ANALYTIC STUDY OF SPACE-TIME AND SPACE-EREQUENCY ADAPTIVE PROCESSING FOR RADIO FREQUENCY INTERFERENCE SUPPRESSION
DISSERTATION
Presented in Partial Fnlfillment of the Reqnirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Thomas Dean Moore, B.S.E.E., M.S.E.E.
*****
The Ohio State University
2002
Dissertation Committee: Approved by
Professor W. D. Burnside, Adviser
Professor I. J. Gupta Adviser Professor J. T. Johnson Department of Electrical Engineering UMI Number: 3081948
UMI UMI Microform 3081948 Copyright 2003 by ProQuest information and Learning Company. Aii rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.
ProQuest information and Learning Company 300 North Zeeb Road PO Box 1346 Ann Arbor, Mi 48106-1346 ABSTRACT
A study of space-time (STAP) and space-frequency adaptive processing (SFAP) techniques for interference suppression in digital spread spectrum receivers is per formed. STAP and SFAP algorithms combine spatial and temporal filters to suppress radio frequency interference (RFI) extending the nulling capability of a conventional adaptive array beyond its sensor-limited spatial degrees of freedom. The STAP/SFAP configured array’s ability to null multiple RFI without previous knowledge of the in terference characteristics or location while simultaneously receiving desired signals makes these techniques well suited for spread spectrum receivers operating in ill- defined interference environments. In the study, analytic models of STAP, SFAP, and the received noise, interference, and desired signals are used to estimate performance for narrowband, wideband, and mixed-bandwidth interference scenarios using various processor configurations.
For STAP, a detailed analysis on the elfect of interference power and bandwidth on the multi-tap STAP is performed. The eigenvalue distribution of the array covariance matrix is used to characterize the interference effects on performance and consumption of the available space-time degrees of freedom. Additionally, the STAP configuration is studied to understand the elfect of varying the time/tap reference within the STAP steering vector with the finding that the location can significantly affect the system’s phase performance.
11 For SFAP, the effect of interference power and bandwidth is also studied using the eigenvalue distribution of the array covariance matrix. As SFAP is implemented in the frequency domain with the weights for each bin determined independently, the total eigenvalue distribution over all bins is accounted for. Windowing of the input time domain samples is included as a means to limit the degrees of freedom consumed.
Varying the time reference location in SFAP is also studied along with window cor rection of the output samples to determine potential benefits and implementation impacts. Additionally, a detailed study on normalization of the frequency domain weights is presented with the finding that normalization affects processor flexibility in the nulling of individual frequency bins overwhelmed by unsuppressed interference.
The study concludes with a comparison of two computationally equivalent STAP and SFAP processors.
Ill To my wife Teresa for her loving support and encouragement, and to my children Michelle, Joseph, Ashley, and Suzanne for the collective patience
extended to me during this work and the journeys which preceded it.
IV ACKNOWLEDGMENTS
I would like to express my sincere thanks to Professor I.J. Gnpta for sharing his experience and insight on adaptive antenna systems and general guidance throughout this research. The encouragement and discussions of Professors W.D.Burnside, J.T.
Johnson, and R.L. Moses were also helpful in the execution of this work. VITA
January 28, 1952 ...... Born - Charlotte, Michigan
1983 ...... B.S.E.E. University of Arizona
1988 ...... M.S.E.E. Air Eorce Institute of Tech- nology 1975-1983 ...... SSGT, United States Air Force.
1983-1998 ...... Major, United States Air Force
1998-present ...... Graduate Research Associate, The ElectroScience Laboratory, The Ohio State University.
PUBLICATIONS
Research Publications
I.J. Gupta and T.D. Moore, "Space-Frequency Adaptive Processing (SFAP) for Inter ference Suppression in GPS Receivers”. Proceedings of ION 2001 National Technical Meeting, Institute of Navigation, Long Beach, CA, Jan. 2001.
T.D. Moore and I.J. Gupta, “Comparison of SFAP and STAP for Interference Supres- sion in GPS Receivers”. Briefing at the ION 57*^ Annual Meeting and the CIGTF 20*^ Biennial Guidance Test Symposium, Albuquerqe, NM Jun. 2001.
T.D. Moore and I.J. Gupta, “Calibration of Range Probe Data for Stray Signal Analysis”. Annual Meeting and Symposium, Antenna Measurement Techniques As sociation, Philidelphia, PA,pp 228-233, Oct. 2000.
VI T.D. Moore and I.J. Gupta, “Low Frequency Compact Range Analysis” Technical Report 737052-2, The Ohio State University, The ElectroScience Laboratory, Nov. 1999.
I.J. Gupta and T.D. Moore, “Time Domain Processing of Range Probe Data for Stray Signal Analysis”. Annual Meeting and Symposium, Antenna Measurement Techniques Association, Monterey, GA, pp 213-218 Oct. 1999.
FIELDS OF STUDY
Major Field: Electrical Engineering
Studies in: Electromagnetics Professor Johnson, and Professor Munk Communications and Signal Processing Professor Gupta and Professor Moses Mathematics Professor Gerlach
Vll TABLE OF CONTENTS
Page
A b stra c t ...... ii
D edication ...... iv
Acknowledgments ...... v
V i t a ...... vi
List of Tables ...... xi
List of Figures ...... xii
Chapters:
1. Introduction ...... 1
1.1 Space-Time and Space-Frequency Adaptive Processing ...... 1 1.2 History of STAP and S F A P ...... 4 1.3 Current W o rk ...... 10
2. Space-Time Adaptive Processing ...... 15
2.1 Analytic M o d e l ...... 15 2.2 STAP W e ig h ts ...... 26 2.2.1 Maximum SIN K ...... 27 2.2.2 Unconstrained M M SE ...... 28 2.2.3 MMSE Beamformer ...... 30 2.2.4 Constrained MMSE ...... 32 2.3 Performance M e tric s ...... 33 2.3.1 Pre-correlation Metrics ...... 34 2.3.2 Post-Correlation M etrics ...... 35
v i i i 3. Reference Tap Selection in ST A P ...... 42
3.1 Antenna Array and Signal Scenarios ...... 43 3.2 Power Minimization Results ...... 45 3.3 MMSE B e a m fo rm e r ...... 55 3.4 Constrained MMSE ...... 59 3.5 C onclusions ...... 64
4. The Effect of Interference Bandwidth and Power on S T A P ...... 66
4.1 STAP Configuration and Signal Scenario ...... 68 4.2 Eigenvalue Distribution ...... 69 4.3 INR and SINR Performance ...... 73 4.4 Conclusions ...... 80
5. Space-Frequency Adaptive Processing ...... 83
5.1 Analytic M o d e l ...... 86 5.2 Weight E s tim a tio n ...... 90 5.3 SFAP O u tp u t ...... 93 5.3.1 Comments ...... 95 5.4 S u m m a ry ...... 96
6. Normalized versus Un-normalized SFAP Weights ...... 97
6.1 SFAP Configuration and Signal Scenario ...... 98 6.2 Pre-Correlation Performance ...... 100 6.2.1 Wideband RFI Scenario without Bin Nulling ...... 101 6.2.2 Wideband RFI Scenario with Bin N ulling ...... 106 6.2.3 Mixed Bandwidth RFI Scenario ...... 110 6.2.4 Summary ...... 112 6.3 Post-correlation Performance ...... 115 6.4 Conclusion ...... 121
7. SFAP Time Reference ...... 123
7.1 Shifting the Time Reference ...... 125 7.2 Results ...... 133 7.2.1 Comments ...... 139 7.3 C onclusions ...... 144
IX 8. The Effect of Interference Power and Bandwidth on SFAP Performance . 146
8.1 SFAP Configuration and Signal Scenario ...... 150 8.2 Eigenvalue Distribution ...... 151 8.3 INR and SINR Performance ...... 157 8.4 Conclusions ...... 164
9. SFAP and STAP Performance Comparison ...... 166
9.1 Processor Configurations and Signal Scenario ...... 168 9.1.1 Computational Cost Estim ates ...... 169 9.1.2 Signal Scenario ...... 172 9.2 Results ...... 175 9.2.1 Broadside Look D ire c tio n ...... 175 9.2.2 Off Broadside Look Direction ...... 185 9.2.3 Comments ...... 189 9.3 C onclusions ...... 190
10. Conclusions and Future W ork ...... 193
10.1 Future Research ...... 199
Appendices:
A. Computational Estimates ...... 200
A.l Definitions ...... 201 A.2 Joint Space-Time Processor ...... 202 A.3 Joint Space-Frequency P r o c e s s o r ...... 203 A.4 STAP & SFAP Computational Cost Summary ...... 205 A.4.1 S p a c e -tim e ...... 205 A.4.2 Space-frequency ...... 206
Bibliography ...... 207 LIST OF TABLES
Table Page
3.1 Interference s c e n a r io ...... 44
4.1 Interference covariance matrix eigenvalue magnitudes {dB) for 10 dB input INR. Number of Taps = 7 ...... 72
4.2 Interference covariance matrix eigenvalues magnitude {dB) for 50 dB input INR. Number of Taps = 7 ...... 72
4.3 Interference covariance matrix eigenvalue magnitudes (dB) for 10 dB input INR. Number of Taps = 1 ...... 73
4.4 Interference covariance matrix eigenvalue magnitudes (dB) for 50 dB input INR. Number of Taps = 1 ...... 73
6.1 Mixed wide and narrow-band RFI scenario ...... 100
6.2 Quiescent post-correlation performance under power minimization . . 115
6.3 Quiescent post-correlation performance under beam form ing ... 116
7.1 Mixed wide and narrow-band RFI scenario ...... 133
8.1 RFI scenario ...... 151
8.2 Eigenvalue magnitudes (dB) for a single RFI source with 10 dB input INR...... 153
8.3 Eigenvalue magnitudes (dB) for a single RFI source with 60 dB input INR...... 153
9.1 RFI scenario ...... 175
x i LIST OF FIGURES
Figure Page
1.1 Conventional array pattern - spatial filter ...... 3
1.2 STAP/SFAP array pattern - spatial-temporal filte r ...... 3
2.1 Plane wave geometry for incoming wave ...... 17
2.2 Space-time adaptive process block diagram ...... 20
2.3 Flat power spectral density and auto-correlation function Fourier trans form p a ir ...... 24
3.1 Linear array g e o m e try ...... 43
3.2 Pre-correlation performance for mixed RFI bandwidth scenario using power minimization ...... 47
3.3 Adaptive pattern with fixed power inversion weights versus frequency and look direction for mixed bandwidth RFI using power minimization 48
3.4 CCF peak-to-mean and peak-to-peak performance for mixed band width RFI scenario using power m inim ization ...... 49
3.5 Peak delay and carrier phase error for mixed RFI bandwidth scenario using power minimization ...... 50
3.6 RMS error in CCF mainlobe shape for mixed bandwidth RFI scenario using power minimization ...... 51
3.7 Cross-correlation function for four CW plus three wideband RFI sce nario using power m inim ization ...... 52
Xll 3.8 SINR and CCF PMR versus look direction using power minimization 53
3.9 CCF PFR and RMS error versus look direction using power minimization 54
3.10 CCF peak delay and carrier phase error versus look direction for power minimization ...... 54
3.11 [Output INR and SINR versus look direction for MMSE beamformer 56
3.12 CCF peak-to-peak and peak-to-mean performance versus look direc tion for MMSE beam fo rm e r...... 58
3.13 Cross-correlation function in broadside and —80° direction for mixed bandwidth RFI scenario for MMSE beamformer ...... 59
3.14 CCF peak delay and carrier phase error versus look direction for MMSE b eam fo rm er ...... 60
3.15 CCF RMS error versus look direction for MMSE beam former .... 61
3.16 Output INR and SINR performance versus look direction for con strained M M SE ...... 62
3.17 CCF peak-to-peak and peak-to-mean performance using constrained MMSE weight adaptation ...... 62
3.18 CCF peak delay and carrier phase error versus look direction using constrained MMSE weight adaptation ...... 63
3.19 CCF RMS error versus look direction using constrained MMSE weight a d a p ta tio n ...... 64
4.1 Quiescent array pattern at 2004 M H z ...... 69
4.2 Eigenvalue distribution and output INR for seven-tap STAP in the presence of a single RFI source ...... 75
4.3 STAP output SINR in the presence of a single RFI source ...... 76
X lll 4.4 Eigenvalue distribution and output INR for a seven-tap STAP in the presence of three RFI sources ...... 77
4.5 STAP output SINR performance in the presence of three RFI sources. 78
4.6 Eigenvalue distribution and output INR performance for a seven-tap STAP in the presence of five RFI sources ...... 79
4.7 Output SINR performance in the presence of five RFI sources ...... 80
5.1 Space-frequency adaptive process block diagram ...... 87
6.1 Linear array geometry for SFAP configuration ...... 99
6.2 Output INR and SINR for wideband scenario using power minimization 103
6.3 Output SNR and SIR for wideband scenario using power minimization 103
6.4 SFAP beamformer output INR and SINR for wideband RFI scenario 105
6.5 SFAP beamformer output SNR and SIR for wideband RFI scenario . 105
6.6 Power minimization INR and SINR performance for the wideband RFI scenario using bin nulling ...... 108
6.7 Power minimization SNR and SIR performance for the wideband RFI scenario using bin nulling ...... 108
6.8 Beamformer INR and SINR performance for the wideband RFI sce nario using bin nulling ...... 109
6.9 Beamformer SNR and SIR performance for the wideband RFI scenario using bin nulling ...... 109
6.10 Power minimization INR and SINR performance for mixed RFI sce nario using bin nulling ...... 113
6.11 Power minimization SNR and SIR performance for mixed RFI scenario using bin nulling ...... 113
XIV 6.12 Beamformer INR and SINR performance for mixed RFI scenario using bin nulling ...... 114
6.13 Beamformer SNR and SIR performance for mixed RFI scenario using bin nulling ...... 114
6.14 CCF peak-to-mean ratio for mixed RFI scenario using bin nulling . . 117
6.15 CCF peak-to-peak ratio for mixed RFI scenario using bin nulling . . 119
6.16 RMS error in CCF mainlobe shape for mixed RFI scenario using bin n u llin g ...... 119
6.17 CCF peak delay for mixed RFI scenario using bin nulling ...... 120
6.18 CCF carrier phase error at peak location for mixed RFI scenario using bin nulling ...... 120
7.1 Output time samples for one data block ...... 125
7.2 Quiescent output signal power and SINR versus window correction . . 129
7.3 Quiescent output signal power and SINR using the 40*^ output index as reference versus window correction ...... 130
7.4 Quiescent output signal power and SINR versus bin nulling ...... 131
7.5 Quiescent output signal power and SINR using the 40*^ output index as reference versus bin nulling ...... 132
7.6 Output signal power and SINR versus bin nulling for a single WB RFI s o u r c e ...... 134
7.7 Output signal power and SINR versus bin nulling for a single WB RFI source with time reference at 40*^ index ...... 135
7.8 Output signal power and SINR versus bin nulling when three WB RFI sources are present ...... 136
7.9 Output signal power and SINR versus bin nulling for three WB RFI sources with time reference at 40*^ in d e x ...... 137
XV 7.10 Output signal power and SINR versus bin nulling when four WB RFI sources are present ...... 140
7.11 Output signal power and SINR versus bin nulling for four WB RFI sources with time reference at 40*^ in d e x ...... 140
7.12 Output signal power and SINR versus bin nulling when four WB plus three CW RFI sources are present ...... 141
7.13 Output signal power and SINR versus bin nulling for four WB plus three RFI sources with time reference at 40*^ in d e x ...... 142
7.14 SFAP frequency domain weights for wideband only and mixed band width scenario ...... 143
7.15 Block processing with 0 % overlap ...... 143
7.16 Block processing with overlapped data blocks ...... 143
8.1 Eigenvalue spread for CW RFI ...... 155
8.2 Eigenvalue distribution and output INR for a single 8 MHz bandwidth RFI source ...... 156
8.3 Eigenvalue distribution and output INR for a single RFI source of various bandswidths ...... 159
8.4 SFAP output SINR for a single RFI so u rc e ...... 160
8.5 Eigenvalue distribution and output INR for three RFI sources of vari ous bandwidths ...... 161
8.6 SFAP output SINR for three RFI sources of various bandwidths . . . 161
8.7 Eigenvalue distribution and output INR for five RFI sources of various b a n d w id th s ...... 163
8.8 SFAP output SINR for five RFI sources of various bandwidths .... 163
9.1 STAP computation e s tim a te ...... 172
XVI 9.2 SFAP computation estim ate ...... 173
9.3 SFAP computation estimate versus data overlap rate { a ) ...... 174
9.4 STAP and SFAP eigenvalue distribution versus the number of interfer ence sources ...... 178
9.5 Output INR versus the number of interference sources ...... 178
9.6 Output SINR and CCF PMR versus the number of interference sources 179
9.7 STAP and SFAP adaptive pattern when four wideband sources are p re se n t ...... 181
9.8 STAP and SFAP adaptive pattern when four wideband plus 5 CW sources are present ...... 183
9.9 CCF PPR and mainlobe RMS error versus the number of interference so u rces ...... 184
9.10 CCF peak delay and versus the number of interference sources .... 185
9.11 STAP and SFAP eigenvalue distribution versus the number of interfer ence sources evaluated for the 45° look direction ...... 186
9.12 Output INR versus the number of interference sources evaluated for the 45° look direction ...... 188
9.13 Output SINR and CCF PMR versus the number of interference sources evaluated for the 45° look direction ...... 188
9.14 CCF PPR and mainlobe RMS error versus the number of interference sources evaluated for the 45° look direction ...... 189
9.15 CCF peak delay and versus the number of interference sources evalu ated for the 45° look direction ...... 190
x v ii CH APTER 1
INTRODUCTION
1.1 Space-Time and Space-Frequency Adaptive Processing
A conventional adaptive array combines a collection of sensors each equipped with an adjustable weight to place a null in the direction of each undesired signal while maintaining gain in other look directions for a desired signal or signals. By sampling the received sensor voltages and estimating the covariance between the various sen sors, the weights are adjusted to automatically steer the nulls and form the desired response in reaction to changes in the relative directions of arrival. Such an array is limited to L - 1 narrowband spatial nulls or degrees of freedom for suppressing the interference where L is the number of sensors. The single weight per element does not have the ability to change with frequency. An adaptive filter can be placed after the sensor array to supplement the “spatial filter” with temporal filtering. However, when accomplished in a serial fashion the combined spatial-temporal filter is still susceptible to interference if the L-1 spatial degrees of freedom are exceeded. In a space-time adaptive processor (STAP) or space-frequency adaptive processor (SFAP) an adaptive filter is placed behind each sensor element of the array and the filter co efficients determined “jointly” from the sample covariance matrix estimated from the various array sensor filter tap voltages. As in the conventional array, the weights are
1 adjusted to minimize the output power subject to various constraints for the desired signals. However, the spatial and temporal degrees of freedom are assigned jointly.
The result is a frequency dependent response which extends the combined degrees of freedom, theoretically, to LN-1. The combined effect is illustrated in Figures 1.1 and
1.2 where four CW signals are incident on a hve-element array. The array is assumed to operate at a center frequency of 2004 MHz over a minimum 20 MHz bandwidth to receive a 20 MHz bandwidth signal. In Figure 1.1, the left hand plot shows the array pattern at the center frequency when the single-tap conventional array is used. Four distinct nulls are formed in the directions of the interference. However, the effect of the non-frequency dependent weights is observed on the right where the spatial nulls attenuate the energy over all the frequencies within the receiver’s bandwidth which originate from the RFI direction of arrival. A white ’X’ is placed at the angle and center frequency of each RFI source in the plot on the right. Figure 1.2 shows the conceptual result for the same array of sensors and RFI sources when conhgured as a STAP or SFAP processor. The joint spatial-temporal nulls for each of the four
RFI sources are now localized in space and frequency. Theoretically, up to LN - 1 spatial-temporal nulls can be generated significantly enhancing the capability of the
L-element sensor array. In practice, the realization is less as RFI sources with finite bandwidths consume multiple degrees of freedom in the multi-tap STAP. Also, the computational burden of the joint processor on the adaptive system is much more significant.
Over the last decade, advances in digital signal processor speed and capacity have made the practical realization of such a system possible. Moreover, the same advances RFI 40 , -30 , -10 , -1 0 -10
-20
-3 0 -30
-4 0 -40
-5 0 -90 -50 45 1994 1998 2002 2006 2010 2014 0 (D eg ) f (MHz) a) Array Pattern at 2004 MHz b) Array Pattern versns Freqnency
Figure 1.1: Conventional array pattern or “spatial” filter in the presence of four CW RFI sources at center frequencies of 2003.1, 1999.3, 2006.6, and 1995.1 MHz are incident from —40°, —30°, —10°, and 0° respectively.
f
1994 1998 2002 2006 2010 2014 f(MHz)
Figure 1.2: Array pattern for the five element array configured as a STAP or SFAP processor in the presence of four CW RFI sources at center frequencies of 2003.1, 1999.3, 2006.6, and 1995.1 MHz are incident from —40°,—30°,—10°, and 0° respec tively. Weights are calculated using 128-bin SFAP w/Blackman window. in digital signal processing have also increased interest in spread spectrum commn- nication and navigation systems in which multiple transmitters and receivers can simultaneously share the same spectrum without experiencing mutual interference.
The popularity of these systems is evident with the growth in code-division mnltiple access systems for mobile communication/ navigation applications. The mobility of these systems, in turn, increases the likelihood of operation in signal environments which are not well defined with respect to either interference characteristics or desired signal direction of arrival. For systems in which reception must be insured, STAP and SFAP offer significant performance enhancement.
1.2 History of STAP and SFAP
In the past, extensive analysis was conducted on conventional arrays or “single tap” STAP. In 1967, Widrow demonstrated a least mean squares (LMS) approach to adapt array weights with no requirement for knowledge of either the desired or inter ference signal’s direction of arrival (DOA) [1]. Widrow also proposed the use of analog tapped delay lines behind each antenna element to provide broadband performance for the desired signal. The broadband performance offered by the tapped delay lines was motivation for Grifhth’s work in 1969 [2] which concentrated on iterative least squares processing versus a direct calculation approach because of the limit imposed by the computational capabilities of the time. Frost [3] derived a constrained LMS algorithm for the tapped delay line adaptive array in order to maintain a chosen fre quency response towards a desired look direction while minimizing the output power over all remaining directions. Brennan and Reed [4] demonstrated the direct method of computing the steady state weights for an adaptive array with a tapped delay line using the input signal-plus-interference-plus-noise array covariance matrix estimated from the input sample tap voltage sequence. Later Reed et al demonstrated that rapid weight convergence is obtained using the interference-plus-noise sample covari ance matrix [5]. The latter is important not only for the rapid convergence of the weight solution but for applications in which the desired signal level is weak or lower than the system thermal noise.
While interest in tapped delay lines for improving bandwidth performance was established early in the history of adaptive array, computational power limited devel opment. Much of the published research concentrated on single-tap or conventional adaptive array performance. Gupta described adaptive array performance in terms of the conventional or non-adaptive array performance and design parameters [6].
Gupta included the effect of mutual coupling on conventional adaptive array perfor mance in [7] and in [8] described the effect of interference power and bandwidth on single-tap adaptive array performance. Through the use of the eigenvalue distribution for the interference plus noise covariance matrix, Gupta analyzed the effect on the array’s output signal-to-interference-plus-noise and interference-to-noise performance versus interference power and bandwidth. In [9], Zatman also utilized the eigenvalue distribution of the array spatial covariance matrix to study the effect of interference power and bandwidth on STAP performance for doppler radar. As in Gnpta’s study
[8], Zatman used the relationship between the number of eigenvalues associated with an interference source as a function of power and bandwidth to relate the performance effect on multi-tap STAP performance in Doppler radar applications. Later, Zatman used the eigenvalue distribution of the array spatial covariance matrix to define the narrowband limit of an array [10]. However, in each of these studies, the eigenvalue distribution of only the spatial covariance matrix was nsed to describe performance.
The joint spatial-temporal eigenvalues of the compete multi-tap STAP was not used in the interference characterization or performance analysis. In recent studies, the eigenvalue distribution based on the spatial covariance matrix of a STAP application has been used to study STAP performance in the presence of multipath. In [11], the performance of a multi-tap STAP configured array mounted on an aircraft platform was reported. In the stndy, the presence of a second eigenvalue above the level of noise was used as a threshold for verifying the presence of multi-path, thus demon strating the successful performance of STAP in a multi-path environment. Again, only the spatial covariance matrix was used in the eigenvalue distribution analysis.
Furthermore, no baseline analysis was provided to demonstrate that the bandwidth of the signal was responsible (or not) for the presence of the second eigenvalue. As
Zatman reported in [10], the second eigenvalue could be due to the signal bandwidth.
While it is probable that multi-path was indeed present, this conld not be verified from the paper.
The tap delays of early publications [1, 2] utilized analog tapped delay lines with delays on the order of the period of the RF center frequency ^ j-. In 1988, Compton pnblished an analysis of a two-element array with tapped delay lines in which he demonstrated that the bandwidth performance of the array could be improved by adding additional taps provided the delay was shorter than the inverse of the signal bandwidth, Tg < demonstrating that tapped delay lines implemented via digital tapped delay lines could improve adaptive array performance. In a companion pa per, Compton showed the eqnivalence of time and freqnency domain approaches for multi-tap adaptive array processes [12]. Given the same number of taps and frequency
6 bins and equal sample rates, Compton showed that the two processes are eqnivalent yielding the same performance. Compton’s implementation of the frequency domain processor was as an optimal process. However, implemented in its optimal form, the freqnency domain process offered no advantage over the time domain processor. In
1995, Godara presented a snb-optimal frequency domain processor offering signih- cant compntational savings [13]. In the sub-optimal approach the frequency domain weights are determined one bin at a time independent of the other frequency bins. To determine the frequency domain weights for each of the N bins solution of an L x L matrix N times is required where L is the number of antenna elements. For an N-tap
STAP, solution of an LN x LN matrix is required for the direct solution method.
As the computational burden to solve the matrix goes up as Ç){LN)^, the potential savings are significant. While performance for the narrow band frequency domain approach was snboptimal, Godara proposed that performance could outperform the broadband time domain performer if computational equivalence was the benchmark for eqnivalent processors. Godara’s work in [13] and [14] established the baseline de scriptions and theory for a narrowband space-frequency adaptive processor complete with the equations necessary to determine the frequency domain covariance matrix given the time domain covariance matrix. While Godara’s work established the base line, the model did not include windowing of the time domain samples prior to the freqnency domain translation nor was it clear that SFAP performance could vary for different output samples at the array output.
Interest in STAP and SFAP increased in the late 90’s with the growth of spread spectrum communications and navigation applications as well as the continued im provements in digital signal processing capacity and compntational speed. In these
7 applications both the receiver and signal transmitters may be mounted on mobile platforms. Additionally, the spread spectrum system is likely to have lower signal strength at the receiver due to the spreading of the signal frequency spectrum. The signal power may also be lowered to limit interference from the spread spectrum sig nal to other signals in the same band. As a result, the spread spectrum receiver may be vulnerable to unintentional interference sources from nearby television sta tions, radar, and other RF sources which may be co-located on the same platform as the receiver [15, 16]. Thus, the need for broadband performance and simultaneous interference suppression in an environment lacking characterization of the incident interference environment and desired signal DOA are necessary.
Faute and Vaccaro have also contributed to the theoretical basis and understand ing of SFAP and STAP with studies comparing processor performance in wideband and multipath environments [17]. Although not generally available to the public, their work in [18] included the description of a narrowband SFAP with windowing. How ever, their performance analysis relied on the average output signal and interference power ratios computed in the frequency domain. In an N-bin SFAP, N-output time domain samples are available. However, due to the non-infinite support given the dis crete or fast Fourier transform by the N-samples per element for an N-bin transform, the output performance may vary for different output time samples. Also, their work did not include window correction of the output samples. As a result, their predic tions may underestimate performance. In 1999, Gupta [19] incorporated windowing into a SFAP simulation and found that performance was significantly improved for scenarios which include narrowband interference. Recognizing that performance for the frequency domain processor can differ at the various output time samples, Gupta based his performance predictions on the central sample of the N available. While
Faute and Vaccaro's work predated Gupta’s incorporation of the window, Gupta’s work appears to be the first time SFAP performance predictions were based on the optimum output sample. More recently, SFAP performance using various windows has been demonstrated and reported by Gupta and Moore in [20, 21]. In Gupta and
Moore’s analysis, performance estimates were based on the optimal output sample in order to determine the best performance which is possible. However, in a practical application, one would like as many of the N available samples as possible. In a performance analysis of windowed SFAP versus the output time sample, Gupta and
Moore demonstrated the variation in performance which can occur [22].
In STAP and SFAP various constrained and power minimization algorithms can be used to adjust the processor weights. A steering or correlation vector is required in the direct computation approach. Within the steering vector a time reference is selected. Some of the reported research uses the first tap of the reference element as reference [23]. In others, the center tap was selected [17]. Neither provide justification for the choice. In SFAP, one can also shift the time reference to provide output time samples based on an equivalent STAP process as reported in Godara’s work of [14].
However, in Godara’s work analysis or results describing the effect on performance versus the output time samples which result from the time reference shift needed to implement the equivalent process is not provided.
While much work has been conducted on conventional adaptive array radar, and recently, research on multi-tap STAP and SFAP has started to become available, there is still much to understand about how these two competitive systems perform. How are they influenced by interference power and bandwidth? Can additional per formance be gained by manipulation of simple weight constraints without impacting the computational or processor implementation? A broad band response in one do main transforms to a localized phenomena in the other. While the various constraint algorithms employed in STAP can be transformed to the frequency domain for use in SFAP there may be unintended consequences. In the current work, the effect of interference power and bandwidth on the STAP and SFAP processes are investigated.
The time reference shift is addressed for both processors and a performance assess ment conducted. Also, in SFAP, the transformed power minimization and constrained beamformer weight adaptation approaches are analyzed In their un-normalized and normalized forms. A comparison analysis of the two processors is also performed which is based on computationally equivalent implementations to understand the performance which can be obtained under similar operating constraints and identical interference scenarios.
1.3 Current Work
STAP and SFAP are studied using analytic models of the processor configura tions and desired, noise, and interference signal scenarios. Performance metrics for the STAP and SFAP models are designed to measure both signal and Interference levels at the processor output as well as predict the resultant processor effect on the receiver cross-correlation between a reference and received signal. The goal of the research is to understand the effects of interference power and bandwidth on perfor mance, analyze and recommend processor configuration based on performance, and ultimately, compare their performance for mixed narrow and wideband interference.
1 0 In a conventional single-tap adaptive array only the antenna reference element needs to be determined. The tap reference is set by defanlt. In mnlti-tap STAP, one needs to consider a reference tap in addition to the reference antenna element. As dis cussed previously, various researchers have used different tap locations as a reference without analysis of the effect on STAP performance. In this work, the reference tap selection is analyzed and shown to affect processor performance. In the past, perfor mance emphasis was often on maximizing the output signal-to-interference-plus-noise power ratio. However, as adative array sytems are considered for use in spread spec trum receivers, processor induced distortion of the signal must also be considered.
The results will show that the reference tap selection primarily affects the signal distortion induced by the processor.
The effect of interference power and bandwidth on conventional adaptive array performance was studied in the past. Recently, the effect of interference power and bandwidth on STAP performance for doppler radar applications was studied by Zat- man [9]. However, Zatman’s analysis used only the eigenvalue distribution of the spatial covariance matrix to characterize the interference effect. In the current work, the eigenvalue distribution of the complete spatial-temporal eigenvalue distribution is used to characterize the interference and relate STAP performance to the degrees of freedom consumed by the RFI. It is found that the bandwidth of the interference relative to the sampling rate of the STAP is as important a factor as the fractional bandwidth relative to the RF center frequency for STAP performance. It is shown that a signal meeting Zatman’s definition of a narrowband signal (a single eigenvalue or single spatial degree of freedom consumed) can consume multiple spatial-temporal
II degrees of freedom in the STAP and is dependent on the interference power and band width relative to the sample frequency. More significantly, it is also shown that the mnlti-tap performance is also dependent on the interference power and bandwidth.
However, the degradation in output performance is minimal, given that the interfer ence is suppressed, in comparison to the conventional adaptive array performance.
In SFAP, one can also determine an equivalent time reference via the Fourier transform. Godara used such a relationship in [14] to generate the array output for equivalent time samples in STAP and SFAP. However, SFAP performance varies over the output time samples available from the N-length SFAP processor. Godara’s work did not include an analysis of the effect of shifting the time reference on SFAP performance. Nor did Godara’s work include windowing of the time domain samples.
The current work incorporates a window and window correction of the output time samples. The effect of varying the time reference on SFAP performance is studied at length. Also, window correction, given the time reference shift, is implemented and demonstrated to significantly increase the number of good quality time samples available in a given data block. It is shown that shifting the time reference changes the location of good and poor signal quality output samples within the N-length block.
As a result, the time reference has a signihcant effect on the amount of overlap of adjacent data blocks which is needed to avoid passing poor signal quality samples to a receiver. The computational burden imposed by SFAP is increased as data overlap increases. Thus, understanding the consequences of time referencing in SFAP is important to the current research in SFAP.
As in the time domain processor, various approaches for adapting the array weights in the frequency domain can be implemented in SFAP. In narrow-band SFAP, the
1 2 frequency domain weights are determined for individual frequency bins independent of adjacent frquency bins. This gives SFAP additional flexibility in that the weights for individual bins can be shut ofl" when the associated degrees of freedom for the bin are exceeded by the incident interference. In this study, normalized and un-normalized versions of the power minimization and constrained beamformer weight algorithms are analyzed. It is found that the normalized and un-normalized implementation of the constraint affects processor flexibility limiting the number of interference sources which can be suppressed when the sources overlap portions of the frequency spectrum.
Thus, the issue should be of interest in current SFAP design considerations.
Narrowband SFAP can be thought of as N narrowband conventional arrays op erating in adjacent frequency bands. Thus one is tempted to extend the bandwidth analysis of the past directly to the SFAP. However, the narrowband limit of each of the N processors is not hardlimited by a front end filter. Instead, the Fourier transform is used to translate the time domain samples to the frequency domain.
Thus, narrowband energy which ideally would be assigned to a single bin can “leak” into adjacent bins. Thus, in SFAP a CW source may consume multiple degrees of freedom whereas in STAP only a single degree of freedom is consumed. Windowing is implemented to limit the effect. Still, multiple degrees of freedom are consumed.
However, the computational efficiency of SFAP allows one to increase the number of frequency bins to compensate. The total eigenvalue distribution over the frequency bins is analyzed together with the SFAP performance to understand the effect of the interference power and bandwidth. Analysis of the effect of the interference power and bandwidth on SFAP is not available in the literature. Thus, the results of the current work should contribute useful information to the field.
13 The study concludes with a performance comparison of a STAP and SFAP pro cessor. The processors are selected to present an equivalent computational burden to the system. Equations are provided in the appendix to support the computational es timate. Then, based on recommendations from the preceding chapters, the processors are configured with the appropriate time references, window correction, weight nor malization, and other processor parameters and the comparative performance analysis conducted.
The rest of this dissertation is organized as follows. Chapters 2-4 describe the
STAP analytic model, reference tap selection in STAP, and the effects of interference power and bandwidth on STAP. Additionally, performance metrics used for both
STAP and SFAP are described in Chapter 2. SFAP analysis, including a complete description of windowing, time reference shifts, and window correction are given in
Chapters 5-8. Chapter 5 provides a description of the SFAP model. Chapters 6-8 contain analysis of the time referencing, window correction, weight normalization, and the effect of interference power and bandwidth on SFAP performance. Chapter 9 concludes the study with a comparative analysis of computationally equivalent STAP and SFAP in a mixed narrow and wideband interference environment.
14 CH APTER 2
SPACE-TIME ADAPTIVE PROCESSING
The STAP analytical model, weight adaptation approaches used, and performance metrics are described in this chapter. The STAP analytic model is based on earlier work by Gupta and Compton in [6, 8, 23, 24, 25, 20] and is provided here for complete ness. The STAP model assnmes that interference and desired signals are mutnally
uncorrelated independent, zero mean, and wide sense stationary random processes.
The voltage at the array taps also contains a thermal noise component which is as
sumed to be a zero mean wide sense stationary Gaussian random process mutually
uncorrelated with the received interference and desired signals.
First, the theoretical description of the STAP model and equations describing the interference, noise, and desired signals are provided. Next, various methods for
adapting the weights are presented. Then metrics for determining STAP performance
are described.
2.1 Analytic Model
Consider an adaptive array with L antenna elements. Each antenna element is followed by an N-length tapped delay line or digital buffer. The voltage output from
each element is converted to baseband, sampled every % seconds, and stored in the
15 digital buffer. The output of each tap is multiplied by a complex weight and summed to form the array output, y it). Let Xinit) represent the complex voltage sample at the tap of the antenna element. Let x{t) represent the complex voltage sample at output of the reference antenna element. With respect to the reference element, the voltage at an arbitrary element and tap number is expressed as
where,
To = the tap delay or time between samples,
Ti{0, (j)) = directional dependent inter-element time delay,
n = tap number.
To compute the directional delays for the various sensors relative to the origin for an incoming signal, consider a plane wave incident at an angle 6 and (j) as shown in
Figure 2.1.
Let the signal be represented by
= (2 .2) where A is the signal amplitude, u = 2nf is the frequency in radians, t is time, [3 is the wave velocity vector, and f is the position vector from the origin. Note that [3 and r can be decomposed into their Cartesian components, ie
= ;2(—Æsin^cosf^ —ÿsin^sin;^ —zcos^), and (2.3)
1 6 Figure 2.1: Plane wave geometry for incoming wave.
(2.4) where [3 = |/3| and {x, y, z) is the Cartesian coordinate of the point in space to determine the phase relative to the origin. Let, the position of the sensor be denoted by Zk). Then one finds that
- ft = —^ (æt sin ^ cos (^ + sin ^ sin (^ + z* cos , (2.5) and the incident field is
E = Qj[<^t-{-l3(xkSmecos(t>+ykSmesin(t>+ZkCose)}] (2 6)
^ [uj t+13{xk sin 0 cos (p+Vk sin 0 sin (p+Zk cos 0)] ( 2 7 )
17 The phase of the incident field at the element relative to the phase at the origin
IS
%A( (^ ) = /) (æt sin ^ cos (^ + sin ^ sin (^ + cos . (2.8)
Assuming free space incidence, (5 can be expressed as ^ where c ( c = ) is the speed of light in a vacuum. Define the directional delay for the element relative to the origin as
r( 0 ) = - {xh sin 9 cos The incident field in terms of r is then % ,T ) = — E(t + t ( 0, (/))) = E(t), (2.10) where t = t + t. With respect to a STAP conhguration, the total delay for at the element and the 77,*^ tap relative to the origin is then t = t + Tk{9^ (/» ) — (n — 1) To, (2.11) where n is the number of delays and To is the delay. The array output at time t, y{t), is formed from the weighted sum of tap voltages such that 18 L N = (2.12) 1=1 n = l where gin is the adaptive complex weight for the element and n*^ tap. Representing the tap voltages and weights as column vectors, one hnds that y = x^g = g^x, (2.13) X = [xii ... XiM X21 ... X2iv,... xlnY-> Eind (2.14) g = [s'il ••• 9iN 921 ••• 92N,-- - 5'ltv ]^. (2.15) Lower case bold is used here to indicate vectors and upper case bold is used to indicate matrices. The time dependence, t, in these and subsequent equations is inferred to implicitly via the tap and element subscripts. It is assumed that time delay is with respect to the most current sample at time t. Note that the STAP processor block diagram shown in Figure 2.2. While specific weight optimization approaches will be discussed later, for now assume that the weights have been adapted to minimize the noise and interference power at the array output and are available to the STAP processor. In the analytic model, one does not have access to the individual voltage samples. Instead, the array covariance matrices as derived from known power spectral density functions for the various signals will be used to estimate the expected power at the array output. Given the weight vector in (2.15), the expected power at the output of the array can be calculated so that p = !/*!/}■ = ^q(gq)*(qg)}. 19 ^2(1) - 4 I, (t-(N-l)V -(N-I)l,^ S2N Figure 2.2: Space-time adaptive process block diagram. 2 0 = = ^ g ^ $ g • (2.16) where $ is the received signal covariance matrix. In this study, it is assumed that the array receives various interference and desired signals which are independent and mutually uncorrelated. The voltage received by each element also includes an independent Gaussian noise component due to the thermal noise in the receiver front end. The various element signal voltages are assumed zero mean and wide sense stationary. Assuming linear operation by the receiver, the voltage vector can be separated into the desired, interference, and noise components such that x = Xrf + Xi + x„, (2.17) where Xj, x,, and x„ are the desired, interference, and noise voltage vectors, re spectively. The signal covariance matrix, can be decomposed into the individual desired, interference, and noise covariance matrices by expanding $. Thus, one finds that $ = E{x*x^} - E{(xj + x,j 4- x„)*(x^ + Xj 4- x„)^}, = -E;{x^x^} + + -Bl{x^xn + + - - -, = + $ , + $». (2.18) As the various signals are mutnally uncorrelated and zero mean, the cross terms in the expansion go to zero. The covariance matrices are both Herniitian and positive definite, a property which will be useful later. To see this, Rrst note that the covariance m atrix $ is Hermitian, i.e. $ = 2 1 $ = E{xx^}, (2.19) = (E{xx^})^ = E{ [xx^]^}, (2.20) = E{ [(x^)V]} = E{xx^} = (2.21) =4> $ is Hermitian. (2.22) $ is also positive semi-definite. Let z be any complex vector of length M. Then, > 0 if $ is positive semi-definite.[26, pp233 — 234] (2.23) z^$z = z^E{xx^}z = E{z^xx^z} (2.24) = E{(z^x)(x^z)}^E{(f*(f}^E{|d|^}, (2.25) d = x^z, a complex scalar, and (2.26) E{|d|^} —> E-f ^ > 0 ^ $ is positive semi-definite (2.27) lm=l J We also observe in this result that (2.28) Lm=l J lm=l J —)■ E I. ^ {Zjn Xm){Xm ^m,)* j —t j ^ (2.29) lm=l } lm=l ) = tr |$ [ z z ^ ] |, (2.30) where “tr” is the trace operator. The covariance matrices used by the STAP analytical model are based on a flat power spectral density (see Figure 2.3 ) and its associated auto-correlation function. The auto-correlation function, R{t), corresponding to the flat power spectral density function, S'(/), is 22 f /-/o+V R(r) = I = p sinc(7T A / r)ej2T/oT (2.31) with. < / - /o < (2.32) 0, Otherwise where S{f) and R{t) are Fourier transform pairs, A / is the signal bandwidth, P is the signal power, and /o is the center frequency. Consider the interference covariance matrix, due to the RFI source. In terms of the tap voltages. :c];2a;i2 = E{x:xf} = (2.33) . _ The LN X LN covariance matrix can be partitioned into sub-matrices of size N x N between associated taps of two antenna elements: ■ ^*22 ^^ = (2.34) ------ - ^iLl ^^LL - The interference sub-matrices are found by considering the and element signal vectors and x,j : 23 ACF = P sinc(7t BW i) X 10 ^ Flat PSD, Af = 20 MHz -15 Lobe width - 2/BW -20 0.8 - 6.6 ^ ------Af ------► 0.6 P/Af we — -30 0.4 -35 0.2 -100 100 200 -20 -15 -1 0 -5 0 5 10 15 20 X (ns) f(MHz) Figure 2.3: Flat power spectral density (PSD) and auto-correlation function (ACF) transform pair for an SNR level of -20 dB and signal bandwidth of 20 MHz = £ { x - .x '} (2.35) Within the sub-matrix, the (m, n*^) entry is the covariance between the and tap voltage of antenna elements k,l. Based on the directional dependent inter-element delays, ti, and tap delays, Tq, the (m, entry is i^iki]mn = E{x*(t + Tk — {m — l)To) Xi(t + Ti — {n — l)To)}, = — T~k) + {m — ïi)Tq}, (2.36) where Ri is the anto correlation fnnction for the RFI signal. Next, substituting the brick-wall auto-correlation function from (2.31), - Tt) -b (m - Tz)?))} = 8inc(7rA/T)e^^''^'"' (2.37) 24 where, T ^ (Tf(g, <;6) - Tt(^, <6)) + (/» - (2.38) Noise is modeled as a brick-wall or flat PSD with constant power density Nq over the bandwidth of the receiver. The thermal noise is uncorrelated between elements. As a result, the off-diagonal snb-matrices in the covariance matrix are zero. ^«11 0 0 ■ 0 ^n-22 0 0 (2.39) 0 0 0 - The elements of the matrix are fonnd from = -Rn{(ni " M):Zo} = 8mc(7T /)n(TM - M)2o) (2.40) Where is the noise bandwidth, Tq is the tap delay spacing, and the inter-element delays are zero. In the study, the noise bandwidth is assumed equal to the inverse of the tap delay. The total covariance matrix is estimated from the snm of the desired, interference, and noise covariance matrices. In this study, the received signal power for the desired signal is assumed to be several orders of magnitude less than the system thermal noise. Unless noted otherwise, the contribution from the desired signal covariance matrix is ignored in the weight calculations and the total covariance matrix is estimated from 25 the sum of the interference and noise covariance matrices The directional dependent inter-element delays are based on the desired and in terference signal direction of arrival. Assuming plane wave incidence from the (0, (j)) direction, Ti = — sin(0) cos(0) 4- — sin(0) sin(0) 4- — cos(0), (2.41) where the position of the element in Cartesian coordinates is designated by {xi, yi, zi). Note that 6, (j) are the usual directional angles with respect to the z and x axis respec tively, and “c” is the speed of light in a vacuum. In the model, ideal antenna elements are assumed. However, one could incorporate non-isotropic element patterns into the expressions by modifying the received voltage vectors: (2.42) (2.43) where A\[6'\ 0*) is the gain pattern at the center frequency. If the pattern varies with frequency, one could instead, incorporate it into the power spectral density expression of (2.32) and obtain the modified auto-correlation function via the inverse Fourier transform. 2.2 STAP Weights In the STAP performance analysis, the tap weights are adapted using power min imization and constrained power minimization approaches. In the constrained power 26 minimization, both beam forming and the minimum mean square error (MMSE) correlation vector approaches are used. The power minimization approach used is commonly referred to in the literature as a “power inversion” steering vector. While not completely unconstrained, the power minimization restricts the weight solution the least-only requiring that the array remain on. Excellent treatments of each ap proach are provided in [3, 24, 27, 28]. To provide completeness, the development for the weight adaptation approaches used is provided in the following section. 2.2.1 Maximum SINR In forming the power ratios of the desired to interference plus noise powers at the array output, a natural approach to weight adaptation is based on the following. From (2.16), the desired signal-to-interference-plus-noise power ratio at the array output is Pé (2,44) Pi + Pn g^(*i + $»)g' The ratio is maximized [17] if the weights are the solution to ($4 + = Ag, (2.45) where A is the largest eigenvalue of the desired signal covariance matrix. Pre-multiplying (2.45) by + *n)Ag = A($) + $»)g. (2.46) Substituting (2.46) in (2.45) P, _ g«A(#. + #„)g ^ ^ (2.47) Pi + Pn g ^($7 + $ » ) g 27 Since A is the largest eigenvalue, the signal ratio is maximized. However, in practice, one may not have access to the individual covariance matrices which would then pre vent one from obtaining the solution to the generalized eigenvalue problem in (2.45). Another approach is to minimize the mean square error between the received signal at the array output and a locally generated reference signal without any constraints. 2.2.2 Unconstrained MMSE In the unconstrained MMSE weight solution, a locally generated reference signal is used to minimize the error between the received signal at the array output and a known reference signal giving the unconstrained Wiener-Hopf equations [27, pp 98- 104] and [pp 1203-1205] [28]. For example, in a receiver processing a spread spectrum signal, the spreading code is known to the receiver and is readily available as a reference. Let the reference signal be denoted by d{t). The output signal, y{t) is subtracted from the reference signal to form an error signal e(t), e(() = d(^) - 2/(^) = (f(() - g^x. (2.48) The adaptive weights, g are adjusted to minimize the mean square error, = -E;{(d(^) - g^x)*(c((t) - = |d(^)|^ - d*(()g"^x - g^x*d(^) + g^x*g^x }, = |c(W 1^ - + (g^x)*) + g^x*g^x }, = (g^x + (g^x)*) } + g^x*x^g }, = (g^x + (g^x)*) } + g^* g, (2.49) 28 where g^x = x^g, and ^ = E{ x*x^ }. Next, to obtain the optimal operation in the mean square sense, (2.49) is differentiated with respect to g and the result set equal to zero. First, it is helpful to note the following relationships [27, pp 794-798], = 0, 9 g * 9 g % 9 g * dg where the differentiation is performed term by term for the vector g and I is the identity vector. If C\ and Cg are the scalars defined by Cl = x ^ g and C2 = g ^x , ^ = ,a „ d 0 = 0 . ^(^2 _ » , acz _ _ — 0 , âncl — X. ag ag* And for the power term (or quadratic form), = $ g , and 9g* #= s " * - Differentiating (2.49) with respect to g*, = 0 - E{ x*d(t) } + $ g = 0. (2.50) Solving for g, g = E{x*d(f)} = (2.51) 29 where s is the correlation between the reference signal and the received signal. The thermal noise contribution to $ guarantees the invertibility of the covariance matrix. The result in (2.51) is the Wiener-Hopf equation for the optimal weight solution to the nnconstrained minimum mean square error [27, 28]. If the desired signal strength is weak, a convenient choice for the correlation vector is to replace it with a vector of zeros and a single nnity weight yielding the power minimization or “power inversion” steering vector [24, 28], ie s^[10 0 0...0]^. (2.52) 2.2.3 MMSE Beamformer One can also choose to minimize the mean square output of the array subject to other constraints. For example, one may wish to constrain the array to point a unity beam in a specified direction to provide a distortion free response for a narrow band beamformer [27, pp 119]. The optimization problem is to minimize the mean square output of the array f =)g"$g. (2.53) subject to a constraint in the desired look direction g u = u g ^ 1, (2.54) 30 where u is the vector of tap voltages induced by a narrowband signal arriving from the desired look angle. Following an approach outlined in the appendix of [27, pp 795-801], let (2.53) be represented by the function /(g), ie /(g) = ^g^^g- (2.55) Introduce a linear function C in g which is set equal to zero. (7(g) = u^g - 1 = 0-f / 0, (2.56) Relate (2.55) and (2.56) via the cost function ;f(g) = /(g) + AC(g), (2.57) where A is a Lagrange multiplier. Next, minimize (2.57) with respect to g, = ig^$ + Au^ = 0. (2.59) Solving for g, g^ = -2 Au^$-\ (2.60) g = -2A*$-^u*. (2.61) Substituting (2.61) into (2.56) and solving for A u^g = 1, (2.62) 31 = 1, (2.63) -2A*u^$-^u* = 1 (2.64) Substituting the result into (2.61 ) yields « = u w - As pointed out by Haykin in [27, pp 117:119], in the case where the system input and output are both zero mean (as is the case here), (2.66) is the minimum mean square error solution. Thus, the above gives the optimal MMSE beamformer weights. 2.2.4 Constrained MMSE The desired signal may not be narrowband. If a reference signal is available, one can constrain the weights in a mean square sense using the correlation between the received signal and reference. The identical approach used in obtaining (2.66) is followed by replacing the the steering vector in (2.66) with s*, s = E{x*d(f)}, (2.67) u —> E { x d(t)} = s*, (2.68) u* —> E {x.*d*{t)} =E {x.*d{t)} = s, and (2.69) ^ ^ (2.70) where d{t) is real. This is the constrained MMSE weight solution. With the exception of the constant in the denominator, (2.66) is identical to the Wiener-Hopf equation 32 of (2.51). In general, the constraint in (2.54) does not need to be unity. Also, one may use multiple constraints as in [3] to specify a frequency response in a given look direction, constrain the response for multiple directions, and/or steer nulls in predetermined directions. However, each additional constraint imposed consumes an additional de gree of freedom. In this study, the minimum mean square error approaches to weight adaptation given in (2.51), (2.66), and (2.70) are used to adapt the weights and min imize the interference power at the array output. However, a slight modihcation is made to simplify performance comparisons at the input and output of the adaptive array. Noise power at the array input is assumed to be unity or 0 dB. If the weight vectors are scaled by their norms, the noise power at the array output power is always unity or 0 dB. This simplifies the metric comparisons at the input and output of the adaptive array. Next, the performance metrics are described. 2.3 Performance Metrics The performance evaluation considers the relative ratios of desired, interference, and noise powers in the array output as well as the effect of the processor on the desired signal. Two sets of metrics are used. The first concentrates on the power ratios of the various signal components at the array input and processor output. These are used frequently in the STAP literature and represent the desired, interference, and noise powers at the array output in terms of the steady state adaptive weights and 33 various signal covariance matrices. The second set of metrics is more unique in that it attempts to measure the distortion induced by the adaptive processor on desired signal. To measure the distortion, the cross-correlation between the desired signal and the received signal at the array output is used. Through the use of the known signal power spectral density and the resultant array transfer function, the shape and phase characteristics of the cross-correlation can be obtained. This allows access to the cross-correlation function shape, phase, and peak delay. In that the second set of metrics rely on the correlation process, they are referred to as “post-correlation” metrics. The first set of metrics is termed “pre-correlation” metrics. In the analysis, both metrics are used to evaluate performance. 2.3.1 Pre-correlation Metrics Given the steady state adaptive weights and using (2.16), the respective desired, interference, and noise powers at the array output are jPd == (2.71) Pi = and, (2.72) Pn = ^ g ^ * » g . (2.73) With (2.71) through (2.73), the interference-to-noise ratio (INR), signal-to-noise ratio (SNR), and signal-to-interference-plus-noise ratio (SINR) are defined as INR — 10 log^Q — , (2.74) SNR = lOlogio^, and (2.75) 34 SIN R = 10 log,, ^ (2.76) ■^n INR and SNR are used at both the array input and output to define the relative signal strengths and performance for the signal scenario under consideration. The post-correlation metrics are based on the cross-correlation between the desired signal and received signal at the adaptive array output. The performance metrics are based on the resulting cross-correlation function peak, sidelobe levels, shape, and carrier phase. These are defined next. 2.3.2 Post-Correlation Metrics The cross-correlation function between the desired signal and the adaptive array output is defined in terms of the signal power spectral density, auto-correlation func tion, and array frequency response. The array frequency response is a function of the adaptive weights and look direction. Let y{t) be the output of the STAP equipped array. The output can be written as ?/(t) = 4- 4- (2.77) where do{t), io{t), and n o {t) are the total desired, interference, and noise signals at the array output respectively. Let d {t) represent the known desired or reference signal. The cross-correlation between the desired signal, d{t), and the output signal is Eyd(T) = ^ ^ - T)d(. (2.78) 35 The desired, thermal, and interfering signals are assnmed mntually uncorrelated. If the integration is carried out over a long period of time, the cross-correlation can be approximated as ^ ^ ^ (2.79) where the cross terms between the desired, interference, and noise signals are zero. At the output of the adaptive array, the received desired signal can be expressed as do(f) = (f(^) = / /i(a) — a )d a , (2.80) J — OC where h{t, dd-, 4>d) is the adaptive filter impulse response in the desired signal direction {Od,4>d) and “0 ” denotes convolution. Substituting (2.80) into (2.79), Rydir) ^ ÿ % { / h{a) d{t - a)da^d*{t - T)dt, = j h ( a ) | — y d{t — a) d*{t - T)dt^ da, / OO h{a)Rdd{T — a)}da, thus, -OO 7^(f(T) % /t(T,gd,<^(()07^((j(T), (2.81) where Rdd{T) is the auto-correlation of the desired signal. Using the Fourier transform (2.81) can be written as 7ü&f(T) = .^cf) (2.82) 36 where Sdd{f) is the power spectral density of the desired signal, H{f,9d,(j)d) is the frequency domain response of the array in the desired signal direction, and de notes the inverse Fourier transform. The frequency domain response as a function of frequency and direction, H{f, (j)d) is calculated from the adaptive antenna weights, L N = E E (2.83) 1=1 n = l where L equals the number of antenna elements, N equals the number of taps per element, gin is the adaptive weight for the tap of the antenna element, and To is the delay between taps. Once (2.83) is calculated, the cross-correlation function (CCF) given by (2.82) is found using the inverse fast Fourier transform. The cross-correlation calculation is much the same process as that performed by a digital receiver. The metrics derived from the CCF are aptly coined “post-correlation” metrics in that they are derived after the correlation is performed. As such, the met rics only have meaning if the receiver can successfully acquire and track the signal. Thus, if severe degradation in SINR occurs, the CCF post-correlation metrics should not be used. While the amount of SINR, degradation which can be tolerated is receiver specific, the current work assumes that SINR degradation of 15 dB from quiescence will result in loss of the signal. Given adequate SINR, the following metrics, peak- to-peak ratio, peak delay, carrier phase error at the peak, peak-to-mean ratio, and root mean square error in mainlobe shape are used to represent the post-correlation performance. Of these, the peak-to-peak ratio, peak delay, carrier phase error and root mean square error (RMSE) are determined from the “noise-less” CCF as calcu lated using (2.82). Using the “noise-less” CCF provides metrics which are sensitive to 37 non-uniform frequency response or phase non-linearity induced by the processor irre spective of noise. Based on the shape of the CCF magnitude, the peak-to-peak ratio and RMSE metrics are particularly sensitive to processor induced distortion resulting from non-uniform frequency response while peak delays and carrier phase error at the peak are sensitive to processor induced phase non-linearities. A peak-to-mean ratio of the CCF is also determined which includes an estimate of the noise in the processor output. The “noisy” CCF is estimated using (2.84) through (2.86) and is discussed in more detail below. The CCF peak-to-mean ratio is closely related to the SINR as it is impacted both by interference power in the array output as well as reductions in the CCF peak due to distortion, broadening of the CCF mainlobe, and reduced signal power at the array output. In all, the post-correlation metrics indicate the amount of signal distortion caused by the adaptive processor. These are summarized below. CCF Performance Metrics • Peak-to-peak ratio (PFR): Ratio of the main peak level to the highest sidelobe level. In practice, the receiver determines the peak in the presence of noise. Thus, PFR levels less than 3 dB may not be adequate. • Peak Delay: The deviation of the CCF peak value from the corresponding quiescent value. Receivers may use different algorithms to track the CCF peak. Typically, peak delays exceeding half of the quiescent CCF mainlobe width are not acceptable. • Carrier Phase Error: Phase of the CCF at the main peak location. Ideally, zero phase is expected at the peak location. 38 • Peak-to-mean ratio (PMR): Ratio of the main peak level to the mean value of the CCF when the central ±2r are excluded. Note that the mainlobe peak is approximately 2r in width. • Root mean square error (RMSE): Essentially the difference in energy between the perturbed CCF and the quiescent mainlobe shape normalized to the energy under the quiescent mainlobe width (null to null). To illustrate, Figure 2.3 shows the CCF obtained using quiescent weights for a five-element seven tap STAP configuration. The CCF peak location, PPR, and main lobe shape are shown. The CCF shown assumes an input SNR of -20 dB and signal bandwidth of 20 MHz. In the various CCF representations used throughout the study, the peak level of the CCF is shown relative to a fixed unity power level (0 dB). For example, under quiescent conditions, with only a single isotropic antenna element active, the array yields unity gain. For an input SNR of -20 dB, the quiescent CCF peak is shown at -20 dB. Depending on the nominal processing gain, the peak signal level after correlation will experience a gain relative to wideband system noise of lOlog^Q Gp dB, where Gp is the nominal processing gain magnitude. To facilitate comparisons between signal scenarios, the system noise and interference levels in the CCF are reduced by the processing gain as shown below in (2.84) and (2.86). In this manner, the peak of the CCF is only shifted upwards or downward by the array antenna response after adaptation. In the above, it is assumed that the integration in (2.78) is performed over a large time interval. In practice, the longest reasonable period of integration is one symbol period, T^.To approximate the the effect of thermal noise and interference in the array output in the CCF, the resultant CCF is approximated as 39 = -Ryd(T) + AT(0, (7^), (2.84) where A^(0, cr^) is Gaussian with zero mean and variance equal to < 7 ^. The noise power at the output is assumed equal to unity or 0 dB. The total variance is approximated from the output INR and nominal processing gain of the spread spectrum signal (2.85) Lrp where INRg is the INR at the array output and Gp is the nominal processing gain for the spread spectrum signal of interest. For example, if a 5 kHz data signal is spread at a 10 MHz chip rate, the processing gain is approximately Gp - 10 logio 33 dB. (2.86) data The modified CCF given in (2.84) is used to define the PMR and RMSE. Addi tionally, the RMSE is defined as I t p - T o \Rdd{T- T p ) \- G \Ryd{T) I dr RMSE = — , (2.87) HTj7^dd(T)PdT where Tp is defined as the peak location of the modified CCF, G is the ratio of the ideal main peak level to the peak level in the presence of interfering signals, and Rdd{T~) is the quiescent or ideal CCF. Note that the ideal CCF in the numerator is shifted in time so that its peak coincides with the peak of the modified CCF. Integra tion is carried out in the mainlobe region and provides a direct measure of waveform distortion. Ideally, the rms error is zero. 40 The above describes metrics for evaluating the pre-correlation (power ratio) and post-correlation (signal distortion) STAP performance. These will be used to evaluate the STAP performance for various reference tap positions. 41 CH APTER 3 REFERENCE TAP SELECTION IN STAP In STAP, the tap weights are adjusted to minimize the mean output power subject to various linear constraints. If linear constraints are selected such that only a single degree of freedom is consumed per desired signal or look direction, the use of a steering vector, correlation vector, or the unconstrained power inversion steering vector all require selection of a reference tap. The ideal solution minimizes the interference without degrading the desired signal. As discussed previously, linear constraints can be imposed in the desired look directions which ensure either a linear phase or all pass response. However, these additional constraints consume additional degrees of freedom which could be used to reduce interference. Is there a simpler way to reduce the possibility of severe distortion while minimizing the interference power yet without using up precious degrees of freedom? In this chapter, it will be shown that reference tap selection effects both the pre- and post-correlation metrics and that a reference choice can be made which offers less signal distortion. Performance is evaluated for the power minimization, minimum mean-squared error (MMSE), and optimum beam former weight adaptation approaches. The results will show that selecting the center tap of the N-length tapped delay line as the reference offers the best overall STAP performance. 42 First the model configuration and signal scenarios are described. Then perfor mance is evaluated as a function of the tap reference tap selection for the power minimization, minimum mean-squared error (MMSE), and optimum beam former weight adaptation approaches. 3.1 Antenna Array and Signal Scenarios An antenna array with co-linear elements is nsed for the stndy. The array consists of five ideal isotropic elements with an inter-element separation of a half-wavelength at the highest desired signal frequency. Fignre 3.1 shows the general antenna geometry. Because the antenna geometry is symmetric with respect to the x-axis origin, signal scenarios are selected which confine the incidence angles to the x-z plane {(j) = 0) and -9 0 < g < 90°. Sensor Locations Figure 3.1: Multi-element equispaced co-linear array. Elements are placed along the x-axis. 43 /o BW (MHz) e° INR (dB) 2004.0 32 65 40 2004.0 32 -75 40 2004.0 32 80 40 2002.4 0.0 -45 40 1994.3 0.0 -25 40 1997.1 0.0 -15 40 2010.3 0.0 0 40 Table 3.1: Interference scenario Performance is evaluated assuming various look directions for the desired signal and are identified in each case for the results under consideration. In each case, the desired signal is modeled as a 20 MHz bandwidth flat power spectral density signal centered at a frequency of 2004 MHz with an input SNR of -20 dB. The PSD is zero outside of this bandwidth. The data bandwidth is 5 KHz giving a nominal processing gain of 33 dB for a spread spectrum signal. The system noise is assnmed Gaussian with zero mean and unity variance. The flat noise spectrum is assumed to occupy the full sampled spectrum or Interference signals are also modeled as having a flat power spectral density with various interference-to-noise (INR) power ratios and bandwidths. A combination of narrow and wideband RFI sources are nsed in the analysis and are listed in Table 3.1. Each of the RFI sources have a fixed INR level of 40 dB at the input to the array. The first three signals occupy the full 32 MHz of sampled spectrum. The remaining signals are modeled as continuous wave (CW) RFI sources with 0 % bandwidth. 44 3.2 Power Minimization Results Figures 3.2 through 3.6 show results for a seven tap STAP when the interference scenario consists of the RFI sources listed in Table 3.1. Each figure contains results for five different scenarios: one through three wideband sources and three wideband with none, two, and four CW RFI sources from the table. The INR level of each source at the array input is 40 dB. Tap delays are set at Tg = ^^mhz ■ Weights are adapted using the unconstrained power minimization algorithm (power inversion steering vector) as described in (2.51) through (2.52). For the linear array of Figure 3.1, the center element is selected as the reference. Figure 3.2 shows the pre-correlation STAP per formance in terms of the output INR and SINR towards the broadside direction when the first through seventh tap is selected as the reference. With only one wideband RFI source present, the output INR is extremely low with the output INR using the center tap as reference somewhat better than the other tap selections. The output INR results are symmetric with respect to the center tap, ie, the output INR when the first tap is selected as the reference produces the same output INR as when the seventh tap is selected. Results for the second tap are the same as the sixth, etc. With the addition of the second wideband source, the output INR increases signif icantly, but is still 10 dB lower than the noise power. Again, there is a tendency for the center tap to yield more RFI suppression. When three wideband plus four CW RFI sources are present, the output INR is relatively insensitive to the reference tap choice. The output SINR, shown in the right hand plot, is less sensitive to the reference tap selection than the INR results. The quiescent SINR output is -20 dB. The output SINR in the presence of one through two wideband RFI sources is close to the quiescent level with only the first and seventh reference tap choice deviating 45 by about 2 dB. In the remaining three scenarios, SINR increases by 4-5 dB indicating that the array pattern is being pushed up in the broadside direction, or more toward the desired signal. Figure 3.3 shows the adaptive pattern for the power inversion weights versus frequency and look angle {6 4 ) when a total of three wideband plus four CW RFI sources are incident. The array outpnt signal magnitude is represented by a 25 dB gray scale. Deep, broad nulls across the frequency band corresponding to the wideband RFI sources (marked with a white “W” in the hgure) are shown in the figure while the CW RFI sources have much narrower frequency nulls toward their DOA (indicated by the white “x”). The quiescent pattern is 0 dB over all frequencies and angles. In the direction and frequency corresponding to the desired signal, the pattern gain is increased over its quiescent value over much of the frequency band of interest. Returning to Figure 3.2, note that the SINR for the center tap reference location with three wideband plus four CW RFI sources is slightly less than that of the D* (7*^) and 2"^ (6*^) reference tap choices. Based on the pre-correlation INR and SINR metrics, the results are not conclnsive. However, any of the reference tap choices provide more or less the same SINR response. Figures 3.4 through 3.6 show the post-correlation performance for the same five RFI scenarios versus the reference tap choice. Figure 3.4 shows the CCF PPR and PMR performance in the broadside look direction. The quiescent PPR is 6.6 dB and the quiescent PMR is 12.5 dB. When only one or two wideband interferers are present, the PPR is insensitive to changes in the reference tap selection. As addi tional interference is added, the change in tap reference makes a difference. This is most apparent in the scenario with three wideband plus four CW RFI sources. The 46 -10 -10 -1 i l CÛ '-20 'Â- -20 3 -30 0_ OZ) 3 O 1 WB -25 2WB -40 3 WB 3WB + 2 CW 3WB + 4 CW -50 -30 3 4 5 6 3 4 5 6 REFERENCE TAP REFERENCE TAP a) Output INR b) Output SINR Figure 3.2: Fre-correlation performance evaluated in the broadside direction for the 7-tap STAP using power minimization or “power inversion” steering vector in the presence of one through three wideband plus zero, two, and four CW RFI sources. center tap improves PPR performance by 2.5 dB over that when the first tap is used as the reference. For the center tap, PPR is within 1 dB of quiescent performance. The PMR results shown in the plot on the right are less conclusive. When one wide band RFI source is present, results are close to quiescent for each of the reference tap choices. With two wideband RFI sources, slight changes from one reference tap selection to another are apparent. When three wideband and two CW interferers are added, the array response in the center of the pattern begins to get pushed up in the center causing the peak value to increase. When additional interference is added, side-lobes increase, and the mean level outside of the main lobe increases which in turn reduces the PMR. In this scenario, performance is better for the central three tap positions. 47 1995 2000 2005 2010 f (MHz) Figure 3.3: Adaptive pattern with fixed power inversion weights versus look angle {6d) and frequency for the three wideband plus four CW RFI scenario using power minimization. The “W” indicates the DOA of the wideband RFI while the “x” indicates a CW RFI source. In Figure 3.5 the CCF peak delay and carrier phase error for the five scenarios are shown. Here the sensitivity to tap position is more dramatic. Using the center tap as reference, no peak delay is present for any of the five scenarios. The remaining six possible reference tap positions show deviation in three of the five scenarios. Only the scenario with one and three wideband interferers seem impervious to delay. The same conclusion is obtained from the carrier phase error results shown on the right. At the center tap position, none of the scenarios induce any carrier phase error. The remain ing tap reference positions have significant carrier phase error. The narrow-band or CW interferers have a more noticeable effect on the phase error than the wideband 48 10 9 8 u I1 I r o — 1 f ^ il 6 \ 5 CL \ < ) H \ =) w ...... 4 / 0 / / W 1 WB > ' v - ^ > - - 2 WB 2 ' a 3 WB 1 -w - 3WB + 2CW -O - 3WB + 4 CW 0 2 3 4 5 6 2 3 4 5 6 REFERENCE TAP REFERENCE TAP al Peak-to-Peak Ratio b) Peak-to-Mean Ratio Figure 3.4: Post-correlation CCF PPR and PMR evaluated in the broadside direction for the 7-tap STAP using the PI steering vector in the presence of one through three wideband plus zero, two, and four CW RFI sources. RFI. This may be due to the concentration of the CW power within a narrow band of the frequency spectrum while the wideband sources spread their power across a much wider spectrum. Next, the distortion in the CCF main lobe as a function of tap position is shown in Figure 3.6. The results when up to three wideband plus 2 CW RFI sources are present are fairly insensitive to the tap reference selection. When three wideband plus four CW RFI sources are incident, the three center tap reference positions give the worst performance with the RMS error twice that of the first and second reference tap positions. This seems counter-intuitive. However, the effect is primarily due to a broadening of the main lobe rather than an asymmetric distortion. This can be seen in Figure 3.7 where a plot of the CCF amplitude for the three wideband plus four 49 LU 30 - -<> LU Qlt- Q ir < -30 1 WB - - 2 WB -60 -W- 3WB + 2CW -O- 3WB + 4 CW -1 0 -90' 7 REFERENCE TAP REFERENCE TAP a) Peak Delay b) Carrier Phase Error at Peak Figure 3.5: Post-correlation CCF peak delay and carrier phase error evaluated in the broadside direction for the 7-tap STAP using the PI steering vector in the presence of one through three wideband plus zero, two, and four CW RFI sources. CW scenario for the first and fourth reference tap positions is shown. Also shown is the quiescent response. When the first tap is used as the reference, the main-lobe is slightly broadened but there is also an asymmetry in the CCF which pushes the mean level in the PMR up. The response using the center tap as reference shows increased broadening in the main lobe. However, the response is very symmetrical and the peak side-lobes are reduced in comparison to that of the first tap. Note that US. the CCF peak when the 4th tap is used as reference occurs at 32 The previous figures were only for the broadside direction. Figures 3.8 through 3.10 show similar results versus look direction {9d ) for the three wideband plus four CW scenario when the first and fourth (center) tap are used as the reference position. Figure 3.8 shows the SINR and CCF PAIR versus look direction. Results were evaluated every 10° from —90° to 4-90°. Both the SINR and PMR results are 50 0.5 1 1 ----- 1 WB — - 2 WB ' a 3 WB - m - 3WB + 2 CW ■ -O- 3WB + 4 CW œ: g or LU w *" ------( ^ - c> 2 N a: \ / \ / \ >■------( » ( ^ ------( 0.1 i______X:— Î:— : L______i {------1 3 4 5 REFERENCE TAP Figure 3.6: Post-correlation rms error in the CCF mainlobe shape evaluated in the broadside direction for the 7-tap STAP using the PI steering vector in the presence of one through three wideband plus zero, two, and four CW RFI sources. similar for both reference tap choices. The array is able to maintain the SINR within 10 dB of the quiescent value (-20 dB) over a % 60° region centered at broadside. The CCF PMR is maintained to within 6 dB of its quiescent value in the same region. Figure 3.9 shows the CCF PPR and rms error for the same scenario. The center tap position is able to maintain the PPR over a broader region than when the first tap is used as the reference. Neither can maintain the PPR to acceptable values for regions where the look direction is in close proximity to the wideband RFI sources. The right hand figure displays the rms error in the main-lobe for the same region. The rms error is higher for the center tap selection than when the first tap is used. As shown in Figure 3.7, this appears to be primarily due to broadening of the main-lobe and may be of less significance to a correlation receiver than the peak-to-peak ratio and phase distortion. Figure 3.10 shows the CCF peak delay and carrier phase error at 51 -15 -15 3WB + 4CW Quiescent -20 -20 — -30 — -30 -35 -35 - 4 0 ------^ ^ -40'------■ - -150 -100 -50 150 200-100 -50100 100 150 200 250 X (ns) x(ns) a) Reference at 1®* Tap b) Reference at 4*^ Tap Figure 3.7: Cross-correlation function magnitnde versus delay evaluated in the broad side look direction in the presence of three wideband pins four CW RFI sources. Re sults using the 1®* tap as reference are displayed on the left. The right hand side displays results for the center tap. In both plots the quiescent results are scaled to match the peak of the 3WB-I-4CW Response CCF. 52 -1 0 15 Tap 1 Tap 1 O Tap 4 O Tap 4 -2 0 12 -30 œ: 9 I -40 s CL Z) g -50 CL 6 I- Z) Z) o o -60 3 -70 -80 0 -90 -60 -30 -90 -60 -30 0 30 60 90 e.d «d SINR b) CCF Peak-to-Mean Ratio Figure 3.8: Output SINR and CCF PMR performance versus look direction {9^) using the PI steering vector in the presence of three wideband plus four CW RFI sources. the peak location versus look direction for the same scenario. Again, STAP performs better when the center tap is selected as reference. However, neither choice produces acceptable delay and phase performance for look directions in close proximity to the wideband RFI. To suppress the interference the system must also suppress the desired signal in these regions. When the power minimization or power inversion approach to weight adaptation is used, the pre- and post-correlation metrics demonstrate that if the primary concern is to suppress the interference while maintaining the desired signal power level and phase characteristics, the selection of the center tap as the reference provides superior performance. By moving the tap reference to the center position, STAP performance with respect to the PPR, peak delay, and carrier phase error are improved signihcantly without any processor cost in terms of degrees of freedom. 53 Tap 1 0.5 O Tap 4 Tap 1 O Tap 4 0.4 O O O « G « G ôi G (> I 0.2 06 O O O O O Ô -90 -60 -30 -90 -60 -30 0 30 60 90 ed a) CCF Peak-to-Peak Ratio b) CCF RMS Error in Mainlobe Shape Figure 3.9: CCF PPR and RMS error performance versus look direction (6d) using the PI steering vector in the presence of three wideband pins fonr CW RFI sources. Tap 1 O Tap 4 LU LUÛ S CL -30 -1 0 -15 -60 -20 Tap 1 O Tap 4 -25 -90 -90 -60 -30 90 -90 -60 -30 ed e.d a) CCF Peak Delay b) CCF Carrier Phase Error Figure 3.10: CCF peak delay and carrier phase error versus look direction {9d) using the PI steering vector in the presence of three wideband plus four CW RFI sources. 54 Next, the pre- and post-correlation performance results nsing a minimnm mean square error beamforming weight adaptation approach is shown for the first and center tap positions. 3.3 MMSE Beamformer In the previous results, the three wideband plus four CW RFI sources proved to be the most challenging of the five scenarios for the seven tap STAP array. In this section, pre- and post-correlation performance is evaluated for the same array configuration nnder this scenario when the power minimization is carried out using a minimum mean square error narrowband beamforming approach of (2.66) and the weights are normalized. In the MMSE beamforming solntions shown below, the adaptive weights are con strained to develop a beam in each of 19 look directions from —90° < O4 < +90°. Only one constraint is applied at a time using np only one degree of freedom. In quiescent mode, only one tap per element is on. Thns 19 weight solutions are found. In each solntion the array is constrained nsing (2.54) such that in the desired look direction g^u = l, (3.1) where g are the adaptive weights and u is the vector of tap voltages induced by a narrowband signal arriving from the desired look direction. Figure 3.11 shows the pre-correlation output INR and SINR versus look direction when the three wideband plus four CW RFI sources are incident on the array. Results 55 -1 0 Tap 1 O Tap 4 -2 0 CO -30 I- -1 0 -40 Z) Q. I- OZ) -20 -60, Tap 1 O Tap 4 -30 -70 -90 -60 -30 90 -90 -60 -30 e.d e.d a) Output INR b) Output SINR Figure 3.11: O utput INR and SINR versus look direction {9a) for the MMSE beam former in the presence of three wideband plus four CW RFI sources. are shown for the first and center (fourth) reference tap choices. Results are similar for both reference tap selections. The interference is suppressed below the system noise for look directions in the range —58° < 9a < 58°. For the power inversion solution, SINR fell below -30 dB for look directions outside of ±30° of broadside. Here, SINR falls to less than -30 dB for look directions outside of approximately ±45° of broadside. Using the constraint broadened the region of acceptable signal level. As in the previous results, the pre-correlation performance is relatively insensitive to the reference tap choice. Figure 3.12 shows the CCF peak-to-peak and peak-to-mean ratio for the same scenario. As in the SINR, the CCF PMR results are relatively insensitive to the tap selection with the center tap slightly outperforming the case when the first tap is selected as the reference. In the PPR results, STAP using the center tap as the 56 reference is clearly performing better. When the first tap is selected as the reference, STAP is unable to maintain the PPR when the desired signal look direction is in close proximity to the CW RFI from the 0° DOA. The PPR results also show that both choices maintain the PPR close to quiescent for look directions outside of ±60°. However, the performance in this region is an artifact of the processing. We note from the INR, SINR, and PMR results in this region that the interference power in the output is not suppressed. In these regions, the receiver would not be able to acquire the signal. Figure 3.13 shows the cross-correlation function for a signal in the broadside and —80° look directions. The solid trace is the CCF for the —80° look direction; while, the dashed line is the CCF for the broadside direction. In the broadside direction, the PPR is reduced from the quiescent PPR of -6.6 dB and the main-lobe is broadened from the quiescent width. However, the signal degradation is not severe. The CCF for the —80° direction has a PPR and main-lobe width close to the quiescent values. However,the plot shows a severely attenuated peak level even though the “shape” is clearly close to quiescent. This example illustrates that multiple metrics must be considered when evaluating the performance. Figure 3.14 shows the peak delay and carrier phase error at the CCF peak for the same scenario. The left hand plot shows the peak delay as a function of look angle when using the first and fourth (center) taps as the reference. The center tap results display no variation with look angle clearly outperforming the results from the first tap. The same is true for the carrier phase results. While the center tap results show no deviation over the look angles, the carrier phase error when the first tap is selected as reference varies significantly versus look angle. 57 15 Tap 1 C Tap 4 12 00 « o Ûl q ; 9 CL CL S CL I- 3 3 0_ CL 3I- ©O 6 O O3 3 Tap 1 -Q - Tap 4 -90 -60 -30 30 60 90 -60 -30 a) CCF Peak-to-Peak Ratio b) CCF Peak-to-Mean Ratio Figure 3.12: CCF RPR and PMR versus look direction {9^) for the MMSE beam former in the presence of three wideband pins four CW RFI sources. Figure 3.15 displays the rms error in the CCF main-lobe shape for the same scenario. As was observed for the power inversion approach, the rms error nsing the center tap as the reference is worse than that when the hrst tap is selected. When taken in consideration with the results from Figure 3.13, this may not be of significance as the increase in rms error appears to be primarily due to broadening of the main lobe. As in the power inversion results, the pre- and post-correlation metrics nsing the MMSE beamformer demonstrate that if the primary concern is to suppress the interference while maintaining the desired signal power level and phase characteristics, the selection of the center tap as the reference provides superior STAP performance. Using the center tap as the reference produces better performance with respect to the post-correlation PPR, PMR, peak delay, and carrier phase error at the peak at little 58 -1 0 ■ 100 ns - -20 15.4 dB — -40 -5 0 -6 0 '------2 0 0 -100 0 100 200 300 400 x(ns) Figure 3.13: CCF in the broadside and —80° ’- -’ look directions for the MMSE beam former in the presence of three wideband plus four CW RFI sources when the center tap is nsed as the reference. cost to the processor in terms of degrees of freedom. Next, results are shown when the weights are adapted nsing the expected correlation between the desired signal and reference signal in the constrained MMSE weight adaptation approach. 3.4 Constrained MMSE In the constrained MMSE weight approach, the adaptive weights are constrained by correlating the input signal vector with a known reference signal as in (2.70). In the results presented, the known reference or desired signal is determined based on the tap selected as the reference time position. The correlation vector for the constrained weight solution is obtained from the column of the estimated signal covariance matrix for the look direction of interest. The column is determined from the reference element and reference tap under consideration. 59 Tap 1 O Tap 4 ~ 60 LU QLU o o 6 o -60 Tap 1 O Tap 4 -1 0 -90 -90 -60 -30 30 60 90 -90 -60 -30 60 90 ed ed a) CCF Peak Delay b) CCF Carrier Phase Error Figure 3.14: CCF peak delay and carrier phase error at the peak versus look direction for the MMSE beam former in the presence of three wideband plus four CW REI sources. Figure 3.16 shows the pre-correlation output INR and SINR versus look direction when the three wideband plus four CW RFI sources are incident on the array. Results are displayed when the first and center (fourth) tap are used as the reference. The results shown are similar for both reference tap selections with the interference being suppressed below the system noise when the look direction is restricted to —58° < 6d < 58°. SINR performance is slightly better than the beam former solution and significantly better than the power inversion approach. However, the pre-correlation performance does not show much sensitivity to the reference tap selection. Figure 3.17 shows the CCF peak-to-peak and peak-to-mean ratio for the same scenario. Unlike the power inversion and MMSE beam former, the CCF PMR results show sensitivity to the tap selection with the center tap outperforming the case when the first tap is selected as the reference. This is also true for the PPR results. Results 6 0 0.5 ----- Tap 1 O Tap 4 G o O û : ■ ê cr LU w S cr y V — '------'------'------90 -60 -30 30 60 90 Figure 3.15: CCF main-lobe rms error versus look direction (0^) for the MMSE beam former in the presence of three wideband plus four CW RFI sources. using the first tap show more degradation in the zenith direction which is in close proximity to one of the CW sources. As in the beam former results, PPR values outside ±60° indicate good performance. However, as in the previous result, this performance is artificial and more an artifact of the processing. We note from the INR, SINR, and PMR results in this region that the interference power in the output is not suppressed. In these regions, the receiver would not be able to acquire the signal. Figure 3.18 shows the peak delay and carrier phase error at the CCF peak for the same scenario. The left hand plot shows the peak delay as a function of look angle when using the first and fourth (center) taps as the reference. For the center tap, the peak delay shows no variation versus look angle clearly outperforming the results when the first tap is used as the reference. The same is true for the carrier phase. 61 -1 0 Tap 1 O Tap 4 -2 0 CO -30 I- -1 0 -40 Z) Q. O p I- oZ) -20 Tap 1 O Tap 4 -30 -70 -90 -60 -30 90 -90 -60 -30 e.d e.d a) Output INR b) Output SINR Figure 3.16: Output INR and SINR versus look direction for the constrained MMSE weight adaptation in the presence of three wideband plus four CW RFI sources. 15 Go Tap 1 O Tap 4 12 p O DU 9 S Q_ Z) Û. 6 oZ) 3 Tap 1 O Tap 4 -90 -60 -30 -60 -30 0 30 60 90 ed a) CCF Peak-to-Peak Ratio b) CCF Peak-to-Mean Ratio Figure 3.17: CCF PPR and PMR versus look direction (6b) using constrained MMSE weight adaptation for the in the presence of three wideband plus four CW REI sources. 62 Tap 1 O Tap 4 LU LU LU o(>ooèooôoooooéoo<>oo<> O O O O bdQ < LU CL -1 0 -15 -60 -2 0 Tap 1 O Tap 4 -25 -90 -90 -60 -30 30 60 90 -90 -60 -30 60 90 e e d d a) CCF Peak Delay b) CCF Carrier Phase Error Figure 3.18: CCF peak delay and carrier phase error at the peak versus look direction (dd) using constrained MMSE weight adaptation for the in the presence of three wideband plus four CW RFI sources. While the center tap choice results in no deviation versus look angle, the carrier phase error when the first tap is selected as reference varies significantly versus look angle. Figure 3.19 displays the rms error in the CCF main-lobe shape for the same scenario. As was observed for the power inversion and beam former approaches, the rms error using the center tap as reference is worse than that when the first tap is selected. When taken in consideration with the results from Figure 3.13, this may not be of significance as the increase in rms error appears to be primarily due to broadening of the main lobe. As in the power inversion and MMSE beam former results, the pre- and post correlation metrics using the constrained MMSE approach demonstrate that if the primary concern is to suppress the interference while maintaining the desired signal power level and phase characteristics, the selection of the center tap as the reference 63 0.5 ----- Tap 1 o O Tap 4 0.4 O \ 0 OO a: ê cr LU w S cr 0.2 “A 7 0.1 i Z / v -90 -60 -30 0 30 60 90 «d Figure 3.19: CCF main-lobe rms error versus look direction (û^i) for the using con strained MMSE weight adaptation in the presence of three wideband plus four CW RFI sources. provides superior performance. Using the center tap as the reference yields superior STAP performance with respect to the post-correlation PPR, PMR, peak delay, and carrier phase error at the peak at little cost to the processor in terms of degrees of freedom. 3.5 Conclusions Pre- and post-correlation performance for a STAP configured array were presented using power minimization, MMSE beam former, and constrained MMSE weight adap tation approaches. Consistent performance trends were observed in all three ap proaches versus the reference tap choice. Pre-correlation metrics showed only slight sensitivity to changes in reference tap location. Of INR and SINR, INR showed the 64 greatest sensitivity when the amount of suppression (deep nulls) is the greatest. Post correlation metrics showed sensitivity to the reference tap selection. CCF peak delay and carrier phase error at the peak location showed the greatest sensitivity. Use of the center tap as a reference signihcantly reduced the peak delay and carrier phase error. When the constrained minimum mean square error adaptation approaches were used, use of the center tap resulted in no peak delay or carrier phase error. In the power inversion or power minimization approach, peak delay and carrier phase error were reduced to zero in areas of adequate signal level. PPR and PMR also benefitted from the use of the center tap. In the case of rms error with respect to the main lobe shape, use of the center tap as reference almost always resulted in greater error. The CCF was generated and displayed for several of these cases. The results showed that the rms error was primarily due to a broadening of the main-lobe shape versus asymmetries or other types of distortion. This may or may not be a concern depend ing on the application. All of the results demonstrated that if peak delay and carrier phase distortion is the primary concern after the interference is suppressed, use of the center tap will significantly reduce errors in the STAP performance at no additional cost to the implementation in terms of degrees of freedom or computational cost. In summary, use of the center tap as the reference in the STAP steering vector improves overall performance: • Significant reduction in cross-correlation function peak delay and carrier phase offset errors. • No additional degrees of freedom consumed to implement approach. • No additional computation burden. Thus, use of the center tap in STAP is recommended. 65 CH APTER 4 THE EFFECT OF INTERFERENCE BANDWIDTH AND POWER ON STAP In this chapter, the analytical STAP model is used to study the effect of interfer ence bandwidth and power on STAP performance. To relate the degrees of freedom consumed in the processor with the interference characteristics, the eigenvalue distri bution of the interference plus noise array covariance matrix is studied along with the output INR and SINR metrics. One of the findings from this study is that a single strong wideband source of interference can, in the absence of multi-path, consume multiple degrees of freedom in an adaptive array. An additional finding of the study is that a signal which meets the narrow-band assumption can consume multiple de grees of freedom in a STAP. Wideband interference is known to limit adaptive array performance and was the original motivation to include tapped delay lines [1]. The effect of interference power and bandwidth on the performance of conventional single-tap adaptive arrays was studied in the past by Gupta in [8]. Gupta demonstrated that a wideband RFI source can consume multiple spatial degrees of freedom (DGF) depending on the in terference power and bandwidth where increasing either the power, bandwidth, or 66 both increase the degrees of freedom consumed. Later, in 1988, Compton [23] studied the performance of a two element array demonstrating that the addition of taps can improve the conventional array bandwidth performance. He found that the addition of just a few taps per element can significantly improve bandwidth performance. The degree of improvement was shown to depend both on the tap delay and number of taps given that < Tq < Recently, Zatman [10] nsed the eigenvalue dis tribution of the interference plus noise spatial array covariance matrix to precisely define the narrow-band limit of an array. Zatman recommended that the point at which the second eigenvalue of the array spatial covariance matrix associated with a signal exceeds the noise eigenvalue magnitude be defined as the narrow-band limit of the array for the incident signal. In recent studies, the eigenvalue distribution of the STAP array has been used to help study the performance of STAP in the presence of wideband multi-path. For example, in-situ measurements taken from an array mounted on a static aircraft were used to study the performance of STAP in the pres ence of multi-path [29]. The presence of multi-path was inferred from the eigenvalue distribution wherein the presence of a second eigenvalue larger than noise power was considered the threshold for multi-path in the system. Based on the independent works by Gupta and Zatman, a second eigenvalue larger than the noise eigenvalues may not be due to multi-path. Instead, the interference bandwidth and power level may increase the eigenvalue magnitude above the noise. Characterizing the response of the antenna array to the interference bandwidth and expected power could provide a baseline for the multi-path determination. 67 First the STAP configuration and signal scenario are described. Next, the eigen value distribution of the interference covariance matrix in the presence of a single RFI source is analyzed. Then the pre-correlation INR and SINR performance in the pres ence of multiple RFI sources using a constrained MMSE weight adaptation approach is presented. 4.1 STAP Configuration and Signal Scenario Results are presented using the five-element linear array of ideal isotropic elements shown earlier in Figure 3.1 with a seven-tap STAP as the adaptive configuration. Tap delays are set at 1/32 jjs. The desired signal has a fiat power spectral density with a 20 MHz bandwidth centered at 2004 MHz and incident from the broadside direction {6 = 0°). Additionally, to compare STAP performance with results published in the past, the SINR performance for a single-tap configuration is also provided. The RFI scenario consists of up to five RFI sources at various incident angles. As the desired signal is located at broadside, the RFI directions of arrival are selected such that they are not located in the vicinity of an array null. The RFI sources are also modeled as flat power spectral density sources. For example, Figure 4.1 shows the array pattern at the center frequency under quiescent conditions. The weight constraint peaks the array response in the direction of the desired signal at broad side. Nulls form at ±25° with side-lobe peaks occurring at ±35°. Broader side-lobe peaks result for angles towards grazing incidence. 68 10 5 0 -5 -15 -20 -25 -30 -50 0 (deg) Figure 4.1: Quiescent array pattern at 2004 MHz 4.2 Eigenvalue Distribution Let a single interference signal be incident on the array at 6 = 30°, center frequency of 2004 MHz, and bandwidths of 0, 10 kHz, 8 MHz, 16 MHz, and 32 MHz. The corresponding RFI fractional bandwidths, for the respective sources are 0.4%, 0.8% and 1.6%. Tables 4.1 and 4.2 show the associated eigenvalue magnitudes in dB for the interference covariance matrix when the input INR is 10 and 50 dB. The first column of Table 4.1 shows the eigenvalues in descending order for a CW interference signal. Since a CW signal is completely correlated between the various taps and elements, only one eigenvalue is active. The table also includes the values of a few eigenvalues beyond the principal eigenvalues. The principal eigenvalue magnitude for a CW signal is equal to the signal power times the number of taps and elements, i.e. l-^cwl = P X L X N = 350 -4- 25.44 dB, (4.1) 69 where P is the CW signal power. The next largest eigenvalue is 14 orders of magni tude less. If the input INR is increased by 40 dB, each of the eigenvalue magnitudes also increase by 40 dB as shown in Table 4.2. The second column shows the eigenvalues when the RFI bandwidth is 10 kHz. The second largest magnitude is -33.5 dB. While still quite small, it is seven orders of magnitude larger than the third — lOOdB). A small fraction of the signal en ergy is now associated with the second eigenvector and eigenvalue pair. When the INR increases to 50 dB, the second eigenvalue increases to 6.5 dB as shown in Table 4.2. Once an interference eigenvalue exceeds the magnitude of the noise eigenvalues, the adaptive weights react-suppressing the interference energy associated with the eigenvalue-eigenvector pair[8]. The through 35*^ eigenvalue magnitudes are still insignificant. It is interesting that a signal with such a small fractional bandwidth excites a second eigenvalue and consumes a second degree of freedom. Does this sig nal power and bandwidth exceed Zatman’s narrow-band limit? Tables 4.2 and 4.2 show the eigenvalue magnitudes for a single-tap STAP. For the 10 kHz bandwidth, only a single eigenvalue is larger than 0 dB in either the 10 or 50 dB cases. This does not exceed Zatman’s criterion for the narrow-band assumption. Thus a signal can meet Zatman’s precise definition of the narrow-band limit yet still consume multiple degrees of freedom in a multi-tap STAP. For a finite bandwidth source multiple eigenvalues are excited in the STAP co- variance matrix. Increasing the interference power increases each of the interference eigenvalue magnitudes by the same amount. For example in Table 4.1, the 8, 16, and 70 32 MHz bandwidth RFI sources have four, five, and seven eigenvalues respectively larger than 0 dB. When the input INR is increased by 40 dB, Table 4.2 shows that in each case, additional eigenvalues are raised above the noise level. The same effect occurs in the single tap array covariance matrix. In Table 4.2, for 8, 16, and 32 MHz bandwidth sources only one eigenvalue exceeds the noise level. Increasing the INR by 40 dB increases a second eigenvalue magnitude above the noise level and the narrow-band limit is exceeded. These examples demonstrate that a finite bandwidth source will cause multiple in terference eigenvalues to exceed the noise eigenvalue magnitudes in STAP. Increasing the interference power, bandwidth, or both increase the number of eigenvalues which exceeding the noise eigenvalue magnitudes. In the above, the RFI fractional band widths were all less than 1.6%. However, when the STAP sample rate is taken into consideration, the 1.6% bandwidth occupies 100% of the sampled spectrum. At low input INR, this “wideband” signal excites a full N eigenvalues-consnming N degrees of freedom in the STAP. Yet, for conventional arrays, the signals meet the narrow band criteria-only one eigenvalue is active. STAP is more sensitive to the interference bandwidth and power improving its ability over the conventional array to react to the interference. In addition, one should note that the above effects result from direct- path interference. Multi-path was not included. Thus, eigenvalue distribution alone is no indication of the presence of multi-path. Next, the effect of interference bandwidth and power on STAP INR and SINR performance in the presence of multiple RFI sources is studied. 71 CW 10k Hz 8 MHz 16MHz 32MHz 25.4 25.4 22.9 20.0 17.0 -110.1 -33.5 21.0 20.0 17.0 -110.1 -100.6 14.5 19.6 17.0 -111.0 -100.1 2.1 17.0 17.0 -110.9 -100.1 -14.4 9.7 17.0 -111.7 -111.2 -34.5 -2.9 17.0 -111.7 -111.2 -58.8 -20.6 17.0 -112.6 -111.7 -97.5 -57.3 -20.4 -112.6 -111.7 -114.1 -68.2 -27.2 -113.2 -112.6 -116.5 -94.6 -37.6 -113.2 -112.6 -116.5 -112.8 -50.6 -114.1 -113.5 -117.2 -120.0 -65.8 -114.5 -113.5 -117.2 -120.0 -83.6 -114.6 -114.2 -118.2 -120.0 -104.6 Table 4.1: Interference covariance matrix eigenvalue magnitudes {dB) for IQdB input INR. Number of Taps = 7 CW lOkHz 8MHz 16MHz 32MHz 65.4 65.4 62.9 60.0 57.0 -70.1 6.5 61.0 60.0 57.0 -70.1 -60.6 54.5 59.6 57.0 -71.0 -70.2 42.1 57.0 57.0 -71.0 -70.2 25.6 49.7 57.0 -71.7 -71.2 5.5 37.0 57.0 -71.7 -71.2 -18.8 19.4 57.0 -72.6 -71.7 -57.5 -17.3 19.6 -72.6 -71.7 -74.1 -28.2 12.8 -73.2 -72.6 -76.5 -54.6 2.4 -73.2 -72.6 -76.5 -72.8 -10.6 -74.1 -73.5 -77.2 -80.0 -25.8 -74.1 -73.5 -77.2 -80.0 -43.5 -74.5 -74.2 -78.2 -80.0 -64.6 Table 4.2: Interference covariance matrix eigenvalues magnitude {dB) for 50 dB input INR. Number of Taps = 7 72 CW 10 kHz 8 MHz 16 MHz 32MHz 17.0 17.0 17.0 17.0 17.0 -143.7 -93.0 -34.9 -28.9 -22.8 -147.5 -148.1 -95.3 -83.3 -71.2 -152.6 -144.9 -144.5 -142.1 -124.6 -142.9 -142.2 -144.8 -156.5 -144.2 Table 4.3: Interference covariance matrix eigenvalue magnitudes {dB) for IQdB input INR. Number of Taps = 1 CW 10 kHz 8 MHz 16 MHz 32 MHz 57.0 57.0 57.0 57.0 57.0 -104.2 -53.0 5.1 11.1 17.2 -108.2 -106.7 -55.3 -43.3 -31.2 -118.2 -118.3 -108.5 -101.2 -84.6 -102.9 -104.7 -111.5 -108.7 -108.0 Table 4.4: Interference covariance matrix eigenvalue magnitudes {dB) for 50 dR input INR. Number of Taps = 1 4.3 INR and SINR Performance Figure 4.2 shows the eigenvalue distribution of the interference plus noise covari ance matrix and output INR for the seven-tap STAP when a single RFI source is incident at 30°. Results are shown for RFI bandwidths of CW, 8 MHz, 16 MHz, and 32 MHz when the input INR is allowed to vary from -20 to 80 dB. Only eigenvalue magnitudes 3 dB larger than the noise eigenvalue magnitudes are “counted”. In the figure, CW interference excites only a single eigenvalue over the input INR range; whereas, the number of eigenvalues excited by the finite bandwidth sources are de pendent on both interference bandwidth and power. The output INR is shown in the right hand plot. At low levels of input INR, the interference power is still below the 73 noise, and STAP is unable to detect the interference. As the interference power ap proaches the noise power level, the adaptive weights react and begin to suppress the interference. Additional increases in the input INR drive the output INR lower. For the CW RFI, theoretically, the decrease in output INR will continue indefinitely [8]. In practice, numerical precision limits the suppression which is obtained. The preci sion limit is reached for the displayed results when the input INR of the CW source is 55 dB. The results were generated on a PC platform using MATLAB. Numbers differing by less than “eps” or ^ —156.5 dB cannot be distinguished and represents the best cancellation that can be obtained. At an input INR of 55 dB, the expected output INR is then 55 + 10logio(TfV) - 156.5 = -86.5 dB, (4.2) which is the minimum output INR observed in the figure. As the input INR increases further, the output INR must also increase as suppression is limited to 156.5 dB. For the finite bandwidth RFI sources the linear decrease in output INR does not continue indefinitely. As additional degrees of freedom are consumed by the interference (as indicated by the increase in the number of eigenvalues in the eigenvalue distribution) the output INR oscillates. This behavior is similar to the one observed for single-tap adaptive arrays [8]. In the past, a strong wide-band interference source was shown to cause sharp degradation in the output SINR of an adaptive array [8]. The left hand plot of Figure 4.3 shows the output SINR for the seven-tap STAP for the single RFI scenario. The quiescent SINR for the seven-tap STAP is ^ —11.1 dB. Due to the 20 MHz bandwidth 74 8 MHz -10 16 MHz 32 MHz -20 m -30 CD 20 z -4 0 ? -50 6 -60 -70 8 MHz -80 16 MHz 32 MHz -90 -20 0 20 40 60 80 -20 0 20 40 60 80 INPUT INR (dB) INPUT INR (dB) a) Eigenvalue distribution b) Output INR Figure 4.2: Eigenvalue distribution and output INR for seven-tap STAP in the pres ence of a single REI source. of the desired signal, the quiescent value is lower than the —20-I-I0 logio(TA^) = —4.56 dB which one might expect for a CW desired signal given the number of taps and elements. However, as seen in the correlation function of (2.31), the hnite bandwidth results in less than full signal power at each tap. If the desired signal bandwidth is increased to 32 MHz, the correlation function is only nonzero when the relative delay in (2.31) is zero-reducing the quiescent gain to that experienced by a single-tap array (-13 dB). For the 20 MHz bandwidth signal, the quiescent SINR is roughly 2 dB higher than that of a single tap array. In the figure, only the finite bandwidth REI sources cause the SINR to degrade from quiescence. The largest drop (0.2 dB) occurs for the 32 MHz bandwidth RFI source. As shown in Figure 4.2, the largest number of eigenvalues, thus, the largest number of degrees of freedom are consumed by the 32 MHz RFI source. In comparison, the SINR performance of a single-tap array is 75 shown in the right hand plot of Figure 4.3. SINR performance is not as good. As noted in the past [8], the SINR for the single tap array is observed to sharply degrade as a result of RFI bandwidth and power level. Here, degradation for the single-tap array is significantly greater than that experienced by the seven-tap STAP. -13 - 11.2 -13.5 Pi 1.6 8 MHz 16 MHz 32 MHz -14.5 8 MHz 16 MHz 1 RFI Sour 32 MHz -1 2 -15 0 20 40 60 80 -20-20 INPUT INR (dB) INPUT INR (dB) a) Seven-tap STAP b) Single-tap STAP Figure 4.3: STAP output SINR in the presence of a single RFI source. Figure 4.4 shows the eigenvalue distribution and ontput INR for the seven-tap STAP when two more RFI sources at incident angles of —35° and 45° are added to the scenario. The eigenvalue distribution shows that the three 32 MHz RFI sources excite seven eigenvalues each in the seven-tap STAP at low levels of INR. As in the previous result, increasing the input INR causes more degrees of freedom to be consumed for the finite bandwidth sources. Because the array degrees of freedom are not totally consumed, the same oscillatory behavior in output INR versus input INR 76 is also observed. However, the output SINR performance for the seven-tap STAP is only minimally affected as shown in the left hand plot of Figure 4.5. The most degradation (0.4 dB) occurs for the 32 MHz bandwidth RFI. As seen in Figure 4.4, the 32 MHz case consumes the most degrees of freedom. In contrast, significant degradation is observed in the SINR performance for the single-tap array as shown in the right hand plot of Figure 4.5. Performance degrades by over 25 dB for the 32 MHz bandwidth RFI. From Table 4.2, each finite bandwidth RFI source consumes two degrees of freedom when the input INR is 50 dB. The single-tap STAP has only four degrees of freedom available. Thus, all the available degrees of freedom are consumed for the hnite bandwidth cases and the interference can not be suppressed. Once the degrees of freedom are consumed, SINR performance degrades quickly. The Rgure clearly shows that the degradation worsens with increasing RFI bandwidth. — CW 8 MHz -1 0 16 MHz 32 MHz -20 m -30 CÛ 2 0 -40 ? -50 -70 8 MHz -80 16 MHz 32 MHz -90 -20 0 20 40 60 80 -20 0 20 40 60 80 INPUT INR(dB) INPUT INR (dB) a) Eigenvalue Distribution b) Output INR Figure 4.4: Eigenvalue distribution and output INR for a seven-tap STAP in the presence of three RFI sources. 77 -10 - 11.2 -15 -20 -25 Ml.6 R -30 CW 8 MHz -35 8 MHz 16 MHz 16 MHz 32 MHz 32 MHz 3 RFI Sources -12 -40 -20 0 20 40 60 80 -20 0 20 40 60 80 INPUT INR{dB) INPUT INR (dB) a) Seven-tap STAP b) Single-tap STAP Figure 4.5: STAP output SINR performance in the presence of three RFI sources. Figure 4.6 shows the eigenvalue distribution and output INR for the seven-tap STAP when two more RFI sources are added at incidence angles of —65° and —50°, respectively. The eigenvalue distribution indicates the effect interference will have on STAP with the wider bandwidth sources consuming all available degrees of freedom as interference power is increased. The plot of the output INR shown on the right still exhibits oscillatory behavior as the input INR increases. However, once all the degrees of freedom are consumed, the output INR ceases to oscillate increasing lin early as the input interference power increases. As shown in the left hand plot of Figure 4.7, STAP output SINR is severely degraded for the 16 MHz and 32 MHz RFI as the degrees of freedom consumed approach the limit. Performance is only sustained over the complete input INR range in the case of the narrow band CW RFI. For intermediate bandwidths, as represented by the 8 MHz bandwidth RFI, 78 performance is sustained up to moderate ranges of input INR. For the single-tap con ventional array, the output SINR is shown on the right hand side of the figure. SINR performance is severely degraded in the presence of three RFI sources. Now, with hve sources incident. Figure 4.7 shows no differentiation between the bandwidth of the four scenarios. With all available degrees of freedom consumed in each case, SINR performance degrades sharply with any increase in interference power regardless of interference bandwidth. 8 MHz 16 MHz 32 MHz "Om Œ: 8 MHz z 16 MHz I- u Z) 15 32 MHz CL ^ -10 o -20 -30 -40 -20 -20 INPUT INR (dB) INPUT INR (dB) a) Eigenvalue Distribution b) Output INR Figure 4.6: Eigenvalue distribution and output INR performance for a seven-tap STAP in the presence of five RFI sources. In the above, the degrees of freedom consumed were shown to be strongly depen dent on the interference power and bandwidth. In a conventional array, the fractional bandwidth, ^ was the primary bandwidth consideration. In STAP, the percentage of bandwidth relative to the tap delay, Tg was shown to be just as important a factor 79 -1 0 -10 CW 8 MHz -15 16 MHz -15 32 MHz -20 cr z -25 3 R -30 o -25 -35 8 MHz 16 MHz 5 RFI Sc un Single Tup 32 MHz -40 -30 -20 0 20 40 60 80 -20 0 20 40 60 80 INPUT INR (dB) INPUT INR (dB) a) Seven-tap STAP b) Single-tap STAP Figure 4.7: Output SINR performance in the presence of five RFI sources. in the degrees of freedom consumed. Output INR performance for STAP showed trends similar to that observed in single-tap arrays with the output INR oscillating as the input INR level increased. However, STAP output INR was shown to be more robust. Another important difference demonstrated in the results is STAP’s ability to sustain the output SINR performance with little sensitivity to interference power or bandwidth given that the interference is suppressed. 4.4 Conclusions An analytic model of STAP using a constrained power minimization approach for weight adaptation was used to demonstrate the effect of interference power and band width on performance. In contrast to the conventional adaptive array, while STAP performance was shown to be sensitive to the interference power and bandwidth, the performance degradation is moderate in comparison. In general, for multi-tap STAP 80 • RFI bandwidth relative to the tap delay, A / Tg, was shown to be as important as fractional bandwidth, Af/fg, to performance. - “Wideband” in STAP need only be as wide in bandwidth as the reciprocal of the tap delay (Tq). • SINR performance is sensitive to interference power and bandwidth with wider bandwidth and stronger interference cansing more degradation. - If the interference is snppressed below the noise, SINR degradation is min imal. • Ontput INR level oscillates with increases in interference power. • A single finite bandwidth direct-path RFI sources excites multiple eigenvalues in the interference plus noise array covariance matrix. - Occurs at low interference power levels (INR > 0 dB). - Increases in interference power increases the nnmber of active eigenvalues thus more degrees of freedom are consumed. - “Wideband” RFI consumes a minimum of N degrees of freedom where N is the number of taps per sensor. Increasing the interference power increases the degrees of freedom consumed to more than N. Thus, an important inference from the findings is that as a single finite band width RFI source can excite multiple eigenvalues in the interference pins noise array covariance matrix, the eigenvalue distribution by itself, is not a good indicator of the presence of multipath in measured or simulated data. Using the analytic model, it 81 was shown that a signal could meet the narrowband criteria for an array as defined by Zatman [10] yet still use multiple degrees of freedom in the STAP. 82 CH APTER 5 SPACE-FREQUENCY ADAPTIVE PROCESSING In SFAP, the adaptive array weights are determined in the frequency domain. In narrowband SFAP, the frequency domain weights are calculated for each frequency bin independently. This approach provides a significant computational advantage- decreasing the size of the array covariance matrices which mnst be estimated and inverted from LN x LN to L x L where L is the number of antenna elements and N is the number of bins or taps (in a STAP processor). With the advent of modern digital signal processors this savings in computations and potential fiexibility of the proces sor has motivated development in the radar, sonar, navigation, and communication communities. Particular theoretical development of SFAP on which this dissertation forms its basis was provided in the past in independent studies by Godara, Faute and Vaccaro, and Gupta. In this chapter, the background theory and equations for the SFAP model are presented. The theoretical basis for narrowband SFAP was established by Godara in [13] and Faute and Vaccaro in [18]. However, Godara A work did not include window ing of the time domain samples. Fante and Vaccaro’s report does include analysis of windowed SFAP. However, their work was not widely known or available to the 83 general public at the time. Moreover, their performance analysis relied on the aver age output SINR and INR computed in the frequency domain. In SFAP an N-bin processor outputs N time domain samples. SFAP performance can vary for different output time samples with significant degradation appearing in some samples due to edge effects. Thus, Fante and Vaccaro’s performance measurements result in sub optimum performance predictions as the average metrics include the performance of all the output time samples. Later, in an open-literature publication including an analysis of windowed SFAP performance, the same averaged metrics are used re sulting in sub-optimal performance for the SFAP [17]. In 1999, Gupta successfully incorporated windowing into a SFAP model and found that performance was signif icantly improved for interference scenarios containing narrowband interference [19]. Recognizing that the end time samples may be corrupted, in Gupta’s analysis, SFAP performance was determined based on the central sample of the N samples available. While Fante’s work predated Gupta’s incorporation of windowing, Gupta’s analysis appears to be the hrst time SFAP performance was correctly analyzed for the opti mum output samples. More recently, SFAP performance using various windows has been demonstrated and reported by Gupta and Moore in [20, 21]. In the analysis by Gupta and Moore, performance estimates were based on the central sample which provides the best possible performance. However, in a practical application, more than a single sample is needed from the data block. A performance analysis of win dowed SFAP as a function of the output time sample was recently submitted to the IEEE [22] by Gupta and Moore. To our knowledge, this represents the first time a complete treatment of windowed narrowband SFAP performance versus time sample has been treated in the open literature. 84 In the current work SFAP dependence on the output time sample is further de veloped to analyze the effect of shifting the SFAP time reference. Similar to the shift in time reference for STAP discussed in Chapter 4, one can also shift the reference in SFAP. The effect of the reference shift on data throughput for block processing and window compensation is analyzed in a subsequent chapter. It will be shown that shifting the time reference can have significant effects on SFAP performance. Additionally, SFAP performance using normalized and un-normalized forms of the weight adaptation algorithms is analyzed. It is found that the form of the weight adaptation can affect SFAP performance-sometimes with substantial consequences for mixed bandwidth RFI scenarios. The rest of this chapter concentrates on a description of the SFAP analytic model. While the basic model, incorporation of windowing, and the output performance dependence on time sample for an unshifted time reference is available in the various studies discussed above, the theoretical description necessary to implement the SFAP model is provided in this chapter for completeness. SFAP performance metrics are also necessary. However, by using “equivalent” time domain weights, SFAP can use the same pre- and post correlation metrics as described in Chapter 2. Thus, these are not repeated here. The rest of this chapter is organized as follows: the SFAP analytic model and frequency domain representation of the array covariance matrices are described first, the narrow-band frequency domain weight adaptation approaches used are described next, and lastly, the SFAP output in terms of the output time samples, window coefficients, and equivalent time domain weights is provided. 85 5.1 Analytic Model In SFAP, the time domain voltages received by each antenna element are converted to baseband, digitized, and stored in an N-length buffer. The buffered voltages are transformed to the frequency domain where constrained (or unconstrained) mini mization of the output power is performed in each frequency bin. By performing the minimization on a bin by bin basis, the size of the array covariance matrix which must be estimated and inverted by SFAP is reduced from LN x LN to an L x L matrix. Thus, significant computational advantages are obtained. However, treating the frequency bins independently results in a sub-optimal process [13]. The claim is that the loss in performance can be more than gained back by increasing the number of bins sufficiently before the computational burden exceeds that of a comparable STAP process. In this work, then, weight adaptation is performed for each frequency bin independent of the adjacent bins. With the adaptive array output formed for each frequency bin in a SFAP, subsequent processing can be performed directly in the frequency domain or the output can be transformed back to the time domain where N-time domain samples are obtained. In the SFAP model, the interference and desired signals are assumed to be mutually uncorrelated independent, zero mean, and wide sense stationary random processes. The voltage samples also contain a thermal noise component which is assumed to be a zero mean, wide sense stationary, Gaus sian random process mutually uncorrelated with the received interference and desired signals. Figure 5.1 shows the block diagram for the SFAP analytical model. Consider an antenna array with L elements. Behind each element is an N-length digital buffer. The digitized voltage samples in the N-length buffer by represented by 86 X,(t) h ,(l) (K points) X (l-(n-l)'l) —^ y(0 (K points) ,(N) X (t -(N-l)T) h(N) Figure 5.1: Space-frequency adaptive process block diagram. xi{n,t) = x{t + Ti{6,(f)) — {n — 1)Tq), n = 1 ... A^, (5.1) I = 1 ... L, where t is the current time sample, Tf(^, 0) is the element to element time delay for a signal received from the {6, (/)) direction, L is the number of antenna elements, N is the number of buffered voltage samples per element, and Tq is the delay time between the buffered samples. At any given time (after the initial buffer fill), each buffer has N samples available. To facilitate the implementation discussion, let “f ’ be replaced by an integer to represent the serial blocks of N-buffered samples behind each antenna element. Letting t —11™, (5.2)becomes 3:f(M,^m)=3;(tm+T((g,(;!))-(M-l)2o)=æ((M,m), M = 1...#, (5.2) 87 I = 1 ... where xi(n,m) is the time sample for the element of the data block. Let X|(m) represent the N-length vector for the buffered voltage samples for the element. (5.3) In SFAP, adaptive processing occurs in the frequency domain. Let transformed vari ables be denoted by a tilde. The frequency domain sample for the bin of the element for the block is given by N m) = m)}, = (5.4) n=l 1 = 1 ... L, k = 1 N, where in the analytic model, provision is made to multiply the buffered voltage sam ples by a window coefficient, Wn (real), prior to the discrete Fourier transform (denoted above by F{-} ). Dehne a vector, d(/c), Wi d(A;) = (5.5) In terms of this windowed DFT operator, the frequency domain sample for the bin and element is xi{k,m) = d^(A:)x|(m) = xf(m)d(A:). (5.6) For L elements, x(/c,m) = ..., ]^. (5.7) In the frequency domain, the Lx L array covariance matrix for the k^^ bin is denoted ^(k) = E{-x*{k,m)i^{k,m)}. (5.8) The and entry of the matrix corresponding to the covariance between the and antenna elements is given by [ê(A;)]fj = Æ'{æXA;,m)æj(A;,m)}, (5.9) = E{[d^(A;)x((m)]*[xj'(m)d(A;)]}, = E{d^(/c)x*(m)xT(m)d(A;)}, = d^(A:)E{x;(m)xT(m)}d(A;), = d^(A;)[$]fjd(A;), (5.10) where is the N x N time domain covariance matrix between the and antenna elements. In practice, the covariance matrix in (5.9) is estimated by averaging the vector product within the expectation operator over M blocks of data samples. In the frequency domain, the L x L covariance matrix over L elements for the bin IS 89 ^ 1 ^ ^ ^ m)x^(A;,m). (5.11) m=l In the analytic model, the individual voltage samples in the time and frequency domain are not available. Instead, the frequency domain covariance matrix ^{k) is obtained by transforming the time domain covariance matrix (5.10) using (2.37) through (2.40) as described in Chapter 2. As the various received signals and noise are uncorrelated and zero mean, as in the STAP model, the array covariance matrix can be decomposed into the individual interference, noise, and desired signal covari ance matrices. With the frequency domain covariance matrix for the bin available, the adap tive weights for the k*^ frequency bin can be estimated. The narrowband frequency domain approaches used to adapt the weights in this work are described next. 5.2 Weight Estimation In a previous chapter, the STAP weights for power minimization, MMSE beam- former, and constrained MMSE weight adaptation were provided. The SFAP fre quency domain weight adaptation methods can be obtained directly from the STAP approaches through the use of the Fourier transform. The SFAP narrowband version of the transformed weight approaches are provided in this section. Consider the narrow-band SFAP weights for the k^^ frequency bin: h(A;) = [^i(A;), ..., ^f(/c), ..., ^^(A;) ]^, (5.12) 90 where hi{k) is the weight for the antenna element. Given the covariance matrix for the bin, ^{k), the frequency domain version of the power minimization weights are obtained directly from (2.51) ie, h(A;) = 0 s(A;), (5.13) where 0 is the convolution operator. In this work, the adaptive weights for each freqnency bin are determined independently. With the bins calcnlated independently, the convolution over all the frequency bins reduces to h(A;)^é-XA;)s(A;), (5.14) As in STAP, in the case of a weak signal, the correlation vector can be replaced by the power inversion vector which simply constrains the array to remain on. Replacing s{k) by VLs{k), the L x 1 power inversion steering vector in the freqnency domain is given by ^ I 1, / = Reference element, “•■W = | o , eke. ■ giving the narrowband frequency domain power minimization weights h(A;) ^ $ ^{k)üs{k), (5.16) For the five element array nsed as an example geometry in the cnrrent work, the third element is selected as the reference. Whereas the time domain power minimization weights minimize the output power subject to the constraint that a single element remain on, the frequency domain power minimization weights minimize the power in 91 each frequency bin independently snbject only to the soft constraint that the refer ence element remain on. Under the same independent frequency bin weight calcnlation assnmption, the freqnency domain version of the MMSE beamformer weights is obtained by Fonrier transforming the narrowband solution of (2.66), where û(&) is the voltage indnced in each element for the frequency bin due to a signal received from the specified look direction, fk is a scalar which conld be tailored to specify a frequency response characteristic in the specified look direction. In this work, /fc, is set to nnity to receive a flat PSD signal from the desired look direction. One could also pre-specify a null in a known interference direction. Replacing the received voltage vector for the frequency with a steering vector ûs(A:) = ü*{k) and setting ~{f)h = 1, (5.17) becomes The constrained MMSE solution is obtained in the same manner. The Fonrier transform of (2.70) yields the narrowband frequency domain weights which is the narrowband frequency domain version of the constrained MMSE weight adaptation approach. In the SFAP model the correlation vector in the frequency domain is readily available. For example, if the third element is the reference antenna 92 element, one can use as the correlation vector the third column of ^d{k). However, one must be careful in the implementation as s{k) is zero outside of the desired signal power spectral density. In these bins, the frequency domain weights should be set to zero. Equations for the narrowband frequency domain versions of power minimization, MMSE beamformer, and constrained MMSE were provided. The equations assume that the weights for each frequency bin are determined independently of all others. In the current work, either power minimization or the MMSE beamformer approach are used to find the weights. The equations for the constrained MMSE weights are provided for completeness. In the model, the equations are straightforward to imple ment. In practice, the covariance matrix for the desired signal is not readily available and SFAP weights are typically found using either the power inversion or narrowband beamformer approaches. Once the frequency domain weights have been determined, the output of the SFAP processor can be calculated. This is described next. 5.3 SFAP Output Given the frequency domain weights and frequency domain samples, the SFAP output in the frequency domain for the bin of the data block is L = /: = !, 2, ...A. (5.20) 1=1 The time domain output can be obtained by inverse Fourier transforming (5.20) 93 1 ^ 2/(g,m) = ---- — ç = l, 2, ...AT, (5.21) ""^9 t = i where the window coefficient, Wq is included to compensate for the window applied to the time domain data in (5.5). Snbstituting (5.5) and (5.20) in (5.21) — E .1 = 1 LN — E .1=1 n = l -r L N -EE ^1 E ^ ^ 9 1=1 n=l k=l ^ L N 1 ^ _ — EE W E ^ 9 1 = 1 n = l k = l L N Wr. N = E E ^ 'r (5.22) 1=1 n = l . ^ 9 ^ t = l Let N (5.23) ' " ^ 9 t = l The time domain array ontput of the SFAP is then L N 2/(g, TM) = E E 2:(n(m)/i(T.(g). (5.24) 1=1 n = l Thus, hin{q) can be recognized as “equivalent” time domain weights. With a slight reordering, (5.22) can be rewritten as N Wr. hlnijl) — (5.25) ^9 ^ t=l 94 Thus, given the frequency domain weights, the equivalent time domain weights can be obtained by a Fourier transform of the intermediate variable h((g, k) A;) = (5.26) The equivalent time domain weights are thus a function of the ontput time sample, q. In the current work, the equivalent time domain weights, hin{q) are calculated and used to determine the SFAP output performance in (5.24). With the equivalent time domain weights, the same metrics used to determine performance for the STAP model can be directly used in the SFAP performance calculations. 5.3.1 Comments Based on the work of Gupta and Moore in [20, 21], in the current work, windowing of the time samples are used to localize the effects of narrowband interference to a few frequency bins. In the reported results, uniform, Hamming, and Blackman windows were studied with the Blackman outperforming the other windows in a scenario of mixed narrow and wideband interferers. As interference scenarios are expected to include a mix of wide and narrowband interferers, the current work utilizes the Blackman windowing scheme. Moreover, in [22], for an nnshifted reference, the center sample was demonstrated to perform the best. If window correction was implemented, performance was reasonably constant over 70% of the output time samples. Thus, with the exception of Chapter 7 where the effect of shifting the reference is examined, SFAP performance is evaluated at the central output time sample unless otherwise noted. 95 5.4 Summary In this chapter, the theoretical basis for the narrowband SFAP analytical model was described. Equations to transform the analytical time domain covariance matrix to the frequency domain were presented. Also, a narrowband frequency domain ap proach for the power minimization, constrained beamformer, and constrained MMSE weight adaptation approach were given. Lastly, equations for the SFAP output were given which show the “equivalent” time domain weights based on the adaptive weights found in the freqnency domain. 96 CH APTER 6 NORMALIZED VERSUS UN-NORMALIZED SFAP WEIGHTS In space-time adaptive processing, the adaptive weights are found simultaneously. In space-frequency adpative processing, the adaptive weights are calculated for each frequency bin independently. As a result, SFAP performance between frequency bins can vary as each bin’s weight attempts to minimize the energy within the bin. As the weights adapt, the output noise power can also vary between frequency bins. One may be able to control the variation in noise power by normalizing the frequency domain weights on a bin by bin basis. In (5.16), the narrow-band frequency domain weights for the constrained power minimization approach was presented in an un normalized form while in (5.18), the MMSE beamformer was given as normalized in order to fix the gain at a specified level over all frequencies in the desired look direc tion. Either approach can be implemented in an un-normalized or normalized version. For either the normalized or un-normalized approach, if all the frequency bins of the sampled spectrum are occupied by interference energy, the frequency domain weights will adapt in order to minimize the bin energy. However, if the the interference only partially fills the spectrum, the noise power in the unoccupied bins can dominate 97 the output power. Normalizing the weights can help to limit this effect. However, normalizing the bins prevents SFAP from completely shutting down an individual bin. Thus normalizing may not be the best solution if the interference environment is narrowband or a mix of narrow and wideband interference. In this chapter, SFAP performance is analyzed for both power minimization and narrow-band beamforming weights in their normalized and un-normalized versions forms. Additionally, nulling of frequency domain weights associated with unused fre quency bins is investigated. First, the SFAP configuration, signal scenario, and equa tions for the normalized and un-normalized weight approaches are provided. Then, results for SFAP in the presence of wideband only and wideband plus narrow-band RFI are presented. The chapter concludes with a recommendation for a preferred weight adaptation approach. 6.1 SFAP Configuration and Signal Scenario In this study, a 128-bin SFAP using a Blackman window, 32 MHz sample rate, and five-element uniform linear array are used. SFAP performance is evaluated using the pre-correlation power ratio metrics evaluated for a desired signal incident from the broadside direction (6 = 0°). The desired signal is a 20 MHz flat power spectral density signal with input SNR = -20 dB. Also, the performance evaluation is based on the center output time sample of the 128 available where performance is expected to be the best [22]. The array elements of the five element UFA configuration are placed along the x-axis and separated by half-wavelengths as shown in Figure 6.1. The RFI scenario is composed of various sources directions of arrival and center frequencies 98 as shown in Table 6.1. The finite bandwidth RFI sources are modeled as fiat power spectral density signals with bandwidth centered at 2004 MHz. Each RFI source has an input INR of 50 dB. Additionally, each element has a thermal noise voltage modeled as a zero-mean Ganssian random process. The desired signals, interference sources, and thermal noise are, as described in previous chapters, are assumed to be mutually independent zero mean random processes treated as wide sense stationary over the processing time interval. Sensor Locations Figure 6.1: Multi-element eqnispaced co-linear array. Elements are placed along the x-axis. Both the power minimization and beamformer weight adaptation approaches can be represented by h(A;) è-X/c) ûX/c), (6.1) è -\k ) üs{k) h(A;) (6.2) è-XA;) ûX/c)' 99 A (MHz) DOA (Degrees) 2004 30 2004 -35 2004 45 2004 -65 2004 -50 2004 38 2004 55 2004 -40 2004 -30 2004 65 Table 6.1: Mixed wide and narrow-band RFI scenario where (6.1) represents un-normaiized weights, (6.2) represents normalized weights, and the power minimization and beamforming steering vectors are given respectively by u,(A;) = [00 10 0]\VA;, A; = l, 2, ...,7V, (6.3) (6.4) where ti{0) is the directionally dependent steering delay for the element as given in (2.10). In (6.4), ti(9) = 0 due to the broadside incidence of the desired signal. Next, results for both normalized and nn-normalized weights with and without bin nulling are presented. 6.2 Pre-Correlation Performance SFAP performance using power minimization and beamforming nsing normalized and un-normalized weight approaches is evaluated using pre-correlation INR, SINR, 1 0 0 SNR, and SIR power ratios. Additionally, the use of bin nulling to control the noise power in frequency bins not occupied by the desired signal is investigated. Two RFI scenarios are nsed. The first assumes that the RFI bandwidth is matched to that of the desired signal with the first six RFI sources in Table 6.1 modeled as 20 MHz fiat PSD sources. Each has an input INR of 50 dB. The second scenario mixes narrow band CW with wideband 20 MHz RFI sources. The first four RFI sources in Table 6.1 are treated as 20 MHz fiat PSD signals while the last five sources are all treated as narrowband CW signals. All have input INRs of 50 dB. In SFAP, the frequency bin weights are found independently. The worst case scenario is when each interference source is located in the same frequency bin. When this occurs, performance, for the most part, is dictated by the SFAP response of the affected bin. For a wideband source, all bins are affected the same. For a narrowband source, only a few bins are affected. First, results are shown when the RFI sources are all “wideband”. For the purposes of this study, “wideband” refers to the RFI when the bandwidth is as great as the desired signal bandwidth. Next, results are shown for the mixed RFI case when both wideband and CW RFI sources are incident. 6.2.1 Wideband RFI Scenario without Bin Nulling Figure 6.2 shows the output INR and SINR using normalized and un-normalized power minimization weights when up to six 20 MHz bandwidth RFI sources listed in Table 6.1 are present. For the first four interference sources, the results are similar in both cases with the interference power suppressed well below the noise. When the next two RFI sources are added, the un-normalized weights suppress the interference lOI well below the noise while the normalized weights fail to suppress the interference. Figure 6.3 shows the output SNR and SIR for the same scenario. Note that the SINR in Figure 6.2 and SNR results in Figure 6.3 for the first four RFI sources are virtually identical for both approaches. When four wideband sources are present, the output INR is suppressed well below the noise level. When this occurs, the output interference plus noise power is dominated by the noise and the output SINR and SNR are virtually identical: (8-5) where Pd is the desired signal power, Pj is the interference power, and P„ is the noise power at the array output. With respect to SINR and SNR, the normalized weights perform slightly better. When the fifth and sixth RFI sources are added to the scenario, the un-normalized weights continue to suppress the interference well below the noise while the normalized weights do not suppress the interference. However, neither approach produces useable SINR. In contrast, the normalized weights continue to yield good SNR performance. On the surface this seems contradictory. The difference is due to the increase in noise power relative to the signal and interference power. When the interference is not suppressed below the noise, the interference plus noise power is dominated by the interference power. Then SINR and SIR approach the same levels, such that 1 0 2 -1 0 -e- u -21 - -10 -40 -30 - e - u -50 -50 6 # OF INTERFERENCE SOURCES # CE WIDEBAND INTERFERENCE SOURCES a) Output INR b) Output SINR Figure 6.2: Output INR and SINR using power minimization for up to six 20 MHz bandwidth RFI sources, each at 50 dB input INR. Results for normalized “N” and un-normalized “U” weights without bin nulling are shown. -1 0 - e - u -12 -16 -20 -30 -18 -40 - e - u -50 6 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Output SNR b) Output SIR Figure 6.3: Output SNR and SIR using power minimization for up to six 20 MHz bandwidth RFI sources, each at 50 dB input INR. Results for normalized “N” and un-normalized “U” weights without bin nulling are shown. 103 where Pj » Thus, if the interference plus noise power in the array output is dominated by the interference power, the SINR and SIR are approximately equal. If the noise power is also high, the SNR ratio also degrades. As the two previous hgures show, when the hfth and sixth RFI sources are added to the scenario, the SINR and SIR for both weight sets are the same-verifying that interference is dominating the output. However, the SNR for the un-normalized weights degrades significantly due to the relative increase in noise power explaining how the interference power can still be well below the noise power yet SINR and SNR degrade. The same performance trends are observed when the narrowband SFAP beam- forming weight solutions are used. Figure 6.4 displays the output INR and SINR for the same wideband RFI scenario. The output INR performance is virtually identical to that of the power minimization. The SINR performance trend is also the same. However, the absolute SINR increases due to the array gain. Figure 6.5 shows the output SNR and SIR for the same scenario. For up to four wideband RFI sources, performance is similar between the normalized and un-normalized weights. With the addition of the fifth source, the normalized weights cannot suppress the interference power below the noise and both SINR and SIR performance degrade quickly. In the un-normalized case, interference is still well below the noise power through six wide band sources. However, this is due to the relative increase in noise power permitted by the un-normalized weights. Thus, SINR, SIR, and SNR all degrade. The above results demonstrate that the un-normalized weights result in relatively higher noise power at the array output compared to the normalized weights. The extra noise power is primarily due to the frequency bins which do not have interference power present. If interference power was present in the unused bins, the adaptive 104 -1 0 -e- u < ' -20 i S' K-30 z - -10 w -40 -30 -e- u N -50 -50 1 0 1 2 3 4 5 6 # OF INTERFERENCE SOURCES # CE WIDEBAND INTERFERENCE SOURCES a) Output INR b) Output SINR Figure 6.4: Output INR and SINR using beamforming weights in the presence of np to six 20 MHz bandwidth RFI sources, each at 50 dB input INR. Results for normalized “N” and un-normalized “U” weights without bin nulling are shown. -1 0 -e- u -12 Curves Overlap -16 -1 0 -18 -30 - e - U -20 -50 6 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Output SNR b) Output SIR Figure 6.5: Output SNR and SIR using beamforming weights in the presence of np to six 20 MHz bandwidth RFI sources, each at 50 dB input INR. Results for normalized “N” and un-normalized “U” weights without bin nulling are shown. 105 weights would minimize the bin energy. This suggests, in turn, that one can further improve performance by zeroing or nulling the unused frequency bin weights. Next, results for the same scenario when the unused frequency bins are nulled is presented. 6.2.2 Wideband RFI Scenario with Bin Nulling Figure 6.6 shows the output INR and SINR based on the normalized and un- normalized power minimization weights when the frequency bins outside of the de sired signal spectrum are nulled. Results are shown for quiescent through the hrst six wideband RFI sources. Normalized and un-normalized results overlap. Note that for the un-normalized results, now, the interference power is no longer suppressed when more than four wideband sources are incident. Instead the output INR is at the same level as the normalized results. By reducing the noise power due to the unused bins, the output SINR for the two approaches is identical. Figure 6.7 shows the output SNR and SIR for the same scenario. Again, the results are identical for the normal ized and un-normalized cases. Close comparison with the results when the bins were not nulled shows that both approaches beneht from the nulling of the unused bins. However, the un-normalized weights show the most improvement in performance with the SNR increasing by about 3 dB when up to four RFI sources are present. When the un-normalized weights have the noise only bins nulled, performance is about the same as the normalized weights without bin nulling. Figure 6.8 shows the output INR and SINR based on the normalized and un- normalized beamformer weights when the frequency bins outside of the desired sig nal spectrum are nulled. As in the power minimization weights, nulling the unused 106 frequency bins lowers the noise in the output benefitting both the normalized and un-normalized approaches. For the un-normalized weights the output INR no longer indicates that the output interference power is suppressed when the fifth wideband source is added. Limiting the relative increase in noise power at the output shows that in reality, the interference power was not suppressed. Interference power dom inates with the addition of the fifth and sixth wideband sources and SINR and SIR performance degrades. C om m ents In the presence of wideband interference, SFAP performance improved when the frequency domain weights were normalized. Both power minimization and beamform ing weight adaptation approaches showed improved performance with normalization. Output noise power is higher when the un-normalized weights are used. The un- normalized weights appear to provide better suppression. One must be careful. The un-normalized weights increased the output noise relative to both the interference and signal power giving the appearance that the system was able to suppress more than L-1 wideband interference sources. However, when the noise bins were nulled, it was clear that the interference was not suppressed. Moreover, it was found that nulling the frequency domain weights which correspond to the noise only bins improved the performance of the un-normalized weights to the level of performance provided by the normalized weights without bin nulling. The above demonstrated performance in the presence of wideband interference. Next, performance is evaluated when the interference scenario is a mixture of narrow and wideband RFI sources. 107 -1 0 - e - u 4) -20 Curves overlapy Curves C'verlap - -10 -40 -30 - e - u -506 -50 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Output INR b) Output SINK Figure 6.6: Power minimization INR and SINR performance using bin nulling for up to six 20 MHz bandwidth RFl sources, each at 50 dB input INR.Results for normalized “N” and un-normalized “U” weights with nulling of the frequency bins outside of the desired signal 20 MHz bandwidth are shown. -1 0 - e - u Curves eve -12 Curves Overlap -16 -1 0 -30 -e- U -20 -50 6 1 2 3 4 5 6 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Output SNR b) Output SIR Figure 6.7: Power minimization SNR and SIR performance using bin nulling for up to six 20 MHz bandwidth RFI sources, each at 50 dB input INR.Results for normalized “N” and un-normalized “U” weights with nulling of the frequency bins outside of the desired signal 20 MHz bandwidth are shown. 108 -1 0 -e- u -20 Cur\ es Overla - -10 -40 -30 -e- u -506 -50 # OF INTERFERENCE SOURCES # CE INTERFERENCE SOURCES a) Output INR b) Output SINR Figure 6.8: SFAP beamformer INR and SINR performance using bin nnlling for np to six 20 MHz bandwidth RFI sources, each at 50 dB input INR.Results for normalized “N” and un-normalized “U” weights with nulling of the frequency bins outside of the desired signal 20 MHz bandwidth are shown. -1 0 -12 Curves iveriap Curves Overlap -16 -1 0 -18 -30 - e - U -e- u -206 -50 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Output SNR b) Output SIR Figure 6.9: SFAP beamformer SNR and SIR performance using bin nulling for up to six 20 MHz bandwidth RFI sources, each at 50 dB input INR.Results for normalized “N” and un-normalized “U” weights with nulling of the frequency bins outside of the desired signal 20 MHz bandwidth are shown. 109 6.2.3 Mixed Bandwidth RFI Scenario In SFAP, the adaptive weights for each irequency bin are processed independently. Thus, the output noise power can vary from bin to bin as seen in the previous re sults where nulling of the noise only bins improved performance. When bin nnlling was used, performance for the un-normalized and normalized results was about the same. However, the scenario was limited to wideband interference. In this section, performance is examined when only the first four sources in Table 6.1 are allowed to be wideband. The remaining six are modeled as CW sources. Also, note that all of the sources are centered at 2004 MHz. This represents a worst case scenario for the narrowband processor. Only a few frequency bins at most are available to null the additional CW interferers. As the bins are processed independently, failure to minimize the output power in a single bin results in the interference power appearing at the output. The previous results showed that both approaches benefitted from bin nulling. Thus in the following, bin nulling is used throughout. Figure 6.10 shows the output INR and SINR using power minimization when the frequency domain weights outside of the desired signal bandwidth are nulled. Results are shown for both normalized and un-normalized for up to 10 RFI sources of which the first four are wideband. Looking first at the output INR in the left hand plot, both approaches result in identical output INR through the first four wideband RFI sources. When a fifth source is added, the output INR for the normalized weights increases by almost 30 dB; while, the output INR for the un-normalized weights increases by only 5-10 dB. The fifth source is CW with 0% bandwidth. In this case, the un-normalized weights are reducing the interference energy due to the fifth source no well below the noise power. Examining the SINR performance on the right, the un- normalized weights are able to maintain the SINR at abont the same level degrading by only about 1 dB from quiescence through ten RFI sources demonstrating that the relative noise power in the output has not increased. The SINR for the normalized weights is severely degraded with the addition of the fifth source as expected due to the sharp rise in output INR. Figure 6.11 shows the output SNR and SIR for the same scenario. The plot of the SNR shows that the output noise and signal power levels are fairly constant for both approaches. The SIR performance on the right shows again that the SFAP performance under the normalized weights is dominated by the interference power with the SIR level severely degraded when the fifth through tenth RFI sources are present. SFAP using un-normalized weights with bin nulling is performing better. Figure 6.12 shows the output INR and SINR using the beamformer weight solution with bin nulling. The performance trends are the same as discussed above for the power minimization with the exception that the beamformer provides higher SNR and SINR levels due to the array gain. Figure 6.13 shows the output SNR and SIR. The un-normalized weights with bin nulling are able to sustain performance through the ten RFI sources. The difference in performance between the normalized and un- normalized approach is due to the few bins occupied by the CW interference. In the un-normalized weights, SFAP merely nulls the affected frequency bins acting much like a frequency excision process. In the case of the normalized weights, SFAP forces the bin to remain on. Since there are only four available degrees of freedom in each of the affected bins to null interference, SFAP is unable to suppress the interference below the noise power and the system is overwhelmed. I l l 6.2.4 Summary With the addition of the mixed narrow band and wideband RFI scenario, the flexibility of the un-normalized approach to weight adaptation is revealed. The nn- normalized weights are free to completely null a bin if interference energy is detected. However, the normalized weights force the bin to remain on limiting its interference suppression ability to the four degrees of freedom associated with the affected bin. Thus, SFAP using an un-normalized weight solution with nulling of the “noise only” frequency bins is performing better than normalized weights with and without bin nulling. However, in improving the performance of the un-normalized weights, the “noise-only” bins of the frequency domain weights were zeroed. To ensure that in the process of improving the signal power ratio that the signal fidelity has not been compromised, the post-correlation metrics will be shown. 1 1 2 -1 0 -e- u -20 m S '-30 z - -1 0 05 )I — j -40 -30 -e- u N -50 -50 1 2 3 4 5 6 7 8 9 10 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Output INR b) Output SINR Figure 6.10: Power minimization INR and SINR performance for mixed RFI scenario using bin nulling for up to four 20 MHz bandwidth and six CW RFI sources, each at 50 dB input INR.Results for normalized “N” and un-normalized “U” weights with nulling of the frequency bins outside of the desired signal 20 MHz bandwidth are shown. -1 0 -e- u -12 Cun es Cveric -16 -1 0 -30 -e- u -2 0 -50 10 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Output SNR b) Output SIR Figure 6.11: Power minimization SNR and SIR performance for mixed RFI scenario using bin nulling for up to four 20 MHz bandwidth and six CW RFI sources, each at 50 dB input INR.Results for normalized “N” and un-normalized “U” weights with nulling of the frequency bins outside of the desired signal 20 MHz bandwidth are shown. 113 -1 0 -e- u ii ( ) ( ) () () ( ) () n N * • t- )t -20 00 m s S '-30 q; z 05 1-----i -40 • - e - u N -50 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Output INR b) Output SINR Figure 6.12: SFAP beamformer INR and SINR performance for mixed RFI scenario nsing bin nulling for np to four 20 MHz bandwidth and six CW RFI sources, each at 50 dB input INR.Results for normalized “N” and un-normalized “U” weights with nulling of the frequency bins outside of the desired signal 20 MHz bandwidth are shown. -1 0 30 s 20 )-----i) -12 10 -14 0 00 a: ^-10 ^ -16 -20 -30 -18 ) ‘ «...... -40■ - e - u n N t-----u -20 -50 0123456789 10 1 2 3 4 5 6 7 8 9 10 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Output SNR b) Output SIR Figure 6.13: SFAP beamformer SNR and SIR performance for mixed RFI scenario using bin nulling for up to four 20 MHz bandwidth and six CW RFI sources, each at 50 dB input INR.Results for normalized “N” and un-normalized “U” weights with nulling of the frequency bins outside of the desired signal 20 MHz bandwidth are shown. 114 Normed/Nulled PMR (dB) PPR (dB) RMS A Peak Carrier (dB) (dB) (Unitless) Delay (ns) Phase Error (Deg) No/No 12.5 6.6 0.000 0.00 0.0 No/Yes I3.I 6.6 0.005 0.01 5.5 Yes/ No 12.6 6.6 0.00 0.00 0.0 Yes/Yes I3.I 6.6 0.005 0.01 5.5 Table 6.2: Quiescent post-correlation performance under power minimization 6.3 Post-correlation Performance Nulling the noise only bins of the un-normalized frequency domain weights im proved performance in both the wideband RFI scenario and in the mixed narrow and wideband scenario. Pre-correlation metrics demonstrated that the performance matched the performance of the normalized weights and exceeded performance in the case that a finite number of frequency bins were overwhelmed by more than L-1 interferers. In this section, the post-correlation performance for the normalized and un-normalized weights with bin nulling is examined for the mixed narrow and wide band scenario to ensure that the signal fidelity is not overly compromised by the bin nulling process. It will be shown that degradation to the cross-correlation metrics does indeed occur. However, the decrease in performance is minimal. For reference, Tables 6.2 and 6.3 list the quiescent values for the post-correlation metrics using the power minimization and beamformer weights under un-normalized and normalized weights with and without bin nulling. The small systematic errors observed for the quiescent peak delay and carrier phase error are the same for both the power minimiza tion and narrowband beamformer and can be corrected in a practical implementation. 115 Normed/Nulled PMR (dB) PPR (dB) RMS A Peak Carrier (dB) (dB) (Unitless) Delay (ns) Phase Error (Deg) No/ No 14.2 6.6 0.000 0.00 0.0 No/ Yes 14.5 6.6 0.005 0.01 5.5 Yes/ No 14.2 6.6 0.000 0.00 0.0 Yes/Yes 14.5 6.6 0.005 0.01 5.5 Table 6.3: Quiescent post-correlation performance under beam forming Figure 6.14 shows the peak-to-mean ratio (PMR) for the cross-correlation func tion for the power minimization and beamforming weights when the “noise-only” bins of the frequency domain weights are nulled versus the total number of inter ference sources incident. The quiescent response is also shown and corresponds to “0” interference sources. The DOA of the RFI sources are listed in Table 6.1. The first four sources are “wideband” interference with 20 MHz bandwidth. Sources #5 through #10 are CW. All have input INR of 50 dB. Results for both normalized and un-normalized versions of the weight approaches are shown. The figure on the left shows results for power minimization and the plot on the right provides results for the beamformer. In the figures, the PMR for both normalized weight sets show severe degradation when the fifth RFI source is added. Although the source is only CW and thus should only affect a few bins, the resnlt in the peak-to-mean ratio is significant. In comparison. Figures 6.10 and 6.12 show that the corresponding SINR drops by about 20 dB when the fifth source is added. The PMR results for the un-normalized weights are able to sustain the PMR ratio through the full ten sources as the bins corresponding to the severe interference are suppressed. 116 é é 4 «—O—» o o 12 m m 9 a: cr 5 5 Q. Q- 6 -e- u -e- u X N 0123456789 10 # 0 F INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Power Minimization b) Beam Forming Figure 6.14: SFAP cross-correlation peak-to-mean performance evaluated in the broadside direction for power minimization and beamformer weights for mixed RFI scenario when bin nulling is used. Results for normalized “N” and un-normalized “U” weights. Figure 6.15 show the CCF peak-to-peak ratio (PPR) for the same scenario and weight sets. Both the normalized and nn-normalized weight sets maintain the quies cent level of PPR through the first four wideband interference sources. When the fifth source is added, the un-normalized weights experience a 1.4 dB drop in PPR level. For the normalized weights, performance appears unaffected by the interference. How ever, this performance is misleading. As shown in the PMR and SINR results, the desired signal level relative to the interference and noise is now severely degraded. The PPR, RMS Error, peak delay and carrier phase error metrics are based solely on the resultant shape and phase fidelity of the resultant cross-correlation function as affected by the adaptive weights. In practice, the severe degradation in SINR and PMR will either prevent formulation of the CCF (lost signal) or severely degrade the 117 shape, as well as other features due to the included noise and interference. Once the array is overwhelemed, the weights begin to appear much like the quiescent weights. As a result, the unsuppressed interference power appears at the output. However, the quiescent-like weights do not introduce any distortion into the signal and the analytic CCF approaches the quiescent shape. In the un-normalized case, the SINR is maintained and the sustained performance in the PPR metric is an indicator of expected performance. The few bins affected by the interference can be snppressed without affecting the rest of the bins such that resultant CCF is minimally degraded. Figure 6.16 shows the CCF RMS deviation in the mainlobe shape from the ideal. Both the normalized and un-normalized weights maintain close to quiescent perfor mance through the first four wideband sources. When the fifth is added, the un- normalized weights experience a jump from 0.004 to 0.04 (4%) deviation from the ideal which is a minimal change in the mainlobe shape. Figure 6.17 shows the delay in location of the CCF mainlobe peak. As shown in Tables 6.2 and 6.3, the shift in the quiesent peak location is almost entirely due to bin nulling. The effect is minimal-less than 0.01 ns. However, the shift does cause a small offset in carrier phase at the peak as indicated previously as well as in Figure 6.18. The post-correlation metrics demonstrate that bin nulling caused only minimal degradation of the post-correlation performance. These errors were small and for the most part systematic. The effect is the same for both power minimization and the narrowband beamformer weights. For the performance gain which was demonstrated, the use of bin nulling with the nn-normalized weights provided superior performance. 118 10 10 o o Ih 1!^ t » )t )t X CQ ÛÛ S 0 0 O (>- B cd a: CL ü_ O- 4 ü_ - e - u X N 1 0123456789 10 0123456789 10 # 0 F INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Power Minimization b) Beam Forming Figure 6.15: SFAP cross-correlation peak-to-peak performance evaluated in the broadside direction for power minimization and beamformer weights for mixed RFI scenario when bin nulling is used. Results for normalized “N” and un-normalized “U” weights. - e - u - e - u 0.08 0.08 fe 0.06 5 0.06 4* q: 0.04 Cd. 0.04 0.02 0.02 T— -H--T 10 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Power Minimization b) Beam Forming Figure 6.16: SFAP cross-correlation RMS error in mainlobe shape evaluated towards the broadside direction for power minimization and beamformer weights for mixed RFI scenario when bin nulling is used. Results for normalized “N” and un-normalized “U” weights. 119 0.05 0.05 Q < -0.05 -0.05 - - u - e - u-e - 0.1 - 0.1 # O F INTERFERENCE SOURCES # OF INTERFERENCE SOURCES ) Power Minimization b) Beam Forming Figure 6.17: SFAP cross-correlation peak delay evaluated towards the broadside di rection for power minimization and beamformer weights for mixed RFI scenario when bin nulling is used. Results for normalized “N” and un-normalized “U” weights. 1Ur 8 ^ (D t h II—II' * K It K a: cn o o <1—e- o n 4 ■ q: rr q: ÛC 9 ■ LU III • (n I ) ■ < < I 1 CL □ _ -2 rr LU -4 ■ DC (r QC rr < - H ■ 5 u . - e - u -8 ■ N 1 . . . . - i q L 0123456789 10 0123456789 10 # OF INTERFERENCE SOURCES # OF INTERFERENCE SOURCES a) Power Minimization b) Beam Forming Figure 6.18: SFAP cross-correlation carrier phase error at peak location evaluated towards the broadside direction for power minimization and beamformer weights for mixed RFI scenario when bin nulling is used. Results for normalized “N” and un- normalized “U” weights. 1 2 0 6.4 Conclusion SFAP performance using normalized and un-normalized weights with and without nulling of “noise only” frequency bins was examined for wideband and mixed band width RFI scenarios. Through the use of the pre- and post-correlation metrics, it was demonstrated that SFAP performs better overall if the weights are left un-normalized. This allows the un-normalized weight solution the flexibility to nnll or zero the fre quency domain weights in which severe interference is present. In the examples, up to ten sources were easily notched by the nn-normalized weights; whereas, the nor malized weight solution could only perform for up to L-1 interference sources within a frequency bin. However, to obtain the superior performance for the un-normalized weights it was necessary to zero the frequency domain weights associated with the noise only bins. Otherwise, the un-normalized weights resulted in significantly higher noise power in the ontput of the array. Thns, use of un-normalized weights with bin nulling is recommended for SFAP. In summary, • Normalized weights outperformed un-normalized weights when bin nulling is not used. • Nulling the frequency domain weights associated with “noise-only” bins signifi cantly improves performance of un-normalized weights by reducing the relative noise power at the array output. • Un-normalized weights with bin nnlling outperform normalized weights in RFI scenario composed of both narrow and wideband interference sources. 1 2 1 Un-normalized weights allow nulling of individnal frequency bins when all available degrees of freedom associated with the bin are consumed. Shutting down individual bins allows more narrowband sources to be suppressed- sustaining SFAP performance. 122 CH APTER 7 SFAP TIME REFERENCE In space-time adaptive processing, it was found that a simple shift of the time reference could improve performance with the largest improvement observed in the post-correlation metrics. By shifting the reference to the center tap, better symmetry was obtained in the weight sets which in-turn resulted in less non-linear phase in the resulting array transfer function. This was accomplished without any increased com putational burden or sacrihce in the power level of the desired signal. In SFAP one can also shift the time reference. Does this provide a commensurate improvement in per formance? In space-time processing, the adaptive weights are found simultaneously. In SFAP, the frequency domain weights are found for each bin independently. SFAP performance is expected to vary as a function of the output sample position. How ever, as each frequency bin weight is found independently from the others, shifting the output-time reference may not have the same effect as in a space-time processor. In this chapter the effect of shifting the time reference in a narrowband SFAP processor is analyzed. In [13], Godara developed the concept of equivalent time domain weights to demonstrate the equivalence of SFAP and STAP in processing the most current time sample. He developed equations allowing the processing of equivalent output time samples in each processor. However, his work did not show 123 the effect of shifting the time reference in SFAP nor did it include the effect of windowing. In a practical implementation of SFAP, contiguous serial blocks of data are processed in order to receive the information bit stream in the spread spectrum signal. For each block of data processed, an N-bin SFAP can output N time samples. Within these N samples, SFAP performance can vary as shown by Gupta and Moore in [22]. Thus to provide a continuous stream of data, some overlap of the samples is needed. The amount of overlap will impact the computational performance. In practice, 50% overlap is used [17]. However, one may be able to overlap the blocks at a lessor rate without loss in performance as shown in [22]. In this chapter, it will be shown that shifting the time reference in SFAP can significantly affect the overlap rate. Additionally, it will be shown that there is no apparent performance benefit obtained if the reference is shifted. It will also be shown that shifting the reference will change the output sample positions in which poor samples are located. Additionally, shifting the time reference also requires a shift in the window coefficient which must be applied for window correction. Results are shown in this chapter using un-normalized power minimization weights with and without bin nulling. Additionally, a Blackman window is used to multiply the time samples prior to transforming them to the frequency domain. This is done to localize the effect of narrow-band interference sources. Application of the window necessitates that the output samples be corrected for the window. The equations required to correctly apply the window correction are provided in this chapter along with the necessary equations to implement the shift in output time index. First, the output time sample equations in (5.20) through (5.26) are modified to include the effect of shifting the time reference and implementing the window correction 124 Output samples for current block Previous blocks n n-1 n-2 n-(N-1 ) Sample Index Figure 7.1: Output time samples for one data block coefficients. Next, SFAP performance is evaluated as a function of the output time sample index for several time reference shifts for quiescent, wideband only, and mixed wideband plus narrowband RFI scenarios. 7.1 Shifting the Time Reference In this section, the equations necessary to shift the time reference in SFAP and correctly apply window correction are developed. In (5.20) through (5.26), the data block reference “m” was retained to emphasize block processing in SFAP. In a given block, for an N-frequency bin SFAP, N output time samples are available: 2/(1,m), 2/(2,m), ..., 2/(-^,n^), (71) where y{l,m) is the index for the first output time sample, 2/(2, m) is the second output time sample, etc., for the data block at the array output as shown in Figure 7.1. In STAP, one selects a reference tap in addition to a reference element for weight adaptation. The effect of the reference tap selection on STAP performance was demonstrated in Chapter 3. In STAP it was found that performance improved if 125 the time reference was shifted to the center tap position. The reference was imple mented by altering the steering and correlation vectors. In SFAP, having only the frequency domain samples available, one does not have direct access to the output time samples. However, the Fourier relationships between the time and frequency domain allow one to vary the time reference in SFAP as well via the relationship F{r(f + 4 )} = (7.2) where F{-} is the forward Fourier transform operator and r{t) and R{f) are Fourier pairs. Using this relationship one can shift the time reference by applying a linear phase shift to the weights. For example, to shift the time reference from the first output time sample index by — 1 positions, a linear phase shift of = (7.3) is applied to the frequency domain weights of (5.26). For example, if F{/if(M,g)} = A;), (7.4) n = 1, ... N; k = 1, ... N, F{/^;(n-|-(AIo-l),g)} = Â((ç, (7.5) Applying the result to (5.23), one finds that N fif(n + (Alo - 1), g) = ^ fif(A;) e 'lf '"^9 t=l WTI N Z , , X . 2% ^9 ^ 126 where q indicates that the index for the window correction coefficient must also be shifted. In (5.21), the window coefficient in the denominator, Wg, was included to compensate the output sample for the window function which was originally ap plied to the time domain data. In (7.6), the correction coefficient is “hatted” to indicate that the correction coefficient must also be shifted if the time reference has been changed. To find the correct shift, one can track the shift of the zero phase reference in the exponent of (7.6). Setting the exponent to zero for the output time sample, q — n — [Ng — 1) = 0, q — (Ng — 1) = n. (7.7) Now for the reference position, the window coefficient, Wq, which normalizes the ratio ^ is obviously Wn = Wq. Therefore the correct window coefficient for the output time sample for which the time reference is shifted to the Ng time index is q = q — Ng — 1. (7.8) However, if ç < Ng^ q is negative. This is rectified by employing the periodic nature of the Fourier sequence. In (7.6) the period of the sequence is N. Adding N to q in the exponent does not change the phase, but does yield the positive index needed as the argument to the vector of window coefficients, ie 127 (7.9) since k is an integer. Adding # to g in (7.8), q = q + N — [Ng — 1). (7.10) In summary, to find the correct window correction coefficient from the N-length vector of coefficients for the q^^ output sample - ^ f Q -{N o - 1), iiq>Ng, (7 9 + ifg<7V« ' For example, let N = 128 and Ng = 1. The ontpnt signal power and SINR under quiescent conditions is shown in Fignre 7.2. In the example, a Blackman window is nsed with nn-normalized power minimization weights to generate the array response. Bin nulling is not used. The left hand plot shows the output signal power with “WC” and without “NWC” window correction while the right hand plot shows the qnies- cent ontput SINR with and withont window correction. If window correction is not applied, the ontput signal power quality degrades quickly as one moves away from the center sample in the left hand plot. Less than half of the ontput signal samples are within 5 dB of the quiescent output power (-20 dB) and half are within 15 dB of quiescence. The remaining samples are severely degraded. When window correction is applied, all the samples are at the quiescent signal power. Clearly window correc tion helps. In the SINR plot the difference cannot be observed. The curve in which window correction is not applied is offset by 3 dB for display purposes. Since window correction is applied to the output sample, it affects the desired signal, noise, and interference power equally. Thus, the samples which have poor signal quality due to 128 -10 -10 NWC ----- NWC WC WC -15 -15 S -2 0 -20 i 5 dB s p -25 ^-25 t z 3dBOff set 1 05 CD -30 -30 47 Bins—► -35 -35 -40 1 20 40 64 90 110 128 20 40 64 90 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.2; Quiescent SFAP output signal power and SINR evaluated in the broad side direction when window correction “WC” and no window correction “NWC” is applied to the output time samples. SINR results are offset by 3 dB for display pur poses. Results are shown using nn-normalized power minimization adaptation with a Blackman window. The time sample index is used as the reference. a lack of window correction cannot be distinguished in the power ratio metrics. In the example, the ontpnt time reference was set to the first ontput time index. No shift of the window coefficients was required. Next, observe the results when the time reference is shifted to the 40*^ ontpnt time index. In Figure 7.3 the output signal power and SINR for the same Blackman window SFAP processor in quiescence is shown when the time reference is shifted to the 40*^ sample. Results are shown with and without window correction as in the previous figure. In these results, the window correction coefficients were shifted using (7.11). Without correction, the signal power in the region around the 40*^ sample is severely degraded. With window correction, the performance is similar for all output samples. 129 -10 -10 NWC ----- NWC WC ----- WC -15 -15 3 2 0 -20 i S' 3 G-25 47 Bins- a; -25 z 3dBOff set Î w 6-30 -30 -35 -35 -40 -40 20 40 64 90 110 128 20 40 64 90 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.3: Quiescent SFAP output signal power and SINR evaluated in the broad side direction when window correction “WC” and no window correction “NWC” is applied to the output time samples. SINR results are offset by 3 dB for display pur poses. Results are shown using nn-normalized power minimization adaptation with a Blackman window. The 40*^ time sample index is used as the reference. In the SINR plot, the results are again offset from each other by 3 dB. The window correction is transparent in the SINR results. If bin nnlling is used, only 80 (||% ) of the 128 bins, corresponding to the de sired signal 20 MHz bandwidth, would be processed. The frequency domain weights corresponding to frequency bins outside of the 20 MHz desired signal bandwidth are zeroed. Since noise occupies each frequency bin, zeroing these weights reduces the relative noise level at the output by about 2 dB. However, the desired signal power is contained within the 20 MHz of the bins which are not nulled and the output power should remain at -20 dB in quiescence. Figure 7.4 shows the output signal power and SINR with and without bin nulling under quiescent conditions. In the figure, the time reference is at the 1** output time sample and window correction is applied for each 130 -10 -10 -15 -15 ------1 1-20 -20 2 dE■t 0-25 ^-25 z < 05 § -30 -30 -35 -35 ----- ALL ----- ALL ----- BN ----- BN -40 -40 20 40 64 90 110 128 20 40 64 90 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.4; Quiescent SFAP output signal power and SINR with window correction evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is used. Results are shown using un-normalized power minimization adaptation with a Blackman window. The 1®* time sample index is used as the reference. case. When bin nulling is not used, the signal power remains at -20 dB for all the output samples. When bin nulling is used, the central 90 or so samples have a flat re sponse and the signal power is at -20 dB. Nulling of the bins has increased the region which suffers from end effects and about 10 samples at either end are lost due to poor signal quality. This effect is also apparent in the output SINR results observed on the right. First, note that the quiescent SINR has increased 2 dB above the nominal -20 dB quiescent SINR level. This is due to a relative decrease in the output noise power of 2 dB due to bin nulling of the “noise only” bins for the frequency domain weights. The output signal power is unaffected. From the degradation observed in the SINR, we can conclude that the noise power is also increasing rapidly for the end samples. 131 -10 -10 17 Bins -15 -15 1-20 -20 2dB -25 ^-25 ~ 19 Bins z 05 -30 -30 -35 -35 ALL ALL BN BN -40 -40 1 20 40 64 90 110 128 1 20 40 64 90 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.5; Quiescent SFAP output signal power and SINR with window correction evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is used. Results are shown using un-normalized power minimization adaptation with a Blackman window. The 40*^ time sample index is used as the reference. The effects noted when the 1^* time index is used, are the same if the reference is shifted to the 40*^. Figure 7.5 shows the output signal power and SINR when the reference is shifted to the 40*^ sample. The affect of the shift is to move the region with the “end effects” to the 40*^ output time sample. Examining the signal power and output SINR, about 20 bins are lost to poor quality samples if a 5 dB threshold is selected. Bin nulling is causing a larger number of the output samples to be discarded due to poor quality. However, if the array operates only in a quiescent environment, the adaptive array is not necessary. Next, the performance for the various output time samples are evaluated in the presence of wideband and narrowband RFL Results are 132 A (MHz) Bandwidth (MHz) DOA (Degrees) 2004 32 30 2004 32 -35 2004 32 45 2004 32 -65 1998.1 0 -50 2001.3 0 38 2004.2 0 55 Table 7.1: Mixed wide and narrow-band RFI scenario shown for varions time reference positions nsing the nn-normaiized power minimiza tion weights with window correction applied to the output time samples. As above, a Blackman window is used to localize the effects of the narrowband RFI. 7.2 Results In this section, SFAP performance is evaluated as a function of the ontput time sample index and reference for a desired signal incident from the broadside (0 = 0°) direction. The desired signal is a 20 MHz bandwidth flat power spectral density signal with a center frequency at 2004 MHz and inpnt SNR of -20 dB. The RFI sources are also modeled as flat power spectral density signals with bandwidths, center freqnencies, and directions of arrival as shown below in Table 7.1. A 128-bin SFAP with sample frequency of 32 MHz using an nn-normalized narrow-band power minimization weight adaptation approach as described in Chapter 6 is used. Previously, the output signal power and SINR under quiescent conditions was shown. Next, consider the response when a single 50 dB wideband (32 MHz band width) RFI source is incident. Figure 7.6 shows the output signal power and SINR versus output sample position. In the figure, the output time reference is at the R* 133 -10 -10 .-1 5 -15 — — — — S00 a l 0-20 S ' -2 0 z 05 o U3 ■25 -25 ----- ALL ----- ALL ----- BN ----- BN -30 -30 20 40 64 90 110 128 20 40 64 90 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.6; SFAP output signal power and SINR with window correction in the pres ence of a single WB RFI source at 50 dB input INR evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is used. Results are shown using un-normalized power minimization adapta tion with a Blackman window. The 1** time sample index is used as the reference. sample index. The ontput power levels for the good samples with and without bin nulling have increased slightly as the array reacts to the interference. The region with poor signal quality is about the same width. The SINR level has also increased by several dB for both cases. However, the number of samples affected by the shift when bin nulling is used is about the same as in quiescence. The number of samples at the end region with poor signal quality in the case when bin nulling is not used has increased slightly. Figure 7.7 shows results for the same scenario when the output time reference is set at the 40*^ time sample. As in the quiescent results, the effect is a shift of the region with poor samples to the new reference position. Other than the shift, the output performance as a function of the output time indices is the same. 134 -10 -10 .-1 5 -15 L______------— ■ S00 a l 0-20 S ' -2 0 z 05 o U3 ■25 -25 ----- ALL ----- ALL ----- BN ----- BN -30 -30 20 40 64 90 110 128 20 40 64 90 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.7; SFAP output signal power and SINR with window correction in the pres ence of a single WB RFI source at 50 dB input INR evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is used. Results are shown using un-normalized power minimization adapta tion with a Blackman window. The 40*^ time sample index is used as the reference. Figures 7.8 and 7.9 show the output signal power and SINR in the presence of the first three wideband RFI sources. The output power levels change slightly as the array reacts to the different scenario. However, the performance as a function of time index has not changed much from that observed in the quiescent and single wide-band RFI examples. The region containing the unusable samples is centered around the reference index. The number of end samples affected is about the same. However, the SINR plot shows that the power and signal levels are affected differently in the area which is impacted by “end effects”. For the SINR performance, the region is considerably enlarged over the previous results for the case when bin nulling is not used. However, if bin nulling is used, the region affected is unchanged. The performance for the different time references is the same, only the regions affected have 135 -10 -10 -15 O -20 -25 -25 ALL ALL BN BN -30 -30 110 128 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.8: SFAP output signal power and SINR with window correction in the pres ence of three WB RFI sources at 50 dB input INR evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is used. Results are shown using un-normalized power minimization adaptation with a Blackman window. The P^* time sample index is used as the reference. been shifted. This example also illustrates that bin nulling does not adversely affect the number of good signal quality samples available at the output when compared with the result where bin nulling is not used. When interference is present, the number of poor signal quality samples is observed to be about the same for either approach. As the “noise-only” bins become hlled with interference energy, the power minimization algorithm reduces the power in these bins. Eventually, the frequency domain weights are nulled when the individual bins are overwhelmed. Thus, bin nulling and power minimization are essentially performing the same action with the result that no fewer output time samples are degraded due to end effects when bin nulling is used. Next, consider the response in the presence of four wideband RFI sources. In Fig ure 7.10 the output signal power and SINR versus output time sample is shown when 136 -10 -10 -15 Bins 5dB 21 Bin: O -20 -25 -25 ALL ALL BN BN -30 -30 110 128 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.9: SFAP output signal power and SINR with window correction in the pres ence of three WB RFI sources at 50 dB input INR evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is used. Results are shown using un-normalized power minimization adaptation with a Blackman window. The 40*^ time sample index is used as the reference. the 1®* output sample is used as the reference. The output signal power level has changed somewhat from the three wideband RFI sources case; otherwise, the results show the same trends. The SINR shows that both cases have about the same number of degraded SINR samples for the end samples. If the time reference is shifted to the 40*^ sample, the levels and effects are the same as shown in Figure 7.11. Only the region affected has changed. In the plot of SINR, a 5 dB threshold, indicating a drop in level of 5 dB from the region of good quality samples, is drawn for both cases. The thresholds show that the number of samples with degraded quality is now the same in both approaches. Also, the SINR degradation is more severe in the case where bin nulling is not used. However, the signal power plot shows that in this region the output signal power is severely degraded for the case where bin nulling is used. Thus, 137 these samples should be discarded. The mixed scenario containing all seven sources of Table 7.1 is shown in Figure 7.12 when the first output time index is used as the reference. The left hand plot shows the signal power. With the narrowband RFI present, the output signal power curves overlap. Also, the region with poor samples has increased for both approaches. The output samples wherein bin nulling is not used are affected the most. Also, the output signal power is lower. When the narrowband sources are added, the frequency bins associated with the narrow-band sources are “over-whelmed” due to the total number of sources present in the narrow frequency band region. The output of the array for each frequency bin is similar to that of a narrow-band five-element single-tap beamformer operating at the bin center frequency. As such, the single-tap system has only four available degrees of freedom to null the interference. When the four wide band sources are incident, the available degrees of freedom are fully consumed. The additional source results in the bin being over- constrained and the bin “shuts off” in an attempt to minimize the power. When these bins are within the bandwidth of the desired signal, the bin nulling also reduces the output power of the desired signal. For example. Figure 7.14 shows the frequency domain weight magnitudes for the ref erence antenna element for the four wideband and four wideband plus three CW RFI sources. When only the four wideband RFI sources are present, the adaptive array is able to cancel the interference energy by adjusting the various element weights and phases generating narrowband nulls in the direction of the interferers. When a fifth source is added, no additional degrees of freedom are available and the associated bin is nulled as shown in the figure on the right in the presence of four wideband 138 plus three CW sources. The frequency bins occupied by the desired signal and the frequency bins which are nulled are labeled. Since the affected bins fall within the bandwidth of the desired signal, the signal power level at the output is also degraded. Returning to Figure 7.12, the right hand plot shows the output SINR for the same scenario. Here, the results can be somewhat deceiving. The region with “poor” SINR shows fewer samples affected in the mixed bandwidth scenario than in the wideband only scenario. The SINR in the questionable region is sustained by a rapid increase in the output signal power due to edge effects and bin nulling. By nulling the bins associated with the CW interference, the overall bandwidth of the signal energy is artificially reduced. As a result, the signal power has a slightly enlarged region due to “edge effects”. Examining both the signal power and SINR, the conclusion is reached that the addition of more interference energy has reduced the number of acceptable samples. 7.2.1 Comments Examining the output signal power and SINR for the un-normalized power mini mization weights, the results demonstrate that the effect of the shift in time reference is to shift the indices affected by poor signal samples. The number of samples with poor or good quality is unaffected by the shift. The specific signal, noise, and in terference power associated with a sample shifts when the time reference is changed, however its level is unchanged in the new position. This is a reasonable effect when the manner in which the frequency domain weights are formed is considered. Each 139 -10 -10 -15 ca p -20 a; -2 0 -25 -25 ALL ALL BN BN -30 -30 110 128 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.10: SFAP output signal power and SINR with window correction in the pres ence of four WB RFI sources at 50 dB input INR evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is nsed. Results are shown using un-normalized power minimization adaptation with a Blackman window. The 1** time sample index is used as the reference. -1 0 -1 0 -15 5dB ca p - 2 0 a; -2 0 -25 -25 ALL ALL BN BN -30 -30 110 128 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.11: SFAP ontput signal power and SINR with window correction in the pres ence of four WB RFI sources at 50 dB input INR evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is nsed. Results are shown using un-normalized power minimization adaptation with a Blackman window. The 40*^ time sample index is used as the reference. 140 -10 -10 -15 p -2 0 -25 -25 ALL ALL BN BN -30 -30 110 128 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.12; SFAP output signal power and SINR with window correction in the presence of four WB plus three RFI sources at 50 dB input INR evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is nsed. Results are shown using nn-normalized power minimization adaptation with a Blackman window. The 1^* time sample index is nsed as the reference. frequency bin represents an independent narrowband beam former which has a single tap per element. The linear phase shift imposed by the time reference does not affect the nulling or beamforming capability of the individual bin since the phase shift is applied equally to each element. Thus, the principle effect is to “place” the good or poor samples into the most advantageous region. What is the most advantageous region? Since the only effect is to move the region of poor samples within the block of N samples, the primary consideration must be implementation. Assuming that a continuous stream of data samples is needed, unusable or poor quality signal samples cannot be passed to the receiver. For example, in Figure 7.15, a conceptual drawing of non-overlapped data blocks is shown. 141 -10 -10 -15 p -2 0 -25 -25 ALL ALL BN BN -30 -30 110 128 110 128 OUTPUT TIME INDEX OUTPUT TIME INDEX a) Signal Power b) SINR Figure 7.13; SFAP output signal power and SINR with window correction in the presence of four WB plus three CW RFI sources at 50 dB input INR evaluated in the broadside direction when bin nulling “BN” and no bin nulling “ALL” of the frequency domain weights is nsed. Resnlts are shown using nn-normalized power minimization adaptation with a Blackman window. The 40*^ time sample index is used as the reference. A region of poor samples which has been shifted by W samples is indicated. To avoid passing these samples to the receiver, Figure 7.16 shows the minimum overlapping of data blocks to prevent the loss of the samples. One can see that as the region with poor samples moves farther into the block, the overlap rate must be increased. As the overlap rate increases, the processor data throughput is reduced. The most advantageous location is to place the time reference and thus the region of poor samples at the ends of each data block where an overlap rate somewhat greater than ^ is needed to prevent distorted samples from reaching the receiver. 142 1.2 1.2 Signal _ Signal Bandv/idth Bandwidth 0.8 0.8 1 Bin Nulling Bin Nulling ' 0.6 ' 0.6 0.4 0.4 0.2 0.2 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 f (MHz) f (MHz) a) Four Wideband RFI Sources b) Four Wideband + Three CW RFI Sources Figure 7.14: SFAP frequency domain weights in the presence of four wideband and four wideband plus three CW interference sources each at 50 dB INR using the un- normalized power minimization weights with bin nulling. Only the weights for the reference antenna element are shown. Weights are scaled to the maximum weight value of the wideband only case for display purposes. Sam ples N-N„ - / • • • V \/ • • • N Samples N Samples N Samples Figure 7.15: N-length data blocks with no overlap. No + A samples overlapped O verlapped S am ples • •• • •• N S a m p l e s N S a m p l e s N S a m p l e s Figure 7.16: N-length data blocks with minimum overlap. 143 7.3 Conclusions In this chapter, SFAP performance was shown to depend on the output time sam ple position, the time reference, and window correction of the output time samples. Equations were provided to implement the shift in time reference and to apply win dow correction. While bin nulling was shown to significantly improve the performance of SFAP when un-normalized weights are used, quiescent results give the impression that bin nulling significantly reduces the number of output samples with good sig nal quality. However, using mixed narrow and wideband scenarios to demonstrate performance, it was demonstrated that in the presence of interference, bin nulling and/or no bin nulling resulted in the same number of bins with poor quality signal samples. Thus, the recommendation of the previous chapter holds. With respect to SFAP performance versus output time sample position and time referencing, one finds that • Window correction should be used in SFAP. With window correction, the num ber of usable samples exceeds the industry standard of 50 %. In the examples, about 70% to 75% of the output samples were usable. • Shifting the time reference in SFAP neither improves nor degrades output signal power and SINR performance. • Shifting the time reference from the R* output time index shifts the region with poor signal quality into the data block from the end regions increasing the overlap rate in block processing. 144 • Performance estimates should exclude the poor data samples as these samples are generally not used in block processing. As a result, it is recommended that the reference be set at the first or last output sample. If the SFAP reference remains at the 1®* output sample index, the best per formance for the data block occurs at the center output sample. Thus, if performance estimates are based on the best performance obtainable, the center sample should be used. If one is evaluating performance based on an averaging over all samples, one should exclude the samples with degraded performance as these will be overlapped in a practical implementation. 145 CH APTER 8 THE EFFECT OF INTERFERENCE POWER AND BANDWIDTH ON SFAP PERFORMANCE In space-time adaptive processing, it was shown that a single interference source with finite bandwidth can consume multiple degrees of freedom. If the bandwidth occupies the full sampled spectrum, a minimum of N degrees of freedom are consumed by the source for an N-tap STAP. As the interference power level increases, the degrees of freedom consumed also increased which caused oscillations in the output INR performance. Similar behavior is expected from SFAP. However, in a narrowband SFAP pro cessor, the frequency domain weights are processed independently for each bin, in effect, forming an L-element narrowband single tap adaptive processor at the bin center frequency. The overall array output is then formed from the coherent sum of the N-arrays where N is the number of frequency bins per element. An important difference is that the front end bandwidth for each of these processors is not hard limited but the result of an N-length discrete Fourier transform with each bin having a bandwidth of ^ where Fg is the sample frequency. Whereas a narrowband CW source would ideally consume a single degree of freedom of the total LN-1 available, in SFAP the narrowband source also consumes degrees of freedom in the adjacent 146 frequency bins due to energy leakage between bins [17, 30]. Thus, the interference effect on SFAP may be different than that observed in STAP. As discussed previously, interest in using adaptive processors for spread-spectrum communication and navigation systems to suppress interference has grown. Recently, several articles describing SFAP performance in the presence of wideband interference have been published with principle contributions by Godara, Fante et al, and Gupta et al [13, 14, 17, 18, 20, 22]. Fante provides some analysis of SFAP performance with respect to interference bandwidth in [17], wherein SFAP INR and SINR performance is reported for various combinations of wide and narrowband interference. The effect of bandwidth on the SFAP processor was discussed in the context of windowing and multiple order processing. Both were used in an attempt to counter energy leakage which can occur in the Fourier process between frequency bins [30]. In Fante’s study, multiple order processing in which several frequency bins are processed simultane ously as well as use of a Blackman window to reduce the amount of spectral leakage were used. However, both attempts had only limited success. One reason may have been due to the limited number of bins used in the study with results for up to 64 bins shown. In a related study by Fante, et al reported in [18], the output signal power ratio metrics are shown to be computed in the frequency domain. As shown in Chapter 7, for an N-length SFAP, N output time samples are produced. SFAP performance varies significantly across the samples requiring overlap of adjacent data blocks to compensate for samples with poor signal quality. From Parseval’s theorem, the total energy is constant whether calculated in the frequency domain over the signal bandwidth or in the time domain over the time extent of the signal. Thus, 147 by calculating the output signal, interference, and noise powers in the frequency do main, sub-optimal performance estimates may have been obtained. Additionally, the study concluded that the use of a window worsens the spectral leakage as the win dow spreads the mainlobe frequency response. While it is true that a window does spread the mainlobe frequency response over several bins, one must be careful. It will be shown in this chapter that windowing actually reduces the overall frequency spread. How can both be true? A simple example is provided to demonstrate the signihcantly different spectral leakage which can occur with two sources with identi cal bandwidths. If the narrowband source falls exactly on a bin center, the spectral energy is well contained within the bin. However, if the frequency deviates from the bin center, the energy leaks into many bins. In practice, whether due to doppler effects from moving platforms or non-coherent interference sources received by the processor, the received interference signal energy is not expected to be localized to the principle frequency bins as a result of an exact match between the frequency bin center and the interference signal frequency. Thus, it will be shown that windowing can significantly help contain the energy of the finite bandwidth source to only a few bins. In this chapter, the effect of interference power and bandwidth on a narrowband SFAP processor is studied. As in the related STAP study, the eigenvalue distribution of the array covariance matrix together with the output INR and SINR are studied to understand the effects of the interference characteristics on the adaptive process in the frequency domain. It will be shown that in SFAP, a wideband source occupying the full sampled spectrum, as in STAP, consumes a minimum of N degrees of freedom. However the bins are processed independently and the total interference bandwidth in 148 a single narrowband bin is mnch lower than the overall bandwidth in the joint STAP. Thus, the incremental increase in degrees of freedom associated with the increase in the input INR is expected to be lower. It will be demonstrated that for severe levels of input INR, more than a single degree of freedom is consumed by the source. Conversely, whereas in STAP a single CW source consumed only a single degree of freedom without regard to the power level of the source, it will be shown that in SFAP, the CW source can consume up to N degrees of freedom. The effect can be signihcantly reduced by windowing. However, due to the spreading of the mainlobe frequency response by the window, the number of degrees of freedom consumed cannot be reduced to a single bin. Additionally, in multi-tap STAP and conventional adaptive arrays, the output INR for a hnite bandwidth RFI source oscillates as the input INR is increased. The results will show that SFAP also exhibits oscillation in output INR as the input interference power is increased. By observing the total degrees of freedom consumed in the SFAP rather than the degrees of freedom in the primary frequency bins associated with the interference, it will be shown that the oscillations are related to the growth in degrees of freedom consumption which occurs as interference power is increased. The rest of the chapter is organized as follows: hrst, the SFAP conhgnration, signal scenario, and performance evaluation metrics to be nsed are described. Next, the eigenvalue distribution of the interference covariance matrix in the frequency domain for a single interference source versus bandwidth and power is examined. Then, SFAP performance is analyzed using the un-normalized narrowband beamforming weight adaptation approach with bin nulling. 149 8.1 SFAP Configuration and Signal Scenario A 128-bin narrowband SFAP with sample rate of 32 MHz operating at a center frequency of 2004 MHz is used together with the five-element array of Figure 3.1 to study the effect of interference power and bandwidth on SFAP performance. For the examples shown, the output performance is demonstrated using the center out put time sample with the time reference set to the 1®* time index as explained in the previous chapter. The adaptive weights are calculated in the frequency domain based on the unnormalized narrowband beamforming approach described in Chapters 5 and 6. Zeroing or bin-nnlling of the frequency domain weights corresponding to the “noise-only” frequency bins is used to limit the relative noise power appearing at the array ontput. Windowing of the time domain weights is also utilized in the study as it directly impacts the total number of degrees of freedom consumed by a source. In the study, both the uniform and Blackman windows are used to look at the initial eigenvalue distributions. However, once the effect of the non-uniform window on the eigenvalue distribution is demonstrated, further performance evaluation is conducted nsing only the Blackman window. Performance evaluation is conducted assuming that a 20 MHz bandwidth flat-power spectral density signal centered at 2004 MHz is incident on the array from the broadside direction. The desired signal has an input SNR of -20 dB. Interference signals are also modeled as fiat power spectral density signals centered at 2004 MHz. Table 8.1 shows the RFI center frequencies and directions of arrival. Various bandwidths and power levels are assumed for the sources and are specified as they are used. The desired and interference signals are assumed mutually uncorrelated, wide sense stationary, zero-mean random processes. In addition, each antenna element has wideband noise present in its associated front 150 A (MHz) DOA (Degrees) 2004 30 2004 -35 2004 45 2004 -65 2004 -50 Table 8.1: RFI scenario end electronics which is also modeled as a flat power spectral density signal over the sampled spectrum. The noise is assumed wide sense stationary, zero-mean, and un correlated between the various antenna elements. The noise is also assumed mutually uncorrelated with the desired and interference signals. 8.2 Eigenvalue Distribution Consider the eigenvalue distribution of the array covariance matrix in the fre quency domain when only the interference signal is included. The first interference signal in Table 8.1 is evaluated at bandwidths of 0, 10 kHz, 8 MHz, 16 MHz, and 32 MHz. Tables 8.2 and 8.3 show the magnitude of the eigenvalues for the frequency bin corresponding to the 2004 MHz center frequency when the input fNR level is 10 and 60 dB, respectively. Results are shown for both uniform and Blackman windowing in SFAP. In the following examples, un-normalized window functions are used. As a result, the magnitude of the noise eigenvalues are 21.1 dB for the uniform window and 15.9 dB for the Blackman. Thus in Table 8.2, one observes that only a sin gle eigenvalue exceeds the noise eigenvalue magnitude for each bandwidth case. The eigenvalues shown result from the estimate of the interference covariance matrix given by the SFAP analytic model when only interference is included. For a given input 151 INR level, the magnitude of the principle eigenvalue decreases as the bandwidth of the signal increases as shown in the tables where the eigenvalue for the CW bandwidth interference has the largest magnitude. The magnitudes of the remaining eigenvalues are well below the magnitude of the principle eigenvalue and are below the magnitude of the noise thresholds for both the uniform and Blackman windows. When the input INR is increased by 50 dB to an input INR level of 60 dB, the eigenvalue magnitudes also are increased by 50 dB. The resulting eigenvalue magnitudes for the center fre quency are shown in Table 8.3. In the case of the uniform window, the value of the 2nd ]^ig] 2est eigenvalue is increased to a level above the noise threshold for each of the 8MHz through 32 MHz bandwidth cases. The input INR would have to be increased an additional 22 dB to bring the 2"*^ eigenvalue magnitude for the 10 kHz case above the noise level. For the Blackman window, only the principle eigenvalue is above the noise threshold. However, an additional 10 dB increase in the input INR would also raise the magnitude of the 2"^^ eigenvalue for the 8 through 32 MHz bandwidth cases above the noise threshold. Thus, in SFAP, as in STAP, a strong interference source with finite bandwidth can excite multiple eigenvalues consuming multiple degrees of freedom per frequency bin. In the tables, only the eigenvalue magnitudes associated with the center frequency bin for the interference power spectral density was shown. For the 32 MHz bandwidth, all of the frequency bins in the 32 MHz sampled spectrum will have similar eigenvalue distributions. In the CW case, ideally, only a single bin should have a large eigenvalue associated with it. However, the non-uniform window broadens the spectral main lobe of the energy causing adjacent frequency bins to also have active eigenvalues. Also, if the center frequency of the signal is not the same as one of the bin center 152 Uniform Window CW 10 kHz 8 MHz 16 MHz 32 MHz 59T 59T 44.1 41.1 38T -6&6 -5R8 -2&0 -25J -2&6 -6&6 -73.1 -84/4 -7&0 -70.0 -7^4 -7^2 -94.5 -10343 -115.7 -7&4 -93T -9&0 -10341 -120.6 Blackman Window CW 10 kHz 8 MHz 16 MHz 32 MHz 5T5 5T5 3&9 35^ %19 -78T -5&4 -3&7 -38T -41.7 -78T -80T -97.1 -104.0 -118.2 -8E7 -8R2 -97.1 -104.0 -121.5 -8E7 -101.5 -115.7 -121.6 -122/7 Table 8.2: Eigenvalne magnitudes (dB) for a single RFI source with 10 dB input INR. Uniform Window CW 10 kHz 8 MHz 16 MHz 32 MHz 109.1 109.1 94T 91.1 88T -19.6 -0.9 25T 2&3 2R4 -19.6 -23T -34.4 -27^ -2R0 -25.4 -23.2 -4T5 -53J -6T2 -25.4 -4&0 -45.1 -5^8 -7^2 Blackman Window CW 10 kHz 8 MHz 16 MHz 32 MHz 101.5 101.5 8&9 8&9 8Z9 -28.1 -8.4 14.3 11.3 8.3 -28.1 -30.1 -47.1 -5T0 -68T -31.8 -30.2 -47.1 -5T0 -71.6 -31.8 -52M -65M -76.1 -75^ Table 8.3: Eigenvalne magnitudes (dB) for a single REI source with 60 dB input INR. 153 frequencies, more terms in the Fourier summation are needed to represent the signal and as a result spectral leakage occurs. Interference energy is then present in all of the frequency bins. For example, in Figure 8.1, the eigenvalue distribution versus frequency bin is shown for a CW signal when its center frequency is 2004 MHz (a bin center) and 2004.1 MHz. The bandwidth of each frequency bin is 0.25 MHz. Thus, the 2000.1 MHz CW is slightly off the bin center. The left hand plot shows the dis tribution for the uniform window and the right hand figure shows the distribution for the Blackman window. When the center frequency is exactly at 2004 MHz the figure on the left shows that the energy is confined to a single frequency bin for the uniform window as identified by the sharp spike at 2004 MHz. In the Blackman window, the energy is spread over the adjacent bins and the peak eigenvalue magnitude at the center frequency is reduced slightly. When the center frequency is slightly offset from 2004 MHz, the situation is significantly different. For the uniform window, the effect is to spread the energy over all the bins at a level that at its lowest, is only about 40 dB less than the peak. The effect on the Blackman window is minimal. While the interference energy is also spread to all of the bins in the Blackman window, the level is attenuated to a much higher degree than in the uniform window case. The benefit of localizing the narrowband energy is significant in reducing the degrees of freedom which are consumed across the sampled spectrum. As shown by Gupta and Moore in [20], the Blackman window shows significant performance improvement over the uniform window. The eigenvalue distribution clearly shows the benefit obtained for a mixed bandwidth RFI scenario. In the remainder of the results shown in this chapter, the Blackman window is used. 154 2004 MHz 2004 MHz 2004.1 MHz 2004.1 MHz 40 40 -20 -20 -40 -40 -60 -60 -80 -80 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 f (MHz) f (MHz) al Uniform Window b) Blackman Window Figure 8.1: Eigenvalue magnitudes versus frequency for a CW RFI signal at 10 dB INR centered at 2004 MHz and 2004.1 MHz in a 128 bin SFAP nsing uniform and Blackman windowing. At the bin corresponding to the center frequency. Table 8.3 shows that the 2"*^ eigenvalue for the Blackman window is close to the noise threshold. Consider the eigenvalue distribution for the 8 MHz bandwidth RFI when the inpnt INR is at 70 dB. Figure 8.2 shows the eigenvalue distribution for the two highest magnitude eigenvalues in each frequency bin. In the figure, the noise threshold of 15.9 dB is indicated by the heavy dashed line. In the bins corresponding to the 8 MHz bandwidth of the fiat power spectral density signal, both of the eigenvalue magnitudes are above the noise threshold. For about 2 MHz on either side of the 8 MHz bandwidth, the adjacent bins also have a single eigenvalne larger than the noise threshold. The eigenvalue distributions illustrate several characteristics of the interference and SFAP processing which affect performance. First, the single source can consume 155 100 Cû T 3 I = 15.9 dB -20 -40 1985 1990 1995 2000 2005 2010 2015 f (MHz) Figure 8.2: SFAP eigenvalue distribution for a single 70 dB INR 8 MHz bandwidth RFI source centered at 2004 MHz. Results are for a 128 bin SFAP nsing a Blackman window with a 32 MHz sample rate. multiple degrees of freedom per bin if the inpnt INR is high enough. Second, ad ditional degrees of freedom may be consumed which are adjacent to the principle frequency bins corresponding to the signal bandwidth. Windowing helps to limit the consumption of degrees of freedom in the adjacent bins. However, as seen here, a strong source will still consume degrees of freedom in some of these bins. Next, the effect of the interference bandwidth and power on the output INR and SINR per formance is shown. Additionally, the total degrees of freedom or eigenvalues which exceed the noise eigenvalues are shown to relate the effect of the bandwidth on per formance. 156 8.3 INR and SINR Performance As seen previously, windowing can improve SFAP performance by localizing the energy of the RFI to the bins corresponding to the signal bandwidth plus a few bins to either side of the principle frequency bins. The number of adjacent bins is af fected by the input INR of the incident source. Additionally, the number of active eigenvalues in the principle bins can increase to more than one per bin. However a large input INR is needed to cause the effect. Unlike STAP in which additional eigenvalues were excited by fairly modest levels of input INR, in SFAP a much larger increase is needed. Conversely, additional degrees of freedom are consumed in the SFAP processor due to the active eigenvalues of the adjacent frequency bins. Thus in the following, the Blackman window is used to minimize the number of bins af fected. Additionally, it was demonstrated in Chapter 6 that un-normalized weights with bin nulling of the “noise-only” bins performed better than normalized weights. Also, from the eigenvalue distributions shown above, nulling of the bins outside of the desired signal bandwidth will also eliminate any interference energy associated with the bins. The performance results shown below rely on the un-normalized beam- former weight adaptation approach with bin nulling as described in Chapters 5 and 6. Figure 8.3 shows the eigenvalue distribution and output INR for SFAP when the first RFI source in Table 8.1 is incident. In the figure, the input INR ranges from -20 dB to 80 dB. Performance is evaluated when the source has bandwidths of 0, 8, 16, and 32 MHz centered at the desired signal center frequency of 2004 MHz. In the CW case, the center frequency is slightly offset to 2004.1 MHz. Bin nulling is used, however, the eigenvalue distribution shown on the left accounts for the total 157 number of eigenvalues in all 128 freqnency bins. When the interference eigenvalue magnitudes equal the magnitude of the noise eigenvalues, the combined interference plus noise eigenvalue magnitude is 3 dB higher than the noise eigenvalues alone. Thus, eigenvalues in the interference pins noise array covariance matrix which exceed the noise threshold by 3 dB were counted as “active”. By accounting for all of the active eigenvalues in all 128 bins, the degrees of freedom consumed in the adjacent bins are also accounted for. This allows one to track the degrees of freedom consumed in the sidelobes of the window function as well as the principle frequency bins as these can also affect performance. Even the CW source consumes additional degrees of freedom as the input INR level increases. In the figure, for input INR levels greater than 60 dB, the 8 through 32 MHz sources are consuming two degrees of freedom per bin. Thus the total number of eigenvalues exceeding the threshold for the 32 MHz bandwidth soruce is greater than 128. The output INR for the same scenario is shown on the right. The output INR shows oscillations as the input INR increases. The same effect was observed in Chapter 4 for the multi-tap STAB and was also observed for single-tap conventional adaptive arrays by Gupta in [8]. However, in SFAP one also observes oscillatory behavior for the CW bandwidth RFI. This is due to the spectral leakage of the interference into the adjacent freqnency bins as discussed previously. The oscillations are less than those observed for the finite bandwidth RFI cases but are consequential enough to be noted. The sharp “knee” in the output INR which occurs at a 60 dB input INR for the CW case is due to the numerical precision of the computational platform as was also noted and discussed for the multi-tap STAB in Chapter 4. The output SINR for the same scenario is shown in Figure 8.4. The output SINR varies only by about 0.5 dB from quiescence (-11 dB) over the range 158 300 8 MHz -1 0 250 16 MHz 32 MHz 200 ûl -40 100 -50 8 MHz -60 16 MHz 32 MHz 0# -70 80 -20 INPUT INR (dB) INPUT INR (dB) a) Number of Eigenvalues b) Output INR Figure 8.3: SFAP eigenvalue distribution and output INR versus RFI bandwidth and INR level for a single RFI source with CW, 8 MHz, 16 MHz, and 32 MHz bandwidths centered at 2004 MHz. Results are for a 128 bin SFAP using a Blackman window and an un-normalized narrowband beamforming weight adaptation approach with bin nulling. of input INR and bandwidth. However, within the variation of SINR performance, a clear progression is present with respect to RFI bandwidth and power with the output SINR experiencing greater degradation for larger bandwidth and increased input INR. Figure 8.5 shows the eigenvalue distribution and output INR when the second and third RFI sources from Table 8.1 are added to the scenario. Again, results are shown for bandwidths of CW, 8, 16, and 32 MHz. For the three 32 MHz sources, three degrees of freedom are consumed per bin when the input INR for each source is between 0 to 60 dB. When the input INR exceeds 60 dB, the number of eigenvalues associated with the 8-32MHz bandwidth sources increase as seen in the eigenvalue distribution on the left. The output INR shown on the right again exhibits oscillatory 159 -1 0 -11 m"O ce - 1 2 S -1 3 Z)O CW -1 4 8 MHz 16 MHz 32 MHz -15 -20 -10 0 10 20 30 40 50 60 70 80 INPUT INR (dB) Figure 8.4: SFAP output SINR versus RFI bandwidth and INR level in the presence of a single RFI sources when its bandwidth is CW, 8 MHz, 16 MHz, and 32 MHz bandwidths centered at 2004 MHz. Results are for a 128 bin SFAP using a Blackman window and an un-normalized narrowband beamforming weight adaptation approach with bin nulling. behavior for the various sources as the input INR increases. Still, SFAP suppresses the interference. The output SINR for the same scenario is shown in Figure 8.6. The overall degradation in SINR is slightly more than when a single RFI source was incident. Still, the degradation in performance is only slight. However, the sensitivity to RFI bandwidth and power is apparent with the drop in SINR increasing both with RFI bandwidth and power level. Figure 8.7 shows the eigenvalue distribution and output INR when the fourth and fifth RFI source from Table 8.1 are added to the scenario. The five wideband sources excite a full 640 eigenvalues at an input INR of 20 dB. At this INR level, a five degrees of freedom per frequency bin are being consumed by the five wideband sources. As shown in the output INR plot on the right, once the degrees of freedom are consumed. 160 650 600 8 MHz 550 16 MHz 500 32 MHz 450 m ■'° S 400 + 350 ^^00 -1 0 j^-250 200 -30 150 CW -40 100 8 MHz -50 16 MHz — 32 MHz 0# -60 80 INPUT INR (dB) INPUT INR (dB) a) Number of Eigenvalues b) Output INR Figure 8.5: SFAP eigenvalue distribution and output INR versus RFI bandwidth and INR level in the presence of three RFI sources with CW, 8 MHz, 16 MHz, and 32 MHz bandwidths centered at 2004 MHz. Results are for a 128 bin SFAP using a Blackman window and an un-normalized narrowband beamforming weight adaptation approach with bin nulling. -10 "OCO K -1 2 Z) HÛ. ■13 OZ) CW -1 4 8 MHz 16 MHz 32 MHz -1 5 -20 -10 0 10 20 30 40 50 60 70 80 INPUT INR (dB) Figure 8.6: SFAP output SINR versus RFI bandwidth and INR level in the presence of three RFI sources with CW, 8 MHz, 16 MHz, and 32 MHz bandwidths centered at 2004 MHz. Results are for a 128 bin SFAP using a Blackman window and an un- normalized narrowband beamforming weight adaptation approach with bin nulling. 161 the SFAP cannot suppress the interference below the noise level. The eigenvalue distibution and output INR for the five CW, 8 MHz, and 16 MHz bandwidth sources are also shown. SFAP is still able to suppress the interference for each of these scenarios as is shown in the figures. Figure 8.8 shows the output SINR for the same scenario. The output SINR for the 32 MHz scenario shows severe degradation once the input INR reaches 20 dB due to a complete consumption of the degrees of freedom. The 16 MHz scenario starts to degrade at the same rate as the wideband RFI until the input INR reaches about 20 dB. As seen in the output INR plot, the output INR peaks at about this point and then begins to decrease. Once the output INR starts to decrease, the output SINR begins to recover as is seen in the figure. The 8 MHz scenario shows a similar trend although with smaller swings in level. The SINR for the CW scenario is not affected until the input INR is approximately 60 dB. Once the input INR is about 60 dB, the eigenvalue distribution shows a rapid increase in the degrees of freedom consumed by the CW source, and as a result, performance degrades. The eigenvalue distributions demonstrated that in narrowband SFAP, as the input INR increases, the number of degrees of freedom consumed also increases. However, the degrees of freedom associated with the primary frequency bins only increase at very high levels of input INR. The additional degrees of freedom consumed at the modest levels of input INR are associated with the “skirts” or spectral sidelobes of the window function used by the SFAP. A Blackman window was employed to reduce the effect. Furthermore, as the input INR increased, the output INR exhibited oscillations similar to that seen in STAP. Oscillations in output INR are related to the additional degrees of freedom consumed in the same manner as that experienced in STAP. A 162 650 600 8 MHz 550 16 MHz 500 32 MHz 450 m ■'° S 400 + 350 -1 0 ^ ^ 0 0 j^-250 200 -30 150 CW -40 100 8 MHz -50 16 MHz — 32 MHz -60 80 INPUT INR (dB) INPUT INR (dB) a) Number of Eigenvalues b) Output INR Figure 8.7: SFAP eigenvalue distribution and output INR versus RFI bandwidth and INR level in the presence of five RFI sources with CW, 8 MHz, 16 MHz, and 32 MHz bandwidths centered at 2004 MHz. Results are for a 128 bin SFAP using a Blackman window and an un-normalized narrowband beamforming weight adaptation approach with bin nulling. -10 ce - 1 2 g- -1 3 -1 4 8 MHz 16 MHz 32 MHz -1 5 INPUT INR (dB) Figure 8.8: SFAP output SINR versus RFI bandwidth and INR level in the presence of five RFI sources with CW, 8 MHz, 16 MHz, and 32 MHz bandwidths centered at 2004 MHz. Results are for a 128 bin SFAP using a Blackman window and an un- normalized narrowband beamforming weight adaptation approach with bin nulling. 163 significant difference is that while in STAP, a narrowband CW source will experience power inversion of the input INR level, in SFAP the CW source will experience small oscillations in output INR. 8.4 Conclusions An analytic model of windowed SFAP using an un-normalized narrowband beam- former weight adaptation approach with bin nulling was used to demonstrate the effect of interference power and bandwidth on performance. As has been shown for both multi-tap STAP in Chapter 4 and for a conventional array by Gupta in [8], SFAP output INR and SINR performance degrade with increase in interference power and bandwidth. SFAP is also sensitive to the RFI bandwidth and power with the degrees of freedom consumed in proportion to the interference bandwidth and power. While the narrowband SFAP weights are determined independently, the energy which should ideally be confined to a specific bin can still be present in the adjacent bins due to the Fourier process. Thus, the degrees of freedom consumed by a source cannot be limited to only the specific frequency bins which the source would ideally occupy. As a result even a CW source will consume multiple degrees of freedom whether a non-uniform or uniform window is used. While published literature reports that a non-uniform window such as a Blackman window degrades the response, it was shown here that the opposite effect is realized if all the bins are considered and the narrowband source is not exactly on a bin center frequency. In practice it is far more unlikely for narrow band sources to fall exactly on the SFAP bin centers. Thus spectral leakage occurs regardless of window type. However, the sidelobe structnre of non-uniform windows snch as Blackman are signihcantly lower than the nniform window, thus the effect of 164 the spectral leakage is limited and performance is improved for most scenarios. The exception is the wideband source which occupies all bins. In this case, windowing cannot reduce the number of bins affected by the source. In summary, • SFAP output INR level oscillates as interference power increases. • SFAP output SINR level decreases as interference power and bandwidth in creases. However, degradation is minimal. • Increasing the interference power and bandwidth increases the total number of degrees of freedom consumed out of the LN -1 available. - A CW source consumes more than a single degree of freedom due to spec tral leakage into adjacent bins. - In general, any finite bandwidth interference consumes more than the degrees of freedom in its principle bins due to spectral leakage. - Increasing the interference power increases the number of adjacent bins affected thus increasing the total degrees of freedom consumed. - Strong levels of interference power can consume more than one degree of freedom per bin. Generally, this only occurs in the principle bins. • Windowing significantly reduces the degrees of freedom consumed in adjacent bins. 165 CH APTER 9 SFAP AND STAP PERFORMANCE COMPARISON In the preceding chapters, STAP and SFAP performance have been analyzed sep arately. Each systems sensitivity to time referencing, interference power, interference bandwidth, as well as implementation considerations for the general low SNR spread spectrum communications/ navigation application were analyzed. Based on the re sults, recommendations for implementation were offered for each approach. In STAP, a straight-forward time reference based on the center tap position was found to sig nificantly improve post-correlation performance without any sacrifices in algorithm complexity or implementation. In SFAP, it was found that shifting the time ref erence had no impact on the received signal performance but had a large impact on data throughput and computational burden. For the narrowband SFAP, it was also determined that use of un-normalized frequency domain weights with bin nulling of the weights corresponding to the “noise-only” bins improved SFAP performance. The performance of both STAP and SFAP was shown to be affected by the power and bandwidth of the incident interference. In the presence of RFI interference with bandwidths less than 100% of the sampled spectrum, improved SFAP performance by windowing of the time domain samples was verified by examining the degrees of freedom consumed in the processor by interference sources with various bandwidths. 166 Both systems are quickly overwhelmed by wideband RFI with the maximum number of wideband signals which can be suppressed determined by the number of sensor el ements in the array. However, narrowband signals, while easily suppressed by either system, introduce sharp transitions in the array frequency response distorting the received signal. The use of post-correlation metrics was introduced to help determine the effect induced on the received signal by the narrowband signals. In this chapter, the performance of the two adaptive systems is compared based on the recommended approaches and implementation considerations from the previous analysis. However, comparing the two processors based on equal filter lengths (taps and bins) is not performed. SFAP is implemented as a narrowband frequency do main processor and performance is known to be sub-optimal in comparison with the same sized equivalent broadband processor [23, 13, 14]. Rather, performance is com pared based on the computational equivalence of the two processors which provides a more practical comparison of the two approaches. A mixed scenario of narrow and wideband RFI sources at moderate input INR levels is chosen in order to stress the de gree of freedom consumption of both systems. Pre- and post-correlation performance metrics are used to evaluate the processors for both broadside and olf-broadside look directions. The RFI DOA’s are selected to give a mix of well separated and close proximity performance regions. Next, the STAP and SFAP configurations and computational equivalence of the two is described. The RFI and signal scenario is summarized next, and then results are presented for the mixed narrow and wideband RFI scenario. The chapter concludes with a summary of the performance observed. 167 9.1 Processor Configurations and Signal Scenario The constrained MMSE weight adaptation approach in STAP and the un-normalized narrowband beamformer in SFAP provide similar SINR performance and are used in this chapter as the basis for performance comparison. Both configurations use the five element uniform linear array of ideal isotropic elements with half-wavelength separation as shown in Figure 3.1. In STAP, the center tap of the center antenna element is selected as the time ref erence in the steering vector for weight adaptation. Use of the center tap was found to significantly reduce non-linear phase error in the post-correlation metrics without any added computational cost or loss in output SINR. In SFAP, the un-normalized narrowband beamformer is used with bin nulling and a Blackman window to limit spectral leakage of narrowband interference. In contrast to STAP, no advantage was found if the SFAP output time reference was shifted from the first time index in the output samples. Thus the U* output time index is used as the reference. SFAP performance evaluation is based on the central output time sample of the N available for each data block. For the time reference at the U* output time index, the center sample represents the performance obtained from the “good” output samples. For the 128-bin SFAP studied, it was found that only about 30 out of 128 (77%) time samples showed significant reduction in signal quality. In the following computational estimates, it is assumed that 75% of the data samples are usable {a = .75) in (A. 12), ie, 75% of the data block samples are passed to the receiver. Window correction of the SFAP output time samples is also assumed. 168 Next, estimates of the computational cost associated with each processor are pro vided. 9.1.1 Computational Cost Estimates Equations to estimate the computational burden of STAP and windowed-SFAP were originally given by Gupta et al in [25]. These are updated in the Appendix to include the effect of window correction, bin nulling, and use of a more conservative estimate for the forward/ inverse Fast Fourier Transform computational costs. From results in a previous study by Gupta et al in [25], computational costs for a seven-element seven-tap STAP were found to be about the same as that of a seven- element, 128-bin SFAP. Reducing the elements to five and incorporating window correction in SFAP is not expected to change the results significantly. Figure 9.1 shows the cost estimate in GFLOPS (billion floating point operations per second) for STAP when the five element array with a sample rate of 32 MHz is assumed. The plot on the left assumes that the STAP has seven taps per element and that 64, 128, and 256 snapshots are used to estimate the array covariance matrix. Based on results provided by Reed et al in [5] at least 2M — 3 snapshots are needed to provide a reasonable estimate (3 dB loss in signal power) for an M x M sample matrix used in the sample matrix inversion. For 7-taps, 64 snapshots are marginal. For the following analysis and examples, 128 snapshots are selected. If the number of snapshots used in the covariance estimate are increased, the number of operations per second required to estimate the covariance matrix and calculate the weights within the update time increases accordingly. At fast update rates, the computation rate is governed by the matrix estimation and weight calculation. At slow update rates corresponding 169 to the right end of the update time-line, the computation rate is dominated by the fixed weight or “combination” costs as given in (A.8) in the Appendix. This can be observed in the left hand figure as the three curves converge on the horizontal asymptote at 9 GFLOPS. Increasing the number of taps results in a higher fixed rate cost as well as higher computational rates for the sample matrix estimation, inversion, and weight calculation. The figure on the right hand side shows the computational burden when the number of taps is allowed to vary from five to nine taps. The number of samples used in matrix estimation is held constant at 128 for all three cases. At the far right side of the update time scale where the update rate is slow, the three curves are dominated by the fixed costs of 6.3, 8.9, and 11.6 GFLOPS for the 5, 7, and 9-tap STAP configurations. As the update rate is increased, the sample matrix estimation and weight costs quickly drive the computation rate upward. Decreasing the update time to less than 0.1 ms causes the the estimated GFLOPS to increase rapidly. Thus, for an update rate of 0.1 ms the 9-tap STAP requires a computational rate of approximately 25 GFLOPS and the 7-tap STAP ^ 17 GFLOPS. Of the three curves displayed, the 7-tap STAP is selected for the subsequent examples. As will be seen in the computational estimate for SFAP, the 7-tap STAP using 128 snapshots imposes a computational burden on the same order as the SFAP processor. The computational estimate for SFAP is shown in Figure 9.2. The estimate as sumes a 128-length forward and inverse FFT per element, 75% usable samples per data block, application of a window, and window correction. Savings which might be realized by bin nulling are not included. The number of GFLOPS is calculated when 64, 128, and 256 snapshots are used to estimate the frequency domain covari ance matrix. In SFAP, the fixed weight costs are higher than STAP’s. In addition to 170 the “combination” costs, in SFAP the fixed weight costs also include application of the window and the forward and inverse FFT transforms. At very fast update rates the weight calculation dominates the computation rate. At slower update rates, the curves converge to the fixed weight cost as indicated by the horizontal asymptote drawn at 11.6 GFLOPS. Overall, the SFAP computational rate is less sensitive to changes in the update rate than STAP. If the number of bins in SFAP is increased, the computational burden increases. The effect of increasing the number of frequency bins from 64 to 128 and then 256 bins is shown on the right. The estimate assumes that 75% of the samples in the data block are usable. Doubling the number of bins raises the fixed weight cost by a little over 1 GFLOPS. Increasing the number of bins in SFAP has less effect on the computational rate than increasing the number of taps. For the 128 bin windowed-SFAP, operating at a sample rate of 32 MHz, the fixed weight cost is ^ ^ GFLOPS. Figure 9.3 shows the change in computational burden when the number of usable samples per block is varied beween 50%, 65% and 75%. The solid trace shows the results when 75% of the data block samples are assumed usable. As the number of usable samples per block decreases from 75%, the computational burden increases. A decrease from 75% to 50% represents an increase in the computation rate by almost 6 GFLOPS. Previously, it was shown that the number of usable samples per data block can be increased over the current industry practice of 50% by applying window correction to the output samples. The results in Figure 9.2 show the significant advantage in computations which are gained. 171 100 100 128 60 256 60 OCl oCL _ i _1 Ou_ OLL Fixed « ghto 10"^ 10"^ 10' ' 10° UPDATE TIME (ms) UPDATE TIME (ms) a) Computations versus # of Snapshots b) Computations versus Taps Figure 9.1: STAP computational estimate assuming a sample rate of 32 MHz and five antenna elements, left hand plot assumes seven taps per element and 64, 128, and 256 snapshots for covariance matrix estimation. Right hand plot assumes 128 snapshots to estimate the covariance matrix and 5, 7, and 9 taps per element. Based on the above, the 7-tap STAP and 128-bin windowed SFAP using 128 snapshots to estimate the sample covariance matrix impose about the same compu tational burden on the system for update rates greater than about 0.1 ms. The two configurations provide the basis for the following comparisons. 9.1.2 Signal Scenario The performance metrics are evaluated assuming a desired signal with flat power spectral density over a 20 MHz bandwidth centered at 2004 MHz and input SNR of -20 dB. Performance is evaluated for two incidence angles, 0° and 45°, respectively. The interference signals are also modeled as fiat power spectral density signals with corresponding bandwidths, center frequencies, and directions of arrival as listed in 172 40 64 Bins 128 — 128 Bins 256 30 256 Bins 25 20 CL m O _ i O 15 LL _ l O OLL :ost -11.6 GFLOI’s 10 Fixed We Costs: d 12.9 GFLOF 5 10'^ lOT' UPDATE TIME (ms) UPDATE TIME (ms) a) Computations versus # of Snapshots b) Computations versus Bins Figure 9.2: SFAP computational estimate assuming a sample rate of 32 MHz and five antenna elements. Left hand plot assumes 128 frequency bins, 75% of samples are usable, and 64, 128, and 256 snapshots for covariance matrix estimation. In the right hand plot the snapshots are fixed at 128, 75% of the bins are assumed usable, and the number of bins is varied from 128 to 256 bins. 173 40 50% 65% 30 75% 20 m O 15 _ i OLL 10 5 UPDATE TIME (ms) Figure 9.3: SFAP computational estimate assuming a sample rate of 32 MHz and five antenna elements, 128 frequency bins, 128 snapshots for matrix estimation, and assuming th at 50%, 65%, and 75% of the 128 samples have good signal quality (a = 0.5, 0.65, and 0.75). Corresponding fixed weight costs are 17.4, 13.4, and 11.6 GFLOPs. Table 9.1.2 for both scenarios. Each of the interference signals has a 50 dB input INR. Note that the first four signals in the table are wideband and occupy the full 32 MHz sampled spectrum. With the first four signals incident, the processor is almost fully constrained. However, each processor has some additional degrees of freedom remaining which can be used to suppress narrowband RFI signals. The next seven interference sources are CW with staggered center frequencies. These eleven signals represent the example RFI scenario. The desired and interference signals are assumed mutually uncorrelated, wide sense stationary, zero-mean random processes. In addition, each antenna element has wideband noise present in its associated front end electronics which is also modeled as a flat power spectral density signal over the 174 A (MHz) Bandwidth (MHz) DOA for D* Scenario DOA for 2"^^ Scenario 2000 32 30 10 2000 32 -35 -35 2000 32 45 75 2000 32 -65 -10 1995.1 0.0 -50 -50 1998.1 0.0 -50 -55 2001.3 0.0 38 5 2004.2 0.0 55 70 2006.15 0.0 -40 -40 1996.4 0.0 -30 -20 2007.3 0.0 60 0 Table 9.1: RFI scenario sampled spectrum. The noise is assumed wide sense stationary, zero-mean, and un correlated between the various antenna elements. The noise is also assumed mutually uncorrelated with the desired and interference signals. 9.2 Results In Table 9.1.2, the DOA of each RFI source was excluded from ±25° of the desired signal look directions to exclude RFI proximity as a performance factor. In the next section, the performance of the two processors is analyzed for the broadside direction first. Then, performance is evaluated when the desired signal is incident from the non-broadside direction. 9.2.1 Broadside Look Direction In the following, the first interference scenario from Table 9.1.2 is used. When four wideband sources are incident, both the STAP and SFAP systems are close to 175 being fully constrained. Figure 9.4 shows the eigenvalue distributions for the STAP and SFAP noise + interference covariance matrices. The number of active eigen values which exceed the noise magnitudes by 3 dB are plotted versus the number of RFI sources from the Table 9.1.2. The STAP distribution is shown on the left and the SFAP distribution is shown on the right. In SFAP, the total number of eigenvalues active in all 128 frequency bins are shown in the plot. For the STAP, a total of 34 degrees of freedom are available to suppress interference without the system becoming overwhelmed. Once the number of active eigenvalues reaches 34, additional interference cannot be suppressed unless it falls within a co-incidental null. In the 128-bin SFAP, 512 eigenvalues (four degrees of freedom per bin) are available to suppress interference with one degree of freedom per bin remaining to maintain the desired signal response. Thus, we see in the eigenvalue distribution for SFAP, that the 512 bin “limit” is exceeded with the addition of the 4*^ wideband RFI source to the scenario. However, when an individual frequency bin is overwhelmed, the SFAP system can null the individual frequency bin and still sustain performance. Only the individual bin is strictly overwhelmed. As a result, the SFAP system should be able to suppress the interference below the noise while STAP may not be able to null the interference beyond the seven sources in this scenario. Figure 9.5 shows the output INR versus the number of incident RFI sources from Table 9.1.2. The INR of each RFI source is 50 dB at the input to the array. In the plot, the output INR for each system is evaluated for up to four wideband sources without CW sources present, and then, four wideband pins one through seven CW RFI sources. With the RFI excluded from the array broadside direction, performance 176 for each processor is best in the broadside look direction. For np to four wideband plus three CW RFI sources incident, both processors perform well, suppressing the interference well below the noise level. If additional RFI is added beyond the 8*^ source, only SFAP suppresses the interference. The STAP eigenvalue distribution matches the INR performance, ie, through eight RFI sources, less than or equal to 34 degrees of freedom are consumed. With the addition of the ninth RFI source, the number of active eigenvalues has reached 35 and performance degrades quickly. In the SFAP eigenvalue distribution, the degrees of freedom consumed per bin exceed four for multiple frequency bins as shown in the eigenvalue distribution. As each additional narrowband source is added to the scenario, the frequency bin associated with the source is overwhelmed. Each bin already has four degrees of freedom out of five consumed by the wideband sources. However, the number of bins affected by the narrowband RFI is limited by the Blackman window. The bins which are affected and overwhelmed, are nulled. Thus, the interference energy is suppressed for SFAP even though the degrees of freedom for the bin are consumed. As a result, the output INR for SFAP remains below 0 dB for all eleven sources. Figure 9.6 shows the output SINR and CCF PMR evaluated at broadside for the same RFI scenario. Both STAP and SFAP perform well throngh the first seven RFI sources, as expected, given the eigenvalue distributions. SINR performance is virtually identical through the first seven sources while STAP PMR performance is slightly better. When the next two CW sources are added, bringing the total number of sources to four wideband plus five CW interferers, the STAP SINR and PMR degrade quickly falling below that of SFAP. SFAP performance also degrades, but the decrease in performance is less than experienced for STAP. With respect to 177 600 ■<> 550 500 450 S 25 ■o 400 -350 '._300 250 200 150 100 1 NUMBER OF RFI SOURCES NUMBER OF RFI SOURCES a) STAP Eigenvalue Distribution b) SFAP Eigenvalue Distribution Figure 9.4: STAP and SFAP eigenvalue distribution for the broadside direction weight solutions versus the number of interference sources. RFI Sources 1 -4 are wideband, sources 5-11 are CW, each at 50 dB input INR. -e- SFAP STAR -10 "OCO 01 -20 z -3 0 -4 0 -5 0 NUMBER OF RFI SOURCES Figure 9.5: Comparison of STAP and SFAP output INR in the broadside direction versus the number of interference sources. RFI Sources 1 -4 are wideband, sources 5 - 11 are CW, each at 50 dB input INR. 178 -1 0 ii K —1 -15 s00 o; z m -2 0 -e- SFAP -e- SFAP % STAP X STAR -25 1 NUMBER OF RFI SOURCES NUMBER OF RFI SOURCES a) Output SINR b) Output PMR Figure 9.6: Comparison of STAP and SFAP output SINR and CCF PMR in the broadside direction versus the number of interference sources. RFI Sources 1 -4 are wideband, sources 5 - 11 are CW, each at 50 dB input INR. SINR performance, SFAP degradation is less than 1 dB compared to a drop in STAP performance of about 13 dB. For the cross-correlation PMR performance, SFAP’s PMR drops by less than 5 dB compared to a drop in STAP PMR of about 8 dB. Technically, both systems are overwhelmed. A number of SFAP frequency bins have had all five degrees of freedom consumed. However, the combination of independent bin processing and use of un-normalized weights have allowed the SFAP system to continue to sustain performance beyond the STAP processor. Before continuing with the results for the remaining metrics, it is worthwhile to examine the adaptive array receive pattern for several cases. Consider the response when four wideband sources are incident. Figure 9.7 shows the receive patterns for STAP and SFAP. In each plot the magnitude of the pattern is shown versus frequency and angle. The magnitudes are represented by the 40 dB gray scale to the right of 179 each pattern. The plot on the left shows the STAP resnlt over the 32 MHz sampled spectrum bandwidth from 1984 - 2016 MHz. The DOA of the wideband sources are marked with an “X” at the center frequency and DOA of the source. A broadband null appears across the full bandwidth at the DOA of each wideband source. The desired response for STAP is formed by the constrained MMSE weight solution. The response shows a nearly flat response over the desired signal 20 MHz bandwidth centered at 2004 MHz for the desired signal look direction at 0°. The resnltant response provides good signal performance and interference suppression for the scenario. The SFAP response is shown on the right. The same 40 dB gray scale is used to represent the magnitude of the response versus frequency and angle. The wideband interference sources have a well defined null at their respective DOA’s and bandwidths. For the desired signal, the narrowband beamformer provides a well defined beam in the direction of the desired signal from 1994-2014 MHz. Outside of the desired signal bandwidth, bin nulling places a deep null from 1984-1993.75 MHz and from 2014.25 MHz = 2015.75 MHz. Thus any interference in these freqnency regions is suppressed. Next, consider the antenna response when five CW sources (RFI sources 5-9 in Table 9.1.2) are added to the scenario. From the previous results, STAP INR, SINR, and PMR metrics show severe degradation for this scenario. Conversely, SFAP INR and SINR performance show only minimal degradation. SFAP PMR has degraded by approximately 3.5 dB, thus distortion of the cross-correlation function shape must be occurring. Figure 9.8 shows the STAP and SFAP array response in the presence of both wide and narrowband RFI. The same gray scale is used as in the previous plots. The four wideband interference sources are marked by the “X” and the narrowband source locations in frequency and angle are marked by a “C”. The STAP pattern shows 180 9 X 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 f(MHz) f (MHz) a) STAP Response b) SFAP Response Figure 9.7: STAP and SFAP adaptive array response versus frequency and angle in the presence of the first four wideband sources in Table 9.1.2. The RFI DOA’s correspond to the 1®* scenario. that broad regions in the pattern are attenuated in response to the interference. From the INR plots the output INR for STAP is about 5 dB. The scale shows only that the null regions are attenuated by greater than 30 dB. Thus, some suppression is occurring (each source has an input INR of 50 dB). However, the nulls are much less effective. Also, in the desired signal region, the array is no longer able to sustain a broadband beam in the direction of the desired signal. The degrees of freedom have been consumed and overall performance suffers. Clear, the frequency response in the direction of the desired signal will distort the received signal. The SFAP response is shown on the right. Again, the effect of bin nulling is evident and forms a sharp spatial-frequency “mask” or broad null region over the noise- only bins. The response versus angle over the desired signal bandwidth is markedly 181 different for SFAP. Instead of placing broad nulls over many angles and frequencies, SFAP is able to localize the impact of the nulls. The frequency bins overwhelmed by the narrowband sources are effectively notched out. The result is a null over all angles for a small set adjacent frequency bins. The broad frequency nulls for the wideband sources are still localized in angle over the complete bandwidth. In the vicinity of the desired signal, the frequency response is no longer flat due to the notched out bins. However, a good portion of the signal energy is still received while interference is suppressed. Thus, SFAP is able to sustain its performance. However, as more narrowband sources are added and new bins are overwhelmed and subsequently nulled, distortion of the signal must occur. However, the compromise between unsuppressed interference and some signal distortion seems acceptable. Returning to the post-correlation metrics for the broadside look direction, the manner in which the two adaptive systems react to the interference helps aid in un derstanding the observed distortion (or lack of) in the wideband and mixed bandwidth scenarios. Figure 9.9 shows the CCF PPR and RMS error in the CCF mainlobe shape. When only wideband RFI sources are present (sources 1-4 in the table), both systems perform about the same. SFAP RMS error is slightly less. When the narrowband sources are added to the scenario, the shape of the CCF is affected and performance degrades. As expected from the array receive patterns, the narrowband sources affect the STAP performance more than SFAP. When up to three CW sources are added to the scenario, STAP PPR has dropped by about 3 dB and the RMS error has increased to 0.6. An RMS error of 1.0 represents a 100% deviation in the cross-correlation mainlobe shape. With 60% error, significant distortion is occurring. In SFAP, the 182 10 90 75 5 60 0 45 30 -5 _.15 -10 a 0 ^ 15 -15 -30 -20 -45 -60 -25 -75 -30 -90 1985 1990 1995 2000 2005 2010 2015 1985 1990 1995 2000 2005 2010 2015 f (MHz) f (MHz) a) STAP Response b) SFAP Response Figure 9.8: STAP and SFAP adaptive array response versus frequency and angle in the presence of the first nine RFI sources in Table 9.1.2. The RFI DOA’s correspond to the R* scenario. PPR has also changed, with increases of several dB over quiescence. Increases are not necessarily desired as they are also due to a distortion of the mainlobe shape. SFAP RMS error has also increased to just under 0.2. Continued increase in the number of RFI sources reduces the STAP PPR to almost zero and RMS error increases to almost 70% showing severe distortion of the CCF shape. The SFAP PPR also degrades. With all 11 RFI sources, SFAP PPR has decreased by about 3 dB from its quiescent value while SFAP RMS error remains below 0.2. Thus, in the presence of wide and narrowband sources, SFAP distorts the CCF shape less than STAP. Figure 9.10 shows the CCF peak delay and RF carrier phase offset at the peak for the same scenario. In SFAP, bin nulling has introduced a slight systematic offset in peak delay and carrier phase. Still, the shift is slight and conld be corrected in a system. In the figure, both systems are performing well throughout the scenario. 183 - e - SFAP 0.9 X STAP 0.7 m 0.6 HI a: ^ 0.5 CL CL 0.4 0.3 0.2 - e - SFAP n STAP NUMBER OF RFI SOURCES NUMBER OF RFI SOURCES a) CCF PPR b) CCF RMS Error Figure 9.9: Comparison of STAP and SFAP cross-correlation function PPR and RMS error in mainlobe shape in the broadside direction versus the number of interference sources. RFI Sources 1 - 4 are wideband, sources 5-11 are CW, each at 50 dB input INR However, in practice, it is doubtful that the peak could be detected once the PPR has degraded to only a dB. For STAP, discounting the peak delay and carrier phase error performance for the RFI scenarios beyond eight sources, it is still significant that no significant delays or phase non-linearities are observed. As observed in Chapter 3, use of a tap other than the center showed significant delays and phase error. Use of the center tap as the STAP reference has successfully reduced the phase error as shown here. One should also note that in both plots the scale is very small. The phase and delay errors introduced by the processors are minimal. Evaluation of the pre- and post correlation metrics for the broadside direction show that SFAP is outperforming STAP. Through the use of independent frequency bins, un-normalized weights, and to a lesser extent, bin nulling, SFAP is able to sustain 184 -©- SFAP -e- SFAP « STAP » STAP 0.6 _ 0.4 LU ^ - 0.2 û- -0.4 m -10 - < -15 0.6 O - 0 ., -2 0 -25 1 NUMBER OF RFI SOURCES NUMBER OF RFI SOURCES a) CCF Peak Delay b) CCF Carrier Phase Offset at Peak Figure 9.10: Comparison of STAP and SFAP cross-correlation function peak delay and carrier phase offset at the peak evaluated in the broadside direction versus the number of interference sources. RFI Sources 1 -4 are wideband, sources 5-11 are CW, each at 50 dB input INR. performance beyond the point at which all of the degrees of freedom are consumed in a portion of the frequency bins. As more of the frequency bins within the desired signal bandwidth are nulled due to the interference, SFAP performance also must fail. By utilizing the nn-normalized weights with bin nulling, as shown in Chapter 6, the number of wideband and narrowband sources which can be suppressed has been extended. Next, performance for the two systems is evaluated for the off-broadside look direction. 9.2.2 Off Broadside Look Direction In the following, performance is evaluated when the desired signal is incident from 9 = 45°. The RFI scenario consists of the first eleven sources in Table 9.1.2 with the DOA’s listed in the last column. The eigenvalue distributions for the scenario have 185 600 a 550 500 450 ■D 400 '._300 250 200 150 100 1 NUMBER OF RFI SOURCES NUMBER OF RFI SOURCES a) STAP Eigenvalue Distribution b) SFAP Eigenvalue Distribution Figure 9.11; STAP and SFAP eigenvalue distribution when the desired signal DOA is 45°. RFI Sources 1 - 4 are wideband, sources 5-11 are CW, each at 50 dB input INR. only minimal differences due to the source DOA and are shown in Figure 9.11 for completeness. Figure 9.12 shows the output INR versus the number of RFI sources from Table 9.1.2. Since the look direction constraint now corresponds to 6 = 45° and the RFI direction of arrival distribution is different, the absolute output INR versus the in cremental addition of sources is slightly different than the broadside case. However, the overall performance trend is unchanged. Both processors suppress the first seven RFI sources of the table well below the noise power. By the eighth source, the 7-tap STAP has no remaining degrees to null the interference. With additional interfer ence added, the output interference power in the STAP system increases accordingly. The figure shows that SFAP continues to suppress the interference well below the noise power similar to the previous example. In Chapter 6, it was observed that for 186 the normalized weight algorithm, the weights reverted to a state close to quiescent and the cross-correlation metrics improved. There, it was necessary to consider the relative interference power at the output. Here, the situation is somewhat different. Interference power is suppressed at the output. It may be possible to acquire the sig nal. However, as more of the desired signal bandwidth is notched ont in response to the interference, signal distortion increases and the cross-correlation function shape will degrade. Figure 9.13 shows the output SINK and cross-correlation fnnction peak- to-mean ratio (PMR) for the same scenario. SINR degrades rapidly for the STAP system with the addition of the eighth RFI source. For the SFAP, the SINR is again sustained through the scenario. With the addition of the narrowband sources, the frequency response towards the desired signal begins to degrade and the CCF is dis torted. As a result, the PMR performance for both STAP and SFAP degrade. STAP performance degrades faster than SFAP, but both are affected. The cross-correlation function peak-to-peak ratio and RMS error in mainlobe shape shown in Figure 9.14 give a direct indication of distortion in the fnnction. The PPR and RMS metrics illustrate that the shape of the cross-correlation is chang ing as the narrowband sources are added. For STAP, the addition of the sixth source initiates a rapid degradation in PPR. In SFAP, this occurs when the eighth source is added. The RMSE error shows the same effect. SFAP performs slightly better with the knee in the performance curve delayed for several narrowband sources. STAP and SFAP both perform extremely well with respect to delay in the cross correlation function peak and carrier-phase offset at the peak. Figure 9.15 illustrates the performance for the two configurations. Even though, the look direction is no longer at broadside, both systems have virtually no peak delay error. STAP performs 187 -e- S F A P -4 # - S T A P -10 -3 0 -4 0 -5 0 1 2 3 4 5 6 7 8 9 10 11 NUMBER OF RFI SOURCES Figure 9.12: Comparison of STAP and SFAP output INR when the desired signal DOA is 45°. RFI Sources 1 - 4 are wideband, sources 5-11 are CW, each at 50 dB input INR. -1 0 ■O -15 m a: a: 10 z s CL CO -2 0 - e - SFAP - e - SFAP * STAP X STAP -25 1 NUMBER OF RFI SOURCES NUMBER OF RFI SOURCES a) Output SINR b) Output PMR Figure 9.13: Comparison of STAP and SFAP output SINR and CCF PMR when the desired signal DOA is 45°. RFI Sources 1 -4 are wideband, sources 5-11 are CW, each at 50 dB input INR. 188 -e- SFAP 0.9 * STAP 0.7 m 0.6 LU q; ^ 0.5 CL CL 0.4 0.3 0.2 -e- SFAP X STAP 0123456789 10 11 NUMBER OF RFI SOURCES NUMBER OF RFI SOURCES a) CCF PPR b) CCF RMS Error Figure 9.14: Comparison of STAP and SFAP cross-correlation function PPR and RMS error in mainlobe shape when the desired signal DOA is 45° versus the number of interference sources. RFI Sources 1 -4 are wideband, sources 5-11 are CW, each at 50 dB input INR. slightly better than SFAP showing less carrier phase offset at the peak. However, even at its largest deviation, the phase error is small. 9.2.3 Comments SFAP provided more interference suppression than STAP with more sources being suppressed to lower output INR levels. SFAP’s better interference suppression can be attributed to a larger number of degrees of freedom and the ability to shut down individual frequency bins which are overwhelmed by interference. This flexibility is not without cost as the desired signal also suffers due to the lost bandwidth of the array transfer function when the individual bins are nulled. Still, this is preferable to allowing the unsuppressed interference energy to pass to the receiver. The results demonstrated that as additional bins were shut down the cross-correlation function 189 -e- SFAP -e- SFAP » STAP » STAP 2 ns 0.6 _ 0.4 4) LU ^ - 0.2 °- -0.4 m -10 - < -15 0.6 O - 0 ., -2 0 -25 1 NUMBER OF RFI SOURCES NUMBER OF RFI SOURCES a) CCF Peak Delay b) CCF Carrier Phase Offset at Peak Figure 9.15: Comparison of STAP and SFAP cross-correlation function peak delay and carrier phase offset at the peak evaluated when the desired signal DOA is 45° versus the number of interference sources. RFI Sources 1 -4 are wideband, sources 5 - 11 are CW, each at 50 dB input INR. waveform distortion increased affecting PMR, PPR, and RMS error. However, the peak of the cross-correlation function remained at its zero location and phase offset was not severe. In STAP, the degradation in the CCF waveform occurred at fewer sources and was more severe. Thus, SFAP outperformed STAP in all metrics except for carrier phase. 9.3 Conclusions STAP and SFAP performance was compared for wideband only and wideband plus narrowband RFI scenarios at moderate interference power levels. The STAP and SFAP weight adaptation algorithms were selected to provide similar SINR per formance at the array output and impose equivalent computational burdens on the 190 system. Thus, the constrained MMSE weights were used for STAP and the un normalized narrowband beamformer weights with bin nulling were used in SFAP. The computational burdens for the 7-tap STAP and 128-bin windowed SFAP were estimated and equivalence established. Given the equivalent computational cost and similar SINR performance overall, SFAP outperformed STAP. Pre-correlation per formance is similar, however, given that the scenario is mixed wide and narrowband interference, SFAP could suppress more narrowband interference. In maintaining the cross-correlation function (CCF) shape, SFAP outperformed STAP. In summary, • SINR performance is virtually identical, given the interference is suppressed. • SFAP provided superior RFI suppression in the mixed bandwidth scenario sup pressing more sources and to lower output INR levels. • SFAP distorted the CCF shape less as measured by the CCF peak-to-peak ratio and RMS error in the mainlobe shape. • While STAP provided better CCF peak-to-mean ratio in the presence of wide band RFI sources, STAP PMR degraded faster in the presence of narrow and wideband RFI sources. • Excellent CCF peak delay and carrier phase offset performance was provided by both systems. As a last comment, signihcant performance improvements were obtained in SFAP from windowing the time domain samples, use of un-normalized weights, bin nulling, window correction of the output time samples, and the avoidance of poor signal quality samples which is assumed due to overlap in block processing. In STAP, 191 improved performance was obtained throngh the nse of the center tap as a reference. Changes in any of these assumptions can have a significant effect on the final result. 192 CH APTER 10 CONCLUSIONS AND FUTURE WORK The performance of STAP and SFAP was studied using analytic representations of the processor configurations and various interference, noise, and desired signals. Pre- and post-correlation metrics based on the output signal powers and cross-correlation between the received signal and ideal reference signal were used to evaluate system performance in terms of the output signal power and output signal distortion. The general findings for STAP and SFAP are summarized below. In STAP, the effect of varying the reference tap location was studied. While having no impact on the computational burden, strong sensitivity to the reference tap selection was noted in the post-correlation metrics-particularly peak delay and carrier phase-offset errors associated with the cross-correlation peak where the effect is most pronounced in the presence of narrowband interference. The effect on pre correlation metrics, ie output power ratios, was less significant-explaining perhaps, why this effect was not studied extensively in the past. The results demonstrated that moving the tap reference to the center tap location significantly reduced peak delay and carrier phase offset errors associated with the desired signal. For systems where signal distortion is important, use of the center tap as reference is strongly recommended. The effect of interference power and bandwidth on STAP was also 193 studied. The eigenvalue distribution of the spatial-temporal array covariance matrix was used to relate interference characteristics to performance for bandwidths between 0 and 100% of the the STAP sample rate (A). As in the conventional adaptive array with only a single tap per element, the multi-tap STAP was found to be sensitive to bandwidth and interference power. However, in the multi-tap STAP, performance degradation is minimal in comparison. As the bandwidth of the source increases, more degrees of freedom are consumed by the RFI sources. At one extreme is the CW RFI source with 0% bandwidth which only consumes a single degree of freedom regardless of the input power. At the other extreme is the “wideband” source whose bandwidth is equal to or greater than the sample rate. The wideband signal consumes a minimum of N degrees of freedom where N is the number of taps per element. As the input INR is allowed to increase the hnite bandwidth interference source (including the wideband case) consumes increasingly more degrees of freedom as the power level increases. In general, for STAP the findings can be summarized as follows: • Use of the center tap as reference is strongly recommended to keep processor induced signal distortion low. • Narrowband interference sources with finite bandwidth consume multiple de grees of freedom which increases as the interference power level is increased. • Wideband RFI with a bandwidth ~ where To is the tap delay, consumes a minimum of N degrees of freedom which increases as the interference power level is increased. 194 • Output INR oscillates as interference power is increased for finite bandwidth in terference sources. Oscillations are related to the increase in degrees of freedom consumed by the source. • Only minimal degradation to SINR performance is observed, given that inter ference is suppressed. • Narrowband sources have a larger impact on signal distortion than wideband sources. In SFAP, the effect of shifting the time reference was also studied. Because SFAP produces N output time samples per data block, where N is the number of frequency bins per element, the shift effect was studied in the context of all N samples. It was found that shifting the reference has no impact on the number of poor or good signal quality samples. However, the shift has a significant impact on the computational burden. Shifting the reference changes the output sample index where the poor and good quality samples are located. Thus, the primary effect is on data throughput affecting the number of samples overlapped between blocks. The smallest overlap results if the the time reference is placed at the 1^* or output time index. The use of un-normalized versus normalized frequency domain weights was also examined. It was found that un-normalized weights give the processor greater flex ibility to null individual frequency bins if the bins are overwhelmed by interference. However to obtain the superior performance for the un-normalized weights, it was nec essary to zero the frequency domain weights associated with the “noise-only” bins. Otherwise, the un-normalized weights resulted in significantly higher noise power at 195 the output of the array. As a result, more narrowband interference sources can be suppressed with sustained performance in the other pre and post-correlation metrics using the un-normalized weights. The effect of interference power and bandwidth on SFAP was also analyzed. In SFAP, the eigenvalue distribution of the frequency domain covariance matrix for each bin was used to relate the interference characteristics to performance. Additionally, because spectral leakage will occur between frequency bins for a Fourier process, the degrees of freedom in all bins were accounted for. SFAP performance, as in STAP, was found to be sensitive to bandwidth and power with greater degradation occurring as bandwidth and power increased. However, given the interference is suppressed, performance degradation is minimal. Unlike STAP, in SFAP, a CW source can consume up to N degrees of freedom due to spectral leakage. However, windowing is used to significantly reduce this effect. Also, while the total degrees of freedom increase with power, strong levels of interference are required to consume a second degree of freedom in a given bin. In general, for SFAP the hndings can be summarized as follows: • The time reference should be set at the U* or output time index to avoid increasing the number of samples overlapped in adjacent data blocks. - Poor signal quality output time samples should be excluded from perfor mance analysis. • Un-normalized frequency domain weights should be used in frequency domain weight approaches. 196 - Bin nulling of “noise-only” frequency bins should be used in conjunction with the un-normalized weights. • Narrowband interference can consume N degrees of freedom in an N-bin nar rowband SFAP due to spectral leakage. - Windowing is recommended to limit the degrees of freedom consumed due to spectral leakage. - If windowing is used, window correction should be applied to the output time samples. • The total degrees of freedom consumed by a source increases with the input power level. • Strong interference can consume more than one degree of freedom per bin. A comparison of SFAP and STAP in the presence of narrow and wideband inter ference was performed using the above recommendations. Generally SFAP outper formed STAP. Also, it was found that one had to be careful in the interpretation of post-correlation metrics when the system’s “shut down”. In the extreme cases when the system shuts down, the post-correlation metrics can provide erroneous results. Thus, meaning is only derived in the cases when the receiver is expected to be able to acquire and track the signal. As a result, use of the post-correlation metrics was avoided when output SINR degraded by more than 10 dB. The processors compared were selected to provide similar output SINR and to present equivalent computational burdens in terms of GFLOPS. In general, it was found that 197 • SINR performance is virtually identical for the two systems given that interfer ence is suppressed. • SFAP can suppress more narrowband interference than STAP. • SFAP distorted the desired signal less than STAP as determined by the output cross-correlation function mainlobe. • Both STAP and SFAP had minimal peak delay and carrier phase offset error associated with the output cross-correlation peak. • Given computationally equivalent processors, SFAP generally outperformed STAP. 198 10.1 Future Research In the current work, STAP and SFAP were analyzed assuming a multi-path free environment, isotropic sensors, and identical sensor channels. While useful for estab lishing baseline performance expectations, in practice, none of these assumptions may be true. System electronics and individual sensor responses will differ causing channel mismatch. However, this may be corrected using channel equalization. Additional mismatch may be caused by mutual coupling between the sensor elements. Mutual coupling between elements is both frequency and angle dependent and may not be correctable. As a result, channel mismatch is expected to degrade STAP and SFAP performance. However, the effect may be more severe for the STAP system than SFAP due to SFAP’s narrowband implementation. In addition to channel mismatch, wideband multi-path sources are also expected. In the current study only direct-path independent sources were allowed. Multi-path sources will be partially correlated. While several empirical studies of STAP performance in the presence of multi-path have been published, a detailed study of the effect of multi-path on STAP and SFAP is not available. Future research should consider both the effects of channel mismatch and wideband multi-path on STAP and SFAP performance. 199 A PPE N D IX A COMPUTATIONAL ESTIMATES In this appendix formulas are provided to estimate the computational cost in floating point operations per second (FLOPS) for the space-time and narrowband space-frequency processor. The original formulas were developed by S. Ellingson as part of the 1998 study by Gupta et al [25]. In the appendix, the estimates for the space-frequency processor are slightly modified to incorporate window correction of the output time samples and bin nulling of the frequency domain samples. Addi tionally, a more conservative estimate of the forward/inverse fast Fourier transform (FFT) process is incorporated as in [17]. In general, the computations assume • L-element array • N-length tapped delay per element for the space-time processor • A^j-length FFT per element for the space-frequency processor • Equal spaced delays of T = ^ seconds for space-time processor • Sampling rate of Fg samples per second for the space-frequency processor 200 A.l Definitions A “FLOP” is defined as one real floating point operation with the number of FLOP per second as “FLOPS”. Basic operations used in analyzing the processor speed are complex adds, multiplies, vector inner products, and outer products. Basic operations: A complex number. (A.l) z = (0 , h) = Complex add: 2 FLOP (a, h) + (c, d) = ((a + c), (6 4- d)), (At.2) Complex multiply: 6 FLOP (0 ,6) X (c, d) = ((ac — M), (cuf -t- 6c)), (A.3) Inner product: 8N - - 2 FLOP < æ, ^ > = -b 3:22/2 ... a:%r2/jv, (A.4) where < x,y > refers to the complex inner product of two vectors. The outer product, X x^, is used in the estimation of the covariance matrix: 1 R = — ^ x x ^ , (A.5) R = array covariance matrix estimated from Ng snapshots, and, X = snapshot vector of tap voltages. The FLOP estimate for the outer product, x x^, is 6 {LNŸ FLOP, where X = [a:ii 3:12 . . . 2:21 . . . (A .6 ) Exploiting the Hermitian property of x x^ avoids computing the lower diagonal ele ments of the complex outer product reducing the computational estimate to 4(L7V)^ FLOP. Computation of R then requires 201 Ns complex outer products of length LN: —> A^LN^Ns FLOP, Ng — 1 additions of LN x LN complex matrices —> 2{LN)‘^(Ns — 1) FLOP, 4- iVg(real) x {L N ^ (complex) multiplications —> 2{LNŸ FLOP, Total: ^ 6(TAr)^#,FL0P (A.7) The Hermitian property of the matrix can also be exploited in the summation of the matrix reducing the estimated computational cost to ~ ^[LN^Ng FLOP. A.2 Joint Space-Time Processor The space-time processor assumes an L-element array, an N-length tapped delay per element, and equal spaced delays of T = ^ seconds. Given the steady state weights,w, the computational cost of multiplying each tap voltage by its respective weight and summing at a rate of Fg is: LN Fixed weight combination: w^x = y^^WjXj, i=l LN Complex multiplications: -4- 6{LN) FLOP, (LN — 1) Complex additions: —> 2{LN — 1) FLOP, and Total computational cost rate: —> Fg {8LN — 2) FLOPS. (A.8) Weights are only updated every T„ samples. Using the power minimization form for weight adaptation. w = R u, (A.9) 202 T-‘- u — Fts h n where The covariance matrix is estimated from Ns snapshots of the array tap voltages. Using an LU decomposition of R with forward/backward substitution, the total FLOP for calculating the weights is estimated as Weight calcnlation cost estimate = -{LNŸ + 8{LN)‘^ + i{LN) FLOP. (A.10) The total computational cost is obtained from the sum of the fixed weight combina tion, covariance matrix calculation, and weight calculation costs: Total — Fs{8LN — 2) + — 5(LA^)^W + (LiV)^ + 8(LA^)^ + 4(LW)]. (A.11) A.3 Joint Space-Frequency Processor The space-frequency processor computations assume an L-element array, a Wy- length FFT per element, a Wy-length window applied to the tap voltages prior to transforming to the frequency domain, and a data throughput of 100 x a %. The “a ” term can be thought of as sample efficiency. If all the samples are used, a = 1. However, if a < 1 then only a portion of the samples are actually used in the output. The remainder are used as padding. Data overlap is used to avoid passing poor quality output samples to the receiver. The Fourier frequency transform uses the same number of operations in either direction. To transform the data to the frequency domain, the window is applied to the time domain samples first and then the FFT is used to transform the data: 203 Two applications of the window = 4 A^j FLOP, Forward/inverse FFT estimate = 5Ny logg N; FLOP, p Transform rate = —^ iterations per second, a Nf p Total transform costs = — (L + 1)(4 + 5 logg fVy) FLOPS,(A.12) where window correction has been added to Ellingson’s original equations and a more conservative estimate for the FLOP required for the FFT computations are used as in [17]. Non-transform related costs include multiplying the frequency samples by the weights, summing the weighted frequency samples to form the frequency domain output, estimating the covariance matrix for each bin, and updating the frequency domain weights every T„ samples. To obtain the frequency domain output, the frequency domain data samples are combined with the frequency domain weights. The cost of combining the output frequency samples with the weights is p Combining (one bin) = —^(8T — 2), dN f All Nj bins = — (8L — 2). (A.13) If bin nulling is used, only AC bins are processed and the above reduces to Ap out of AT bins = ^ — (8L —2). (A.14) JS f Q . 204 The covariance matrix is estimated one bin at a time from “Ng” snapshots. The cost estimate for the covariance matrix is 5 One bin = Tu a Aj F r 2 ;V All bins = (A.15) 1 ?/ Ck, which can also be reduced if bin nulling is utilized to AT,ontofAry = & ÿ^^. (A.16) f ^ U O i The FLOPs estimate to solve Rw = u every T„ samples in the frequency domain is Weights for one bin = ^ ( -Ü" + 8L^ + 4 l| FLOPS, Weights for Nf bins = Nf ^ I + 8Û + 4 l| FLOPS, (A.17) I O J in which Nf is replaced by Np if nulling is used. A.4 STAP & SFAP Computational Cost Summary The resultant equations for both the space-time and space-frequency processor are: A.4.1 Space-time Fixed weight cost = Fs{8LN — 2) FLOPS (A.18) Covariance matrix = ^5(LW)^W FLOPS (A.19) ^ U Weight calculations = ^(^(T A )^ + 8(TA)^ + 4(TA)l FLOPS (A.20) 205 Total Cost = F,(8T7V - 2) + + 8(TAr)^ + 4(TÆ) j ^ U ^ U ^ ^ J (A.21) A.4.2 Space-frequency p Fixed weight cost = — {(8L — 2) + (L + 1)(4 + 5logg A^/)} (A.22) a Covariance matrix = p^Qpg (A.23) T» « Calculating the weights = ^ |^(L)^ + 8(L)^ + 4(L)| A^/FLOPS (A.24) I q , K O J Total Cost — — { (8L — 2) + (L + 1) (4 + 5 logg A)y)} + A^/ F I + m " - + 4(1)1 I . (A^25) If bin nulling is used in the computation of the frequency domain weights, Total Cost = (8T-2) + —(T + l)(4 + 51ognAf) Ay a ^ a ^ ^ ; ; + Ë lL A i + [Ë(L)S + S{LŸ + 4(L)| I . (X26) 206 BIBLIOGRAPHY [1] B. Widrow, P. Mantey, L. Griffiths, and B. Goode, “Adaptive antenna systems,” Proc. IEEE, vol. 55, pp. 2143-2159, Dec. 1967. 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