 0.6505 23 0.0072 1.7628 0.5450 0.5728 0.6436 24 0.0074 1.7565 0.5421 0.5828 0.6477 25 0.0072 1.7956 0.5413 0.5888 0.6514 26 0.0072 1.8388 0.5477 0.5835 0.6563 27 0.0068 1.7969 0.5350 0.5876 0.6417 28 0.0064 1.7846 0.5295 0.5847 0.6308 29 0.0068 1.8494 0.5413 0.5906 0.6529 30 0.0064 1.8363 0.5310 0.5904 0.6384 31 0.0063 1.8375 0.5284 0.6094 0.6516 32 0.0066 1.8753 0.5350 0.5982 0.6509 33 0.0065 1.8542 0.5295 0.5958 0.6411 34 0.0064 1.8589 0.5215 0.6136 0.6455 35 0.0066 1.7816 0.5250 0.6005 0.6387 36 0.0066 1.8970 0.5240 0.6098 0.6457 37 0.0067 1.8722 0.5177 0.6163 0.6427 38 0.0067 1.8050 0.5290 0.5870 0.6325 39 0.0065 1.8788 0.5216 0.6028 0.6359 40 0.0066 1.8300 0.5118 0.6151 0.6330 41 0.0062 1.8672 0.5244 0.6005 0.6378 42 0.0062 1.8194 0.5115 0.6038 0.6218 43 0.0061 1.8352 0.5116 0.6135 0.6311 44 0.0062 1.8593 0.5115 0.6027 0.6206 45 0.0060 1.9013 0.5092 0.5988 0.6134 46 0.0059 1.8639 0.5015 0.6259 0.6282 47 0.0062 1.8949 0.5124 0.6123 0.6313 381
48 0.0057 1.8764 0.5016 0.6269 0.6295 49 0.0057 1.9382 0.4966 0.6254 0.6205 50 0.0056 1.9147 0.4990 0.6213 0.6199 51 0.0057 1.9081 0.5015 0.6215 0.6239 52 0.0059 1.9071 0.5153 0.6180 0.6409 53 0.0061 1.8970 0.5124 0.6021 0.6215 54 0.0060 1.9104 0.5140 0.6132 0.6345 55 0.0058 1.9121 0.5111 0.6092 0.6264 56 0.0059 1.9411 0.5113 0.6257 0.6425 57 0.0058 1.9050 0.5010 0.6185 0.6201 58 0.0059 1.8818 0.4989 0.6162 0.6146 59 0.0060 1.9156 0.5092 0.6147 0.6288 60 0.0059 1.9044 0.5121 0.6159 0.6342 61 0.0063 1.8697 0.5226 0.5918 0.6272 62 0.0062 1.8721 0.5218 0.5926 0.6265 63 0.0062 1.8926 0.5197 0.5920 0.6231 64 0.0064 1.9229 0.5152 0.6156 0.6384 65 0.0060 1.8680 0.5045 0.6086 0.6157 66 0.0061 1.8443 0.5119 0.6081 0.6262 67 0.0064 1.8472 0.5047 0.6145 0.6218 68 0.0063 1.8314 0.5152 0.6097 0.6327 69 0.0067 1.7549 0.5150 0.6124 0.6351 70 0.0060 1.8705 0.5095 0.6153 0.6298 71 0.0058 1.8264 0.4952 0.6291 0.6220 72 0.0064 1.8603 0.5116 0.6040 0.6222 73 0.0059 1.8830 0.5035 0.6214 0.6268 74 0.0061 1.8317 0.4910 0.6413 0.6283 75 0.0061 1.8655 0.5032 0.6149 0.6201 76 0.0062 1.8439 0.5102 0.6101 0.6259 77 0.0060 1.8642 0.5032 0.6188 0.6239 78 0.0058 1.8846 0.4973 0.6269 0.6230 79 0.0064 1.7536 0.5073 0.6257 0.6366 80 0.0062 1.8012 0.5060 0.6285 0.6374 81 0.0061 1.8419 0.5002 0.6189 0.6194 82 0.0060 1.8401 0.5032 0.6068 0.6121 83 0.0064 1.8234 0.5065 0.6209 0.6308 84 0.0059 1.8664 0.5068 0.6220 0.6323 85 0.0063 1.8232 0.5000 0.6126 0.6128 86 0.0061 1.8653 0.5152 0.6084 0.6317 87 0.0064 1.8418 0.5024 0.6219 0.6257 88 0.0060 1.8751 0.5063 0.6127 0.6225 89 0.0066 1.7958 0.5211 0.5914 0.6248 90 0.0071 1.6285 0.5131 0.6014 0.6218 382
91 0.0074 1.5905 0.5160 0.6055 0.6301 92 0.0078 1.5673 0.5271 0.6018 0.6429 93 0.0076 1.5980 0.5290 0.5829 0.6289 94 0.0076 1.6477 0.5324 0.5757 0.6276 95 0.0075 1.6348 0.5226 0.5851 0.6212 96 0.0076 1.6799 0.5319 0.5744 0.6254 97 0.0083 1.5022 0.5513 0.5550 0.6377 98 0.0083 1.5272 0.5534 0.5708 0.6537 99 0.0080 1.5953 0.5534 0.5470 0.6345 100 0.0078 1.6409 0.5652 0.5326 0.6406
Table C.11 [M1, Large Airport, NLOS-S, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.9251 3.6531 1.0000 Na 1.0000 2 0.0626 1.6233 0.8843 0.2043 0.8958 3 0.0079 1.79 0.5244 0.5862 0.6245 4 0.0032 1.62 0.3323 0.7641 0.5256 5 0.0020 1.67 0.2690 0.8157 0.4991
Table C.12 [M2, Large Airport, NLOS-S, 20] Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0025 1.3071 0.0083 0.9970 0.6400 2 0.2341 0.9405 0.9997 0 0.9997 3 0.4427 1.3891 0.9797 0.0325 0.9799 4 0.1394 1.1629 0.8084 0.3123 0.8370 5 0.0341 1.2856 0.5343 0.5761 0.6305 6 0.0149 1.4339 0.3584 0.7402 0.5346 7 0.0104 1.4866 0.3065 0.7805 0.5035 8 0.0085 1.4956 0.2564 0.8310 0.5093 9 0.0071 1.5602 0.2357 0.8414 0.4860 10 0.0062 1.6656 0.2314 0.8451 0.4850 11 0.0064 1.5427 0.2238 0.8534 0.4919 12 0.0069 1.5159 0.2248 0.8485 0.4779 13 0.0060 1.5698 0.2194 0.8456 0.4506 14 0.0058 1.5399 0.1967 0.8670 0.4571 15 0.0049 1.5793 0.1809 0.8819 0.4653 383
16 0.0043 1.6563 0.1604 0.8950 0.4505 17 0.0040 1.6300 0.1518 0.8975 0.4270 18 0.0030 1.7415 0.1435 0.8977 0.3894 19 0.0028 1.6438 0.1370 0.9015 0.3788 20 0.0027 1.8410 0.1311 0.9102 0.4048 21 0.0022 1.8307 0.1119 0.9119 0.3013 22 0.0023 1.6995 0.1184 0.9115 0.3394 23 0.0024 1.7990 0.1213 0.9125 0.3665 24 0.0021 1.6880 0.1056 0.9194 0.3177 25 0.0020 1.6526 0.0984 0.9250 0.3126 26 0.0026 1.3930 0.0929 0.9275 0.2918 27 0.0028 1.2916 0.0922 0.9268 0.2796 28 0.0013 1.9401 0.0914 0.9245 0.2495 29 0.0010 2.3937 0.0865 0.9280 0.2395 30 0.0018 1.7417 0.0979 0.9239 0.2990 31 0.0021 1.5796 0.0949 0.9278 0.3118 32 0.0013 1.9178 0.0870 0.9310 0.2757 33 0.0014 1.9774 0.0903 0.9297 0.2912 34 0.0013 2.0568 0.0846 0.9286 0.2285 35 0.0009 2.1465 0.0794 0.9373 0.2729 36 0.0011 1.9008 0.0676 0.9429 0.2127 37 0.0009 2.0249 0.0656 0.9467 0.2393 38 0.0010 2.1451 0.0679 0.9468 0.2701 39 0.0008 1.7595 0.0555 0.9533 0.2054 40 0.0008 2.0060 0.0547 0.9528 0.1843 41 0.0006 2.2367 0.0514 0.9556 0.1801 42 0.0005 2.4354 0.0474 0.9571 0.1394 43 0.0005 2.3967 0.0512 0.9547 0.1613 44 0.0006 2.2791 0.0466 0.9596 0.1738 45 0.0005 2.5538 0.0494 0.9560 0.1538 46 0.0005 2.5622 0.0468 0.9598 0.1802 47 0.0004 2.8013 0.0479 0.9580 0.1655 48 0.0004 3.0022 0.0473 0.9592 0.1783 49 0.0004 2.7030 0.0427 0.9627 0.1628 50 0.0003 2.9953 0.0387 0.9646 0.1197 51 0.0004 2.7165 0.0387 0.9637 0.0983 52 0.0003 3.1720 0.0354 0.9685 0.1402 53 0.0004 2.9551 0.0565 0.9469 0.1140 54 0.0004 2.9697 0.0603 0.9437 0.1233 55 0.0005 2.8671 0.0650 0.9404 0.1425 56 0.0005 2.9267 0.0620 0.9425 0.1307 57 0.0004 2.7463 0.0423 0.9608 0.1133 58 0.0003 2.3295 0.0354 0.9669 0.0981 384
59 0.0003 2.9406 0.0415 0.9619 0.1195 60 0.0005 3.2786 0.0708 0.9340 0.1332 61 0.0011 2.9218 0.1395 0.8826 0.2761 62 0.0010 3.1372 0.1172 0.8899 0.1709 63 0.0012 2.8837 0.1385 0.8770 0.2351 64 0.0008 3.0539 0.0863 0.9240 0.1954 65 0.0004 2.6044 0.0433 0.9599 0.1107 66 0.0003 2.7866 0.0403 0.9636 0.1352 67 0.0003 2.5907 0.0365 0.9670 0.1312 68 0.0002 2.9569 0.0349 0.9674 0.0995 69 0.0003 2.7246 0.0336 0.9689 0.1034 70 0.0002 2.9420 0.0339 0.9689 0.1122 71 0.0003 2.8314 0.0364 0.9677 0.1455 72 0.0003 2.8109 0.0354 0.9674 0.1121 73 0.0002 3.0367 0.0354 0.9673 0.1075 74 0.0002 2.9681 0.0331 0.9702 0.1300 75 0.0002 3.2059 0.0352 0.9680 0.1221 76 0.0002 3.1654 0.0307 0.9719 0.1129 77 0.0002 2.9496 0.0357 0.9659 0.0787 78 0.0002 2.8424 0.0319 0.9693 0.0674 79 0.0002 3.3635 0.0291 0.9722 0.0739 80 0.0002 3.1485 0.0283 0.9729 0.0702 81 0.0002 3.4130 0.0274 0.9742 0.0843 82 0.0002 3.1928 0.0271 0.9743 0.0793 83 0.0002 3.1440 0.0266 0.9752 0.0932 84 0.0002 3.2090 0.0299 0.9715 0.0773 85 0.0002 2.9922 0.0294 0.9716 0.0618 86 0.0002 3.3238 0.0281 0.9721 0.0353 87 0.0002 3.2082 0.0291 0.9722 0.0739 88 0.0002 3.4926 0.0303 0.9710 0.0710 89 0.0002 3.1158 0.0375 0.9636 0.0661 90 0.0002 3.5244 0.0324 0.9701 0.1071 91 0.0002 2.9900 0.0306 0.9724 0.1243 92 0.0002 2.6960 0.0319 0.9701 0.0933 93 0.0002 3.2103 0.0322 0.9703 0.1077 94 0.0002 2.8574 0.0298 0.9717 0.0778 95 0.0002 3.0568 0.0317 0.9703 0.0938 96 0.0002 3.4997 0.0341 0.9678 0.0874 97 0.0002 2.8675 0.0360 0.9666 0.1055 98 0.0002 2.9455 0.0415 0.9612 0.1036 99 0.0003 3.0213 0.0503 0.9537 0.1250 100 0.0006 2.6965 0.1035 0.9021 0.1518
385
Table C.13 Correlation coefficient matrix for [M1, Large Airport, NLOS-S NLOS, 10], lower triangular part for NLOS and upper triangular part for NLOS-S.
i, 1 2 3 4 5 6 7 8 9 10 11 12 13 j 1 1 0.31 0.36 0.23 — — — — — — — — — 2 0.47 1 - - — — — — — — — — — 0.23 0.21 3 0.28 0.25 1 0.28 — — — — — — — — — 4 0.25 0.09 0.17 1 — — — — — — — — — 5 0.15 0.07 0.15 0.23 1 — — — — — — — — 6 0.27 0.17 0.08 0.03 0.12 1 — — — — — — — 7 0.17 0.10 0.06 0.18 0.20 0.12 1 — — — — — — 8 0.13 0.10 0.08 0.09 0.10 0.11 0.29 1 — — — — — 9 0.12 0.12 0.08 0.15 0.05 0.06 0.16 0.24 1 — — — — 10 0.12 0.08 0.12 0.00 0.07 0.12 0.09 0.24 0.24 1 — — — 11 0.13 0.04 0.06 0.12 0.07 0.08 0.09 0.14 0.04 0.17 1 — — 12 0.10 0.07 0.06 0.01 0.12 0.08 0.07 0.17 0.04 0.18 0.20 1 — 13 0.07 0.07 0.06 0.02 0.02 0.03 0.10 0.07 0.04 0.05 0.15 0.01 1 14 0.00 - 0.02 0.10 0.05 0.02 0.03 0.12 0.05 0.06 0.06 - 0.11 0.04 0.03
Table C.14 Correlation coefficient matrix for [M1, Large Airport, NLOS-S NLOS, 5], lower triangular matrix for NLOS and upper triangular part for NLOS-S.
i, 1 2 3 4 5 6 7 j 1 1 - - — — — — 0.31 0.07 2 0.40 1 0.02 — — — — 3 0.31 0.19 1 — — — — 4 0.22 0.07 0.14 1 — — — 5 0.14 0.07 0.14 0.12 1 — — 6 0.13 0.03 0.10 0.10 0.13 1 — 7 0.11 0.00 0.08 0.09 0.07 0.12 1 8 0.11 0.11 0.07 0.06 0.03 0.03 0.07
386
C.2 [M1 M2, Medium Airport, All regions, 5 10 20]
This section in the appendix presents the 5, 10 and 20 MHz models for different regions of the medium airport. M1 for 50 MHz and [M1 M2 and M3] for 25 MHz have already been presented in Chapter 6.
Table C.15 [M1, Medium Airport, NLOS, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.8309 1.64 1.0000 na 1.0000 2 0.0683 1.32 0.8086 0.3018 0.8356 3 0.0444 1.34 0.7379 0.3454 0.7684 4 0.0297 1.40 0.6974 0.3989 0.7389 5 0.0267 1.36 0.6578 0.4710 0.7257
Table C.16 [M2, Medium Airport, NLOS, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0483 0.9368 0.2647 0.8427 0.5635 2 0.2358 0.8870 0.9853 0.1765 0.9877 3 0.1637 0.9850 0.9000 0.2414 0.9156 4 0.0850 1.0979 0.8069 0.2768 0.8278 5 0.0643 1.1538 0.7569 0.4291 0.8176 6 0.0503 1.1972 0.7103 0.4792 0.7874 7 0.0317 1.2704 0.6560 0.4536 0.7145 8 0.0275 1.3328 0.6319 0.4871 0.7022 9 0.0221 1.3224 0.6233 0.5400 0.7230 10 0.0229 1.2628 0.6224 0.5274 0.7143 11 0.0190 1.3196 0.5672 0.5717 0.6743 12 0.0193 1.2692 0.5560 0.5786 0.6646 13 0.0192 1.3219 0.5931 0.5360 0.6827 14 0.0188 1.4126 0.5716 0.5573 0.6677 15 0.0191 1.3377 0.5698 0.5351 0.6500 16 0.0146 1.3755 0.5534 0.5734 0.6568 17 0.0152 1.4791 0.5741 0.5628 0.6752 387
18 0.0145 1.4235 0.5457 0.6167 0.6820 19 0.0135 1.4381 0.5233 0.6184 0.6535 20 0.0160 1.3882 0.5440 0.5728 0.6429 21 0.0162 1.4317 0.5414 0.5639 0.6316 22 0.0170 1.3850 0.5655 0.5972 0.6901 23 0.0154 1.4012 0.5431 0.5981 0.6630 24 0.0139 1.4032 0.5517 0.5769 0.6557 25 0.0166 1.4289 0.5638 0.5672 0.6662
Table C.17 [M1, Medium Airport, NLOS-S, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.9728 4.04 1.0000 na 1.0000 2 0.0272 1.55 0.6011 0.5532 0.7028
Table C.18 [M2, Medium Airport, NLOS-S, 5]
Tap Energy Weibull P1,k P00,k P11,k N(707) Index Shape k Factor (bk) 1 0.0380 1.2302 0.0651 0.9742 0.6304 45 2 0.6314 1.1115 0.9873 0.4444 0.9928 697 3 0.2388 1.0047 0.7100 0.4244 0.7645 501 4 0.0412 1.2005 0.4215 0.7188 0.6128 297 5 0.0155 1.3582 0.2984 0.8085 0.5524 210 6 0.0108 1.4290 0.2631 0.8442 0.5645 185 7 0.0052 1.7429 0.2122 0.8237 0.3467 149 8 0.0033 1.5961 0.1938 0.8348 0.3139 136 9 0.0049 1.5276 0.1796 0.8705 0.4094 126 10 0.0020 1.6786 0.1429 0.8826 0.2871 100 11 0.0016 1.9365 0.1485 0.8686 0.2476 104 12 0.0008 2.5820 0.0934 0.9187 0.1970 65 13 0.0009 1.5910 0.1018 0.9243 0.3194 71 14 0.0008 1.1163 0.0948 0.9139 0.1791 66 15 0.0011 1.2919 0.1287 0.8862 0.2308 90 16 0.0008 1.2370 0.1132 0.8978 0.2000 79 17 0.0009 3.4703 0.1655 0.8540 0.2650 116 18 0.0005 2.8129 0.0778 0.9264 0.1296 54 19 0.0003 2.4862 0.0509 0.9522 0.1111 35 388
20 0.0003 1.1304 0.0438 0.9556 0.0323 30 21 0.0002 2.7122 0.0453 0.9585 0.1250 31 22 0.0002 1.0371 0.0325 0.9693 0.0870 22 23 0.0002 1.1975 0.0382 0.9661 0.1481 26 24 0.0003 1.0228 0.0523 0.9552 0.1622 36 25 0.0002 3.4900 0.0297 0.9723 0.0952 20
Table C.19 [M1, Medium Airport, LOS-O, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.9893 5.83 1.0000 na 1.0000 2 0.0107 1.59 0.4458 0.6667 0.5830
Table C.20 [M2, Medium Airport, LOS-O, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0612 1.3393 0.0818 0.9700 0.6667 2 0.7344 1.1029 0.9818 0 0.9813 3 0.1982 0.9329 0.6091 0.5238 0.6866 4 0.0046 1.3056 0.1273 0.8737 0.1429 5 0.0008 1.6624 0.0636 0.9510 0.2857 6 0.0004 1.2607 0.0636 0.9314 0 7 0.0002 1.5284 0.0727 0.9406 0.2500 8 0.0001 1.3293 0.0273 0.9720 0
Table C.21 [M1, Medium Airport, NLOS, 10]
Tap Energy Weibull P1,k P00,k P11,k Index k Shape Factor (bk) 1 0.8323 1.62 1.0000 na 1.0000 2 0.0660 1.28 0.7867 0.3399 0.8208 3 0.0220 1.45 0.6265 0.5034 0.7035 4 0.0198 1.42 0.5767 0.5398 0.6618 5 0.0224 1.36 0.5725 0.5779 0.6858 6 0.0148 1.40 0.5582 0.6031 0.6868 389
7 0.0107 1.49 0.5017 0.5956 0.5993 8 0.0122 1.42 0.5034 0.5823 0.5889
Table C.22 [M2, Medium Airport, NLOS, 10]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.0230 0.9348 0.1290 0.9390 0.5882 2 0.2198 0.8867 0.9907 0.1818 0.9923 3 0.1935 0.9710 0.9039 0.2018 0.9150 4 0.0771 1.0797 0.7378 0.3859 0.7815 5 0.0392 1.2197 0.6669 0.4354 0.7177 6 0.0317 1.2153 0.6088 0.5862 0.7337 7 0.0282 1.3177 0.6054 0.5385 0.7001 8 0.0199 1.3134 0.5346 0.5743 0.6303 9 0.0174 1.3242 0.5337 0.5967 0.6487 10 0.0159 1.3730 0.5202 0.5870 0.6201 11 0.0145 1.4206 0.5008 0.6199 0.6223 12 0.0115 1.3824 0.4941 0.6400 0.6325 13 0.0087 1.3816 0.4536 0.6512 0.5810 14 0.0112 1.3955 0.4469 0.6326 0.5463 15 0.0110 1.5015 0.4384 0.6532 0.5568 16 0.0109 1.3468 0.4654 0.6325 0.5789 17 0.0092 1.3837 0.4224 0.7080 0.6000 18 0.0084 1.3673 0.4233 0.6798 0.5649 19 0.0082 1.3345 0.4039 0.7082 0.5699 20 0.0073 1.4281 0.3836 0.7315 0.5692 21 0.0077 1.4427 0.3997 0.6812 0.5201 22 0.0078 1.3571 0.4098 0.6971 0.5649 23 0.0070 1.3641 0.4039 0.6973 0.5523 24 0.0085 1.4665 0.4106 0.6810 0.5432 25 0.0068 1.4820 0.3895 0.7210 0.5640 26 0.0088 1.4179 0.3862 0.6951 0.5164 27 0.0081 1.3899 0.4013 0.6897 0.5378 28 0.0092 1.4712 0.4216 0.6574 0.5311 29 0.0078 1.5372 0.4073 0.6866 0.5445 30 0.0079 1.4472 0.4047 0.7035 0.5646 31 0.0068 1.4303 0.3820 0.7176 0.5420 32 0.0077 1.4352 0.3997 0.6826 0.5243 33 0.0068 1.3888 0.3702 0.7336 0.5457 34 0.0061 1.4898 0.3634 0.7520 0.5661 390
35 0.0070 1.4624 0.3853 0.7037 0.5285 36 0.0068 1.4811 0.3904 0.7165 0.5584 37 0.0069 1.4494 0.3938 0.7368 0.5953 38 0.0071 1.4173 0.3820 0.7271 0.5597 39 0.0079 1.3524 0.3769 0.7182 0.5347 40 0.0070 1.3544 0.3592 0.7013 0.4682 41 0.0072 1.4660 0.3853 0.7188 0.5504 42 0.0074 1.4198 0.3617 0.7213 0.5093 43 0.0080 1.4094 0.3685 0.7059 0.4966 44 0.0083 1.3253 0.3761 0.7388 0.5673 45 0.0073 1.5285 0.4073 0.6923 0.5528 46 0.0070 1.4785 0.3718 0.7195 0.5250 47 0.0072 1.4198 0.3879 0.6997 0.5272 48 0.0100 1.3539 0.4165 0.6777 0.5497 49 0.0110 1.2593 0.4056 0.6936 0.5521 50 0.0104 1.4204 0.4191 0.6749 0.5504
Table C.23 [M1, Medium Airport, NLOS-S, 10]
Tap Energy Weibull P1,k P00,k P11,k Index k Shape Factor (bk) 1 0.9479 4.0 1.0000 na 1.0000 2 0.0521 1.48 0.7249 0.3862 0.7666
Table C.24 [M2, Medium Airport, NLOS-S, 10]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.0166 1.2236 0.0408 0.9772 0.4643 2 0.4870 0.9678 0.9971 0 0.9971 3 0.3700 1.1290 0.8748 0.2674 0.8950 4 0.0506 1.1748 0.5080 0.5888 0.6006 5 0.0148 1.4182 0.3435 0.7422 0.5042 6 0.0120 1.4699 0.2809 0.8032 0.4974 7 0.0114 1.2226 0.2242 0.8402 0.4481 8 0.0047 1.6379 0.1907 0.8667 0.4351 9 0.0066 1.4985 0.1907 0.8613 0.4046 10 0.0040 1.4715 0.1645 0.8743 0.3628 11 0.0023 1.4747 0.1033 0.9008 0.1408 391
12 0.0013 1.8732 0.1106 0.8934 0.1447 13 0.0017 2.4986 0.1310 0.8859 0.2444 14 0.0013 1.5920 0.0844 0.9220 0.1552 15 0.0015 1.5467 0.0888 0.9232 0.2131 16 0.0026 1.6153 0.1077 0.9199 0.3378 17 0.0014 1.3933 0.1135 0.9013 0.2308 18 0.0008 2.2783 0.0742 0.9433 0.2941 19 0.0008 1.4563 0.0801 0.9239 0.1273 20 0.0008 2.1080 0.0684 0.9421 0.2128 21 0.0006 2.8298 0.0684 0.9421 0.2128 22 0.0005 0.9964 0.0437 0.9573 0.0667 23 0.0005 1.7766 0.0640 0.9486 0.2273 24 0.0005 2.4146 0.0422 0.9635 0.1724 25 0.0002 1.3566 0.0320 0.9729 0.1818 26 0.0003 1.0033 0.0437 0.9558 0.0333 27 0.0005 1.4032 0.0509 0.9539 0.1143 28 0.0006 1.2556 0.0728 0.9324 0.1400 29 0.0006 1.5168 0.0757 0.9259 0.0962 30 0.0003 0.8932 0.0393 0.9636 0.1111 31 0.0003 4.6489 0.0539 0.9538 0.1622 32 0.0006 3.6070 0.1194 0.8992 0.2593 33 0.0003 4.3046 0.0524 0.9492 0.0833 34 0.0003 2.5570 0.0451 0.9603 0.1290 35 0.0001 2.4703 0.0233 0.9776 0.0625 36 0.0001 2.7207 0.0204 0.9792 0 38 0.0001 3.3656 0.0218 0.9791 0.0667 39 0.0001 0.9162 0.0204 0.9807 0.0714 40 0.0001 6.6691 0.0189 0.9837 0.1538 42 0.0002 5.0186 0.0291 0.9715 0.0500 43 0.0001 0.8471 0.0189 0.9807 0 44 0.0001 4.0198 0.0247 0.9821 0.2941 45 0.0002 1.0924 0.0277 0.9760 0.1579 46 0.0001 5.0590 0.0247 0.9746 0 50 0.0001 2.9676 0.0233 0.9776 0.0625
392
Table C.25 [M1, Medium Airport, LOS-O, 10]
Tap Energy Weibull P1,k P00,k P11,k Index k Shape Factor (bk) 1 0.9778 4.54 1.0000 na 1.0000 2 0.0222 1.67 0.6405 0.4467 0.6886
Table C.26 [M2, Medium Airport, LOS-O, 10]
Weibull Tap Index k Energy Shape P1 P00,k P11,k N Factor (bk) 1 0.0293 0.9529 0.0268 0.9722 0 24 2 0.5680 1.0086 0.9911 0 0.9909 543 3 0.3721 0.8947 0.7946 0.2273 0.7978 442 4 0.0218 1.0634 0.3839 0.6618 0.4651 186 5 0.0062 0.9833 0.1161 0.8878 0.1538 73 6 0.0010 1.8152 0.0804 0.9118 0 42 7 0.0004 2.1420 0.0357 0.9720 0.2500 25 12 0.0001 5.2677 0.0625 0.9327 0 13
Table C.27 [M1, Medium Airport, NLOS, 20]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.837 1.5867 1.0000 na 1.0000 2 0.076 1.3342 0.7701 0.3811 0.8150 3 0.019 1.4443 0.5812 0.5221 0.6551 4 0.013 1.4912 0.5064 0.5814 0.5914 5 0.010 1.4350 0.4711 0.6210 0.5734 6 0.007 1.4641 0.4220 0.6825 0.5657 7 0.007 1.4470 0.4132 0.6493 0.5029 8 0.008 1.5378 0.4220 0.6996 0.5878 9 0.012 1.3594 0.4558 0.6736 0.6113 10 0.007 1.6166 0.4164 0.7025 0.5841 11 0.005 1.4417 0.3770 0.6929 0.4936
393
Table C.28 [M2, Medium Airport, NLOS, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0110 0.8456 0.0780 0.9555 0.4742 2 0.1580 0.9266 0.9944 0 0.9943 3 0.2190 1.0038 0.8987 0.2381 0.9141 4 0.1029 1.0841 0.7677 0.4360 0.8291 5 0.0471 1.1482 0.6294 0.5119 0.7123 6 0.0237 1.2211 0.5474 0.5382 0.6176 7 0.0171 1.3023 0.4863 0.6088 0.5861 8 0.0110 1.4173 0.4461 0.6352 0.5477 9 0.0101 1.3987 0.4325 0.6638 0.5595 10 0.0115 1.4055 0.4486 0.6662 0.5889 11 0.0147 1.3968 0.4421 0.6686 0.5829 12 0.0091 1.4431 0.4051 0.6905 0.5467 13 0.0064 1.4815 0.3730 0.7295 0.5464 14 0.0076 1.5554 0.3947 0.6959 0.5347 15 0.0081 1.3561 0.3674 0.7061 0.4945 16 0.0068 1.5644 0.3706 0.7088 0.5065 17 0.0055 1.4463 0.3521 0.7097 0.4668 18 0.0061 1.4631 0.3465 0.7565 0.5419 19 0.0050 1.4485 0.3143 0.7538 0.4615 20 0.0067 1.3701 0.3352 0.7316 0.4688 21 0.0060 1.4569 0.3392 0.7336 0.4822 22 0.0047 1.5767 0.3135 0.7611 0.4781 23 0.0039 1.5842 0.2854 0.7728 0.4322 24 0.0037 1.4458 0.2878 0.7810 0.4594 25 0.0041 1.4265 0.2894 0.7885 0.4819 26 0.0051 1.4590 0.2934 0.7668 0.4396 27 0.0046 1.4852 0.3143 0.7570 0.4706 28 0.0045 1.4085 0.2982 0.7709 0.4622 29 0.0055 1.5582 0.2950 0.7811 0.4754 30 0.0038 1.4785 0.2749 0.7936 0.4561 31 0.0043 1.4367 0.3071 0.7610 0.4593 32 0.0042 1.4180 0.2789 0.7924 0.4640 33 0.0038 1.4761 0.2701 0.7874 0.4269 34 0.0038 1.4828 0.2588 0.8111 0.4596 35 0.0033 1.4597 0.2556 0.8153 0.4637 36 0.0041 1.4483 0.2902 0.7916 0.4917 37 0.0040 1.3762 0.2532 0.8267 0.4904 38 0.0037 1.5106 0.2765 0.8044 0.4898 394
39 0.0034 1.5012 0.2580 0.8015 0.4299 40 0.0034 1.4559 0.2548 0.8045 0.4259 41 0.0037 1.4698 0.2677 0.8024 0.4608 42 0.0034 1.3829 0.2637 0.8002 0.4434 43 0.0034 1.4014 0.2613 0.8148 0.4738 44 0.0045 1.3991 0.2605 0.8074 0.4537 45 0.0033 1.4388 0.2452 0.8284 0.4721 46 0.0034 1.4773 0.2516 0.8161 0.4537 47 0.0038 1.4991 0.2548 0.8013 0.4196 48 0.0029 1.4823 0.2395 0.8214 0.4343 49 0.0038 1.4168 0.2452 0.8222 0.4539 50 0.0034 1.4693 0.2476 0.8045 0.4072 51 0.0035 1.4470 0.2540 0.8177 0.4652 52 0.0038 1.4738 0.2677 0.7978 0.4474 53 0.0046 1.4545 0.2685 0.7956 0.4444 54 0.0047 1.5001 0.2749 0.7871 0.4399 55 0.0045 1.3724 0.2709 0.8004 0.4643 56 0.0036 1.5282 0.2637 0.7991 0.4373 57 0.0036 1.4975 0.2613 0.7998 0.4352 58 0.0038 1.3637 0.2661 0.8015 0.4502 59 0.0038 1.4478 0.2749 0.7894 0.4457 60 0.0038 1.5514 0.2516 0.8151 0.4473 61 0.0029 1.4558 0.2516 0.8118 0.4409 62 0.0038 1.5178 0.2508 0.8036 0.4148 63 0.0035 1.4739 0.2637 0.8068 0.4587 64 0.0033 1.4999 0.2580 0.8037 0.4361 65 0.0033 1.5255 0.2540 0.8209 0.4747 66 0.0033 1.4382 0.2420 0.8333 0.4751 67 0.0031 1.3981 0.2379 0.8342 0.4696 68 0.0034 1.4626 0.2548 0.8058 0.4335 69 0.0038 1.5673 0.2653 0.8050 0.4606 70 0.0026 1.4070 0.2355 0.8326 0.4573 71 0.0035 1.5640 0.2540 0.8082 0.4381 72 0.0036 1.4577 0.2484 0.8180 0.4498 73 0.0036 1.3676 0.2331 0.8291 0.4394 74 0.0031 1.4581 0.2307 0.8368 0.4564 75 0.0035 1.3887 0.2492 0.8178 0.4516 76 0.0037 1.4342 0.2492 0.8276 0.4822 77 0.0045 1.4132 0.2524 0.8256 0.4841 78 0.0041 1.4792 0.2379 0.8186 0.4203 79 0.0035 1.4051 0.2452 0.8060 0.4000 80 0.0033 1.4149 0.2412 0.8229 0.4433 81 0.0030 1.5354 0.2540 0.8026 0.4209 395
82 0.0030 1.4056 0.2428 0.8153 0.4252 83 0.0039 1.3962 0.2395 0.8192 0.4276 84 0.0034 1.4084 0.2605 0.8085 0.4568 85 0.0033 1.4906 0.2476 0.8225 0.4610 86 0.0031 1.4582 0.2395 0.8169 0.4195 87 0.0041 1.4100 0.2725 0.7967 0.4586 88 0.0034 1.4141 0.2621 0.8061 0.4554 89 0.0039 1.4681 0.2629 0.8024 0.4465 90 0.0040 1.4341 0.2588 0.7939 0.4081 91 0.0042 1.4205 0.2492 0.8060 0.4161 92 0.0037 1.3608 0.2524 0.8052 0.4236 93 0.0047 1.3676 0.2669 0.7980 0.4458 94 0.0043 1.3665 0.2797 0.7810 0.4368 95 0.0052 1.3625 0.2685 0.7943 0.4401 96 0.0051 1.3698 0.2693 0.7941 0.4418 97 0.0055 1.1687 0.2669 0.7859 0.4127 98 0.0048 1.3589 0.2757 0.7980 0.4708 99 0.0054 1.4394 0.3111 0.7608 0.4715 100 0.0064 1.4745 0.3199 0.7317 0.4307
Table C.29 [M1, Medium Airport, NLOS-S, 20]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.917 3.73 1.0000 na 1.0000 2 0.071 1.53 0.8188 0.3071 0.8482 3 0.012 1.53 0.5007 0.5743 0.5743
Table C.30 [M2, Medium Airport, NLOS-S, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.2754 0.8579 0.9986 0 0.9986 2 0.4594 1.0548 0.9572 0.0667 0.9582 3 0.1395 0.9859 0.7546 0.3198 0.7803 4 0.0389 1.2421 0.5107 0.5860 0.6022 5 0.0152 1.2651 0.3210 0.7521 0.4777 6 0.0072 1.4331 0.2397 0.8105 0.4012 7 0.0038 1.6737 0.2068 0.8324 0.3586 396
8 0.0056 1.4191 0.1954 0.8544 0.4015 9 0.0030 1.6850 0.1427 0.8719 0.2323 10 0.0056 1.5496 0.1398 0.9037 0.4082 11 0.0046 1.3456 0.1312 0.8947 0.3043 12 0.0028 1.4104 0.0999 0.9208 0.2899 13 0.0014 1.7993 0.1056 0.9217 0.3378 14 0.0023 1.9936 0.1084 0.9071 0.2368 15 0.0022 1.6064 0.0999 0.9286 0.3571 16 0.0018 2.0988 0.0913 0.9324 0.3281 17 0.0016 1.9186 0.0942 0.9196 0.2273 18 0.0009 2.3414 0.0642 0.9450 0.2000 19 0.0014 1.5473 0.0785 0.9271 0.1455 20 0.0012 1.4420 0.0628 0.9390 0.0909 21 0.0009 1.7728 0.0685 0.9356 0.1250 22 0.0007 2.8317 0.0728 0.9384 0.2157 23 0.0007 2.8902 0.0628 0.9361 0.0465 24 0.0007 1.3297 0.0542 0.9456 0.0526 25 0.0009 1.7130 0.0599 0.9422 0.0952 26 0.0009 1.5494 0.0528 0.9532 0.1622 27 0.0008 1.5835 0.0528 0.9532 0.1622 28 0.0010 2.1322 0.0585 0.9530 0.2439 29 0.0013 1.5854 0.0599 0.9468 0.1667 30 0.0007 1.8451 0.0613 0.9498 0.2326 31 0.0005 1.2553 0.0485 0.9595 0.1765 32 0.0004 4.2969 0.0385 0.9703 0.2593 33 0.0006 2.0086 0.0499 0.9579 0.2000 34 0.0004 1.6042 0.0428 0.9582 0.0667 35 0.0004 1.8199 0.0357 0.9630 0 36 0.0003 2.3213 0.0385 0.9629 0.0741 37 0.0003 2.5579 0.0314 0.9720 0.1364 38 0.0003 2.7051 0.0357 0.9659 0.0800 39 0.0003 2.8072 0.0342 0.9675 0.0833 40 0.0002 2.5848 0.0300 0.9735 0.1429 41 0.0004 2.6035 0.0228 0.9781 0.0625 42 0.0001 2.7610 0.0243 0.9751 0 43 0.0003 2.8336 0.0371 0.9629 0.0385 44 0.0002 2.7040 0.0257 0.9780 0.1667 48 0.0002 2.2993 0.0243 0.9795 0.1765 49 0.0002 2.1383 0.0214 0.9781 0 50 0.0003 0.9926 0.0342 0.9734 0.2500 51 0.0002 1.6579 0.0257 0.9765 0.1111 52 0.0002 1.0603 0.0342 0.9675 0.0833 53 0.0003 2.3092 0.0442 0.9537 0 397
54 0.0003 2.0919 0.0371 0.9659 0.1154 55 0.0002 2.9245 0.0328 0.9675 0.0435 56 0.0002 0.8061 0.0214 0.9781 0 57 0.0001 6.9412 0.0185 0.9825 0.0769 58 0.0001 1.0077 0.0200 0.9796 0 59 0.0001 3.7140 0.0200 0.9796 0 60 0.0003 4.3646 0.0699 0.9324 0.1020 61 0.0004 3.8219 0.0685 0.9371 0.1458 62 0.0002 4.0893 0.0428 0.9597 0.1000 63 0.0002 7.3575 0.0399 0.9673 0.2143 65 0.0001 2.26 0.0200 0.9796 0 99 0.0003 1.24 0.0799 0.9224 0.1071
Table C.31 [M1, Medium Airport, LOS-O, 20]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.968 4.83 1.0000 na 1.0000 2 0.032 1.63 0.6735 0.4194 0.7121
Table C.32 [M2, Medium Airport, LOS-O, 20]
Tap Energy Weibull P1,k P00,k P11,k N Index Shape k Factor (bk) 1 0.0165 0.8614 0.0306 0.9681 0 13 2 0.3040 0.9068 1.0000 na 1.0000 472 3 0.5393 1.0581 0.8776 0.2500 0.8941 439 4 0.1048 0.7840 0.6327 0.4286 0.6613 333 5 0.0239 0.8843 0.4082 0.7018 0.5500 183 6 0.0040 1.5943 0.1837 0.8228 0.2222 85 7 0.0032 1.5166 0.1429 0.8554 0.1429 50 8 0.0017 1.1859 0.0510 0.9457 0 26 9 0.0005 2.4750 0.0510 0.9457 0 23 10 0.0004 2.5538 0.0510 0.9565 0.2000 18 11 0.0003 2.4283 0.0102 0.9896 0 13
398
Table C.33 Correlation coefficient matrix for [M1, Medium Airport, NLOS-S NLOS, 10], lower triangular matrix for NLOS and upper triangular matrix for NLOS-S.
i, 1 2 3 4 5 6 7 j 1 1 0.75 — — — — — 2 0.44 1 — — — — — 3 0.45 0.37 1 — — — — 4 0.50 0.33 0.42 1 — — — 5 0.43 0.37 0.33 0.35 1 — — 6 0.47 0.40 0.50 0.49 0.47 1 — 7 0.47 0.39 0.40 0.47 0.43 0.53 1 8 0.47 0.39 0.41 0.28 0.34 0.45 0.46
Table C.34 Correlation Coefficient matrix for [M1, Medium Airport, NLOS-S NLOS, 5], lower Triangular matrix for NLOS and upper Triangular matrix for NLOS-S
i, j 1 2 3 4 1 1 0.09 — — 2 0.51 1 — — 3 0.48 0.39 1 — 4 0.48 0.42 0.40 1 5 0.43 0.36 0.50 0.39
Table C.35 Correlation Coefficient matrix for [M1, Medium Airport, LOS-O, 5 10], lower Triangular matrix for 10 MHz and upper Triangular matrix for 5 MHz
i, j 1 2 1 1 0.8155 2 0.95 1
C.3 [M1 M2, Small Airport, All regions, 5 10 20]
This section in the appendix presents the 5, 10 and 20 MHz models for different regions of the small airport. M1 for 50 MHz and [M1 M2 and M3] for 25 MHz have already been presented in Chapter 6. 399
Table C.36 [M1, Small Airport, NLOS, 5]
Tap Energy Weibull P1,k P00,k P11,k Index k Shape Factor (bk) 1 0.730 2.12 1.0000 na 1.0000 2 0.098 1.57 0.9162 0.2426 0.9306 3 0.064 1.49 0.8704 0.2824 0.8935 4 0.040 1.67 0.8004 0.3168 0.8300 5 0.037 1.68 0.7935 0.2608 0.8081 6 0.029 2 0.7639 0.3515 0.8000
Table C.37 [M2, Small Airport, NLOS, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0246 1.2611 0.2144 0.8983 0.6276 2 0.1371 0.9680 0.9936 0.4615 0.9965 3 0.1256 1.0803 0.9724 0.2364 0.9782 4 0.0998 1.1592 0.9320 0.3650 0.9540 5 0.0757 1.2526 0.9049 0.3177 0.9286 6 0.0579 1.3058 0.8728 0.3191 0.9012 7 0.0457 1.3584 0.8452 0.3770 0.8863 8 0.0395 1.4028 0.8236 0.3631 0.8641 9 0.0324 1.5130 0.8073 0.3897 0.8541 10 0.0305 1.4702 0.7881 0.3837 0.8348 11 0.0262 1.5338 0.7541 0.4659 0.8255 12 0.0251 1.5212 0.7516 0.4433 0.8164 13 0.0233 1.5570 0.7334 0.4426 0.7977 14 0.0248 1.5950 0.7388 0.4302 0.7991 15 0.0227 1.6669 0.7496 0.4181 0.8060 16 0.0259 1.5836 0.7383 0.4132 0.7917 17 0.0221 1.6430 0.7452 0.4225 0.8029 18 0.0232 1.5337 0.7240 0.4580 0.7931 19 0.0207 1.6254 0.6974 0.4992 0.7830 20 0.0203 1.6425 0.7003 0.5041 0.7882 21 0.0182 1.6990 0.7013 0.4810 0.7793 22 0.0197 1.6405 0.7043 0.4958 0.7887 23 0.0187 1.7582 0.6939 0.5081 0.7834 24 0.0184 1.7033 0.6841 0.4875 0.7637 25 0.0220 1.4951 0.7023 0.4328 0.7600 400
Table C.38 [M1, Small Airport, NLOS-S, 5]
Tap Energy Weibull P1,k P00,k P11,k Index k Shape Factor (bk) 1 0.959 4.22 1.0000 na 1.0000 2 0.041 1.41 0.6028 0.6996 0.8016
Table C.39 [M2, Small Airport, NLOS-S, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0796 1.3280 0.1962 0.9589 0.8320 2 0.6777 2.0100 0.9749 0.3750 0.9839 3 0.1386 1.1637 0.6107 0.7097 0.8144 4 0.0308 1.2034 0.3548 0.8537 0.7301 5 0.0064 1.4556 0.2182 0.8853 0.5899 6 0.0039 1.3897 0.1444 0.9210 0.5326 7 0.0079 1.6290 0.2135 0.9240 0.7206 8 0.0057 1.6974 0.1805 0.9213 0.6435 9 0.0066 1.4578 0.2072 0.9028 0.6288 10 0.0087 1.5166 0.2229 0.9130 0.6972 11 0.0047 1.7759 0.2418 0.8921 0.6623 12 0.0044 1.6200 0.1711 0.9184 0.6055 13 0.0016 2.0995 0.0848 0.9364 0.3148 14 0.0007 2.1558 0.0628 0.9581 0.3750 15 0.0012 2.8411 0.1476 0.9098 0.4839 16 0.0008 2.7581 0.1020 0.9212 0.3077 17 0.0043 1.6558 0.2480 0.8577 0.5633 18 0.0028 2.2055 0.1115 0.9575 0.6620 19 0.0017 2.6254 0.0895 0.9568 0.5614 20 0.0003 2.6608 0.0298 0.9806 0.3684 24 0.0028 0.8565 0.0251 0.9823 0.3125 25 0.0077 1.2929 0.0361 0.9853 0.6087
Table C.40 [M1, Small Airport, LOS-O, 5]
Tap Energy Weibull P1,k P00,k P11,k Index k Shape 401
Factor (bk) 1 0.986 5.15 1.0000 na 1.0000 2 0.014 1.58 0.5092 0.7744 0.7810
Table C.41 [M2, Small Airport, LOS-O, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0084 1.3091 0.0185 0.9810 0 2 0.8290 2.4179 1.0000 na 1.0000 3 0.1546 0.9859 0.7685 0.6000 0.8780 4 0.0068 1.4659 0.2407 0.8415 0.5200 5 0.0004 1.7099 0.0926 0.9381 0.4000
Table C.42 [M1, Small Airport, NLOS, 10]
Tap Energy Weibull Shape P1,k P00,k P11,k Index k Factor (bk) 1 0.714 2 1.0000 na 1.0000 2 0.073 1.42 0.8779 0.2549 0.8963 3 0.048 1.42 0.8123 0.3708 0.8550 4 0.041 1.42 0.7605 0.3848 0.8067 5 0.033 1.48 0.7419 0.4164 0.7973 6 0.022 1.59 0.6873 0.4509 0.7505 7 0.020 1.6 0.6662 0.4540 0.7268 8 0.017 2 0.6595 0.4563 0.7197 9 0.019 1.66 0.6753 0.4195 0.7213 10 0.014 2 0.6509 0.4547 0.7079
Table C.43 [M2, Small Airport, NLOS, 10]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.0100 1.2187 0.1154 0.9388 0.5311 2 0.1368 0.9426 0.9957 0.4444 0.9976 3 0.1414 1.0191 0.9569 0.1778 0.9629 4 0.0793 1.1667 0.9004 0.3237 0.9255 5 0.0617 1.2350 0.8420 0.4073 0.8891 6 0.0519 1.2879 0.8123 0.3939 0.8603 7 0.0380 1.3180 0.7744 0.4085 0.8281 402
8 0.0319 1.3420 0.7524 0.4128 0.8071 9 0.0264 1.3809 0.7079 0.4401 0.7693 10 0.0200 1.5280 0.7107 0.4312 0.7689 11 0.0185 1.5069 0.6935 0.4351 0.7507 12 0.0154 1.6665 0.6863 0.4694 0.7579 13 0.0141 1.6436 0.6576 0.4930 0.7363 14 0.0137 1.6009 0.6470 0.4858 0.7200 15 0.0118 1.6583 0.6327 0.4896 0.7033 16 0.0110 1.7291 0.5963 0.5439 0.6908 17 0.0110 1.6452 0.5795 0.5496 0.6736 18 0.0110 1.6634 0.6039 0.5417 0.7000 19 0.0103 1.6586 0.5824 0.5539 0.6798 20 0.0098 1.6963 0.5690 0.5951 0.6936 21 0.0089 1.7855 0.5680 0.5771 0.6788 22 0.0097 1.7354 0.5790 0.5513 0.6741 23 0.0100 1.6630 0.5589 0.6185 0.6992 24 0.0094 1.6765 0.5618 0.5602 0.6573 25 0.0093 1.7361 0.5642 0.5710 0.6689 26 0.0097 1.7789 0.5541 0.5639 0.6497 27 0.0082 1.8025 0.5565 0.5719 0.6592 28 0.0089 1.8145 0.5565 0.5795 0.6652 29 0.0086 1.8100 0.5455 0.5812 0.6506 30 0.0088 1.7524 0.5321 0.5953 0.6445 31 0.0095 1.8142 0.5651 0.5402 0.6466 32 0.0097 1.8265 0.5718 0.5599 0.6709 33 0.0096 1.7746 0.5575 0.5905 0.6744 34 0.0083 1.8352 0.5393 0.5994 0.6581 35 0.0080 1.8525 0.5249 0.6206 0.6569 36 0.0081 1.7162 0.5110 0.6176 0.6345 37 0.0080 1.7846 0.5196 0.6248 0.6535 38 0.0087 1.7561 0.5369 0.6170 0.6699 39 0.0087 1.7234 0.5192 0.6301 0.6577 40 0.0086 1.8678 0.5340 0.5916 0.6439 41 0.0088 1.6945 0.5192 0.6032 0.6328 42 0.0086 1.7379 0.5354 0.6017 0.6547 43 0.0093 1.6852 0.5220 0.6169 0.6495 44 0.0084 1.8015 0.5057 0.6343 0.6430 45 0.0094 1.7373 0.5005 0.6344 0.6354 46 0.0099 1.7205 0.5254 0.6081 0.6463 47 0.0098 1.6519 0.5340 0.5967 0.6484 48 0.0099 1.5828 0.5431 0.5708 0.6393 49 0.0121 1.5100 0.5570 0.5790 0.6655 50 0.0112 1.5686 0.5565 0.5492 0.6411 403
Table C.44 [M1, Small Airport, NLOS-S, 10]
Tap Energy Weibull Shape P1,k P00,k P11,k Index k Factor (bk) 1 0.959 4.2 1.0000 na 1.0000 2 0.041 1.37 0.7276 0.5202 0.8200
Table C.45 [M2, Small Airport, NLOS-S, 10]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.0288 1.0998 0.0709 0.3666 0.6222 2 0.5119 1.3249 0.9969 0.9711 0.9968 3 0.3098 1.1404 0.8630 0 0.8958 4 0.0320 1.1809 0.4866 0.3448 0.6623 5 0.0320 1.2885 0.3811 0.6779 0.7635 6 0.0097 1.3556 0.2598 0.8524 0.6424 7 0.0025 1.5490 0.1528 0.8742 0.3814 8 0.0024 1.5421 0.1354 0.8883 0.4767 9 0.0016 1.7951 0.0803 0.9179 0.4510 10 0.0027 1.2717 0.0945 0.9520 0.4333 11 0.0028 1.4682 0.1134 0.9408 0.5000 12 0.0051 1.6988 0.1449 0.9359 0.6304 13 0.0035 1.5861 0.1102 0.9373 0.5143 14 0.0023 1.6132 0.1165 0.9397 0.5811 15 0.0024 1.5368 0.1008 0.9446 0.5469 16 0.0031 1.3410 0.1244 0.9491 0.4937 17 0.0029 1.5190 0.0961 0.9279 0.5574 18 0.0016 1.8799 0.1197 0.9529 0.4868 19 0.0047 1.6044 0.1669 0.9301 0.5943 20 0.0033 1.5407 0.1417 0.9186 0.4889 21 0.0023 1.7322 0.1449 0.9154 0.5217 22 0.0025 1.5792 0.1008 0.9188 0.5938 23 0.0010 1.9661 0.0693 0.9544 0.4773 24 0.0004 3.8925 0.0409 0.9610 0.1538 25 0.0004 2.8115 0.0315 0.9638 0.4000 26 0.0004 2.8394 0.0409 0.9805 0.1923 28 0.0003 4.0527 0.0583 0.9888 0.2703 29 0.0006 3.4229 0.1039 0.9548 0.4462 404
30 0.0006 2.4250 0.0504 0.9350 0.2500 31 0.0005 3.9325 0.0945 0.9601 0.3500 32 0.0024 1.9395 0.1260 0.9338 0.4625 33 0.0020 1.7230 0.1354 0.9224 0.5465 34 0.0023 2.0984 0.0677 0.9288 0.4651 35 0.0010 2.6302 0.0598 0.9611 0.3421 36 0.0010 2.6911 0.0488 0.9581 0.4839 37 0.0007 2.6259 0.0362 0.9735 0.3043 49 0.0088 1.7536 0.0362 0.9872 0.5217 50 0.0017 1.0495 0.0315 0.9820 0.2000
Table C.46 [M1, Small Airport, LOS-O, 10]
Tap Energy Weibull Shape P1,k P00,k P11,k Index k Factor (bk) 1 0.985 5.3 1.0000 na 1.0000 2 0.016 1.58 0.6593 0.6237 0.8045
Table C.47 [M2, Small Airport, LOS-O, 10]
Weibull Tap Index k Energy Shape P1 P00,k P11,k Factor (bk) 1 0.5786 1.3370 1.0000 Na 1.0000 2 0.3911 1.1289 0.9074 0.2000 0.9175 3 0.0185 1.4135 0.6667 0.6389 0.8310 4 0.0094 1.6354 0.4722 0.7719 0.7600 5 0.0009 3.4562 0.1759 0.9091 0.5789 6 0.0005 2.2414 0.1204 0.9158 0.4167
Table C.48 [M1, Small Airport, NLOS, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.716 1.9015 1.0000 na 1.0000 2 0.064 1.4460 0.8425 0.2696 0.8634 3 0.028 1.6016 0.7262 0.3967 0.7723 4 0.026 1.5266 0.6901 0.4410 0.7493 5 0.019 1.4751 0.6340 0.4906 0.7063 405
6 0.017 1.5234 0.6184 0.5401 0.7166 7 0.021 1.4673 0.6262 0.5208 0.7143 8 0.015 1.6056 0.5961 0.5509 0.6960 9 0.015 1.5371 0.5883 0.5316 0.6726 10 0.012 1.5970 0.5792 0.5440 0.6690 11 0.010 1.6856 0.5454 0.5749 0.6460 12 0.009 1.6861 0.5235 0.5820 0.6199 13 0.008 1.7003 0.5066 0.5796 0.5910 14 0.008 1.6990 0.4911 0.6086 0.5948 15 0.007 1.6969 0.4874 0.6114 0.5918 16 0.007 1.7822 0.4820 0.6049 0.5758 17 0.007 1.7225 0.4902 0.6228 0.6080 18 0.007 1.8088 0.5048 0.6061 0.6139 19 0.006 1.8115 0.4601 0.6371 0.5744
Table C.49 [M2, Small Airport, NLOS, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0055 1.1002 0.0470 0.9727 0.4466 2 0.0958 0.9908 0.9959 0.1111 0.9963 3 0.1722 1.0234 0.9603 0.2184 0.9677 4 0.0821 1.1530 0.8722 0.2750 0.8937 5 0.0441 1.2849 0.7978 0.3431 0.8334 6 0.0312 1.3161 0.7325 0.4427 0.7969 7 0.0251 1.3616 0.6919 0.5089 0.7817 8 0.0244 1.3447 0.6600 0.5296 0.7580 9 0.0235 1.2839 0.6308 0.5136 0.7156 10 0.0183 1.4598 0.6504 0.5085 0.7361 11 0.0157 1.4844 0.6294 0.5314 0.7244 12 0.0134 1.4901 0.6066 0.5436 0.7043 13 0.0116 1.5270 0.5696 0.5361 0.6498 14 0.0097 1.6118 0.5431 0.5660 0.6353 15 0.0094 1.6278 0.5390 0.5719 0.6342 16 0.0083 1.6126 0.5098 0.5993 0.6150 17 0.0069 1.7418 0.5025 0.6134 0.6176 18 0.0076 1.7048 0.5021 0.5927 0.5964 19 0.0075 1.6460 0.4879 0.6066 0.5875 20 0.0068 1.8636 0.5075 0.5955 0.6079 21 0.0068 1.7745 0.4906 0.6152 0.6009 22 0.0060 1.8548 0.4633 0.6374 0.5803 406
23 0.0060 1.7747 0.4523 0.6480 0.5742 24 0.0059 1.8226 0.4541 0.6519 0.5819 25 0.0063 1.6811 0.4569 0.6325 0.5634 26 0.0056 1.8207 0.4509 0.6481 0.5719 27 0.0052 1.8065 0.4263 0.6545 0.5353 28 0.0049 1.8301 0.4354 0.6594 0.5577 29 0.0052 1.8126 0.4359 0.6559 0.5539 30 0.0051 1.7797 0.4144 0.6934 0.5661 31 0.0052 1.7658 0.4190 0.6895 0.5697 32 0.0055 1.6373 0.4226 0.6859 0.5713 33 0.0050 1.8593 0.4217 0.6919 0.5779 34 0.0046 1.8226 0.4149 0.6893 0.5622 35 0.0055 1.7109 0.4194 0.6963 0.5800 36 0.0046 1.8544 0.3975 0.6998 0.5442 37 0.0044 1.8634 0.3943 0.6961 0.5324 38 0.0045 1.8739 0.3957 0.7060 0.5513 39 0.0047 1.8085 0.3765 0.7209 0.5382 40 0.0041 1.7567 0.3683 0.7296 0.5366 41 0.0045 1.8424 0.3934 0.7018 0.5406 42 0.0050 1.7963 0.3939 0.7159 0.5632 43 0.0044 1.8519 0.3930 0.6877 0.5180 44 0.0037 1.8449 0.3670 0.7302 0.5348 45 0.0046 1.7478 0.3756 0.7125 0.5225 46 0.0046 1.9001 0.3957 0.7082 0.5548 47 0.0044 1.8183 0.3943 0.6991 0.5382 48 0.0045 1.8527 0.3884 0.6990 0.5264 49 0.0044 1.8429 0.3706 0.7250 0.5333 50 0.0044 1.9281 0.3775 0.7256 0.5478 51 0.0042 1.8844 0.3583 0.7274 0.5121 52 0.0038 1.8937 0.3619 0.7294 0.5233 53 0.0040 1.8931 0.3825 0.7256 0.5573 54 0.0040 1.9688 0.3820 0.7088 0.5293 55 0.0038 1.8353 0.3752 0.7032 0.5061 56 0.0038 1.9817 0.3775 0.7111 0.5230 57 0.0038 1.9528 0.3743 0.7343 0.5561 58 0.0048 1.8392 0.3761 0.7313 0.5546 59 0.0046 1.8671 0.3711 0.7226 0.5301 60 0.0044 1.8193 0.3870 0.7086 0.5389 61 0.0046 1.8952 0.4194 0.6640 0.5354 62 0.0043 1.9000 0.3870 0.6990 0.5236 63 0.0047 1.8579 0.3880 0.7104 0.5435 64 0.0046 1.9064 0.3738 0.7150 0.5220 65 0.0043 1.8446 0.3638 0.7408 0.5471 407
66 0.0035 1.8919 0.3619 0.7509 0.5612 67 0.0039 1.8753 0.3651 0.7496 0.5650 68 0.0040 1.8444 0.3633 0.7382 0.5415 69 0.0038 1.9078 0.3665 0.7390 0.5492 70 0.0036 1.8673 0.3409 0.7491 0.5154 71 0.0038 1.8727 0.3533 0.7458 0.5349 72 0.0038 1.8389 0.3533 0.7401 0.5245 73 0.0045 1.8219 0.3733 0.7376 0.5599 74 0.0040 1.8914 0.3660 0.7327 0.5374 75 0.0043 1.7442 0.3642 0.7284 0.5263 76 0.0047 1.8056 0.3501 0.7470 0.5306 77 0.0041 1.8311 0.3628 0.7376 0.5396 78 0.0039 1.8403 0.3501 0.7498 0.5359 79 0.0041 1.8270 0.3391 0.7450 0.5034 80 0.0043 1.7952 0.3555 0.7371 0.5237 81 0.0046 1.8391 0.3738 0.7360 0.5580 82 0.0043 1.7659 0.3715 0.7217 0.5295 83 0.0038 1.8609 0.3697 0.7362 0.5506 84 0.0042 1.7625 0.3555 0.7442 0.5366 85 0.0038 1.8620 0.3510 0.7502 0.5384 86 0.0036 1.8573 0.3583 0.7530 0.5580 87 0.0042 1.9133 0.3610 0.7377 0.5348 88 0.0043 1.8872 0.3597 0.7482 0.5520 89 0.0046 1.7049 0.3692 0.7263 0.5328 90 0.0048 1.8263 0.3624 0.7285 0.5227 91 0.0047 1.7035 0.3806 0.7301 0.5612 92 0.0055 1.6525 0.3843 0.7307 0.5689 93 0.0049 1.8085 0.3761 0.7269 0.5473 94 0.0046 1.8607 0.3852 0.7080 0.5344 95 0.0061 1.5100 0.3994 0.7065 0.5589 96 0.0072 1.4461 0.4149 0.6877 0.5600 97 0.0083 1.4503 0.4322 0.6605 0.5544 98 0.0081 1.4282 0.4441 0.6311 0.5385 99 0.0081 1.4218 0.4605 0.6283 0.5649 100 0.0079 1.5794 0.4642 0.6329 0.5768
Table C.50 [M1, Small Airport, NLOS-S, 20]
Tap Energy Weibull Shape P1,k P00,k P11,k Index k Factor (bk) 1 0.934 3.97 1.0000 na 1.0000 2 0.066 1.54 0.8493 0.3229 0.8815 408
Table C.51 [M2, Small Airport, NLOS-S, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.0136 1.1996 0.0377 0.9886 0.7083 2 0.2541 1.0023 0.9984 0 0.9984 3 0.4782 1.5679 0.9655 0.1364 0.9691 4 0.0994 1.1535 0.7410 0.3818 0.7856 5 0.0204 1.1638 0.4349 0.6750 0.5797 6 0.0120 1.3337 0.3611 0.8059 0.6594 7 0.0171 1.2567 0.3391 0.8266 0.6651 8 0.0186 1.3779 0.3203 0.8845 0.7586 9 0.0090 1.3472 0.2512 0.8574 0.5786 10 0.0032 1.4320 0.1805 0.8983 0.5391 11 0.0023 1.3675 0.1381 0.8978 0.3636 12 0.0012 1.9788 0.1193 0.9125 0.3553 13 0.0013 1.4262 0.0926 0.9307 0.3220 14 0.0012 2.1748 0.0942 0.9323 0.3500 15 0.0005 1.8496 0.0518 0.9635 0.3333 16 0.0004 1.7586 0.0345 0.9756 0.3182 17 0.0010 1.6462 0.0565 0.9733 0.5556 18 0.0010 1.2179 0.0455 0.9638 0.2414 19 0.0016 1.4459 0.0769 0.9557 0.4694 20 0.0017 1.5744 0.0706 0.9577 0.4444 21 0.0028 1.7587 0.0879 0.9707 0.6964 22 0.0030 1.5629 0.0895 0.9447 0.4386 23 0.0026 1.6796 0.0706 0.9577 0.4444 24 0.0014 2.0409 0.1083 0.9383 0.4928 25 0.0015 1.9149 0.0801 0.9573 0.5098 26 0.0006 1.7938 0.0502 0.9669 0.3750 27 0.0009 1.5805 0.0502 0.9652 0.3438 28 0.0016 1.4651 0.0706 0.9628 0.5111 29 0.0012 1.4033 0.0644 0.9563 0.3659 30 0.0012 1.3712 0.0691 0.9510 0.3409 31 0.0010 1.5073 0.0455 0.9687 0.3448 32 0.0013 1.6780 0.0565 0.9650 0.4167 33 0.0012 1.7452 0.0754 0.9592 0.5000 34 0.0010 1.7042 0.0628 0.9530 0.3000 35 0.0006 1.8979 0.0628 0.9597 0.4000 36 0.0020 1.5413 0.1005 0.9371 0.4375 409
37 0.0032 1.5645 0.1162 0.9431 0.5676 38 0.0016 1.6218 0.0895 0.9361 0.3509 39 0.0014 1.8227 0.0911 0.9412 0.4138 40 0.0013 1.8784 0.0518 0.9602 0.2727 41 0.0014 1.8457 0.0801 0.9521 0.4510 42 0.0008 2.2431 0.0659 0.9646 0.5000 43 0.0008 1.8160 0.0345 0.9723 0.2273 44 0.0005 1.9375 0.0377 0.9755 0.3750 45 0.0003 3.2827 0.0251 0.9823 0.3125 46 0.0002 3.3060 0.0220 0.9823 0.2143 50 0.0002 2.8802 0.0267 0.9790 0.2353 51 0.0001 2.4675 0.0204 0.9872 0.3846 55 0.0002 4.3612 0.0502 0.9619 0.2813 56 0.0004 2.9538 0.0706 0.9493 0.3409 57 0.0002 2.6071 0.0283 0.9790 0.2778 58 0.0002 3.1710 0.0235 0.9823 0.2667 59 0.0002 2.8878 0.0235 0.9823 0.2667 60 0.0005 3.9113 0.0471 0.9703 0.4000 61 0.0003 3.5798 0.0518 0.9701 0.4242 62 0.0025 1.3371 0.0848 0.9519 0.4815 63 0.0012 1.7654 0.1146 0.9236 0.4110 64 0.0013 1.7663 0.0863 0.9552 0.5273 65 0.0007 1.8818 0.0424 0.9721 0.3704 66 0.0010 1.8105 0.0408 0.9770 0.4615 67 0.0007 2.2075 0.0408 0.9738 0.3846 68 0.0006 2.7924 0.0361 0.9804 0.4783 69 0.0005 2.2548 0.0314 0.9773 0.3000 71 0.0002 3.6216 0.0220 0.9807 0.1429 97 0.0044 1.8048 0.0267 0.9855 0.4706 98 0.0024 1.1824 0.0283 0.9806 0.3333 99 0.0016 1.0820 0.0408 0.9656 0.1923 100 0.0012 1.2840 0.1005 0.9056 0.1563
Table C.52 [M1, Small Airport, LOS-O, 20]
Tap Energy Weibull Shape P1,k P00,k P11,k Index k Factor (bk) 1 0.978 6 1.0000 na 1.0000 2 0.022 1.98 0.7800 0.5455 0.8701
Table C.53 [M2, Small Airport, LOS-O, 20]
410
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.2350 0.9213 1.0000 na 1.0000 2 0.6124 1.7285 0.9500 0.4000 0.9681 3 0.0923 1.0731 0.7100 0.4828 0.7857 4 0.0389 1.0016 0.5000 0.6735 0.6800 5 0.0107 1.3447 0.4500 0.8000 0.7727 6 0.0037 1.4887 0.3100 0.7536 0.4667 7 0.0018 2.3760 0.2800 0.8472 0.6296 8 0.0038 1.4318 0.2300 0.8701 0.5909
Table C.54 Correlation Coefficient matrix for [M1, Small Airport, NLOS-S NLOS, 10], lower Triangular matrix for NLOS and upper Triangular matrix for NLOS-S
i, j 1 2 3 4 5 6 7 8 9 1 1 0.01 — — — — — — — 2 0.27 1 — — — — — — — 3 0.24 0.03 1 — — — — — — 4 0.34 0.26 0.13 1 — — — — — 5 0.24 0.21 0.11 0.41 1 — — — — 6 0.18 0.20 0.03 0.13 0.28 1 — — — 7 0.24 0.16 0.13 0.15 0.23 0.05 1 — — 8 0.09 - 0.06 0.13 0.10 0.05 0.14 1 — 0.03 9 0.18 - 0.14 0.14 0.13 0.09 0.11 0.04 1 0.01 10 0.17 0.06 -0.02 0.11 0.14 0.13 0.11 0.07 0.07
Table C.55 Correlation Coefficient matrix for [M1, Small Airport, NLOSS NLOS, 5], lower Triangular matrix for NLOS and upper Triangular matrix for NLOSS
i, j 1 2 3 4 5 1 1 0.04 — — — 2 0.25 1 — — — 3 0.29 0.26 1 — — 4 0.09 0.20 0.16 1 — 5 0.12 0.04 0.14 0.16 1 6 0.13 0.12 0.08 0.10 0.11
411
Table C.56 Correlation Coefficient matrix for [M1, Small Airport, LOSO, 5 10], lower Triangular matrix for 10 MHz and upper Triangular matrix for 5 MHz
i, j 1 2 1 1 -0.0646 2 0.0392 1
412
Appendix D: AFS models for [M1, MIA JFK, All regions, 20]
This appendix presents the 20 MHz M1 model for the different regions of MIA and JFK while transmitting from AFS locations. M1 for 50 MHz for both the airports have already been presented in Chapter 7.
D.1 [M1, MIA, All regions, 20]
Table D.1 [M1, MIA, NLOS, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.610 2.4719 1.0000 na 1.0000 2 0.062 1.7761 0.8598 0.2435 0.8764 3 0.053 1.4996 0.7902 0.3567 0.8287 4 0.021 1.9346 0.7061 0.3900 0.7457 5 0.017 1.9491 0.6634 0.4036 0.6967 6 0.016 2.0578 0.6573 0.5160 0.7472 7 0.016 1.9566 0.6354 0.5034 0.7140 8 0.016 1.8133 0.6280 0.5377 0.7257 9 0.014 1.9166 0.6134 0.5158 0.6938 10 0.015 1.8658 0.6159 0.5350 0.7089 11 0.013 1.9880 0.5866 0.5444 0.6778 12 0.019 1.8884 0.6476 0.4861 0.7194 13 0.012 2.1043 0.6073 0.5358 0.6988 14 0.015 1.7350 0.5854 0.5752 0.6979 15 0.008 2.1476 0.5415 0.5813 0.6441 16 0.008 2.0459 0.5329 0.6021 0.6499 17 0.007 2.0403 0.5049 0.6222 0.6280 18 0.008 2.1137 0.5354 0.6079 0.6583 19 0.009 1.9793 0.4939 0.6280 0.6173 20 0.008 2.1595 0.5134 0.6055 0.6247 21 0.009 1.7926 0.5256 0.6186 0.6543 22 0.006 2.2229 0.4902 0.6523 0.6368 23 0.010 1.7714 0.5220 0.6368 0.6659 413
24 0.006 2.1520 0.4720 0.6759 0.6357 25 0.005 2.2892 0.4585 0.6817 0.6223 26 0.009 1.8686 0.4817 0.6557 0.6278 27 0.009 1.9265 0.5171 0.6557 0.6769
Table D.2 [M1, MIA, NLOS-S, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.826 3.7050 1.0000 na 1.0000 2 0.089 1.6216 0.8938 0.3711 0.9252 3 0.068 1.5492 0.7388 0.5744 0.8499 4 0.010 1.7512 0.5237 0.6480 0.6807 5 0.004 1.6433 0.4135 0.7392 0.6311 6 0.003 1.4328 0.2826 0.8101 0.5190
D.2 [M1, JFK, All regions, 20]
Table D.3 [M1, JFK, NLOS, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.405 1.8276 1.0000 na 1.0000 2 0.041 1.5010 0.8971 0.2685 0.9160 3 0.027 1.7209 0.8575 0.3359 0.8896 4 0.026 1.7560 0.8497 0.3529 0.8855 5 0.025 1.7841 0.8465 0.3726 0.8862 6 0.024 1.7622 0.8378 0.3816 0.8802 7 0.023 1.8214 0.8340 0.3947 0.8795 8 0.023 1.7920 0.8310 0.3933 0.8766 9 0.023 1.7940 0.8358 0.4040 0.8829 10 0.022 1.7681 0.8298 0.4075 0.8784 11 0.022 1.8526 0.8298 0.4050 0.8779 12 0.021 1.8488 0.8240 0.4168 0.8754 13 0.021 1.8215 0.8234 0.4080 0.8730 14 0.021 1.7751 0.8140 0.4409 0.8722 15 0.020 1.8325 0.8171 0.4207 0.8703 414
16 0.022 1.7717 0.8212 0.4339 0.8767 17 0.020 1.8014 0.8125 0.4491 0.8728 18 0.020 1.8484 0.8126 0.4218 0.8667 19 0.020 1.8077 0.8110 0.4250 0.8659 20 0.020 1.8339 0.8074 0.4300 0.8639 21 0.019 1.8400 0.8063 0.4472 0.8672 22 0.020 1.7884 0.8071 0.4062 0.8580 23 0.020 1.8149 0.8107 0.4015 0.8602 24 0.019 1.8360 0.7985 0.4342 0.8572 25 0.019 1.8564 0.8021 0.4267 0.8585 26 0.019 1.8589 0.8016 0.4164 0.8555 27 0.019 1.8467 0.7977 0.4208 0.8531 28 0.018 1.8298 0.7977 0.4081 0.8498
Table D.4 [M1, JFK, NLOS-S, 20]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.930 4.0862 1.0000 na 1.0000 2 0.055 1.5525 0.8079 0.3501 0.8459 3 0.009 1.6644 0.5086 0.6485 0.6609 4 0.006 1.8322 0.4782 0.6630 0.6329
415
Appendix E: VTV models for [M2, Small UIC OLT, 5]
This appendix presents the 5 MHz M2 model for UIC, OLT and Small regions. [M1 M2 and M3] models of 10 MHz for all regions have already been presented in Chapter 8. We have also presented the [M1 M2 and M3] models for 5 MHz for UOC and OHT in
Chapter 8.
Table E.1 [M2, UIC, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.7733 2.42 1.0000 na 1.0000 2 0.1159 1.65 0.9438 0.1026 0.9465 3 0.0506 1.69 0.8314 0.3333 0.8646 4 0.0247 1.63 0.6153 0.5880 0.7418 5 0.0132 1.91 0.4265 0.7261 0.6305 6 0.0093 2 0.3501 0.8022 0.6337 7 0.0065 1.88 0.2752 0.8489 0.6053 8 0.0065 2.06 0.2493 0.8654 0.5954
Table E.2 [M2, Small, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.9219 3.91 1.0000 na 1.0000 2 0.0681 1.77 0.8442 0.3400 0.8815 3 0.0088 2.89 0.5078 0.5796 0.5951 4 0.0008 4.74 0.0685 0.9430 0.2273
416
Table E.3 [M2, OLT, 5]
Tap Energy Weibull P1,k P00,k P11,k Index Shape k Factor (bk) 1 0.9479 4.71 1.0000 na 1.0000 2 0.0382 1.64 0.7643 0.3803 0.8084 3 0.0081 1.83 0.3400 0.7566 0.5280 4 0.0025 2.23 0.1290 0.9202 0.4615 5 0.0016 2.44 0.0753 0.9490 0.3736 6 0.0012 2.3 0.0596 0.9595 0.3611