Array-Based Localization in DTV Passive Radar
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Huimin Huang
Graduate Program in Electrical and Computer Engineering
The Ohio State University
2019
Dissertation Committee:
Graeme Smith, Advisor
Emre Ertin
Niru Kamrun Nahar
Gerald Kosicki
Copyrighted by
Huimin Huang
2019
Abstract
This dissertation investigates the use of array-based localization to passively locate a target. Target localization in passive radar typically employs multilateration where a set of hyperbolic equations from range differences or elliptic equations from bistatic ranges are solved to determine the position of the target. To provide an unambiguous target location in three dimensions, there is another way, which is to use an efficient direction finder capable of providing a precise measurement of the target’s bearing. This direction finder can be implemented using an array of antenna elements and the intersection of ellipsoid and bearing vector gives the three-dimensional (3D) position of the target.
The array-based localization method is developed by noting that the stable range resolutions afforded by digitally modulated signals, like digital television (DTV), and
Doppler resolution from long integration times of passive radar can sufficiently resolve returns from multiple targets. The phase information from each array element from a resolved target can then be used with the Bartlett method of bearing estimation. Thus, target bearing can be estimated with far greater accuracy than the beamwidth resolution of the array alone. This was demonstrated in experiments conducted using a DTV-based passive radar with an electrically small array. The procedure of the in-field calibration of the array using opportunistic air targets is also elaborated.
ii
Expressions to calculate the location accuracies from the array-based localization method were derived and an analysis was performed on the location accuracy as a function of bistatic geometry. The developed theory was verified through simulation and experiments that made use of an aircraft, which traversed a wide range of bistatic geometries, and whose position was known.
Finally, the performance of array-based localization was compared with that of multilateration. In multistatic radar, target location estimates from both multilateration and array-based localization would be available. Based on insights from the comparative analysis of both methods as a function of geometry, a method of selecting the best target location estimate was proposed that makes use of the Kalman filter for data fusion.
This dissertation demonstrates that with the array-based localization method,
DTV-based passive radar is able to locate and track a target in 3D Cartesian space using an electrically small array.
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Dedication
To God, Sejun, and my parents for the sacrifices they made for me out of love.
iv
Acknowledgments
I thank Chris Baker and Graeme Smith for starting me off and guiding me through the
PhD program; Landon Garry, for teaching me about passive radar and aspects of doing research; Danny Tan, Andrew O’Brien, and Matthew Barr for their help in the passive radar experiments; Domenic Belgiovane and Teh-Hong Lee for their help with antenna characterization; Emre Ertin, Niru Kamrun Nahar, Inder Jeet Gupta, Philip Schniter, and
Gerald Kosicki for being on my committee; and my group members in the Cognitive
Sensing Group: Moayad Alslaimy, Peter John-Baptiste, Jakob DeLong, Saif Alsaif,
Ahmed Balakhder, Adam Mitchell, Zachary Cammenga, and Seung Ho Doo, for their encouragement to keep going.
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Vita
2009 ...... B.S. Electrical Engineering, Purdue
University
2010 ...... M.S. Electrical Engineering, Stanford
University
2010 to present ...... Senior Member of Technical Staff, DSO
National Laboratories
Publications
J. H. Huang, J. L. Garry, G. E. Smith and D. K. Tan, "In-field calibration of passive array receiver using detected target," in 2018 IEEE Radar Conference (RadarConf18),
Oklahoma City, 2018.
J. H. Huang, M. N. Barr, J. L. Garry and G. E. Smith, "Subarray processing for passive radar localization," in Radar Conference (RadarConf), 2017 IEEE, Seattle, 2017.
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J. H. Huang, J. L. Garry, G. E. Smith and C. J. Baker, "Array based passive radar target localization," in Radar Conference (RadarConf), 2016 IEEE, Philadelphia, 2016.
Fields of Study
Major Field: Electrical and Computer Engineering
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Table of Contents
Abstract ...... ii
Dedication ...... iv
Acknowledgments ...... v
Vita ...... vi
List of Tables...... xiv
List of Figures ...... xvi
Chapter 1 Introduction ...... 1
1.1 Localization with Passive Radar ...... 1
1.2 Aims of the Dissertation ...... 3
1.3 Overview of the Dissertation ...... 4
Chapter 2 Background Theory ...... 5
2.1 Bistatic Radar Fundamentals ...... 5
2.1.1 North-Referenced Coordinate System and Definitions of Bistatic Terms ...... 6
2.1.2 Bistatic Radar Range Equation ...... 10
2.1.3 Bistatic Range and Doppler Resolutions ...... 12
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2.1.4 Operational Differences between Monostatic and Bistatic Radar ...... 17
2.2 Bearing Estimation ...... 18
2.2.1 Linear Array ...... 19
2.2.2 Bearing Estimation Algorithms ...... 21
2.3 Evaluation of Error in Estimates ...... 25
2.3.1 General Formula for Error Propagation ...... 25
2.3.2 Cramer-Rao Lower Bound (CRLB) ...... 26
2.4 Sensor Fusion ...... 27
2.4.1 The Kalman Filter ...... 27
2.4.2 The Extended Kalman Filter ...... 31
Chapter 3 Literature Review ...... 32
3.1 Waveforms for Use in Passive Radar ...... 32
3.2 Passive Localization Methods ...... 33
3.2.1 Passive Source Localization Methods ...... 33
3.2.2 Multistatic Passive Radar Localization Methods ...... 34
3.3 Array-Based Localization ...... 35
Chapter 4 DTV-Based Passive Radar ...... 38
4.1 Detection Range of DTV Passive Radar ...... 38
4.2 Specifications of DTV Signals...... 42
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4.3 Signal Processing Stages in DTV-Based Passive Radar ...... 44
4.3.1 Waveform Conditioning Processing ...... 44
4.3.2 Direct Signal Interference Suppression...... 47
4.3.3 Short-Time Cross-Correlation ...... 51
Chapter 5 Array-Based Localization ...... 58
5.1 Received Signal Model...... 59
5.2 Bearing Estimation ...... 64
5.3 Conversion to Cartesian Space ...... 67
Chapter 6 System Overview ...... 70
6.1 The MUTERA Testbed ...... 70
6.2 Transmitters ...... 72
6.3 Antennas ...... 73
6.3.1 Reference Antenna ...... 73
6.3.2 Surveillance Antenna: 1D Array ...... 75
6.3.3 Surveillance Antenna: 2D Array ...... 77
6.3.4 Managing Ambiguities from Grating Lobes ...... 81
6.3.5 Mismatch Loss ...... 81
6.4 RF Front End ...... 82
6.5 Digital Backend...... 83
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6.6 Air Truth Data ...... 85
6.7 Conclusions ...... 87
Chapter 7 In-Field Calibration of Passive Array ...... 88
7.1 Calibration Using a Fixed Transmitter ...... 91
7.1.1 Calibration Results ...... 94
7.2 Passive Array Calibration in Literature ...... 96
7.3 Calibration Using Detected Targets ...... 96
7.3.1 Calibration Results ...... 99
7.3.2 Validation Results ...... 100
7.4 Conclusions ...... 103
Chapter 8 Analysis of Passive Localization Accuracies...... 104
8.1 Passive Localization Accuracies in Bistatic Radar ...... 104
8.1.1 Bistatic Range Sum Measurement ...... 109
8.1.2 Bearing Estimation ...... 112
8.1.3 Bistatic Velocity Measurement ...... 117
8.1.4 Target Location Estimate Using Array-Based Method ...... 121
8.2 Passive Localization Accuracies in Multistatic Radar ...... 140
8.2.1 Target Location Estimate Using Range Multilateration ...... 142
8.2.2 Comparison with Array-Based Method ...... 145
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8.2.3 Multilateration Accuracy as a Function of Target Altitude ...... 148
8.3 Conclusions ...... 150
Chapter 9 Array-Based Localization Experiments ...... 152
9.1 Azimuth Only Localization across Bistatic Geometries ...... 152
9.1.1 Experiment Settings ...... 153
9.1.2 Observations on Detection ...... 155
9.1.3 Simulation as an Analysis Tool ...... 157
9.1.4 Observations on Localization ...... 158
9.2 Azimuth and Elevation Localization ...... 163
9.2.1 Experiment Settings ...... 163
9.2.2 Observations on Detection ...... 164
9.2.3 Observations on Localization ...... 165
9.3 Comparison of Array-Based Localization with Multilateration ...... 166
9.3.1 Experiment Settings ...... 166
9.3.2 Observations on Detection ...... 168
9.3.3 Observations on Localization ...... 169
9.4 Fusion of Localization Estimates ...... 169
9.4.1 Kalman Filter ...... 171
9.4.2 Initialization of Kalman Filter ...... 173
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9.4.3 Filtered Localization Results ...... 174
9.5 Conclusions ...... 176
Chapter 10 Summary and Conclusions...... 177
Bibliography ...... 180
Appendix A: Plotting Contours of Constant SNR in Bistatic Radar ...... 189
Appendix B: Array-Based Localization Accuracy for T2 and T3 ...... 193
Appendix C: Localization Results from Experimental Datasets for Investigation 1 .. 196
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List of Tables
Table 4.1 Parameters of a typical DTV passive radar ...... 40
Table 6.1 Parameters of DTV transmitters for a receiver at the Electro-Science
Laboratory ...... 73
Table 6.2 Specifications of reference antenna ...... 75
Table 6.3 Specifications of antenna element in 1D surveillance array ...... 76
Table 6.4 Specifications of antenna element in 2D surveillance array ...... 79
Table 7.1 Transmitter properties for calibration using fixed transmitter experiment ...... 94
Table 7.2 Transmitter properties for calibration using detected target experiment ...... 99
Table 7.3 Bearing estimation results from test/validation data ...... 102
Table 8.1 Settings for bistatic and approximately monostatic configurations used in analysis of errors in the specific case where the bistatic plane is coincident with the x-y plane ...... 108
Table 8.2 Settings for bistatic configuration in the specific and general cases ...... 135
Table 8.3 Settings for multistatic configurations used in analysis of errors in a general case ...... 141
Table 9.1 Collected datasets from WWHO transmitter for investigation 1...... 154
Table 9.2 Collected datasets from WOSU transmitter for investigation 1 ...... 154
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Table 9.3 Localization results for datasets from WWHO transmitter for investigation 1
...... 162
Table 9.4 Localization results for datasets from WOSU transmitter for investigation 1 162
Table 9.5 Collected datasets from WWHO transmitter for investigation 2...... 163
Table 9.6 Localization results for datasets from WWHO transmitter for investigation 2
...... 165
Table 9.7 Collected dataset for investigation 3 ...... 167
Table 9.8 Localization results for datasets from WWHO transmitter for investigation 2
...... 169
Table 9.9 Localization results from KF and EKF implementations ...... 175
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List of Figures
Figure 1.1 (a) Monostatic radar; (b) Bistatic radar; (c) Passive radar ...... 1
Figure 2.1 Sample bistatic radar geometry ...... 6
Figure 2.2 Contours of constant bistatic angle ...... 7
Figure 2.3 Velocity relationships in bistatic radar geometry ...... 9
Figure 2.4 Ovals of Cassini ...... 11
Figure 2.5 Geometry for bistatic range resolution ...... 12
Figure 2.6 Bistatic range cell width as a function of bistatic angle...... 14
Figure 2.7 Geometry for bistatic Doppler ...... 15
Figure 2.8 Variation of required difference between target velocities with bistatic angle 16
Figure 2.9 Operational differences between monostatic and bistatic radar ...... 17
Figure 2.10 The bearing estimation problem ...... 19
Figure 2.11 Path difference as a function of target bearing ...... 21
Figure 4.1 Simulated elevation pattern of DTV transmit antenna in elevation ...... 39
Figure 4.2 Ovals of Cassini plotted in azimuth plane for target altitudes of 2000, 5000, and 10000 m (from left to right) for (top) directional DTV beam pattern in elevation and
(bottom) omnidirectional reference ...... 41
Figure 4.3 NTSC and 8-VSB systems ...... 43
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Figure 4.4 (a) AF surface of DTV signal; (b) Zero-Doppler offset range cut; (c) Zero-time offset Doppler cut ...... 45
Figure 4.5 (a) AF surface of DTV signal after removal of pilot tone; (b) Zero-Doppler offset range cut; (c) Zero-time offset Doppler cut ...... 46
Figure 4.6 DSI suppression block diagram ...... 48
Figure 4.7 FIR filter model ...... 49
Figure 4.8 (a) RD map before DSI suppression using LS filter; (b) After DSI suppression
...... 51
Figure 5.1 Computed RD maps for all 6 antenna elements on receiver array ...... 58
Figure 5.2 Passive radar set-up ...... 59
Figure 5.3 Linear array ...... 65
Figure 5.4 Array-based localization data flow ...... 67
Figure 6.1 MUTERA passive radar system ...... 71
Figure 6.2 Map showing locations of MUTERA and DTV Transmitters ...... 72
Figure 6.3 Configuration of reference antennas ...... 74
Figure 6.4 Gain of reference antenna (a) in azimuth; (b) in elevation ...... 75
Figure 6.5 Configuration of 1D surveillance array ...... 76
Figure 6.6 Gain of antenna element in 1D surveillance array (a) in azimuth; (b) in elevation ...... 77
Figure 6.7 Configuration of 2D surveillance array ...... 78
Figure 6.8 Gain of antenna element in 2D surveillance array (a) in azimuth; (b) in elevation ...... 79
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Figure 6.9 Subarray architecture ...... 80
Figure 6.10 MUTERA RF front end block diagram ...... 82
Figure 6.11 Digital down conversion block diagram ...... 84
Figure 6.12 Kinetic Avionics SBS-3 receiver system ...... 86
Figure 6.13 Basestation display ...... 87
Figure 7.1 Sources of measurement and modeling errors to the output of an antenna array
...... 88
Figure 7.2 (a) Calibration geometry; (b) Results of calibration: Top is Bartlett method spectrum output before calibration and bottom is the spectrum after calibration ...... 94
Figure 7.3 Calibration using detected target ...... 97
Figure 7.4 Calibration and validation flight paths ...... 98
Figure 7.5 Phases and magnitudes of calculated correction factors from calibration data
...... 100
Figure 8.1 2D bistatic planes within 3D coordinate system: One with the bistatic plane coincident with the x-y plane and the other general case, which is not ...... 105
Figure 8.2 Bistatic radar configuration (푅푅 = 5.7 km) for analysis in the case where the bistatic plane is coincident with the x-y plane: (a) Top-down view and (b) Side view .. 107
Figure 8.3 Approximately monostatic radar configuration (푅푅 = 460 km) for analysis in the case where the bistatic plane is coincident with the x-y plane: (a) Top-down view and
(b) Side view ...... 107
Figure 8.4 Uniform linear array geometry ...... 112
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Figure 8.5 Example showing DOA 휃ℎ표푟 of target with azimuth 퐴푡 in frame of reference of horizontal antenna array ...... 114
Figure 8.6 SD of 퐴푡 error as a function of 퐴푡 ...... 116
Figure 8.7 SD of 푉퐵 error as a function of 휃푅 ...... 120
Figure 8.8 SD of 휃푅 error as a function of 휃푅/퐴푡 ...... 123
Figure 8.9 Components of SD of 휃푅 error (Bistatic and approximately monostatic) ..... 124
Figure 8.10 SD of 푅푅 error as a function of 휃푅/퐴푡 ...... 126
Figure 8.11 Components of SD of 푅푅 error: In (a) bistatic configuration and (b) approximately monostatic configuration ...... 128
Figure 8.12 Components of SD of 푥푡 error: In (a) bistatic configuration and (b) approximately monostatic configuration ...... 130
Figure 8.13 Components of SD of 푥푡 error: In monostatic configuration for (a) near target at 5.7 km and (b) far target at 460 km from receiver ...... 131
Figure 8.14 Components of SD of 푦푡 error: In (a) bistatic configuration and (b) approximately monostatic configuration ...... 132
Figure 8.15 Components of SD of 푦푡 error: In monostatic configuration for (a) near target at 5.7 km and (b) far target at 460 km from receiver ...... 133
Figure 8.16 SD of target location error as a function of 휃푅/퐴푡 in bistatic configuration in
(a) specific case when bistatic plane is coincident with x-y plane and (b) general case when it is not ...... 136
Figure 8.17 Nonlinear relationship between 휃푅 and 퐴푡 in the general case ...... 137
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Figure 8.18 Components of SD of 푥푡 error in bistatic configuration for the general case
(where the bistatic plane does not coincide with x-y plane) ...... 138
Figure 8.19 SD of target location estimate errors using range multilateration: (a) x- coordinate, (b) y-coordinate, and (c) z-coordinate ...... 144
Figure 8.20 SD of target location estimate errors using array-based localization with transmitter T1: (a) x-coordinate, (b) y-coordinate, and (c) z-coordinate ...... 146
Figure 8.21 Selection of the best localization method in: (a) x-coordinate, (b) y- coordinate, and (c) z-coordinate ...... 148
Figure 8.22 SD of target location estimate errors in y-z plane using range multilateration:
(a) x-coordinate, (b) y-coordinate, and (c) z-coordinate ...... 150
Figure 9.1 Detection results from WWHO transmitter datasets for investigation 1 ...... 155
Figure 9.2 Detection results from WOSU transmitter datasets for investigation 1...... 157
Figure 9.3 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 5A for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results ...... 158
Figure 9.4 SDs of 퐴푡 error from dataset 5A for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results ...... 159
Figure 9.5 Localization error results from dataset 5A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 161
Figure 9.6 Percentiles for the SD of 퐴푡 ...... 161
Figure 9.7 Detection results from WWHO transmitter datasets for investigation 2 ...... 164
Figure 9.8 Experimental set-up for investigation 3 ...... 166
Figure 9.9 Detection results from multistatic dataset for investigation 3 ...... 168
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Figure 9.10 Multistatic radar information processing scheme ...... 170
Figure 9.11 Filtered target location estimates (KF implementation): (a) x-coordinate; (b) y-coordinate; (c) z-coordinate ...... 175
Figure A.1 Geometry for converting North-referenced coordinates into polar coordinates
...... 190
Figure B.1 SD of target location estimate errors using array-based localization with transmitter T2: (a) x-coordinate, (b) y-coordinate, and (c) z-coordinate ...... 194
Figure B.2 SD of target location estimate errors using array-based localization with transmitter T3: (a) x-coordinate, (b) y-coordinate, and (c) z-coordinate ...... 195
Figure C.1 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 1A for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results ...... 197
Figure C.2 SD of 퐴푡 error from dataset 1A for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results ...... 197
Figure C.3 Localization error results from dataset 1A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 198
Figure C.4 Percentiles for the SD of 퐴푡 from dataset 1A ...... 198
Figure C.5 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 2A for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results ...... 199
Figure C.6 SD of 퐴푡 error from dataset 2A for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results ...... 199
Figure C.7 Localization error results from dataset 2A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 200
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Figure C.8 Percentiles for the SD of 퐴푡 from dataset 2A ...... 200
Figure C.9 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 3A for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results ...... 201
Figure C.10 SD of 퐴푡 error from dataset 3A for investigation 1: (a)
Theoretical/Simulation results; (b) Experimental results ...... 201
Figure C.11 Localization error results from dataset 3A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 202
Figure C.12 Percentiles for the SD of 퐴푡 from dataset 3A ...... 202
Figure C.13 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 4A for investigation 1: (a)
Theoretical results; (b) Simulated results; (c) Experimental results ...... 203
Figure C.14 SD of 퐴푡 error from dataset 4A for investigation 1: (a)
Theoretical/Simulation results; (b) Experimental results ...... 203
Figure C.15 Localization error results from dataset 4A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 204
Figure C.16 Percentiles for the SD of 퐴푡 from dataset 4A ...... 204
Figure C.17 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 1B for investigation 1: (a)
Theoretical results; (b) Simulated results; (c) Experimental results ...... 205
Figure C.18 SD of 퐴푡 error from dataset 1B for investigation 1: (a)
Theoretical/Simulation results; (b) Experimental results ...... 205
Figure C.19 Localization error results from dataset 1B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 206
Figure C.20 Percentiles for the SD of 퐴푡 from dataset 1B ...... 206
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Figure C.21 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 2B for investigation 1: (a)
Theoretical results; (b) Simulated results; (c) Experimental results ...... 207
Figure C.22 SD of 퐴푡 error from dataset 2B for investigation 1: (a)
Theoretical/Simulation results; (b) Experimental results ...... 207
Figure C.23 Localization error results from dataset 2B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 208
Figure C.24 Percentiles for the SD of 퐴푡 from dataset 2B ...... 208
Figure C.25 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 3B for investigation 1: (a)
Theoretical results; (b) Simulated results; (c) Experimental results ...... 209
Figure C.26 SD of 퐴푡 error from dataset 3B for investigation 1: (a)
Theoretical/Simulation results; (b) Experimental results ...... 209
Figure C.27 Localization error results from dataset 3B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 210
Figure C.28 Percentiles for the SD of 퐴푡 from dataset 3B ...... 210
Figure C.29 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 4B for investigation 1: (a)
Theoretical results; (b) Simulated results; (c) Experimental results ...... 211
Figure C.30 SD of 퐴푡 error from dataset 4B for investigation 1: (a)
Theoretical/Simulation results; (b) Experimental results ...... 211
Figure C.31 Localization error results from dataset 4B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 212
Figure C.32 Percentiles for the SD of 퐴푡 from dataset 4B ...... 212
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Figure C.33 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 5B for investigation 1: (a)
Theoretical results; (b) Simulated results; (c) Experimental results ...... 213
Figure C.34 SD of 퐴푡 error from dataset 5B for investigation 1: (a)
Theoretical/Simulation results; (b) Experimental results ...... 213
Figure C.35 Localization error results from dataset 5B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡 ...... 214
Figure C.36 Percentiles for the SD of 퐴푡 from dataset 5B ...... 214
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Chapter 1 Introduction
1.1 Localization with Passive Radar
Conventional radar systems have collocated transmitters and receivers, and are known as monostatic radar. A bistatic radar is a system in which its transmitter and receiver are at separate locations [1]. These kind of systems can operate with their own dedicated transmitters, designed for bistatic operation, or with transmitters of opportunity, which are designed for other purposes but suitable for radar operation. When the transmitter is non-cooperative, the bistatic system is referred to as a passive radar [2]. illustrates their differences.
(a) (b) (c)
Figure 1.1 (a) Monostatic radar; (b) Bistatic radar; (c) Passive radar
1
Without a dedicated transmitter, a passive radar is comparatively low cost because it avoids spectrum licensing altogether, requires less maintenance, and can be constructed using relatively cheap components like universal radio receivers and a general purpose personal computer [3] [4]. Passive radar provides high data refresh rates of typically 0.1 –
1 s [3]. Moreover, simultaneous unambiguous range and velocity measurements are possible with passive radar, which are problematic in classical active pulse radars [3] [5].
Compared to conventional microwave radar, the lower bandwidth of broadcast signals causes poor range and bearing measurement accuracies; however, Doppler is more accurate due to the extended integration times possible with passive radar [5].
While active radars quickly become targets due to their energy emission, which can be used by the target under track to estimate the radar’s transmitter location, in passive radar there is almost no risk of being detected and jammed as the transmission sources are already out there in the environment for their intended purposes. However, this also means that passive radar is not useful in rural or remote areas, where there are no transmitters of opportunity, or in emergency scenarios where the broadcast transmitter can be destroyed or switched off [6]. This makes them more suitable for general, peacetime surveillance purposes. In populated areas where commercial signals are widely available, passive radar can be used to fill the blind zones of existing radar since they can detect low-flying stealthy targets due to its wide area coverage and frequency diversity
[7] [8].
A number of recent improvements in technology help to address the extra complexities of bistatic radar. These include high-speed digital signal processors (DSP),
2 phased array antennas, and the deployment of GPS satellite navigation systems that can be used for synchronization [9]. This makes the benefits of localization with passive radar accessible even with a modest budget.
1.2 Aims of the Dissertation
This dissertation presents research utilizing the DTV broadcast signal available in
North America in recent years as a signal of opportunity for the purpose of passive target localization. The following are addressed in the research:
• The suitability of DTV waveform for the purpose of passive localization is
investigated. A preliminary evaluation of the DTV transmitter was made and a
detailed study of its ambiguity function including the use of signal-processing
algorithms at the receiver to remove the inherent ambiguity in the transmitted
DTV signal was done.
• A novel localization method that relaxes the angular resolution requirement was
developed, making localization using a bistatic passive radar (BPR) configuration
possible.
• A novel method of in-field calibration of the passive array using a detected target
as a calibration target was proposed and validated with experimental data.
• Integration of said localization method into a multistatic passive radar (MPR)
configuration was performed.
• A theoretical analysis of the accuracy of said localization method was performed,
and this was validated with both simulation and experimental data.
3
1.3 Overview of the Dissertation
Chapter 2 starts with background theory relevant to the subsequent discussions in the dissertation. The basics of bistatic radar and bearing estimation are useful for understanding the array-based localization method; the calculation of error bounds for its performance evaluation; and state estimation and sensor fusion methods for its integration into a multistatic passive radar system. To provide context for this work, recent publications in passive radar systems and passive localization research are surveyed in Chapter 3. Chapter 4 is concerned with the evaluation of DTV-based passive radar, which is the system selected to carry out this research work. The signal model and signal processing involved in the developed localization method is covered in Chapter 5.
Chapter 6 outlines the experimental testbed and hardware, which allowed for experimental validation using actual air targets. Since the experiments required the use of arrays, their calibration is detailed in Chapter 7. Two in-field calibration methods are presented: An initial method using fixed transmitters, and a subsequent, novel method using detected targets as calibration targets of opportunity. Chapter 8 gives a theoretical analysis of the localization accuracy of the developed array-based localization method, and compares it to the conventional range multilateration method. This is subsequently validated with experiments carried out using 1D and 2D arrays in bistatic and multistatic radar configurations, which are detailed in Chapter 9. Finally, Chapter 10 concludes with a discussion of the results of the investigations. Important observations are made and the scope for future work is mapped out.
4
Chapter 2 Background Theory
The array-based localization work using passive radar described in this dissertation depends on fundamental bistatic radar theory, target bearing estimation, calculation of theoretical error bounds, and state estimation. For this reason, this chapter provides a brief overview of each of the four topics. Except when cited otherwise, the content in 2.1 can be attributed to Bistatic Radar [1] and Ambiguity Function for a
Bistatic Radar [10]; those in 2.2 to Introduction to Direction-Of-Arrival Estimation [11] and those in 2.3 to An Introduction to Error Analysis [12].
This chapter assumes the reader is familiar with basic radar theory. For readers unfamiliar with the area, a textbook introduction to radar is given in Introduction to
Airborne Radar [13] and Principles of Modern Radar [14].
2.1 Bistatic Radar Fundamentals
In a monostatic radar configuration, estimation of the time delay and Doppler shift directly provides information on target range and velocity. This information can also be retrieved in a bistatic radar configuration, even if the relations between measured time delay and target distance, and Doppler frequency and target velocity, are not linear. The target parameters that the receiver can measure are its bistatic range, bistatic velocity, and
5 bearing. When made with a certain update rate, these measurements can be used to establish a target track by methods similar to those of monostatic radar.
2.1.1 North-Referenced Coordinate System and Definitions of Bistatic Terms
The North-referenced coordinate system in Figure 2.1 is useful for specifying target position relative to the receiver or transmitter and is described here to represent bistatic radar geometries. It is a two-dimensional coordinate system confined to the bistatic plane formed by the transmitter, the receiver, and the target.
Figure 2.1 Sample bistatic radar geometry
This coordinate system is represented with a north-up orientation. Connecting the transmitter and the receiver is the bistatic baseline whose length is 퐿푇. The triangle formed by the transmitter, the receiver and the target is called the bistatic triangle. The two quantities 푅푇 and 푅푅 are the range from transmitter to target and the range from
6 receiver to target respectively, their sum being the bistatic range or range sum, 푅퐵 =
푅푇 + 푅푅. The look angle 휃푇 of the transmitter and the look angle 휃푅 of the receiver are measured positive clockwise from the north of the coordinate system and are restricted to the interval [−휋/2, 3휋/2]. The bistatic angle 훽 is the angle at the apex of the bistatic triangle, at the vertex which represents the target, so that 훽 = 휃푇 − 휃푅. 훽 is in the interval
[0, 휋] when the target is to the north of the baseline and is negative and in the interval
[−휋, 0] when the target is to the south of the baseline. The line that bisects the bistatic angle is called the bistatic bisector. Constant bistatic angle contours for four values of 훽 are plotted in Figure 2.2.
Figure 2.2 Contours of constant bistatic angle
Each contour passes through the transmitter and receiver location, and is a circle of radius
퐿푇 푟 = (2.1) 훽 2 sin(훽)
7
The center of the circle is located on the perpendicular bisector of the baseline, displaced from the baseline by a distance
퐿 푑 = 푇 (2.2) 훽 2 tan(훽) The bistatic radar geometry can be completely specified in terms of any three of the four parameters, 휃푇, 휃푅, 퐿푇, and 푅퐵. Assume that 푅퐵, 휃푅, and 퐿푇 are known. The law of cosines applied to the bistatic triangle gives 휋 푅 = 푅2 + 퐿 2 − 2 푅 퐿 cos ( + 휃 ) (2.3) 푇 푅 푇 푅 푇 2 푅
Substituting 푅푇 = 푅퐵 − 푅푅 and simplifying gives
2 2 푅퐵 − 퐿푇 푅푅 = (2.4) 2[푅퐵 + 퐿푇 sin(휃푅)]
This is a significant derivation because it means that if 푅퐵, 휃푅, and 퐿푇 are known, then using (2.4), we have the polar coordinates of the target from the receiver (푅푅, 휃푅), which is equivalent to the information measured by a monostatic radar at the receiver location.
Next, we consider the relationship between range rate and target velocity. It is assumed that the transmitter and the receiver are both stationary and only the target is moving. It is also assumed that the target is located north of the baseline; by symmetry, similar conclusions can be drawn for the case of a target located to the south of the baseline.
8
Figure 2.3 Velocity relationships in bistatic radar geometry
Consider the first-order time-derivative of the total range
푑 푑 푑 푅 = 푅 + 푅 (2.5) 푑푡 퐵 푑푡 푇 푑푡 푅 From Figure 2.3,
푑 훽 푅 = 푉 cos (훿 + ) 푑푡 푇 푡 2 (2.6)
푑 훽 푅 = 푉 cos (훿 − ) 푑푡 푅 푡 2 (2.7)
Where 푉푡 is the velocity of the target and 훿 is the angle between the target velocity vector and the bistatic bisector, measured in a clockwise direction from the bisector.
Then we have
푑 푑 푑 훽 푅̇ = 푅 = 푅 + 푅 = 2푉 cos(훿) cos ( ) 퐵 푑푡 퐵 푑푡 푇 푑푡 푅 푡 2 (2.8)
9
The factor cos(훽/2) causes the range rate to be smaller in bistatic radar than in the equivalent monostatic case, where 훽 = 0.
2.1.2 Bistatic Radar Range Equation
The bistatic radar range equation is a modified version of the monostatic radar range equation because of the difference in propagation paths as shown in Figure 2.1 and is given by
2 2 2 푃푇퐺푇퐺푅휆푇휎퐵퐹푇 퐹푅 SNR = 3 2 2 퐺푃 (2.9) (4휋) 푘푏푇0퐵푁퐹퐿푇퐿푅푅푇푅푅 Where
SNR − Signal-to-noise power ratio 푃푇 푊 Power output of transmitter 퐺푇 − Gain of transmitting antenna 퐺푅 − Gain of receiving antenna 휆푇 푚 Wavelength of transmitter’s signal 2 휎퐵 푚 Bistatic radar cross section (RCS) of target 퐹푇 − Pattern propagation factor for transmitter-to-target- path 퐹푅 − Pattern propagation factor for target-to-receiver path −23 푘푏 = 1.38 × 10 퐽/퐾 Boltzmann’s constant 푇0 = 290 퐾 Standard temperature reference 퐵 퐻푧 Bandwidth of receiving system 푁퐹 − Noise figure of receiving system 퐿푇 − Transmitting system losses (>1) not included in other parameters 퐿푅 − Receiving system losses (>1) not included in other parameters 푅푇 푚 Range between transmitter and target 푅푅 푚 Range between receiver and target 퐺푃 − Gain from signal processing
The system loss terms account for inefficiencies such as cable losses and signal processing losses. The pattern propagation factors are the catch-all patterns that model
10 the spatial variations of antenna beam shape, multipath, diffraction and refraction, on the transmitter-target and target-receiver paths.
The propagation path in bistatic radar has implications on its contours of constant
SNR, which differs from the monostatic case. These contours, shown in Figure 2.4, are known as Ovals of Cassini. The equations used for plotting the contours can be found in
Appendix A.
Figure 2.4 Ovals of Cassini
The regions with the highest SNRs, and hence easiest detections, are the receiver- centered region, the small oval around the receiver and the transmitter-centered region, the small oval around the transmitter. The latter allows a passive receiver to monitor activity near an uncooperative transmitter that can be very far away.
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2.1.3 Bistatic Range and Doppler Resolutions
The definition of bistatic target resolution is the same as that of monostatic target resolution: the degree to which two or more targets (of approximately equal amplitude and arbitrary constant phase) may be separated in one or more dimensions. In the monostatic case, target separation is referenced to the radar-to-target line-of-sight (LOS).
In the bistatic case, target separation can conveniently be referenced to the bistatic bisector, which is indicated with a dashed line in Figure 2.5.
Figure 2.5 Geometry for bistatic range resolution
For both monostatic and bistatic range resolution, an adequate degree of separation between two target echoes at the receiver is conventionally taken to be [15]
[16] 푐 훥푅 = (2.10) 2퐵
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Where
훥푅 푚 Range resolution 푐 = 3 × 108 푚/푠 Speed of light 퐵 퐻푧 Bandwidth of transmitted signals
To generate this separation at a bistatic receiver, two point-scattering targets must lie on bistatic isorange contours having a separation 훥푅퐵, as illustrated in Figure 2.5. In [1], an exact, though implicit expression for the bistatic range resolution was derived. This is frequently approximated by 푐 훥푅 = 퐵 훽 (2.11) 2퐵 cos (2) Where
훥푅퐵 푚 Bistatic range resolution 훽 ° Bistatic angle
The approximation results in large errors when the bistatic angle is large. This is shown in Figure 2.6. The blue line is the bistatic range cell width calculated with the exact expression, and the red dotted line was calculated using the approximate expression. The two lines converge to the monostatic range cell width, which is also the best range resolution, when the bistatic angle is 0°. This corresponds to the pseudo-monostatic mode of bistatic operation. As the bistatic angle increases from 140°, the two lines start to diverge more and the width of the bistatic range cell, or the bistatic resolution, increases rapidly when the bistatic angle is greater than 160°. This corresponds to the start of the forward scatter mode of bistatic operation.
13
Figure 2.6 Bistatic range cell width as a function of bistatic angle
For monostatic and bistatic Doppler resolution, an adequate degree of Doppler frequency separation between two target echoes at the receiver is conventionally taken to be [15] [16]
1 훥휈 = (2.12) TCPI Where
훥휈 퐻푧 Doppler frequency resolution TCPI 푠 Coherent processing interval
The definition of bistatic target Doppler also uses the bistatic bisector as a reference, as shown in Figure 2.7.
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Figure 2.7 Geometry for bistatic Doppler
Bistatic target Doppler is defined as
푅̇퐵 2푉푡 훽 휈퐵 = = ( ) cos(훿) cos ( ) 휆푇 휆푇 2 (2.13)
Where
휈퐵 퐻푧 Bistatic Doppler shift 푅̇퐵 푚/푠 Bistatic range rate (bistatic velocity) of target 휆푇 푚 Wavelength of transmitter’s signal 푉푡 푚/푠 Velocity of target 훿 ° Angle between the target velocity vector 푉 and the bistatic bisector, measured in a clockwise direction from the bisector
Assuming that the two targets are collocated so that they share a common bistatic bisector, taking
1 | 휈퐵1 − 휈퐵2| = (2.14) TCPI
15
Where
휈퐵1 퐻푧 Bistatic Doppler shift corresponding to target 1 휈퐵2 퐻푧 Bistatic Doppler shift corresponding to target 2
And combining it with (2.13) gives the required difference between the two target velocity vectors, projected onto the bistatic bisector. The bistatic velocity resolution is
휆 훥푉 = 푇 퐵 훽 (2.15) 2 푇퐶푃퐼 cos (2) Figure 2.8 shows this for different values of bistatic angles.
Figure 2.8 Variation of required difference between target velocities with bistatic angle
It can be seen, again, that the bistatic velocity resolution approaches that of the monostatic case when the bistatic angle approaches 0° (corresponding to the pseudo- 16 monostatic mode of bistatic operation) and increases rapidly when the bistatic angle is greater than 160° (corresponding to the start of the forward scatter mode of bistatic operation).
2.1.4 Operational Differences between Monostatic and Bistatic Radar
At this point, there are three differences worth highlighting here between monostatic and bistatic radar due to the separation of the transmitter and receiver in the latter. They are illustrated in Figure 2.9.
Figure 2.9 Operational differences between monostatic and bistatic radar
Firstly, as indicated with black curves in Figure 2.9, bistatic thermal noise-limited detection contours are defined by ovals of Cassini, rather than by circles for the monostatic case. These ovals define regions where bistatic radar can operate. Secondly, the bistatic constant range contours, indicated with dashed blue ellipses, are not collinear with bistatic constant detection contours, defined by the ovals of Cassini. In contrast, in the monostatic case, they are collinear circles. This causes the target’s SNR to vary as a 17 function of its position on a constant range sum contour for bistatic radar. Thirdly, there are three different operational modes in bistatic radar, indicated by the red, green, and white areas in Figure 2.9. This is in contrast with monostatic radar with a single operational mode (white area only).
The main operational mode (white area) for target detection and tracking is bistatic, where 5° ≤ 훽 < 160°. The forward-scatter radar (FSR) mode (red area) defines bistatic operation on the baseline (훽 = 180°) [17] [18], or in practice near the baseline
(160° ≤ 훽 < 180°). From the equations for the resolution of bistatic radar, both the range resolution and the Doppler resolution become very poor when this case is approached.
The pseudo-monostatic mode (green area) defines bistatic operation on either end of the extended baseline where 0° < 훽 ≤ 5°. It is of interest because the bistatic radar operates much like a monostatic radar, except that the target’s range from the transmitter is not equal to its range from the receiver for these geometries.
2.2 Bearing Estimation
K signals, each with direction 휃푘, impinge on a linear array with Ne elements that are equally spaced. The goal of bearing estimation is to use the data received at the array to estimate 휃푘 where 푘 = 1, … , 퐾. The bearing estimation problem set up is shown in
Figure 2.10. In practice, the estimation is made difficult by the fact that there are usually an unknown number of signals impinging on the array simultaneously, each from an unknown direction and with an unknown amplitude. Also, the received signals are always corrupted by noise.
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Figure 2.10 The bearing estimation problem
2.2.1 Linear Array
A directional antenna is one designed to have a gain in one direction and a loss in others. A simple directional antenna consists of a linear array of small radiating antenna elements. When the elements are spaced equally apart, the antenna is called a uniform linear array (ULA). By varying the signal amplitude and phase of each element in a linear array, its main beam and nulls can be steered and side lobe levels can be adjusted. The combined relative amplitude and phase shift for each antenna is called a complex weight
and is represented by a complex constant 푤푛푒 (for the 푛푒th antenna element). In digital beamforming, the operations of phase shifting and amplitude scaling are done digitally.
2.2.1.1 Grating Lobes in ULA
If the spacing between the elements is greater than half a wavelength, large side lobes begin to appear in the antenna radiation pattern. These are known as grating lobes.
Grating lobes are generally undesirable as they remove energy from the main beam. They also produce responses that are ambiguous with those produced with the main beam,
19 making it difficult to accurately calculate a target’s bearing. Applying phase shift for beam steering can cause grating lobes to form as it reduces the element spacing required for the currents produced by an incident wave to combine in phase [13]. The relationship between the element spacing 푑푚푎푥 and the maximum scan angle 휃푚푎푥 for a ULA is given by [19]
휆푇 휆푇 푑푚푎푥 = ≤ (2.16) 1 + |sin(휃푚푎푥)| 2
Where
푑푚푎푥 푚 Element spacing corresponding to maximum scan angle 휆푇 푚 Wavelength of transmitter’s signal 휃푚푎푥 ° Maximum scan angle of ULA
If the elements of an array are spaced less than half a wavelength apart, the array can scan to any angle without grating lobes [14].
2.2.1.2 Coning Angle Error
The coning angle error is present in a linear array when the array and the target are not in the same plane. As shown in Figure 2.11, in a horizontal linear array, the path difference between element 푛푒 and the reference element 1 is (푛푒 − 1)푑 cos(휃). When the target is at an angle of elevation 퐸푡 relative to the array, the path difference is actually
(푛푒 − 1)푑 cos(휃) cos(퐸푡). This gives an apparent bearing which differs from the true azimuth bearing and the difference increases with increasing angle 퐸푡. Since this same error occurs for all angles in a cone of semi-angle 퐸푡, this is called the coning angle error
[20]. The coning error limits the selection of a linear array for bearing in azimuth up to elevation ≤ 10° or nearly at the horizon [21]. It can be removed by using a planar array
20 antenna or two orthogonal linear arrays capable of measuring the elevation angle [21]
[22].
Figure 2.11 Path difference as a function of target bearing
2.2.2 Bearing Estimation Algorithms
In general, bearing estimation techniques can be broadly classified into conventional beamforming techniques, subspace-based techniques, and maximum likelihood techniques. From the conventional beamforming techniques and subspace- based techniques, the most popular algorithms – Bartlett method and Multiple signal classification (MUSIC) algorithm respectively [23], were used in this work. Maximum likelihood techniques were not used because they are more computationally intensive and hence less attractive in a system that will eventually be required to achieve real-time operation.
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2.2.2.1 Bartlett Method – A Beamforming Technique
The basic idea behind beamforming techniques is to steer the array in one direction at a time and measure the output power. When the steered direction coincides with a bearing of a signal, the maximum output power will be observed. An array can be steered electronically just as a fixed antenna can be steered mechanically. However, the array pattern can change shape in addition to changing orientation. A weight vector 풘 can be designed and then used to linearly combine the data received by the array elements to form a single output signal 푦(푡)
푦(푡) = 풘퐻풙(푡) (2.17) Where
푦(푡) 푉 Output signal 풘 − Weight vector (∙)퐻 − Hermitian transpose 풙(푡) 푉 Noise-added received signal, also called data
The averaged output power out of an array estimated over 푁 snapshots is expressed as
푁 푁 1 1 푃(풘) = ∑|푦(푡 )|2 = ∑ 풘퐻풙(푡 )풙퐻(푡 )풘 = 풘퐻 푹̂ 풘 (2.18) 푁 푛 푁 푛 푛 푥푥 푛=1 푛=1 Where
푁 − Number of snapshots 푡푛 푠 Continuous time at nth instant 퐻 푹̂푥푥 = 퐸[풙풙 ] 푉 Estimate of data covariance matrix
Different beamforming techniques have been developed with different choices of the weighting vector 풘. In the Bartlett method,
푇 [ ( )] ( ) (2.19) 풘퐵푎푟푡푙푒푡푡 = 풂(휃푠) = [1 exp 푗휙2 휃푠 ⋯ exp[푗휙푁푒 휃푠 ]] Where 22
풂(∙) − Steering vector function 휃푠 ° Scanning spectrum angle 푗 = √−1 − Unit imaginary number ( ) 휙푛푒 휃 푟푎푑 Differential phase at the 푛푒th array element with respect to a reference element to steer array to the 휃 direction
For each scanned direction 휃푠, the average power output 푃(휃푠) of the steered array is then computed with (2.18). When 휃푠 = 휃푘, the bearing of the 푘th impinging signal, the output power 푃(휃푠) will reach a peak or maximum point. At this moment, 풘 = 풂(휃푠 − 휃푘) aligns the phases of the signal components received by all the elements of the array, causing them to add constructively and produce a maximum power. The weight vector
(2.19) can be interpreted as a spatial filter; it is matched to the impinging spatial angles of the incoming signal to produce a peak but attenuate the output power for signals not coming from the angles of the incoming signals. While the Bartlett method is easy to implement, the width of the beam associated with a peak, or its resolution is limited by the number of array elements. For an electrically small array, this technique has poor resolution.
2.2.2.2 Multiple Signal Classification (MUSIC) Algorithm – A Subspace-Based
Technique
Subspace-based techniques rely on the following proven properties of the matrix space defined by 푹푥푥:
1. The space, spanned by its eigenvectors, can be partitioned into two
orthogonal subspaces, namely, the signal subspace and noise subspace.
2. The steering vectors correspond to the signal subspace.
23
3. The signal subspace is spanned by the eigenvectors associated with the
larger eigenvalues.
4. The noise subspace is spanned by the eigenvectors associated with the
smaller eigenvalues of the correlation matrix.
The MUSIC algorithm is summarized as follows:
• Step 1: Collect input samples 풙(푡푛), 푛 = 1,2, … , 푁 and estimate the input
covariance matrix
푁 1 푹 ≈ 푹̂ = ∑ 풙(푡 ) 풙퐻(푡 ) (2.20) 푥푥 푥푥 푁 푛 푛 푛=1 Where
푹푥푥 푉 Data covariance matrix 푹̂푥푥 푉 Estimate of data covariance matrix 푁 − Number of snapshots 풙(푡) 푉 Noise-added received signal, also called data 푡푛 푠 Continuous time at nth instant (∙)퐻 − Hermitian transpose
• Step 2: Perform Eigen-decomposition on 푹̂푥푥
(2.21) 푹̂푥푥푽 = 푽휦 Where
푹̂푥푥 푉 Estimate of data covariance matrix
휦 = diag([휆1 휆2 ⋯ 휆푀푒푖푔 ]) − 푀푒푖𝑔 × 푀푒푖𝑔 square matrix with diagonal elements of eigenvalues
with 휆1 ≥ 휆2 ≥ ⋯ ≥ 휆푀푒푖푔 푽 − Matrix containing all corresponding eigenvectors of 푹̂푥푥 as columns
• Step 3: Estimate the multiplicity 퐾푚 of the smallest eigenvalue 휆푚푖푛 and then the
number of signals 퐾 from 24
퐾 = 푀푒푖𝑔 − 퐾푚 (2.22) Where
퐾 − Number of signals 푀푒푖𝑔 − Number of eigenvalues 퐾푚 − Multiplicity of smallest eigenvalue
• Step 4: Compute the MUSIC spectrum
1 ( ) (2.23) 푃푀푈푆퐼퐶 휃푠 = 퐻 퐻 풂 (휃푆) 푽푛표푖푠푒 푽푛표푖푠푒 풂(휃푠) Where
푃(∙) 푊 Power function 휃푠 ° Scanning spectrum angle 풂(∙) − Steering vector function (∙)퐻 − Hermitian transpose operator 풒 ⋯ 풒 푽푛표푖푠푒 = [ 퐾+1 푀푒푖푔] − Matrix containing all noise eigenvectors corresponding to the smallest eigenvalue 휆푚푖푛
• Step 5: Find the 퐾 largest peaks of 푃푀푈푆퐼퐶 (휃푠) to obtain DOA estimates
2.3 Evaluation of Error in Estimates
In this section, we present some performance measures that describe the accuracy of an estimation scheme. This would allow us to evaluate the error in our location estimates.
2.3.1 General Formula for Error Propagation
Most physical quantities are usually not measured in a single direct measurement but are instead found in two distinct steps. First, we measure one or more quantities that can be recorded directly and from which the quantity of interest can be calculated.
25
Second, we use the measured values of these quantities to calculate the quantity of interest. When a measurement involves these two steps, the estimation of uncertainties also involves two steps. We must first estimate the uncertainties in the quantities measured directly and then determine how these uncertainties propagate through the calculations to produce an uncertainty in the final parameter estimate.
Suppose we have measured one or more quantities 푥, … , 푧 with corresponding uncertainties 훿푥, … , 훿푧 and that we now wish to use the measured values of 푥, … , 푧 to calculate the quantity of interest, 푞(푥, … , 푧). The problem we are interested in is how the uncertainties 훿푥, … , 훿푧 propagate through the calculation and lead to an uncertainty 훿푞 in the value of 푞. If the uncertainties in 푥, … , 푧 are independent and random, then the uncertainty in 푞 is
2 2 휕푞 휕푞 (2.24) 훿푞 = √( 훿푥) + ⋯ + ( 훿푧) 휕푥 휕푧 Where
훿푞 − Uncertainty in function output 푞 휕푞/휕푥 − Partial derivative of 푞 with respect to 푥 훿푥 − Uncertainty in first input 푥
훿푞 is never larger than the ordinary sum
휕푞 휕푞 훿푞 ≤ | | 훿푥 + ⋯ + | | 훿푧 (2.25) 휕푥 휕푧
2.3.2 Cramer-Rao Lower Bound (CRLB)
The Cramer-Rao lower bound (CRLB) provides a theoretical lower limit to the error covariance matrix 푷 of any unbiased estimator of coordinate vector x such that [24]
26
푷 = 퐸{(풙̂ − 풙) (풙̂ − 풙)퐻} ≥ 푱−1 (2.26) Where
푷 − Error covariance matrix 퐸{∙} − Expectation operator 풙̂ − Coordinate vector estimate 풙 − Coordinate vector of true value to be estimated 푱 − Fisher information matrix (FIM) 푨−1 − Inverse of matrix 푨
Where the Fisher information matrix (FIM) is given by
휕 휕 퐻 푱 = −퐸 { ( ln 푃 (풓)) } (2.27) 휕휽 휕휽 푞 Where
휽 Parameter vector 푃푞(풓) Joint probability density distribution of observation vector 풓 (∙)퐻 − Hermitian transpose operator
2.4 Sensor Fusion
Multiple bistatic radar pairs could take measurements of the same target and we wish to incorporate information from all the bistatic pairs into a single location estimate.
This problem, often referred to as sensor fusion, can be handled with a wide variety of centralized or decentralized architectures [25]. In this work, the fusion center receives the measurements from all available bistatic pairs, and the final state estimate can be easily determined using a Kalman filter.
2.4.1 The Kalman Filter
The Kalman Filter addresses the problem of state estimation where the state of an evolving system 풙 is estimated based on a discrete set of noisy measurements of that
27 system. If the underlying system state we wish to estimate at the discrete time step 푛 is
풙푛, the value of 풙푛 will depend on the previous state 풙푛−1 according to
(2.28) 풙푛 = f(풙푛−1, 풗푛−1) Where
풙푛 − System state at discrete time step 푛 f(∙) − State transition model 풙푛−1 − System state at previous discrete time step 푛 − 1 풗푛−1 − Process noise
The measurements 풛푛 depend on the state 풙푛 according to the measurement model
(2.29) 풛푛 = h(풙푛, 풘푛) Where
풛푛 − Measurement at discrete time step 푛 h(∙) − Measurement model 풘푛 − Measurement noise
The true system states must be estimated by combining the incoming measurements with the underlying models of the system.
In the particular case where both f(풙푛−1, 풗푛−1) and h(풙푛, 풘푛) are linear functions and both 풗푛−1 and 풘푛 are additive Gaussian random variables, the Kalman filter produces the optimal estimate of 풙푛. In this case, the state transition model 푭 can be expressed as
풙푛 = 푭풙푛−1 + 풗푛−1 (2.30)
And the measurement model H as
풛푛 = 푯풙푛 + 풘푛 (2.31)
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The process noise 풗푛 and measurement noise 풘푛 have covariances 푸푛 and 푹푛 respectively, so
풗푛~풩(0, 푸푛) (2.32)
풘푛~풩(0, 푹푛) (2.33)
The Kalman filtering process can be broken down into two stages: the prediction and the measurement update. In the prediction stage, the previous state estimate is used to form an estimate of the current state based on the state transition model
− 풙̂푛 = 푭풙̂푛−1 (2.34)
Where
− 풙̂푛 − Estimate of system state at discrete time step 푛, predicted 푭 − State transition matrix 풙̂푛−1 − Estimate of system state at previous discrete time step 푛 − 1
To quantify the uncertainty regarding the state estimate, the Kalman filter tracks the covariance of the estimates, and the covariance matrix 푷 must be updated during the prediction stage as well
− 푻 푷푛 = 푭푷푛−1푭 + 푸푛−1 (2.35)
Where
푷푛−1 − Covariance matrix at discrete time step 푛 − 1 푸푛−1 − Process noise covariance matrix
The measurement update forms a posterior estimate of the system state by balancing the information received in the measurement with the predicted state
− − 풙̂푛 = 풙̂푛 + 푲푛(풛푛 − 푯풙푛) (2.36)
29
− − 푷푛 = 푷푛 − 푲푛푯푷푛 (2.37)
Where
풙̂푛 − Estimate of system state at discrete time step 푛 (∙)− − Prediction 푲푛 − Kalman gain 풛푛 − Measurement at discrete time step 푛 푯 − Measurement matrix 푷푛 − Covariance matrix at discrete time step 푛
Kn is known as the Kalman gain and defined as
− 푻 −1 푲푛 = 푷푛 푯 푺푛 (2.38)
− 푇 푺푛 = 푯푷푛 푯 + 푹푛 (2.39)
Where
푷푛 − Covariance matrix at discrete time step 푛 푯 − Measurement matrix (∙)푇 − Transpose operator 푨−1 − Inverse of matrix 푨 푹푛 − Measurement noise covariance matrix
By repeating this process of prediction and measurement update at every time step, the
Kalman filter is an efficient means of estimating the system state.
To begin the Kalman filtering process, it is also necessary to initialize the state estimate and covariance matrix to some set of initial values 풙̂0 and 푷0 respectively.
These values may come from an initial detection process which starts off the Kalman filtering, or they may be set to values based on a priori knowledge of the observed system.
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2.4.2 The Extended Kalman Filter
When either f(풙푛−1, 풗푛−1) or h(풙푛, 풘푛) are nonlinear functions, approximations can be used which allow us to apply the Kalman filter to nonlinear state estimation. An example of which is the extended Kalman filter (EKF), which maintains much of the simplicity that makes the Kalman filter appealing and has seen widespread adoption.
The EKF approximates the linear matrices 푭 and 푯 in (2.30) and (2.31) by forming local linearizations of the nonlinear functions f(풙푛−1, 풗푛−1) and h(풙푛, 풘푛). The local linearizations are calculated by evaluating the Jacobian of each nonlinear function:
푑푓(풙) 푭̂ = | 푛 푑풙 (2.40) 푥=푥̂푛−1
푑ℎ(풙) ̂ 푯푛 = | (2.41) 푑풙 − 푥=푥̂푛
Where
푭̂ − Estimate of state transition matrix f(∙) − State transition model 풙 − System state 풙̂푛 − Estimate of system state at discrete time step 푛 푯̂ − Estimate of measurement matrix h(∙) − Measurement model
The standard Kalman filter prediction and measurement update equations in (2.34) –
(2.39) are used to form estimates of the system state, with 푭̂푛 and 푯̂ 푛 replacing 푭 and 푯.
While the EKF relaxes the Kalman filter’s requirement for linear functions, it is important to note that it still assumes both the process and measurement noise are
Gaussian random variables.
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Chapter 3 Literature Review
To provide context for this work, this chapter reviews and discusses recent publications that cover the development of passive radar systems and passive localization algorithms.
3.1 Waveforms for Use in Passive Radar
Passive radar exploits transmissions of opportunity normally used for other applications. In an increasingly congested spectral environment, many transmissions are available. These include analogue frequency-modulated (FM) radio [5], high definition
FM (HDFM) [26] and digital audio broadcast (DAB) [27], analogue television (TV) [28] and digital TV (DTV) [29], global system for mobile communication (GSM) [30], Wi-Fi
[31] and wireless regional area network (WRAN) signals [6]. The properties of the illuminator are a major determinant of passive radar performance. For example, digitally modulated waveforms typically present a stable power spectrum and bandwidth. In contrast, FM radio has a time-varying bandwidth because of its modulating signal content that affects its range resolution and SNR [32]. The illuminator’s properties also determine the passive radar’s application. For example, the potentialities of digital video broadcasting-terrestrial (DVB-T) based passive radar for short-to-medium [29] and long
32 range maritime surveillance [33] have been demonstrated. The DVB-T waveform is a
European standard for the broadcast transmission of DTV and its North American counterpart is the advanced television systems committee (ATSC) DTV. Barott presented autocorrelation properties of the ATSC DTV in [34] and Garry presented an ambiguity diagram analysis in [35]. The favorable noise-like thumbtack ambiguity surface of ATSC
DTV after removal of its fixed structures motivates experimental verification of its localization accuracy.
3.2 Passive Localization Methods
Passive localization methods are classified based on the knowledge available to the radar system, which results in different types of available measurements. The two main groups are passive source localization (PSL) methods, where no information about the transmitted signal is known; and multistatic passive radar (MPR) localization methods, where information about the transmitted signal is known to varying degrees.
3.2.1 Passive Source Localization Methods
Distributed passive source localization (PSL) networks use multiple receivers to intercept signals transmitted by the target itself, and then estimates the emitting target’s position. Unlike active radar where the exact forms of the transmitted signals are completely known, the exact forms of the transmitted signals are completely unknown in
PSL.
Applied in these networks are localization techniques that are based on received signal strength [36], direction of arrival [37], and time difference of arrival [38] [39] [40].
Of these, the performance of TDOA localization is better than the others in terms of
33 accuracy. This is equivalent to taking the range difference measurement between the target and the two transmit or receive sensors. The range differences form a set of hyperbolic equations from where the position of the target is determined. Hyperbolic positioning has been widely used in PSL problems with various closed-form solutions proposed [41] [42] [43]. Application of Taylor-series estimation to the problem of target localization is presented in [44] and extended in [45]. The maximum likelihood (ML) estimation for bearing-only target localization (triangulation) was considered in [46].
3.2.2 Multistatic Passive Radar Localization Methods
In multistatic passive radar (MPR) systems, generally, there exist multiple transmitters and receivers located at different spatial coordinates. The reflecting target’s location is then estimated by fusing measurements from the different transmitter-receiver pairs. MPR is intermediate to active radar and PSL in terms of knowledge possessed by the sensor network about the transmitted signals – the direct-path signals provide a varying degree of knowledge about the transmitted signals that depends on direct-path signal strength [47].
The most general case configuration in MPR is multiple-transmitter and multiple- receiver. Most MPR literature, however, looks at the case for single-transmitter and multiple-receiver or multiple-transmitter and single receiver configuration. In such systems, the relative delay can be measured by comparing the reference signal with the reflected target echo. The delay is then converted to the bistatic range if the transmitter’s position is known, which is the sum of transmitter-target and target-receiver ranges.
These bistatic ranges form a set of elliptic equations through which the target position is
34 obtained. In [48] and [49], the accuracy of bistatic range-based method is shown to be better than the range difference-based method. In [50], it was suggested that bistatic range-based method performs better than range difference-based ones because at large target ranges, range difference values are smaller than bistatic range values, so noise is more influential on range difference values. It also compared the performance of bistatic range-based method with three receivers and with four receivers, and found that they are almost the same. For the case of one receiver with multiple transmitters, [51] presents two closed-form solutions for the target location utilizing least-square technique. Some recent literature that has looked at the general case of multiple-transmitter and multiple- receiver topology are [52], which also developed another closed-form two-step weighted least squares (WLS) method from the minimization of the weighted equation error energy to estimate target location. Further improvements were made in [53] and [50]. A localization method using all the raw receiver measurements to directly localize targets, in contrast to reducing measurement data to target measurements and then combining targets to localize targets, was proposed in [54], by taking inspiration from a centralized approach in MIMO radar networks.
3.3 Array-Based Localization
Most of the passive localization methods in MPR focused exclusively on the ellipsoid intersection approach, which uses bistatic range measurements (or range difference measurements as a variant). To provide an unambiguous target location in three dimensions (3D), there is another way, which is to use an efficient direction finder capable of providing a precise measurement of the target’s bearing. The intersection of
35 ellipsoid and bearing vector leads then to the 3D position of the target. Compared to the direction finding approach, the ellipsoid intersection approach has been deemed to be more robust and accurate, though it requires adequate algorithms and sufficient processing power to cope with multiple-target scenarios and run in real-time. This preference is apparent in literature: in [51], the angle information is assumed to be unavailable due to poor accuracy. In [55] and [56], both angle and range measurements are assumed to be unavailable, and Doppler-only tracking is performed. In [5], angle measurements were used only to assist in target ambiguity resolution. In [57], using a 2D array, a target’s azimuth and elevation were tracked using DVB transmitters in a single frequency network (SFN). The use of these measurements in localization, however, was not elaborated.
This dissertation investigates the use of the direction finding approach in order to fill the gap in the knowledge of performance of such a localization method in passive radar systems. The direction finding approach also has its unique advantages. Bearing information, along with bistatic range, allows for any single receiver to provide a target position estimate from a detection, i.e. it is the only method of 3D localization that works in a bistatic radar configuration. This is useful for assisting the multilateration mode of a system by resolving position ambiguities that can be significant when there are many targets in the surveillance space. A track update or initialization can therefore also be made when only a single receiver within the network detects a target. Bearing information therefore trivializes the target location and provides robustness in tracking when added as an input parameter.
36
The proposed array-based localization method, which will be described in
Chapter 5, relaxes the requirement of a direction finder to be high-resolution by making use of the high range and Doppler resolution afforded through cross-correlation processing of DTV signals, examined in greater detail in Chapter 4.
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Chapter 4 DTV-Based Passive Radar
From the literature review, DTV-based passive radar, with a frequency band from
470 – 700 MHz, looks to be a promising system for passive localization of air targets because of DTV’s high transmit powers and digital modulation. In this section, a quantitative analysis of its detection range is given. A summary of the specifications and structure of the ATSC DTV signal is also presented. Designed for communication systems rather than radar, the signal contains features for synchronization that degrade radar performance if not mitigated. Therefore, the last section describes the signal conditioning performed for removing these features. The remaining signal processing stages are direct signal interference suppression and short-time cross-correlation [35], which are fundamental to the operation of passive radar.
4.1 Detection Range of DTV Passive Radar
With equivalent isotropically radiated powers (EIRP) up to 1 MW, DTV has the highest transmit powers amongst other transmitters; the next highest are FM radio and
DVB-T that have EIRPs up to 100 kW. Unfortunately, like DVB-T, DTV has a small elevation beam width and the use of an additional down tilt of the antenna beam makes its illumination of targets at high altitudes near the transmitter rather weak [58]. Figure
38
4.1 shows a simulated elevation pattern that is typical of a DTV transmitter [59]. It has a
3 dB beamwidth of 3.8° and a downtilt of 0.5°.
Figure 4.1 Simulated elevation pattern of DTV transmit antenna in elevation
We use the ovals of Cassini – constant sensitivity contours for a Swerling 5 target
[58], to illustrate SNR variations at distances from the transmitter and receiver. This gives us an idea of the detection range of the passive radar in a bistatic configuration. To evaluate the bistatic radar range equation (2.9), we use information from the WOSU DTV
[60] along with typical values for a passive radar at this band. The parameters are tabulated in Table 4.1.
39
Parameter Unit Description
3 푃푇 = 251.5 × 10 푊 Power of DTV transmitter, 3 dB below maximum EIRP of 503 kW 퐺푇 = 16 − 12 dB; Gain of DTV transmitting antenna at boresight (12 dB) 퐺푅 = 6 − 4 dB; Gain of receiving antenna (6-element uniform linear array) at boresight 휆푇 = 0.5 푚 Wavelength of DTV signal with center frequency 푓푇 of 617 MHz 2 휎퐵 = 10 푚 Bistatic radar cross section (RCS) 퐹푇 = 1 − Pattern propagation factor for transmitter-to-target- path (0 dB) 퐹푅 = 1 − Pattern propagation factor for target-to-receiver path (0 dB) −23 푘푏 = 1.38 × 10 퐽/퐾 Boltzmann’s constant 푇0 = 290 퐾 Standard temperature reference 퐵 = 6 × 106 퐻푧 Bandwidth of DTV signal 푁퐹 = 4 − Noise figure of receiving system (6 dB) 퐿푇퐿푅 = 10 − System losses (10 dB) 5 퐺푃 = 퐵푇퐶푃퐼 = 6 × 10 − Processing gain obtained with bandwidth B = 6 MHz and coherent processing interval time 푇퐶푃퐼 = 0.1 s Table 4.1 Parameters of a typical DTV passive radar
Ovals of Cassini are routinely plotted in a horizontal/azimuth plane in literature.
In Figure 4.2, we include target altitude in the analysis by plotting the Cassini ovals for three altitudes (from left to right): 2000 m, 5000 m, and 10000 m (approximately cruising altitude). The scale for the SNR variation on all the plots are from 10 to 70 dB. The upper plots correspond to the ovals of Cassini for a directional beam pattern in elevation, while the lower plots provide an omnidirectional reference.
40
Figure 4.2 Ovals of Cassini plotted in azimuth plane for target altitudes of 2000, 5000, and 10000 m (from left to right) for (top) directional DTV beam pattern in elevation and (bottom) omnidirectional reference
In each of the plots, the receiver is located at (0 km, 0 km, 300 m) and the transmitter is at (14 km, 18 km, 330 m). The effect of the directional elevation pattern is most obvious when we compare the top and bottom plots of the first column of Figure 2. Since the target altitude is constant, the locus of constant target elevation from the transmitter is a circle centered at the transmitter. Thus, the effect of the null in elevation beam pattern at about 6° elevation can be seen as a dark ring around the transmitter in the top plot. This dark ring moves outwards as the target altitude increases from 2000 m to 10000 m (as we move from left to right on the top row) because the target needs to be further in ground range from the transmitter to achieve the same elevation from the transmitter. The 41 difference in the SNR as we move from left to right is much more pronounced in the case of the directional elevation beam pattern (top row) than the omnidirectional reference
(bottom row). In the omnidirectional case, the difference is due to increased target range as the target’s altitude increases. In the directional case, there is an additional loss due to the elevation beam pattern. This restricts target detection at medium- (5000 m) and cruising-level (10000 m) altitudes.
While Figure 4.2 gives a predicted coverage of a bistatic radar using DTV as the transmitter, it is not a complete picture since simplifying assumptions have been made.
This include the assumption that an adequate line of sight (LOS) exists between the transmitter-to-target path and receiver-to-target path, and that the pattern propagation factors and bistatic target radar cross section (RCS) are invariant with range and aspect angle, which is usually not the case. For that, the effects of a curved earth, diffraction of the radar energy around the earth, and RCS analysis on different targets and bistatic angles need to be considered. The details of bistatic scattering modeling, however, are beyond the scope of this work.
4.2 Specifications of DTV Signals
DTV broadcast signals employed in North America are specified by the
Advanced Television Systems Committee (ATSC) A/53 standard [61]. The modulation used by DTV is 8-VSB (vestigial sideband) and designed to operate over the same 6
MHz channel bandwidth as the NTSC system without interfering with it [62]. This is illustrated in Figure 4.3.
42
Figure 4.3 NTSC and 8-VSB systems
8-VSB is an 8-level amplitude-shift-keyed (ASK) signal that has been filtered at
Nyquist rate to a single sideband. The waveform consists of a signal with 8-levels (3-bit) symbols of [±1, ±3, ±5, ±7] at a rate of 10.76 MHz. Filtering is done using a root-raised- cosine filter having a roll off factor of α = 0.1152 that passes the upper sideband (5.38
MHz) and limits the total bandwidth to the channel size of 6 MHz [34]. Randomization and interleaving processes result in data that have correlation properties similar to a pseudo random sequence. However, there are some fixed structures. The pilot carrier is generated by adding a constant value of 1.25 to the symbols, which makes up 7% of the total signal power. There is also a segment synchronization of 4 bits, a [+5, -5, -5, +5] sequence, that repeats every 77.3 μs (828 symbols). Finally, there is a longer field synchronization that occurs at 24.2 ms intervals [34].
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4.3 Signal Processing Stages in DTV-Based Passive Radar
4.3.1 Waveform Conditioning Processing
The fixed structures in DTV signal cause some side peaks in the corresponding ambiguity function (AF), and these unwanted peaks could be responsible for masking weak targets in the vicinity of strong ones, increasing the false alarm rate and degrading the detection performance of the passive radar [4] [63]. Different techniques have been proposed to mitigate the side peaks of the AF, and these pre-processing steps of both the reference and surveillance signals used in passive radar are mainly based on synchronization and suppression or elimination of the frame structure and pilots [63] [64]
[65].
Figure 4.4 shows the AF surface of a collected DTV signal integrated over 0.5 seconds, along with a range cut of the surface at zero-Doppler offset and a Doppler cut at zero-time offset. The presence of the pilot carrier results in a zero-Doppler ridge 11 dB below the autocorrelation peak, which can be seen in Figure 4.4a and Figure 4.4b.
44
(a)
(b) (c)
Figure 4.4 (a) AF surface of DTV signal; (b) Zero-Doppler offset range cut; (c) Zero-time offset Doppler cut
To remove the pilot carrier, a spectral mask was applied to the data by finding the frequency bin corresponding to the maximum power spectral density of the received signal, then zeroing all frequency bins at ±1 kHz. Figure 4.5 shows the AF surface after the pilot carrier has been removed along with its range and Doppler cuts at zero-Doppler and zero-time offset respectively.
45
(a)
(b) (c)
Figure 4.5 (a) AF surface of DTV signal after removal of pilot tone; (b) Zero-Doppler offset range cut; (c) Zero-time offset Doppler cut
The segment synchronization results in a -22.5 dB ambiguity that occurs every
77.3 μs. This can be seen most clearly in Figure 4.5b. The long field synchronization that occurs every 24.2 ms exceeds the relevant duration of target observation and is not shown. There is also another unknown range ambiguity at 27.2 μs. Thus, the AF is not an ideal thumbtack; however, the range ambiguities are sufficiently low at 22.5 dB below the main peak. The zero-range Doppler cut in Figure 4.5c shows that it can separate well
46 moving target echoes from the stationary clutter and direct signal components because the autocorrelation floor of noise-like digital waveform is at -35 dB below the main peak.
After application of the spectral mask, an inverse Fourier transform was calculated for the next stage of time-domain signal processing.
4.3.2 Direct Signal Interference Suppression
A passive radar system consists of at least two channels – the reference channel, which collects the direct signal emitted from transmitters, and the surveillance channel, which receives the echo signal reflected from potential targets. Since the signals are typically 100% duty cycle, there are strong interferences in the passive radar receiver’s surveillance channel from the signal that propagates from the transmitter to the receiver directly and scattered signals from stationary objects in the environment. While clutter is significant since the transmitting antenna is often pointed towards the ground, the direct signal interference (DSI) is the greatest limiting factor. DSI can be up to 90 dB greater than the target echo [5].
DSI suppression aims to estimate the terms that comprise the DSI using knowledge of the transmitted waveform acquired through the reference channel 푠푟[푛].
This estimate of the DSI component can then be coherently subtracted from the surveillance waveform 푠푠[푛] such that the residual leaves only the target response 푠푡푎푟[푛] and noise. This is represented in Figure 4.6.
47
Figure 4.6 DSI suppression block diagram
The reference waveform passes through an FIR filter with impulse response ℎ[푖], which represents the delay and complex scattering coefficients of the DSI.
푀−1 ∗ 푠푑푠푖[푛] = ∑ ℎ [푚]푠푟[푛 − 푚] (4.1) 푚=0 Where
푠푑푠푖 푉 DSI contributions 푠푟 푉 Reference waveform ℎ − Impulse response of filter representing delay and complex scattering coefficients of DSI ′ 푀 = ⌈푅퐵/푐푇푠⌉ − Number of discrete time delay coefficients to model the maximum bistatic range of the DSI components greater than the noise power ⌈∙⌉ − Ceiling operator ′ 푅퐵 푚 Bistatic range past baseline 푇푠 푠 Sampling interval/period
The DSI removal process first estimates the unknown clutter and direct path coefficients
ℎ̂[푖]. Then the result is convolved with the reference channel waveform to estimate
푠푑푠푖[푛]. This output is subtracted from the surveillance channel waveform 푠푠[푛] to leave only target responses and thermal noise, 푠푐[푛] – a ‘clean’ surveillance channel waveform that is free of DSI. 48
Results of the study by Garry [66] showed that the Extensive Cancellation
Algorithm (ECA) generally exhibited the best suppression performance, but with significant runtimes due to high computational cost. In practice, either the Least Squares
(LS) or Fast Block Least Mean Squares (FBLMS) should be employed, due to their generally good performance with very low computational costs and thus fast runtimes. Of these two algorithms, the LS filter was used for the localization experiments because the targets of interest observed were under situations where the bistatic geometries produced relatively low bistatic velocities.
The procedure for LS is derived in detail by Haykin [67], and in this method, the tap weights of the FIR filter are chosen so as to minimize a cost function that consists of the sum of error squares. The FIR filter model is shown in Figure 4.7.
Figure 4.7 FIR filter model
By forming inner scalar products of the tap inputs 푠푟[푛], 푠푟[푛 − 1], …, 푠푟[푛 − 푀 + 1] and the corresponding tap weights 푤0, 푤1, …, 푤푀−1, and by utilizing 푠푠[푛] as the desired
49 response, the estimation error or residual 푠푐[푛] is defined as the difference between the desired response 푠푠[푛] and the filter output 푠̂푑푠푖[푛]
푠푐[푛] = 푠푠[푛] − 푠̂푑푠푖 [푛] (4.2) 푀−1 ∗ 푠̂푑푠푖 [푛] = ∑ 푤푘푠푟[푛 − 푚] (4.3) 푚=0 Where
푠푐[푛] − Estimation error or residual: ‘Clean’ surveillance channel waveform 푠푠[푛] − Desired response: Surveillance channel waveform 푠̂푑푠푖[푛] − Filter output: Estimate of DSI contributions
Substitution gives
푀−1 ∗ 푠푐[푛] = 푠푠[푛] − ∑ 푤푘푠푠[푛 − 푖] (4.4) 푖=0 The tap weights are chosen to minimize the sum of error squares
푛2 2 휉(푤0, … , 푤푀−1) = ∑ |푠푐[푛]| (4.5)
푛=푛1 Where
휉(∙) − Cost function 푛1, 푛2 − Index limits at which error minimization occurs
Figure 4.8 shows the range-Doppler (RD) map outputs from cross-correlation, before and after DSI suppression using the LS filter was applied. The DSI suppression revealed not only the target (circled in red), but also residual clutter components that were also previously masked.
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(a) (b)
Figure 4.8 (a) RD map before DSI suppression using LS filter; (b) After DSI suppression
4.3.3 Short-Time Cross-Correlation
Computation of the RD maps shown in Figure 4.8 is a fundamental stage of passive radar processing, upon which target detections can be performed. This surface is calculated via Doppler-shifted cross-correlations of the reference and surveillance waveforms in the form of the cross ambiguity function (CAF) as
푇 ∗ 휒(휏, 휈) = ∫ 푠푠(푡)푠푟(푡 − 휏) exp(−푗2휋휈푡) 푑푡 (4.6) 푡=0 Where
휒 퐽 Cross ambiguity function (CAF) 휏 푠 Time delay 휈 퐻푧 Doppler frequency shift 푠푠(푡) 푉 Surveillance waveform 푠푟(푡) 푉 Reference waveform
Efficient implementation of the CAF is done in software after digitization. The calculation of 휒(휏, 휈) will now be discrete so
51
푡 = 푛푇푠 (4.7)
푁 = 푓푠푇 (4.8) Where
푡 푠 Continuous time 푛 − Index corresponding to discrete time 푇푠 푠 Sampling interval/period 푓푠 = 1/푇푠 퐻푧 Sampling frequency 푁 − Number of samples 푇 푠 Observation duration
The equivalent form of (4.6) in discrete time is given as
푁−1 푚 χ[푙, 푚] = ∑ 푠 [푛]푠 [푛 − 푙]∗ exp (−푗2휋 푛) (4.9) 푠 푟 푁 푛=0 Where
푙 − Index corresponding to discrete time delay 푚 − Index corresponding to discrete Doppler frequency shift 푖 푁×1 풔푠 ∈ ℂ 푉 Signal received at the surveillance channel of the 푖th bistatic pair 푖 푁×1 풔푟 ∈ ℂ 푉 Signal received at the reference channel of the 푖th bistatic pair
Exact implementation of (4.9) can be achieved using correlation fast Fourier transform (FFT) and the direct FFT method of [68]. The correlation FFT method implements the CAF sequentially for each Doppler shift by first Doppler shifting the reference or surveillance waveform, followed by cross-correlation implemented as frequency domain multiplication. The Doppler shift operation is also implemented via a circular shift in the frequency domain to ease the computational burden. The second, direct FFT approach fixes the delay, then multiplies the waveforms in the time domain.
52
The remaining expression to calculate the CAF is in the form of the FFT, to generate the desired Doppler points. Although both approaches make use of the efficiency of the FFT, they are still computationally costly and would be difficult to implement in real time.
Both algorithms involve calculation of 푁 correlation points for each Doppler or range bin, most of which are discarded because the values are far greater than the effective detection range or expected target velocities.
In order to significantly reduce the computation burden, an approximation was made to both simplify the calculation and reduce the points which are subsequently discarded as they lie outside the range and Doppler values of interest due to the finite detection range of a particular passive radar system and limits of realistic target velocities
[35]. (4.9) is rewritten so that Q short-time cross correlations of reduced length L are calculated (where QL = N)
푄−1 퐿−1 푚 푚 χ[푙, 푚] = ∑ exp (−푗2휋 푞) ∑ 푠 [푛 + 푞퐿]푠 [푛 + 푞퐿 − 푙]∗ exp (−푗2휋 푛) (4.10) 푄 푠 푟 푄퐿 푞=0 푛=0 Where
푄 − Number of short-time cross correlations of length 퐿 퐿 = 푁/푄 − Reduced cross correlation length 푁 − Original number of samples in a coherent processing interval
The correlation repetition rate (or new sampling frequency) should be more than twice the maximum expected Doppler frequency 휈푚푎푥
1 ≥ 2휈푚푎푥 (4.11) 퐿푇푠
53
If the Doppler index is limited to Q Doppler bins, then the final phase term in (4.10) can be ignored with minimal impact on the RD map. The approximation is then written as
푄−1 퐿−1 푚 χ[푙, 푚] ≈ ∑ exp (−푗2휋 푞) ∑ 푠푖 [푛 + 푞퐿]푠푖 [푛 + 푞퐿 − 푙]∗ (4.12) 푄 푠 푟 푞=0 푛=0 This approximation is analogous to the stop-and-hop approximation [69] made by pulsed
Doppler radar system.
(4.12) is implemented using the batch FFT algorithm in [35]. The correlation of each batch is implemented in the frequency domain, utilizing zero padding such that the circular correlation output is identical to that of linear correlation. This is followed by
Doppler processing via a second DFT across the Q batches, and finally an IFFT along each batch to the Range-Doppler domain. First, the reference and surveillance batch matrices 푺푟 and 푺푠 are constructed for L+∆ delay bins and Q Doppler bins. The reference matrix is constructed from Q non-overlapping reference batches of length L, zero-padded by ∆ samples
풔 [0] 풔 [퐿] ⋯ 풔 [(푄 − 1)퐿] 푟 푟 푟 [ ] [ ] [( ) ] 풔푟 1 풔푟 퐿 + 1 ⋯ 풔푟 푄 − 1 퐿 + 1 ⋮ ⋮ ⋱ ⋮ 푺푟 = 풔푟[퐿 − 1] 풔푟[2퐿 − 1] ⋯ 풔푟[푄퐿 − 1] (4.13) 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ [ 0 0 ⋯ 0 ]
The surveillance channel is similarly constructed from batch segments of L+∆ samples, that overlap by ∆ points.
54
풔 [0] 풔 [퐿] ⋯ 풔 [(푄 − 1)퐿] 푠 푠 푠 [ ] [ ] [( ) ] 풔푠 1 풔푠 퐿 + 1 ⋯ 풔푠 푄 − 1 퐿 + 1 ⋮ ⋮ ⋱ ⋮ 푺푠 = 풔푠[퐿 − 1] 풔푠[2퐿 − 1] ⋯ 풔푠[푄퐿 − 1] (4.14) 풔푠[퐿] 풔푠[2퐿] ⋯ 풔푠[푄퐿] ⋮ ⋮ ⋱ ⋮ [풔푠[퐿 + 훥 − 1] 풔푠[2퐿 + 훥 − 1] ⋯ 풔푠[푄퐿 + 훥 − 1] ]
Constructed in this manner, the first ∆ points of interest will be equivalent to the linear correlation of each batch. This is calculated through column-wise FFT’s, piece-wise multiplication using the Hadamard product, followed by an IFFT along the columns, and then truncating to the first ∆ points
−1 푹 = 퐅 (퐅푺푟 ∘ 퐅푺푠) (4.15) 푹(1,1) 푹(1,1) ⋯ 푹(1, 푄) 푹(2,1) 푹(2,2) ⋯ 푹(2, 푄) 푹′ = [ ] (4.16) ⋮ ⋮ ⋱ ⋮ 푹(훥, 1) 푹(훥, 2) ⋯ 푹(훥, 푄) Where
푹 − Range-slow time for (퐿 + 훥) batch correlation points 푹′ − Range-slow time matrix for linear correlation of each batch 퐅 − Fourier transform matrix 퐅−1 − Inverse Fourier transform matrix
To obtain the RD map, the range-slow time matrix is then multiplied by the Fourier transform matrix along the rows (slow-time)
흌 = 푹′퐅 (4.17)
For a typical coherent processing interval of 0.1 s or above (N = 1,000,000 or more), Q =
200, L = 5,000, and ∆ = 4,000 are used.
55
4.3.3.1 Selection of Coherent Processing Interval for use with DTV
When detecting a moving target like aircraft, the choice of the coherent processing interval (CPI) is very important. The reason for this is that choosing a CPI that is too long may lead to range cell migration or velocity cell migration effect. During the
CPI, the target moving with a range rate 푅̇ changes its range by 푅̇ 푇퐶푃퐼. If this distance is greater than the range resolution cell, range cell migration occurs. Since the bistatic range resolution cell is larger or equal to the monostatic range resolution cell, we use the latter to calculate the limitation of the CPI:
훥푅 푐 푇퐶푃퐼 < = (4.18) 푅̇푚푎푥 2퐵푅̇푚푎푥 Where
푇퐶푃퐼 푠 Coherent processing interval time at receiver 훥푅 푚 Range resolution 푅̇ 푚푎푥 푚/푠 Maximum range rate 푐 = 3 × 108 푚/푠 Speed of propagation 퐵 퐻푧 Bandwidth of signal
The second limitation of the CPI is from the velocity cell migration. Again, the bistatic velocity resolution cell is larger or equal to the monostatic velocity resolution cell, so the latter is used
휆 훥푉 = 푇 (4.19) 2푇퐶푃퐼 Where
훥푉 푚/푠 Radial velocity resolution 휆푇 푚 Wavelength of signal
56
The target with radial acceleration A changes radial velocity during the CPI by A푇퐶푃퐼. If the echo remains in one velocity resolution cell, the change should be smaller than the resolution cell, which leads to the following limitation of the CPI:
훥푉 (4.20) 푇퐶푃퐼 < 퐴푚푎푥
Substituting (4.19) into (4.20) and making 푇퐶푃퐼 the subject,
휆 (4.21) 푇퐶푃퐼 < √ 2퐴푚푎푥
Assuming a maximum radial velocity of 250 m/s and maximum radial acceleration of 5 m/s2, a CPI of 0.1 s or less will satisfy (4.18) and (4.21) for DTV.
57
Chapter 5 Array-Based Localization
Each of the antenna elements in the surveillance array receives signals from the surveillance area of interest. Thus, there are 푁푒 surveillance channels corresponding to the number of elements in the array. After the signal processing steps described in
Chapter 4 have been performed for each surveillance channel, we obtain 푁푒 Range-
Doppler (RD) maps as shown in Figure 5.1. In this section, we describe the data processing stages that are performed on this array of RD maps to estimate the target’s 3D
Cartesian location.
Figure 5.1 Computed RD maps for all 6 antenna elements on receiver array 58
5.1 Received Signal Model
Here, we develop the received signal model in a multistatic configuration made up of multiple transmitters and a receiver with a linear array of 푁푒 elements. A bistatic pair i is made up of the receiver and transmitter i. Figure 5.2 illustrates this.
Figure 5.2 Passive radar set-up
We assume that all transmitter and receiver positions are time-independent and known. We also assume that the direct signal and clutter have been sufficiently mitigated, leaving only the target-path signal to be incident on the array. Let
(5.1) 퐿푇푖 = ‖풓푅 − 풓푇푖‖ (5.2) 푅푇푖 = ‖풓푡 − 풓푇푖‖ (5.3) 푅푅 = ‖풓푅 − 풓푡‖
59
Where
퐿푇푖 푚 Range between transmitter 푖 and reference element of the receiver array known as the baseline 푅푇푖 푚 Range between transmitter 푖 and target 푅푅 푚 Range between reference element of the receiver array and target 풓푅 푚 Position of the reference element of the receiver array 풓푇푖 푚 Position of transmitter 푖 풓푡 푚 Position of target
Then the propagation delay on the target path is given by
(5.4) 휏푖 = (푅푇푖 + 푅푅)/푐 Where
휏푖 푠 Propagation delay along target path of 푖th bistatic pair 푐 = 3 × 108 푚/푠 Speed of propagation
Similarly, Doppler shift on the target path is
(5.5) 휈푖 = −(푅̇ 푇푖 + 푅̇푅)/휆푇푖 Where
휈푖 퐻푧 Doppler shift along target path of 푖th bistatic pair 푅̇ 푇푖/푅̇푅 푚/푠 Time derivative of 푅푇푖/푅푅 휆푇푖 푚 Wavelength of signal from transmitter 푖
The transmitted signals are non-overlapping in frequency with a common bandwidth 퐵. The received signal is channelized in frequency, and each channel is demodulated to baseband and sampled at a rate 푓푠 ≥ 퐵 for 푇 seconds, resulting in signals of length
(5.6) 푁 = 푓푠푇
60
Where
푁 − Number of samples 푓푠 = 1/푇푠 퐻푧 Sampling frequency 푇 푠 Observation duration Let the sampled complex baseband signal of the 푖th bistatic pair at element 푛푒 of the receiver array be given by
( ) (5.7) 풔푛푒,푖 = 훾푛푒,푖 exp[푗휙푛푒,푖 풓푡 ] 풟푖풖풊 + 풏푛푒,푖 Where
푁×1 풔푛푒,푖 ∈ ℂ 푉 Sampled complex baseband signal of the 푖th bistatic pair on element 푛푒 of the receiver array 푁×1 풖푖 ∈ ℂ 푉 Sampled complex baseband signal emitted by the 푖th transmitter
훾푛푒,푖 − Complex scaling of 풖풊 along the target path 푗 = √−1 − Imaginary number ( ) 휙푛푒,푖 풓푡 푟푎푑 Differential phase at the 푛푒th array element with respect to a reference element due to propagation from the target 푁×푁 풟푖 = 풟(휏푖, 휈푖) ∈ ℂ − Delay-Doppler operator that accounts for the delay and Doppler shift imparted to the 푖th transmit signal as it propagates to the receiver along the target path 푁×1 Circular Gaussian noise distributed as 풏푛푒,푖 ∈ ℂ 푉 2 2 풩(ퟎ푁, 휎 퐈푁) with known variance 휎 ; ퟎ푁 is a length-푁 zero vector and 퐈푁 is an 푁 × 푁 identity matrix
The differential phase at the 푛푒th array element with respect to a reference element due to plane wave propagation from a direction 풙 is defined by
2휋 ( ) ̂ (5.8) 휙푛푒,푖 풙 = ( ) 풌(풙) ∙ 풅푛푒 휆푇푖 Where
풌̂(풙) 푚 Unit vector pointing from 풓푅 toward 풙
풅푛푒 푚 Position of element 푛푒 of the receiver, relative to
풓푛푒
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The unitary linear delay and Doppler shift operator is defined as
휈 퐻 −휏푓푠 풟(휏, 휈) = 퐃푁 ( ) 퐖 퐃푁 ( ) 퐖 (5.9) 푓푠 푁
푗2휋(0)푥 푗2휋(푁−1)푥 푁×푁 (5.10) 퐃푁(푥) = diag([푒 , … , 푒 ]) ∈ ℂ Where
풟(휏, 휈) ∈ ℂ푁×푁 − Unitary linear delay-by- 휏 and Doppler shift-by- 휈 operator diag(풙) − 푁 × 푁 square matrix with diagonal elements 풙, 풙 ∈ ℂ푁×1 퐖 ∈ ℂ푁×푁 − Unitary Discrete Fourier Transform matrix
Finally, noise is assumed to be independent across transmit bands and array elements, i.e.
퐻 2 (5.11) E {풏푛푒,푖(풏푚푒,푘) } = 휎 훿푛푒−푚푒훿푖−푘퐈푁 Where
(∙)퐻 − Hermitian transpose 훿푥 − Kronecker delta
Since 훾푛푒,푖 is multiplicative with 풖풊 in (5.7), the transmitted signal 풖풊 is defined
‖ ‖2 such that 풖풊 = 푁 and the composite scaling of 풖풊 is accounted for by 훾푛푒,푖. Therefore,
훾푛푒,푖 accounts for various effects including antenna gains, spreading losses, non-isotropic target reflectivity, and any unknown phase offsets between signals measured at different receivers due to the use of receivers that are not calibrated to a common phase reference.
The linear signal model in (5.7) is only valid under certain conditions. First, it requires that the effect of the relative propagation delay between elements of a given array can be modelled as a phase shift [70]. This means that the complex envelope of a received signal is approximately constant across the elements of the array, which holds if
62
퐵훥휏 ≪ 1 (5.12) Where
퐵 퐻푧 Bandwidth of signal 훥휏 푠 Maximum relative delay 훥휏 between array elements
Next, it requires that the complex envelope of each received signal experiences negligible distortion (i.e. amplitude scaling and time dilation) due to relative motion between transmitters, receivers, and the target [71]. This holds when
(5.13) 퐵푇퐶푃퐼 < 푐/(10푅̇푚푎푥) Where
퐵푇퐶푃퐼 − Time-bandwidth product 푅̇ 푚푎푥 푚/푠 Maximum target range rate
Finally, it requires the effects of range acceleration and higher-order range derivatives to be negligible [72]. This holds with respect for range acceleration when
2 ̈ (5.14) 푇퐶푃퐼 < 휆푇/(10푅푚푎푥) Where
TCPI 푠 Coherent processing interval time 휆푇 푚 Wavelength of signal 2 푅̈푚푎푥 푚/푠 Maximum target range acceleration
These conditions are satisfied in many passive radar scenarios that involve narrowband signals and slowly maneuvering targets.
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5.2 Bearing Estimation
A third measurement that can be obtained after the cross-correlation operation is the differential phase at array element 푛푒 due to propagation from the target. The received signal model (5.7) at element 푛푒 is reproduced below for convenience
풔 = 훾 exp[푗휙 (풓 )] 풟 풖 + 풏 푛푒,푖 푛푒,푖 푛푒,푖 푡 푖 풊 푛푒,푖 (5.15)
The cross-correlation operation (4.9) between the received signal 풔푛푒,푖 and the reference signal 풔푟,푖 is given by
푁−1 푚 χ[푙, 푚] = ∑ 푠 [푛]푠 [푛 − 푙]∗ exp (−푗2휋 푛) (5.16) 푛푒,푖 푟,푖 푁 푛=0 Where
χ 퐽 Cross ambiguity function (CAF) 푙 − Index corresponding to discrete time delay 푚 − Index corresponding to discrete Doppler frequency shift 푁×1 풔푛푒,푖 ∈ ℂ 푉 Signal received at the 푛푒th surveillance channel of the 푖th bistatic pair 푁×1 풔푟,푖 ∈ ℂ 푉 Signal received at the reference channel of the 푖th bistatic pair
Upon inspection, the complex scaling and differential phase in (5.15) can be shifted outside of the summation operator in (5.16). Therefore, the same phase information present in the time domain signal could be obtained in the Doppler domain for phase comparison direction finding. Compared to the raw time-domain signal, the Doppler domain signal has a much better SNR due to the processing gain after cross-correlation.
The complex scaling 훾푛푒,푖 in (5.15) accounts for various effects including antenna gains, spreading losses, non-isotropic target reflectivity, and any unknown phase offsets between signals measured at different receivers due to the use of receivers that are not
64 calibrated to a common phase reference. This will be elaborated on in Chapter 7, which discusses array calibration. We take a closer look now at the differential phase term in
(5.15) assuming that array calibration has been done correctly.
The differential phase at array element 푛푒 with respect to a reference element due to propagation from the target’s direction 풓푡 is defined by
2휋 ( ) ̂ (5.17) 휙푛푒,푖 풓푡 = ( ) 풌(풓푡) ∙ 풅푛푒 휆푇푖 Where
̂ 풌(풓푡) 푚 Unit vector pointing from 풓푅 toward 풓푡
풅푛푒 푚 Position of element 푛푒 on the receiver array, relative to 풓푅 (position of the reference element on receiver array)
This is applied to a linear array as shown in Figure 5.3.
Figure 5.3 Linear array
65
We obtain
2휋 2휋 ( ) ( ) ( ) ( ) ( ) 휙푛푒,푖 풓푡 = 푛푒 − 1 푑 cos 훼 = 푛푒 − 1 푑 cos 180 − 휃 휆푇푖 휆푇푖 2휋 = (푛푒 − 1) 푑[− cos(−휃)] (5.18) 휆푇푖 2휋 = − (푛푒 − 1) 푑 cos(휃) 휆푇푖
Where 휃 is the bearing of the impinging signal. A horizontal linear array will give azimuth and a vertical linear array, elevation.
With the differential phase available at the array, the target’s azimuth and elevation can be estimated with different bearing estimators. We use (5.18) in the Bartlett method weights:
푇 [ ( )] ( ) 풘퐵푎푟푡푙푒푡푡 (휃푠) = [1 exp 푗휙2 휃푠 ⋯ exp[푗휙푁푒 휃푠 ]] (5.19)
This weight vector is then used to linearly combine the data received by the array
풔 ⋯ 풔 푇 elements 풙 = [ 1 푁푒 ] to form a single output signal 푦(휃푠)
퐻 푦(휃푠) = 풘 (휃푠)풙 (5.20)
By scanning across a range of angle 휃푠 using 풘퐵푎푟푡푙푒푡푡 (휃푠), we obtain a power spectrum
2 푃(휃푠) = |푦(휃푠)| (5.21)
The 휃푠 value that corresponds to the peak of 푃(휃푠) is the target’s bearing estimate
휃̂ = arg max[푃(휃푠)] 휃푠 (5.22)
The conventional beamformer using the Bartlett weights and MUSIC estimator give similar performances since the differential phases came from a single target isolated using the RD map.
66
By using the good bistatic range and Doppler resolution accorded by DTV signal to resolve multiple targets, it relaxes the requirement for the direction finder to have high angular resolution. Thus, the Bartlett method of bearing estimation can be used. In this manner, angular position can be estimated with far greater accuracy than the 3 dB angular resolution of the array alone.
5.3 Conversion to Cartesian Space
Figure 5.4 summarizes the data flow so far:
Figure 5.4 Array-based localization data flow
After detection, and isolation, of the return from a single target in each of the 푁푒 RD
′ maps, 푁푒 sets of bistatic range past baseline 푅퐵푖, bistatic velocity 푉퐵푖, and differential
′ phases 휙푛푒,푖 are available. The 푁푒 sets of 푅퐵푖 and 푉퐵푖 are effectively identical due to the negligible distance between the closely-spaced array elements relative to the target’s
range from receiver. Only the differential phases 휙푛푒,푖 are able to capture the relationship of plane wave propagation between each array element with respect to the reference array element. These differential phases are used to estimate the azimuth and elevation of the
67 target and along with one set of bistatic range past baseline measurement, conversion of these measurements to Cartesian space can be done to give an estimate of the target’s location in 3D.
The relevant measurements for array-based localization are the target azimuth and elevation angles, and bistatic range. The bistatic range is defined as the sum of the transmitter-target and target-receiver distances.
(5.23) 푅퐵푖 = 푅푇푖 + 푅푅 Where
푅퐵푖 푚 Bistatic range corresponding to transmitter 푖 푅푇푖 푚 Range between target and transmitter 푖 푅푅 푚 Range between target and receiver
After cross-correlation processing, we obtain the target’s bistatic range past baseline from the RD map. This is related to the bistatic range by
′ (5.24) 푅퐵푖 = 푅퐵푖 − 퐿푇푖 Where
′ 푅퐵푖 푚 Bistatic range past baseline of 푖th bistatic pair 퐿푇푖 푚 Range between receiver and transmitter 푖
Since the positions of receiver and transmitters are known, the bistatic range past baseline can be easily converted to the bistatic range. The receiver look angle is calculated from the target’s and transmitter’s azimuth and elevation angles using [1]
68
−1 (5.25) 휃푅푖 = − sin [cos(퐸푡) cos(퐸푇푖) cos(퐴푡 − 퐴푇푖) + sin(퐸푡) sin(퐸푇푖)] Where
휃푅푖 ° Receiver look angle 퐸푡 ° Elevation angle of target 퐸푇푖 ° Elevation angle of transmitter 푖 퐴푡 ° Azimuth angle of target 퐴푇푖 ° Azimuth angle of transmitter 푖
This is used to calculate the target’s range from the receiver
2 2 푅퐵푖 − 퐿푇푖 푅푅 = (5.26) 2[푅퐵푖 + 퐿푇푖 sin(휃푅푖)]
We now have the polar coordinates of the target from the receiver (푅푅, 휃푅), which is equivalent to the information measured by a monostatic radar at the receiver location.
Target localization in 3D Cartesian coordinates is then achieved by the following conversion
(5.27) 푥푡 = 푅푅 cos(퐸푡) cos(퐴푡) (5.28) 푦푡 = −푅푅 cos(퐸푡) sin(퐴푡) (5.29) 푧푡 = 푅푅 sin(퐸푡) Where
푥푡 푚 x-coordinate of target 푦푡 푚 y-coordinate of target 푧푡 푚 z-coordinate of target 푅푅 푚 Range between target and receiver 퐸푡 ° Elevation angle of target 퐴푡 ° Azimuth angle of target
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Chapter 6 System Overview
The multistatic digital television passive radar (MUTERA) [35] was used extensively for the experimental testing discussed in this dissertation. For this reason, this chapter is devoted to outlining the hardware of MUTERA.
6.1 The MUTERA Testbed
MUTERA is a passive radar system consisting of multiple transmitters and a single receive site. It is situated on the rooftop of the Electro Science Laboratory (ESL) in
Columbus, Ohio. The radar collects a 150 MHz portion of the DTV spectrum on eight separate receive channels, allowing for multistatic observations of a single target over a range of diverse geometries, frequencies, and power levels. It is equipped with a GPS receiver for precise time-stamping of the ADC data, and an automatic dependent surveillance-broadcast (ADS-B) receiver is integrated with the system to provide air truth data, used in this research to validate the accuracy of target location estimates. As illustrated in Figure 6.1, the system is controlled with a TALON 2746 data recorder, a turnkey system produced by Pentek Inc. that is Windows PC-based and equipped with two PCI express (PCIe) cards for ADC capability and an additional PCIe card as the GPS receiver. The ADS-B receiver, a Kinetic Avionics SBS-3, is connected via USB and
70 processes the 1090 MHz extended squitter before transferring to the TALON 2746 recorder for cataloging.
Figure 6.1 MUTERA passive radar system
The MUTERA passive radar system designed at the Ohio State University meets the following requirements:
• Each receive channel consist of a separate antenna, RF down-conversion, and
ADC with storage capability
• Each channel can collect multiple illuminating signals concurrently
• Reference channels have adequate line-of-sight (LOS) to the transmitter to
provide a delayed copy of the transmit waveform to use in direct signal
interference (DSI) suppression and Range-Doppler (RD) map processing
• Surveillance channels have sufficient dynamic range to receive, down-convert,
and sample to preserve weak target echoes in the presence of strong DSI
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6.2 Transmitters
The location of the MUTERA testbed and transmitter locations, center frequencies, and transmit power levels are shown in Figure 6.2.
Figure 6.2 Map showing locations of MUTERA and DTV Transmitters
Only transmitters whose center frequencies are within the passband and roll-off of the system’s bandpass filter from 530 – 680 MHz are shown in the map. Notice that the transmitters are located north, east, and south of the MUTERA site, indicated by the radome icon. Parameters of these illuminators specific to a receiver located at the ESL 72 are listed in Table 6.1. They include their center frequencies, effective isotropic radiated powers (EIRPs), bistatic baselines, and estimated receive power levels for the direct path.
The received powers were calculated by the Longley-Rice algorithm [73], which incorporates knowledge of the transmitter height and EIRP, terrain between the transmitter and receiver, as well as receive antenna height.
Call Sign Frequency EIRP [kW] Received Power Baseline [MHz] [dBm] [km]
W23BZ 527 15 -39.4 11.7 WSFJ 533 1000 -36.3 31.1 WCSN 587 15 -29.8 2.9 WTTE 605 1000 -15.1 7.1 WOSU 617 503 -23.8 20.6 W44DC 653 15 -40.8 3.4 WWHO 665 1000 -34.4 45.9 WSYX 677 1000 -16.1 7.1 Table 6.1 Parameters of DTV transmitters for a receiver at the Electro-Science Laboratory
6.3 Antennas
The MUTERA system is made up of the reference antenna to collect the direct signal emitted from transmitters, and the surveillance antenna to receive the echo signal reflected from potential targets.
6.3.1 Reference Antenna
For a multistatic system with illuminators at various angular positions, a proportionate number of receive antennas are required to gather the various reference signals. In cases where LOS to the transmitter is available, a narrow beam width antenna
73 is desired to minimize the multipath in the reference waveform, whose presence can degrade the effective SNR of the radar system. As shown in Figure 6.3, 5 high-gain Yagi antennas are available to be used for the reference channel, pointing at different directions for each group of illuminators. During experiments, usually one or two of them are connected at a time to collect the reference signals.
Figure 6.3 Configuration of reference antennas
The antennas are set in horizontally-polarized (H-pol) and vertically-polarized (V- pol) positions, but typically the H-pol antenna is used because DTV transmitters are also
H-pol. This helps to maximize the SNR of the LOS reference signal and to minimize the strength of any multipath bounces, whose polarization state is likely altered upon reflection off nearby objects and clutter around the bistatic baseline. The properties of the antenna element are summarized in Table 6.2, with its gain in azimuth and elevation shown in Figure 6.4. 74
Antenna model Antennas Direct 43XG Band UHF (470 – 806 MHz) Impedance 75 ohm Dimensions 50 in by 18 in by 19 in (127 cm by 45.72 cm by 48.26 cm)
Table 6.2 Specifications of reference antenna
(a) (b)
Figure 6.4 Gain of reference antenna (a) in azimuth; (b) in elevation
6.3.2 Surveillance Antenna: 1D Array
A set of wide-angle antennas are closely spaced to form a digitally steered array.
This results in flexible beamforming for target tracking and null steering for DSI reduction but requires calibration (see Chapter 7). A linear array made with small bow-tie elements was fabricated to have a fine beam width in a limited angular section. It is pictured in Figure 6.5.
75
Figure 6.5 Configuration of 1D surveillance array
The antenna elements are set in H-pol positions, with equal spacing of 35 cm between them. The length of this horizontal array is 2.3 m. The properties of each antenna element are summarized in Table 6.3, with its gain in azimuth and elevation shown in Figure 6.6.
Due to a lack of technical datasheet for this antenna, measurements were made in the compact range at the ElectroScience Laboratory (ESL) to obtain the gain information.
Antenna model Winegard FreeVision FV-30BB Band VHF, UHF Impedance - Dimensions 20.57 in by 12.98 in by 6.20 in (52.25 cm by 32.97 cm by 15.75 cm)
Table 6.3 Specifications of antenna element in 1D surveillance array
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(a) (b)
Figure 6.6 Gain of antenna element in 1D surveillance array (a) in azimuth; (b) in elevation
The separation of the antenna elements are fixed and uniform at 35 cm, and the range of center frequencies of the transmitters used with this array is from 605 MHz to 665 MHz.
Applying the maximum scan angle (MSA) formula (2.16) , we obtain 휃푚푎푥 = 17°
(corresponding to the 665 MHz transmitter).
6.3.3 Surveillance Antenna: 2D Array
To enable elevation measurements, a 2D array was fabricated. It is made up of a horizontal and vertical linear array in a T-shaped pattern, populated with subarray elements that are made up of 2 dual-bowtie elements each. This configuration is shown in
Figure 6.7.
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Figure 6.7 Configuration of 2D surveillance array
The subarray elements (boxed in black) are set in V-pol positions. In the vertical linear array section, the subarray elements are spaced 60 cm equally apart and in the horizontal linear array section, the subarray elements are spaced 80 cm equally apart. The dimension of the array is 2.4 m vertically and 3.2 m horizontally. The properties of the dual-bowtie elements are summarized in Table 6.4, with its gain in azimuth and elevation shown in Figure 6.8.
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Antenna model Antennas Direct DB2E Band UHF (470 – 698 MHz) Impedance 75 ohm Dimensions 23 in by 16.25 in by 7 in (58.42 cm by 41.28 cm by 17.78 cm)
Table 6.4 Specifications of antenna element in 2D surveillance array
(a) (b)
Figure 6.8 Gain of antenna element in 2D surveillance array (a) in azimuth; (b) in elevation
The 2D array employs a subarray architecture, which is illustrated in Figure 6.9 [19]. The subarray architecture allows for an increased number of antenna elements to increase gain without the costs associated with increasing the number of RF and digitization channels.
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Figure 6.9 Subarray architecture
The nomenclature used in the subarray architecture is built from the bottom up, beginning with a single individual element and ending with the final radiation pattern of the array.
Each summation site in between is denoted as an array level: for example, the first summation site after the element pattern is referred to as “array level 1” and the array factors that are representative of the structure of the array level is denoted as 퐴퐹1. Two array levels are present in the experimental system photographed in Figure 6.7: The subarray element that is boxed in black forms 퐴퐹1, and the horizontal/linear array made up of subarray elements forms 퐴퐹2.
The MSA of the array is primarily limited by a grating lobe of the second array level moving into the main beam of the first [19]. Thus, for the experimental array, MSA in azimuth is limited by the spacing between the horizontal subarrays, and MSA in elevation by the spacing between the vertical subarrays. For the 665 MHz transmitter, the
80 fixed separation of the horizontal elements of 80 cm and of the vertical elements of 60 cm is more than a wavelength, and so grating lobes already appear when the main lobe is at boresight (without steering), i.e. the MSA is zero. This is due to the size of the subarrays, which have already been placed as closely as possible both horizontally and vertically.
6.3.4 Managing Ambiguities from Grating Lobes
If we use the 2D array, or use the 1D array to scan angles beyond the MSA, then ambiguous responses could arise if noise or another target of similar signal strength lies in the vicinity of the grating lobes. For our localization experiments, we remove the ambiguities, if any, by selecting the peak response closest to the detected target’s bearing using information from the air truth, and use that as the bearing estimate.
6.3.5 Mismatch Loss
All the antennas are off-the-shelf, low-cost, commercially available DTV antennas. Note that DTV antennas are designed for cable TV coaxial cable with a characteristic impedance of 75 Ω, in contrast to 50 Ω of most RF equipment. The mismatch loss experienced by this is 0.177 dB, calculated as
2 푑퐵 2 푍2 − 푍1 퐿푚푖푠푚푎푡푐ℎ = −10 log10(1 − 훤 ) = −10 log10 (1 − ( ) ) 푍2 + 푍1 (6.1) 50 − 75 2 = −10 log (1 − ( ) ) = 0.177 dB 10 50 + 75 Where
푑퐵 퐿푚푖푠푚푎푡푐ℎ 푑퐵 Mismatch loss 훤 − Reflection coefficient 푍1 = 75 훺 Source (antenna) impedance 푍2 = 50 훺 Load (transmission line) impedance
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By placing the interface as close to the antenna as possible, it minimizes the ringing effect. If the interface is placed directly at the antenna, any reflections from mismatch are simply re-radiated out of the receiving antennas.
6.4 RF Front End
The RF front end of the radar system has the important task of amplifying and translating the RF waveforms received by the antennas to a lower intermediate frequency scaled such that the ADC can properly sample the output signal. The design of the RF front end was such that the desired signal is preserved throughout this process. This requires image rejection during the mixing process, and proper anti-aliasing filtering prior to digitization with the ADC. Figure 6.10 shows the general block diagram and filtering scheme of the RF down-conversion chain for MUTERA.
Figure 6.10 MUTERA RF front end block diagram
Eight such RF chains connect to the ADCs of the TALON recorder system. A local oscillator (LO) distribution network, which consists of a tunable Agilent RF signal generator, high-powered amplifier and eight-way splitter, feeds the mixers of each channel at the proper drive level.
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Signals with 푓푐 < 푓퐿푂 are called “high-side converted” and result in a flipped spectrum. If uncompensated, this manifests as a mirrored RD surface around the Zero-
Doppler axis. They are corrected in software by inverting the Doppler axis of the final
RD map, or by conjugating the baseband complex data stream. The IF center frequencies are calculated with
(6.2) 푓퐼퐹 = |푓푐 − 푓퐿푂 | Where
푓퐼퐹 퐻푧 IF center frequency 푓푐 퐻푧 RF center frequency 푓퐿푂 퐻푧 LO frequency
6.5 Digital Backend
The ADCs that make up the digital backend require a sampling frequency satisfying the Nyquist sampling criteria for the signal of interest, and a sufficient number of bits that the dynamic range can sample the strong DSI as well as the target echoes.
DTV transmitters possess a range of EIRPs in addition to different bistatic baselines relative to the passive radar receiver. This further exacerbates the already large dynamic range requirement for a single illuminator passive radar. The dynamic range of a particular ADC is set by the effective number of bits (ENOB) as follows [74]
DRdB = 1.76 + 6.02 × ENOB (6.3) Where
DR푑퐵 dB Dynamic range ENOB − Effective number of bits
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MUTERA has a 16-bit ADC with an effective bit count of approximately 12.5. Using
(6.3), the dynamic range of the back-end is 77 dB.
Single channel receiver systems often do not require digital down conversion
(DDC) but uses an I/Q sampling architecture and analog down conversion using a bandwidth matched to the ADC. This provides a complex, zero-centered frequency representation to the receiver for further processing. For a wideband system consisting of multiple signals of interest at various center frequencies, the ADC samples multiple transmitters of opportunity simultaneously. These individual channels must be extracted with DDC individually to generate the complex analytic representation of the signal required for further processing. Figure 6.11 illustrates the operations for extracting the baseband signal 푠퐵퐵[푛] for a particular transmitter from the wideband intermediate frequency (IF) waveform 푠퐼퐹[푚].
Figure 6.11 Digital down conversion block diagram
푠퐼퐹[푚] is first mixed with a complex sinusoid at the center frequency of the signal of interest, 푓퐼퐹. A finite impulse response (FIR) filter with impulse response ℎ퐿푃[푚] is then used to band-limit the signal to
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퐵 퐵 − < 푓 < + (6.4) 2 2 Where
푓 퐻푧 Frequency 퐵 퐻푧 Bandwidth
The signal can then be down sampled by an integer factor 퐹푑푠 to output the complex analytic signal representation at a sampling frequency of approximately 1.25B. A polyphase filter bank implementation was used to significantly reduce the computational requirements of this operation. The samples indices are different for the baseband and IF representations because of the down sampling operation such that
퐼퐹 퐵퐵 (6.5) 푡 = 푚푇푠 = 푛푇푠 Where
푡 푠 Continuous time 푚 − Sample index for IF signal 퐼퐹 푇푠 푠 Discrete sampling period for IF signal 푛 − Sample index for BB signal 퐵퐵 푇푠 푠 Discrete sampling period for BB signal
6.6 Air Truth Data
Information regarding a target’s aircraft type, position, and velocity is extremely valuable when evaluating the performance of the passive radar. The automatic dependent surveillance-broadcast (ADS-B) is a good source of this data, as it is installed on a significant percentage of commercial and private aircraft. It utilizes a GPS receiver on board the aircraft, then reports the plane’s call sign along with latitude and longitude positioning information and flight altitude level. By 2020, it is mandated that ADS-B be in use for the majority of controlled airspaces [75].
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The Kinetic Avionics SBS-3 receiver system, shown in Figure 6.12, records extended squitter ADS-B transmissions in the 1090 MHz band shared with secondary surveillance radar systems.
Figure 6.12 Kinetic Avionics SBS-3 receiver system
This device is fed by a monopole antenna and parses the ADS-B messages ready for transfer to the TALON PC via a USB link. Basestation software, bundled with the SBS-3, aggregates these packets to provide a display similar to air traffic control, as shown in
Figure 6.13. Mode-S call sign, altitude, latitude, longitude, speed, track and vertical rate information are logged at a rate of approximately 2 Hz. This information is time-stamped, and interpolated to match the middle of the coherent processing interval (CPI) during experiments to form the air truth data used for the evaluation of tracking and localization performances.
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Figure 6.13 Basestation display
6.7 Conclusions
MUTERA is equipped with antenna arrays that can measure target azimuth and elevation, supporting array-based localization in 2D and 3D. Simultaneous observations of a single target using different transmitters can also be made, which supports range multilateration. By processing the same multistatic target data, the two localization methods can be directly compared. The availability of air truth data is very useful for validating the accuracy of the location estimates.
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Chapter 7 In-Field Calibration of Passive Array
Bearing estimates can be good if the calibration of the bearing measurement system is extensive enough [76]. As illustrated in Figure 7.1, there are various sources of measurement and modeling errors that contribute to the eventual outputs from the receiving array channels [76] [77].
Figure 7.1 Sources of measurement and modeling errors to the output of an antenna array
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This makes the receiving array channel outputs significantly different from the idealized outputs assumed and used in bearing estimation algorithms. The objective of array calibration is to correct the received signal so that we obtain the idealized form. We refer to the received signal model (5.7), reproduced below.
풔 = 훾 exp[푗휙 (풓 )] 풟 풖 + 풏 푛푒,푖 푛푒,푖 푛푒,푖 푡 푖 풊 푛푒,푖 (7.1)
Where
푁×1 풔푛푒,푖 ∈ ℂ 푉 Sampled complex baseband signal of the 푖th bistatic pair on element 푛푒 of the receiver array 푁×1 풖푖 ∈ ℂ 푉 Sampled complex baseband signal emitted by the 푖th transmitter
훾푛푒,푖 − Complex scaling of 풖풊 along the target path 푗 = √−1 − Imaginary number ( ) 휙푛푒,푖 풓푡 푟푎푑 Differential phase at the 푛푒th array element with respect to a reference element due to propagation from the target 푁×푁 풟푖 = 풟(휏푖, 휈푖) ∈ ℂ − Delay-Doppler operator that accounts for the delay and Doppler shift imparted to the 푖th transmit signal as it propagates to the receiver along the target path 푁×1 Circular Gaussian noise distributed as 풏푛푒,푖 ∈ ℂ 푉 2 2 풩(ퟎ푁, 휎 퐈푁) with known variance 휎 ; ퟎ푁 is a length-푁 zero vector and 퐈푁 is an 푁 × 푁 identity matrix
The complex scaling 훾푛푒,푖 accounts for various effects including antenna gains, spreading losses, non-isotropic target reflectivity, and any unknown phase offsets between signals measured at different receivers due to the use of receivers that are not calibrated to a
common phase reference. Ideally, with 훾푛푒,푖 removed and ignoring the noise term, this gives us the idealized sampled complex baseband signal
풓 = exp[푗휙 (풓 )] 풟 풖 푛푒,푖 푛푒,푖 푡 푖 풊 (7.2)
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To explain how (7.1) could be obtained from (7.2) allowing to factors such as mutual coupling, attenuation and phase changes introduced by cables, we use a model where the augmented matrix C’ is the product of C and T [78]
퐂′ = 퐂퐓 (7.3) Where
퐂′ − Augmented antenna coupling and cable transmission coefficients matrix 퐂 − Antenna coupling matrix − Diagonal matrix of cable transmission 퐓 = diag([푇1, … , 푇푁푒 ]) coefficients 푇푛푒
Let
퐒 = 퐂퐓퐑 = 퐂′퐑 (7.4) Where
풔1,푖 푐11 ⋯ 푐1푁푒 풓1,푖 [ ⋮ ] = [ ⋮ ⋱ ⋮ ] [ ⋮ ] 풔 푐 ⋯ 푐 풓 푁푒,푖 푁푒1 푁푒푁푒 푁푒,푖 (7.5) 푐 ⋯ 푐 11 1푁푒 exp[푗휙1,푖(풓푡)] = [ ⋮ ⋱ ⋮ ] [ ⋮ ] 풟푖풖풊 푐푁 1 ⋯ 푐푁 푁 ( ) 푒 푒 푒 exp[푗휙푁푒,푖 풓푡 ]
Looking at the 푛푒th entry of S, we have
( ) ( ) 풔푛푒,푖 = {푐푛푒1 exp[푗휙1,푖 풓푡 ] + ⋯ + 푐푛푒푁푒 exp[푗휙푁푒,푖 풓푡 ]}풟푖풖풊 ( ) (7.6) = 훾푛푒,푖 exp[푗휙푛푒,푖 풓푡 ] 풟푖풖풊
Which is equivalent to (7.1) without the noise term.
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7.1 Calibration Using a Fixed Transmitter
A method was developed for calibration based upon the strong direct path signal from a transmitter. The array is mechanically steered to face the transmitter, so that it can be assumed that the direct, line-of-sight path signal is the dominant received signal. The calibration algorithm then goes like this:
1. The returns from all array elements channels are scaled by the norm of the
return from channel 1, which serves as the reference channel. Then
conjugate-multiply the scaled signal from channel 푛푒 with that from
channel 1.
2. The correction factor is calculated for each array element (other than
element 1, which has a correction factor of 1) using the known bearing of
the transmitter.
3. The output for each array element (other than element 1) is scaled by the
output from channel 1, and divided by the correction factor.
4. The output from element 1 is scaled by the output from itself
The objective of array calibration is to correct the received signal (7.1) so that we obtain the idealized form (7.2) . This is because most algorithms such as beamformers or bearing estimators assume the idealized form (7.2) as their inputs. Here, we show how the calibration algorithm described above achieves this.
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The signal on element ne is given by
풔 = 훾 exp[푗휙 (풓 )] 풟 풖 + 풏 푛푒,푖 푛푒,푖 푛푒,푖 푡 푖 풊 푛푒,푖 (7.7)
For ease of analysis, we represent complex numbers in their exponential forms, i.e.
푧 = 퐴푒푗훼 (7.8) Where
푧 ∈ ℂ1×1 − Complex number 퐴 푉 Real amplitude 푗 = √−1 − 훼 푟푎푑 Real phase
( ) We represent the known amplitude and phase (from exp[푗휙푛푒,푖 풓푡 ] 풟푖) using A and 훼;
and unknown amplitude and phase (from 훾푛푒,푖) using B and 훽. We assume that only one transmitter is used in the system, so we omit the subscript 푖 for this derivation. The signal from element 푛푒, ignoring the noise term, is then
푗훼 푗훽 풔 = 퐴 푒 푛푒 퐵 푒 푛푒 풖 푛푒 푛푒 푛푒 (7.9)
And the signal from reference element 1 is
푗훼1 푗훽1 풔1 = 퐴1푒 퐵1푒 풖 (7.10)
The norm of the signal from reference element 1 is
퐻 ‖풔1‖ = √〈풔1, 풔1〉 = √풔1 풔1 (7.11) −푗훼 −푗훽 푗훼 푗훽 = √(퐴1푒 1 퐵1푒 1 풖) ∙ (퐴1푒 1퐵1푒 1 풖) = 퐴1퐵1√푁
Where N is the number of samples in 풖.
We scale the signals from element 푛푒 and reference element 1 by this norm, and then conjugate-multiply them
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퐻 풔 풔1 푛푒 1 퐻 = 2 풔1 풔푛푒 ‖풔1‖ ‖풔1‖ ‖풔1‖ 1 −푗훼1 −푗훽1 퐻 푗훼푛푒 푗훽푛푒 = 2 2 [(퐴1푒 퐵1푒 풖 )(퐴푛푒푒 퐵푛푒푒 풖)] 퐴1퐵1 푁 1 (7.12) 푗(훼푛푒−훼1) 푗(훽푛푒−훽1) = 2 2 [퐴1퐴푛푒퐵1퐵푛푒푒 푒 푁] 퐴1퐵1 푁 퐴 퐵 푛푒 푛푒 푗(훼 −훼 ) 푗(훽 −훽 ) = 푒 푛푒 1 푒 푛푒 1 퐴1퐵1
The correction factor 휖 is calculated using the product from (7.12) along with the known
−푗훼 푛푒 ( ) bearing of the transmitter (in the form of 푒 = exp[−푗휙푛푒,푖 풓푇 ], where 풓푇 is the position of the transmitter)
퐴 퐵 퐴 퐵 푛푒 푛푒 푗(훼 −훼 ) 푗(훽 −훽 ) −푗훼 푛푒 푛푒 −푗훼 푗(훽 −훽 ) 휖 = ( 푒 푛푒 1 푒 푛푒 1 ) 푒 푛푒 = 푒 1 푒 푛푒 1 (7.13) 퐴1퐵1 퐴1퐵1
We correct the output from each array element (other than element 1) by scaling it by the output from channel 1, and divide it by the correction factor
푗훼 푗훽 풔 퐴 푒 푛푒 퐵 푒 푛푒 풖 푛푒 1 푛푒 푛푒 퐴1퐵1 푗훼 ( ) = ( ) = 푒 푛푒 풖 푗훼1 푗훽1 푗(훽 −훽 ) (7.14) 풔1 휖 퐴1푒 퐵1푒 풖 −푗훼1 푛푒 1 퐴푛푒퐵푛푒푒 푒
The output from reference element 1 is scaled by itself
풔1 = 풖 (7.15) 풔1
Now, the relative phase and magnitude between all the antenna elements is as desired
(without the unknown phase and amplitudes) as in (7.2).
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7.1.1 Calibration Results
The calibration algorithm was implemented using actual data collected by pointing the boresight of a 6-element, 1D linear array (for details on the array hardware, refer to Chapter 6) towards a transmitter as in Figure 7.2a. The properties of the transmitter used are summarized in Table 7.1.
(a) (b)
Figure 7.2 (a) Calibration geometry; (b) Results of calibration: Top is Bartlett method spectrum output before calibration and bottom is the spectrum after calibration
Call Sign WSFJ Center Frequency 533 MHz Effective Isotropic Radiated Power (EIRP) 1000 kW Baseline 31.1 km Bearing relative to boresight 0° Table 7.1 Transmitter properties for calibration using fixed transmitter experiment
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The results in Figure 7.2b show the successful implementation of the calibration algorithm, where the output is a sinc2(휃) curve with sidelobes at -13 dB from the peak.
Before implementing the calibration routine, multiple peaks with angular offsets far from the true bearing of the transmitter appear. After normalizing the magnitude and phase, the proper 0° bearing estimate of the direct signal is achieved.
However, there are three practical limitations with this method of calibration in passive radar. Firstly, while the array boresight is rotated towards the transmitter for calibration, this puts the passive radar in the forward scatter operation mode, which is not optimal for detection and localization due to increased DSI and poor resolutions in range and Doppler. Thus, the array boresight needs to subsequently be rotated away to point to another direction of general surveillance to put the passive radar in either bistatic or pseudo-monostatic operational modes. This can give rise to an error in the array pointing due to manual rotation and also cable movement, which can undo some of the effects of correct calibration. For example, during the calibration experiment, the array boresight was pointed at 3 different directions: directly at the transmitter, 20° to the left, and 20° to the right of the transmitter. Using the correction factors from one case on the other two cases gave a fixed bias of 2.6 – 3.7°. Secondly, since calibration is frequency dependent, the array boresight will need to be rotated to point at the various transmitters utilized in the localization if this method was used, which would again introduce the error due to
manual rotation. Finally, this method assumes that the complex scaling 훾푛푒,푖 is fixed for each element 푛푒. However, it has been noted in literature that array element pattern changes due to mutual coupling are dependent on the bearing of the impinging signal [79]
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[80]. Therefore, it is likely that the complex scaling and the calculated correction factors have a bearing-dependence as well. Therefore, a more rigorous calibration method needs to be (and was) developed to be used in subsequent work. Before this, we look to literature for suitable methods.
7.2 Passive Array Calibration in Literature
In [78], the augmented matrix C’ in (7.3) is estimated by using multiple measurements of the signal originating from fixed transmitters located in different positions. The elements of C’ are then calculated to fit to measurements from all sources at the same time. This was done for a circular array made up of 8 elements, although it was not stated how many fixed transmitter positions were used for the calibration. The authors in [78] chose this method of determining C’ over the two methods of measuring the coupling coefficients described in [81]. One of them is based on Fourier decomposition of the measured array element pattern. The second involves measuring coupling between array ports. In the first method, accuracy is the main concern when spacing between array elements is more than half a wavelength, which is the case in their system (and ours). The second method requires using antennas in transmit and receive modes and measuring the parameters of the cables. This is complicated by difficulties connected with receivers and antennas mismatch.
7.3 Calibration Using Detected Targets
The method in [78], which involves measurement of the signal originating from fixed transmitters located in different positions is unsuitable for our system since we utilize a linear array instead of a circular array. To make measurements from different
96 transmitters requires rotation of the array boresight, which will introduce the mechanical rotation error as described earlier. The limitations of the methods in [81] apply to our system as well.
Rather than rely on the use of a formal calibration target or a known transmitter, we propose a new method that uses an opportunistic high signal-to-noise ratio (SNR) target detected by the passive radar is used. Figure 7.3 illustrates this concept.
Figure 7.3 Calibration using detected target
It is assumed that the location of this target is known a priori, perhaps from its automatic dependent surveillance-broadcast (ADS-B) transponder signal, or, as is used for this demonstration, from GPS information, which facilitates the calibration. As the detected target travels across its flight path, its data is processed by the passive radar at regular intervals. By ensuring that the passive radar’s clock is synchronized to its air truth, we get
97 an accurate value of its known bearing. This is used to calculate the correction factors for each bearing as in (7.13). Thus, R sets of calibration vectors corresponding to R known bearings of the detected target are obtained.
The experimental set-up is shown in Figure 7.4.
Figure 7.4 Calibration and validation flight paths
We have a cooperative aircraft – a Piper Saratoga, flying two paths: the blue path, which gives us the calibration data, and the red path, which gives us the test data that allows us to perform validation of the calibration procedure. Ground truth data is supplied in the form of an on-board GPS receiver and inertial measurement unit (IMU). The same 6- element, 1D linear array is used, and it is not moved between the calibration and
98 validation test collections so as to not introduce a bias in the angle estimate caused by misalignment of the array boresight. Properties of the transmitter located in the South of
Figure 7.4 that was used are summarized in Table 7.2.
Call Sign WWHO Center Frequency 665 MHz Effective Isotropic Radiated Power (EIRP) 1000 kW Baseline 45.9 km Table 7.2 Transmitter properties for calibration using detected target experiment
7.3.1 Calibration Results
The aircraft takes 2 minutes to traverse its (blue, calibration) flight path to generate 2 minutes’ worth of calibration data. 0.1 s of data was processed at 1.2 s intervals, thereby generating 100 processed points. Of these 100 processed points, only the points with a reliable target detection (above 13 dB on each array channel output) was used in the calculation of the correction factors. This is so that the assumption that the target path signal is the dominant received signal is valid. 78 sets of correction factors corresponding to 78 known angles of the calibration target were thus obtained. Figure 7.5 plots the magnitudes (in red) and phases (in blue) of these sets of correction factors. Since the first element of the array was used as a reference, its results are not shown and the five rows in Figure 7.5 correspond to the relative phases and magnitudes of the correction factors for elements two to six. The data points plotted in Figure 7.5 correspond to the calculated phases (blue dots) and magnitudes (red dots). Overlaid on the dots are lines of fit to the data. The order of the polynomial fit is two. Since it is not a linear fit, it
99 corroborates our hypothesis that the calculated correction factors have a bearing- dependence.
Figure 7.5 Phases and magnitudes of calculated correction factors from calibration data
7.3.2 Validation Results
The calculated correction factors are now applied to the test data for validation.
These can be applied in a number of ways, which are listed below:
• Clairvoyant method: We assume that the bearing of the test target as it moves
with time (along the red, test path) is known. The correction factors corresponding
to the bearing is calculated from the data using interpolation.
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• Mean method: One set of correction factors is obtained by taking the mean of the
R sets of calculated correction factors. This single set of correction factor is used
for calibration.
• Iterative method: The single set of correction factors from the Mean method is
first used. After an initial estimate of the test target’s bearing is obtained, another
set of correction factors that correspond to this DOA estimate is interpolated and
used. This is iterated for n times.
• Estimated C’ method: The elements of the augmented matrix C’ in (7.3) are
calculated by using the R measurements from the detected target. The R
measurements are represented as
′ 퐒1 = 퐂 퐑1 ⋮ (7.16) ′ 퐒푅 = 퐂 퐑R Where
풔1,푖 푉 Matrix of received array outputs 퐒푟 = [ ⋮ ] corresponding to calibration target 풔 푁푒,푖 푟 at position r 푐 ⋯ 푐 11 1푁푒 − Augmented antenna coupling and 퐂′ = [ ⋮ ⋱ ⋮ ] cable transmission coefficients
푐푁푒1 ⋯ 푐푁푒푁푒 matrix 풓1,푖 푉 Matrix of idealized array outputs 퐑R = [ ⋮ ] corresponding to calibration target 풓 푁푒,푖 푟 at position r
C’ needs to be estimated and this is done by minimizing the cost function
푹 ′ ′ ퟐ 퐉(퐂 ) = ∑(퐒푟 − 퐂 퐑r) (7.17) 풓=ퟏ
Multiplying S with the inverse of the estimated C’ gives the corrected array
outputs. 101
The results of the bearing estimates after the test data has been calibrated using the various methods are summarized in Table 7.3. The measurement/estimate error 흃 is defined as the difference between the measurement/estimate of a quantity and its true value. 흃 is a random variable vector and if it is made up of 푁푠 snapshots or observations then its standard deviation (SD) is defined as [82]
푁푠 1 2 휎흃 = √ ∑|흃[푖] − 휇흃| (7.18) 푁푠 − 1 푖=1
The mean of the error, or the bias, is defined as
푁푠 1 (7.19) 휇흃 = ∑ 흃[푖] 푁푠 푖=1
Method Bias [°] Error SD [°]
No Calibration 13.75 0.73 Clairvoyant 0.00 0.66 Mean -0.07 0.65 Iterative (n = 3) -0.06 0.60 Estimated C’ 0.08 0.64 Table 7.3 Bearing estimation results from test/validation data
In the Clairvoyant case, the calibration is effective (bias in the bearing estimates is zero) since the correct calibration vector corresponding to the true bearing of the target is used.
However, when tracking a non-cooperative target, the bearing of the target will be unknown and this method cannot be applied. Using the Mean method is thus the next reasonable step since it is simple to calculate and although the phase and magnitude of the correction vectors are known to change as a function of angle, the change is small in 102 the range of 0 – 30 degrees’ azimuth [79]. Thus, calibration using the Mean method works well, bringing the bias close to zero. The non-zero SD can be attributed to target- cased errors and other external sources of error [83] when the beamforming method is used as the bearing estimator. The Iterative method has similar performance but it requires the number of iterations times of the computation involved in the Mean method.
Also, it is not guaranteed to converge – the bias actually increases with larger values of iterations n. Finally, the estimated C’ method also gives a good calibration performance.
7.4 Conclusions
In-field calibration using detected targets was selected to be used with our passive radar system since it overcomes the limitations of other calibration methods. By simply using the mean of R sets of calculated correction factors, we are able to achieve good calibration performance and so this method was selected for implementation. The selected calibration method was used on both the 1D and 2D arrays for azimuth and elevation measurements, in experiments that utilized GPS and ADS-B information respectively. The localization results of these experiments are shown in Chapter 9.
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Chapter 8 Analysis of Passive Localization Accuracies
The target parameters that a passive receiver can measure are the bistatic range, the bistatic velocity, and the bearing. These measurements include errors which depend upon the signal-to-power ratio (SNR). Different localization algorithms make use of different sets of measurements, combining them in various ways to estimate the target location. In this chapter, the noise-limited errors associated with measuring the bistatic range, DOA, and bistatic velocity are evaluated. This is then followed up by the error associated with the location estimate derived from these measurements.
8.1 Passive Localization Accuracies in Bistatic Radar
In the bistatic configuration, both geometry factors and transmitted waveform play an important role in the estimation accuracy [84]. In our analysis, we look at the standard deviation (SD) of the various measurement/estimation errors from the radar receiver as a function of the target azimuth 퐴푡 at fixed receiver ranges because this is reminiscent of plan position indicator (PPI) displays widely used in air traffic control and typically what people think of as a radar display [13]. As before, the SD of the measurement/estimation error 흃 is defined as (7.18). We use a constant SNR of 21 dB across the range of azimuth angles because we are interested in finding out how well we
104 can estimate target location upon successful detection. An SNR of 17.1 to 21 dB is
−6 needed to achieve reliable detection (90% 푃퐷) with a reasonable 푃퐹퐴 (10 ) for a target signal that fluctuates (Swerling models 1 to 4), which is the case for most real targets
[85].
퐴푡 coincides with the receiver look angle 휃푅 defined in the bistatic plane in the specific case where target altitude is zero and the transmitter is located along the positive y-axis also at zero altitude, with the receiver at the origin. In this case, the bistatic plane coincides with the x-y plane, which is depicted in Figure 8.1.
Figure 8.1 2D bistatic planes within 3D coordinate system: One with the bistatic plane coincident with the x-y plane and the other general case, which is not
The receiver look angle 휃푅 is the angle between North of the bistatic plane and the line connecting receiver to target, measured positive clockwise from the North and restricted to the interval [−90°, 270°] [10]. The region where 휃푅 ranges from −90° to 90° is 105 termed the “Northern hemisphere” of the bistatic plane and where 휃푅 ranges from 90° to
270°, the “Southern hemisphere” [1]. Since many of the effects of bistatic geometry can be observed as a function of 휃푅, a set of results from this specific case is presented.
However, all the equations presented can be applied to the general case where the bistatic plane does not coincide with the x-y plane (as is the case for the experiments performed; see Chapter 9). The same conclusions on the effects of bistatic geometry still hold in the general case.
Keeping the receiver range 푅푅 (distance of the target from the receiver) fixed and varying 퐴푡 gives a circular locus of target positions around the receiver (see Figure 8.1 –
Figure 8.3). We analyze the results for two different 푅푅 – one at 5.7 km, which is shorter than the 46 km baseline, thus giving us a general bistatic configuration; and the other at
460 km, which is one order of magnitude (101 times) larger than the baseline, thereby giving us an approximately monostatic configuration [86]. These configurations are shown in Figure 8.2 and Figure 8.3. The linear array boresight is indicated with a red dotted line and fixed at −60° in azimuth and 20° in elevation. Additional details of the configurations settings are tabulated in Table 8.1. We also make a comparison to the actual monstatic case as a benchmark (where applicable), and this is done through assuming that the monostatic radar site is coincident with the receiver site of the corresponding bistatic radar.
106
(a) (b)
Figure 8.2 Bistatic radar configuration (푅푅 = 5.7 km) for analysis in the case where the bistatic plane is coincident with the x-y plane: (a) Top-down view and (b) Side view
(a) (b)
Figure 8.3 Approximately monostatic radar configuration (푅푅 = 460 km) for analysis in the case where the bistatic plane is coincident with the x-y plane: (a) Top-down view and (b) Side view
107
Bistatic Configuration Approximately Monostatic Configuration (푅푅 = 460 km) (푅푅 = 5.7 km)
푥푡 푉푎푟푖푒푠, 푠푒푒 푙표푐푖 Coordinates, [푦푡] = [푉푎푟푖푒푠, 푠푒푒 푙표푐푖] 푧푡 0
Range from receiver, 푅푅 = 푅푅 = 460 푘푚 5.7 푘푚 Target settings Azimuth, 퐴푡 varies from −90° to 90°
Elevation, 퐸푡 = 0°
푥̇푡 50 푚/푠 Velocity, [푦̇푡] = [ 0 ] 푧푡̇ 0 푥푅 0 Coordinates, [푦푅] = [0] 푧푅 0
Receiver Array boresight azimuth, 퐴푏표푟푒푠푖𝑔ℎ푡 = −60° settings Horizontal configuration: 6 elements uniformly spaced by 80 cm
Array boresight elevation, 퐸푏표푟푒푠푖𝑔ℎ푡 = 20° Vertical configuration: 6 elements uniformly spaced by 60 cm
푥푇 0 Coordinates, [푦푇] = [46 푘푚] 푧푇 0
Baseline range, 퐿푇 = 46 푘푚 Transmitter settings Azimuth, 퐴푇 = −90°
Elevation, 퐸푇 = 0°
Center frequency, 푓푇 = 665 푀퐻푧 Bandwidth, 퐵 = 6 푀퐻푧 Table 8.1 Settings for bistatic and approximately monostatic configurations used in analysis of errors in the specific case where the bistatic plane is coincident with the x-y plane
108
8.1.1 Bistatic Range Sum Measurement
The propagation delay along the target path (from the 푖th transmitter to the target to the receiver) is related to the bistatic range sum 푅퐵푖 by
푅 푅 + 푅 휏 = 퐵푖 = 푇푖 푅 (8.1) 푖 푐 푐 Where
휏푖 푠 Propagation delay along target path corresponding to transmitter 푖 푅퐵푖 = 푅푇푖 + 푅푅 푚 Bistatic range corresponding to transmitter 푖 푅푇푖 푚 Range between 푖th transmitter and target 푅푅 푚 Range between receiver and target 푐 = 3 × 108 푚/푠 Speed of propagation
Estimation of the bistatic range sum is therefore equivalent to estimation of the delay. A simple model for the received continuous waveform is
(8.2) 푠푠푖(푡) = 푢푖(푡 − 휏푖) + 푤푖(푡) Where
푠푠푖(푡) 푉 Received surveillance waveform corresponding to transmitter 푖 푢푖(푡) 푉 Signal emitted by transmitter 푖 휏푖 푠 Propagation delay along target path corresponding to transmitter 푖 푤푖(푡) 푉 Noise waveform
109
The received continuous waveform is sampled at the Nyquist rate to form the observed data
(8.3) 푠푠푖(푛푇푠) = 푢푖(푛푇푠 − 휏푖) + 푤푖(푛푇푠) Where
푠푠푖(푡) 푉 Received surveillance waveform corresponding to transmitter 푖 푛 = 0, 1, … , 푁푠 − 1 − Index corresponding to discrete time 푁푠 − Number of observed samples 푇푠 = 1/2퐵 푠 Sampling interval/period 퐵 퐻푧 Bandwidth of transmitter’s signal 푢푖(푡) 푉 Signal emitted by transmitter 푖 Propagation delay along target path corresponding to 휏 푠 푖 transmitter 푖 푤푖(푡) 푉 Noise waveform
With the discrete data model
(8.4) 푠푠푖[푛] = 푢푖(푛푇푠 − 휏푖) + 푤푖[푛] Where
푠푠푖[푛] 푉 Sampled received surveillance waveform corresponding to transmitter 푖 푤푖[푛] 푉 Sampled noise waveform
The general Cramer-Rao lower bound (CRLB) for signals in white Gaussian noise is [87]
휎2 푉푎푟(훩) ≥ 2 (8.5) 푁 −1 휕푠[푛; 훩] ∑ 푠 ( ) 푛=0 휕훩 Where
푉푎푟(∙) − Variance function 훩 Unknown parameter 휎2 푉 Variance of Gaussian distributed noise 푤 푁푠 − Number of observed samples 푠[푛] 푉 Sampled signal
110
The CRLB for time delay is evaluated by applying (8.4) to (8.5), eventually yielding [87]
1 푉푎푟(휏 ) ≥ (8.6) 푖 SNR × 퐵2 Where
SNR − Signal-to-noise power ratio 퐵 퐻푧 Bandwidth of transmitter’s signal
We substitute (8.1) into (8.6) and make use of the following basic property
2 푉푎푟(푎푋) = 푎 푉푎푟(푋) (8.7)
Where 푎 is a constant and 푋 is a random variable.
Finally, taking square root of the variance gives the standard deviation (SD). We obtain the SD of the bistatic range sum error corresponding to transmitter 푖 as
푐 휎 푖 ≥ 푅퐵 퐵√SNR (8.8)
From the expression in (8.8), aside from the effect on SNR, the SD in bistatic range sum error is independent of the bistatic geometry and is a constant value as a function of receiver look angle. It is a function of the signal bandwidth 퐵. Using DTV’s bandwidth
of 6 MHz, at 21 dB SNR, 휎푅퐵 = 2.62 m for the bistatic, approximately monostatic, and monostatic cases.
111
8.1.2 Bearing Estimation
We assume that a narrowband signal from a target impinges on a linear array with planar wavefronts because it is located far away enough. This is shown in Figure 8.4.
Figure 8.4 Uniform linear array geometry
Due to the extra propagation distance, the wavefront at element (푛푒+1) lags that at element 푛푒 by
푑 cos(휃)
푐 (8.9)
Where 푑 is the uniform spacing between consecutive antenna elements and 휃 is the bearing of the impinging signal on receiver array.
Thus, the propagation time to element 푛푒 is
푑 푡 = 푡 + 푛 cos(휃) 푛푒 1 푒 푐 (8.10)
Where 푡푛푒 and 푡1 are the propagation times to element 푛푒 and reference element 1 respectively. 112
Assuming that the transmitted signal is sinusoidal with the form
(8.11) 푢(푡) = 퐴푇푖 cos(2휋푓푇푖푡 + 휙푇푖) Where
퐴푇푖 푉 Amplitude of 푖th transmitter’s signal 푓푇푖 퐻푧 Center frequency of 푖th transmitter’s signal 휙푇푖 푟푎푑 Phase of 푖th transmitter’s signal
Then the observed signal at element 푛푒 at a single snapshot in time 푡 = 푡푠 is
( ) 푠푛푒,푖 푡 = 퐴푇푖 cos[2휋푓푇푖(푡 − 푡푛푒) + 휙푇푖] 푑 = 퐴 cos [2휋푓 (푡 − 푡 − 푛 cos(휃)) + 휙 ] 푇푖 푇푖 1 푒 푐 푇푖 (8.12) 푑 = 퐴 cos [2휋 (−푓 cos(휃)) 푛 + 휙′ ] 푇푖 푇푖 푐 푒 푇푖
′ Where 휙푇푖 = 휙푇푖 + 2휋푓푇푖(푡 − 푡1).
By inspecting (8.12), it can be seen that the spatial observations are sinusoidal with frequency
푑 푓 = 푓 cos(휃) 푒푖 푇푖 푐 (8.13)
Where 푓푇푖 is the center frequency of 푖th transmitter’s signal.
(8.13) shows the relationship between the bearing 휃 and the element spacing frequency
푓푒. If the CRLB for the frequency 푓푒 is known, the transformation of parameters formula can be used to find the CRLB for the bearing 휃. The CRLB for the frequency of a sinusoid embedded in white Gaussian noise (WGN) has been derived in [87] and its spatial analog is
12 ( ) 푉푎푟 푓푒푖 ≥ 2 2 (8.14) (2휋) (SNR)푁푒(푁푒 − 1)
113
Thus the CRLB for bearing error is given by
2 휕푔(푓푒푖) 푉푎푟(휃) ≥ [ ] 푣푎푟(푓푒푖) 휕푓푒푖 푐2 12 (8.15) = 2 2 2 × 2 2 (푓푇푖) 푑 sin (휃) (2휋) (SNR)푁푒(푁푒 − 1)
And its SD is obtained by taking the square root of (8.15). (8.15) gives the bearing estimate error assuming that the ambiguity from grating lobes, if any, have already been resolved. The bearing and (8.15) are defined in the frame of the antenna array as shown in Figure 8.4. When calculating the SD of the azimuth angle 퐴푡 measurement error for a horizontal array, the (known) impinging angle in 퐴푡 is converted to the equivalent DOA in the frame of reference of the antenna array 휃ℎ표푟, and then (8.15) is used. This involves a straight-forward translation of angle since the horizontal element array is in the azimuth plane. For example, if the target azimuth 퐴푡 is −18° for the analysis configuration where the array boresight azimuth 퐴푏표푟푒푠푖𝑔ℎ푡 is −60°, its DOA in the frame of the horizontal antenna array 휃ℎ표푟 is 132°. This is illustrated in Figure 8.5 below.
Figure 8.5 Example showing DOA 휃ℎ표푟 of target with azimuth 퐴푡 in frame of reference of horizontal antenna array 114
Similarly, when calcuating the SD in elevation angle 퐸푡 measurement error for a vertical array, the angle in elevation 퐸푡 is converted to the equivalent DOA in the frame of reference of the vertical antenna array 휃푣푒푟푡, and then (8.15) is used.
The SD in the elevation measurement error is constant across different target azimuths since it is independent of the azimuth measurement. In the specific case (where the bistatic plane is coincident with the x-y plane), it has the same value for all three configurations because the target altitude is zero. In the general case, we would expect the values to be different because at the same target altitude, the elevation angle becomes smaller at further ranges.
The SD in the azimuth measurement error as a function of target azimuth is plotted using a blue curve for a target at 푅푅 of 5.7 km (for a bistatic configuration) and a red dotted curve for a target at 푅푅 of 460 km (for an approximately monostatic configuration) in Figure 8.6. A logarithmic scale on the y-axis was selected to allow a large range to be displayed without small values being compressed down into the bottom of the graph. (8.15) is not a function of 푅푅 and hence both bistatic and approximately monostatic configurations give the same results and their curves overlap.
115
Figure 8.6 SD of 퐴푡 error as a function of 퐴푡
In Figure 8.6, vertical lines that indicate the (angular) location of the array boresight and ends have been added . They represent bearings at 90° (along the boresight axis) and 0° and 180° (left and right ends of element array) respectively. They show that bearing estimation is most accurate at array boresight (marked with a red vertical dashed line) and impossible at the ends of the linear array (marked with a black vertical dashed line).
Therefore, it is useful to tilt the array in such a way so as to place the array boresight as close to the expected location of targets as possible. The monostatic configuration would give the same results as the bistatic configuration since the array is located at the same receiver site in both bistatic and monstatic configurations.
116
8.1.3 Bistatic Velocity Measurement
The bistatic Fisher information matrix (FIM) can be used to find the accuracy of bistatic velocity measurements since the inverse of the FIM gives the CRLB [87]. The
FIM is obtained using the received data log-likelihood function (LLF) and [84] observes that the AF is the LLF excluding the effect of signal attenuation and noise. The bistatic
FIM is given as
휕2훩(푅 , 푉 ) 휕2훩(푅 , 푉 ) 푅 퐵푖 푅 퐵푖 2 휕푅 휕푅푅휕푉퐵푖 푱 (푅 , 푉 ) = −2 ∙ SNR ∙ 푅 (8.16) 퐵 푅 퐵푖 휕2훩(푅 , 푉 ) 휕2훩(푅 , 푉 ) 푅 퐵푖 푅 퐵푖 2 [ 휕푉퐵푖휕푅푅 휕푉퐵푖 ] Where
푱퐵 − Bistatic Fisher Information Matrix (FIM) 푅푅 푚 Range between receiver and target 푉퐵푖 푚/푠 Bistatic velocity of target corresponding to the 푖th bistatic pair SNR − Signal-to-noise power ratio 훩(∙) − Parameter function
The CRLB for bistatic velocity then follows
−1 (8.17) 퐶푅퐿퐵(푉퐵푖) = [푱퐵 (푅푅, 푉퐵푖)]2,2 Where
퐶푅퐿퐵(∙) − Cramer Rao Lower Bound function 푱퐵 − Bistatic Fisher Information Matrix (FIM) 푨−1 − Inverse of matrix 푨 [푨]푚,푛 − Entry at row 푚 and column 푛 of matrix 푨
117
The calculation of the CRLB in the bistatic domain makes use of the following result in the monostatic configuration [84]
휕2훩(휏, 휈) 휕2훩(휏, 휈)
휕휏2 휕휏휕휈 푱푀(휏, 휈) = −2 SNR ∙ 2 2 = −2 SNR ∙ 푱0 (8.18) 휕 훩(휏, 휈) 휕 훩(휏, 휈) [ 휕휈휕휏 휕휈2 ]
푘2휋2푇2 푘휋2푇2 – 3 3 (8.19) 푱0 = 푘휋2푇2 휋2푇2 휋2푇2(1 − 푁2) − + 푅 [ 3 3 3 ] Where
푱푀 − Monostatic Fisher Information Matrix (FIM) 휏 푠 Time delay 휈 퐻푧 Doppler shift 푘 = 퐵/푇 − 푇 푠 Duration of each pulse 푇푅 푠 Pulse repetition interval (PRI) 퐵 퐻푧 Bandwidth of signal 푁 − Number of sub-pulses
The waveform used in (8.18) was a burst of linear frequency modulated (LFM) pulses, with the signal embedded in white Gaussian noise. This expression can also be used for the analysis of DTV signals, since they can be modeled as a sequence of pulses, with the following parameters:
푁 = 2 푇푅 = 푇 = 푇퐶푃퐼
It is shown in [88] that when the SNR is high, the CRLB is dependent on both the SNR and the second derivatives of the AF (sharpness of the AF main lobe). Thus, the second partial derivatives in the bistatic FIM (8.16) are found using the derivative chain rule:
118
2 2 2 휕 훩(푅푅, 푉퐵푖) 휕휏푖 휕휏푖 휕휈푖 휕휈푖 2 = [푱0]1,1 ( ) + 2[퐉0]1,2 + [퐉0]2,2 ( ) (8.20) 휕푅푅 휕푅푅 휕푅푅 휕푅푅 휕푅푅
2 2 휕 훩(푅푅, 푉퐵푖) 휕휈푖 2 = [퐉0]2,2 ( ) (8.21) 휕푉퐵푖 휕푉퐵푖
2 2 휕 훩(푅푅, 푉퐵푖) 휕 훩(푅푅, 푉퐵푖) 휕휏푖 휕휈푖 휕휈푖 휕휈푖 = = [퐉0]1,2 + [퐉0]2,2 (8.22) 휕푉퐵푖휕푅푅 휕푅푅휕푉퐵푖 휕푅푅 휕푉퐵푖 휕푅푅 휕푉퐵푖
Where
휏푖 푠 Time delay corresponding to the 푖th bistatic pair 휈푖 퐻푧 Doppler shift corresponding to the 푖th bistatic pair
The first partial derivatives are
휕휏 1 푅 + 퐿 sin(휃 ) 푖 = (1 + 푅 푇푖 푅푖 ) 휕푅 푐 2 2 (8.23) 푅 √푅푅 + (퐿푇푖) + 2푅푅퐿푇푖 sin(휃푅푖)
휕휈푖 푓푇푖 = 푉퐵푖 × 휕푅푅 2푐 (퐿 )2 cos2(휃 ) 푇푖 푅푖 1 푅 + 퐿 sin(휃 ) (8.24) [ 2 2 ( )]3/2 푅 푇푖 푅푖 푅푅 + 퐿푇푖 + 2푅푅퐿푇푖 sin 휃푅푖 2 + √ 2 2 2√푅푅 + 퐿푇푖 + 2푅푅퐿푇푖 sin(휃푅푖)
휕휈푖 2푓푇푖 1 푅푅 + 퐿푇푖 sin(휃푅푖) = + (8.25) 휕푉퐵푖 푐 √2 2 2 2√푅푅 + 퐿푇푖 + 2푅푅퐿푇푖 sin(휃푅푖) Where
퐿푇푖 푚 Baseline of 푖th bistatic pair 휃푅푖 ° Receiver look angle of 푖th bistatic pair 푓푇푖 퐻푧 Center frequency of 푖th transmitter’s signal Bistatic velocity of target corresponding to the 푖th bistatic 푉 푚/푠 퐵푖 pair
119
The SD in the bistatic velocity 푉퐵 error is strongly dependent on the bistatic geometry, as denoted by the blue curve in Figure 8.7. It is a minimum at 휃푅 = 90° (pseudo-monostatic region) and approaches the vertical asymptote at 휃푅 = −90° (forward scatter region). The graph is zoomed in to −90° ≤ 휃푅 < 90° (the Northern hemisphere) because bistatic effects are symmetrical about the baseline [1].
Figure 8.7 SD of 푉퐵 error as a function of 휃푅
Note that for this analysis, the velocity of the target was fixed at 50 m/s in the x-direction and 0 m/s in the y- and z-direction (see Table 8.1). The SD of 푉퐵 error is a constant value for the approximately monostatic configuration (indicated with a red curve) and the monostatic configuration (orange dashed curve) and they overlap. It is independent of
120 geometry, which is expected since in monostatic radar, errrors in measuring radial velocity is dependent on the observation time and (frequency and shape of the) waveform only [1].
8.1.4 Target Location Estimate Using Array-Based Method
The array-based localization method estimates target position from the receiving site in terms of the receiver range 푅푅 and receiver look angle 휃푅. 푅푅 cannot be measured directly, but can be calculated by solving the bistatic triangle of Figure 8.1 [1]
2 2 푅퐵푖 − 퐿푇푖 (8.26) 푅푅 = 2[푅퐵푖 + 퐿푇푖 sin(휃푅푖)] Where
푅푅 푚 Range between target and receiver 푅퐵푖 푚 Bistatic range corresponding to transmitter 푖 퐿푇푖 푚 Baseline of 푖th bistatic pair 휃푅푖 ° Receiver look angle
휃푅 is also not measured directly and is calculated from target azimuth and elevation bearing measurements using [1]
−1 (8.27) 휃푅푖 = − sin [cos(퐸푡) cos(퐸푇푖) cos(퐴푡 − 퐴푇푖) + sin(퐸푡) sin(퐸푇푖)] Where
퐸푡 ° Elevation angle of target 퐸푇푖 ° Elevation angle of transmitter 푖 퐴푡 ° Azimuth angle of target 퐴푇푖 ° Azimuth angle of transmitter 푖
To get the target position in Cartesian coordinates requires the following coordinate conversions:
(8.28) 푥푡 = 푅푅 cos(퐸푡) cos(퐴푡)
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푦푡 = −푅푅 cos(퐸푡) sin(퐴푡) (8.29)
푧푡 = 푅푅 sin(퐸푡) (8.30)
From equations (8.26) and (8.27), it can be seen that the array-based localization method makes use of the set of bistatic range sum and bearing measurements. We are interested in how the errors in these measurements propagate through the calculations and lead to an error in the final value of the target position as given in (8.28) – (8.30).
8.1.4.1 Receiver Look Angle Estimate
Applying (2.24) to (8.27), the SD of the receiver look angle 휃푅 estimation error due to individual measurement errors is
2 2 휕휃푅푖 휕휃푅푖 (8.31) 푑휃푅푖 = √( 푑퐸푡) + ( 푑퐴푡) 휕퐸푡 휕퐴푡 Where
푑휃푅푖 ° Error in receiver look angle of 푖th bistatic pair 푑퐸푡 ° Error in elevation angle of target 푑퐴푡 ° Error in azimuth angle of target
With the following partial derivatives
휕휃 sin(퐸 ) cos(퐸 ) − cos(퐸 ) sin(퐸 ) cos(퐴 − 퐴 ) 푅푖 = − 푇푖 푡 푇푖 푡 푇푖 푡 휕퐸 2 (8.32) 푡 √1 − [cos(퐸푇푖) cos(퐸푡) cos(퐴푇푖 − 퐴푡) + sin(퐸푇푖) sin(퐸푡)]
휕휃 cos(퐸 ) cos(퐸 ) sin(퐴 − 퐴 ) 푅푖 = − 푇푖 푡 푇푖 푡 (8.33) 휕퐴 2 푡 √1 − [cos(퐸푇푖) cos(퐸푡) cos(퐴푇푖 − 퐴푡) + sin(퐸푇푖) sin(퐸푡)] Where
휃푅푖 ° Receiver look angle of 푖th bistatic pair 퐴푡 ° Azimuth angle of target 퐸푇푖 ° Elevation angle of transmitter 푖 퐸푡 ° Elevation angle of target 퐴푇푖 ° Azimuth angle of transmitter 푖 122
The SD in the receiver look angle 휃푅 error is denoted by the blue curve for the bistatic case and the red dashed curve for the approximately monostatic case in Figure 8.8. The monostatic case is not applicable to the 휃푅 error expression since it does not have the baseline term 퐿푇 and with the transmitter and receiver site being coincident, 퐸푇 and 퐴푇
(the elevation and azimuth angle of the transmitter from the location of the receiver at the origin) becomes undefined.
Figure 8.8 SD of 휃푅 error as a function of 휃푅/퐴푡
The curves for the bistatic and approximately monostatic configurations overlap. We analyze this by looking at the components that make up the SD in 휃푅 error. From (8.31),
123 the SD in 휃푅 error depends on the SD in azimuth and elevation measurement errors, as well as their respective partial derivatives. These have been plotted in Figure 8.9 to help in the analysis. Since the error expression in (8.31) is the sum of the product of the partial derivatives with their respectively errors squared (so that the products eventually become positive), the absolute values of the partial derivatives are plotted. This helps to keep all the data information visible in the graph (since the logarithm of negative numbers are undefined).
Figure 8.9 Components of SD of 휃푅 error (Bistatic and approximately monostatic)
From Figure 8.9, the thick solid black line shows the SD of 휃푅 error, which is made up of two components: the product of 휕휃푅/휕퐸푡 with 푑퐸푡 indicated with blue dashed and solid
124 lines respectively; and the product of 휕휃푅/휕퐴푡 with 푑퐴푡, indicated with green dashed and solid lines respectively. As discussed in the previous section, the direction of arrival measurement errors 푑퐴푡 and 푑퐸푡 are the same for both the bistatic and approximately monostatic cases because their azimuth and elevation measurements are equivalent (−90° to 90° for azimuth; and 0° for elevation). The partial derivative expressions (8.32) and
(8.33) are equivalent for the same reason. Therefore, Figure 8.9 applies to both configurations. The partial derivatives are constant over 휃푅, which shows they are independent of bistatic geometry. Rather, the SD of 휃푅 error depends strongly on the limitations of the linear array, with the highest accuracy at array boresight direction
(marked with a red vertical dashed line) and lowest accuracy at the end regions of the array (marked with a black vertical dashed line) for the azimuth measurement. It is also strongly dependent on elevation, which in this analysis is a constant value since the target elevation is fixed (at 0 m).
8.1.4.2 Receiver Range Estimate
Similarly, applying (2.24) to (8.26), the SD of the receiver range estimate error is
2 2 2 휕푅푅 휕푅푅 휕푅푅 (8.34) 푑푅푅 = √( 푑푅퐵푖) + ( 푑퐿푇푖) + ( 푑휃푅푖) 휕푅퐵푖 휕퐿푇푖 휕휃푅푖 Where
푑푅푅 푚 Error in range between receiver and target 푑푅퐵푖 푚 Error in bistatic range sum corresponding to transmitter 푖 푑퐿푇푖 푚 Error in baseline of 푖th bistatic pair 푑휃푅푖 ° Error in receiver look angle of 푖th bistatic pair
With the following partial derivatives
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2 휕푅푅 1 + (푒푖) + 2푒푖 sin(휃푅푖) = 2 (8.35) 휕푅퐵푖 2[1 + 푒푖 sin(휃푅푖)]
2 휕푅푅 [(푒푖) + 1] sin(휃푅푖) + 2푒푖 = − 2 (8.36) 휕퐿푇푖 2[1 + 푒푖 sin(휃푅푖)]
[ 2] ( ) 휕푅푅 퐿푇푖 1 − 푒푖 cos 휃푅푖 (8.37) = − 2 휕휃푅푖 2[1 + 푒푖 sin(휃푅푖)] Where
푅푅 푚 Range between receiver and target 푅퐵푖 푚 Bistatic range sum corresponding to transmitter 푖 퐿푇푖 푚 Baseline of 푖th bistatic pair 휃푅푖 ° Receiver look angle of 푖th bistatic pair 푒푖 = 퐿푇푖/푅퐵푖 − Eccentricity corresponding to transmitter 푖
The SD in the receiver range 푅푅 error is plotted in Figure 8.10.
Figure 8.10 SD of 푅푅 error as a function of 휃푅/퐴푡
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The SD is denoted by the blue curve for the bistatic case, red dashed curve for the approximately monostatic case, and the orange dashed curve for the monostatic case. All three curves are different: In the bistatic case, large SDs occur in the forward scatter region (when 휃푅 = −90°) and at the ends of the linear array (when 퐴푡 = 30°), both indicated with vertical black dashed lines. In the approximately monostatic case, large
SDs occur only at the ends of the array. And in the monostatic case, the SD of 푅푅 error is a constant value.
All three curves in Figure 8.10 are different because of the partial differential expressions (8.35) – (8.37), where the eccentricity term 푒 is the ratio of the baseline 퐿푇 to the bistatic range 푅퐵. The bistatic case has largest eccentricity, and the monostatic case has eccentricity 0, which corresponds to a circle. From (8.34), the terms that make up the
SD in 푅푅 error include the bistatic range and baseline measurement errors 푑푅퐵 and 푑퐿푇, error in the receiver look angle estimates 푑휃푅, as well as their respective partial derivatives. 푑퐿푇 is 0 since we assume accurate knowledge of the transmitter and receiver locations, thus we exclude the curves for it and its partial derivative. For the monostatic configuration, (8.37) is 0 since 퐿푇 is 0 and this also agrees with our previous observation that 푑휃푅 would not be applicable to the monostatic case. The SD for 푅푅 error in the monostatic case reduces to the SD for 푅퐵 error, which is a constant value, as analyzed previously. The other curves for the component errors and their partial derivatives have been plotted in Figure 8.11a for the bistatic configuration and Figure 8.11b for the approximately monostatic configuration.
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(a) (b)
Figure 8.11 Components of SD of 푅푅 error: In (a) bistatic configuration and (b) approximately monostatic configuration
The thick solid black line in both figures shows the SD of 푅푅 error, which is essentially made up of two components: the product of 휕푅푅/휕푅퐵 with 푑푅퐵 indicated with blue dashed and solid lines respectively; and the product of 휕푅푅/휕휃푅 with 푑휃푅, indicated with green dashed and solid lines respectively. The partial derivative 휕푅푅/휕푅퐵 is dependent on bistatic geometry, thus it is more apparent in the bistatic case than the approximately monostatic case, as can be seen by comparing the blue dashed curves in Figure 8.11a and
Figure 8.11b, where it approaches a vertical asymptote at 휃푅 = −90° in the former. The dependence of 푑푅푅 on 푑휃푅 shows in both configurations, with large SD at the end of the linear array (at 퐴푡 = 30°). In the approximately monostatic configuration, although eccentricity is a small value (0.05), it is non-zero and hence still form a large positive product with 푑휃푅, differing from the true monostatic case, which has a small constant value across 휃푅.
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8.1.4.3 Target X-Coordinate Estimate
The SD in the target x-coordinate estimate error is
2 2 2 휕푥푡 휕푥푡 휕푥푡 (8.38) 푑푥푡 = √( 푑푅푅) + ( 푑퐸푡) + ( 푑퐴푡) 휕푅푅 휕퐸푡 휕퐴푡 Where
푑푥푡 푚 Error in x-coordinate of target 푑푅푅 푚 Error in range between receiver and target 푑퐸푡 ° Error in elevation angle of target 푑퐴푡 ° Error in azimuth angle of target
With the following partial derivatives
휕푥푡 = cos(퐸푡) cos(퐴푡) (8.39) 휕푅푅
휕푥푡 = −푅푅 sin(퐸푡) cos(퐴푡) (8.40) 휕퐸푡
휕푥 푡 (8.41) = −푅푅 cos(퐸푡) sin(퐴푡) 휕퐴푡 Where
푥푡 푚 x-coordinate of target 푅푅 푚 Range between receiver and target 퐸푡 ° Elevation angle of target 퐴푡 ° Azimuth angle of target
Figure 8.12a shows trends of the components that make up the SD of 푥푡 error in the bistatic configuration (where the target is 5.7 km away) and Figure 8.12b shows that for the approximately monostatic configuration (where the target is 470 km away). In the specific case, 퐸푡 is zero and hence (8.40) is zero. Thus, the SD of 푥푡 error, plotted using a thick solid black line in Figure 8.12, is made up of two components: the product of
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휕푥푡/휕푅푅 with 푑푅푅 indicated with blue dashed and solid lines respectively; and the product of 휕푥푡/휕퐴푡 with 푑퐴푡, indicated with green dashed and solid lines respectively.
The partial derivative expressions (8.40) and (8.41) are scaled by 푅푅, so targets that are further in range will have larger SDs in 푥푡 error. The larger derivative values can be seen by comparing Figure 8.12a and Figure 8.12b, where (8.41) has been plotted using a green dashed curve, and which is much higher in Figure 8.12b for the target at 470 km away.
(a) (b)
Figure 8.12 Components of SD of 푥푡 error: In (a) bistatic configuration and (b) approximately monostatic configuration
Figure 8.13a and Figure 8.13b shows results for the monostatic configuration for the same targets at 푅푅 of 5.7 km and 470 km respectively. Results from the bistatic configuration approaches that of the monostatic configuration at 휃푅 = 90° (the pseudomonostatic mode of operation). It is also where the error is lowest, mainly due to the contribution of the 휕푥푡/휕푅푅 with 푑푅푅 (blue dashed and solid lines). 130
(a) (b)
Figure 8.13 Components of SD of 푥푡 error: In monostatic configuration for (a) near target at 5.7 km and (b) far target at 460 km from receiver
8.1.4.4 Target Y-Coordinate Estimate
For the y-coordinate estimate
2 2 2 휕푦푡 휕푦푡 휕푦푡 (8.42) 푑푦푡 = √( 푑푅푅) + ( 푑퐸푡) + ( 푑퐴푡) 휕푅푅 휕퐸푡 휕퐴푡 Where
푑푦푡 푚 Error in y-coordinate of target 푑푅푅 푚 Error in range between receiver and target 푑퐸푡 ° Error in elevation angle of target 푑퐴푡 ° Error in azimuth angle of target
휕푦푡 = − cos(퐸푡) sin(퐴푡) (8.43) 휕푅푅
휕푦푡 = 푅푅 sin(퐸푡) sin(퐴푡) (8.44) 휕퐸푡
휕푦푡 (8.45) = −푅푅 cos(퐸푡) cos(퐴푡) 휕퐴푡
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Where
푦푡 푚 y-coordinate of target 푅푅 푚 Range between receiver and target 퐸푡 ° Elevation angle of target 퐴푡 ° Azimuth angle of target
Figure 8.14a shows trends in the components that make up the SD of 푦푡 error in the bistatic configuration (where the target is 5.7 km away) and Figure 8.14b shows that for the approximately monostatic configuration (where the target is 470 km away). The analysis is similar for 푥푡: (8.44) and (8.45) are scaled by 푅푅, resulting in larger values for a further target (in Figure 8.14b). The SD in bistatic configuration approaches that of the monostatic configuration at 휃푅 = 90° (the pseudomonostatic mode of operation), where the error is lowest, mainly due to the contribution of the 휕푦푡/휕푅푅 with 푑푅푅 (blue dashed and solid lines).
(a) (b)
Figure 8.14 Components of SD of 푦푡 error: In (a) bistatic configuration and (b) approximately monostatic configuration
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(a) (b)
Figure 8.15 Components of SD of 푦푡 error: In monostatic configuration for (a) near target at 5.7 km and (b) far target at 460 km from receiver
8.1.4.5 Target Z-Coordinate Estimate
And for the z-coordinate estimate
2 2 휕푧푡 휕푧푡 (8.46) 푑푧푡 = √( 푑푅푅) + ( 푑퐸푡) 휕푅푅 휕퐸푡 Where
푑푧푡 푚 Error in z-coordinate of target 푑푅푅 푚 Error in range between receiver and target 푑퐸푡 ° Error in elevation angle of target
휕푧푡 = sin(퐸푡) (8.47) 휕푅푅
휕푧 푡 (8.48) = 푅푅 cos(퐸푡) 휕퐸푡 Where
푧푡 푚 z-coordinate of target 푅푅 푚 Range between receiver and target 퐸푡 ° Elevation angle of target 133
Graphs for the SD of 푧푡 error as a function of 휃푅/퐴푡 have not been plotted because it is a constant value. (8.47) is zero since 퐸푡 is zero, and (8.48) is a constant value for a fixed 퐸푡 that is scaled by the range of the target from the receiver 푅푅.
8.1.4.6 Target Location Estimate in General Case
Some consideration was needed on how to present the joint effects of bistatic geometry (dependent on 휃푅) and linear array limits (dependent on 퐴푡) on location errors since 퐴푡 and 휃푅 are nonlinear functions of each other (see (8.27)). Thus, the specific case with 퐸푡 = 0° and 퐸푇 = 0° was used to analyze the trend of the location error SD when both 퐴푡 and 휃푅 coincides. However, 퐸푡 and 퐸푇 are also factors in the location error SD
(see (8.31) – (8.33) and (8.38) – (8.48)) and excluding it entirely in our analysis will not give us a complete picture.
Therefore, the location results for a general case in the bistatic configuration, with
퐸푡 = 4.73° (corresponding to a target altitude of 470 m) and 퐸푇 = 0.35° (corresponding to transmitter altitude of 281 m) will be discussed next. Additional details of the settings for the two cases are tabulated in Table 8.2.
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Specific case General case
푥푡 푉푎푟푖푒푠 푥푡 푉푎푟푖푒푠 Coordinates*, [푦푡] = [푉푎푟푖푒푠] Coordinates, [푦푡] = [푉푎푟푖푒푠] 푧푡 0 푧푡 470 푚
*The 푥푡, 푦푡 values in the specific case differs from those in the general case
Target Range from receiver, 푅푅 = 5.7 푘푚 settings Azimuth, 퐴푡 varies from −90° to 90°
Elevation, 퐸푡 = 0° Elevation, 퐸푡 = 4.7°
푥̇푡 50 푚/푠 Velocity, [푦̇푡] = [ 0 ] 푧푡̇ 0 푥푅 0 Receiver Coordinates, [푦푅] = [0] settings 푧푅 0
Array boresight azimuth, 퐴푏표푟푒푠푖𝑔ℎ푡 = −60° Horizontal configuration: 6 elements uniformly spaced by 80 cm
Array boresight elevation, 퐸푏표푟푒푠푖𝑔ℎ푡 = 20° Vertical configuration: 6 elements uniformly spaced by 60 cm
푥푇 0 푥푇 0 Transmitter Coordinates, [푦푇] = [46 푘푚] Coordinates, [푦푇] = [45.9 푘푚] settings 푧푇 0 푧푇 281 푚
Baseline range, 퐿푇 = 46 푘푚
Azimuth, 퐴푇 = −90°
Elevation, 퐸푇 = 0.35°
Center frequency, 푓푇 = 665 푀퐻푧 Bandwidth, 퐵 = 6 푀퐻푧 Table 8.2 Settings for bistatic configuration in the specific and general cases
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Figure 8.16 places the results from the specific case and general case side-by-side for an easy comparison. The SDs of 푥푡, 푦푡 and 푧푡 errors are presented in a single graph, color coded using blue, red, and green respectively. The results of their monostatic equivalents are also included using dashed lines in their respective colors.
(a) (b)
Figure 8.16 SD of target location error as a function of 휃푅/퐴푡 in bistatic configuration in (a) specific case when bistatic plane is coincident with x-y plane and (b) general case when it is not
First, now that 퐸푡 is no longer zero, the SD of 푧푡 error in both bistatic and monostatic configurations (green solid and dashed lines) increases. Just like our prior analysis of the
SD of 푥푡 and 푦푡 errors, the SD of 푧푡 error follows similar patterns of being influenced by the bistatic geometry (increasing as 휃푅 approaches −90°) and linear array limits
(increasing as 퐴푡 approaches 30°). Next, we look at the SD of 푦푡 error in the bistatic configuration indicated with a red solid line. The difference is most obvious at 퐴푡 = −90° because there is a change in the bistatic geometry at 퐴푡 = −90° for the two cases. In the 136 specific case (Figure 8.16a), 휃푅 and 퐴푡 are equivalent. However, for the general case
(Figure 8.16b), 휃푅 and 퐴푡 are different because the bistatic plane no longer lies in the x-y plane. Figure 8.17 shows the actual relationship between 휃푅 and 퐴푡 in the general case.
Figure 8.17 Nonlinear relationship between 휃푅 and 퐴푡 in the general case
From Figure 8.17, we see that when 퐴푡 = −90°, 휃푅 = −85.6° and thus, in Figure 8.16b, the red curve peaks at a finite value of 825 m at 퐴푡 = −90° instead of an infinitely large value as in Figure 8.16a. (The SD value of 825 m corresponds to 휃푅/퐴푡 = −85.6° in
Figure 8.16a.) The monostatic curve (red dashed curve) for the SD of 푦푡 error is similar across Figure 8.16a and Figure 8.16b since in the monostatic configuration, errors are independent of the bistatic geometry. Finally, we look at the SD of 푥푡 error, indicated
137 with blue lines. The most interesting result is that there is a local dip at 퐴푡 = −90°, where we usually expect a large error. By looking at Figure 8.18, which shows the components that make up the SD of 푥푡 error, the reason for this is mainly due to a finite 푑푅푅 along with a deep null in 휕푥푡/휕푅푅.
Figure 8.18 Components of SD of 푥푡 error in bistatic configuration for the general case (where the bistatic plane does not coincide with x-y plane)
The monostatic curve (blue dashed curve) for the SD of 푥푡 error is similar across Figure
8.16a and Figure 8.16b since in the monostatic configuration, errors are independent of the bistatic geometry. For the same reason, results from the approximately monostatic configurations are not shown here. In Figure 8.16b, the local minimums of the SD of 푥푡 and 푦푡 errors occur at different 퐴푡. This suggests that in multistatic radar settings (see 138 next section), it would be more beneficial to combine data from multiple bistatic radar pairs by selecting the best solution for 푥푡, 푦푡, and 푧푡 separately rather than jointly.
Finally, the error in the target location, specified by the x-, y-, and z-coordinates, is the root-sum-square error
2 2 2 √(푑푥푡) + (푑푦푡) + (푑푧푡) (8.49) where
푑푥푡 푚 Error in x-coordinate of target 푑푦푡 푚 Error in y-coordinate of target 푑푧푡 푚 Error in z-coordinate of target
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8.2 Passive Localization Accuracies in Multistatic Radar
The previous section concluded our analysis for a bistatic radar configuration made up of one receiver and one transmitter. Now we expand the analysis for a multistatic radar configuration where there is one receiver and multiple transmitters.
Since we are dealing with a network made up of multiple bistatic radar pairs with transmitters spaced far apart, a grid map display in the x-y plane was selected to present the results. We do this first for target location estimate using multilateration (which is the prevalent method in literature), and then compare it with the array-based method (that is the main topic of this dissertation).
As before, a constant SNR of 21 dB across the range of target x- and y-positions was used and we fix the target altitude at 470 m. This lets us see how the accuracy upon detection of a target at a given altitude changes according to its position relative to the multistatic radar. The positions of the transmitters are modeled after the opportunistic
DTV transmitters used in our experiments and hence this presents the general case
(where the bistatic plane does not coincide with the x-y plane). Details of the analysis settings are tabulated in Table 8.3.
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푥푡 −50 푘푚 푡표 50 푘푚 Coordinates, [푦푡] = [−50 푘푚 푡표 50 푘푚] Target 푧푡 470 푚 settings 푥̇푡 50 푚/푠 Velocity, [푦̇푡] = [ 0 ] 푧푡̇ 0 푥푅 0 Coordinates, [푦푅] = [0] 푧푅 0
Receiver Array boresight azimuth, 퐴푏표푟푒푠푖𝑔ℎ푡 = −60° settings Horizontal configuration: 6 elements uniformly spaced by 80 cm
Array boresight elevation, 퐸푏표푟푒푠푖𝑔ℎ푡 = 20° Vertical configuration: 6 elements uniformly spaced by 60 cm T1 T2 T3 (WTTE) (WOSU) (WWHO) Coordinates, Coordinates, Coordinates, 1 2 3 푥푇 2.04 푘푚 푥푇 10.39 푘푚 푥푇 −5.77 푘푚 1 2 3 [푦푇 ] = [−6.89 푘푚] [푦푇] = [17.76 푘푚] [푦푇] = [−45.57 푘푚] 1 2 3 푧푇 0.26 푘푚 푧푇 0.26 푘푚 푧푇 0.28 푘푚 Baseline range, Baseline range, Baseline range, 퐿1 = 7.19 푘푚 퐿2 = 20.58 푘푚 퐿3 = 45.93 푘푚 Transmitter 푇 푇 푇 settings Azimuth, Azimuth, Azimuth, 1 2 3 퐴푇 = 73.5° 퐴푇 = −59.7° 퐴푇 = 97.2° Elevation, Elevation, Elevation, 1 2 3 퐸푇 = 2.1° 퐸푇 = 0.7° 퐸푇 = 0.3° Frequency, Frequency, Frequency, 1 2 3 푓푇 = 605 푀퐻푧 푓푇 = 617 푀퐻푧 푓푇 = 665 푀퐻푧 Bandwidth 퐵 = 6 푀퐻푧 Table 8.3 Settings for multistatic configurations used in analysis of errors in a general case
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8.2.1 Target Location Estimate Using Range Multilateration
The range multilateration method makes use of the set of bistatic range sum measurements, requiring a minimum of three unique bistatic range sums simultaneously.
This makes it suitable for use only in multistatic radar. Again, the goal here is to calculate the accuracy of the position estimate in Cartesian coordinates based on the known accuracy of the bistatic range sum measurements. The equations for calculating the error bounds of the location estimates using the range multilateration method have been developed in [51] and are summarized below.
First, the covariance matrix R of the measurement error of the bistatic parameters is defined as
푇 푹 = 퐸{풛풛 } = diag([푉푎푟(푅퐵1) ⋯ 푉푎푟(푅퐵퐾)]) (8.50)
Where
푹 − Covariance matrix of measurement error 퐸{∙} − Expectation operator Vector of bistatic range measurements 풛 = [푅 푅 ⋯ 푅 ]푇 푚 퐵1 퐵2 퐵퐾 from 퐾 transmitters (∙)푇 − Transpose operator 퐾 × 퐾 square matrix with diagonal diag(풙) − elements 풙, 풙 ∈ ℂ퐾×1 where 퐾 is the number of transmitters used 푉푎푟(∙) − Variance function Bistatic range sum corresponding to 푅 푚 퐵푖 transmitter 푖, where 푖 = 1, … , 퐾
By using a first-order Taylor series expansion, the covariance matrix of the position estimate can be approximated as
푇 휕풙푡 휕풙푡 푷 = 퐸{풙̂ 풙̂푇} ≈ ( ) 푹 ( ) (8.51) 푡 푡 휕풛 휕풛
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Where
푷 푚 Covariance matrix of position estimate 퐸{∙} − Expectation operator 풙̂푡 푚 Coordinate target position vector estimate 풙푡 푚 Coordinate target position vector
The Jacobian in (8.51) is expressed using the following equations
휕풙 푡 = (휟푇휟)−1휟푇(푰푅 − 휦) 휕풛 푅 (8.52)
휦 = diag([푅퐵1 푅퐵2 ⋯ 푅퐵퐾]) (8.53)
푇 풙푡 휟 = 푺 − 풛 (8.54) ‖풙푡‖
푥푇1 푦푇1 푧푇1 푥푇2 푦푇2 푧푇2 (8.55) 푺 = [ ⋮ ⋮ ⋮ ] 푥푇퐾 푦푇퐾 푧푇퐾 Where
휟 − 3 × 퐾 푡emporary matrix variable, defined as in (8.54) 푨−1 − Inverse of matrix 푨 푰 − 퐾 × 퐾 identity matrix 푅푅 푚 Range between receiver and target 휦 푚 퐾 × 퐾 diagonal matrix, defined as in (8.53) 푺 푚 3 × 퐾 matrix of transmitter position, defined as in (8.55) 푥푇푖 푚 x-coordinate of transmitter 푖 푦푇푖 푚 y-coordinate of transmitter 푖 푧푇푖 푚 z-coordinate of transmitter 푖
The values on the diagonal of P in (8.51) correspond to the variances of the position error in the x, y, and z directions.
Figure 8.19a – Figure 8.19c show contour plots of the SDs of 푥푡, 푦푡 and 푧푡 estimate errors obtained using the range multilateration method. 143
(a) (b)
(c)
Figure 8.19 SD of target location estimate errors using range multilateration: (a) x- coordinate, (b) y-coordinate, and (c) z-coordinate
The target’s position in both x- and y-coordinates varies from -50 km to 50 km in steps of
0.5 km and its position in z-coordinate is fixed at 470 m. The SD is represented by the intensity, marked out at certain values with the contour lines, and is in logarithmic scale
(dB) in meters to allow a large range to be displayed. The locations of the three
144 transmitters T1, T2, and T3 are marked with red triangles and the receiver is marked with a blue square at the origin. The boresight direction of the linear array is marked out using a red dashed line.
When comparing similar areas, errors in 푦푡 has the smallest SD; this is followed closely by errors in 푥푡 (about 0.5 dB larger); and then by a further stretch by errors in 푧푡
(about 2 dB larger). This is due to the difference in separation of the transmitters in the x-
, y-, and z-axis (see Table 8.3). Specifically, the standard deviations in the y-, x-, and z- location of the transmitters are 32 km, 8 km, and 11 m respectively.
8.2.2 Comparison with Array-Based Method
We can directly compare the localization performance of the multilateration method with the array-based method by plotting similar contour plots for the latter. This is shown in Figure 8.20 for the bistatic pair made up of transmitter T1. The contour plots for transmitters T2 and T3 are largely similar to that of T1 and so have been included in
Appendix B to keep this section compact (the interested reader should look at the graph for T2, which shows what happens when the array boresight is close to the forward scatter (FS) region where 휃푅 = −90°).
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(a) (b)
(c)
Figure 8.20 SD of target location estimate errors using array-based localization with transmitter T1: (a) x-coordinate, (b) y-coordinate, and (c) z-coordinate
First, by looking at the intensity scale on the right of Figure 8.20a and Figure
8.20b, it can be seen that the SDs in 푥푡 and 푦푡 errors are comparable. In Figure 8.20c, the
SD in 푧푡 error is smaller. This is because the target’s altitude is restricted to 470 m in the z direction, compared to 50 km in the x and y directions. Second, local dips in the SD of
푥푡 and 푦푡 errors that have been observed in our previous analysis show up again. The
146 former along the y-axis (when 푥 = 0) in Figure 8.20a; and the latter along the x-axis
(when 푦 = 0) in Figure 8.20b. Third, the sectors with large SDs are at the ends of the linear array (the region which is perpendicular to the array boresight indicated with a red dashed line) and in the FS region (region between the receiver at origin and transmitter where 휃푅 = −90°) for all Figure 8.20a – c. This is also consistent with conclusions from the earlier analysis. Finally, Figure 8.20a – c show again that the areas of lowest errors for 푥푡, 푦푡, and 푧푡 do not overlap exactly with one another and so it is better to optimize for them separately.
Now we compare this with the multilateration results in Figure 8.19a – c. The first observation is that the location SDs in multilateration do not have the dependence on array and FS region limitations. Thus, assuming that it is possible to detect a target in the
FS condition, multilateration is advantageous in these regions. The next observation is that for 푧푡, the performance of the array-based method is much better than multilateration due to the limited diversity in transmitters’ locations in altitude.
We summarize these observations into a single “selection” graph, shown in Figure
8.21a – c, which selects the best localization method in the x-, y- and z-coordinates separately. The best localization method is defined as that exhibiting the lowest error bound at each point of the analyzed grid map area. The intensity scale of Figure 8.21a – c has been quantized into 4 levels: referring to array-based localization utilizing transmitters T1, T2, and T3; and range multilateration (MLAT) utilizing all three transmitters.
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(a) (b)
(c)
Figure 8.21 Selection of the best localization method in: (a) x-coordinate, (b) y- coordinate, and (c) z-coordinate
8.2.3 Multilateration Accuracy as a Function of Target Altitude
We can perform an analysis of the SD of multilateration estimation errors as a function of target altitude using the same equations (8.50) – (8.55). Because it is desired to find out its relationship as a function of target altitude, a grid map display in the y-z plane was selected to present the results. A constant SNR of 21 dB across the range of
148 target y- and z-positions was used and we fix the target’s x-position at 100 m. The remaining settings are as in Table 8.3.
Figure 8.22a – c show contour plots of the SDs of 푥푡, 푦푡 and 푧푡 estimate errors in the y-z plane, obtained using the range multilateration method. The target’s position in the y-coordinate varies from -50 km to 50 km and its position in the z-coordinate from 0 km to 50 km m, both in steps of 0.5 km. The SD is represented by the intensity, marked out at certain values with the contour lines, and is in logarithmic scale (dB) in meters to allow a large range to be displayed. The locations of the three transmitters T1, T2, and T3 are marked with red triangles and the receiver is marked with a blue square at the origin.
The shape of the SD of 푥푡 and 푦푡 error contours are similar. Within the confines of the transmitters, their SD increases with altitude, reaches a maximum, and then decreases. For the same target location in the y-z plane, the SD of 푦푡 error is smaller than the SD of 푥푡 error by approximately 0.5 dB. The SD of 푧푡 error is largest when altitude is
0 m and generally decreases with increasing altitude.
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(a) (b)
(c)
Figure 8.22 SD of target location estimate errors in y-z plane using range multilateration: (a) x-coordinate, (b) y-coordinate, and (c) z-coordinate
8.3 Conclusions
In this section, we have detailed all the expressions that allow us to evaluate the accuracy of the array-based localization method as a function of geometry. We have also introduced a way of optimizing localization performance in multistatic radar through selection of the method that exhibits the lowest error bound. By comparing its performance with that of range multilateration, we have made a case for array-based
150 localization to be included in multistatic radar because it is easier and cheaper to install a vertical array than to introduce large diversities in transmitters’ altitudes.
In our analysis of the passive localization performance using array-based and range multilateration methods, we are interested in the relative values of the location accuracy instead of the absolute values for two reasons: First, the relative value is sufficient for selection between the different methods. Next, the values calculated are theoretical lower bounds and in reality, it will certainly be higher.
To complete our study however, we would still like to know the accuracy of the array-based localization method in real life. The next section documents the experiments used to demonstrate the viability of array-based localization in DTV passive radar.
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Chapter 9 Array-Based Localization Experiments
We present the results of four investigations that make use of experimental data in this section. The first investigation looks at the accuracy of the array-based localization method across different bistatic geometries. For this, reflections from two transmitters off of a low-flying aircraft were captured with the 1D array for a wide range of bistatic angles (5.8° ≤ 훽 ≤ 176.3°). The second investigation looks at the accuracy of the array- based localization when elevation angles are estimated: For this, a larger 2D array that was used to capture data from targets at low and high altitudes (up to 10 km) and elevation estimates were made. The third investigation compares performance of the array-based localization with range multilateration. Here, a dataset where there were simultaneous detections of the target from at least three transmitters was used so that range multilateration could be performed. The fourth investigation is an extension of the third, generating some preliminary insights about fusing estimates from different localization methods in multistatic radar.
9.1 Azimuth Only Localization across Bistatic Geometries
The first investigation makes use of dataset consisting of reflections from two transmitters off of a low-flying aircraft for a wide range of bistatic angles. The
152 cooperative aircraft, a Piper Saratoga airplane, was equipped with an on-board GPS receiver and IMU that provide its air truth data. Its reflected signals from the WWHO and
WOSU transmitter were collected simultaneously. The duration of the observations ranged from 60 – 120 seconds, and the array boresight was moved between some collections to ensure that the anticipated target position was as close to boresight as possible for better performance. A 1D array was used so only azimuth measurements were measured, and the elevation of the target was assumed to be known in order to complete the localization process.
9.1.1 Experiment Settings
Information about the collected datasets for the WWHO transmitter is summarized in Table 9.1 and for the WOSU transmitter, in Table 9.2. The range of bistatic angles studied highlights the extensiveness of the investigation in terms of geometry diversity. The last column of each table shows the mean SNR for each collected dataset. The mean SNR was calculated on the linear scale before converting to dB. The SNR for a given target was calculated as the target’s peak response in the range
Doppler (RD) map divided by the average power in a rectangular region of the RD map that excludes any significant clutter contributions [66].
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Dataset Aircraft Mean Mean Elevation Range Bistatic angle Mean # type velocity altitude from from range [°] SNR [m/s] [m] transmitter receiver [dB] [°] [m]
1A 440 0.2 8260 8.9 – 41.0 13.0 2A 481 0.2 2764 7.0 – 74.8 13.6 Piper 3A 50 440 0.1 – 0.2 14590 5.8 – 11.7 12.2 Saratoga 4A 473 0.2 – 0.3 1463 8.3 – 139.8 15.7 5A 470 0.2 – 0.3 5700 52.6 – 99.8 11.3 Table 9.1 Collected datasets from WWHO transmitter for investigation 1
Dataset Aircraft Mean Mean Elevation Range Bistatic angle Mean # type velocity altitude from from range [°] SNR [m/s] [m] transmitter receiver [dB] [°] [m]
1B 440 0.8 – 0.9 8260 128.8 – 176.3 10.4 2B 481 0.6 – 0.8 2764 120.4 – 166.5 19.6 Piper 3B 50 440 0.8 – 1.0 14590 96.1 – 111.9 11.6 Saratoga 4B 473 0.6 – 0.7 1463 57.8 – 154.6 20.0 5B 470 0.6 – 0.7 5700 79.0 – 133.6 14.2 Table 9.2 Collected datasets from WOSU transmitter for investigation 1
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9.1.2 Observations on Detection
Figure 9.1 shows the detection results out of 100 processed points for the first transmitter, WWHO. Manual detection was undertaken to ensure that all detections belong to the target of interest and not spurious noise or some other target. Subsequent analysis of the localization accuracy as a function of bistatic geometry would thus be uncomplicated by any ambiguous detections.
Figure 9.1 Detection results from WWHO transmitter datasets for investigation 1
A top-down view of the experimental scenario is presented. The receiver is marked with a blue square and is located at the origin, and the WWHO transmitter is marked with a red triangle. The area circled in blue near the receiver has been zoomed in on the right of
Figure 9.1 so that the detection results are more visible. Three dashed lines (purple, orange, and green) radiate out from the receiver and these indicate the positions of the
155 linear array boresight, which was moved in between collections in order to place the boresight as close to the expected target position as possible to obtain good angle estimates. The flight paths corresponding to the collection of datasets 1A – 5A have been annotated, and an arrow shows the direction of flight. The paths are color-coded such that the purple paths (2A and 3A) correspond to the purple array boresight direction, the orange paths to the orange boresight direction, and so on. The detections, indicated with dots, have been overlaid on the paths (drawn with a thin line). The number of detected points as a percentage of the total number of points processed (100 points) for each dataset has been tabulated.
Figure 9.2 shows the detection results for the second transmitter, WOSU. While the target paths taken are the same for both transmitters (WWHO and WOSU), the detection results are different because the use of the WOSU transmitter leads to an independent bistatic channel. WOSU has cases with particularly poor detections –
Datasets 1B (28%) and 4B (18%) because of the forward scatter (FS) geometry. In this geometry, the bistatic range (distance from transmitter to target to receiver) is not much different from the baseline (distance from transmitter to receiver) and this causes the target return to be removed by time-domain DSI suppression, making it hard to detect amongst low bistatic velocity clutter. This effect is also evident in the flight path that makes up dataset 2B.
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Figure 9.2 Detection results from WOSU transmitter datasets for investigation 1
After detection, we move on to the next stage of localization. In particular, we are interested in the experimental validation of localization accuracy as a function of bistatic geometry – the theory developed in Chapter 8.
9.1.3 Simulation as an Analysis Tool
The location of the target is estimated from inherently noisy measured data.
Hence, its performance can only be described statistically. While experimental data offers a valuable validation of the localization method, each measurement of a moving aircraft cannot realistically be repeated for a large number of times. Hence, simulation is used to bridge the theoretical bounds introduced in Chapter 8, along with the experimental results of this chapter. All three results are then presented together in the next section. In the
Monte Carlo simulation, the bistatic range and angle measurements used are corrupted by an additive zero-mean Gaussian error with its SD taking on the value of their calculated 157 theoretical error values at the SNRs measured in the experiment. This is repeated for
50,000 runs.
9.1.4 Observations on Localization
Figure 9.3 shows the location error SD in the x-, y- and z-coordinates as a function of azimuth angle for the theoretical (Figure 9.3a), simulation (Figure 9.3b), and experimental (Figure 9.3c) cases for dataset 5A. Figure 9.4 shows the azimuth error SD for the theoretical/simulation (Figure 9.4a) and experimental (Figure 9.4b) cases. The results for the other datasets have been included in Appendix C.
Figure 9.3 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 5A for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
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Figure 9.4 SDs of 퐴푡 error from dataset 5A for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
The theoretical location and azimuth error SDs are calculated using the expressions developed in Chapter 8. Here, the target’s SNR and altitude were not kept constant over the limited range of azimuth angles. Instead, the target’s actual altitude and measured SNR during the experiment were used. Since these varies from CPI to CPI, the calculated SD values fluctuates in the scatter plots in Figure 9.3a and Figure 9.4a, as opposed to being a smooth function of bistatic geometry as seen in the analysis in
Chapter 8. The simulation location error SD is obtained by taking the SD of the errors from 50,000 Monte Carlo runs. This have been included to verify the correctness of
(8.31) – (8.48), i.e. the correct propagation of bistatic range and angles measurement errors to the final value of the location estimate errors. Comparing Figure 9.3a (the
159 theoretical subplot) and Figure 9.3b (the simulation subplot), it can be seen that they are consistent with each other. The experimental location and azimuth errors are calculated from experimental data; they are the magnitudes of the differences between the target’s location and azimuth from GPS air truth and their estimates. Even though the errors are only a single realization of random variables and hence their values plotted in Figure 9.3c and Figure 9.4b occur only by chance, they have been included for completeness. They also give an idea of the location and azimuth accuracies achieved in an actual passive radar.
Figure 9.5 and Figure 9.6 quantify the scatter plots in Figure 9.3 and Figure 9.4 by plotting the percentiles of the SD of errors in the 푥푡, 푦푡, and 푧푡 estimates and 퐴푡 estimates respectively. As an example, we look at the blue dashed curve on Figure 9.5a, which shows the percentile of the SD of errors in 푥푡 for the theoretical case. 0% or none of the points have an SD of more than 40 m (maximum SD); About 10% of the points have an
SD greater than 20 m; and 100% or all of the points have an SD greater than about 4 m
(minimum SD). The distribution of the points for the theoretical and simulation cases are very similar and they use the same (blue) scale on the left. On the other hand, SD values in the experimental case are about a magnitude or two larger, and uses the (red) scale on the right.
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Figure 9.5 Localization error results from dataset 5A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure 9.6 Percentiles for the SD of 퐴푡
The experimental location and azimuth root-mean-square errors (RMSE) are tabulated in
Table 9.3 for the WWHO transmitter and Table 9.4 for the WOSU transmitter. In the last column, the error in location is the root-sum-square (RSS) error of the errors in 푥푡, 푦푡, and 푧푡.
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Dataset # RMSE [°] RMSE [m] RSS error in
퐴푡 푥푡 푦푡 푧푡 location [m]
1A 0.6 75 35 1 82 2A 9.7 211 98 22 234 3A 0.4 99 6 0 100 4A 2.0 41 14 7 44 5A 0.6 46 60 4 76 Table 9.3 Localization results for datasets from WWHO transmitter for investigation 1
Dataset # RMSE [°] RMSE [m] RSS error in
퐴푡 푥푡 푦푡 푧푡 location [m]
1B 3.6 938 1583 75 1841 2B 13.2 613 1276 461 1488 3B 0.4 100 144 4 175 4B 27.4 610 345 301 763 5B 0.8 120 120 12 170 Table 9.4 Localization results for datasets from WOSU transmitter for investigation 1
Again, it is highlighted that the experimental values are averages of a small number (<
100) and cannot be used to determine trends closely unlike the theoretical and simulation results. However, we can see that there are regions that cause significantly larger errors
(in datasets 1B, 2B, and 4B) and this is due to the forward scatter (FS) geometry.
In this section, we have verified, through simulation, the correctness of (8.31) –
(8.48), which describe the array-based localization accuracy as a function of geometry.
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Experimental results also support the acceptable performance of passive radar in the
bistatic and pseudo-monostatic regions of operation, giving accuracy within 250 m for
targets at receiver ranges out to 15 km.
9.2 Azimuth and Elevation Localization
The second investigation looks at the accuracy of the array-based localization
when elevation angles are also estimated. Using a 2D array, datasets from a number of
different opportunistic targets reflecting signals from the WWHO transmitter were
obtained. These included high altitude targets at receiver ranges out to 20 km. Their air
truth data comes in the form of ADS-B data logging from a Kinetic Avionics SBS-3. The
record length of each dataset is 30 s. The 2D array enables elevation estimates to be made
and the array boresight was set so that the anticipated targets would be as close to
boresight as possible in both azimuth and elevation planes.
9.2.1 Experiment Settings
Dataset Aircraft type Mean Mean Elevation Range Bistatic Mean # velocity altitude from from angle range SNR [m/s] [m] transmitter receiver [°] [dB] [°] [m]
1 Embraer 175LR 259 10405 9.4 – 9.6 17859 25.3 – 28.8 10.4 2 Boeing 737-823 206 10062 8.3 – 8.7 22430 24.5 – 26.0 8.1 Cirrus SR22 60 556 0.2 21850 9.2 – 11.9 8.7 3 CSA SportCruiser 54 370 0.0 – 0.1 22628 7.8 – 8.3 8.2 Table 9.5 Collected datasets from WWHO transmitter for investigation 2
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9.2.2 Observations on Detection
Figure 9.7 shows the detection results. The area circled in blue near the receiver has been zoomed in on the right of Figure 9.7 so that the detection results are more visible. The dashed red line indicates the position of the linear array boresight and the flight paths corresponding to the collection of datasets 1 – 3 have been annotated, with an arrow showing the direction of flight. The number of detected points as a percentage of the total number of points processed for each dataset has been tabulated. The cases with particularly poor detection are datasets 1 (36%) and 2 (20%). These involve high altitude targets at significant elevations (> 8°) from the illuminating transmitter where the effects of beam tilt and narrow elevation main lobes – radiation characteristics of typical broadcast antenna, on detection become significant [58].
Figure 9.7 Detection results from WWHO transmitter datasets for investigation 2
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9.2.3 Observations on Localization
The experimental location, azimuth, and elevation RMSEs are tabulated in Table 9.6 for investigation 2.
Dataset # RMSE [°] RMSE [m] RSS error
퐴푡 퐸푡 푥푡 푦푡 푧푡 in location [m]
1 1.3 0.8 330 170 235 440 2 0.4 0.3 140 97 117 206 3A 0.5 0.5 174 24 174 247 3B 0.3 0.4 131 8 158 205 Table 9.6 Localization results for datasets from WWHO transmitter for investigation 2
Column six of Table 9.6 shows the error in 푧푡, which is larger than the values in Table
9.3 and Table 9.4. This is expected since actual elevation measurement is used, rather than elevation derived from air truth in the case of the 1D array. The errors in 푥푡 and 푦푡 are comparable to those in Table 9.3 and Table 9.4. The localization performance in dataset 1 is the worst because the target path is closer to the end region of the array. The other three datasets demonstrate location accuracies within 250 m for targets at ranges out to 20 km. The error in 푧푡 is comparable to the magnitudes of errors in 푥푡 and 푦푡 as indicated by theory.
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9.3 Comparison of Array-Based Localization with Multilateration
The third investigation compares performance of the array-based localization with range multilateration. Range multilateration requires simultaneous detections of the target from at least three transmitters and so a wideband collection was made, followed by digital down-conversion to obtain the target’s data corresponding to each transmitter. The cooperative aircraft is a Piper Saratoga airplane with an on-board GPS receiver and IMU that provide its air truth data. Because only a 1D array was available for this collection, only azimuth measurements were measured, and the elevation of the target is assumed to be known in order to complete the localization process.
9.3.1 Experiment Settings
Figure 9.8 shows the experimental set-up.
Figure 9.8 Experimental set-up for investigation 3
166
The receiver is marked with a blue square and is located at the origin, and the three transmitters (T1, T2, and T3) are marked with red triangles. The dashed red line indicates the position of the linear array boresight and the single flight path is shown in blue.
Information about the collected dataset is summarized in Table 9.7.
Aircraft type: Piper Saratoga Mean velocity: 50 m/s Mean altitude: 470 m Target Elevation from transmitter: 0.2° – 0.3° settings Range from receiver: 5700 m Bistatic angle range: 52.6° – 99.8° Mean SNR: 22.6 dB T1 T2 T3 Coordinates, Coordinates, Coordinates, 1 2 3 푥푇 2.04 푘푚 푥푇 10.39 푘푚 푥푇 −5.77 푘푚 1 2 3 [푦푇 ] = [−6.89 푘푚] [푦푇] = [17.76 푘푚] [푦푇] = [−45.57 푘푚] 1 2 3 푧푇 0.26 푘푚 푧푇 0.26 푘푚 푧푇 0.28 푘푚 Baseline range, Baseline range, Baseline range, 1 2 3 퐿푇 = 7.19 푘푚 퐿푇 = 20.58 푘푚 퐿푇 = 45.93 푘푚 Transmitter Azimuth, Azimuth, Azimuth, settings 1 2 3 퐴푇 = 73.5° 퐴푇 = −59.7° 퐴푇 = 97.2° Elevation, Elevation, Elevation, 1 2 3 퐸푇 = 2.1° 퐸푇 = 0.7° 퐸푇 = 0.3° Frequency, Frequency, Frequency, 1 2 3 푓푇 = 605 푀퐻푧 푓푇 = 617 푀퐻푧 푓푇 = 665 푀퐻푧 Bandwidth 퐵 = 6 푀퐻푧 Table 9.7 Collected dataset for investigation 3
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9.3.2 Observations on Detection
Figure 9.9 shows detections of the target’s reflections from Transmitter 1, 2, and 3 individually and simultaneous detections of reflections from all three transmitters. The number of detected points as a percentage of the total number of points processed (100 points) for each dataset has been tabulated.
Figure 9.9 Detection results from multistatic dataset for investigation 3
Detections from individual transmitters vary from 67% to 92%. From Table 9.7, the bistatic angle range is between 52.6° – 99.8°, and so this is in a general bistatic mode of operation. Simultaneous detections require detections from all three transmitters at the same time, thereby giving the lowest detection (46%). The requirement for simultaneous detection makes multilateration less advantageous than array-based localization in scenarios where frequent update of a target location is desired. 168
9.3.3 Observations on Localization
The experimental location RMSEs are tabulated in Table 9.8 for the different methods of localization: Array-based localization using T1, T2 and T3; and range multilateration.
Method RMSE [m] RSS error in
푥푡 푦푡 푧푡 location [m]
Array-based T1 36 53 3 64 Array-based T2 120 120 12 170 Array-based T3 46 60 4 76 Multilateration 149 46 745 761 Table 9.8 Localization results for datasets from WWHO transmitter for investigation 2
The array-based localization methods have much lower location errors than multilateration mainly due to the difference in 푧푡 error. This is consistent with earlier analysis of multilateration accuracy, where estimates of 푧푡 are poor due to the lack of diversity in the transmitters’ altitude.
9.4 Fusion of Localization Estimates
As the location errors depend heavily on the geometry that changes while the target is moving along its trajectory, a multistatic passive radar system that fuses measurements from multiple localization methods to form its final location estimates would be expected to have more robust performance. In the previous chapter, we saw that the local minimums of SD of 푥푡, 푦푡, and 푧푡 errors occur separately at different
169 geometries. Therefore, we select for the best localization method that gives the lowest theoretical error for 푥푡, 푦푡, and 푧푡 separately. A multistatic radar information processing scheme that does this is shown in Figure 9.10.
Figure 9.10 Multistatic radar information processing scheme
The first subsystem is the rectangular block performing array-based localization and range multilateration. From each bistatic pair, the complete set of target measurements include its azimuth 퐴푡, elevation 퐸푡, bistatic range 푅퐵 and bistatic velocity 푉퐵. In array- based localization, only a subset of this measurement set (퐴푡, 퐸푡, 푅퐵) is needed to obtain the initial target location estimates (푥푡, 푦푡, 푧푡). In range multilateration, measurements
푖 from at least 3 bistatic pairs are needed, utilizing only bistatic range information 푅퐵 (for
푖 ≥ 3). The second subsystem performs the selection of estimates. This selection is done separately for 푥푡, 푦푡, and 푧푡. The best localization method is defined as that exhibiting the lowest error bound for the target’s current position. Since the target’s location is
170 unknown, we use the current estimate of the target’s location. The last subsystem is the
Kalman filter (KF), which filters out the effects of measurement noise.
9.4.1 Kalman Filter
We are interested in forming associated tracks in the x-, y-, and z-coordinates. So we define the state vector as a 6 by 1 vector containing the positional data (푥푡, 푦푡, 푧푡) with its first derivative/velocity (푥̇푡, 푦̇푡, 푧푡̇ ).
푇 풙 = [푥푡 푥̇푡 푦푡 푦̇푡 푧푡 푧푡̇ ] (9.1)
The prediction of what the next state will be given the last best estimate of its state is found using
− 풙̂푛 = 푭풙̂푛−1 (9.2)
Where 푭 is the state transition matrix.
Assuming a constant velocity (CV) model for the target dynamics, which holds for the short time interval spanned by those measurements, we implement (9.2) using
1 훥푡 0 0 0 0 0 1 0 0 0 0
0 0 1 훥푡 0 0 푭 = 0 0 0 1 0 0 (9.3) 0 0 0 0 1 훥푡 [0 0 0 0 0 1 ]
Where Δ푡 is the time interval between samples (and does not have to be uniform). The measurement update is done using
− − (9.4) 풙̂푛 = 풙̂푛 + 푲푛(풛푛 − 푯풙푛 ) Where
푲푛 − Kalman gain 풛푛 − Measurement at discrete time step 푛 푯 − Measurement matrix
171
Since we are providing the target’s estimated position directly as inputs, we have
푥푡(푛) 풛푛 = [푦푡(푛)] (9.5) 푧푡(푛)
1 0 0 0 0 0 푯 = [0 0 1 0 0 0] (9.6) 0 0 0 0 1 0
If we want to exploit also the bistatic velocity 푉퐵푖 information available from the measured bistatic Doppler from the 푖th transmitter, then we utilize the extended Kalman filter (EKF), which linearizes all non-linear transformations and substitutes Jacobian matrices for the linear transformations in the KF equations. In this case, the state-to- measurement matrix 푯 is calculated as a linearization of the function ℎ(풙푛, 풘푛) at the
− predicted state 풙̂푛
1 0 0 0 0 0
푑ℎ(풙) 0 0 1 0 0 0 ̂ 푯푛 = | = 0 0 0 0 1 0 (9.7) 푑풙 − 푥=푥̂푛 휕푉퐵푖 휕푉퐵푖 휕푉퐵푖 휕푉퐵푖 휕푉퐵푖 휕푉퐵푖 [ 휕푥푡 휕푥̇ 푡 휕푦푡 휕푦̇ 푡 휕푧푡 휕푧̇ 푡 ]
푖 The expression for 푉퐵 as a function of the target’s position and range rate is [89]
푖 푖 푖 푖 (푥푡 − 푥푇) 푥̇푡 + (푦푡 − 푦푇) 푦̇푡 + (푧푡 − 푧푇) 푧푡̇ 푉퐵 = 2 2 2 √(푥 − 푥푖 ) + (푦 − 푦푖 ) + (푧 − 푧푖 ) 푡 푇 푡 푇 푡 푇 (9.8) (푥 − 푥 ) 푥̇ + (푦 − 푦 ) 푦̇ + (푧 − 푧 ) 푧̇ + 푡 푅 푡 푡 푅 푡 푡 푅 푡 2 2 2 √(푥푡 − 푥푅) + (푦푡 − 푦푅) + (푧푡 − 푧푅)
Where
훽푖 푚/푠 Bistatic velocity of target corresponding to 푉푖 = 2푉 cos(훿푖) cos ( ) 퐵 푡 2 the 푖th bistatic pair (푥푡 푦푡 푧푡) 푚 Coordinates of target 푖 푖 푖 (푥푇 푦푇 푧푇) 푚 Coordinates of transmitter (푥푅 푦푅 푧푅) 푚 Coordinates of receiver 172
The partial derivatives of (9.8) can then be found and used in (9.7).
9.4.2 Initialization of Kalman Filter
The system covariance matrix 푷, which maintains an estimate of the errors in the
KF’s estimate of the target’s state, was initialized to the identity matrix 푰. The initial state estimate was set to
푇 풙̂1 = [푥̂푡 0 푦̂푡 0 푧푡̂ 0] (9.9)
Where 푥̂푡, 푦̂푡, and 푧푡̂ are the first location estimate inputs to the KF.
The covariance matrix 푸 models the process noise. The choice of 푸 is based on the white noise constant velocity dynamic model, which is the limiting form of the Singer model (a standard target manoeuvre model) [90], given by [91]
3 2 (훥푡) ⁄ (훥푡) ⁄ 3 2 0 0 0 0 2 (훥푡) ⁄ 2 훥푡 0 0 0 0 3 2 (훥푡) ⁄ (훥푡) ⁄ 0 0 3 2 0 0 푸 = 2 푞 (9.10) (훥푡) ⁄ 0 0 2 훥푡 0 0 3 2 (훥푡) ⁄ (훥푡) ⁄ 0 0 0 0 3 2 2 (훥푡) ⁄ [ 0 0 0 0 2 훥푡 ]
Where 푞 was set to 0.18, the value for a lazy maneuvering target [92].
The covariance matrix of the measurement vector is 푹 and changes with each measurement sample, calculated using the relevant error bound expressions with actual
SNR and estimated target location values
2 ( ) 휎푥푡 푛 0 0 2 ( ) (9.11) 푹 = [ 0 휎푦푡 푛 0 ] 2 ( ) 0 0 휎푧푡 푛 173
Where
2 휎푥푡 푚 Variance of 푥푡 error 2 휎푦푡 푚 Variance of 푦푡 error 2 휎푧푡 푚 Variance of 푧푡 error
If bistatic velocity measurements are used, then
휎2 (푛) 0 0 0 푥푡 2 0 휎푦 (푛) 0 0 푹 = 푡 0 0 휎2 (푛) 0 (9.12) 푧푡 2 ( ) [ 0 0 0 휎푉퐵 푛 ]
9.4.3 Filtered Localization Results
The filtered localization results based on the processing scheme given in Figure
9.10 using the KF (without the bistatic velocity information as input) are shown in Figure
9.11. In Figure 9.11a, each blue triangle indicates the 푥푡 estimate selected amongst the four different localization methods (array-based localization using T1, T2, or T3 and multilateration) at each time index. These selected raw estimates were sent as inputs into the Kalman filter and the magenta line shows the filtered estimates. The filtered estimates are compared with the 푥푡 from the air truth (black dashed line) to calculate the RMSE.
Figure 9.11b and Figure 9.11c show the results for 푦푡 and 푧푡 respectively.
174
Figure 9.11 Filtered target location estimates (KF implementation): (a) x-coordinate; (b) y-coordinate; (c) z-coordinate
The RMSE values are summarized in Table 9.9 for two implementations: the KF implementation using Cartesian position only and the EKF implementation that uses also bistatic velocity.
Implementation RMSE [m] RSS error in
푥푡 푦푡 푧푡 location [m]
KF 48 54 5 72
EKF (with 푉퐵) 49 54 5 73 Table 9.9 Localization results from KF and EKF implementations
175
Comparing the RMSEs of the KF and EKF implementations, the velocity information does not improve the localization accuracy significantly, as seen from the similar RMSE values, so a linear KF implementation could suffice for this scenario. The filtered estimates have accuracy within 100 m for a target at range of 6 km. However, a more extensive analysis through simulation for specific flight dynamics of targets of interest will need to be done in order to optimize localization performance for a given application.
9.5 Conclusions
In this chapter, we verified the equations used in the propagation of error in the bistatic measurements to the final location estimate when the array-based localization method is used. This was done using data collected from a cooperative target over a wide range of bistatic angles. We also demonstrated the achievable accuracy of a passive radar system using data collected from non-cooperative targets, and where their elevation angles were estimated. We performed range multilateration using data from a multistatic radar configuration with multiple transmitters, and compared this with array-based localization. Finally, we introduced a way of fusing estimates from these different localization methods. We selected the inputs to the Kalman filter based on the estimated target location error bound, and demonstrated the achievable accuracy of the filtered location estimates.
176
Chapter 10 Summary and Conclusions
An analysis of the accuracy of array-based localization in DTV-based passive radar and its experimental validation has been made in this dissertation.
Chapter 1 introduced the readers to the problem of passive target localization, with the distinguishing feature of passive radar being that it uses a non-cooperative transmitter located at a separate location as its illuminator of opportunity. This has become a sought-after area of research because of improvements in technology and as elaborated in Chapter 3, due to the increasingly congested spectral environment. While any illuminator can be a candidate for passive radar, the properties of the transmitter and its waveform determines the passive radar’s performance and applications. In Chapter 4, power budget analysis and ambiguity function analysis of the DTV transmitter were performed. Chapter 4 also detailed the core processing needed to treat fixed structures in the DTV waveform, suppress direct signal interference (DSI), and generate range-
Doppler (RD) maps.
In Chapter 3, a review of passive localization literature found that compared to ellipsoid intersection (multilateration) methods, the use of the intersection of an ellipsoid with a bearing vector to find the 3D position of a target has not been investigated in
177 sufficient depth or detail. Thus, the array-based localization method was proposed and elaborated in Chapter 5. It makes use of the good range and Doppler resolutions afforded by DTV to resolve multiple targets, thereby relaxing the requirement of the direction finder to be high-resolution. Thus, an electrically small array of six elements or less could be used in the passive radar system to locate a target at receiver range of 20 km with an accuracy of approximately 250 m, as demonstrated in experiments detailed in Chapter 9.
The array-based localization method requires calibration of an array. In Chapter 7, the procedures of in-field calibration using fixed transmitters and opportunistic air targets were developed. The latter was favored and used in the experiments because it does not introduce errors due to movement of the array between calibration and collection of data.
Multiple known angles provided by the moving target also allows for calibration that is more extensive. Taking the mean of the calculated correction factors (a function of target bearing) and using it for calibration has been shown to give good calibration performance.
The equations to predict the accuracy of the array-based localization method as a function of bistatic geometry were developed in Chapter 8. This was used to compare the localization performance between array-based localization and multilateration for typical geographical distributions of DTV transmitters. It was found that the location accuracies in multilateration do not have a dependence on array and forward-scatter region limitations that array-based localization has. However, for estimation of the target’s altitude, the performance of the array-based method is much better than multilateration because of the limited diversity in transmitters’ locations in altitude.
178
Further observations made in Chapter 9 builds the case for the use of array-based localization in multistatic passive radar (MPR) to increase robustness in performance. In an experiment where simultaneous detections of the target from three transmitters were collected, detections from individual transmitters was as high as 92%, whereas simultaneous detections from all three transmitters was about half, at 46%. In this case, array-based localization could be used almost all the time, while multilateration could only be performed half the time.
A preliminary method of fusing localization estimates from both methods was also shown in Chapter 9, which makes use of the developed accuracy bounds as a metric to select for the best location estimate given the estimated target’s location. This is used as an input into the Kalman filter (KF), which then filters out the effects of noise. The filtered location estimates have an accuracy of less than 100 m for a target at a receiver range of 6 km.
Future work could investigate different designs of architecture for fusing the localization estimates. For example, the Kalman filter can be used to provide the prior estimate of target location required for emitter selection. The estimated target location output from the Kalman filter can also be used further upstream, in the selection of which transmitter to use and dynamically control the center frequency for down conversion, etc.
This paves the way towards dynamically controlled passive radar for optimized localization performance.
179
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Appendix A: Plotting Contours of Constant SNR in Bistatic Radar
The bistatic range equation is given by [1]
2 2 2 푃푇퐺푇퐺푅휆푇휎퐵퐹푇 퐹푅 (A.1) SNR = 3 2 2 (4휋) 푘푏푇푠퐵퐿푇퐿푅푅푇푅푅 Where
푃푇 푊 Power output of transmitter 퐺푇 − Power gain of transmitting antenna 퐺푅 − Power gain of receiving antenna 휆푇 푚 Wavelength of transmitter signal 2 휎퐵 푚 Bistatic radar cross section (RCS) 퐹푇 − Pattern propagation factor for transmitter- to-target-path 퐹푅 − Pattern propagation factor for target-to- receiver path −23 푘푏 = 1.38 × 10 퐽/퐾 Boltzmann’s constant 푇푠 퐾 Noise temperature of receiving system 퐵 퐻푧 Bandwidth of receiving system 퐿푇 − Transmitting system losses (>1) not included in other parameters 퐿푅 푚 Receiving system losses (>1) not included in other parameters 푅푇 푊 Range from transmitter to target 푅푅 − Range from receiver to target
Ovals of Cassini occur when the product of the transmit and receive range is constant, assuming all other parameters are unchanged. To plot these, simplify (A.1) as
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퐾 SNR = 2 2 (A.2) 푅푇푅푅
Where 퐾 is the bistatic radar constant.
Figure A.1 illustrates how to convert the North-referenced coordinates into polar coordinates
Figure A.1 Geometry for converting North-referenced coordinates into polar coordinates
From the figure,
퐿2 푅2 = (푟2 + ) − 푟퐿 cos(휃) (A.3) 푅 4 퐿2 푅2 = (푟2 + ) + 푟퐿 cos(휃) (A.4) 푇 4 Where
푟 푚 Range in Polar coordinates 퐿 푚 Range from transmitter to receiver/baseline 휃 ° Angle in Polar coordinates
This gives the constant transmit and receive range product
2 2 2 2 2 2 2 2 (A.5) 푅푇푅푅 = (푟 + 퐿 /4) − 푟 퐿 cos (휃)
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Substituting (A.5) into (A.2) gives
퐾 SNR = (푟2 + 퐿2/4)2 − 푟2퐿2 cos2(휃) (A.6)
In mathematics, the general equation of a Cassini oval is
(x2 + 푦2 + 푎2)2 − 4푎2푥2 = 푏4 (A.7) Where
푎 − Half the distance between the two fixed points that describe the curve 푏 − Square root of the product of the distances between each of the points and any point on the curve
The general shape of the graph is determined by the values of a and b
• If a < b, the graph is a single loop that is the shape of an ellipse
• If a = b, the graph is the shape of the infinity symbol
• If a > b, the graph is two loops in the shape of two eggs with their narrow ends
facing each other
(A.6) can be rearranged in the form of (A.7) to determine the values of a and b:
퐿2 2 퐿2 퐾 (x2 + 푦2 + ) − 4 푥2 = (A.8) 4 4 SNR where
퐿 푚 Half the distance between the two fixed 푎 = 4 points that describe the curve 푚 Square root of the product of the distances 4 퐾 푏 = √ between each of the points and any point on SNR the curve 퐾 − Signal-to-noise power ratio SNR = 푏4
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The parametric form of (A.8) is then used directly to plot the curves
4 푏 (A.9) 푟 = ±푎√cos(2휃) ± √( ) − sin2(2휃) 푎
푥 = 푟 cos(휃) (A.10) 푦 = 푟 sin(휃) (A.11) Where
푟 푚 Range in Polar coordinates 휃 ° Angle in Polar coordinates 푎 푚 Half the distance between the two fixed points that describe the curve 푏 푚 Square root of the product of the distances between each of the points and any point on the curve
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Appendix B: Array-Based Localization Accuracy for T2 and T3
The array-based localization accuracy results are shown in a grid map display in the x-y plane in order to compare it with multilateration accuracy results in a multistatic radar configuration in Chapter 8. A constant SNR of 21 dB across the range of target x- and y-positions was used and we fix the target altitude at 470 m. The remaining settings are as in Table 8.3.
Figure B.1 shows the results for the bistatic pair made up of transmitter T2 and
Figure B.2 for the bistatic pair made up of transmitter T3.
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(a) (b)
(c)
Figure B.1 SD of target location estimate errors using array-based localization with transmitter T2: (a) x-coordinate, (b) y-coordinate, and (c) z-coordinate
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(a) (b)
(c)
Figure B.2 SD of target location estimate errors using array-based localization with transmitter T3: (a) x-coordinate, (b) y-coordinate, and (c) z-coordinate
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Appendix C: Localization Results from Experimental Datasets for Investigation 1
Investigation 1 makes use of dataset consisting of signals originating from two transmitters scattered from a low-flying aircraft at known locations for a wide range of bistatic angles.
Figure C.1 – Figure C.16 shows the SD of the estimated 푥푡, 푦푡, 푧푡 and 퐴푡 errors and their percentiles for Datasets 1A – 4A, when the WWHO transmitter was used. The settings of which are shown in Table 9.1.
Figure C.17 – Figure C.36 shows the SD of the estimated 푥푡, 푦푡, 푧푡 and 퐴푡 errors and their percentiles for Datasets 1B – 5B, when the WOSU transmitter was used. The settings of which are shown in Table 9.2.
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Figure C.1 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 1A for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
Figure C.2 SD of 퐴푡 error from dataset 1A for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
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Figure C.3 Localization error results from dataset 1A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure C.4 Percentiles for the SD of 퐴푡 from dataset 1A
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Figure C.5 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 2A for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
Figure C.6 SD of 퐴푡 error from dataset 2A for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
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Figure C.7 Localization error results from dataset 2A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure C.8 Percentiles for the SD of 퐴푡 from dataset 2A
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Figure C.9 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 3A for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
Figure C.10 SD of 퐴푡 error from dataset 3A for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
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Figure C.11 Localization error results from dataset 3A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure C.12 Percentiles for the SD of 퐴푡 from dataset 3A
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Figure C.13 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 4A for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
Figure C.14 SD of 퐴푡 error from dataset 4A for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
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Figure C.15 Localization error results from dataset 4A for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure C.16 Percentiles for the SD of 퐴푡 from dataset 4A
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Figure C.17 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 1B for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
Figure C.18 SD of 퐴푡 error from dataset 1B for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
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Figure C.19 Localization error results from dataset 1B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure C.20 Percentiles for the SD of 퐴푡 from dataset 1B
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Figure C.21 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 2B for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
Figure C.22 SD of 퐴푡 error from dataset 2B for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
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Figure C.23 Localization error results from dataset 2B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure C.24 Percentiles for the SD of 퐴푡 from dataset 2B
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Figure C.25 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 3B for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
Figure C.26 SD of 퐴푡 error from dataset 3B for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
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Figure C.27 Localization error results from dataset 3B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure C.28 Percentiles for the SD of 퐴푡 from dataset 3B
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Figure C.29 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 4B for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
Figure C.30 SD of 퐴푡 error from dataset 4B for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
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Figure C.31 Localization error results from dataset 4B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure C.32 Percentiles for the SD of 퐴푡 from dataset 4B
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Figure C.33 SDs of 푥푡, 푦푡 and 푧푡 error from dataset 5B for investigation 1: (a) Theoretical results; (b) Simulated results; (c) Experimental results
Figure C.34 SD of 퐴푡 error from dataset 5B for investigation 1: (a) Theoretical/Simulation results; (b) Experimental results
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Figure C.35 Localization error results from dataset 5B for investigation 1: (a) Percentiles for the SD of 푥푡; (b) 푦푡; (c) 푧푡
Figure C.36 Percentiles for the SD of 퐴푡 from dataset 5B
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