SpaceTime Adaptive Processing for Radar
For a listing of recent titles in the Artech House Radar Library, turn to the back of this book.
SpaceTime Adaptive Processing for Radar
J. R. Guerci
Artech House Boston • London www.artechhouse.com
Library of Congress CataloginginPublication Data Guerci, J. R. Spacetime adaptive processing for radar / J. R. Guerci. p. cm. — (Artech House radar library) Includes bibliographical references and index. ISBN 1580533779 (alk. paper) 1. Radar. 2. Adaptive signal processing. 3. Adaptive antennas. 4. Space and time. I. Title. II. Series. TK6580.G84 2003 621.3848—dc21 2003052296
British Library Cataloguing in Publication Data Guerci, J. R. Spacetime adaptive processing for radar. — (Artech House radar library) 1. Adaptive signal processing 2. Radar I. Title 621.3’848 ISBN 1580533779
Cover design by Yekaterina Ratner
2003 ARTECH HOUSE, INC. 685 Canton Street Norwood, MA 02062 All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. International Standard Book Number: 1580533779 Library of Congress Catalog Card Number: 2003052296 10 9 8 7 6 5 4 3 2 1
Contents Preface References
ix x
Acknowledgments References
xiii xiv
1
Introduction
1
1.1
The Need for SpaceTime Adaptive Processing in Moving Target Indicator Radar
1
1.2
STAP for MTI Radar
3
1.3
Book Organization References
7 10
2
Adaptive Array Processing
11
2.1
Introduction
11
2.2 2.2.1 2.2.2 2.2.3
Optimum Spatial (Angle) Beamforming Derivation of the Optimum Beamformer Case I: Additive White Noise Case II: Additive Colored Noise
11 18 19 23
v
vi
SpaceTime Adaptive Processing for Radar
2.3
Optimum Temporal (Doppler/Pulse) Processing
29
2.4
Adaptive 1D Processing
31
2.5
Adaptivity in Nonstationary Environments
39
2.6
Summary References
42 42
Appendix 2A: ULA Antenna Pattern Response
44
Appendix 2B: Derivation of the Maximum Likelihood Sample Covariance Matrix
46
3
SpaceTime Adaptive Processing
51
3.1
Introduction
51
3.2 3.2.1
Need for Joint Space and Time Processing Joint Clutter and Jamming Characteristics
52 63
3.3
Optimum SpaceTime Processing for MTI Radar
65
3.4
STAP
71
3.5
Summary References
71 72
4
Other Important Factors Affecting STAP Performance
75
4.1
Introduction
75
4.2 4.2.1 4.2.2
Channel Mismatch AngleIndependent Channel Mismatch AngleDependent Channel Mismatch
77 78 92
Contents
vii
4.3
Other Interference Subspace Leakage Effects
95
4.4
Antenna Array Misalignment
99
4.5
Nonlinear Arrays
103
4.6
Interference Nonstationarity and the Iceberg Effect
103
4.7
Summary References
106 108
5
STAP for Radar: Methods, Algorithms, and Performance
111
5.1
Introduction
111
5.2 5.2.1
DataIndependent ReducedRank STAP PreDoppler (SignalIndependent) ReducedRank STAP: DPCA and Adaptive DPCA PostDoppler (SignalDependent) ReducedRank STAP Other RankReducing Linear Transformations
114
5.2.2 5.2.3 5.3 5.3.1 5.3.2 5.3.3
114 122 125
DataDependent ReducedRank STAP SignalIndependent Methods SignalDependent Methods Comparison of DataDependent RankReduction Methods
147
5.4.1 5.4.2
StructuredCovariance and ModelBased Methods Covariance Matrix Tapers Other StructuredCovariance Methods
149 151 156
5.5 5.5.1 5.5.2
Illustrative Design Examples SignalIndependent Approach SignalDependent Approach
158 159 161
5.4
129 130 136
viii
SpaceTime Adaptive Processing for Radar
5.6
Summary References
163 165
6
Other Topics
169
6.1
Introduction
169
6.2
Statistical Basis for STAP
169
6.3
STAP Implementation
173
6.4
KnowledgeAided STAP
178
6.5
Summary References
179 179
About the Author
181
Index
183
Preface The burgeoning popularity of spacetime adaptive processing (STAP) is easily demonstrated with a quick keyword search. Although originally coined for airborne multichannel moving target indicator (MTI) radar [1, 2], the acronym has been adopted in many disciplines in which joint adaptive sensor temporal and spatial processing are performed (e.g., multidimensional adaptive filtering). Although a widely published topic, there is a void in bookform coverage at the introductory to intermediate level—a niche which this book is designed to address. Multichannel spacetime array processing is an extremely rich topic area in and of itself (see, for example, the recent authoritative text by H. L. Van Trees [3]). When coupled with the modern marvel of a radar system, it is doubtful that any single source could come close to providing comprehensive coverage. In selecting both the scope and treatment for this book, I had the benefit of having taught both introductory and advanced courses on STAP for radar for several years in industry, academia, and the IEEE Radar Conference series. The audience for such courses is extremely diverse, ranging from pure theoretical academicians to working engineers trying to implement STAP on time and on budget (and most occupation categories in between). Moreover, such forums were generally quite time constrained, placing a premium on efficient teaching methods. SpaceTime Adaptive Processing for Radar is my first attempt to distill the many lessons learned from not only my teaching experiences, but from my interactions with many esteemed colleagues and my own research. My overarching goal is to take the reader equipped with a basic foundation ix
x
SpaceTime Adaptive Processing for Radar
in radar, electromagnetics, and signal processing from basic adaptive array principles to the state of the art in STAP algorithm design. Echoing in my psyche during the creation of this treatment was the oftencited student goal, how do I make STAP work in the real world? The source of such pleas was often the disappointing results achieved when STAP methods were blindly applied to challenging, albeit realistic interference environments. A forensic analysis of such results invariably revealed that the root cause was an extreme oversimplification of the nature of the underlying interference. For example, in the derivation of the optimum spacetime beamformer, stationarity assumptions are imposed that are generally not met in practice (except, of course, in highly contrived scenarios—e.g., a flat uninhabited desert). Thus, after establishing basic firstorder spacetime models for clutter and jamming in Chapter 3, many important secondorder and higher effects are detailed in Chapter 4 before modern STAP algorithms are introduced in Chapter 5. The perhaps obvious reason for this is simply that many algorithms perform very well on firstorder models, but can produce disappointing results when higherfidelity models or real data are employed. This effect is exacerbated as the strength of the interference increases (see the iceberg effect in Chapter 4)—precisely the situation one wants to remedy with STAP. At the time this book was written, the only other books available devoted to STAP for radar were by Klemm [2, 4]. I have copies of both and refer to them often (as evidenced throughout this book). However, these are advanced treatments and, thus, not readily accessible to the newly initiated. Moreover, due to their enormous breadth of coverage, particular topics could not always be explored in great depth. Thus, to come even close to accomplishing my aforementioned objective in a single source, sacrifices in breadth had to be made. For example, I have chosen not to cover explicitly bistatic STAP or newly emerging application areas in synthetic aperture radar (SAR)—topics which are covered in [2, 4]. However, many of the robust STAP algorithms developed in Chapter 5 are applicable to these more exotic applications. Indeed, it is my hope that after completing Chapter 5, the motivated reader could simply learn the particulars of another application and then readily adopt a suitable STAP algorithm from his or her toolkit.
References [1]
Ward, J., SpaceTime Adaptive Processing for Airborne Radar, MIT Technical Report 1015, MIT Lincoln Laboratory, December 1994.
Preface
xi
[2]
Klemm, R., Principles of SpaceTime Adaptive Processing, London, England: IEEE Press, 2002.
[3]
Van Trees, H. L., Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, New York: Wiley Interscience, 2002.
[4]
Klemm, R., SpaceTime Adaptive Processing: Principles and Applications, London, England: IEEE Press, 1998.
Acknowledgments I have been extremely fortunate to have had firsthand interactions and collaborations with some of the founding ‘‘fathers’’ of both the theory of STAP (e.g., Professors Irving S. Reed and Dante C. Youla, who derived the colorednoise multichannel matched filter in continuous form in 1959— the year I was born! [1]) and its practical implementation (e.g., Marshall Greenspan). Their alacrity and clarity are truly inspirational. Other colleagues with whom it has been my privilege to collaborate include E. H. Feria, S. U. Pillai, A. O. Steinhardt, J. S. Goldstein, P. M. Techau, and J. S. Bergin (the latter two of whom were students of Harry Van Trees and were acknowledged in his recent array processing book [2]). Lee Moyer (RPN rules!), Jamie Bergin, Paul Techau, Allen Adler, Michael Zatman, and Amy Alving provided substantial feedback on earlier drafts of this book, and I gratefully acknowledge their support. Also, the anonymous reviewer for Artech House provided many excellent suggestions for improving the technical presentation. I’d also like to acknowledge the gentle yet firm encouragement and assistance of my editor Ms. Barbara Lovenvirth of Artech House. STAP is a vast field and I have benefited over the years from so many generous colleagues that have also directly and indirectly influenced the material in this book. I cannot possibly mention them all, but feel especially compelled to acknowledge M. Zatman, M. Wicks, P. Zulch, E. Baranoski, J. Ward, A. Farina, W. Melvin, and J. Carlini. Last, but by no means least, I would like to acknowledge the support and understanding of my wife, Evelyn, and four children, Sophia, Raymond, xiii
xiv
SpaceTime Adaptive Processing for Radar
John, and Diana. I can never truly make up for the time together lost while I was writing this book, but I will spend the rest of my life trying.
References [1]
Youla, D. C., Theory and Design of MultipleChannel ‘‘Matched’’ Filters, Polytechnic University Report prepared for the Atlantic Research Corporation, June 1959.
[2]
Van Trees, H. L., Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, New York: Wiley Interscience, 2002.
1 Introduction 1.1 The Need for SpaceTime Adaptive Processing in Moving Target Indicator Radar A quick literature search on the keyword ‘‘STAP’’ will quickly serve to illustrate the ubiquitous nature of this rich topic area. Although originally coined for airborne multichannel moving target indicator (MTI) radar [1, 2], spacetime adaptive processing (STAP) has been adopted in many disciplines in which joint adaptive sensor temporal and spatial processing are performed (e.g., multidimensional adaptive filtering). The need for joint space and time processing in either airborne (or spaceborne) MTI radar arises from the inherent twodimensional (2D) nature of ground clutter. Figure 1.1 illustrates the distribution of ground clutter (power spectral density arising from the twoway antenna pattern) for a sidelooking radar employing a uniform linear array (ULA; see Chapter 2) as a function of normalized angle (i.e., the spatial dimension) and normalized Doppler (temporal dimension)—as defined in Chapter 2. Notice that in the absence of both ownship platform motion (v = 0 case in Figure 1.1) and internal clutter motion (ICM; see Chapter 4), the groundclutter returns have no Doppler shift and can thus be characterized by a simple onedimensional (1D) process; that is, filtering of ground clutter can be achieved by simply rejecting signals from the 0Hz Doppler filter [3, 4]. Moving targets are thus detected by looking for the presence of signals (of sufficient strength) in the remaining Doppler filters. 1
2
SpaceTime Adaptive Processing for Radar
Figure 1.1 Illustration of (a) clutter isorange ring and angleDoppler dependence for a ULA radar in constantvelocity motion, and (b) corresponding angleDoppler clutter ‘‘ridge’’ arising from main lobe and sidelobe antenna patterns. Note that a target of interest generally must compete with either sidelobe or main lobe clutter leakage.
Introduction
3
However, when ownship motion is present, ground returns are Doppler shifted according to the relationship [1, 2] f d = 
(1.1)
where and f d are the normalized angle and Doppler respectively (see Chapter 3) and  is a proportionality constant that depends on ownship speed, pulse repetition frequency (PRF), and array interelement spacing (see Chapter 3). As illustrated in Figure 1.1, ownship velocity gives rise to 2D clutter ‘‘ridges’’ whose slope is governed by (1.1). Moreover, due to the Nyquist frequency of ±PRF/2 [3], there may be more than one distinct ridge due to Doppler aliasing [1, 2]. As a consequence, it is likely that for every target of interest, there is a region of ground clutter with the same Doppler frequency—but offset in angle—that potentially could mask the target through either sidelobe or main lobe leakage [1, 2]. The objective of spacetime processing is to place a null in the angleDoppler beampattern where clutter may compete. Due to the aforementioned 2D nature of clutter, this null will depend on the target angleDoppler of interest [1, 2]. While (1.1) establishes the basic need for spacetime (angleDoppler) processing, it in no way implies the need for adaptivity. Since it is conceivable to know the relative orientation of the radar and ground surface through ownship inertial navigation systems (INS) and global positioning systems (GPS), as well as basic ground topology and clutter prediction maps [5], one could simply steer a deterministic null in the predicted clutter direction [see (1.2)]. Unfortunately, a multitude of practical realworld considerations severely limits the efficacy of this technique. Indeed, many of these considerations constitute both the impetus and subject matter of this book. Examples of such considerations include channel mismatch, clutter heterogeneity/ nonstationarity, ICM, and other interference subspace leakage (ISL) mechanisms (see Chapters 2 through 4). It is these effects that give rise to the inherent need for adaptivity—hence STAP.
1.2 STAP for MTI Radar Figure 1.2 shows the basic schematic architecture of a STAP beamformer. For ease of analysis we will nominally consider an N element ULA and a coherent processing interval (CPI) consisting of M pulses with a fixed pulse repetition interval (PRI) (as notionally depicted in Figure 1.2)—although
4
SpaceTime Adaptive Processing for Radar
Figure 1.2 Spacetime (angleDoppler) beamformer consisting of N independent antenna channels and M pulses comprising the CPI. A specific angleDoppler pattern is obtained by judicious selection of the complex linear combiner weights.
many of the concepts and techniques discussed are readily extensible to nonuniform/nonlinear [6] arrays and variable PRFs (see Chapters 4 and 5). As with ordinary onedimensional (1D) spatialonly beamforming, a 2D angleDoppler (spacetime) beampattern can be formed by a judicious selection of the complex linear combiner weights {w i } [1, 2] (see Chapter 3 for further details). As discussed in Chapter 2 (and elsewhere [7]), to maximize the response [and/or signaltonoise ratio (SNR)] to a uniform narrowband plane wave corresponding to a given angle and Doppler, the linear combiner weight vector w = vec (w 1 , . . . , w NM ) (where vec ( ) is the vector operator that simply forms an NM dimensional column vector from the NM elements w 1 , . . . , w NM ) should be set equal to the anticipated structure of the desired signal s; that is, w = s (see Chapter 2 for further details). For the ideal ULA example under consideration, the result is the 2D sinc pattern shown in Figure 1.3. Note the presence of a peak main lobe and both angle and Doppler sidelobes, which for a sinc pattern may be unacceptably high. Although tapering, or ‘‘windowing,’’ can be used to reduce these sidelobes, they come at the expense of an increase in main lobe width and a decrease in SNR (see Chapter 2) [7]. Moreover, channel mismatch places further limitations on the amount of tapering possible (see Chapter 2). As detailed in Chapter 2, a linear combiner of this type has NM degrees of freedom (DoFs). Thus, for a desired angleDoppler look direction s, we
Introduction
5
Figure 1.3 Example spacetime (angleDoppler) beamformer response for N = M = 16 sidelooking ULA. Note the presence of a finitewidth main lobe and significant angle and Doppler sidelobes.
can also specify up to NM − 1 ‘‘nulls.’’ Specifically, if the set of NM angleDoppler vectors {s, j 1 , . . . , j NM − 1 } is linearly independent [8], then a deterministic set of beamformer weights can be obtained from the linear set of equations (see Chapter 2) w′s = 1 w′j 1 = 0
(1.2)
⯗ w′j NM = 0 Unfortunately, due to the relative strength of potential interference (clutter and/or jamming), and the aforementioned practical limitations on deterministic nulling, this approach is untenable in practice [1, 2]. Instead, the radar community has looked towards statistical signal processing for a means of addressing the limitations of deterministic nulling. In particular, clutter and jamming are treated as stochastic processes and the optimum spacetime beamformer is derived via a statistical optimization procedure
6
SpaceTime Adaptive Processing for Radar
[1, 2]. Although there are many variations on the theme (many of which are described in this text), the basic underlying optimum beamformer result is given by w = R −1 s
(1.3)
where as before, s is the desired signal, and w is the optimal set of combiner weights. R is the positivedefinite NM × NM dimensional covariance matrix [1, 2] associated with the total interference (clutter plus jamming plus receiver noise) for the given range cell under test [1, 2]. Since we have already stipulated that it is not possible to characterize the spacetime interference a priori with sufficient precision to effect adequate nulling, it may at first appear that (1.3) is a mere tautology. However, under conditions discussed in Chapters 3 and 4, and elsewhere [9], it may be possible to estimate R from sample data obtained in the normal course of radar operation. Indeed, it is the estimation of R on the fly that is the true basis for the inclusion of the word ‘‘adaptive’’ in STAP [1, 2]. However, as will be made clear in Chapters 2 through 5, great care may be required in practice to glean the essential information from the environment since the amount of data available for estimating R is fundamentally limited not only by the resolution properties of the radar, but by the environment itself [10]. This latter consideration is particularly acute in highly heterogeneous and nonstationary terrain that can be expected to occur in a variety of potential realworld operating conditions [5, 10, 11]. It is the intention of this book to introduce the reader to an important cross section of these issues and to introduce STAP methods which have the potential of meeting those challenges. A clear illustration of the conspicuous differences between realworld and synthetic, or mathematically idealized, ground clutter is afforded by examination of the returns from the Defense Advanced Research Projects Agency (DARPA) mountaintop radar [12], shown in Figure 1.4. Although groundbased, airborne MTI radar operation was emulated by a socalled inverse displaced phase center array (IDPCA) technique in which successive pulses were transmitted from a different phase center along the ULA [12]. Thus, as far as ground clutter was concerned, the illumination appeared to be moving. Figure 1.5 shows the processed returns as a function of range and Doppler for both the actual radar and a simulation based on a homogenous clutter assumption [5]. The radar was situated at the White Sands Missile Range (WSMR). It is clearly not hard to imagine the mismatch pitfalls that
Introduction
7
Figure 1.4 Photograph and specifications of the DARPA mountaintop radar. The IDPCA array was used on transmit to emulate platform motion.
await the unsuspecting radar engineer if care is not taken to properly segment the returns for both training (estimating the covariance—see Chapter 5) as well as establishing a constant falsealarm rate (CFAR) detection scheme [13].
1.3 Book Organization Chapter 2 introduces the basic concepts of adaptive array processing by first considering the 1D ULA case. Surprisingly, many of the issues that are at the forefront in advanced STAP research, such as convergence and robust adaptivity in heterogeneous and nonstationary environments [10], can be introduced with the 1D case. After introducing the basic operation of a 1D phased array, the optimum beamformer is derived that maximizes the total signaltointerferenceplusnoise ratio (SINR). This formulation introduces the concept of the interference covariance matrix R . Next, since R is
8
SpaceTime Adaptive Processing for Radar
Figure 1.5 (a) Actual processed clutter returns for the DARPA mountaintop radar; and (b) simulated returns assuming homogenous clutter. (Courtesy of Information Systems Laboratories (ISL), Vienna, Virginia.)
not known a priori in practice, the concept of an adaptive array is introduced as a practical attempt to estimate R on the fly from the assumedtobeavailable sensor data [1, 2]. Properties of the sample covariancebased adaptive beamformer are examined via an eigen analysis of the adaptive beamformer. This leads to the introduction of the method of principal components to help alleviate the potentially deleterious effects of finite sample estimates [14]. Finally, a covariance augmentation technique [7], referred to as covariance matrix tapers (CMTs) [15], is introduced to illustrate a potential solution to the socalled stale weights problem. Chapter 3 introduces the basic underlying signals and interference models for STAP for airborne MTI radar. Again, beginning with a ULA radar, the basic 2D properties of ground clutter are developed along with a derivation of Brennan’s rule that provides an analytical estimate of the underlying rank of the clutter covariance matrix [1, 2]. The optimum spacetime beamformer is then derived and applied to a notional N = M = 16
Introduction
9
sidelooking ULA radar. As with the 1D case, the optimum beamformer requires knowledge of the total interference covariance matrix. However, the dimensionality of the spacetime covariance is dramatically greater (due to the Kronecker relationship of angle and Doppler)—thus exacerbating the convergence/implementation issues introduced with 1D adaptive array processing. Next, in Chapter 4, important refinements to the signal models introduced in Chapter 3 are detailed. These factors can have a profound deleterious impact on STAP performance in practice and must therefore be properly accounted for in the beginning design stages. Factors considered include channel mismatch (narrow and finite bandwidth) [16], ICM [17], antenna crabbing, other ISL effects [11], as well as clutter heterogeneity [10]. In Chapter 5, a broad class of STAP algorithms is presented and taxonomized based on their fundamental formulation. Emphasis is placed on those techniques that make the most efficient use of the adaptive DoFs of the STAP processor. The primary objective of Chapter 5 is to provide a pallet of techniques that can be combined and tailored to a specific application in order to maximize STAP efficiency—loosely defined as maximizing convergence rate (minimizing requisite sample support), yet closely preserving nearoptimal performance. The chapter culminates with a design example in which the requisite sample support for a sample matrixbased STAP processor is reduced by an order of magnitude with commensurate computational reductions in the adaptation process. Finally in Chapter 6, some important additional topics are discussed. In Section 6.2, the statistical basis for STAP is introduced. For the additive Gaussian noise (AGN) case, it is shown that the max SINR [or minimum meansquared error (MSE)] spacetime beamformer based on the total interference covariance matrix is indeed optimal in the statistical sense (e.g., maximum likelihood sense [7]). References are provided for the reader interested in exploring the latest research trends related to nonGaussian extensions of relevance to STAP. In Section 6.3, basic realtime implementation issues are discussed with an emphasis on the popular datadomain QR factorization approach [18]. This formulation leads to an extremely efficient parallelprocessing implementation of the sample matrix inverse (SMI). Due to its ubiquitous adoption, we have extended it to include both diagonal loading and CMT—the latter covariance augmentation being of value in implementing a class of robust STAP algorithms introduced in Chapter 5. Lastly, we introduce the concept of knowledgeaided STAP (KASTAP), which has recently emerged to address the difficulties encountered when attempting to adapt to extremely complex clutter environments [5].
10
SpaceTime Adaptive Processing for Radar
References [1] [2] [3] [4] [5]
[6] [7] [8] [9]
[10]
[11]
[12]
[13] [14]
[15]
[16] [17]
[18]
Ward, J., SpaceTime Adaptive Processing for Airborne Radar, MIT Technical Report 1015, MIT Lincoln Laboratory, December 1994. Klemm, R., Principles of SpaceTime Adaptive Processing, London, England: IEEE Press, 2002. Barton, D. K., Radar Systems Analysis, Dedham, MA: Artech House, 1976. Schleher, D. C., MTI and Pulsed Doppler Radar, Norwood, MA: Artech House, 1991. Guerci, J. R., ‘‘KnowledgeAided Sensor Signal Processing and Expert Reasoning,’’ Proc. of 2002 Workshop on KnowledgeAided Sensor Signal Processing and Expert Reasoning (KASSPER), Washington, D.C., April 3, 2002 (CDROM). Zatman, M., ‘‘Circular Array STAP,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 36, No. 2, April 2000, pp. 510–517. Van Trees, H. L., Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, New York: Wiley Interscience, 2002. Strang, G., Introduction to Linear Algebra, Wellesley, MA: WellesleyCambridge Press, 1998. Reed, I. S., J. D. Mallet, and L. E. Brennan, ‘‘Rapid Convergence Rate in Adaptive Arrays,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 10, No. 6, November 1974, pp. 853–863. Melvin, W., (ed.), ‘‘SpaceTime Adaptive Processing and Adaptive Arrays: Special Collection of Papers,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 36, No. 2, April 2000, pp. 508–509. Guerci, J. R., and J. S. Bergin, ‘‘Principal Components, Covariance Matrix Tapers, and the Subspace Leakage Problem,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 38, No. 1, January, 2002, pp. 152–162. Titi, G. W., and D. F. Marshall, ‘‘The ARPA/NAVY Mountaintop Program: Adaptive Signal Processing for Airborne Early Warning Radar,’’ Proc. of 1996 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vol. 2, Atlanta, GA, May 7–10, 1996, pp. 1165–1168. Nitzberg, R., Radar Signal Processing and Adaptive Systems, 2nd ed., Norwood, MA: Artech House, 1999. Tufts, D. W., R. Kumaresan, and I. Kirsteins, ‘‘Data Adaptive Signal Estimation by Singular Value Decomposition of a Data Matrix,’’ Proc. of the IEEE, Vol. 70, No. 6, June 1982, pp. 684–685. Guerci, J. R., ‘‘Theory and Application of Covariance Matrix Tapers for Robust Adaptive Beamforming,’’ IEEE Transactions on Signal Processing, Vol. 47, No. 4, April 1999, pp. 977–985. Billingsley, J. B., Radar Clutter, Norwood, MA: Artech House, 2001. Zatman, M., ‘‘The Effect of Bandwidth on STAP,’’ Proc. of IEEE Antennas and Propagation International Symposium, Vol. 2, Baltimore, MD, July 21–26, 1996, pp. 1188–1191. Farina, A., and L. Timmoneri, ‘‘RealTime STAP Techniques,’’ Electronics and Communication Engineering Journal, Vol. 11, Issue 1, February 1999, pp. 13–22.
2 Adaptive Array Processing 2.1 Introduction In this chapter we introduce both optimum and adaptive, spatial (angle) and temporal (Doppler) processing. Many of the key concepts and issues in STAP can be readily grasped by first successively examining the 1D constituent angle and Doppler processing chains. In Chapter 3, we consider the joint spatiotemporal (2D) process required for advanced clutter suppression in airborne radars.
2.2 Optimum Spatial (Angle) Beamforming Consider the effect of a unitamplitude, narrowband electromagnetic (EM) plane wave impinging on an N element ULA with interelement spacing d , as depicted in Figure 2.1. In this context, the term narrowband refers to a signal whose modulation bandwidth, B, is such that c /B >> Nd [1]. This condition insures that propagation delay across the array is manifested as a simple phase shift [see (2.1)]. The impact of finite bandwidth will be addressed in Chapter 4. If we define the plane wave angle of arrival (AoA), 0 , relative to boresight as shown in Figure 2.1, the complex envelope phasor at baseband observed at the n th antenna element as a function of 0 is sn = e
j 2 (n − 1)
d sin 0
11
, n = 1, . . . , N
(2.1)
12
SpaceTime Adaptive Processing for Radar
Figure 2.1 A ULA adaptive beamformer. The spatial (angle) response is controlled by the complex weighting factors.
where is the operating wavelength (units consistent with d ), and 0 is the AoA in radians [1]. Note that the phase progression of a plane wave is linear across a ULA. The impact of a nonlinear array geometry for airborne MTI radar is discussed in Chapter 4. By introducing multiplicative complex weighting factors, w n , in each receive channel of the array (as shown in Figure 2.1), the output response can be maximized for any desired AoA. More specifically, let y denote the scalar beamformer output defined as N
y=
∑ w n* s n = w′s
(2.2)
n =1
where an asterisk (*) denotes complex conjugation, the prime (′ ) denotes vector complex conjugate transposition (i.e., Hermitian transpose [2]), and
Adaptive Array Processing
13
the vectors s ∈ ⺓N and w ∈ ⺓N (⺓N denotes the space of N dimensional complex vectors) are defined as
⌬ s( 0 ) =
冤冥冤 s1 s2 s3 ⯗ sN
e j0 e
=
e
j 2
d sin 0
j 2 (2)
d sin 0
⯗
e
d j 2 (N − 1) sin 0
冥
(2.3)
and
w=
冤冥 w1 w2 w3 ⯗ wN
(2.4)
To maximize the response of the beamformer to a plane wave arriving at an AoA of 0 , we have the following elementary optimization problem: max  y  = max  w′s  2
{w}
2
(2.5)
{w}
subject to  w  = constant < ∞ 2
where the constant gain constraint is imposed to insure a finite solution. Since w′s is simply an inner (or dot) product of the two nonzero norm vectors w and s [2], Schwarz’s inequality can be applied [2, 3]; that is,  w′s  2 ≤  w  2  s  2, with equality if (and only if) the vectors are colinear. This yields the result w = s
(2.6)
where is a scalar chosen to satisfy the normalization constraint. This result is intuitive inasmuch as it states that the optimum beamformer applies phase corrections to each channel to compensate for the
14
SpaceTime Adaptive Processing for Radar
time delays associated with the plane wave traveling across the array. More specifically, at the n th channel, the beamformer forms the product w n* s n ⬀ e −j␣ n e j␣ n = 1, thereby canceling the phase term. The beamformer thus coherently integrates the signal outputs from each channel [1]. Without this compensation, destructive interference would occur with a commensurate decrease in output signal strength. An important and fundamental limitation of linear beamformers is that they will also, in general, respond to signals arriving from other angles. This can lead to many practical problems, as strong unwanted signals from other directions can interfere with the signal of interest. To visualize this effect, consider the response of the above beamformer to plane waves arriving from −90° to +90° with w = s, where s is chosen to be a plane wave with AoA of 0 . If we let x n denote the output of the n th receive channel (and x ∈ ⺓N the corresponding vector of received values), then the total beamformer output, steered to angle 0 , is given by N
y = w′x = ∑ x n e
−j 2 (n − 1)
d sin 0
(2.7)
n =1
which has the form of a discrete Fourier transform (DFT) [1]. If we set d / = 0.5 (halfwavelength element spacing), N = 16, 0 = 30°, and xn = e
j 2 (n − 1)
d sin
, n = 1, . . . , N
(2.8)
while varying from −90° to +90°, the beamformer response of Figure 2.2 results (with = 1). Note that for a ULA, the beamformer response can be obtained via a fast Fourier transform (FFT) [1]. For this particular example, an analytical expression for the normalized beamformer response (  y  ≤ 1) exists and is given by [1] (see Appendix 2A)
1
 y = N

冋 冋
册 册
d (sin ( ) − sin ( 0 )) d sin (sin ( ) − sin ( 0 ))
sin N

(2.9)
The beamformer response of Figure 2.2 has several interesting features. First, note that signals close to 30° also produce a significant response. This region is generally referred to as the main lobe. The lobing structures outside
Adaptive Array Processing
15
Figure 2.2 ULA beamformer response steered to 0 = 30° (with respect to boresight).
of the main lobe region are referred to as sidelobes. For the ULA considered, the first nearin sidelobe is approximately 13 dB down from the peak of the main lobe [1]. The nulltonull width of the main lobe for a ULA depends on the number of elements, N (i.e., the antenna length), and the scan angle, d 0 . It is easily obtained by setting N (sin ( ) − sin ( 0 )) = , and solving for MB , which is the first null. The nulltonull width is thus given by 2 MB = 2 sin−1
冋
− sin ( 0 ) (Nd )
册
(2.10)
which, for modest scan angles [1], can be approximated by
2 MB = 2 sin−1
冋 册 (Nd )
cos ( 0 )
(2.11)
16
SpaceTime Adaptive Processing for Radar
In modern radar systems, the presence of strong targets, clutter, and jamming in the main lobe or sidelobes can mask the presence of a potentially weaker target of interest. There are basically two approaches to combat these effects: array tapering [1] and adaptive beamforming [4]. Tapering is accomplished by applying an amplitude (positive and real) weighting vector t ∈ ⺢ N+ across the array, such as the 40dB Chebyshev taper illustrated in Figure 2.3 (for more details on antenna tapering, see [1]). Mathematically, this is tantamount to replacing the untapered beamformer weight, w = s, with w = s 䊊 t, where 䊊 denotes the Hadamard product or elementwise matrix multiplication [2], which in general is defined as
c 11 c 21 = ⯗ c n1
c 12 c 22
…
冥冤 冥
a 1m
䊊
…
a 12 a 22
a nm …
c 1m
…
c nm
b 11 b 21 ⯗ b n1
b 12 b 22
…
b 1m
…
冤 冤
a 11 a 21 A䊊B= ⯗ a n1
b nm
冥 (2.12)
=C where A , B, C ∈ ⺓n × m, and the ij th element of C is given by c ij = a ij b ij . In words, given two conformal matrices of dimension n × m , the Hadamard product A 䊊 B = C is an n × m matrix whose elements are the pairwise products of the corresponding elements of A and B. Note that the output of the beamformer y in (2.7) can be expressed as a Hadamard product of the form y = [1 1 … 1] (w* 䊊 s)
(2.13)
where * denotes conjugation without transposition, and [1 1 … 1] is a 1 × N row vector (Hadamard identity row vector) that acts as the summer operator. Note that the Hadamard product in parenthesis must be performed first for the expression to have meaning. Figure 2.3(b) shows the resulting tapered patterns, normalized to a peak gain of unity, for two Chebyshev weighting schemes. Although sidelobe levels are significantly reduced, there is an increase in the main lobe width and a corresponding loss in gain [1] (see discussion on impact of tapering
Adaptive Array Processing
17
Figure 2.3 (a) Example of a 40dB Chebyshev taper function for a 16element ULA, and (b) resulting beam patterns for two different Chebyshev tapers. Note the broadening of the main lobe.
18
SpaceTime Adaptive Processing for Radar
on SNR). Also, there are practical limitations to the amount of tapering possible due to everpresent channelmismatch errors [1]. Further details on the impact of various forms of channel mismatch are discussed in Chapter 4. As we will see, adaptive beamforming has the potential to overcome some of these disadvantages. Adaptive beamforming attempts to take advantage of the spatial DoFs afforded by a phased array by adapting the pattern (i.e., the weights, w n ) in real time. The adaptive beamformer thus attempts to steer pattern nulls in the direction of the interference. The nulls must be determined adaptively as, in general, the direction of the interference is not known a priori. In practice, adaptive beamforming techniques are the practical attempt to implement optimum beamforming methods. We thus begin by examining the following optimization problem. 2.2.1 Derivation of the Optimum Beamformer Consider the response of a ULA beamformer to signal s ∈ ⺓ N plus additive interfering signals which we will simply refer to as noise, n ∈ ⺓ N. There are many potential sources of noise: clutter, jamming (intentional or not), and everpresent thermal or receiver noise [5]. However, for the moment, we will simply lump these signals into a single term n. Since the array response is linear, the output is given by y = w′s + w′n = y s + y n
(2.14)
where n is a zeromean, finitevariance, vector random variable (RV) [6]. Our objective is to maximize, in some appropriate sense, the signal response while simultaneously minimizing the response due to noise. A ubiquitously employed metric in radar (and many other sensor) applications is SNR [7], which, for the known signalinnoise case, is defined as
 y s 2 SNR = 2 E 再  yn  冎 ⌬
(2.15)
where E 再  y n  冎 is the expected value of the magnitude squared noise response [6], which formally, is given by [6] 2
E 再 yn 
∞
2
冕
冎 Ⳏ  y 2 f y −∞
n
( y ) dy
(2.16)
Adaptive Array Processing
19
where f y n ( y ) is the probability density function (pdf) associated with the RV y n . SNR is thus a measure of the signal energy to the average noise energy. Since w is not random, the expectation operation E {⭈} only acts on n as follows [6]: E 再 yn 
2
冎 = E 再  w′n  2 冎 = w′ E {nn′ } w
(2.17)
= w′R w where ⌬ E {nn′ } ∈ ⺓ N × N R=
(2.18)
is the covariance matrix associated with n (assumed to be zeromean). In Chapter 6, we will see that maximizing SNR is also optimum in a rigorous statistical setting for the additive Gaussian noise case. 2.2.2 Case I: Additive White Noise We first consider the case when the only competing interference is socalled white noise, which is typical of thermal receiver noise [5]. For this case, R = 2I, where I is the N × N identity matrix, and the scalar 2 is the receiver noise variance, which is assumed to be the same in each channel. For thermal receiver noise, 2 = kT eff B, where k is Boltzmann’s constant W 1.38 × 10−23 , T eff is the effective temperature in absolute degrees Hz ⭈ K° Kelvin, and B is the receiver bandwidth in hertz [5]. For this case, we have
冉
冊
 w′s  2
 w′s  2 = max 2 max 2 {w} w′ R w {w}  w 
(2.19)
subject to: w′w = 1 The unity norm constraint is chosen for both convenience and to insure a finite solution. Substitution of the constraint into the primary objective function yields the equivalent optimization problem
20
SpaceTime Adaptive Processing for Radar
max {w}
 w′s  2 w′ R w
= max {w}
 w′s  2 2
= max  w′s 
(2.20)
2
{w}
which is mathematically identical to the maximumoutput deterministic beamformer of (2.5); that is, w = s
(2.21)
where to meet the unity gain constraint in (2.19), = 1/√N (although this scalar does not effect the SNR as shown below). This is also known as the matched filter solution for the whitenoise case [7]. It is certainly interesting to note that the solution to (2.9), which accounts for noise, is identical to the solution obtained for simply maximizing the beamformer response. Intuitively this makes sense since white noise has no spatial structure and is completely uncorrelated. Consequently, the DoFs are best utilized maximizing the signal response and allowing the summation operation to average out the noise. Moreover, from (2.7), we see that for the whitenoise ULA case, the optimum matched filter can be implemented with an FFT. This is certainly an important benefit of the ULA since the FFT greatly facilitates realtime implementation. The corresponding optimum SNR is obtained by substituting the optimum weight solution into (2.19), and is given by
 w′s  2 SNR opt = 2 2  w   s′s  2 = 2 2 2  s  =
(2.22)
 s  2 2
which is independent of the constant . The gain in output SNR achieved by the optimum beamformer is the ratio of output to input SNR
Adaptive Array Processing
SNR gain =
21
SNR opt SNR in
 s  2 =
2
(2.23)
 sn  2 2
 s  2 =  sn  2 where SNR in is the single channel SNR (assumed to be equal for all channels, n = 1, . . . , N ). Note that the expression is independent of 2, since the gain in SNR is a relative measure and is based on the whitenoise assumption (not its strength). In our previous ULA example, N = 16, s and s n are given by (2.3), and yield
 sn  =  e 2
j 2 (n − 1)
d sin 0 2

=1
and
(2.24)
 s  2 = s′s = 16 which corresponds to SNR gain = 16 = N (or, in decibels, 10 log 16 = 12.04 dB). Thus, in general, for an N element ULA, the whitenoise coherent integration gain of a matched filter is given by SNR gain_dB = 10 log N (dB)
(2.25)
If a taper or window function is employed [1], there is a loss in SNR since the weight vector w is no longer perfectly matched to the signal s of interest. It is straightforward to calculate this loss in SNR as follows. First, define the taper efficiency ratio, eff , as the ratio of the tapered output SNR (SNR tap ) to optimal SNR (SNR opt ); that is,
22
SpaceTime Adaptive Processing for Radar
eff = =
SNR tap SNR opt
(2.26)
 w′s  2 N
where (2.24) was invoked. For the tapered case, w = s 䊊 t, where to insure the unity gain constraint w′w = 1, t must satisfy w′w = (s 䊊 t)′ (s 䊊 t) = (s* 䊊 s)′ (t* 䊊 t) = [1 … 1] (t* 䊊 t)
(2.27)
= t′t =  t  2 = 1 2
where  t  2 denotes the usual Euclidean (L 2norm) vector norm  t  2 =⌬ √t′t [2]. Substituting w = s 䊊 t into (2.26) yields
eff = = = =
 (s 䊊 t)′ s  2 N
=
 (s* 䊊 s)′ t  2 N
 s′ (s 䊊 t)  2 N (2.28)
 [1 … 1]t  2 N
 t  12 N
where  t  1 denotes the L 1norm (sum of the absolute values of t [2]—recall that [1 … 1]t is the summation operator, which for a positive taper is also the L 1 norm). Note that an elementary property of Hadamard products was also invoked, namely, s′(s 䊊 t) = (s* 䊊 s)′ t (which is readily verified by an elementwise comparison of both sides of the equality [8]). For the 16element 40dB Chebyshev taper shown in Figure 2.3(a),  t  1 = 3.5, which when substituted into (2.28) yields eff = 0.7642 (−1.17 dB). Note that for the uniform taper case (i.e., no taper),  t  1 = √N (recall
Adaptive Array Processing
23
normalization constraint) which yields, as expected, eff = 1 (0 dB). It is interesting that for the uniform case,  t  1 =  t  2 . 2.2.3 Case II: Additive Colored Noise In general the additive noise term, n, consists of both white noise and colored noise (has correlation and/or spatial structure) due to the presence of structured interference sources such as clutter and jamming [7]. Thus, in general the total noise (interference plus white noise) covariance matrix R will not typically be diagonal, but will have a more complex structure. However, assuming only that R is positivedefinite, a property guaranteed in practice since receiver noise is always present [7], a closed form solution for the optimum beamformer can still be obtained. Again, our objective is to maximize the SINR:
max {w}
 w′s  2 w′ R w
(2.29)
subject to: w′w = 1 Noting that R 1/2 R −1/2 = I, and again applying Schwarz’s inequality, we have the following inequality:
 w′s  2 w′ R w
= ≤
 w′ R 1/2 R −1/2 s  2 w′ R w (w′ R w) (s′ R −1 s) w′ R w
(2.30)
= s′ R −1 s or more simply
 w′ R 1/2 R −1/2 s  2 w′ R w
≤ s′ R −1 s
(2.31)
Equality is achieved when R 1/2 w = R −1/2 s (colinearity requirement), or equivalently,
24
SpaceTime Adaptive Processing for Radar
w = R −1 s
(2.32)
where the Hermitian property (complex conjugate transpose symmetry [2]) of R has been invoked. Note that since the upper bound in (2.31) is independent of w, (2.32) is strictly optimum, and achieves the upper bound; that is, SINR opt = s′ R −1 s
(2.33)
which reduces to (2.22) for the whitenoise case (R = 2I ). Since R is of the form R = R I + 2I
(2.34)
where R I is a generally positivesemidefinite covariance matrix [7] associated with the colorednoise source (i.e., clutter and/or jamming) that is uncorrelated with the white noise, SINR opt is bounded by the whitenoise SNR opt result (2.22) (i.e., thermal noise limited performance). To illustrate the colorednoise case, consider the effect that a persistent interfering point source (i.e., a jammer) has on our example ULA considered earlier. For this case, n = n J + n Rx
(2.35)
where n J ∈ ⺓N and n Rx ∈ ⺓N are the jammer and receiver noise vector RV signals, respectively [4]. The simplest model for a jammer (or strong cochannel interferer) is n J = ␥˜ s J , where ␥˜ is a zeromean, complex scalar RV, assumed to be uncorrelated with the receiver noise (i.e., E 再 n J n R′x 冎 = ∅), with a variance (jammer power) E再 ␥ 
2
冎 = J2
(2.36)
at the receiver input (per element), and s J is the jammer steering vector associated with a plane wave AoA of J [see (2.3)] [4]. Therefore, for the single jammer case, R = E {nn′ } = E {n J n J′ } + E {n Rx n R′x } = J2 s J s J′ + 2 I
(2.37)
Adaptive Array Processing
25
Note that the matrix J2 s J s J′ is rank one and is thus singular (but positivesemidefinite since J2 > 0). In general, if there are N J uncorrelated jammers, R has the form [4] R=
NJ
∑ J2 s i s′i + 2 I
i =1
i
(2.38)
If, furthermore, the jammer steering vectors are linearly independent [2], then
冢∑ NJ
rank
i =1
冣
J2i s i s′i = N J
for N J ≤ N
(2.39)
As an example, consider the effect of six uncorrelated jammers impinging on the aforementioned 16element ULA, shown in Figure 2.4. Assuming each interferer produces an element jammertonoise ratio (JNR) of 50 dB (i.e., 10 log 10 ( J2 / 2 ) = 50, for j = 1, . . . , N J ), and that the single element
Figure 2.4 Illustration of the angular distribution of the sixjammer example.
26
SpaceTime Adaptive Processing for Radar
SNR is 0 dB (or equivalently, s′s = 16), the resulting quiescent beamformer output SINR q with 0 = 0° (i.e., s = [1 1 … 1]T ) is SINR q = =
 w′s  2 w′ R w
 s′s  2
(2.40)
s′ R s
= 6.3 × 10−4 (−32 dB) This implies that any signal of interest at 0 = 0° would be 32 dB lower than the output response due to interference—a generally intolerable condition. If, on the other hand, we implement optimum beamforming, there is a dramatic improvement. Specifically, SINR opt = s′ R −1 s
(2.41)
= 15.7 or 12.0 dB, a loss of only ∼0.1 dB relative to the whitenoise case. Even this simple example highlights the significant potential of optimum beamforming. Of course, if even one of the jammers had an AoA of 0 = 0°, the optimum beamformer would be unable to null out the jammer and maximize the signal response simultaneously. There would thus be no net gain in SINR relative to that jammer. Figure 2.5 shows the corresponding optimum antenna response for the sixjammer case. Note that nulls or notches have been introduced into the pattern at precisely the angles the jammers are present. Additionally, the pattern still has a distinct and wellformed main lobe to insure maximum response to desired signals at boresight. As mentioned previously, it is possible to combine aperture tapering (e.g., Chebyshev windowing) with optimum beamforming by including in (2.32) the desired aperture taper t as follows w = R −1 (s 䊊 t)
(2.42)
Figure 2.6 illustrates an example of (2.42) and corresponds to the optimum pattern of Figure 2.5 modified by the presence of a 30dB Chebyshev taper. Of course, as previously discussed, the inclusion of a taper mismatch
Adaptive Array Processing
Figure 2.5 Optimum beam pattern for the sixjammer case of Figure 2.4.
Figure 2.6 Example of combining tapering with optimal beamforming.
27
28
SpaceTime Adaptive Processing for Radar
introduces additional SINR loss (in this case resulting output SINR is 11.3 dB, a loss of 0.7 dB relative to the optimum case). However, in practice, the practical benefits of maintaining low sidelobes is often well worth the modest loss. An important fundamental question is, how many jammers can be nulled with an N element array? If we let s J i denote the steering vector associated with the i th jammer, then a null is achieved if s′J i w = 0. To insure that a target of interest is not inadvertently nulled, we need to also invoke a socalled mainbeam constraint of the form s′w = 1 (sometimes referred to as a unity gain constraint). Assuming that there are N J linearly independent jammer steering vectors (independent of s as well), the following linear system of equations results: s′w = 1 s′J 1 w = 0 s′J 2 w = 0
(2.43)
⯗ s′J N w = 0 J
which is a linear system of N J + 1 equations in N unknowns. From elementary linear algebra [3], we know that there exists a w (not necessarily unique) that satisfies the above set of equations provided that N J + 1 ≤ N. This, in turn, immediately implies that the number of jammers that can be nulled is bounded by NJ ≤ N − 1
(2.44)
The above optimum beamformer calculations were highly idealized as they assumed exact knowledge of the total interference covariance matrix, as well as perfectly matched channels and idealized jammer characteristics— conditions not met in practice. Much of this book is devoted to strategies designed to preserve these desirable gains in more realistic operating environments. We next turn our attention to the other significant measurement dimension, namely time. More specifically, we will consider pulsetopulse time processing manifested as Doppler filtering, which plays a fundamental role in MTI radar.
Adaptive Array Processing
29
2.3 Optimum Temporal (Doppler/Pulse) Processing Besides AoA, another key physical observable for separating moving targets from noise is Doppler frequency [5, 9, 10]. Fortunately, the mathematical framework employed in modern Doppler radar systems is essentially identical to that already introduced in our discussion of the ULA; specifically, a spatial point target produces a linear phase progression across a ULA. We will see that a Dopplershifted point target similarly produces a linear phase progression in time and can thus be processed in a manner directly analogous to the beamformer case. Consider the effect of a Dopplershifted return propagating through a single channel, M tap delayline filter as shown in Figure 2.7. For a pulsed Doppler radar [5], the delay T is chosen to match the PRI. The output of the m th tap is then given by s m = e j 2 (m − 1)f d , m = 1, . . . , M , where f d is the normalized Doppler frequency given by [5] fd =
fd PRF
= fd T =
(2.45)
2T (v − v Rx ) ⭈ ˆi Rx tgt
Figure 2.7 Uniform tapped delayline linear combiner for processing Dopplershifted returns. Note that the combiner output has the same mathematical form as the ULA beamformer of Figure 2.1.
30
SpaceTime Adaptive Processing for Radar
where PRF is the pulse repetition frequency (PRF = 1/PRI = 1/T ), f d is the Doppler frequency of the point target in units of 1/time (e.g., hertz), vtgt and vRx are the target and receiver velocity vectors (Cartesian coordinates), ˆi Rx is a unit direction vector pointing from the receiver to the target 冠  ˆi Rx  = 1冡, is the operating wavelength, and ⭈ denotes the vector ‘‘dot’’ product; that is, a ⭈ b =  a  b  cos , where is the angle between a and b. Note for a fixed PRF, the unambiguous Doppler region for complex sampling based on the Nyquist criterion is from −PRF/2 to +PRF/2 [5]. Thus, the unambiguous normalized Doppler region is −0.5 ≤ f d ≤ +0.5
(2.46)
Note that (2.45) is only valid (in general) for the monostatic case (transmitter and receiver collocated). As with the optimum ULA beamformer, an optimized Doppler filter response can also be constructed by a judicious selection of the complex weighting factors w m , m = 1, . . . , M , in Figure 2.7. Let
s=
冤冥冤 s1 s2 ⯗ sM
=
1 e
j 2 f d
⯗
e j 2 (M − 1)f d
冥
(2.47)
denote the M tap Doppler steering vector for a signal (Doppler) of interest and w = [w 1 w 2 … w M ]T denote the vector of weights. Then, as with the ULA case, we desire to choose w to maximize SINR. Since the math is identical, the optimum SINR solution is also given by (2.32); namely, w = R −1 s
(2.48)
where R is the M × M total noise covariance matrix, and is a scalar constant that does not affect SINR. We will defer examples and rationale of optimum Doppler processing until Chapter 3, where ground clutter will be discussed. In this context, we will also introduce the notion of 2D angleDoppler (or spacetime) processing. Next, the notion of adaptive, viceoptimum processing is discussed for the previously introduced 1D beamformer.
Adaptive Array Processing
31
2.4 Adaptive 1D Processing Unfortunately, in much of the radar and signal processing literature the expression for the optimum linear combiner (beamformer) is referred to as an adaptive linear combiner (or adaptive beamformer, and so forth). This equivalence is not only potentially misleading and confusing, it is in general incorrect—and this is not just pedantry. In realworld systems, adaptivity is used to adjust radar processing on the fly. The reasons for this are obvious. In the case of jamming, for example, one cannot know a priori both the strength and exact angular location. Similar arguments are true for airborne radar clutter. This lack of prior information implies that the essential total interference covariance matrix R of (2.32) must be estimated concurrently with the beamforming operation. To accomplish this, some form of sample estimate, derived from actual radar measurements, must be either explicitly or implicitly formed. Referred to somewhat generically as training (the term can be traced to Bernard Widrow’s pioneering work on adaptive neural networks [11]), it is precisely this adaptation process that is a major impetus for this book, many of the techniques described herein, and much of the research literature [12]. Historically, adaptation to interference was accomplished by the explicit or implicit estimation of the local noise statistics. Beginning with cell averaging constant falsealarm rate (CACFAR) techniques, developed concurrently with the very first airborne MTI radars [13], adaptation has been accomplished by identifying a subset of the radar measurements from which noise statistics are estimated. Figure 2.8 presents a pictorial representation of this process. Let x i denote the N dimensional output vector of a ULA corresponding to the i th range bin. To process x i according to (2.32), that is, to look for a target in a given direction, s, while simultaneously minimizing the influence of all other interfering signals, the covariance matrix for that i th range bin R i is required. A reasonable approach is to compute a sample estimate of R i (which we will denote as Rˆ i ) using surrounding range bins (without the cell under test to prevent socalled selfnulling) that are likely to contain similar interference; that is, 1 Rˆ i = L
∑ x l x l′
l ∈⍀i
(2.49)
32
SpaceTime Adaptive Processing for Radar
Figure 2.8 Conventional training strategy for estimating interference covariance matrix.
where L is the total number of samples used in the summation, and ⍀i denotes the set of training samples used for the i th range bin. It is easy to show that if the data samples {x l }, l = 1, . . . , L , are uncorrelated and have identical covariance R , (2.49) is an unbiased estimate of R , since E {Rˆ i } = E = =
1 L 1 L
冦
1 L
∑ x l x l′
l ∈⍀i
∑ E {x l x l′ }
冧 (2.50)
l ∈⍀i
∑R
l ∈⍀i
=R If, additionally, {x l }, l = 1, . . . , L , are Gaussian and independent identically distributed (i.i.d.), than (2.49) corresponds to the maximum likelihood estimate of R i [7] as shown in Appendix 2B. Of course in practice, there is no guarantee that the interference in the i th range bin will have statistics identical with the training data. Nonetheless, it is a reasonable first choice assuming that only range bins from surrounding, and thus hopefully similar, regions are utilized. Another practical constraint on the training data selection process is the presence of a buffer region around the cell under test comprised of socalled guard cells, as shown
Adaptive Array Processing
33
in Figure 2.8. This is sometimes included to prevent socalled selfnulling, since a target response may straddle more than one range bin or have temporal sidelobes present in other range bins, a common problem in pulse compression radars [14]. It is important to realize that if (2.49) is used in (2.32), then a minimum of N i.i.d. samples are required to insure that Rˆ i is nonsingular (almost surely [6]). However, under certain circumstances, as we will see later in this and subsequent chapters, very useful adaptive solutions can be obtained with significantly fewer than N samples. This can be a very desirable practical feature since, in the case of clutter, statistics can vary significantly with range, thereby invalidating the necessary stationarity assumption tacitly employed in (2.49). Reed, Mallet, and Brennan (RMB) [15], in a seminal paper, were able to rigorously characterize the impact of replacing the actual covariance, R i , with its sample estimate, Rˆ i , under certain conditions. Specifically, if the L training samples ⍀i = {x l }i are free of target signal contamination and are independent i.i.d. Gaussian vector RVs with statistics identical to the i th test cell (not included in the training data), then the following are true: (1) the elements of Rˆ i are complex Wishartdistributed [15], and (2) a closed form expression for the ratio of the expected SINR using (2.23) to optimum SINR using (2.32) is betadistributed and is given by [15]
=
L−N +2 ,L≥N L+1
(2.51)
This is the important RMB result, which states that the SMI method produces an SINR loss that is ∼3 dB if L ∼ 2N. Note that (2.51) is also extremely useful for estimating the amount of loss due to finite L . Referred to colloquially as RMB loss, this equation has become an integral part of a radar systems engineer’s SINR budget spreadsheets. As expected, → 1 as L → ∞. An elegant and accessible derivation of this result can be found in the chapter by Steinhardt in [16]. Although 2N i.i.d. Gaussian samples yield an SINR that is within ∼3 dB of optimum, the corresponding adapted pattern may not be suitable for most situations due to pattern distortions. For example, Figure 2.9 displays the adapted pattern based on 2N samples for the N = 16 element ULA example of Figure 2.6, along with the optimum tapered pattern. Note the presence of extremely high sidelobes and shapedistortion of the main lobe. Although the adapted SMI pattern can be improved by increasing L (note the response in Figure 2.9 corresponding to 5N samples), there are
34
SpaceTime Adaptive Processing for Radar
Figure 2.9 Effect of covariance estimation error due to a finite data sample on adapted beam patterns.
many practical limits to doing so due to interference nonstationarity and/ or computational burden. Instead, it is worth examining the cause of the adapted pattern distortion at this stage by conducting an eigen analysis of the underlying adaptation process. In so doing, we will also lay the groundwork for an important class of robust adaptive signalprocessing techniques based on optimum subspace estimation and principal components [17, 18]. We begin by first considering an eigendecomposition of a general Hermitian, positivedefinite, N dimensional covariance matrix R ; that is, N
R = U ⌳U ′ =
∑ n u n u n′
(2.52)
n =1
where U = [u 1 u 2 … u N } ∈ ⺓ N × N
(2.53)
is the N × N matrix whose columns are the orthonormal eigenvectors of R , and
Adaptive Array Processing
⌳ = diag { 1 2 … N } ∈ ⺢ +N × N
35
(2.54)
is the N × N diagonal matrix of corresponding strictly positive eigenvalues [8]. Substituting (2.52) into (2.32) and recalling that R −1 = (U ⌳U ′ )−1 = U ⌳−1 U ′ N
=
∑
n =1
(2.55)
u n u n′ n
(U is a unitary matrix; that is, U −1 = U ′ [8]), and rearranging terms yields w = R −1 s = ˜ min U ⌳−1 U ′s ˜ −1 U ′s) = ˜ (s − s + U ⌳
(2.56)
˜ −1 U ′ )s] = ˜ [s − (I − U ⌳ ˜ −1 ) U ′s] = ˜ [s − U(I − ⌳ where ˜ = (1/ )⌳ ˜ = / min and ⌳ min
(2.57)
and min denotes the minimum eigenvalue of R . Thus, ˜ = diag {˜ ˜ … ˜ } ⌳ 1 2 N
(2.58)
where ˜ n = n / min is the n th normalized eigenvalue. Finally, expressing (2.56) as a summation yields a very useful equivalent representation of the optimum beamformer result expressed in eigen form, namely,
36
SpaceTime Adaptive Processing for Radar
冤 冉 冤 ∑冉 N
w = ˜ s −
= ˜ s −
冊
冥
∑
1 1− ˜ (u n′ s ) u n n
N
n − min (u n′ s ) u n n
n =1
n =1
冊
(2.59)
冥
Equation (2.59) shows that the optimum beamformer weight vector is formed by subtracting a weighted sum of the interference eigenvectors from the quiescent steering vector s. Note that only the eigenvectors with corresponding eigenvalues greater than min may contribute to the sum. The exact amount they contribute is proportional to the difference between their eigenvalues and the minimum eigenvalue term, as well as the amount of colinearity [i.e., (u n′ s )]. To explain the distorted pattern results of Figure 2.9, consider first the eigendecomposition of the exact covariance for the sixjammer, 16element ULA example. Figure 2.10 shows the eigenvalue distribution; there are exactly six large eigenvalues (due to the jammers). The remaining 10 eigenvalues are often referred to as the noisefloor components. To see why
Figure 2.10 Comparison of exact and estimated eigenvalues for the sixjammer example.
Adaptive Array Processing
37
this is so, consider the structure of the exact covariance described by (2.38); that is, R=
NJ
∑ J2 s i s′i + 2 I = R J + 2 I i
i =1
(2.60)
where ⌬
RJ =
NJ
∑ J2 s i s′i
i =1
i
(2.61)
For the current example N J = 6. Thus, the eigendecomposition of R J is of the form R J = U J ⌳ J U J′
(2.62)
= U J diag { J 1 J 2 … J 6 0 … 0 } U J′ where there are six nonzero eigenvalues (the rank of R J is exactly six since the jammers are linearly independent). Although the remaining 10 eigenvalues are zero, we can still use the GramSchmidt method to construct a complete 16dimensional orthogonal basis. Thus, we can assume that U J is full rank −1 and, therefore, that U J = U J′ . Since any unitary matrix diagonalizes a diagonal matrix [8], the eigendecomposition of the total covariance matrix, R , is given by R = U ⌳U ′ = U J ⌳U J′
(2.63)
The eigenvalues of R are thus given by ⌳ = U J′ RU J = U J′ (R J + 2 I ) U J
(2.64)
= U J′ R J U J + 2 I = diag { J 1 + 2 … J 6 + 2 2 … 2 } From (2.64) we see that min = 2 and that only the first six eigenvalues are greater than min . Thus, there are only six terms that contribute to the sum in (2.59) for the ideal case (knowncovariance).
38
SpaceTime Adaptive Processing for Radar
Now consider the eigenvalues of the estimated covariance Rˆ , shown in Figure 2.10, for an example in which L = 2N i.i.d. Gaussian samples are used in (2.49). Note that while there are certainly six large distinct eigenvalues, there is no definitive noise floor. More specifically, since the first 15 estimated eigenvalues {ˆ 1 , … , ˆ 15 } are greater than the minimum eigenvalue ˆ 16 , they may all contribute in (2.59). Unfortunately, except for the very strong jammer eigenvalues, the eigenvalues/vector pairs associated with the noise floor are not well estimated with only 2N samples. The biggest impact, as illustrated in Figure 2.9, is significant corruption of the adapted pattern sidelobes. Although we will defer an indepth discussion of principal components methods until Chapter 4, it is clear from the above discussion that if we only used the six dominant eigenvalues/vector pairs in (2.59) and set ˆ min = 2 (since the receiver noise floor is reasonably wellknown in a calibrated radar), we should have a much better pattern. Indeed, as shown in Figure 2.11, the adapted pattern using only the six principal components in (2.59) is extremely close to the optimum. The corresponding SINR loss relative to the optimum tapered response is only 0.8 dB (for L = 2N ).
Figure 2.11 Adapted patterns using principal components and diagonal loading. Note that the principal components pattern is virtually identical to the optimum pattern.
Adaptive Array Processing
39
Figure 2.11 also displays the adapted pattern obtained by adding a diagonal loading term [19] to Rˆ ; that is, Rˆ DL = Rˆ + ␦ 2 I
(2.65)
where ␦ 2 = 10 2 was chosen (i.e., 10 dB of diagonal loading). Again the adapted pattern is quite good compared with SMI. The reason diagonal loading works somewhat similarly to the principal components method is clear from an examination of the eigenvalues of Rˆ DL , which are 2 ˆ 2 2 ˆ ˆ ˆ ⌳ DL = diag {1 + ␦ 2 + ␦ … 16 + ␦ }
(2.66)
Since, in the example considered, ˆ n << ␦ 2, for n = 7, . . . , 16, we again see that the 10 smallest eigenvalues/vectors will not appreciably contribute to the sum of (2.59). However, the depth of the nulls is not the same for diagonal loading as it is for principal components because the weighting factors in (2.59) have been decreased (uniformly). This resulting decrease in null depth, which can be an issue in some applications, highlights some of the limitations of diagonal loading (not to mention the issue of choosing a correct value of loading factor [20]). Note also that the corresponding SINR loss relative to the optimum tapered response is slightly worse (0.9 dB) than that for principal components, but is still quite good compared to SMI.
2.5 Adaptivity in Nonstationary Environments The availability of a sufficiently large training set of i.i.d. samples is highly suspect in stressing applications involving large numbers of adaptive DoFs. Variability in clutter terrain and other interference nonstationarities place practical limits on the size and quality of the training data available [19]. Thus, an extremely important issue is how many adaptive DoFs can be utilized or supported in a given interference environment. The answer, as we have already seen, is highly dependent on the specific adaptation algorithm employed. For example, in the principal components method, we saw that instead of supporting 16 adaptive DoFs, we only needed to estimate the subspace associated with the interference, which was sixdimensional. Of course, we somehow needed to know that only six eigenvalues were required. For this example, a reasonably simple criterion could
40
SpaceTime Adaptive Processing for Radar
be used based on the size of the estimated eigenvalue relative to the known noise floor. When clutter interference is introduced in the next chapter, we will see that there is often a continuum of eigenvalues with monotonic decreasing amplitudes down to the noise floor, thus complicating the determination of a simple threshold approach. Nonetheless, by a judicious incorporation of prior structural information, an order of magnitude or greater reduction in sample support requirements can be achieved while preserving near optimal performance. While minimizing sample support requirements is an indirect way of addressing nonstationarity, there are other methods that attempt to directly model or parameterize the nonstationarity. Examples of this approach include the socalled time varying weights technique [21], which has been proposed for both bistatic and circular array STAP applications, and CMT, which models the subdominant rank inflation mechanisms that give rise to nonstationarity in interference rank [22]. Although we will discuss CMTs at much greater length in the context of clutter cancellation and STAP in Chapters 4 and 5, they have also proved straightforward to apply in spatialonly adaptivity for the socalled stale weights problem [23–27]—a form of nonstationarity. For a multitude of practical reasons, the adaptive weight vector derived from the necessarily finite training set (secondary data) may need to be applied to data outside of that training region [24]. If a jammer is present, it may be at a slightly different angle if the training data is old relative to the dynamics of the jammer [24]. Although derivative constraints have been proposed to remedy this [24] by producing wider jammer nulls, a much simpler and more straightforward approach that essentially produces the same effect (that is, readily extensible to nonlinear arrays) is via the application of a CMT to the sample covariance matrix (also referred to as covariance augmentation in [28]); that is, Rˆ → Rˆ 䊊 T
(2.67)
where Rˆ is the sample covariance matrix (which may also include diagonal loading [23]), 䊊 denotes the Hadamard matrix multiplication operator (simply conformal elementwise multiplication analogous to matrix addition and subtraction [2]), and T ∈ ⺓N × N is a CMT matrix [23]. Mailloux [26] and Zatman [27] independently showed that by setting T equal to a correlation matrix of the form [23] [T ]i , j = sinc 冠 ⌬  i − j  / 冡
(2.68)
Adaptive Array Processing
41
⌬ sin ( x )/ x , [T ]i , j denotes the (i, j )th element of T, and where sinc (x ) = ⌬ is a positive scalar used to set the amount of notch widening (dimensions of radians [23]) and can also be interpreted as the amount of uniform random dither [23] or bandwidth [26]. The specific CMT of (2.68) is referred to as a MaillouxZatman CMT in [23]. Figure 2.12 shows an example of the
Figure 2.12 Example of the application of a CMT to produce a wider jammer null to help address the stale weights problem. Adapted patterns for a 16element halfwavelength ULA in the presence of a 60dB per element JNR at a normalized angle of −0.25 (−30° off boresight): (a) optimum adapted pattern derived from exact (ideal) covariance with a −35dB sidelobe level Chebyshev taper, (b) pattern obtained with a 2N sample covariance matrix (i.e., SMI method), (c) pattern obtained if 10 dB of diagonal loading is added to the sample covariance matrix, and (d) pattern obtained if a diagonally loaded (10 dB) MaillouxZatman CMT is employed with a ⌬ = 0.01. Note the preservation of good sidelobes associated with diagonal loading as well as a substantially widened (and more robust) jammer null.
42
SpaceTime Adaptive Processing for Radar
application of a MaillouxZatman CMT; the jammer notch widening is quite readily apparent, as are the preservation of the main lobe and average sidelobe levels [23]. We will defer a detailed taxonomical discussion of general methods for addressing other forms of nonstationarity that arise in spacetime clutter cancellation, including additional techniques for minimizing sample support requirements, to Chapter 5.
2.6 Summary In this chapter we began our journey into STAP with a sojourn into its origins in 1D adaptive beamforming and Doppler processing. After deriving the equations for optimal beamforming and illustrating its application to a multijammer scenario, we introduced the concept of adaptive beamforming. An important distinction was drawn between optimal and adaptive beamforming, and some fundamental differences were highlighted, particularly for the finite sample covariance estimation case. For this case, an eigen analysis was conducted to gain key insights into the origins of the degradation from optimality for the finite sample case. In particular, the occurrence of erratic adapted sidelobes and main lobe distortion were shown to be the result of estimation errors associated with the socalled noisefloor eigenvalues/vectors. Both principal components and diagonal loading were introduced to remedy this problem. Finally, a brief discussion of issues arising when applying adaptive processing in nonstationary environments was introduced, along with an outline of strategies that can be brought to bear, including minimal sample support methods (indirect approach) and nonstationary modeling (direct approach). The concept of CMT covariance augmentation was also introduced to simply address the socalled stale weights problem. A detailed survey of these methods will be presented in Chapter 5.
References [1]
Mailloux, R. J., Phased Array Antenna Handbook, Norwood, MA: Artech House, 1994.
[2]
Horn, R. A., and C. R. Johnson, Topics in Matrix Analysis, Cambridge, England: Cambridge University Press, 1991.
[3]
Strang, G., Introduction to Linear Algebra, 2nd ed., Wellesley, MA: WellesleyCambridge Press, 1998.
Adaptive Array Processing
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[4]
Farina, A., AntennaBased Signal Processing Techniques for Radar Systems, Norwood, MA: Artech House, 1992.
[5]
Barton, D. K., Radar System Analysis, Dedham, MA: Artech House, 1976.
[6]
Papoulis, A., and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 3rd ed., New York: McGrawHill, 2001.
[7]
Van Trees, H. L., Detection, Estimation, and Modulation Theory, Part I, New York: John Wiley & Sons, 1968.
[8]
Horn, R. A., and C. R. Johnson, Matrix Analysis, Cambridge, England: Cambridge University Press, 1985.
[9]
Schleher, D. C., MTI and Pulsed Doppler Radar, Norwood, MA: Artech House, 1991.
[10]
Morris, G. V., and L. Harkness, Airborne Pulsed Doppler Radar, 2nd ed., Norwood, MA: Artech House, 1996.
[11]
Widrow, B., and M. A. Lehr, ‘‘30 Years of Adaptive Neural Networks: Perceptron, Madaline, and Backpropagation,’’ Proc. of IEEE, Vol. 78, No. 9, 1990, pp. 1415–1442.
[12]
Melvin, W., (ed.), ‘‘SpaceTime Adaptive Processing and Adaptive Arrays: Special Collection of Papers,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 36, No. 2, April 2000, pp. 508–509.
[13]
Nitzberg, R., Radar Signal Processing and Adaptive Systems, 2nd ed., Norwood, MA: Artech House, 1999.
[14]
Cook, C. E., and M. Bernfeld, Radar Signals: An Introduction to Theory and Application, Norwood, MA: Artech House, 1993.
[15]
Reed, I. S., J. D. Mallet, and L. E. Brennan, ‘‘Rapid Convergence Rate in Adaptive Arrays,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 10, No. 6, November 1974, pp. 853–863.
[16]
Steinhardt, A., ‘‘Adaptive Multisensor Detection,’’ in Adaptive Radar Detection and Estimation, S. Haykin and A. Steinhardt, (eds.), New York: John Wiley & Sons, 1992.
[17]
Guerci, J. R., J. S. Goldstein, and I. S. Reed, ‘‘Optimal and Adaptive ReducedRank STAP,’’ IEEE Trans. on Aerospace and Electronic Systems, Special Section on STAP, Vol. 36, No. 2, April 2000, pp. 647–663.
[18]
Tufts, D. W., R. Kumaresan, and I. Kirsteins, ‘‘Data Adaptive Signal Estimation by Singular Value Decomposition of a Data Matrix,’’ Proc. of IEEE, Vol. 70, No. 6, June 1982, pp. 684–685.
[19]
Carlson, B. D., ‘‘Covariance Matrix Estimation Errors and Diagonal Loading in Adaptive Arrays,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 24, No. 3, July 1988, pp. 397–401.
[20]
Guerci, J. R., Y. L. Kim, and S. U. Pillai, ‘‘Optimal Loading Factor for Minimal Sample Support SpaceTime Adaptive Radar,’’ Proc. of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vol. 4, Seattle, WA, May 12–15, 1998, pp. 2505–2508.
[21]
Zatman, M., ‘‘Circular Array STAP,’’ IEEE Trans. on Aerospace and Electronic Systems, Special Section on STAP, Vol. 36, No. 2, April 2000, pp. 510–517.
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SpaceTime Adaptive Processing for Radar
[22]
Guerci, J. R., and J. S. Bergin, ‘‘Principal Components, Covariance Matrix Tapers, and the Subspace Leakage Problem,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 38, No. 1, January 2002.
[23]
Guerci, J. R., ‘‘Theory and Application of Covariance Matrix Tapers for Robust Adaptive Beamforming,’’ IEEE Trans. on Signal Processing, Vol. 47, No. 4, April 1999, pp. 977–985.
[24]
Gershman, A. B., U. Nickel, and J. F. Bohme, ‘‘Adaptive Beamforming Algorithms with Robustness Against Jammer Motion,’’ IEEE Trans. on Signal Processing, Vol. 45, No. 7, July 1997, pp. 1878–1886.
[25]
Zatman, M., and J. R. Guerci, ‘‘Comments on ‘Theory and Application of Covariance Matrix Tapers for Robust Adaptive Beamforming’ [and Reply],’’ IEEE Trans. on Signal Processing, Vol. 48, No. 6, June, 2000, pp. 1796–1800.
[26]
Mailloux, R. J., ‘‘Covariance Matrix Augmentation to Produce Adaptive Array Pattern Troughs,’’ Electronics Letters, Vol.31, No. 10, 1995, pp. 771–772.
[27]
Zatman, M., ‘‘Production of Adaptive Array Troughs Through Dispersion Synthesis,’’ Electronics Letters, Vol. 31, No. 25, December 1995, pp. 2141.
[28]
Van Trees, H. L., Optimum Array Processing: Part IV of Detection Estimation and Modulation Theory, New York: Wiley Interscience, 2002.
[29]
Swokoski, E. W., Calculus with Analytic Geometry, Boston, MA: Prindle, Weber, and Schmidt, 1975.
[30]
Pillai, S. U., Array Signal Processing, New York: SpringerVerlag, 1989.
[31]
Muirhead, M., Introduction to Multivariate Analysis, Englewood Cliffs, NJ: Prentice Hall, 1985.
[32]
Pierre, D. A., Optimization Theory with Applications, New York: Dover, 1986.
Appendix 2A: ULA Antenna Pattern Response In this appendix, we derive the expression of (2.9) for the normalized antenna pattern of a ULA, which is tantamount to evaluating the output response in (2.7); that is, N
y = w′x =
∑ xn e
−j 2 (n − 1)
d sin 0
(2A.1)
n =1
where for convenience, we have set = 1. For a plane wave impinging on the array from an angle with respect to boresight, we have xn = e
j 2 (n − 1)
d sin
Substituting (2A.2) into (2A.1) yields
, n = 1, . . . , N
(2A.2)
Adaptive Array Processing N
y=
∑e
j 2 (n − 1)
45
d (sin − sin 0 )
n =1
N −1
∑e
=
j 2 n
d (sin − sin 0 )
(2A.3)
n =0
N −1
=
∑ rn
n =0
where r=e
j 2
d (sin − sin 0 )
(2A.4)
From the Geometric Sum Formula [29], N −1
∑
rn =
n =0
1 − rN rN − 1 = 1−r r−1
(2A.5)
we thus have y=
e
j 2 N
e =
e
d j 2 (sin − sin 0 )
j N
e e
d (sin − sin 0 )
d (sin − sin 0 )
j
= e
−1
冠e
d (sin − sin 0 )
d j N (sin − sin 0 )
d j (sin − sin 0 )
−1
j N
冠e
j
冉 冉
d (sin − sin 0 )
d (sin − sin 0 )
sin N sin
−e −e
−j N
−j
d (sin − sin 0 )
d (sin − sin 0 )
d (sin − sin 0 )
冊
冊
冡
冡 (2A.6)
d (sin − sin 0 )
where we have used Euler’s Identity, 2j sin ␣ = e j␣ − e −j␣, to arrive at the final form. Taking the absolute value of (2A.6) and dividing by N (for normalization) yields (2.9).
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SpaceTime Adaptive Processing for Radar
Appendix 2B: Derivation of the Maximum Likelihood Sample Covariance Matrix The derivation in this appendix of the maximum likelihood estimate (MLE) of the sample covariance matrix follows that of Steinhardt [16] and Pillai [30]. The reader is encouraged to consult [16, 30] for further statistical details and properties of the adaptive beamformer including an elegant derivation of the important RMB result of (2.51) in [16]. Let x 1 , x 2 , . . . , x L ∈ ⺓N denote L observations (array snapshots) that are assumed to be complex zeromean i.i.d. and Gaussian, each with a marginal pdf given by [28]
f x i (x) =
1
R  N
e −x′R
−1
x
, i = 1, . . . , L
(2B.1)
where R ∈ ⺓N × N is an unknowncovariance matrix (assumed to be positivedefinite) with nonzero determinant  R  . The associated likelihood function f (x 1 , . . . , x L  R ) (joint density of x 1 , . . . , x L conditioned on R [7]) is thus given by f 冠x1 , . . . , xL  R 冡 = =
写 L
i =1
f xi 冠xi  R 冡
(2B.2)
L
1
NL  R 
L
e
− ∑ x i′ R −1 x i i =1
Letting ⌬ [x 1 x 2 … x L ] ∈ ⺓N × L X=
(2B.2) can be rewritten as
(2B.3)
Adaptive Array Processing
47
L
f 冠X  R 冡 = = = =
1
NL  R 
L
e
L
e −Tr(X ′ R
L
e −Tr(XX ′ R
L
e −Tr(R
1
NL
NL
R 
1
R 
1
NL
R 
− ∑ x i′ R −1 x i i =1
−1
= 冋 −N  R  −1 e −Tr(R
−1
X)
−1
)
(2B.4)
XX ′ )
−1 ˆ
册
R) L
where Tr(⭈) denotes the trace operator (sum of the diagonal elements of a square matrix [2]) and 1 ⌬ 1 XX ′ = Rˆ = L L
L
∑ x i x i′
(2B.5)
i =1
which is recognized as the N × N sample covariance matrix (SCM). Use of the identity Tr (AB ) = Tr (BA ) was utilized in arriving at the final form in (2B.4). The MLE of the unknowncovariance matrix R is obtained by maximizing the likelihood function with respect to R or minimizing the negative loglikelihood [7]; specifically, Rˆ ML ⌬ = arg max 冠 f 冠 X  R 冡冡 {R }
= arg min 冠− ln f 冠 X  R 冡冡
(2B.6)
{R }
= arg min 冠ln  R  + Tr (R −1 Rˆ )冡 {R }
where use was made of the monotonicity of the natural logarithm, and only the terms involving R were retained. Since it is assumed that L ≥ N, Rˆ is
48
SpaceTime Adaptive Processing for Radar
positivedefinite (almost surely [31]) and therefore possesses a matrix square root Rˆ 1/2 [2] (such that Rˆ = Rˆ 1/2Rˆ 1/2 ). This allows us to rewrite (2B.6) as Rˆ ML = arg min 冠−ln  R −1  + Tr (R −1 Rˆ )冡 {R }
= arg min 冠−ln  R −1 Rˆ −1 Rˆ  + Tr (R −1 Rˆ )冡 {R }
= arg min 冠ln  Rˆ −1  − ln  R −1 Rˆ  + Tr (R −1 Rˆ )冡
(2B.7)
{R }
= arg min 冠−ln  R −1 Rˆ 1/2 Rˆ 1/2  + Tr (R −1 Rˆ 1/2 Rˆ 1/2 )冡 {R }
= arg min 冠−ln  Rˆ 1/2 R −1 Rˆ 1/2  + Tr (Rˆ 1/2 R −1 Rˆ 1/2 )冡 {R }
= arg min 冠−ln  ⌿  + Tr (⌿)冡 {R }
where 1/2 −1 1/2 ⌿⌬ = Rˆ R Rˆ
(2B.8)
and use was made of the following identities at arriving at the final form in (2B.7): det (AB ) = det (BA ), det (A −1 ) =
1 , Tr (AB ) = Tr (BA ) det (A ) (2B.9)
Note also that additive terms not involving R were dropped in (2B.7) since they do not affect the result. Since there is a onetoone mapping between R and ⌿ [i.e., (2B.8)] and both are positivedefinite Hermitian matrices, we can first minimize with respect to ⌿, then solve for R to obtain the final result. This process is further facilitated by expressing ⌿ in diagonal form; that is, ⌿ = U ⌳U ′
(2B.10)
where U ∈ ⺓N × N is a unitary matrix [2] (i.e., UU ′ = I ) and ⌳ ∈ ⺢ +N × N is a diagonal matrix of strictly positive eigenvalues of ⌿; that is, ⌳ =
Adaptive Array Processing
49
diag { 1 , 2 , . . . , N }. Substituting (2B.10) into (2B.7) yields the equivalent minimization problem ⌿ML = arg min 冠−ln 冠  ⌿  冡 + Tr (⌿)冡 {⌿}
= arg min 冠−ln 冠  U ⌳U ′  冡 + Tr (U ⌳U ′ )冡 {U ⌳U ′ }
(2B.11)
= arg min 冠−ln 冠  ⌳  冡 + Tr (⌳)冡 {⌳}
冢∑ N
=
arg min
{ 1 , 2 , . . . ,
N }
−
n =1
N
ln ( n ) +
∑ n
n =1
冣
Due to the convexity of the original likelihood function [32], a necessary and sufficient condition for the minimization of (2B.11) is that each of the derivatives of the argument with respect to each eigenvalue vanish; that is,
冢
N
d − ∑ ln ( n ) + n =1
N
∑ n
n =1
冣
d n
=0
(2B.12)
∀n : n = 1, . . . , N which yields −
1 + 1 = 0 → n = 1 n
(2B.13)
∀n : n = 1, . . . , N which in turn implies that ⌳ML = I → U ML = I → ⌿ML = I, and thus −1 ˆ 1/2 R I = Rˆ 1/2 R ML
(2B.14)
or R ML which is the desired result.
1 1 = Rˆ = XX ′ = L L
L
∑ x i x i′
i =1
(2B.15)
3 SpaceTime Adaptive Processing 3.1 Introduction The term STAP was first applied to multidimensional adaptive filtering of clutter and jamming in airborne MTI radars [1, 2]. Unlike groundbased (or groundstationary) MTI radars, clutter returns manifest themselves as fully 2D (nonfactorable) structures due to the motioninduced Dopplerspreading effect described in Section 3.2. As a consequence, the traditional factored, or decoupled, approaches of beamforming followed by Doppler filtering (or vice versa) are not optimal [1]. Instead, as we will see, a better approach is to perform multidimensional filtering that accounts for angleDoppler coupling. As we will shortly see, spacetime clutter is generally colored noise, that is, a nondiagonal covariance matrix. This is a good news/bad news situation: On the one hand, the fact that it has structure suggests that there may be an opportunity for separating the clutter subspace from the signal subspace via spacetime filtering—as described later. On the other hand, this can only be accomplished if we have an accurate model for the clutter structure—therein lies the rub. The requisite accuracy of this model is, of course, highly dependent on the particular application. For example, in a high PRF (Doppler unambiguous [3]) airborne MTI (AMTI) radar with good twoway antenna sidelobes, the targets of interest will be well removed (in Doppler) from the strongest main lobe clutter interference [3]. This situation only requires a sidelobe notch that is well removed in angleDoppler from the target main lobe. 51
52
SpaceTime Adaptive Processing for Radar
Since widening and deepening this notch width (e.g., with CMTs [4] or power selected training (PST) [5]) will have virtually no effect on the target main lobe response, the clutter model need not be very complex. On the other hand—in stark contrast to the AMTI case—ground MTI (GMTI) radars can often encounter slow targets that are very close (in angleDoppler) to mainbeam clutter. In this case, a much higherfidelity clutter model and multidimensional filtering scheme are required [6]. Since ground clutter can be exquisitely complex—comprised of all sorts of terrain, surface reflectivities, and internal motion—this modeling problem can be extremely challenging [6]. To make matters worse, this model must be implementable in real time. Indeed, much of the current STAP research is aimed at ever more refined and robust modeling (explicit or implicit) of ground clutter for just such applications [6–11]. In this chapter, we introduce a basic model of clutter combining both basic Doppler physics and stochastic modeling. This firstorder 2D clutter model will be useful for establishing basic performance characteristics and properties of both optimum and adaptive STAP filtering algorithms. While adequate for some nonstressing applications, many necessary realworld refinements to this model will be described in Chapter 4. Only then will a comprehensive survey of many current STAP techniques be introduced and applied to higherfidelity simulations.
3.2 Need for Joint Space and Time Processing When a radar emits a pulse, it propagates outward at the speed of light with a directional intensity distribution governed by the transmit antenna pattern [12]. In addition to potentially illuminating a target of interest, everything else in the radar’s field of regard will likewise be radiated. Of particular concern are the reflections from ground clutter whose aggregate return can be many orders of magnitude greater than a potential target of interest. In addition to angle and range separation, for MTI radars, the Doppler shift induced by a moving target is pivotal in separating it from ground clutter (which is assumed to be stationary [3]). For a nonmoving radar (e.g., a fixed groundbased or statically deployed airborne system such as an aerostat [13]), groundclutter returns should not have appreciable Doppler shifts and can thus be eliminated by notching out the zeroDoppler filter [3]. Unfortunately, this straightforward 1D processing strategy is insufficient when the radar is moving—such as in airborne or even spaceborne MTI systems [14].
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53
To see why this is the case, consider Figure 3.1, which depicts the N element ULA radar of Chapter 2 in uniform motion. Neglecting for the moment any small elevation angle and curved Earth effects [1, 2], we see that ground clutter contained in an isorange ring centered at the radar will have a Dopplershift distribution due to the motion of the radar. The width of the ring, which is governed by the bandwidth of the radar pulse, is approximately ⌬R ∼ c /(2B ) [2], where c is the speed of light, and B is the pulse bandwidth (consistent units assumed). For the sidelooking case illustrated in Figure 3.1, we see that the normalized Doppler shift induced on a differential clutter patch located at an angle relative to the array boresight is given by [1] fd =
2vT sin
(3.1)
where v is the speed of the radar in units consistent with and T. Note that (3.1) is consistent with (2.23) if we recognize that for the geometry of
Figure 3.1 Illustration of a clutter isorange ring for a ULA radar in uniform constantvelocity motion relative to the ground.
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SpaceTime Adaptive Processing for Radar
Figure 3.1, vtgt = 0 (i.e., the clutter patch is stationary), vRx = v ˆi y , where ˆi y is a unit vector in the y direction, and ˆi y ⭈ ˆi Rx = −sin . In addition to a spatially dependent Doppler shift, each clutter patch will have a spatially dependent reradiating intensity that depends on the transmit antenna pattern and the intrinsic reflectivity of the clutter [15]. In general, this pattern consists of both front lobe (main lobe and sidelobes) and back lobe radiation (see Figure 3.1). Most airborne MTI radars are designed to insure that the back lobe radiation is significantly attenuated and can often be ignored [16]. However, since this may not always be the case (especially at low frequencies), we will explicitly account for it in Chapter 4. Figure 3.2 illustrates the impact of uniform radar motion. In the absence of platform motion (v = 0), the stationary clutter is concentrated along the zeroDoppler contour for all . However, when v ≠ 0, there is a linear relationship between Doppler and sin (or ) governed by (3.1). As a result, the clutter energy is distributed along a line, or ‘‘clutter ridge,’’ as shown in Figure 3.2. Note that for the case illustrated, the antenna is aligned with direction of motion. The effect of antenna crabbing (array misalignment with the groundreferenced velocity vector) will be considered in Chapter 4. The slope  of the clutter ridge is easily obtained by introducing the d normalized angle, = sin , in (3.1), which yields
Figure 3.2 AngleDoppler (spacetime) structure of airborne clutter due to ownship platform motion.
SpaceTime Adaptive Processing
fd = =
冉 冊冉 冊 冉 冊
2vT d
d
55
sin
2vT d sin d
(3.2)
=  from which we see that [1]
=
2vT 4vT = d

d=
(3.3)
2
Thus, the case illustrated in Figure 3.2 corresponds to  = 1. Figure 3.2 also clearly illustrates that a simple 1D filtering scheme is not optimum for separating target from clutter since competing clutter can potentially come from several possible directions (depending on  ) [1], hence the need for joint space and time (i.e., 2D) processing. We are now in a position to quantify the total clutter return from a given isorange ring. Assuming that the i th clutter patch at a normalized angle, i , in the isorange ring is far enough away so that elevation angle is negligible [1], its spatial N dimensional steering vector a i is given by a i = 冋1 e j 2i e j 2 (2)i … e j 2 (N − 1)i 册
T
(3.4)
Since this i th clutter patch also has a normalized Doppler shift f di , each vector of array outputs (i.e., a ‘‘snapshot’’) from successive pulses due to the i th clutter patch will have a temporal linear phase progression; that is, at the m th PRI, the clutter patch snapshot is given by e j 2 (m − 1)f di a i
(3.5)
If M pulses are to be processed in a CPI [1], the total N × M matrix of spacetime steering vectors for the i th clutter patch is of the form
冋a i
e j 2 f di a i … e j 2 (M − 1) f di a i 册
(3.6)
This corresponds to an observation vector consisting of a total of NM measurements that can conveniently be represented by the Kronecker product [17] of the Doppler and spatial steering vectors, namely,
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SpaceTime Adaptive Processing for Radar
vi = b i ⊗ a i
(3.7)
where vi is the NM dimensional spacetime steering vector for the i th clutter patch, and b i is its corresponding M dimensional Doppler steering vector given by b i = 冋1 e j 2 f di … e j 2 (M − 1) f di 册
T
(3.8)
⊗ denotes the Kronecker (or tensor) product, which is defined as follows [17]: Let A and B denote matrices of dimensions m × n and p × q , respectively. Then A ⊗ B is an (mp ) × (nq )dimensional matrix of the form a 12 B a 22 B
… …
冤
a 11 B a 21 B ⯗ a m1 B
a m2 B
…
a 1n B a 2n B ⯗ a mn B
冥
(3.9)
where a ij denotes the (i , j )th element of A. Note that (3.7) is obtained from the concatenation of the columns of (3.6). Figure 3.3 displays a sample normalized angleDoppler signal of the form (3.7); as was the case with the 1D angle response, main lobe and sidelobe features are evident. The pattern in Figure 3.3 can be obtained via a 2D FFT since both the spatial and temporal sampling are uniform [1]. The total spacetime clutter return from a given isorange is thus an NM dimensional random vector X c ∈ ⺓NM of the form Xc =
Nc
∑ ␥˜ i vi
(3.10)
i =1
where ␥˜ i is a complex scalar RV that accounts for the amplitude and phase of the i th clutter patch, and N c is the total number of clutter patches in the isorange ring. Note that (3.10) is essentially a Riemann sum approximation to the actual continuous clutter integral [1, 15]. However, due to the finite spatial and temporal resolution (or bandwidth) of the radar, this approximation is accurate provided that N c and the corresponding clutter patches are chosen properly. For a radarcentric treatise of the validity of integral approximations like (3.8), the reader is referred to Brennan and Mallett [18].
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Figure 3.3 Example of a normalized 2D angleDoppler steeringvector pattern achieved via a 2D DFT (sidelooking case with a normalized Doppler of +0.25). Note the presence of both angle and Doppler sidelobes.
In general, ␥˜ i will depend not only on the transmit antenna pattern, but also on the intrinsic nature of the clutter and can be quite complex in its scattering mechanisms [15, 19–21]. Several of these complicating factors will be considered in later chapters (e.g., inhomogeneities, internal clutter motion). At present, we will assume that the underlying clutter is homogenous and can thus be modeled as a wide sense stationary (wss) process [22] with uncorrelated complex reflectivity factors; that is, E { ␥˜ i ␥˜ j* } = 0, ∀i , j : i ≠ j
(3.11)
Moreover, the average (or expected) strength of the i th scatterer is assumed to be proportional to the transmit antenna gain [1]; that is, E 再  ␥˜ i 
2
冎 = G i , for i = 1, . . . , N c
(3.12)
where G i is a strictly positive realvalued constant that is proportional to the transmit antenna gain in the direction i . For this case, the resulting NM dimensional spacetime clutter covariance matrix R c ∈ ⺓NM × NM is given by
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SpaceTime Adaptive Processing for Radar
R c = E {X c X c′ } =E
i =1
=
′
冦冢 ∑ 冣 冢 ∑ 冣 冧 Nc
Nc
␥˜ i vi
Nc
i =1
␥˜ i vi
(3.13)
Nc
∑ ∑ E { ␥˜ i ␥˜ j* } vi vj′
i =1 j =1
=
Nc
∑ G i vi vi′
i =1
The total spacetime covariance matrix R ∈ ⺓NM × NM, due to both clutter and receiver noise, is of the form R = R c + 2I
(3.14)
where it is assumed that clutter is uncorrelated with receiver noise. The cluttertonoise ratio (CNR) [1] is defined as the ratio of the clutter power to the receiver noise power on a single element and a single pulse; that is, CNR =
R c (1, 1)
2
(3.15)
where the first diagonal element of R c was arbitrarily chosen since each channel is assumed to have the same receiver noise level and elemental antenna pattern. Throughout this text, we will set the noise floor to 0 dB ( 2 = 1) for notational and graphical convenience and, thus, adjust the clutter (and possibly jammer) level to achieve a specified CNR (and/or JNR). We are now in a position to examine the spacetime, or equivalently, the angleDoppler structure of R . One method for characterizing the clutter is by examining the 2D (angleDoppler) power spectrum corresponding to R [2]. The socalled Fourierbased power spectral density (psd) is the 2D positive function defined as [2] P F ( f d , ) = s′ ( f d , ) R s ( f d , )
(3.16)
where s( f d , ) is the NM dimensional target steering vector specifying the normalized Doppler and angle of interest, and R is the total interference covariance matrix. Equation (3.16) can be interpreted as the expected value
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59
of the 2D discrete Fourier transform of the total clutter plus noise signal out of the spacetime beamformer [2, 23]; that is, P F ( f d , ) = E 冠  s′ ( f d , )x 
2
冡
(3.17)
= s′ R s where x ∈ ⺓NM is the received NM dimensional total clutterplusnoise vector RV. It is assumed in (3.16) that the receiver noise floor is set to 0 dB (i.e., 2 = 1) and that the steering vectors are normalized to unity (i.e., s′s = 1), which insures that P F ( f d , ) = 1 if only receiver noise is present. Later in this chapter, a much higher resolution estimate will be obtained that is far more representative of the information exploited by optimum spacetime processors. Figure 3.4 shows the psds for four different Doppler spreads (  = 0, 0.5, 1, and 2) for a 16element, 16pulse example (i.e., N = M = 16) with a CNR set to 40 dB. Note the presence of Doppler aliasing when  > 1 [3]. In this case, for a given target Doppler of interest, competing sidelobe clutter can exist in multiple directions. In certain low Earth orbit (LEO) spacebased radar (SBR) scenarios, Doppler aliasing can occur in the mainbeam due to the extremely high ground speeds (∼7 km/sec) [16, 24], unless a very large antenna is employed. It is apparent from Figures 3.2 and 3.4 that there can be a substantial reduction in the percentage of the unambiguous angleDoppler observation space occupied by clutter when 2D processing is employed. This colorednoise subspace compression is best understood and quantified by conducting an eigen analysis of the total interference covariance matrix to determine the extent of the clutter subspace. Figure 3.5 shows the eigenvalue distributions corresponding to the four different  values considered in Figure 3.4. Note the increase in the effective rank of the clutter portion of the total interference eigenspectrum with increasing  , where the effective rank is defined as NM
rank eff {R c } =
∑ U + 冠 c
i =1
i
− 2 冡 ≤ NM
where U + (z ) =
再
1, z > 0 0, z ≤ 0
(3.18)
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SpaceTime Adaptive Processing for Radar
Figure 3.4 Power spectral densities (Fourierbased) for four different values of  : (a)  = 0, (b)  = 0.5, (c)  = 1, and (d)  = 2.
and R c is the clutteronly covariance matrix. Equation (3.18) defines the effective rank as the number of eigenvalues that exceed the receiver noise floor [25]. This is often a more meaningful definition of rank for radar engineers since eigenvalues below the noise floor can be ignored (recall from Chapter 2 that the eigencanceller form of the optimum weight vector only contains contributions from eigenvectors whose corresponding eigenvalues are greater than min = 2 [25]). A useful expression for estimating the rank (under ideal conditions) of the clutteronly covariance matrix, R c , has been obtained by Brennan for the case of a sidelooking ULA with fixed PRF, constant velocity, and no crab [1, 26]. Specifically
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Figure 3.5 Eigenvalue distributions corresponding to the four different  values of Figure 3.4.
rank {R c } ≈ N + (M − 1) 
(3.19)
Equation (3.19) is an exact equality for integer values of  . Since the effective rank depends on CNR according to (3.18), in general rank eff {R c } ≤ rank {R c }, especially for weak clutter cases. It is worth examining the origins of (3.19) as it sheds light on the issues associated with spacetime clutter cancellation, as well as on an historically important spacetime clutterrejection scheme known as displaced phase center array (DPCA) processing [27]. To accomplish this, we recall from (3.13) that R c is of the form Rc =
Nc
∑ G i vi vi′ = VGV ′
(3.20)
i =1
where ⌬ V= 冋v1 … vNc 册
(3.21)
that is, an NM × N c matrix whose columns are the N c clutter steering vectors, and
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SpaceTime Adaptive Processing for Radar
G = diag 再G 1 … G N c 冎
(3.22)
that is, an N c × N c diagonal matrix of the relative clutter patch strengths (including transmit antenna pattern effects). Since G i represents the average power from each clutter patch, we may assume that G i > 0, ∀i [if a particular G i = 0, simply remove its corresponding steering vector from the summation in (3.20)]. Thus, the rank {R c } is given by rank {R c } = min (rank {[v1 v2 … vN c ]}, NM )
(3.23)
= min (rank {V }, NM ) where it is assumed that N c ≥ NM and vi ≠ vj for i ≠ j (where is an arbitrary complex scalar). In other words, the rank of R c is determined by the number of linearly independent spacetime clutter steering vectors, vi , i = 1, . . . , N c : N c ≥ NM , but may be less than NM (indeed, this is desirable). For the conditions stated, and assuming that  is an integer, the i th clutter steering vector has a Vandermonde structure [1] of the form −1  w i w i + 1 … w i + N − 1  … v i = 冋1 w i … w N i − 1)  + 1 … w i(M − 1)  + N − 1 册 …  w i(M − 1)  w (M i
(3.24)
T
where w i = e j 2 i . Inspection of (3.24) (with  an integer) reveals that there are only N + (M − 1)  unique elements; the rest are redundant. For example, if  = 1, (3.24) becomes −1  w 1i w 2i … w iN  … vi = 冋1 w i … w N i − 1) − 1) + 1 w (M … w i(M − 1) + N − 1 册 …  w (M i i
(3.25) T
Starting with the first N entries from the first PRI (first block), we see that only one element from the second PRI (second block) is distinct from the first N, namely w N i . This trend continues for the third and subsequent pulses, with only one distinct new element being added for each new PRI. Thus, after M pulses, the total number of distinct elements is N + (M − 1), which agrees with (3.19) exactly for  = 1. Since the rank of a matrix is unaffected by an interchange of rows, the above observations imply that, in general, V can be rearranged as follows:
SpaceTime Adaptive Processing
1 w1 w 21 ⯗
…
1 wN
c
w 2N c
=
⯗
… (N − 1) +  (M − 1) w 1(N − 1) +  (M − 1) wN c ———————— NM − [N +  (M − 1)] Repeated Rows
63
冤
⌶ ———— Repeated Rows
冥
(3.26)
where ⌶ is an [N +  (M − 1)] × N c , matrix. Since the rank of a matrix is equal to the number of linearly independent rows or columns (which ever is smaller) [28], we see that rank {V } = rank {⌶} = min {N +  (M − 1), NM }
(3.27)
since w i ≠ w j , for i ≠ j , which is the desired result. It is particularly illuminating to examine the  = 1 case, which corresponds to the situation when the antenna moves precisely one interelement spacing per PRI. If only the first N − 1 elements are used for reception on the first pulse, and only the last N − 1 elements are likewise used for the next (second) pulse, then the ground appears stationary to the antenna (assuming perfect channel match) [27]. This is the socalled DPCA case [27]. Clutter cancellation can then be ‘‘simply’’ accomplished by coherently subtracting the returns from successive pulses on adjacent antennas (i.e., the twopulse MTI canceller [2]). Apart from the challenges of matching antenna channels and eliminating crab angle of the array relative to the true groundtrack flight path, DPCA requires that the PRI (or PRF) be chosen in (3.3) to satisfy  = 1—an often demanding restriction in practice. DPCA will be studied in more detail in Chapter 5. 3.2.1 Joint Clutter and Jamming Characteristics If clutter and jamming are both present, the total spacetime interference covariance matrix has the form R = R c + R J + 2I
(3.28)
where it is assumed that the jamming is uncorrelated with clutter and receiver noise [22]. The single pulse, spatialonly jammer covariance matrix was
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SpaceTime Adaptive Processing for Radar
derived in Chapter 2. Assuming that the jammer signal is uncorrelated from pulsetopulse, the offdiagonal block terms (corresponding to temporal cross correlation) are null matrices. Thus, the NM dimensional jammeronly covariance, R J , has the following blockdiagonal form:
RJ =
∑ j2 s j s j′
j =1
∅ NJ
∑ j2 s j s j′
j =1
…
冤
NJ
NJ
∑ j2 s j s j′
∅
j =1
冥
(3.29)
where ∅ is used to indicate that the offblockdiagonal entries are zero. Since the rank of a blockdiagonal matrix is equal to the sum of the ranks of the constituent blocks [28], we see that the rank of R J is given by
∑
冦∑ NJ
M
rank {R J } =
rank
m =1
= M rank
j =1
冦∑ NJ
j =1
j2 s j s j′
j2 s j s j′
冧
(3.30)
冧
For example, if there are three uncorrelated noise jammers, and M = 16, the rank of R J is 3 × 16 = 48. If N = 16, the fraction of the total available observation space occupied by the jammers is 48 ÷ 256 = 3/16, which is precisely the same ratio if spatialonly processing is performed (N J = 3, N = 16). In other words, there is no advantage to using spacetime processing against conventional noise jammers—spatialonly processing will suffice. The advantage of spacetime processing is realized against clutter (although simultaneous jammer nulling is readily accommodated—as will be shown later). Figure 3.6 displays a spacetime eigenspectrum with and without jamming. Note that the presence of jamming generally increases the effective rank of the combined clutter and jamming subspace.
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Figure 3.6 Impact of jamming on the total interference eigenspectrum.
3.3 Optimum SpaceTime Processing for MTI Radar We next turn our attention to deriving the optimal spacetime beamformer for rejecting both clutter and jamming. Consider the spacetime linear beamformer of Figure 3.7, consisting of N identical antenna elements (spatial DoFs) and M PRI time taps (temporal DoFs). Analogous to the 1D beamformer of Figure 2.1, the output y is in general the linear superposition of the desired signal response y s and an undesired noise/interference response y n . Our objective will thus be to choose an optimal set of complex spacetime weights, w, so as to maximize SINR. By adopting the vector RV notation of Chapter 2, we have an optimization problem identical to that already solved in Chapter 2. Specifically, the weight vector that maximizes SINR is given by w = R −1 s
(3.31)
where R ∈ ⺓NM × NM is the NM × NM total interference (clutter and/or jamming) plus receiver noise covariance matrix, is a scalar that does not affect the SINR, and s ∈ ⺓NM is the NM dimensional spacetime (angleDoppler) steering vector of the desired signal which is given by
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SpaceTime Adaptive Processing for Radar
Figure 3.7 Spacetime (angleDoppler) beamformer and ‘‘data cube’’ illustration.
s=b⊗a
(3.32)
where b denotes the desired Doppler steering vector and a the spatial component; that is, b = 冋1 e j 2 f d … e j 2 (M − 1) f d 册
T
(3.33)
and a = 冋1 e j 2 0 … e j 2 (N − 1) 0 册
T
(3.34)
where 0 and f d denote the desired angle and Doppler, respectively, and the superscript T denotes transposition without conjugation. The corresponding optimal SINR is given by (see Chapter 2), SINR opt = s′ R −1 s
(3.35)
As with the 1D optimal beamformer, tapering can be incorporated into (3.31) to reduce both angle and Doppler sidelobes at the expense of
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67
mainbeam broadening and additional SINR loss (see Section 2.2). For 2D beamforming, tapering is incorporated as follows: w = R −1 (s 䊊 t)
(3.36)
t = td ⊗ t
(3.37)
where
t d ∈ ⺢ +M is the M dimensional Doppler taper, and t ∈ ⺢ +N is the N dimensional angle taper (thus t ∈ ⺢ +NM is NM dimensional as expected). Figure 3.8 provides an example optimum spacetime beam pattern (with and without tapering) for an N = M = 16 example, with look angle and Doppler given by 0 = 0 (sidelooking case), f d = + 0.25. The  is set to 1, the CNR is 40 dB, and all JNRs are 50 dB (at the element level). Note the simultaneous presence of both clutter and jamming nulls. Since for a given radar pointing angle there may be a range of Doppler frequencies of interest, it is often convenient to plot SINR versus Doppler for a fixed angle. Figure 3.9 shows an example of this for the sidelooking case ( 0 = 0°); the SINR dropout due to main lobe clutter is clearly evident. It is evident from an examination of Figure 3.8 that the optimum spacetime beamformer is utilizing very highresolution information regarding the interference, which is manifested as sharp nulls in the angleDoppler patterns.
Figure 3.8 Optimum spacetime (angleDoppler) beam pattern illustrating the simultaneous nulling of both clutter and jamming: (a) without tapering, and (b) with 30dB Chebyshev angle and Doppler tapers.
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SpaceTime Adaptive Processing for Radar
Figure 3.9 SINR versus Doppler for 0 = 0° (sidelooking case) of Figure 3.7. Note the SINR dropout due to mainbeam clutter.
Indeed, a very highresolution (i.e., a socalled superresolution [29]) spectral estimator underlies the beamformer and is given by P MF ( f d , 0 ) =
1 s′ ( f d , o ) R −1 s ( f d , o )
(3.38)
which is simply the reciprocal of the optimal SINR given by (3.35). Note that consistent with (3.16), it is assumed that 2 = 1 and s′s = 1, which insures that P MF ( f d , 0 ) = 1 when only receiver noise is present. Equation (3.38) is generally referred to as the minimum variance or Capon superresolution spectral estimator [23, 29, 30] and can be interpreted as the expected value of the output power of the optimum colorednoise matched filter (optimum spacetime beamformer), when a unity gain on signal constraint is invoked. The reason (3.38) achieves a much higher resolution than the Fourierbased estimator of (3.16) is that the weight vector minimizes leakage from all other angleDoppler, while maintaining a mainbeam constraint [23]. To see this explicitly, consider the Capon estimator (weight vector), which satisfies the following optimization problem:
SpaceTime Adaptive Processing
min: E 冠  w′x  {w}
2
冡 = w′R w
69
(3.39)
subject to: w′s = 1 Since R is assumed to be positivedefinite, the objective function is a convex quadratic form subject to a linear equality constraint [31]. Thus, a necessary and sufficient condition for an extremum is that the first (vector) derivative of the following augmented objective function vanish: d (w′R w + (1 − w′s)) =0 dw
(3.40)
where is the Lagrange multiplier associated with the unity gain constraint [31]. From the vector derivative identities [31] d w′R w = 2R w dw
(3.41)
d w′s = 2s dw
(3.42)
and
we have, upon substitution into (3.40), the following optimum Capon spectral estimator: w = R −1 s
(3.43)
where to satisfy the unity gain constraint w′s = 1,
=
1 sR −1 s
(3.44)
Note that (3.43) is the same expression as the max SINR beamformer. The only difference is the choice of normalization parameter. The beamformer output power corresponding to (3.43) is thus given by
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SpaceTime Adaptive Processing for Radar
E 再  w′x 
2
冎 = w′E {xx′ } w = w′R w = ( *s′R −1 ) R ( R −1 s)
(3.45)
=   s′R −1 s 2
= =
s′R −1 s
 s′R −1 s  2 1 s′R −1 s
which is the expression in (3.38). Figure 3.10 contains a comparison of the Fourierbased [i.e., (3.16)] and Capon powerspectral density plots; the superresolution properties of the spacetime matchedfilterbased method are clearly evident. Finer resolution is often very useful for better understanding the underlying properties of the interference, as well as the performance of optimum spacetime beamformers. This will be made evident in Chapter 4, when clutter subspace leakage phenomena (e.g., internal clutter motion) are introduced.
Figure 3.10 Comparison of (a) Fourierbased and (b) minimum variance (MV) power spectral estimators. Note the high resolution of the minimum variance (MV) technique implicitly utilized by the optimal spacetime beamformer.
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71
3.4 STAP With Chapter 2 as background, it is clear that STAP is simply the practical attempt to implement optimum spacetime processing [i.e., (3.31)]. Since, in practice, the interference environment is generally not known a priori, it is necessary to approximate the ‘‘exact’’ or ‘‘ideal’’ covariance required in (3.31) with a finite sample estimate. Again, this is assumed to be accomplished by the availability of a suitable training region in which the interference is present (without any targetlike signals)—as was the assumption in Chapter 2. Unfortunately, the sample support requirements are much greater for STAP than for the relatively simple 1D adaptive beamforming examples considered in Chapter 2. For example, from the RMB result [see (2.24)], we see that at least 2NM = 512 samples are required for the N = M = 16 case to insure that the resulting SINR is within about 3 dB of optimum, compared with only 32 samples for the N = 16 1D case. Assuming that the interference is stationary over hundreds of range bins is simply not realistic in many applications. Thus, from the very onset, we are faced with the inherent requirement to develop techniques that retain the benefits of 2D filtering, but reduce sample support requirements. Whatever methods are considered must also be robust to other realworld effects (internal clutter motion, channel match, and so forth) described in Chapter 4. Thus, we will defer a discussion of potential candidate algorithms to Chapter 5. Figure 3.11(a) shows the impact on the adapted pattern of replacing the ideal covariance with a finite sample estimate for the case considered in Figure 3.8. Note the presence of very high sidelobes and a distorted main lobe (even though 30dB Chebyshev angle and Doppler tapers were applied). However, as previously analyzed in Section 2.4, a potential remedy is the use of principal components or diagonal loading. Figure 3.11(b) shows the adapted pattern resulting from the addition of 10 dB of diagonal loading to the sample estimate (along with 30dB Chebyshev angle and Doppler tapers). It is important to note that these results still required a significant number of i.i.d. training data and, thus, may still not be practical.
3.5 Summary In this chapter, the rationale for joint 2D space and time (angleDoppler) processing was established. The detailed nature of the angleDoppler structure of clutter was thoroughly examined from a variety of perspectives. In particu
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Figure 3.11 Impact of finite sample support on 2D adapted patterns: (a) adapted pattern resulting from L = 2NM = 512 samples, and (b) result if 10 dB of diagonal loading is added. Both patterns include 30dB Chebyshev angle and Doppler tapers.
lar, a detailed eigen analysis of the spacetime clutter covariance matrix was performed and an expression (i.e., Brennan’s rule) for its rank as a function of  was derived. Next, an expression for the total spacetime clutterplusjammingplusnoise covariance was derived for the case of uncorrelated noise jamming. Utilizing the optimization framework established in Chapter 2, an expression for the optimal SINR spacetime beamformer was derived and illustrated with a multiple jammerplusclutter scenario. Finally, STAP was introduced via the substitution of the ideal covariance matrix (unknown a priori) with an estimate obtained from sample data. Some adapted patterndistortion issues due to finite sample support (estimation errors) were also illustrated along with some rudimentary basic fixes (i.e., diagonal loading).
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Ward, J., SpaceTime Adaptive Processing for Airborne Radar, MIT Technical Report 1015, MIT Lincoln Laboratory, December 1994.
[2]
Klemm, R., SpaceTime Adaptive Processing: Principles and Applications, London, England: IEEE Press, 1998.
[3]
Schleher, D. C., MTI and Pulsed Doppler Radar, Norwood, MA: Artech House, 1991.
[4]
Guerci, J. R., ‘‘Theory and Application of Covariance Matrix Tapers for Robust Adaptive Beamforming,’’ IEEE Trans. on Signal Processing, Vol. 47, No. 4, April 1999, pp. 977–986.
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[5]
Rabideau, D. J., and A. O. Steinhardt, ‘‘Improved Adaptive Clutter Cancellation Through Data Adaptive Training,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 35, No. 3, July 1999, pp. 879–891.
[6]
Guerci, J. R., ‘‘KnowledgeAided Sensor Signal Processing and Expert Reasoning,’’ Proc. of 2002 Workshop on KnowledgeAided Sensor Signal Processing and Expert Reasoning (KASSPER), Washington, D.C., April 3, 2002 (CDROM).
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Melvin, W. L., ‘‘SpaceTime Adaptive Radar Performance in Heterogeneous Clutter,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 36, No. 2, April 2000, pp. 621–633.
[8]
Bergin, J. S., et al., ‘‘GMTI STAP in TargetRich Environments: SiteSpecific Analysis,’’ Proc. of IEEE 2002 Radar Conference, Long Beach, CA, April 22–25, 2002, pp. 391–396.
[9]
Melvin, W. L., and J. R. Guerci, ‘‘Adaptive Detection in Dense Target Environments,’’ Proceedings of the IEEE 2002 Radar Conference, Atlanta, GA, May 1–3, 2001, pp. 187–192.
[10]
Melvin, W., (ed.), ‘‘SpaceTime Adaptive Processing and Adaptive Arrays: Special Collection of Papers,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 36, No. 2, April 2000, pp. 508–509.
[11]
Klemm, R., (ed.), Special Issue on SpaceTime Adaptive Processing, Electronics and Communication Engineering Journal, Vol. 11, February 1999.
[12]
Barton, D. K., Radar Systems Analysis, Dedham, MA: Artech House, 1976.
[13]
‘‘Tethered Aerostat Radar System (TARS),’’ http://www2.acc.af.mil/library/factsheets/ tars.html.
[14]
Davis, M. E., ‘‘Technology Challenges in Affordable Space Based Radar,’’ Record of the 2000 IEEE International Radar Conference, Alexandria, VA, May 7–12, 2000, pp. 18–23.
[15]
Long, M. W., Radar Reflectivity of Land and Sea, 3rd ed., Norwood, MA: Artech House, 2001.
[16]
Mailloux, R. J., Phased Array Antenna Handbook, Norwood, MA: Artech House, 1994.
[17]
Horn, R. A., and C. R. Johnson, Topics in Matrix Analysis, Cambridge, England: Cambridge University Press, 1991.
[18]
Brennan, L. E., and J. D. Mallett, ‘‘Efficient Simulation of External Noise Incident on an Array,’’ IEEE Trans. on Antennas and Propagation, Vol. 24, 1976, pp. 740–741.
[19]
Knott, F. K., J. F. Shaeffer, and M. T. Tuley, Radar Cross Section, 2nd ed., Norwood, MA: Artech House, 1993.
[20]
Billingsley, J. B., Exponential Decay in Windblown Radar Ground Clutter Doppler Spectra: Multifrequency Measurements and Model, Technical Report 997, MIT Lincoln Laboratory, Lexington, MA, July 29, 1996.
[21]
Billingsley, J. B., Low Angle Radar Land Clutter: Measurements and Empirical Models, Raleigh, NC: SciTech Publishing, 2002.
[22]
Papoulis, A., and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 3rd ed., New York: McGrawHill, 2001.
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[23]
Pillai, S. U., Array Signal Processing, New York: SpringerVerlag, 1989.
[24]
Rabideau, D., and S. Kogon, ‘‘A Signal Processing Architecture for SpaceBased GMTI Radar,’’ Proc. of IEEE Radar Conference, Waltham, MA, 1999, pp. 96–101.
[25]
Guerci, J. R., and J. S. Bergin, ‘‘Principal Components, Covariance Matrix Tapers, and the Subspace Leakage Problem,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 38, No. 1, January 2002.
[26]
Brennan, L. E., and F. M. Staudaher, Subclutter Visibility Demonstration, Technical Report RLTR9221, Adaptive Sensors Inc., 1992.
[27]
Skolnik, M., Radar Handbook, 2nd ed., New York: McGrawHill, 1990.
[28]
Horn R. A., and C. R. Johnson, Matrix Analysis, Cambridge, England: Cambridge University Press, 1985.
[29]
Gabriel, W. F., ‘‘Spectral Analysis and Adaptive Array Superresolution Techniques,’’ Proc. of IEEE, Vol. 68, No. 6, 1980, pp. 654–666.
[30]
Capon, J., ‘‘HighResolution, FrequencyWavenumber Spectral Analysis,’’ Proc. of IEEE, Vol. 57, No. 8, 1969.
[31]
Pierre, D., Optimization Theory with Applications, New York: Dover, 1986.
4 Other Important Factors Affecting STAP Performance 4.1 Introduction Due to a multitude of practical considerations, the idealized performance presented in Chapter 3 represents a generally unachievable upper bound on performance—for both the known and iid sample estimate covariance cases. In this chapter, we will examine some of the salient factors that contribute to this degradation and must be considered when both designing a STAP system, as well as predicting realized performance. Most, if not all, of the factors described in this chapter are present to a greater or lesser degree in any STAP radar. Thus, when comparing various STAP algorithms (such as those described in Chapter 5 and elsewhere), careful attention should be paid to how they perform when one or more of the generally deleterious effects described herein is present. This chapter is not meant to serve as a comprehensive treatise on factors that can affect STAP performance. It is, however, designed to illustrate an important phenomenon of realworld STAP systems: The effective clutter/ interference rank is almost certainly greater than would be the case in idealized, simplified signal and systems models, and the rank increases with increasing interferencetonoise ratio (INR). This latter dependency on INR is referred to as the iceberg effect (see Section 4.6), since increasing INR tends to ‘‘lift’’ the colored interference eigenvalues further above the ‘‘ocean’’ noise floor—thereby exposing more of the ‘‘iceberg’’ (effective interference rank). 75
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In varying clutter terrain situations, this can be a significant source of nonstationarity and often will result in wider Doppler clutter notches (and poorer minimum detectable velocities). Advanced methods for addressing these issues will be discussed and analyzed in Chapter 5. Although a multitude of factors contribute to increasing the effective rank (as defined in Chapter 3) of the interference subspace, the net effect is essentially a decorrelation between spatial and temporal measurements (channels and/or pulses). The phenomena that give rise to this rankincreasing effect are referred to as interference subspace leakage, or ISL, since the decorrelation tends to cause a smearing or leakage of the interference subspace [1]. For example, in the derivation of Brennan’s rule (Chapter 3), it was assumed that if the clutter was sampled at a later time (e.g., one PRI later) with an identical antenna channel observing from exactly the same spot, the total clutter return would be identical (i.e., perfectly correlated). If, however, ICM is present, the signals will not be the same, and the net effect is a decorrelation between samples. Moreover, any mismatches between antenna channels could likewise cause differences in the observed signals, again potentially resulting in decorrelation. As we will see later in this chapter, a convenient mathematical representation of decorrelating ISL (and nondecorrelating channel mismatch) is via CMT [1–4]. Specifically, if R ∈ ⺓NM × NM denotes the total interference covariance matrix, then the resultant covariance including all ISL effects has the form [1] ⌬ ˜ ISL R ∈ ⺓ NM × NM R 䊊 T1 䊊 T2 䊊 . . . 䊊 Tk = R –––––→
(4.1)
where T 1 , T 2 , . . . , T k ∈ ⺓NM × NM are positive (semi) definite Hermitian matrices associated with uncorrelated ISL phenomenon (such as ICM and decorrelating channel mismatch—see below), and 䊊 denotes Hadamard (elementwise) matrix multiplication (see Chapter 2). This mathematical form turns out to be more than just a convenient compact representation of realistic covariance structure. For example in Chapter 5, the CMT structure is explicitly incorporated into two STAP algorithms (PCCMT and SMICMT) that retain the desirable minimal sample support properties of principalcomponent and diagonalloading SMI techniques (i.e., nonISL environments) respectively, yet account for ISL effects with an increase in sample support requirements [1]. The remainder of this chapter is organized as follows: In Section 4.2 we examine channelmismatch effects, which are delineated into two major
Other Important Factors Affecting STAP Performance
77
categories: bandwidth and angledependent effects. For example, in the narrowband signal model case, channel mismatch results in an amplitude and phase offset between channels that can be modeled as an unknown (but fixed over the CPI) multiplicative complex scalar for each channel. This form of mismatch is nondecorrelating and is shown not to result in an increase in the effective interference rank, whereas in the finite (nonzero) bandwidth case, both bandwidth dispersion effects [5] and transfer function differences [6] result in decorrelating mismatch. Angledependent channelmismatch effects arise when the amount of channel mismatch depends on the AoA. For example, element position errors or mutual coupling [7] tend to have an angle dependency. However, as discussed in Section 4.2, since these errors are small to begin with, the dominant effects are associated with the look direction of the radar. Thus, the angledependent CMT T ( ) can be approximated by the fixed CMT T ( ) → T ( 0 ) = T 0 and then incorporated into (4.1). Section 4.3 discusses other ISL effects such as ICM and rangewalk. Again, it is shown that these effects result in a spatial and/or temporal decorrelation that can be conveniently modeled by (4.1) (under certain general conditions). Section 4.4 examines the impact of antenna misalignment with the groundtrack velocity vector (i.e., crabbing [8]) on the effective clutter rank and derives an expression for the angleDoppler clutter locus. A brief discussion of nonlinear array geometries is contained in Section 4.5, which also includes a circular array example. Section 4.6 contains a discussion and illustration of the mechanism by which the aforementioned ISL phenomenon couple into CNR nonstationarity (which occurs in nature due to terrain variability [9]) to produce a more complex nonstationary vector stochastic process whose rank is likewise nonstationary ([1, 10]). This iceberg effect is fundamental, omnipresent, and must be accounted for in the STAP design stage (as illustrated in Chapter 5)—otherwise clutter under or overnulling will result [1]. More exotic radar applications such as bistatic spacetime adaptive radar [11] generally only exacerbate this effect.
4.2 Channel Mismatch An important and everpresent factor affecting STAP performance is socalled channel mismatch (referred to colloquially as channel match ). The N separate antenna channels employed in our ULA model have been assumed
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SpaceTime Adaptive Processing for Radar
up to this point to be ideal; that is, the output response of a ULA was assumed to have a perfect linear phase (Vandermonde) response and no mismatchinduced decorrelation between channels. In reality, it is impossible to have each channel pathway perfectly matched from an inputoutput perspective for all AoAs. For analysis purposes, it is convenient to taxonomize each contributing channelmismatch source based on whether or not it is dependent on AoA and whether it is narrowband (or not). Angledependent channelmismatch errors are often due to array manifold and possibly radome effects (e.g., element position errors and pattern differences, near field scattering and multipath) and are often quite complex and difficult to model and/or predict accurately [1, 6, 12]. Angleindependent sources are generally due to mismatches between the channels after the antenna elements [6, 12]. To facilitate the analysis of channel mismatch, we will adopt the CMT framework that provides a convenient and unifying mathematical framework for a broad class of channel mismatch [1, 2]. Moreover, this framework will also prove useful when examining other subspace leakage phenomenon such as ICM, as well as when considering modelbased minimal sample support STAP methods in Chapter 5.
4.2.1 AngleIndependent Channel Mismatch Angleindependent channel mismatch is due to variations between the transfer functions, H n ( ), n = 1, . . . , N, associated with each receive channel (see Figure 4.1) without angledependent effects (see following discussion). We will first consider the ideal narrowband case in which each transfer function simplifies to a complex gain multiplier.
Figure 4.1 Angleindependent channel mismatch for narrowband arrays arises from variations in the signal paths in each receive channel generally after the array manifold.
Other Important Factors Affecting STAP Performance
79
4.2.1.1 Narrowband Case
The impact of angleindependent narrowband channel mismatch that is stable over a CPI (normally the case) is the introduction of unknown but constant gain and phase differences (i.e., mismatch) between channels. This is conveniently modeled as a constant tapering of all signal vectors impinging on the array. Specifically, if a i denotes the steering vector associated with the i th incoming signal, then the resulting actual array response vector aˆ i is given by aˆ i = a i 䊊 t
(4.2)
where t is an N dimensional vector (or taper) of the form t = 冋⑀ 1 e j 1 . . . ⑀ N e j N 册
T
(4.3)
In (4.3), ⑀ 1 , . . . , ⑀ N and 1 , . . . , N denote the amplitude and phase errors, respectively. The amplitude errors represent the differences in gain through each channel, and thus 0 ≤ ⑀ i ≤ 1 for i = 1, . . . , N. Typically, the variation in gain is quite small and is usually less than 0.1 dB (i.e., ∼0.99 ≤ ⑀ i ≤ 1). Typical values for phase errors are 5° or less. Note that in general  t  ≤ 1. Also, it has been tacitly assumed that the noise floor in each channel is identical. As discussed in Chapter 2, these errors introduce a signal mismatch that limits achievable tapered sidelobe levels and signal integration gain. For a given mismatch taper t, the SNR loss ratio, defined as the ratio of the realized SNR to optimum SNR [see (2.10)] is given by
 s′ (s 䊊 t)  2 SNR loss =
SNR act = SNR opt
2
 s′s  2 2
=
s′ (s 䊊 t) (s 䊊 t)′s
=
s′ [(ss′ ) 䊊 (tt)′ ]s
=
s′ [(ss′ ) 䊊 T ]s
 s′s  2  s′s  2  s′s  2
(4.4)
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SpaceTime Adaptive Processing for Radar
where T = tt′ ∈ ⺓ N × N
(4.5)
is a rankone matrix referred to as a CMT [2]. The final factored form made use of the Hadamard matrix multiplication identity (A 䊊 B ) (C 䊊 D )′ = (AC ′ ) 䊊 (BD ′ )
(4.6)
where A , B are m × n and C, D are n × p matrices [13]. Note that in the absence of channel mismatch, T is the Hadamard identity matrix (i.e., matrix with all unity entries) [13], and (4.4) reduces to unity (i.e., no loss). In a wellcalibrated radar, the SNR loss due to signal mismatch is quite small (∼1 dB or less). Of greater concern in general is the impact of channel mismatch on interference suppression. For the angleindependent narrowband case, each signal impinging on the array is tapered according to (4.2). Thus, the total received spacetime interference signal is of the form X = Xc + XJ + n =
(4.7)
Nc
NJ
i =1
j =1
∑ ␥˜ i vi 䊊 t + ∑ ˜ j sj 䊊 t + n
where X c , X J , n ∈ ⺓NM denote the spacetime clutter, jamming, and receiver noise RVs, respectively. Note that the mismatch taper only affects the structured (colored) noise signals and not the thermal or white noise. This is a consequence of assuming that each channel has the same noise floor (differences in receiver noise levels can be viewed as differences in SNR, which ultimately can be represented as differences in signal gain for a fixed noisefloor level). If only spatial channel mismatch is present, that is, (4.2), then the corresponding spacetime mismatch taper, t, in (4.7) is of the form
冧
M
t = [1 … 1]T ⊗ 冋⑀ 1 e j 1 . . . ⑀ N e j N 册
T
(4.8)
Examining (4.7) in further detail, we see that the first sum is due to clutter (see Chapter 3) with the inclusion of mismatches, while the second sum is due to N J uncorrelated jammers (see Chapters 2 and 3). Assuming that
Other Important Factors Affecting STAP Performance
81
the clutter, jamming, and receiver noise obey the aforementioned statistical models described in Chapters 2 and 3, and that they are mutually uncorrelated, the resulting total interference covariance in the presence of simple channel mismatch is given by ⌬ R cov (X) =
=
Nc
∑ G i (vi vi′ ) 䊊 (tt′ ) +
i =1
=
冢∑ Nc
i =1
G i (vi vi′ ) +
Nc
Nc
∑ j2 (vj vj′ ) 䊊 (tt′ ) + 2 I
j =1
冣
∑ j2 (vj vj′ )
j =1
䊊
(tt′ ) + 2 I
(4.9)
= R I 䊊 T + 2I = R˜ I + 2 I where R˜ I = R I 䊊 T, and R I is the total interference covariance matrix (clutter plus jamming) in the absence of mismatch. T is the rankone CMT associated with the mismatch; that is, T = tt′. An important result regarding the impact of simple channel mismatch on STAP performance can be immediately inferred from (4.9) based on the properties of Hadamard products. Specifically, since rank (A 䊊 B ) ≤ rank (A ) rank (B ), we have [13] rank (R˜ I ) = rank (R I 䊊 T ) = rank (R I 䊊 (tt′ ))
(4.10)
≤ rank (R I ) rank (T ) = rank (R I ) rank (tt′ ) However, since rank (T ) = rank (tt′ ) = 1, we immediately see that rank (R˜ I ) ≤ rank (R I )
(4.11)
That is, the rank of the total interference covariance matrix cannot be increased by simple angleindependent narrowband channel mismatch. The practical consequence of this result is that optimum nulling performance is
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SpaceTime Adaptive Processing for Radar
essentially unaffected (there is slight degradation when amplitude mismatch is present—see following discussion). However, there will still be a small amount of SNR loss due to steeringvector mismatch as described by (4.4). Also, limits on nominal tapered sidelobe levels will also be present as described in Chapter 2. Stronger statements regarding the exact impact of simple channel mismatch can be derived for certain special cases. For example, for phase only mismatch, the eigenvalues of the total interference covariance matrix can be shown to be invariant to the phase errors. Coupled with the rank property of (4.11), this implies that no loss in nulling performance is incurred (there is still a small signalmismatch loss). To see this explicitly, we begin by assuming that t is of the form
冧
M
t = [1 … 1]T ⊗ 冋e j 1 . . . e j N 册
T
(4.12)
that is, phaseonly mismatch. Expressing the resulting covariance in terms of the eigenbased representation of the original covariance (no mismatch), we have R = R I 䊊 T + 2I
冢∑
NM
=
i =1
i u i u i′
冣
䊊
T + 2I
NM
=
∑ i u i u i′ 䊊 T + 2 I
(4.13)
i =1 NM
=
∑ i (u i u i′ ) 䊊 (tt′ ) + 2 I
i =1 NM
=
∑ i (u i 䊊 t) (u i 䊊 t)′ + 2 I
i =1
where R I is the total colorednoise interference covariance matrix (clutter plus jamming), and 2 I is the receiver noise (white noise). Correspondingly, { i , u i } denotes the i th eigenvalue–eigenvector pair for R I . To prove the eigenvalue invariance property, we need to show that the set of modified vectors {u i 䊊 t} retains the orthonormality property (orthogonal and unit norm). Orthonormality is proved as follows
Other Important Factors Affecting STAP Performance
83
(u i 䊊 t)′ (u j 䊊 t) = (u *i 䊊 u j )T (t* 䊊 t) = (u *i 䊊 u j )
T
冤冥 1 ⯗ 1
(4.14)
= u i′u j = ␦ ij where ␦ ij denotes the Kronecker delta function. Note that t* 䊊 t = [1 1 … 1]T since each element of t is of the form e j i (unity modulus). Thus, the eigenvalues of R I 䊊 T + 2 I are the same as R I + 2 I, and the eigenvectors of R I 䊊 T + 2 I are those of R I + 2 I modified by the phaseonly mismatch taper; that is, u i → u i 䊊 t. Note, however, that if amplitude errors are present, the eigenvalues are not invariant (which impacts SINR— although very modestly in practice). Figure 4.2 shows the optimum beam pattern (with a 30dB Chebyshev taper) with channel mismatch for the sixjammer case described in Section 2.2. The phaseonly channelmismatch taper, t, was selected from a 5° (onesigma) Gaussian randomnumber generator. In generating the adapted pattern, modified DFT vectors are required since plane waves impinging on the array no longer obey a perfect linear phase response. If [s 0 . . . s j . . . ] denotes the DFT vectors used to generate the optimum response in the absence of mismatch, then modified DFT vectors of the form [s 0 䊊 t . . . s j 䊊 t . . . ] are required to generate the correct response when mismatch is present. As expected, the adapted pattern has nulls placed at the jammer locations and an average sidelobe level of approximately 30 dB (variations due to channel mismatch). The corresponding SNR loss computed from (4.4) is 0.6 dB—an extremely modest loss. Indeed, as emphasized in Chapter 2, the ability of adaptive beamforming to effect deep nulls even in the presence of certain channel mismatches is a major practical benefit. 4.2.1.2 Finite (Nonzero) Bandwidth Case
For the finite (nonzero) bandwidth case, the N receive channel transfer functions, H n ( ), n = 1, . . . , N, can no longer be simply modeled as unknown scalar complex gain multipliers (i.e., a rankone CMT). Instead, the instantaneous frequency content of a PRI (pulse spectrum) is subtly altered between channels. The net effect for a stochastic signal (i.e., clutter and jamming) is a decorrelation of the signals between channels. To see why this is the case, we first recall the form of the received total interference
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SpaceTime Adaptive Processing for Radar
Figure 4.2 Optimum pattern (with 30dB Chebyshev taper) with 5° (onesigma) phaseonly channel mismatch for the sixjammer case of Chapter 2. Although there is modest 0.6dB SNR loss due to signal mismatch, the adapted pattern places sufficiently deep nulls at the jammer locations to effectively cancel the interference.
signal with channel mismatch given by (4.7), but with t a vector RV (as opposed to a fixed, but unknown, constant vector). Since we are focusing on spatial decorrelation effects resulting from channeltochannel random mismatch [6, 12], it suffices to analyze the pairwise cross correlation between channels. Specifically, E再x i x k* 冎 = E再冠冠x c i + x J i 冡 ⭈ t i + n i 冡冠冠x c k + x J k 冡 ⭈ t k + n k 冡* 冎 (4.15) where (x i , x k ) denotes the complex (inphase and quadrature) interferenceonly outputs of the i th and k th channels for a single PRI, respectively. Under the assumption that the clutter, jamming, thermal (white) noise, and channel amplitude/phase mismatch gains are all mutually uncorrelated (physically justifiable assumptions), (4.15) can be written as
Other Important Factors Affecting STAP Performance
E再x i x k* 冎 = 冠E再x c i x c*k 冎 + E再x J i x J*k 冎冡 ⭈ E再t i t k* 冎 + 2 ␦ ik
85
(4.16)
R i , k = R I i,k ⭈ T i , k + 2 ␦ ik where R i , k and R I i,k denote the (i , k )th elements of the total and colorednoiseonly (clutter plus jamming) spatial covariance matrices, respectively. T i , k = E再t i t k* 冎 is the crosscorrelation coefficient due to channel mismatch and has the property 0 ≤  T i , k  ≤ 1, although in practice it is much closer to unity than to zero (see the following discussion). In matrix form, (4.16) becomes R = R I 䊊 T + 2I
(4.17)
where R is the spatialonly covariance matrix, and T = E{tt′ } is the CMT associated with the channel mismatch. Equation (4.17) is of the same mathematical form as (4.9) with one major exception: T is generally full rank. Thus, unlike the simple rankone narrowband CMT, the presence of decorrelating channel mismatch will alter the eigenvalues and eigenvectors of the interference covariance matrix and may increase the effective rank of the colorednoise subspace (see the following discussion) [1]. This increased effective rank, in turn, will often result in a widening of the angleDoppler nulls— thereby reducing the amount of observation space available for target detection. To illustrate the impact of decorrelating random amplitude and phase channel mismatch on the interference signal, consider a simple uncorrelated amplitude and phase mismatch model: 1 − ␦ ⑀ i ≤ ⑀ i ≤ 1, where
p 冠␦ ⑀ 冡 = i
冦
1 , ⌬⑀ 0,
(4.18) for 0 ≤ ␦ ⑀ i ≤ ⌬⑀ , ∀i elsewhere
and ⌬ −⌬ ≤ i ≤ 2 2 where p ( i ) =
冦
1 , ⌬ 0,
(4.19) −⌬ ⌬ ≤ i ≤ , ∀i 2 2 elsewhere for
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SpaceTime Adaptive Processing for Radar
where p 冠␦ ⑀ i 冡 and p ( i ) are the pdfs (uniform) associated with the amplitude and phase errors respectively. Note that other authors have used different stochastic representations to model decorrelating channel mismatch (e.g., [6, 12, 14, 15])—all of which are but approximations to the actually underlying mismatch mechanisms. In practice, the correlation coefficients required by (4.16) would be measured experimentally as part of the calibration process. Once known, a CMT comprised of those coefficients completely characterizes the effects from an SINR standpoint. That said, we continue with the analytical illustration: With (4.18) and (4.19), we can now calculate the elements of the spatialonly CMT as follows: [T ]i , k = E再t i t k* 冎 = E再⑀ i e j i ⑀ k e −j k 冎 = E{⑀ i } E{⑀ k } E再e j i 冎 E再e −j k 冎
冉
⌬ = 1− ⑀ 2
(4.20)
冊 冉 冊 2
sinc2
⌬ 2
for i ≠ k with diagonal elements [T ]i , i = E再⑀ i2 e j i e −j i 冎 = E再⑀ i2 冎 = 1 − ⌬⑀ +
(4.21) 1 3 ⌬ 3 ⑀
Thus, T is of the following linear matrix form: T = 11 + 2I
(4.22)
where 1 is the rankone Hadamard identity matrix [13] (all elements equal unity), I is the usual identity matrix, and
1 = [T ] i , k
(4.23)
2 = [T ]i , i − [T ] i , k = [T ]i , i − 1
(4.24)
i≠k
i≠k
Other Important Factors Affecting STAP Performance
87
Applying (4.22) to R I yields RI 䊊 T = RI 䊊 (11 + 2I ) = 1RI 䊊 1 + 2RI 䊊 I
(4.25)
= 1 R I + 2 diag {R I } where the Hadamard product identities R I 䊊 1 = R I and R I 䊊 I = diag {R I } were invoked (diag {R I } is a diagonal matrix whose entries are the diagonal elements of R I ). From (4.25) we see that the net effect of uniform decorrelating channel mismatch is to effectively raise the whitenoise floor (increased diagonal entries). Thus, for this particular case, the spatial colorednoiseonly covariance rank is unaffected (although the effective rank (Chapter 3) may change since the INR has changed). However, this is not generally the case for the spacetime clutter rank as illustrated below. Nor is it generally the case when the mismatch is nonuniform, that is, when the offdiagonal terms of T are not identical—which, in other words, means the pairwise correlation between channels is nonuniform. Note also that in the absence of amplitude modulation, that is, ⌬⑀ = 0, 2 = 1 − 1 , and thus (4.22) is a convex sum of the Hadamard and ordinary identity matrices. The resulting covariance is of the form R = R I 䊊 T + 2I = R I 䊊 ( 1 1 + 2 I ) + 2I
(4.26)
= R I 䊊 ( 1 1 + ( 1 − 1 )I ) + 2 I
冤
1 1
…
1
…
= RI 䊊
1 1 ⯗ 1
1
冥
+ 2I
The expression in (4.22) is for the spatialonly CMT. The corresponding fulldimensional spacetime CMT is simply given by T = 1 M × M ⊗ Tspace
(4.27)
where 1 M × M is the M × M Hadamard identity matrix, and Tspace denotes the spatialonly CMT. Equation (4.27) is a special case of a more general expression
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SpaceTime Adaptive Processing for Radar
T = T time ⊗ Tspace
(4.28)
when there are also temporal mismatches and/or decorrelating effects that are uncorrelated with the spatial factors (e.g., ICM). A proof of (4.28) is relatively straightforward: Let t t ∈ ⺓ M, t s ∈ ⺓ N denote the uncorrelated temporal and spatial vector RVs, respectively, with corresponding correlation matrices cov (t t ) = Ttime ∈ ⺓ M × M, cov (t s ) = Tspace ∈ ⺓ N × N, then
cov (t t ⊗ t s ) ⌬ = T is given by
T⌬ = E((t t ⊗ t s ) (t t ⊗ t s )′ ) = E((t t t t′ ) ⊗ (t s t s′ ))
(4.29)
= E(t t t t′ ) ⊗ E(t s t s′ ) ⌬
= Ttime ⊗ Tspace where use was made of the Kronecker identity (A ⊗ B )(C ⊗ D )′ = (AC ′ ) ⊗ (BD ′ ) [13] (where conformality of the constituent matrix dimensions is assumed). We will make use of (4.28) later in this book when a multitude of spatial and temporal effects are present simultaneously. To gain insight into the impact decorrelating channel mismatch can have on the spacetime clutter covariance matrix, recall from Chapter 3 the Riemann sum approximation for the total spacetime clutter RV X c ∈ ⺓NM, Xc = =
Nc
∑ ␥˜ i vi
(4.30)
i =1 Nc
∑ ␥˜ i b i ⊗ a i
i =1
where {␥˜ i } represents mutually uncorrelated zeromean complexvalued scalar RVs with E 再  ␥˜ i  2 冎 = G i , and vi = b i ⊗ a i ∈ ⺓NM is the spacetime steering vector corresponding to the i th clutter patch (see Chapter 3) with corresponding temporal b i ∈ ⺓M and spatial a i ∈ ⺓N steering vectors. Uniform spatialonly decorrelating channel mismatch alters (4.30) as follows: Nc
Nc
i =1
i =1
∑ ␥˜ i b i ⊗ a i → ∑ ␥˜ i b i ⊗ (a i 䊊 t s )
(4.31)
Other Important Factors Affecting STAP Performance
89
where t s ∈ ⺓N is the vector RV associated with the decorrelating channel mismatch. The corresponding resultant covariance matrix, assuming {␥˜ i } and t s are uncorrelated, is thus given by
冢∑ Nc
cov
i =1
␥˜ i b i ⊗ (a i
冣
䊊 ts )
=
Nc
∑ cov ( ␥˜ i b i ⊗ (a i 䊊 t s ))
i =1
=
Nc
∑ cov ( ␥˜ i (b i ⊗ a i ) 䊊 (1 M ⊗ t s ))
i =1
=
Nc
∑ cov ( ␥˜ i ) cov ((b i ⊗ a i ) 䊊 (1 M ⊗ t s ))
i =1
=
Nc
∑ G i (vi vi′ ) 䊊 cov ((1 M ⊗ t s ))
(4.32)
i =1
=
Nc
∑ G i (vi vi′ ) 䊊 (1 M × M ⊗ Tspace )
i =1
=
冢∑ Nc
i =1
冣
G i (vi vi′ ) 䊊 (1 M × M ⊗ Tspace )
= R c 䊊 (1 M × M ⊗ Tspace ) which is of the factored form in (4.17) with a CMT of the form given by (4.27). Note that the above derivation depends on the angleindependence of the decorrelating channel mismatch. However, as shown below, even for angledependent mismatch, it is still possible to approximate the resultant covariance with the factored form. To illustrate the impact of decorrelating channelmismatch errors on the spacetime clutter eigenspectrum, consider the following three cases: (1) ⌬⑀ = 0, ⌬ = 0°, that is, no errors; (2) ⌬⑀ = 0.01, ⌬ = 2°; and (3) ⌬⑀ = 0.02, ⌬ = 5°. To make a fair comparison, we will hold the original CNR constant regardless of the amount of mismatch. This is tantamount to adjusting the original whitenoise floor in (4.17) to preserve CNR as defined in Chapter 3. Figure 4.3(a) shows the total interference eigenvalues for the three cases. Note that the effective rank of the clutter is increased due to the
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Figure 4.3 Impact of decorrelating channel mismatch on (a) the clutter eigenspectrum and (b) SINR loss for (1) ⌬⑀ = 0, ⌬ = 0°, that is, no errors; (2) ⌬⑀ = 0.01, ⌬ = 2°; and (3) ⌬⑀ = 0.02, ⌬ = 5°.
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presence of decorrelating channel mismatches, which will have a deleterious impact on SINR performance. Figure 4.3(b) shows the impact on SINR loss. Note the increased width in the mainbeam clutter notch—resulting in poorer lowDopplershift signal detection. Also evident is the effective increase in the noise floor as a consequence of the structure of (4.22) and (4.26). In general, both the simple rankone CMT of (4.9), denoted as T 1 , and the (generally) fullrank CMT of (4.22), denoted as T 2 , will be present. The net effect, assuming that the errors are uncorrelated, is a composite CMT of the form T = T1 䊊 T2
(4.33)
A proof of the Hadamard multiplicative property can be found in [1] and is based on the fact that the expected value of the product of two uncorrelated RVs is simply the product of the their expected values [16]. Specifically, consider the (i , k )th element of (4.33); that is, [T ]i , k = [E((t 1 䊊 t 2 ) (t 1 䊊 t 2 )′ )]i , k = E冠t 1 i t 2 i t 1*k t 2*k 冡
(4.34)
= E冠t 1 i t 1*k 冡 E冠t 2 i t 2*k 冡 = [T 1 䊊 T 2 ]i , k where t 1 , t 2 ∈ ⺓NM are the vector RV modulations with associated correlation matrices T 1 , T 2 ∈ ⺓NM × NM, respectively. Note that the above argument can be repeated to accommodate any number of uncorrelated random effects, resulting in the Hadamard factored form of (4.1). An interesting fact associated with the Hadamard product of two matrices is that if T 1 and T 2 are both positivedefinite, then T = T 1 䊊 T 2 is also guaranteed to be positivedefinite as a direct consequence of the Schur Product theorem [13]. While the reader can consult [13] for a general proof (and extensions), it is obvious from (4.34) that this should be the case since T 1 䊊 T 2 can be associated with a finite covariance stochastic process t 1 䊊 t 2 [1]. Before we leave this section, mention should be made of how the colloquial term cancellation ratio, or CR [6, 12], usually specified in decibels, relates to the preceding discussion. CR is defined as the expected value of the normalized energy residue (in decibels) resulting from subtracting signals
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SpaceTime Adaptive Processing for Radar
from two different channels [6]. Thus, if the correlation coefficient between two adjacent channels is , then the cancellation ratio (in decibels) is defined as ⌬ 10 log 10 (1 − )−1 CR (dB) =
(4.35)
For example, if = 0 (no correlation between channels), then the CR is 0 dB (no cancellation). If = 0.9, the CR is 10 dB. The CR is simply a measure of the pairwise correlation between channels. The CMT framework of this section shows explicitly how to relate the correlation between channels (and thus the CR between channels) to its impact on interference cancellation. Namely, if the pairwise correlation between channels is uniform and equal to , (which could be experimentally determined), then the corresponding spatialonly CMT is simply given by
冤
1
…
…
T=
1 ⯗
1
冥
(4.36)
4.2.2 AngleDependent Channel Mismatch Due to the potential presence of a multitude of array manifold effects (mutual coupling, nearfield multipath, element position errors, bandwidth dispersion [3], and so forth), T of (4.33) may be a function of AoA; that is, T = T ( )
(4.37)
which can significantly complicate the analysis of the impact on the interference covariance matrix with no mismatch errors. Specifically, when present, the total (clutter plus jamming plus noise) covariance matrix R is of the form R = cov (X c ) + cov 冠 X J 1 冡 + . . . + cov 冠 X J N
J
冡 + 2I
(4.38)
where cov 冠 X J i 冡 = R J i 䊊 T 冠 J i 冡 ∀i : i = 1, . . . , N J
(4.39)
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93
where R J 1 is the covariance associated with the i th jammer in the absence of channel mismatch, and T 冠 J 1 冡 is the total CMT for jammer AoA J 1 . The expression for cov (X c ) is more complicated due to the fact that clutter originates from all angles (in general). Specifically, from Chapter 3 we have cov (X c ) = cov
=
冢∑ Nc
i =1
␥˜ i vi
t( i )
䊊
冣
(4.40)
Nc
∑ G i (vi v i′ ) 䊊 T ( i )
i =1
where we have assumed that the clutter and random RV mismatch error are uncorrelated. Notice that since T ( i ) is inside the summation, cov (X C ) is not of the previously encountered factored form R 䊊 T. However, in a normally operating radar, the transmit and receive antenna patterns are pointing in the same direction. This implies that G i will have a significant maximum in the look direction. Since the effect of a CMT is the introduction of sideband energy that is generally several (if not many) orders of magnitude down from the primary (mainbeam) unmodulated eigenvectors (see [1] and the discussion on the PCCMT technique in Chapter 5), only the dominant eigenvectors will generally play a role in introducing subspace leakage. The practical consequence of these observations is that the unfactored result of (4.40) can reasonably be approximated by the factored result cov (X c ) ≈ R c 䊊 T ( 0 )
(4.41)
where R c is the covariance in the absence of angledependent channel mismatch, and T ( 0 ) is the CMT associated with the radar transmitreceive look direction 0 . Combining (4.41) with (4.38) and (4.39) yields a simplified, yet useful, result for the impact of angledependent channel mismatch when both clutter and jamming are present; that is, R = cov (X c ) + cov 冠 X J 1 冡 + . . . + cov 冠 X J N
J
冡 + 2I
≈ R c 䊊 T ( o ) + R J 1 䊊 T 冠 J 1 冡 + . . . + R J N 䊊 T 冠 J N J
(4.42) J
冡
+ 2I
One potential source of angledependent channeltochannel spatial decorrelation is antenna dispersion due to finite bandwidth [7, 17]. Referring
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SpaceTime Adaptive Processing for Radar
back to Figure 2.1, we see that the absolute time delay between the k th and n th channels in a ULA is
n −k  =  n − k 
d sin c
(4.43)
where c is the speed of light, d is the element separation (consistent units assumed), and is the angle off boresight (as defined in Chapter 2). For the finite (nonzero) bandwidth case, the receiver (pulse) will have an autocorrelation function associated with it [18]. For an ideal bandpass, with bandwidth B (hertz), the corresponding autocorrelation function, r( ), is given by [16]: r ( ) = sinc (B )
(4.44)
where to be consistent, is expressed in seconds. Other correlation functions are of course possible, such as that corresponding to a linear frequency modulation (LFM) with a temporal taper for range sidelobe reduction. The corresponding angledependent CMT is thus given by
冤
r ( 1 ) 1 r ( 1 )
r ( 2 ) r ( 1 ) 1
…
r ( N − 1 )
…
T ( ) =
1 r ( 1 ) r ( 2 ) ⯗ r ( N − 1 )
r ( 1 ) 1
r ( 1 )
冥
(4.45)
where
m = m
d sin c
(4.46)
Although for a sidelooking radar the peak of the mainbeam is nominally pointing at = 0°, for which m = 0, it is not advisable to ignore the above dispersion since mainbeam clutter is still spread in angle. A reasonable approximate angleindependent CMT for this case [i.e., (4.41)] is T (⌬ ), where ⌬ is some suitable measure of mainbeam width. Figure 4.4 illustrates an example of the impact of antenna dispersion on the clutter rank.
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Figure 4.4 Impact of antenna array dispersion on the clutter eigenspectrum for the finite bandwidth case (10% bandwidth example).
4.3 Other Interference Subspace Leakage Effects ISL [1] refers generally to situations in which an increase in the rank of the colored noise (clutter and/or jamming) is present. As we saw in the previous section, decorrelating channel mismatch can cause an increase in rank and is thus a special case of ISL. There are, in general, many other ISL mechanisms. For example, random amplitude and/or phase modulation of the clutter signal will generally cause an increase in its rank. One common example is ICM [4, 9, 19] (e.g., windblown foliage— an extensive research and measurement study was recently conducted by Billingsley of MIT Lincoln Laboratory [19]). A product of this investigation is a very useful and accurate empirical exponential model that captures the salient characteristics of ICM and is colloquially referred to as the Billingsley model. The only parameters required to specify the clutter Doppler power spectrum are essentially the operating wavelength and wind speed. Specifically, r 1 b − e ␦( f ) + Pc ( f ) = r+1 r+1 4
b f  2
(4.47)
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SpaceTime Adaptive Processing for Radar
where Pc ( f ) is the clutter Doppler psd, is the operating wavelength, f is the Doppler frequency (hertz), and ␦ (⭈) is the Dirac delta function. The shape parameter b depends primarily on wind conditions and has been tabulated [19]. The ratio r , 0 ≤ r ≤ ∞, between the dc [first term in (4.47)] and ac components (second term) was found to be a function of both the carrier frequency and wind speed [19], namely, 10 log r = −15.5 log w − 12.1 log f c + 63.2
(4.48)
where w is the wind speed in miles per hour (mph), f c is the carrier frequency in megahertz, and log (⭈) is the base 10 logarithm. The corresponding correlation (temporal) function is given by the inverse Fourier transform of (4.48) [4], that is, ∞
r c ( ) =
冕
Pc ( f ) e j 2 f
(4.49)
−∞
=
1 (b )2 r + r + 1 r + 1 (b )2 + (4 )2
The CMT corresponding to this ICM model is easily obtained by sampling r c ( ) at multiples of the PRI; that is, r c (0) (first pulse), r c (PRI) (correlation between first and second pulses), . . . , r c ((M − 1)PRI) (correlation between first and last (M th) pulses [4]. The fullup spacetime CMT is thus given by T = T ICM ⊗ 1 N × N
(4.50)
[T ICM ]i , j = r c 冠  i − j  PRI冡
(4.51)
where
Note that ICM is a temporal modulation effect, as evidenced by (4.50), which is a special case of (4.28). Figure 4.5 shows an example of the impact of ICM on both the eigenspectrum and SINR for a modest average wind speed of 10 mph. Note that similar to the decorrelating channelmismatch case, the most significant impact is an increase in the mainbeam clutter notch—with a commensurate decrease in MDV.
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Figure 4.5 Impact of ICM on (a) eigenspectrum and (b) SINR loss. Note that even a modest average wind speed can have an impact on performance in high CNR environments.
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Another potential source of ISL is socalled range walk, which can occur due to rangebin migration of the clutter during a CPI (see [20] and references cited therein). Figure 4.6 illustrates this effect for the general case when the look direction is not broadside. Since, to a reasonable approximation, the amount of decorrelation is proportional to the amount of area overlap ⌬A (see Figure 4.6), we see that this effect is in general angledependent—with a minimum at broadside and a maximum in the forwardlooking direction. However, for the same reason cited above, a useful approximation can be obtained by considering the amount of decorrelation in the look direction thereby resulting in an approximate, but angleindependent, CMT. Assuming a relatively short CPI with a constant velocity, we see from Figure 4.6 that the correlation between successive pulses is approximately given by
Figure 4.6 Illustration of the cause of clutter decorrelation due to rangebin migration.
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≈ ≈ =
99
⌬A A ⌬A ⌬R ⭈ ⌬
(4.52)
⌬A
冉冊 c B
⭈ ⌬
where ⌬ is the mainbeam width, and ⌬R is the rangebin size—which is taken to be c /B, where c is the speed of light and B is the bandwidth of the compressed pulse (consistent units assumed). For a short CPI, it is reasonable to assume that the change in area overlap is approximately constant between successive pulses. Thus, if the decorrelation between successive pulses is , the temporal decorrelation between the m th and n th pulses, denoted by m , n , is then simply given by
m,n = m −n 
(4.53)
The corresponding temporal CMT is therefore of the form
冤
1
…
M −1
…
T=
1 ⯗
M −1
1
冥
(4.54)
Note that this has a similar (not identical) structure to the ICM case inasmuch as there is a monotonic decrease in correlation with increasing PRI separation. Thus, the resulting clutter eigenspectrum will have a similar appearance to that of the ICM case.
4.4 Antenna Array Misalignment Brennan’s rule, as derived in Chapter 3, assumed that the ULA antenna was aligned with the ground velocity vector. In fact, this was crucial for showing that the clutter rank was generally significantly less than the total spatiotemporal DoFs [i.e., the redundancy argument demonstrated in (3.18)]. Moreover,
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when aligned, the back lobe clutter exactly coincides with the front lobe clutter as a function of angleDoppler as evidenced by an examination of Figure 3.1 and (3.1) [8]. In practice, it is not possible to maintain antenna alignment to ground track (e.g., aircraft crabbing and implementation constraints). Consequently, the clutter rank (and angleDoppler spectrum) is altered. The impact of velocity misalignment on the angleDoppler relationship of ground clutter (at long ranges) is the introduction of a socalled crab angle c in (3.1) as follows [8]: fd =
2vT sin ( + c )
(4.55)
Note that the symmetry between back lobe and front lobe has generally been lost. Specifically, with c = 0 (i.e., no crab) there is even Doppler +␣ = frequency symmetry about the nose of the aircraft; that is, sin 2 − ␣ , for  ␣  ≤ , where ␣ is the angle of a given clutter patch sin 2 with respect to the nose of the aircraft. Front and back lobe clutter will thus +␣ ≠ lie along the same angleDoppler contour. With c ≠ 0, sin 2 − ␣ , for  ␣  ≤ . This will result in a socalled back lobe ridge, sin 2 which may be of concern in strong clutter with poorly attenuated back lobes [8]. Another observation gleaned from (4.55) is that there is no longer a strictly linear relationship between normalized angle and Doppler [i.e., (3.2)] no longer holds. Instead, the clutter locus forms an ellipse [8]. To see this explicitly, we manipulate (4.55) as follows:
冉 冉
冊 冊
fd = =
冊
冉
冊
2vT sin ( + c ) 2vT [sin ( ) cos ( c ) + cos ( ) sin ( c )]
=  [sin ( ) cos ( c ) + cos ( ) sin ( c )] where
冉
(4.56)
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=
2vT d ,= d
101
(4.57)
Rearranging (4.56) and employing a trigonometric identity, we have f d −  d cos c =  cos sin c
(4.58)
= ±  √1 − sin2 sin c where the expression for normalized angle d = sin is inserted and the trigonometric identity cos = ± √1 − sin2 is employed. Squaring both sides of (4.58) and rearranging terms yields f d2 +  2 d2 − 2 cos c f d d −  2 2 sin2 c = 0
(4.59)
Equation (4.59) yields a rotated ellipse in the normalized angleDoppler coordinates. The standard form for a rotated (and generally translated) ellipse is given by [21] Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
(4.60)
Comparing this with (4.59) yields the following assignments x = d y = fd A = 2 B = −2 cos c
(4.61)
C=1 D= E=0 F = − 2 2 sin2 c from which it is evident that the ellipse is centered at the origin, but is rotated from the d axis by an angle given by [21]
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cot 2 = =
A−C B 1 − 2 2 cos c
(4.62)
B≠0 Note that for  2 = 1, cot 2 = 0, which implies that =
, that is, an 4
ellipse oriented at 45°. Figure 4.7 displays angleDoppler contours for different values of crab angle, along with corresponding clutterplusnoise eigenspectrum and Capon
Figure 4.7 Effect of crabbing and finite back lobe rejection ratio on (a) angleDoppler clutter loci, (b) power spectra, and (c) eigenspectra, respectively. Back lobe rejection ratio is fixed to 30 dB.
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powerspectral densities. Note the appearance of a second clutter ridge whose strength is determined by the back lobe rejection level of the array.
4.5 Nonlinear Arrays1 Brennan’s rule, as derived in Chapter 3, assumed that the sidelooking ULA antenna was aligned with the groundreferenced velocity vector. Indeed, it was because of this assumption (and several others) that a generally lower rank for the clutter eigenspectrum was realized. When this assumption is violated—as is the case in Section 4.4 due to crabbing—the rank of the clutter is generally increased. Unfortunately, this rank inflation also arises when nonlinear array geometries are employed (even if the geometrical axis of the array is aligned with the groundreferenced velocity vector) [8]. As with the crabbing case, the violation of Brennan’s rule is easily established since the requisite linear phase relationship and redundancy in (3.16) no longer hold. There are obviously too many possible nonlinear array geometries to contemplate them all in this venue. However, an increasingly popular configuration is the socalled circular array [22, 23]. A static (nonrotating), electronically scanned, circular array can eliminate the need for bulky, heavy, and energyconsuming mechanically scanned servos, as well as provide enhanced trackwhilescan (TWS) capabilities [22]. Figure 4.8 illustrates an example UHF linear and circular array configuration. To insure the formation of proper transmit and receive patterns, only a subsection of the total circular array is active [23]. Note that to make a fair comparison with the linear case, a few extra elements are included in the circular subsection to maintain an approximately equivalent aperture size. Figure 4.9 shows the impact of circulararray geometry on the clutterplusnoise eigenspectrum or the case of 30dB and 100dB back lobe rejection. Note that as with the crabbing array, low back lobe rejection can have a major impact on clutter spread.
4.6 Interference Nonstationarity and the Iceberg Effect Due to both manmade and natural variations in terrain, realworld clutter is heterogeneous [24]. Thus, the statistics (e.g., pdf, CNR, and correlation 1. Simulation material in this section was provided by Mr. Jamie Bergin, Information Systems Laboratories, Inc.
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Figure 4.8 UHF linear and circular arrays illustrating the impact of nonlinear array geometry. (Data provided courtesy of Information Systems Laboratories, Vienna, Virginia.)
Figure 4.9 Impact of nonlineararray geometry on clutter eigenspectrum for back lobe rejection ratios of (a) 30 dB and (b) 100 dB. First 250 eigenvalues displayed. (Data provided courtesy of Information Systems Laboratories, Vienna, Virginia.)
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properties) will vary with location. This fact obviously presents a severe challenge to any adaptive cluttercancellation scheme that must estimate requisite statistical information on the fly. To begin to appreciate the impact nonstationarity can have on STAP performance, we first consider perhaps the simplest form of heterogeneity, namely CNR variation. Figure 4.10 shows the impact of a change in CNR for the ideal ULA case (for which Brennan’s rule holds) and for the more realistic case in which subspace leakage is present (10mph ICM). Note that for the low CNR case, the effective ranks of the clutter eigenspectra are approximately equal. However, for high CNR, there is a significant increase in the rank when subspace leakage is present. If one thinks of the whitenoise floor (0 dB on graph) as the surface of a perfectly calm ‘‘ocean,’’ and the clutteronly eigenspectrum as an ‘‘iceberg,’’ one can see that the increase in effective clutter rank with increasing CNR is analogous to an iceberg rising above the ocean surface. Although ICM was used to provide the additional subspace leakage, any of the decorrelating effects described in this chapter will produce the same qualitative result.
Figure 4.10 Illustration of the iceberg effect, which describes the increase in interference rank (eigenvalues above noise floor) with increasing CNR when realistic eigenspectra are considered (i.e., Brennan’s rule no longer holds). Note that for low CNR the effect is negligible, while for large CNR it can be quite significant.
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This iceberg effect can have a significant impact on sample covariance(adaptive) based beamforming. As a simple illustration, consider an ideal sidelooking ULA (no crab) where the CNR decreases from 70 to 20 dB over a range support of 6,000 samples (bins), where N = M = 16, and there is 10mph ICM. This power variation can simply be the result of decreasing CNR with increasing range from the radar [24], or it can be due to gradual variations in terrain elevation or coverage [24]. Figure 4.11 shows SINR versus Doppler and range for both the ideal optimum (knowncovariance) and estimated covariance (2NM SMI centered at range bin 3,000) cases. The additional loss in the nearin ranges is due to clutter undernulling (i.e., the notch width and depth are inadequate), while the loss in the far range bins is due to clutter overnulling (i.e., notch width and depth are too large—resulting in unnecessary target signal loss). In both cases, the biggest impact is on the minimum detectable velocity (MDV), which is defined as the velocity at which SINR loss is 10 dB (see [11] for further discussion of MDV and alternate definitions). For GMTI radar, this severe loss in MDV (mainbeam clutter notch width) may be undesirable [25].
4.7 Summary In this chapter, we touched upon a few of the many realworld effects that can have a significant impact on STAP performance, as compared with socalled ideal performance. Although one cannot reduce the effective rank of the coloredinterference subspace when ISL and other rankinflation mechanisms are present, judicious algorithm design can minimize its impact. This is a primary consideration in the next chapter.
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Figure 4.11 Impact of clutter nonstationarity on SINR: (a) ideal SINR performance achieved when exact covariance is used for each range bin, and (b) actual SINR performance when a 2NM sample (centered at range bin 3,000) covariance is used. The additional SINR loss at nearin range bins is due to clutter undernulling, while the additional loss at far ranges is due to overnulling. This example illustrates the importance of tracking clutter statistics for optimal performance.
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Guerci, J. R., and J. S. Bergin, ‘‘Principal Components, Covariance Matrix Tapers, and the Subspace Leakage Problem,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 38, No. 1, January 2002.
[2]
Guerci, J. R., ‘‘Theory and Application of Covariance Matrix Tapers for Robust Adaptive Beamforming,’’ IEEE Trans. on Signal Processing, Vol. 47, No. 4, April 1999, pp. 977–986.
[3]
Zatman, M., ‘‘Production of Adaptive Array Troughs by Dispersion Synthesis,’’ Electronics Letters, Vol. 31, No. 25, 1995, pp. 2141–2142.
[4]
Techau, P. M., J. S. Bergin, and J. R. Guerci, ‘‘Effects of Internal Clutter Motion on STAP in a Heterogeneous Environment,’’ Proc. of 2001 IEEE Radar Conference, May 2001.
[5]
Zatman, M., ‘‘How Narrow Is Narrowband?’’ IEE Proc. of Radar, Sonar and Navigation, Vol. 145, April 1998, pp. 85–91. Monzingo, R. A., and T. W. Miller, Introduction to Adaptive Arrays, New York: John Wiley & Sons, 1980, pp. 56–64. Mailloux, R. J., Phased Array Antenna Handbook, Norwood, MA: Artech House, 1994. Ward, J., SpaceTime Adaptive Processing for Airborne Radar, MIT Technical Report 1015, MIT Lincoln Laboratory, December 1994. Billingsley, J. B., Radar Clutter, Norwood, MA: Artech House, 2001. Melvin, W. L., ‘‘SpaceTime Adaptive Radar Performance in Heterogeneous Clutter,’’ IEEE Transactions on Aerospace and Electronic Systems, Vol. 36, No. 2, pp. 621–633. Melvin, W. L., M. J. Callahan, and M. C. Wicks, ‘‘Bistatic STAP: Application to Airborne Radar,’’ Proc. of 2002 IEEE Radar Conference, Long Beach, CA, April 22–25, 2002, pp. 1–7.
[6] [7] [8] [9] [10] [11]
[12] [13] [14] [15] [16] [17] [18] [19]
Compton, R. T., Jr., Adaptive Antennas: Concepts and Performance, Englewood Cliffs, NJ: Prentice Hall, 1988. Horn, R. A., and C. R. Johnson, Topics in Matrix Analysis, Cambridge, England: Cambridge University Press, 1991. Hudson, J. E., Adaptive Array Principles, London, England: Peter Peregrinus Ltd. on behalf of the IEE: (reprint) 1991. Farina, A., AntennaBased Signal Processing Techniques for Radar Systems, Norwood, MA: Artech House 1992. Papoulis, A., and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, 3rd ed., New York: McGrawHill, 2001. Zatman, M., ‘‘Production of Adaptive Array Troughs Through Dispersion Synthesis,’’ Electronics Letters, Vol. 31, No. 25, December 1995, pp. 2141. Techau, P. M., ‘‘Effects of Receiver Filtering on Hot Clutter Mitigation,’’ Proc. of the 1999 IEEE Radar Conference, Waltham, MA, April 20–22, 1999, pp. 84–89. Billingsley, J. B., Exponential Decay in Windblown Radar Ground Clutter Doppler Spectra: Multifrequency Measurements and Model, Technical Report 997, MIT Lincoln Laboratory, Lexington, MA, July 29, 1996.
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[20]
Klemm, R., Principles of SpaceTime Adaptive Processing, London, England: IEEE Press, 2002.
[21]
Swokoski, E. W., Calculus with Analytic Geometry, Boston, MA: Prindle, Weber and Schmidt, 1975.
[22]
Zatman, M., ‘‘Circular Array STAP,’’ Proc. of IEEE National Radar Conference, Waltham, MA, 1999.
[23]
Guerci, J. R., et al., ‘‘Optimal ReducedRank STAP for Circular Adaptive Arrays,’’ Proc. of SpaceTime Processing Methods for Circular Ring Arrays with Application to Navy Airborne Surveillance Radar, George Mason University, Fairfax, VA, February 22–23, 1999.
[24]
Long, M. W., Radar Reflectivity of Land and Sea, 3rd ed., Norwood, MA: Artech House, 2001.
[25]
Guerci, J. R., ‘‘KnowledgeAided Sensor Signal Processing and Expert Reasoning,’’ Proc. of the 2002 Workshop on KnowledgeAided Sensor Signal Processing and Expert Reasoning (KASSPER), Washington, D.C., April 3, 2002 (CDROM).
5 STAP for Radar: Methods, Algorithms, and Performance 5.1 Introduction Although in theory we have already specified the theoretically optimum strategy for spacetime filtering of interference—that is, the ubiquitous WienerHopf equation w = R −1 s—many practical considerations (described in previous chapters and elsewhere [1–4]) preclude its direct implementation. Most notable is the fact that the interference covariance matrix R is unknown a priori and must be estimated on the fly, then utilized in an appropriate realtime spacetime filtering architecture. Even in socalled stationary Gaussian environments (which, in fact, never truly exist) when a sample covariance estimate is indicated (see Section 2.4), care must be taken to avoid erratic sidelobe levels introduced due to estimation errors and potentially limited sample support. Even greater care is required when confronted with nonstationary environments—particularly when ISL and other rankinflating phenomenon (crabbing, nonlinear array geometries, and so forth) are present. In this chapter, we introduce a broad range of STAP algorithms that have been developed (many in recent years) to combat one or more of the aforementioned performance degrading effects. Although it is not practical to consider every current STAP algorithm that has been proposed (a keyword search on ‘‘STAP’’ should suffice to make the point), a representative sampling of algorithms drawn from a STAP taxonomy are analyzed in this chapter. It should be mentioned that our emphasis here is on performance and not 111
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specifically on implementation complexity. Certain key aspects of realtime STAP will be explored in Chapter 6. Figure 5.1 illustrates a STAP algorithm taxonomy. All of these algorithms either explicitly or implicitly attempt to reduce the dimensionality of the unknown interference parameters via a rankreducing transformation or an explicit modelbased method (or potentially both), while still preserving acceptable performance [1]. Thus, although the STAP CPI dimension NM might be quite large, the dimension of the unknown parameters vector or matrix could be substantially less—since clutter is generally restricted in angleDoppler space [1, 2, 5]. This reduceddimension adaptivity is only
Figure 5.1 Taxonomy of reduceddimension adaptivity STAP algorithms. It is important to note that many of the methods can be combined, thus greatly increasing the palette of techniques from which to choose.
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possible if additional prior knowledge of the interference is assumed. Indeed, it is critical to know what prior knowledge is explicitly or implicitly assumed in order to select an algorithm appropriate to a given radar scenario. For example, in postDoppler methods [for an example of a reducedrank linear transformation (RLT) dataindependent method, see Figure 5.1], an assumption on the Doppler extent of ground clutter is exploited to subband the rejection problem [1, 2, 5]. Thus, a different, but substantially reducedrank (or reduceddimension) adaptive problem is solved for each Doppler bin. Since a much smaller covariance matrix is utilized in these methods, it can provide significant advantages in nonstationary environments and also be substantially easier to implement. Under certain conditions, which may or may not be sufficiently satisfied in any given application, postDoppler methods can closely approximate the performance of optimum fullDoF STAP [1, 2]. An example of an explicit modelbased algorithm that can reduce the effective adaptive DoFs is Toeplitz matrix fitting [6, 7]. For an ideal sidelooking ULA (no crab or channel mismatch) and uniform PRF in the presence of uniform uncorrelated clutter scatterers (such as considered in Chapter 3), the ideal STAP covariance matrix has a ToeplitzblockToeplitz (TBT) structure [1, 2, 6, 7]. The redundancy in the covariance structure effectively reduces the dimension of the unknown parameter vector (unknown elements of covariance matrix) which, in turn, reduces requirements on sample support for estimation—a highly desirable property in nonstationary environments or where sample support is limited by other factors (e.g., radar horizon). However, the Toeplitz assumption is an approximation at best and thus must be applied judiciously. The remainder of Chapter 5 will be spent exploring the STAP taxonomy tree of Figure 5.1 and comparing the different methods in a variety of simulated, but nonetheless realistic scenarios. Though this treatise is not exhaustive, the reader should gain a foundation from which to craft a custom STAP solution well suited to his or her application. A unique pedagogical feature of this chapter is the explicit mathematical I , for each derivation of an effective covariance inverse (ECI), denoted by R eff STAP algorithm. That is, each reduceddimension STAP algorithm can be shown to result in a mathematical form I s w = R eff
(5.1)
I is always an NM × NM matrix. Adopting this approach not only where R eff significantly aids in the mathematical analysis of various methods, but also
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provides insight into precisely how a particular algorithm works—an invaluable aid when attempting to choose among the many different methods. It is important to note that the ECI may be singular and thus may not have I = R. an inverse. Obviously, for the fullDoF knowncovariance case, R eff
5.2 DataIndependent ReducedRank STAP The first branch point in the STAP taxonomy of Figure 5.1 differentiates between RLT and structuredcovariance (modelbased) methods. An RLT is either dataindependent (nonadaptive) or datadependent (adaptive) and can be thought of as a nonsquare matrix transformation (with more columns than rows) applied to the concatenated spacetime snapshot data vectors (column vectors). The most extreme example of a dataindependent RLT is the socalled factored approach that consists of nonadaptive Doppler filtering and beamforming. Since there is no joint spacetime (or joint angleDoppler) processing, this technique is also referred to as 1D factored spacetime processing [1, 2]. This method is extremely suboptimal for airborne radar in the presence of ground clutter since it makes no provision for the Doppler dependency on angle as described in Chapter 3. Succinctly stated, this method is mathematically tantamount to simply w = s, which clearly shows the complete lack of interference structure (i.e., interference covariance). Figure 5.2 shows the extreme suboptimality of this approach and serves as a reminder of why STAP for clutter cancellation was invented in the first place. 5.2.1 PreDoppler (SignalIndependent) ReducedRank STAP: DPCA and Adaptive DPCA The first practical step, both historically and pedagogically, towards joint spacetime processing is the DPCA method [1]. Taxonomically, it is a preDoppler dataindependent RLT. The DPCA method attempts to operate a ULA in such a manner that (1) the elements (or subarrays) of the ULA are extremely well matched (just how well will be discussed momentarily); (2) the axis of the ULA is aligned with the groundtrack velocity vector (not necessarily the aircraft orientation due to crabbing); (3) the PRI is precisely chosen to insure that the array advances one interelement spacing per pulse; and (4) the trailing and leading elements are blanked on receive, as shown in Figure 5.3, which makes the indicated subarray appear stationary relative to the ground. If all four of these steps are performed to requisite tolerances,
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Figure 5.2 Comparison of fullDoF optimal spacetime processing with a nonadaptive 1D factored approach.
Figure 5.3 Illustration of the DPCA concept for an ideal ULA perfectly aligned with the groundtrack velocity vector. The PRI is chosen so that the array advances exactly one interelement spacing per pulse (i.e.,  = 1). By using only the subarrays indicated above, the array appears to be stationary relative to the ground, vastly simplifying the cluttercancellation process.
the ground clutter will appear stationary (without ICM, and so forth), thus vastly simplifying the cluttercancellation process—at the cost of aperture efficiency. More precisely, consider the N dimensional DPCA ULA ‘‘snapshot’’ returns from two successive pulses, x 1 and x 2 , with a corresponding concate
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nated twopulse spacetime data vector x′ = [x 1′ x 2′ ]. Since the clutter will appear to be stationary (i.e., no Doppler shift), simply subtracting the returns from the two pulses should suffice in cancelling ground clutter [1, 2]. If we assume that we are interested in returns emanating from boresight (0°, that is, sidelooking), then mathematically the entire twopulse MTI DPCA canceller is equivalent to w=
冋 册 冋 册 w1
w2
=
s1
(5.2)
−s 2
where s 1 , s 2 are the N dimensional spatialonly steering vectors for the first and second pulses, respectively. Thus, to achieve the DPCA effect we must have 0 1 1 ⯗ s 1 = ⯗ (1st element off ), s 2 = 1 1 0
冤冥
冤冥
(N th element off )
(5.3)
Equation (5.2) can also be cast into the ECI framework as follows I s w = R eff
= (I − e 1 e 1′ − e 2N e 2′N )s = (I − e 1 e 1′ − e 2N e 2′N )
冢冤
冢冋 册 冤 冥冣 1
−1
冥冤
⊗
1 ⯗ 1
…
1 0 … 0 0 … 0 ⯗ 0 0 ⯗ I− ⯗ − ⯗ 0 0 0 … 0 0 … 0 1 …
=
(5.4)
冥冣冢冋
1 −1
册 冤 冥冣 ⊗
1 ⯗ 1
where I is the 2N × 2N identity matrix, and e 1 , e 2N are the first and 2N th Euclidean basis vectors (e i is a vector of zeros except for a unity entry in the i th position). Equation (5.4) is worthy of further examination. It is left as an exercise for the reader to prove equivalence between (5.4) and (5.2) (i.e., that the weight vectors are identical, which can be accomplished by comparing both
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vectors on an element basis). The implied spacetime steering vector for the twopulse DPCA system is
s=
冋 册 冤冥 1
−1
⊗
1 ⯗ 1
(5.5)
This corresponds to a sidelooking (boresightaligned) spatial steering vector [second vector in (5.5)] and a Doppler steering vector of [1 −1]′—which in turn corresponds to a simple twopulse MTI canceller [8]. Since there are only two pulses being processed for clutter cancellation and since clutter in a perfect DPCA system occupies the Doppler space corresponding to dc (0 Hz), the only Doppler space left for signal detection is that which is orthogonal to [1 1]′, namely [1 −1]′. Further inspection of (5.4) reveals the effective matrix inverse for twopulse DPCA I = (I − e 1 e 1′ − e 2N e 2′N ) R eff
(5.6)
which is recognized to be a projection operator [9] that eliminates the 2D subspace spanned by the Euclidean basis vectors e 1 , e 2N . This is the operator responsible for alternately shutting off the leading and trailing elements on the first and second pulse, respectively. Comparing (5.6) with the eigenbased form for the inverse of the covariance [see (2.59)] implies that for an ideal twopulse DPCA system, the clutter rank after the DPCA operation is only two—regardless of the number of antenna elements. If the antenna were not operated in this DPCA fashion, Brennan’s rule would set the clutter rank at N +  (M − 1) = N + 1, since  = 1 and M = 2. This apparent violation of Brennan’s rule can be explained by the fact that it is only applicable to a linear timeinvariant (LTI) spacetime ULA. A DPCA antenna is timevarying (since the array manifold is different on each pulse) and is thus not covered by Brennan’s rule. Before we analyze the performance of DPCA, we need to extend it to the processing of an entire STAP CPI, that is, more than just two pulses. This is easily accomplished by simply Doppler filtering the scalar (but complex) outputs from the twopulse DPCA MTI filter as shown in Figure 5.4. The scalar outputs from the MTI canceller, y i , that are the inputs to the Doppler filter bank are related to the original NM dimensional STAP data by a rankreducing (nonsquare) linear transformation ⍀ of the form
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Figure 5.4 A twopulse DPCA MTI canceller extensible to an arbitrarily long CPI. The scalar residues from the twopulse DPCA MTI canceller (first stage) are sequentially fed into a Doppler filter bank for further temporal integration gain.
冤 冥冤 yM −1
=
w 1′ 0′ ⯗ 0′
w 2′ w 1′
0′ w 2′
… …
…
y=
y1 y2 ⯗
0′
…
w 1′
0′ 0′ ⯗ w 2′
x1 x2 ⯗ xM
冥冤 冥
= ⍀x
(5.7)
where 0′ is an N dimensional row vector, and the DPCA weight vectors w 1 and w 2 are given by (5.2). Thus, the original spacetime data has been transformed into a clutterwhitened space spanned by y. For twopulse DPCA, the dimension of y is M − 1, as indicated by the index subscript in (5.7). This corresponds roughly to a factor of N rank reduction. The final stage of filtering is thus accomplished by matched filtering in this transformed space to a steering vector of interest. If we are interested in detecting a signal with original NM dimensional spacetime steering vector s, the corresponding steering vector in the transformed space is s y = ⍀s. We are now in a position to state explicitly what the equivalent NM dimensional spacetime weight vector is (along with its ECI) for a twopulse DPCA followed by Doppler filtering. This is accomplished by working backwards from the final scalar output of the DPCA filtered CPI; that is, z = s y′ y = s′ ⍀′⍀x
(5.8)
= w′DPCA x from which we readily ascertain w DPCA = ⍀′⍀s I s = R eff
(5.9)
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and thus I = ⍀′⍀ R eff
(5.10)
Figure 5.5 displays the ideal performance for both fullDoF STAP and DPCA for an N = M = 16 sidelooking ULA example. Also shown is the performance of DPCA when errors are present, namely, simple channelmismatch errors (5° rms phase) and/or PRI mismatch (which is equivalent to  ≠ 1). The loss can be as much as 20 dB or more, especially for lower Doppler shifts. As we demonstrated in Chapters 2 and 4, adaptive processing can be robust to certain channelmismatch errors. Thus, it is natural to introduce adaptivity into the DPCA process and hopefully restore performance in the presence of channel mismatch and allow for operation when  ≠ 1. Adaptive DPCA (ADPCA) replaces the nonadaptive DPCA MTI canceller weight vector in (5.2) with an adaptive one, namely,
Figure 5.5 Performance of twopulse DPCA for both ideal and realistic (with errors) conditions. The extreme sensitivity of DPCA to simple channel and  mismatches is one of the prime reasons for considering STAP.
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w=
冋 册 w1
w2
= R −1 s = R −1
(5.11)
冋册 s1
s2
where R is the 2N × 2N interference covariance matrix associated with two successive array snapshots (i.e., two pulses). This twopulse ADPCA canceller is then followed by Doppler filtering in precisely the same manner as DPCA (see Figure 5.4). Indeed, ADPCA has a similar mathematical structure to DPCA, namely, I s w ADPCA = ⍀′⍀s = R eff
(5.12)
where s is the fullDoF steering vector, and, in contrast to DPCA, ⍀ is comprised of the adaptive weights from (5.11). More insight into the rankreducing mechanism employed by ADPCA can be gained by examining the final scalar ADPCA output for the entire CPI as was done in (5.8). Specifically, z = w′ADPCA x −1 x = s′R eff
冤
0 2N × 2N
…
0 2N × 2N
0 2N × 2N
−1 2 R 2N × 2N
0 2N × 2N
…
0 2N × 2N
0 2N × 2N
0 2N × 2N
…
0 2N × 2N
−1 M − 1 R 2N × 2N
1 R 2N × 2N
0 2N × 2N
0 2N × 2N
…
0 2N × 2N
0 2N × 2N
2 R 2N × 2N
0 2N × 2N
…
0 2N × 2N
0 2N × 2N
…
0 2N × 2N
⯗
⯗ 0 2N × 2N
…
= s′ ⌫′
冤
0 2N × 2N
…
= s′ ⌫′
−1 1 R 2N × 2N
M − 1 R 2N × 2N
冥 冥
⌫x
−1
⌫x
= s′ ⌫′ (⌫R ⌫′ )−1⌫x (5.13)
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where R is the fullDoF NM × NM covariance matrix, m R 2N × 2N is the m th 2N × 2N covariance submatrix associated with the m th and (m + 1)th pulses, and the 2N (M − 1) × NM subCPI selection transformation matrix ⌫ is given by
冤
0 2N × N
0 2N × N
…
0 2N × N
0 2N × N
2 I 2N × 2N
0 2N × N
…
0 2N × N
0 2N × N
…
⯗ 0 2N × N
…
⌫=
1 I 2N × 2N
0
M − 1 I 2N × 2N
冥
(5.14)
where 0 2N × N is a 2N × N matrix of zeros and I 2N × 2N is the 2N × 2N identity matrix. Equation (5.14) explicitly shows the rankreduction mechanism employed by ADPCA: Only consider the spacetime covariance matrix of successive pulses. Thus, an alternative equivalent (yet more revealing) expression for the ECI of twopulse ADPCA is I s w ADPCA = R eff
(5.15)
= ⌫′ (⌫R ⌫′ )−1 ⌫s from which I = ⌫′ (⌫R ⌫′ )−1 ⌫ R eff
(5.16)
Although we have only considered two pulses in the clutterrejection filter, the form of (5.16) is the same for an arbitrary number of pulses in the first MTI stage with a suitable modification of (5.14). For example, threepulse ADPCA has a transformation matrix ⌫ given by
冤
0 3N × N
0 3N × N
…
0 3N × N
0 3N × N
2 I 3N × 3N
0 3N × N
…
0 3N × N
0
M − 2 I 3N × 3N
⯗ 0 3N × N
…
⌫=
1 I 3N × 3N
0 3N × N
…
冥
(5.17)
However, when the number of pulses equals the entire CPI (i.e., M ), ADPCA becomes equivalent to fullDoF STAP and can thus no longer be considered a rankreducing linear transformation.
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Figure 5.6 demonstrates the potential of ADPCA to retain performance in the presence of both channelmismatch and nonunity  conditions. Of course, a sample covariance is required in practice, and this can reduce the effectiveness of ADPCA from what is shown (although all STAP techniques must work with sample statistics). However, we will defer these considerations for the moment and will revisit them later in this chapter when a comparison can be drawn with other STAP techniques. 5.2.2 PostDoppler (SignalDependent) ReducedRank STAP PostDoppler rankreducing methods apply adaptivity after Doppler filtering has first been applied. The primary rationale for this is the spatial (angle) dependency of clutter Doppler as described in Chapter 3. This Doppler subbanding approach typically allows for significantly smaller (reducedrank) adaptivity, with a commensurate reduction in sample support and implementation requirements [1, 2, 5]. The simplest form of postDoppler rank reduction is the singlebin method, which transforms an NM dimensional spacetime filtering problem
Figure 5.6 Theoretical performance of ADPCA in the presence of both channel and  mismatch.
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into (typically) M separate N dimensional spatialonly adaptive beamforming problems as shown in Figure 5.7. Mathematically, the first 1D Doppler filtering stage can be represented by
=
I
*w 10 I
*w 20 I
I
*w 11 I
*w 21 I
⯗ I
*w 1M − 1 I f 0′ ⊗ I
=
−1 *w M I 0
…
−1 *w M I 1
⯗
*w 2M − 1 I
…
冤 冥冤 冥 f 1′ ⊗ I ⯗
f M′ − 1 ⊗ I
=
…
…
冤冥冤 y1 y2 ⯗ yM
x1 x2 ⯗ xM
−1 *w M M −1 I
冥冤 冥 x1 x2 ⯗ xM
(5.18)
⍀0 ⍀2 x ⯗ ⍀M − 1
冤 冥
where the N × NM linear rankreducing transformation is given by ⍀ m = f m′ ⊗ I, and I is the N × N identity matrix. The M × 1 vector f m is readily recognized as the m th Doppler DFT steering vector given by
Figure 5.7 Singlebin postDoppler STAP beamformer.
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w 0m
fm =
冤 冥 w 1m
(5.19)
⯗
−1 wM m
where w km = e j 2 kf m and f m is the m th normalized Doppler frequency defined in Chapter 2 (Doppler frequency divided by PRF). Also, *w km is the scalar complex conjugate of w km . Rank reduction is achieved by solving a different spatialonly adaptive beamforming problem for each of the M Doppler bins. Thus, M different N × N adaptive problems are solved in contrast to M different NM × NM fullDoF STAP problems (assuming a single look direction and M Doppler bins), which can be a dramatic computational savings. The m th Doppler bin, y m , is thus processed as follows: z m = w′m y m
= 冠R y−m1y m s y m 冡 y m −1
= s′⍀′m R y−m1y m ⍀ m x
(5.20)
= s′⍀′m 冠⍀ m R ⍀′m 冡 ⍀ m x −1
= s′R mI eff x where R y m y m is the N × N (spatialonly) covariance matrix for the m th Doppler bin, and s y m = ⍀ m s , that is, the steering vector corresponding to the full dimensional s in the reduced rank subspace (m th Doppler filter). Note that the Doppler frequency selected in the full dimensional s does not necessarily have to agree exactly with that implied by ⍀ m . For example, if the cost associated with estimating R y m y m is high, one may use a coarse Doppler grid (i.e., M subbands) for R y m y m , even though many more Doppler bins may be processed. From (5.20), we readily infer that the effective covariance inverse for singlebin postDoppler STAP (m th Doppler bin) is R mI eff = ⍀′m 冠⍀ m R ⍀′m 冡 ⍀ m −1
(5.21)
The reader is encouraged to contrast (5.21) with (5.16). In particular, note that the effective inverse in (5.21) is dependent on the desired Doppler
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(signal), in contrast to the signalindependent effective inverse for ADPCA in (5.16). As will shortly be made apparent, socalled singlebin postDoppler STAP performance is usually poor for most applications. A much better approach is to apply spacetime adaptivity to two or more Doppler bins. For example, the effective matrix inverse for threebin (adjacent bins on either side of the bin of interest) postDoppler STAP is I = 冋⍀′m − 1 ⍀′m ⍀′m + 1 册 R eff
⍀m − 1 ⍀ m R 冋⍀′m − 1 ⍀′m ⍀′m + 1 册 ⍀m + 1
冢冤 冥
⍀m − 1 ⍀m ⍀m + 1
冤 冥
−1
冣
(5.22)
= 3 ⍀′m ( 3 ⍀ m R 3 ⍀′m )−1 3 ⍀ m where ⍀m − 1
3⍀m
=
冤 冥 ⍀m
(5.23)
⍀m + 1
Note that, similarly to multipulse ADPCA, in the limit, as the number of bins processed equals M, fullDoF STAP is realized—and thus no rank reduction is achieved. Figure 5.8 demonstrates the performance of singlebin postDoppler STAP for several different tapering levels. The need for a heavy Doppler taper is particularly acute in the singlebin approach since mainbeam clutter can leak through the Doppler sidelobes (nominally a sinc response). This problem is significantly alleviated by the multibin approach (see Figure 5.9) at the expense of a substantially increased computational burden. 5.2.3 Other RankReducing Linear Transformations As will become apparent throughout this chapter, there are many combinations and variations possible in designing a STAP algorithm—indeed too many to address adequately in any one source. Nonetheless, our objective
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Figure 5.8 Theoretical performance of singlebin postDoppler STAP. Note the need for heavy Doppler tapering and the presence of significant loss in the mainbeam clutter region.
in this chapter is to ‘‘span the space’’ of salient methods in sufficient detail as to allow the reader to design his or her own tailored solution. An example of a natural extension to the above pre and postDoppler techniques is the socalled beamspace method [1, 5]. In this approach N B beams are first formed from the N antenna elements (either with phase shifts or time delays [10]), such that N B < N. The beams are then substituted for the ULA outputs. Beamspace rank reduction can be represented by a linear N B × N matrix operator, B, applied to each spatialonly array snapshot x i , whose output, x B i , is given by x Bi = B x i
(5.24)
where to be of value as an RLT, dim 冠 x B i 冡 < dim (x i ) = N. The corresponding fullDoF spacetime beamspace RLT is thus given by
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127
Figure 5.9 Theoretical performance of multibin (threebin) postDoppler STAP.
冤
… 0 B
⯗ 0
…
xB =
B
B
冥
x
(5.25)
= B ST x where B ST is an N B × N B M block diagonal matrix comprised of M repetitions of B. Due to the linear nature of beamspace rank reduction, other rankreducing methods (e.g., DPCA, postDoppler) can be extended to include beamspace via a composite linear operator. For example, the effective covariance inverse for singlebin postDoppler modified to include beamspace rank reduction, has the general form R mI eff = B S′T ⍀′m (⍀ m B ST RB S′T ⍀′m )−1 ⍀ m B ST
(5.26)
where it is understood that since the beamspace rankreduction operator is applied first, the dimensions of ⍀ m are now given by N B × N B M instead
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of N × NM , as was the case for the element space formulation. It is also tacitly assumed in (5.26) that the beams are linearly independent (likely orthogonal in practice) [10], thus insuring that the indicated inverse exists. The reader is encouraged to derive similar expressions for the ADPCA expression in (5.16) and the multibin postDoppler approach, along with a dimensional analysis of the constituent matrices. If the competing clutter of concern is of limited spatial extent and its AoA is generally known a priori, a substantially reduced number of beams could be substituted for the N omnibeams. An example of such an approach for ground clutter cancellation is the point beams at clutter (PBC) approach [11]. In one potential manifestation of this approach, separate beams are formed for each Doppler bin (based on the angleDoppler relationship described in Chapter 3) and then adaptively combined in either a single or multibin approach. However, this approach requires precise knowledge of ownship orientation and velocity along with terrain characteristics to maximize beamspace rank reduction. Although an indispensable tool for rank reduction in practice (particularly at higher frequencies), we will not explicitly include beamspace methods in the examples considered in this chapter because their efficacy is extremely dependent on specific implementation details. For example, poor sidelobe control can lead to interference leakage. Also, beampointing errors can have a significant deleterious effect. Nonetheless, it is easily accommodated as described above and can thus be readily incorporated into any analyses or examples considered in this book. One last reducedrank method worthy of discussion is the socalled PRIstaggered [1, 2] method of rank reduction, which is in some sense a cross between timedomain (e.g., ADPCA) and Dopplerdomain approaches. Similar to postDoppler methods, a rankreducing linear transformation is performed that subbands the fullDoF spacetime CPI into distinct Doppler bins. However, unlike the previously discussed single or multibin postDoppler methods, a single Doppler bin is formed from multiplePRIstaggered windows or subCPI segments. For example, the rankreducing transformation, 2⌿m , for the m th Doppler bin, twoPRIstaggered case, is given by
2⌿m
=
冋
f m′ 1 ⊗ I N × N
0N × N
0N × N
f m′ 2 ⊗ I N × N
册
(5.27)
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where 2⌿m is a 2N × NM matrix, and f m 1 , f m 2 are both M − 1 dimensional vectors comprised of the first and last M − 1 elements, respectively, of f m defined in (5.19). Thus, the effective inverse for the twoPRIstaggered case is given by I = 2⌿′m ( 2⌿m R 2⌿′m )−1 2⌿m R eff
(5.28)
Again the reader is encouraged to extend (5.27) to the threePRIstaggered (or higher) case. The performance of the PRIstaggered approach is similar to that achieved by the multibin postDoppler approach for like adaptive dimensionality (e.g., the threebin postDoppler and the threePRIstaggered approaches have the same size reduceddimension covariance matrices, and thus to first order, also have similar complexity).
5.3 DataDependent ReducedRank STAP Recently, to achieve even more efficient use of the available adaptive DoFs in a STAP beamformer, advanced methods of rank reduction have been developed whose exact behavior is dependent on the received data (generally sample statistics associated with the incoming data)—in other words adaptive rank reduction [12]. Indeed, we have already encountered one of the more popular methods in Chapter 2, namely principal components (PCs). Analogous to the deterministic or dataindependent case, these methods also can be delineated into signaldependent or independent categories, which is to say, the rankreduction method may or may not (respectively) choose its reducedrank subspace based on the particular steering vector of interest (see the taxonomy of Figure 5.1). Another subtle, yet important, discriminant is whether the spacetime snapshot that is being interrogated for the presence of targets is included in the datadependent rank reduction. For example, the PC method performs (directly or indirectly) an eigendecomposition of a sample covariance matrix (as was the case in Chapter 2). If the sample covariance was computed without the current resolutioncellundertest data, then the PC method is still a linear transformation, because it is tantamount to a linear matrix transformation (nonsquare) whose elements are independent of the data being transformed (see below). If, on the other hand, the sample covariance was computed with the current resolutioncellundertest data (along with other cells in general),
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then technically this is a nonlinear rankreduction method. A nonlinear operation can be a significant complicating factor if not carefully controlled and understood. Nonlinearities can introduce spurious false targets (violation of linear superposition) and can transform the statistics of the interference data. For example, the application of a nonlinear transformation to a Gaussian stochastic process results in a generally nonGaussian output, which can significantly impact the design of the detector (e.g., CFAR detector) [13]. However, reasons to include the cell under test, despite these concerns, might be to detect the presence of discretes (i.e., strong interferers that only exist in a single resolution cell [11]), or if there is a scarcity of representative training data due to extreme nonstationarity [14]. Methods have been developed that allow for the inclusion of the test cell in the training data without significantly altering the linear behavior of the filter. One such method is PC and its variants [15]. Section 5.3.1 will examine the signalindependent variants of the datadependent rankreduction methods. These methods choose a linear subspace solely on the basis of the data and not on what particular steering vector will be applied for filtering. In contrast, Section 5.3.2 will introduce signaldependent methods. Again, we emphasize that these methods can be combined with the deterministic rankreducing methods previously described. This fact will be illustrated later on in this chapter. 5.3.1 SignalIndependent Methods The most fundamental and important method in the signalindependent (datadependent) category is principal components. Here, the rankordering metric (ROM) is simply the expected energy, or variance, along an eigenvector dimension of the interference covariance matrix (sample or ideal). Rank reduction is achieved by truncating the eigenbased expansion of the covariance. For an illustration of this process, consider Figure 5.10, which shows a block diagram for the KarhunenLoeve transform (KLT) analysis filter bank [16, 17]. The input NM dimensional randomvector process (spacetime array output) is projected onto an orthonormal basis in which the ROM is the variance along an eigenvector dimension. Thus, for example, at the first stage u 1 (an NM dimensional vector) is chosen to maximize the objective function J given by J = E 冠  u 1′x 
2
冡 = u 1′ R u 1
Subject to: u 1′u 1 = 1
(5.29)
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Figure 5.10 KLT analysis filter bank illustrating the decomposition employed in the PC methods. The ROM is simply the expected value of the energy (or variance) at each stage. Rank reduction is achieved by truncating the analysis tree.
Since R is positivesemidefinite, J is a quadratic form and convex functional [18]. Thus, a necessary and sufficient condition for a maximum is that the vector derivative of the Lagrange multiplier augmented objective function vanish; that is, d (u ′ R u − 1 (u 1′u 1 − 1)) = 2(R u 1 − 1 u 1 ) = 0 NM × 1 d u1 1 1 (5.30) where 1 is the Lagrange multiplier associated with the unit norm constraint in (5.29) [18]. The result is an eigenvalue/vector problem of the form R u1 = 1u1
(5.31)
which for a positivedefinite R is known to have exactly NM solutions (not all necessarily corresponding to distinct eigenvalues [16]). Thus, at the first stage, we choose that solution u 1 which corresponds to a maximum eigenvalue 1 . Note that 1 is also the variance of y 1 , that is, the variance of the output of the first stage of the KLT analysis tree:
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E冠 y1
2
冡 = E 冠  u 1′x  2 冡 = u 1′ R u 1
(5.32)
= 1 u 1′u 1 = 1 where (5.31) and the unity norm constraint were invoked in the second and third steps, respectively, of (5.32). Except for the case when no clutter is present (and at most only one rankone jammer), a 1D subspace is inadequate for representing the colorednoise process. Consequently, the above procedure is repeated for k stages, where for rankreduction k < NM (typically k << NM ). Referring to Figure 5.10, we see that the second stage of the KLT consists of selecting that second orthonormal basis vector (eigenvector) u 2 such that max u 2′ R 1 u 2 u2
(5.33)
Subject to: u 2′u 2 = 1 where R 1 = E (x 1 x 1′ ) = E (B 1 xx′B 1′ )
(5.34)
= B 1 RB 1′ = (I − u 1 u 1′ ) R (I − u 1 u 1′ ) Thus, R 1 is the covariance (positivesemidefinite, even if R was fullrank) associated with the subspace that is orthogonal to u 1 since B 1 is a projection operator that effectively removes the u 1 subspace [16]. The mathematical procedure for finding u 2 is identical to that for finding u 1 . The solution is again that eigenvector that maximizes the eigenvalue 2 associated with R 1 u2 = 2 u2
(5.35)
Principal components rank reduction is achieved by truncating the above procedure at k stages where k < NM . To be useful, however, k should
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be << NM . Recall that for the jammeronly case considered in Chapter 2, the number of independent jammers, N J , was shown to be equal to the rank of the colorednoise covariance matrix. Moreover, N J < N , and thus only a few eigenvectors were required to form the adaptive weights. Resulting in a wellbehaved adapted pattern and commensurately reduced sample support. For the clutter case (or clutterplusjamming case), Brennan’s rule establishes a lower bound for the clutter rank and can thus be used to aid in the selection of a practical stopping rule. However, as was detailed in Chapter 4, Brennan’s rule may significantly underestimate the true clutter rank due to a multitude of important factors (e.g., crabbing, nonlinear arrays, ISL, and other decorrelating effects). This can lead to significantly suboptimal performance and/or a loss of the desirable minimal sample support property. However, a method for extending the PC method in the presence of ISL is considered in Section 5.4. As derived in Chapter 2, the effective covariance inverse for the rankk PC method is k
I
R eff = I −
∑
i =1
i − min u i u i′ i
(5.36)
Note that if k >> min , then (5.36) can be approximated by I
R eff = I −
k
∑ u i u i′
(5.37)
i =1
which is a projection matrix [16]; that is, 冠R eff 冡 = R eff (which is left as an exercise for the reader to verify). In this case, the PC method projects the data out of the subspace spanned by the k dominant eigenvectors. Figure 5.11 shows the performance of the PC method relative to both the straight SMI and diagonally loaded SMI. In this example, Brennan’s rule holds (ideal case) and the PC method can produce results with only 62 training samples that SMI can only achieve with 2NM or 512 samples. However, as detailed in Chapter 2, the proper choice of diagonal loading can approximate the PC method. As discussed in Chapter 4 (and elsewhere [1, 2, 4]), Brennan’s rule never strictly holds in practice due to a multitude of potential rankincreasing effects. As a consequence, PC methods lose their appeal in practical applications as they require the inclusion of more subdominant eigenvalues to I
2
I
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Figure 5.11 Performance of the PC method for an ideal ULA for which Brennan’s rule holds. The efficiency of the PC method results in a substantial reduction in sample support requirements compared with the SMI method. Note that a properly loaded SMI (10 dB in this case) achieves similar performance since it mathematically approximates the PC method for this case (see Chapter 2 for an explanation of this effect).
account for the additional interference subspaces. For example, in Figure 5.12, the same example is repeated, but with the inclusion of ICM. If Brennan’s rule is employed as a PC rankselection criterion, it will result in significant SINR loss. This can be alleviated by including more components, but at the expense of a commensurately increased sample support (and computational burden). A method of extending the PC approach to more realistic environments—without increasing the requisite sample support— will be discussed in Section 5.4. Another lessknown, signalindependent, rankreduction method somewhat related to principal components, is the HungTurner projection [19]. This method has the form I
R eff = I − YY ′ where
(5.38)
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Figure 5.12 Impact when interference subspace leakage (10 mph ICM) is present. Note that for this case, Brennan’s rule is not a good choice for rank selection. A method for addressing this problem is described in Section 5.4.
Y = [x 1 x 2 … x L ]
(5.39)
is a NM × L , L << NM , matrix comprised of spacetime array snapshots (i.i.d. training data) generally selected from surrounding rangebins. If the colorednoise interference is both strong (relative to the noise floor) and low rank (i.e., k << NM ), then the PC method can be approximated by (5.38) if L ≈ k [19]. This is because k i.i.d. samples will, in the limit as CNR → ∞, exactly span the k dimensional colorednoise subspace (almost surely [20]). Hence, projecting orthogonal to that subspace [i.e., (5.38)] will result in an optimum filter. Zatman has analyzed the optimum choice of L for finite CNR [19]. Although not generally considered a robust practical technique, it is relatively simple to implement (no need to perform an eigendecomposition) and, for the dominant lowrank colorednoise case, can yield essentially equivalent performance as PC (i.e., same SINR for properly selected training data)—hence its appeal. The reader is encouraged to experiment with this
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technique and compare it to PC. However, for the reasons previously cited, we will not consider it further in this book. 5.3.2 SignalDependent Methods As was the case for the dataindependent rankreducing methods, the main reason for considering signaldependency in the rankreduction process is to make more efficient use of the available adaptive DoFs. PostDoppler methods (an example of a signaldependent dataindependent rankreduction method) solved a different adaptive problem for each Doppler bin of interest, in contrast to ADPCA, which solved only one adaptive cluttercancellation problem for all Doppler. If the number of signals (i.e., steering vectors) is large, signaldependent methods can have dramatically greater overall computational burdens. However, they also enjoy a high degree of parallelism since in most cases the signaldependent solutions can be derived independently of each other. However, this feature can only be exploited if a large number of parallel CPUs is available in the realtime signal processor. Perhaps the most straightforward method for introducing signal dependency (i.e., steeringvectorspecific dependency) into the subspace selection process is to modify the ROM of principal components. For example, the standard signalindependent ROM for PC is simply the strength of the eigenvalue. There are several relatively straightforward methods for modifying the ROM to include a desired steering vector [12]. 5.3.2.1 Relative Importance of Eigenbeam Method
One method is based on examining the amount of influence a particular eigenvector has on the weight vector associated with a particular steeringvector solution. For example, recall from Chapter 2 the expression for the optimum weight vector expressed in terms of the eigenvalues/vectors of the interference covariance matrix; that is,
冢
NM
w= I−
∑
i =1 NM
=s−
∑
i =1
冣
i − min u i u i′ s i
i − min (u i′ s) u i i
An examination of (5.40) revels a natural choice of ROM; namely
(5.40)
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max {i }
i − min  u i′ s  i
137
(5.41)
Thus, for example, {˜ 1 , u˜ 1 } is the eigenvalueeigenvector pair that maximizes (5.41), which may or may not correspond to the PC ROM; that is, { 1 , u 1 }. Assuming, without loss of generality, that s′s = u i′ u i = 1, (5.41) represents the normalized amplitude of the eigenbeam (i.e., u i ) subtracted from s. For convenience, we will refer to (5.41) as the relative importance of eigenbeam, or RIE, metric of rank ordering. There are three ways in which the metric in (5.41) could be zero: (1) i = min , which implies (in practice) that the eigenvalue is equal to the whitenoise floor and thus should not be subtracted (recall that the optimum filter for the whitenoise case is w = s); (2)  u i′ s  = 0, which implies that u i is orthogonal to s and can thus be ignored (regardless of the strength of the eigenvalue); and (3) both conditions are met. In practice, one should be very careful about condition 2, since errors in the assumed steering vector can cause erroneous results. A remedy is to place a lower bound on  u i′ s  to insure that it is not completely ignored (a natural choice is to set the lower bound equal to the expected steeringvector mismatch residue error). Indeed, sensitivity to steeringvector mismatches is a serious concern in all signaldependent reducedrank methods. From (5.40) and (5.41), it follows that the ECI for the relative importance of eigenbeam (RIE) method is I R eff
冢
= I−
k
∑
i =1
˜ i − min u˜ i u˜ i′ ˜ i
冣
(5.42)
where {˜ i , u˜ i } denotes the eigenvalueeigenvector pair that ranked in the i th position based on the RIE rankordering metric. Thus, in general, ˜ i ≠ i (although this is entirely possible). Again for meaningful rank reduction, it is tacitly assumed that k << NM . 5.3.2.2 Cross Spectral Metric
Another technique for modifying the PC approach to include signal dependency is based on viewing the STAP problem from a sidelobe canceller (SLC) perspective [21]. From Figure 5.13, we see that the spacetime filtering problem can reformulated from a max SINR problem (which is the framework adopted so far in this book) to a minimum variance estimation problem [21]. Specifically, the objective is to choose the auxiliary spacetime weight
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Figure 5.13 Spacetime sidelobe canceller structure.
vector w o [which is of dimension (NM − 1) × 1)] so as to minimize the scalar MSE; that is, 2 min E 冠  d o − dˆ o  冡 wo
(5.43)
Since dˆ o is a linear estimator, a necessary and sufficient condition for minimizing the quadratic convex functional (5.43) is setting its vector derivative equal to zero (on an element basis): d d 2 E 冠  d o − dˆ o  冡 = E 冠冠d o − wo′ x o 冡′ 冠d o − wo′x o 冡冡 d wo d wo = −2E 冠 x o d o* 冡 + 2E 冠 x o x o′ 冡 wo
(5.44)
= −2R x o d o + 2R x o x o wo =0 which yields wo = R x−o1x o r x o d o
(5.45)
R x o x o = B o RB o′
(5.46)
where
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is the (NM − 1) × (NM − 1) covariance matrix associated with the auxiliary data x o and r xo do = B o R s
(5.47)
is the (NM − 1) × 1 crosscorrelation matrix (actually a vector). B is a (NM − 1) × (NM ) blocking [21] or null [10] matrix whose (NM − 1) rows are orthonormal to s, the desired steering vector (also referred to as the mainbeam response [21]). To obtain the form of the effective covariance inverse for the SLC formulation, we again examine the output and work back as follows: z = s′x − wo′ B o x
(5.48)
= (s′x − wo′ B o ) x from which the NM dimensional spacetime weight vector w is readily extracted and further reduced w = s − B o wo′ = s − B o R x−o1x o r x′ o d o
(5.49)
= s − B o R x−o1x o B o R s = (I − B o R x−o1x o B o R ) s and thus I = I − B o R x−o1x o B o R R eff
(5.50)
While seemingly a different approach than the max SINR method, the above SLC formulation is equivalent and yields identical output SINR. One way to see this is first to observe that the output from the mainbeam channel (scalar) concatenated with the (NM − 1) outputs from the auxiliary channels formed by the (NM − 1) × 1 blocking matrix is the result of applying an NM × NM unitary (and thus invertible) matrix Q given by Q=
冋 册 s′
Bo
(5.51)
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The unitary property of Q is easily established since, by definition, the rows of B are pairwise orthonormal and orthonormal to s (and, without loss of generality, we can assume s′s = 1). Since the application of a unitary matrix does not alter the SINR [21], a completely equivalent (and optimal from a max SINR perspective) filter is given by w eq = (QRQ ′ )−1 y s = (QRQ ′ )−1 Q s
(5.52)
= (QR −1 Q ′ ) Q s where y s is the steering vector corresponding to s in the transformed space, and the unitary property of Q was invoked in obtaining the final inverse form. Note, as expected from Figure 5.13, y s = [1 0 … 0]′ since the first channel is the mainbeam response. The corresponding SINR is given by SINR = y s′ QR −1 Q ′y s′ = [1 0 … 0]
=
冋
2d o
r′x o d o
r xo do
R xo xo
册 冤冥 −1
1 0 ⯗ 0
(5.53)
1
2d o − r′x o d o R x−o1x o r x o d o
where ⌬ 冠 2d o = E  do 
2
= E 冠  s′x 
冡
2
冡
(5.54)
= s′R s and use was made of the partitioned matrix inverse lemma [9], that is,
冋
A 11
A 12
A 21
A 12
册 冋 −1
=
−1
−A 11 A 12 B 2
B1 −1
−B 2 A 21 A 11
B2
册
(5.55)
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where −1 ⌬ (A 11 − A 12 A 22 A 21 )−1 B1 =
(5.56)
and −1
B 2 = (A 22 − A 21 A 11 A 12 )−1
(5.57)
Note that only B 1 needed to be computed in (5.53) to obtain the SINR. Again, (5.53) is the max SINR for the transformed (via Q ) colorednoise matched filter, which is also equal to the original optimum SINR, s′ R −1 s, since a unitary transformation cannot alter the SINR. Showing that this is also the SINR corresponding to the minimum MSE SLC is straightforward. From an examination of the signal paths in Figure 5.13 we have SINR SLC =
 s′ s  2
2 E 冠  d o − dˆ o  冡
1
=
E 冠  s′x − w o′ x o 
=
1
= = =
2
冡
E 冠冠 s′x − w o′ x o 冡冠 s′x − w o′ x o 冡′ 冡
(5.58)
1
E (s′xx′s) − w o′ E 冠 x o x o′ 冡 w o 1
2d o
− w o′ R x o x o w o 1
2d o − r′x o d o R x−o1x o r x o d o
which is identical to (5.53), thus establishing the equivalence between the SLC and max SINR (colorednoise match filter) approaches. Note that the orthogonality between s and the rows of B o was invoked to eliminate the cross terms in line four of the above proof.
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The crossspectral metric (CSM) rankordering metric as developed in [22] is a method for prioritizing the eigenvalues/vectors of the covariance matrix associated with the auxiliary channels. This is first accomplished by expressing the SINR in terms of these eigenvalues; that is, SINR =
1
2d o − r′x o d o R x−o1x o r x o d o
(5.59)
1
=
2d o
NM − 1
−
∑
 u i′ r x o d o  2 i
i =1
where ( i , u i ) denotes the i th eigenvalue/vector pair associated with R x o x o , and use was made of the eigenbased form of the inverse of R x o x o ; that is, R x−o1x o
NM − 1
=
u i u i′ i
∑
i =1
(5.60)
From (5.59) it is apparent that the eigenvectors that contribute the most to maximizing SINR (thus minimizing MSE) are those for which the following quantity is relatively large:
 u i′ r x o d o  2
(5.61)
i
Using this CSM as the rankordering metric yields the following reducedrank SLC beamformer: wo = R x−o1x o r x o d o ≈
k
冠 u˜ i′ r x d 冡
i =1
˜ i
∑
o o
(5.62) u˜ i′
where (˜ i , u˜ i ) denotes the eigenvalue/vector pair that ranked in the i th position based on the CSM. For effective rank reduction, it is again assumed that k << NM − 1.
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Finally, the corresponding effective covariance inverse for the CSM method is obtained by examining the effective fullDoF spacetime weight vector; specifically, w CSM = s − B o′ w o CSM = s − B o′ w o CSM k
= s − B o′
∑
冠 u˜ i′ s 冡
i =1
冢
= I − B o′
(5.63)
k
u˜ i ˜ i
冣
∑
u˜ i u˜ i′ s ˜ i
k
u˜ i u˜ i′ ˜ i
i =1
from which we readily ascertain I R eff
= I − B′
∑
i =1
(5.64)
Equation (5.64) has a similar form as the PC methods previously described, but with a different rankordering metric and unitary transformation Q . 5.3.2.3 Multistage Wiener Filter
The above signaldependent methods (RIE and CSM) for modifying the datadependent PC method are relatively straightforward methods for modifying the basis selected for rank reduction. However, both methods utilize an existing basis (eigenvectors of R or R x o x o ) that was developed without any knowledge of s. An entirely different method for selecting a useful basis that explicitly takes s into account was developed by Goldstein et al. and is referred to as the multistage Wiener filter (MWF) [23]. To see how the MWF selects suitable basis vectors, consider the first stage depicted in Figure 5.14. The objective is to choose a rankone NM − 1 dimensional basis vector, h 1 , that is useful for minimizing the MSE. The optimal choice for h 1 is, of course, the minimum MSE weight vector w o given by (5.45). But this solution requires knowledge of the full covariance matrix R x o x o , and is thus counter to the objective of rank reduction. We thus seek a basis vector that is useful for estimating d o , but does not require full knowledge of R x o x o .
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Figure 5.14 First stage of the MWF.
One candidate, which is the rationale of the MWF, is to choose that rankone basis vector h 1 such that the crosscorrelation energy between d 1 = h 1′x o and d o is maximized [12]; that is, max : E 冠  h 1′x o d o* 
2
h1
冡 =  h 1′r x d  2 o o
(5.65)
Since (5.65) is the magnitude squared of an inner product, a direct application of Schwarz’s inequality yields the optimum h 1 , namely, h1 =
r xo do
 r x o d o 
(5.66)
where a unity norm constraint was imposed on the basis vector h 1 ; that is,  h 1  = (h 1′ h 1 )1/2 = 1. Note that as desired, (5.66) does not depend directly on R x o x o . Equation (5.66) is an intuitively appealing result: Set h 1 equal to the cross correlation between x o and d o . With this choice of h 1 , the optimal rankone MWF (scalar) weight vector w 1 is given by −1
w 1 = R d1 d1 rd1 do =
(5.67)
d1 do d21
where d 1 d o is the cross correlation between d 1 and d o , and d2 is the 1 variance of d 1 .
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Since h 1 is not in general colinear with the optimal Wiener vector w o (as this requires the full covariance and crosscorrelation information), additional basis vectors will be required to adequately span the colorednoise subspace that is interfering with the mainbeam response d o . To remedy this, we can add more stages to the above process—hence the meaning of ‘‘multistage’’ in MWF. Figure 5.15 shows a twostage (ranktwo) MWF. The space spanned by x 1 can be seen to be orthogonal to span (s, h 1 ) due to the blocking matrices B o and B 1 . Our objective at this stage is to choose a basis vector h 2 , which, reapplying the criterion from the first stage, is maximally correlated with the coefficient from the first stage d 1 , [12, 23], that is, 2 2 max : E 冠  h 2′ x 1 d 1*  冡 =  h 2′ r x 1 d 1 
(5.68)
h2
subject to the usual unit norm constraint h 2′ h 2 = 1. Again from Schwarz’s inequality we have h2 =
r x1 d1
 r x 1 d 1 
(5.69)
The corresponding optimal SLC weight for the twostage MWF is thus −1
w2 = R d2 d2 rd2 do
Figure 5.15 Twostage MWF.
(5.70)
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where R d 2 d 2 = E (d 2 d 2′ ) =E
冉冋 册 冋 d1
d2
d 1* d 2* 册
冊
(5.71)
and r d 2 d o = E 冠 d 2 d o* 冡
(5.72)
w 2 can be related back to the original covariance and crosscorrelation quantities of (5.46) and (5.47) as follows: R d 2 d 2 = E (d 2 d 2′ ) = E (L 2′ xx′L 2 )
(5.73)
= L 2′ RL 2 where L 2′ =
冋
h 1′ B o h 2′ B 1 B o
册
(5.74)
with r d 2 d o = E 冠 d 2 d o* 冡 = E (L 2′ xx′s)
(5.75)
= L 2′ R s From (5.50) and (5.70), the corresponding NM dimensional spacetime weight vector w for the twostage MWF is w = s − L 2 w2 −1
= s − L 2 R d2 d2 rd2 do = s − L 2 (L 2′ RL 2 )−1 L 2′ R s = (I − L 2 (L 2′ RL 2 )−1 L 2′ R ) s
(5.76)
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Thus, the ECI for the twostage MWF is readily seen to be I = I − L 2 (L 2′ RL 2 )−1 L 2′ R R eff
(5.77)
In general, for the k stage case, we have I = I − L k (L ′k RL k )−1 L ′k R R eff
(5.78)
where
L ′k =
冤
h 1′ B o h 2′ B 1 B o ⯗ h ′k B k − 1 … B 1 B o
冥
(5.79)
Recently, Weippert et al. [24] have shown the relationship between the above sequential decomposition and the method of conjugate gradients. This observation is of more than just theoretical interest since many previously established convergence analyses can be readily adapted to the special MWF case [24]. 5.3.3 Comparison of DataDependent RankReduction Methods Before applying these methods to the STAP problem, we will examine the relative rankcompression performance of the datadependent methods for the 1D beamforming example first considered in Chapter 2. This simple illustration will aid in understanding the differences between the various methods without extraneous complications. Section 5.5 will illustrate application of these methods to STAP. Figure 5.16 shows the SINR performance for the sixjammer example of Figure 2.4 as a function of the rank k of the adaptive beamformer (number of principal components or MWF stages) for each of the datadependent reducedrank methods. Figure 5.16(a) is for the ideal case (exact covariance); Figure 5.16(a) shows the finite sample case (2k samples, where k is the jammer rank). Note that in the ideal case, the MWF enjoys a significant rankcompression advantage over the other methods. However, this advantage is essentially removed when a finite amount of training data is available. In fact, all six basis vectors are required to achieve good nulling performance— exactly the rank required by the PC methods. This last behavior is due to
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Figure 5.16 Rankcompression performance as a function of filter rank: (a) knowncovariance case, and (b) finite sample covariance case.
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the need to fully characterize the colorednoise subspace before an optimal choice of basis can be determined. This characterization is fundamentally limited by the amount of sample support (as well as the estimation procedure). The interested reader is referred to [12] for further details regarding SINR performance as a function of filter rank and sample support. To gain further insight into how the MWF can achieve higher rankcompression (for the known or large samplesupport covariance case), consider the eigenbeam responses of Figure 5.17. Depicted is the PC response, along with the principal MWF response. Since PC is based strictly on energy and not the steering vector of interest (in this case broadside), it simply selects an eigenvector that is essentially aligned with the maximum jammer signal. In contrast, the MWF uses the shaping of the quiescent steering vector to select responses corresponding to jammers that have a greater influence, that is, stronger JNR after beamforming. Of course, if there are steeringvector mismatches, the efficiency of the MWF will be reduced. However, this effect is highly scenariodependent, and it is difficult to make general statements regarding MWF robustness.
5.4 StructuredCovariance and ModelBased Methods Structuredcovariance methods, also referred to sometimes as modelbased methods, refer to techniques that explicitly incorporate specific mathematical constraints or structures into the covariance estimation process. Implicit constraints, such as Doppler or beamspace binning, are almost always present in any MTI radar and are generally not considered explicit modelbased methods. Instead, they can be considered dataindependent/signaldependent methods in accordance with the taxonomy of Figure 5.1. One of the primary reasons for considering structuredcovariance methods is their potential for significantly reducing samplesupport requirements. This can be of considerable practical value in nonstationary environments where there may be a dearth of auxiliary data available for covariance estimation (or weight training) [12]. Section 5.4.1 will consider a broadly applicable class of structured covariance methods based on the observation that many realistic clutter/ interference scenarios involve a multitude of ISL phenomena, such as ICM, dispersion, and so on, as described in Chapter 4. In particular, a method that combines the CMT structure with either PC or diagonal loading will be explored. Due to the ubiquitous nature of ISL, this class of methods enjoys particularly broad applicability [12].
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Figure 5.17 Eigenbeams corresponding to the primary (first stage) basis functions associated with the PC and MWF methods: (a) eigenbeam patterns, and (b) quiescent pattern with incident jammer locations.
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Other modelbased methods with far less broad applicability include Toeplitz and TBT [6, 7], forwardbackward smoothing (FBS) [25, 26], and parametric multichannel signal modeling [27–29]. Due to their narrow scope, we will only briefly discuss their basic modeling assumptions in Section 5.4.2 and provide references for the interested reader. 5.4.1 Covariance Matrix Tapers As discussed in Chapter 4, many realworld phenomena give rise to a specific covariance structure involving a Hadamardmultiplicative matrix operation referred to as a CMT [30]. Specifically, the structure is of the form R 䊊 T, where R is the underlying spacetime covariance matrix in the absence of any ISL or random modulation effects, and T is the generally composite CMT that accounts for all ISL effects. To see how this CMT structure can be exploited to provide a more efficient covariance estimation process, consider the spacetime eigenspectrum of our standard N = M = 16 ULA with and without ICM (see Figure 5.18). Notice that the spectrum of the dominant eigenvalues (k = 31 for this example) appears essentially unaffected by the ICM. As discussed in Chapter 4, ICM is well modeled as a CMT and only requires a few simple parameters to characterize (e.g., Billingsley’s model [31]). This suggests a hybrid estimation approach in which the dominant eigenvalues/vectors are estimated in the usual manner using the PC method (with Brennan’s rule serving as a nominal stopping criterion), and then a final estimated covariance is obtained by applying a CMT and reestablishing the noise floor [4]; that is,
冢∑ k
Rˆ PCCMT =
i =1
ˆ i uˆ i uˆ i′
冣
䊊
Tˆ + ˆ n2
(5.80)
where the set of k eigenvalues/vectors {ˆ i , uˆ i } is estimated in the normal manner (e.g., an SVD [9]) with the usual (e.g., 2k ) sample support, Tˆ is the estimated CMT, and ˆ n2 is the estimated noise floor. Before commenting on the methods for obtaining Tˆ and ˆ n2 , a bit more justification for (5.80) is in order, other than just a simple visual examination of an eigenspectrum invariance. Note that the corresponding ECI for the PCCMT of (5.80) is simply its inverse. Consider the KarhunenLoeve [21] representation of the unmodulated spacetime clutter signal, which is based on a stochastic expansion of the clutteronly eigenvectors; that is,
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Figure 5.18 Illustration of the relative invariance of the dominant eigenvalues to small amounts of ISL. This property can be exploited by a structuredcovariance estimation procedure that models the subdominant eigenvalues/vectors by a CMT. k
xc =
∑ ␥ c uc
i =1
i
i
(5.81)
where x c is an NM dimensional vector RV corresponding to the clutteronly signal in the absence of random modulation, u c i is the i th eigenvector associated with the clutteronly covariance matrix (see Chapter 3), and the k complex scalars and mutually uncorrelated RVs 再␥ c i 冎 have the property E 冠␥ c i ␥ c*q 冡 = c i ␦ iq
(5.82)
where ␦ iq is the Kronecker delta function. As described in Chapter 4, the presence of a CMT implies that a vector random modulation t is present; that is,
STAP for Radar: Methods, Algorithms, and Performance
冢∑ k
xc 䊊 t =
i =1
␥ ci u ci
冣
䊊
䊊
t)
t
153
(5.83)
k
=
∑ ␥ c (u c i
i =1
i
where the distributive property of Hadamard products has been invoked [9]. In practice, the amount of modulation is small; otherwise, the colored noise (clutter/jamming) would be effectively whitened, precluding the need for structured spacetime filtering as a simple whitenoise matched filter would suffice. Indeed, it is this characteristic of ISL that results in the seemingly unperturbed eigenvalues of Figure 5.18. To see this, we first explicitly invoke the small modulation approximation; namely t≈1+⑀
(5.84)
where 1 is an NM dimensional vector with unity entries, and ⑀ is a zeromean, NM dimensional complexvalued vector RV such that E 冠 ⑀ 冡 << 1. Substituting this approximation into (5.83) yields k
xc 䊊 t ≈ =
∑ ␥ c uc
i =1 k
i
∑ ␥ c uc
i =1
i
i
䊊
(1 + ⑀ )
(5.85)
k
i
+
∑ ␥ c uc
i =1
i
i
䊊
⑀
where the vector Hadamard identity property was invoked; that is, u c i 䊊 1 = u c i . Equation (5.85) shows that the introduction of a small ISL modulation can be viewed as a small complex modulation of a large carrier signal (the eigenvectors). From elementary communication systems theory [32], it is known that this will result in a large carrier (original eigenvectors) with small sideband signals (subdominant eigenvectors) due to the modulation. The corresponding covariance matrix is given by [9] cov (x c 䊊 t) ≈ cov (x c 䊊 (1 + ⑀ )) = R c 䊊 (1 NM × NM + R ⑀ ) = Rc + ␦R
(5.86)
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where R c = cov (x c ), R ⑀ = cov (⑀ ), and ␦ R = R c 䊊 R ⑀ . While by definition  ␦ R  << 1, it is in general of full rank [9]. Thus, it can be decomposed as follows
␦ R = Pc ␦ RPc′ + (I − Pc ) ␦ R (I − Pc )′
(5.87)
where Pc is an NM × NM projection matrix onto the subspace spanned by the k clutter eigenvectors, and, thus, I − Pc is the projection onto the orthogonal compliment [16]. Since, again by definition,  Pc ␦ RPc′  << 1, we see that the subspace spanned by R c is essentially undisturbed—which explains our observations in Figure 5.18. The subdominant eigenvalues/ vectors introduced into the subspace orthogonal to the unmodulated clutter spectrum may or may not be of concern. As discussed in Chapter 4, these subdominant components only have an impact if their eigenvalues exceed the elemental noise floor (see the discussion on the iceberg effect in Chapter 4). The preceding argument justifies the use of (5.80) under the conditions (normally encountered) of modest ISL. Thus, the indicated eigenvalues/ vectors in (5.80) can be estimated in the usual manner with—and this is the important part—a sample support based on k , the nominal clutter rank (for which Brennan’s rule can be used as a guide). Of course, it is tacitly assumed that the CNR is high; otherwise, the effective rank of the clutter may be even less than that predicted by Brennan’s rule (and the subdominant eigenvalues due to ISL and so on are below the noise floor—and thus of no concern). The noisefloor estimate ˆ n2 is easily obtained from routine calibration procedures and thus will not be further discussed. However, the choice of CMT Tˆ is a different matter. There are a number of different ways in which Tˆ may be estimated, such as using knowledge of average channelmismatch specifications, bandwidth dispersion, array manifold calibration tables, and realtime equalization procedures. Also, knowledge of the general clutter type and wind state can be used. However, since the presence of some form of space and/or time decorrelating mechanism is a virtual certainty, having some reasonable estimate of Tˆ is far better than none at all. To illustrate this point, Figure 5.19 compares a nominal PC method (rank = k ) and several PCCMT methods with varying degrees of CMT accuracy for a fixed sample support of 2k = 62 (two times the Brennan rank). Even with imperfect knowledge of Tˆ, the PCCMT method consistently outperforms direct PC, as evidenced by the ± 5mph mismatch cases in Figure 5.19.
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Figure 5.19 Performance of the PCCMT and SMICMT approaches in the presence of ICM. Note the robustness of mismatches in the CMT.
Figure 5.19 also shows a diagonally loaded sample matrix method augmented with a CMT. As described in Chapter 2, diagonal loading approximates PC methods and can thus be used as a surrogate for the eigendecomposition indicated in (5.80). This can be of considerable practical implementation value since this SMICMT algorithm is amenable to conventional systolic array implementation (see Chapter 6). Specifically, the SMICMT estimator is of the form Rˆ SMICMT = (Rˆ SMI ) 䊊 Tˆ + ␦ I
(5.88)
where Rˆ SMI is the usual sample matrix estimate (based on 2k samples, and thus generally positivesemidefinite), and ␦ is the loading factor [33]. Note that the effective covariance inverse is simply the inverse of (5.88). Since ISL in its many forms is a ubiquitous realworld phenomenon, the CMT structure should likewise be widely adopted to account for subdominant interference. The above methods are merely two simple illustrations of how one might incorporate the CMT structured covariance. Many others will undoubtedly be developed. For example, a more formal estimation procedure might entail a composite joint MLE approach. However these
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approaches are generally only efficient asymptotically [34] and thus might require significantly larger sample supports than the divideandconquer approach considered here. 5.4.2 Other StructuredCovariance Methods Due to the ubiquitous presence of spatial and/or temporal decorrelating effects and channel mismatch, the above CMT structuredcovariance methods are fairly general and are thus not overly constraining. Other structuredcovariance methods have been suggested that are far more restrictive and, thus, less generally applicable. For example, from Section 3.2, under the assumptions of a perfectly matched ULA in the presence of mutually uncorrelated clutter point scatterers in the absence of antenna crabbing and any ISL, it can be deduced that the spacetime covariance matrix of an ideal ULA with constant PRF has a TBT structure of the form [1]
R=
… R M −1 … R M −2 … R M −3 ⯗
…
冤
Ro R1 R2 * Ro R1 R1 * * R1 Ro R2 ⯗ * −2 R M * −3 * −1 R M RM
…
Ro
冥
(5.89)
which is an NM × NM blockToeplitz matrix comprised of M , N × N block matrices, R o , R 1 , . . . , R M − 1 , which in turn are Toeplitz (hence the term ToeplitzblockToeplitz ); that is,
冤
r m1
r m2
…
r mN −1
r m −1
r mo
r m1
…
r mN −2
r m −2
r m −1
r mo
…
r mN −3
…
Rm =
r mo
⯗ r mo
⯗ r m −N + 1
r m −N + 2
r m −N + 3
…
冥
(5.90)
where the entries are complex scalars. Thus, the spacetime matrix is comprised of O (NM ) distinct elements, dramatically reducing the number of unknowns from the general case that involves O (N 2 M 2 ) unknowns. This implies that there exists significant potential for reducing samplesupport requirements [6, 7].
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Unfortunately, fitting a TBT matrix to sample data is a highly nonlinear process (see, for example, [6, 7], which discuss the MLE and EM approaches) that involves iterative processes that are not guaranteed to converge—a significant drawback for practical applications. Since the efficacy of this technique is tightly coupled to the specifics of any given application, it is difficult to draw any general conclusions regarding its utility. The interested reader is thus referred to the research literature for further details. Another very closely related technique that applies under the same general ULA constraints cited for Toeplitz is the method of FBS [25, 26, 35]. Consider the linear flipupdownthenconjugate operator J (⭈): ⺓ NM → ⺓ NM, given by
冢冤
1 …
J (x) = conj
0 1
冥冣 x
0
(5.91)
It is straightforward to show that a linear phase Vandermonde steering vector, v, is, to within a complex scalar phase term, invariant to J (⭈), that is, J (v) = e j v [25, 26]. However, if n is an NM dimensional vector RV comprised of complex, zeromean, and uncorrelated entries, then E( J (n)n′ ) = ∅. This suggests that if the colorednoise portion of the total interference signal is a linear superposition of Vandermonde steering vectors, then additional data samples can be constructed whose whitenoise components are uncorrelated. For the Gaussian case, this further implies that the extended samples have additive white noise that is independent of the original samples—thus effectively increasing the available sample support. Again, the efficacy of this technique in practice has been found to be extremely scenariodependent, and it is thus not considered further in this text. Another class of structuredcovariance methods effectively reduces the number of independent dimensions of the full spacetime stochastic process by imposing a vector autoregressive multichannel signal model [27–29]. For example, the output of the ULA can be thought of as a vector AR time series of the form x(m + 1) = A (m ) x(m ) + v(m )
(5.92)
where x(m ) ∈ ⺓ N is the array snapshot vector associated with the m th pulse, and v(m ) is assumed to be a whitenoise vector RV, that is, E(v(m )v′ ( q )) = ␦ mq D, where D is a diagonal matrix. Thus, all of the correlation structure
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is contained in the state transition matrix A (m ) ∈ ⺓ N × N (and the implied autoregressive structure). Although certainly a simplifying structural constraint, its effectiveness is also highly scenariodependent and thus not amenable to generalizations. The interested reader should consult [27–29] and the references cited therein for further details.
5.5 Illustrative Design Examples In this section, STAP design examples are presented that illustrate ways in which various reducedrank STAP methods can be combined to yield good SINR performance, but with dramatically reduced samplesupport (and computational) requirements. Specifically, we consider both signalindependent (ADPCA with datadependent CMT rank reduction) and signaldependent (multibin postDoppler with MWF rank reduction). Our motivation is simple: to minimize sample support to insure reliable performance and/or graceful degradation in nonstationary heterogeneous clutter environments. While this design process does not explicitly include realtime embedded computing constraints, they are indirectly, but significantly, alleviated due to the presence of a deterministic rankreducing prestage transformation [1, 2]. Further discussions of realtime implementation issues and methods can be found in Chapter 6 (and references cited therein). A somewhat stressing clutter scenario is chosen to magnify the impact of each of the design stages and to provide a better gauge of algorithm robustness. Figure 5.20 shows the fullDoF eigenspectrum for an N = M = 16 ULA with a CNR of 60 dB,  = 1, ICM of 15 mph, 5° rms of simple channel mismatch (rankone CMT), 2° of crab with 40dB back lobe rejection, and decorrelating channel mismatch (fullrank CMT) corresponding to ⌬⑀ = 0.01, ⌬ = 1° (see Chapter 4). Note that the effective rank of the clutter (∼100; see Chapter 4) is substantially greater than that predicted by Brennan’s rule, which for this example is 31. At this stage, the reader might be wondering why we have chosen not to illustrate the design methods on real data. There are actually several reasons: First, there is a paucity of publicdomain multichannel STAP data that is truly representative of diverse and complex clutter environments. Although some data does exist (e.g., [24, 36–39]), it is by no means enough to draw definitive conclusions regarding the best algorithm for a given operating condition. Second, the reader may very well be interested in selecting a STAP algorithm for a radar configuration and operation that is dissimilar to the publicly available data. So rather than potentially bias the
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159
Figure 5.20 Eigenspectrum for the design example involving both strong clutter and a multitude of realworld deleterious effects.
reader with information based on sparse experimentation, this book has opted to develop the salient constitutive signal and system models that allow for the exploration of countless scenarios—thereby affording the reader the ability to build a comprehensive experience base. 5.5.1 SignalIndependent Approach Our first design step is the choice of a dataindependent (deterministic) rankreducing transformation. Even at this stage, there is a significant number of options available, including beamspace versus element space, and preDoppler (PRI space) versus postDoppler. As beamspace is usually indicated when there is a large number of independent receive elements or when severe implementation constraints exist, we will not consider it in this particular design example. We are therefore left with a choice of pre or postDoppler. Figure 5.21 shows the ideal SINR loss curves for the fullDoF optimum, threepulse ADPCA, and threebin postDoppler cases. While both reducedrank methods perform well, the postDoppler method involves substantially greater overall computations since a different adaptive problem must be
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Figure 5.21 Comparison between threepulse ADPCA and threebin postDoppler reducedrank linear transformations relative to fullDoF optimum performance.
solved for each Doppler bin (see Section 5.5.2). In this example, we select threepulse ADPCA. The threebin postDoppler case is considered in Section 5.5.2. Although we have reduced the adaptive DoFs from 256 to 3 × 16 = 48, with a commensurate reduction in samplesupport requirements, we would still like to achieve further reductions to boost robustness. To accomplish this, we will next consider datadependent methods. Figure 5.22 shows the eigenspectrum for the threepulse ADPCA covariance matrix. Brennan’s rule predicts a nominal clutter rank of N +  (M − 1) = N +  2 = 18, but the actual effective rank is again substantially higher due to the composite ISL effects and crabbing. However, by modeling the additional clutter subspace by a CMT, we can theoretically reduce the sample support to that associated with only 18 DoFs using PCCMT— which is more than an orderofmagnitude reduction from the original fullDoF case. In other words, the requisite 3N × 3N ADPCA covariance matrix R is replaced with the corresponding PCCMT estimate
冢∑ 18
Rˆ PCCMT =
i =1
ˆ i uˆ i uˆ i′
冣
䊊
Tˆ + ˆ n2
(5.93)
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Figure 5.22 Eigenspectrum of the threepulse ADPCA covariance.
Shown in Figure 5.23 is the performance of the hybrid ADPCA/PCCMT approach and a alternate method that substitutes a 10dB diagonally loaded SMICMT for the PCCMT (easier to implement since no eigendecomposition is required). The performance was achieved with only 2k = 36 training samples—as compared with 2NM = 512. Figure 5.24, which shows windspeed errors of ±10 mph, demonstrates the robustness of this approach to errors in the estimated CMT. Notice that even in the presence of mismatch, the hybrid approach still provides performance commensurate with that associated with a far greater sample support. In general, it has been observed that it is better to apply a more pessimistic CMT (greater ISL) to impart robustness and prevent undernulling [4]. The reader is encouraged to create his or her own recipe for achieving similar or perhaps better results. 5.5.2 SignalDependent Approach As described previously in this chapter, a popular and effective firststage signaldependent and deterministic rankreduction transformation is Doppler filtering. Since the spatial distribution of clutter is highly dependent on Doppler, there exists an opportunity to dramatically reduce the number of
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Figure 5.23 Relative performance of the hybrid threepulse ADPCA + PCCMT approach. Also shown is a similar approach with a 10dB diagonally loaded SMICMT substituted for PCCMT.
adaptive DoFs without significant loss [1, 2, 5]. For example, Figure 5.25 shows the eigenspectrum corresponding to the threebin postDoppler covariance (3N × 3N ) for a normalized Doppler of 0.125 (a relatively low Doppler, thus a stressing case). Note that the effective rank of the clutter has been reduced from ∼100 (see Section 5.5.1) to ∼25. At this stage there remain (M − 2, 3N )dimensional adaptive filtering problems which could be solved with either signalindependent adaptivity (e.g., SMI, PC, or PCCMT) or signaldependent methods—which is the course pursued in this example. If SMI is used, then 6N ∼ 96 or more samples are required to reduce SINR losses to within 3 dB or so (as discussed in Chapter 2). However, as we have already seen in the previous example, this sample support can be dramatically reduced if we exploit the fact that the effective rank of the clutter can be substantially less than the total number of DoFs (in this case 3N ). As previously described in this chapter, a highly efficient signaldependent adaptive rankreduction method is the multistage Wiener filter (MWF). The primary design variables in the MWF implementation are the number
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163
Figure 5.24 Illustration of the CMT’s robustness to errors.
of stages and the sample support [12]. To illustrate the interplay of these two design parameters, Figure 5.26 plots the results of a Monte Carlo simulation of SINR loss versus the number of MWF stages and sample support for the selected normalized Doppler of 0.125. Notice that as sample support increases, the number of stages can be reduced and approaches a theoretical optimum [12]. Since our objective in this design example is to minimize sample support while preserving good performance, we might select the (stages, samples ) combination of (20, 40) with a corresponding SINR loss of approximately 4 dB relative to the ideal (knowncovariance) threebin postDoppler case (see Figure 5.21). Thus, again we see that it is possible to significantly reduce samplesupport requirements, in this case from 512 samples to 40.
5.6 Summary In this chapter, we surveyed key algorithms of a STAP taxonomy with an eye towards their application to stressing realworld scenarios. In particular, both dataindependent (nonadaptive or deterministic) and datadependent
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Figure 5.25 Eigenspectrum for threebin postDoppler rank reduction.
(or adaptive) reducedrank STAP techniques were discussed and contrasted. The need for either deterministic or adaptive reducedrank methods is inherent in realworld applications due to a multitude of potential sources of nonstationarity and deviation from idealized conditions—which, therefore, places a premium on minimizing sample support. Rank reduction (especially deterministic methods) can also dramatically reduce realtime computational burdens [1, 2]. Deterministic and adaptive reducedrank methods were further delineated into either signaldependent or independent methods; that is, whether or not they depended explicitly on the desired target steering vector. Illustrative design examples were then presented that highlighted the methods of creating hybrid STAP algorithms that combine various rankreduction mechanisms. Specifically considered were both a purely signalindependent example (ADPCA with CMTaugmented PC or diagonally loaded SMI) and purely signaldependent example (threebin postDoppler with MWF). The results illustrated that robust solutions were possible that preserve good performance with dramatically reduced samplesupport (and computational) requirements compared with a fullDoF STAP implementa
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Figure 5.26 SINR loss relative to ideal (knowncovariance) threebin postDoppler case versus the number of MWF stages and samplesupport size.
tion. The statistical basis for STAP, further computational considerations, and emerging areas of research are addressed in the next chapter.
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Melvin, W. L., M. C. Wicks, and R. D. Brown, ‘‘Assessment of Multichannel Airborne Radar Measurements for Analysis and Design of SpaceTime Processing Architectures and Algorithms,’’ Proc. of 1996 IEEE National Radar Conference, Ann Arbor, MI, May 13–16, 1996, pp. 130–135.
6 Other Topics 6.1 Introduction In this chapter, we briefly discuss several important topics in STAP applications and current research. The statistical basis for STAP is presented in Section 6.2, along with a brief discussion of current research trends in this area. Section 6.3 considers implementation issues. An important class of realtime QR factorization computing architectures is extended to include the CMT approach introduced in Chapter 5—thereby expanding the class of robust STAP algorithms amenable to current realtime implementation architectures. Finally, Section 6.4 introduces some concepts in the newly emerging area of research known as knowledgeaided STAP (KASTAP).
6.2 Statistical Basis for STAP The spacetime beamformer derived in Chapters 2 and 3 was optimum in both a max SINR sense and, if viewed from an SLC perspective, a minimum MSE sense as well. However, nothing was said concerning its statistical (i.e., probabilistic) optimality [1]. For the additive Gaussian interference case, the max SINR beamformer can be shown to be statistically optimal. To see this, consider the basic target detection binary hypothesis testing problem [1, 2]: H0 : x = n
(target absent)
H 1 : x = s + n (target present) 169
(6.1)
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SpaceTime Adaptive Processing for Radar
where x ∈ ⺓NM is the received spacetime array snapshot corresponding to a given range bin, s ∈ ⺓NM is the target steering vector (assumed to be known, or as is done in practice, hypothesized), and n ∈ ⺓NM is the total interference RV (receiver noise, clutter, jamming, and so on). Our objective is to construct a decision function that is, in some sense, optimal in terms of maximizing the probability of detection, while simultaneously maintaining the probability of false alarms. For many sensor surveillance systems, the universally adopted framework for constructing an optimum detection test is the NeymanPearson criterion [1, 2], which maximizes the probability of detection for a fixed falsealarm rate. The corresponding decision criterion is the intuitively appealing likelihood ratio test given by
冠 冡 ⌬ p x  H1 ⌳(x) = p 冠x  H 0 冡
再
>␥,
say H 1
<␥,
say H 0
(6.2)
where p 冠x  H 0 冡 and p 冠x  H 1 冡 are the conditional probabilities for the observation RV x under the targetabsent and targetpresent conditions, respectively [1, 2]. The strictly positive threshold ␥ is chosen to fix the probability of false alarm [1]. Note that a completely equivalent test based on the loglikelihood (due to the monotonicity of the log function for positivevalued arguments) is given by ⌬ ln p 冠x  H 1 冡 − ln p 冠x  H 0 冡 ln ⌳(x) =
再
> ln ␥ ,
say H 1
< ln ␥ ,
say H 0
(6.3)
For the complex Gaussian case (see Chapter 2), we can make further progress in specifying the exact nature of the test in (6.3). Based on (6.1), p 冠x  H 0 冡 is given by p 冠x  H 0 冡 =
1
R  N
e −x′R
−1
x
(6.4)
where R is the total NM × NM dimensional interference covariance matrix, which is assumed to be nonsingular, and  R  is its corresponding determinant. The corresponding targetpresent conditional probability, p 冠x  H 1 冡, is given by
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p 冠x  H 0 冡 =
1
R  N
171
e −(x − s)′R
−1
(x − s)
(6.5)
Substituting (6.5) and (6.4) into (6.3) yields ln ⌳(x) = − [(x − s)′R −1 (x − s) + x′R −1 x]
(6.6)
= 2 Re {s′R −1 x} − s′R −1 s The only term in (6.6) that depends on the observed data (and thus cannot be precomputed) is the quantity z = s′R −1 x
(6.7)
z = w′x
(6.8)
w = R −1 s
(6.9)
which we can write as
where
Thus, an equivalent optimum detector for the additive (known statistics) Gaussian case is precisely the max SINR spacetime beamformer, whose residue is compared to a threshold. In the statistical literature, (6.7) is known as a sufficient statistic [1] since it can be substituted for all of the data so far as detection is concerned. Of course in practice, as discussed in Chapter 4, the total interference covariance matrix is generally not known a priori (apart from the additional Gaussian assumption). Thus, in reality, the simple binary detection problem is a composite hypothesis testing problem [1]. A wellknown approach in this case is the generalized likelihood ratio test (GLRT), which consists of a modified likelihood ratio test in which the unknown quantities are replaced with their MLEs under each of the assumed conditions [1, 2], that is, max p 冠x  H 1 , Rˆ 1 冡 ⌬ ˆ (x) = ⌳
Rˆ 1
max p 冠x  H 0 , Rˆ 0 Rˆ 0
冡再
>␥,
say H 1
<␥,
say H 0
(6.10)
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As mentioned in Chapter 2 for the Gaussian case under the targetabsent hypothesis H 0 , under the assumption that there are L ≥ NM i.i.d. targetfree samples available, the MLE of Rˆ 0 is given by the SCM [3], that is, 1 Rˆ 0 = L
L
∑ x l x l′
(6.11)
l =1
where L ≥ NM . If the presence of a target is assumed not to affect the interference statistics (assumed identical for each test cell), then Rˆ 0 can be substituted for Rˆ 1 . The above highly restrictive derivations should motivate the reader to appreciate the theoretical difficulties associated with extending optimal statistical procedures to the generally nonstationary, nonGaussian case in the presence of unknown signalmismatch statistics [4–6], which of course is the real world. Some progress in developing detectors that preserve the CFAR property has been made for the class of elliptically contoured distributions (ECDs), which is a generalization of the Gaussian case [5, 6]. Also, a more effective detection scheme than the GLRT for the finite sample case has been developed by Kelly [7] (popularly known as the Kelly test). At the time of this writing, the most common practical approach to the detection problem is a twostage method in which the scalar residue from a robust STAP beamformer is then fed to some form of a cellaveraging CACFAR detector [8], as shown in Figure 6.1. Due to any or all of the aforementioned practical STAP shortcomings, the noiseonly residue from the STAP beamformer is generally greater than the theoretical optimum (i.e., the whitenoise floor) and is also nonstationary [8]. Thus, the job of
Figure 6.1 Illustration of conventional detector for a spacetime beamformer. The insertion of the CACFAR circuit after the STAP processor is due to inevitable statistical mismatches leading to improperly nulled interference.
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173
the CACFAR circuit is to estimate the local noisefloor residue (generally accomplished via a sliding window mean or median estimator [8]) and set a detection threshold based on this estimate so as to maintain a prescribed falsealarm rate. Designing a robust CACFAR algorithm is a challenging problem in and of itself. Indeed, one of the major motivations for considering the advanced minimalsamplesupport STAP techniques of Chapter 5 is to lessen the burden on the CACFAR designer—since the only DoF is essentially a scalar threshold. The reader is referred to the authoritative text by Nitzberg [8] for further details on CACFAR design.
6.3 STAP Implementation Consider a hypothetical radar with a 1kHz PRF and a bandwidth of 10 MHz. It has approximately 10,000 unambiguous range bins to process. If each range bin has N spatial and M temporal DoFs, then a single CPI could involve as many as 10,000 NM × NM covariance matrix estimation/inversion STAP operations, easily resulting in a compute throughput measured in the hundreds of GFLOPS (hundreds of billions of floatingpoint operations). Thus, from the very onset, a radar designer must look for very efficient parallelprocessing implementations. Many such architectures have been developed for SMIbased techniques [9, 10]. One of the more popular is the systolic array implementation of the QRfactorized SMI. It results from recasting the SMI problem in terms of an equivalent QR decomposition as follows. First, recall the SMI formulation for the STAP weight vector w:
冢∑ 冣 L
l =1
x l x l′ w = (YY ′ ) w = s
(6.12)
where by definition, Y = [x 1 , . . . , x L ] is the NM × L data matrix (matrix of spacetime array snapshots), and the 1/L factor has been dropped from the sample covariance for convenience since it does not affect the SINR (it could easily be absorbed into Y ). A QR factorization of Y ′ is of the form QY ′ =
冋册 r ∅
(6.13)
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where Q is an L × L unitary matrix (i.e., Q ′Q = I ), r is an NM × NM upper triangular positivedefinite matrix, and ∅ is an (L − NM ) × NM null matrix. Substituting Q ′Q into (6.12) yields YY ′w = YQ ′QY ′w = [r ′ ∅′ ]
冋册
r w ∅
(6.14)
= r ′r w = s which can be solved in two steps involving straightforward backsubstitutions (due to the triangular nature of r ) [10] as follows: r ′a = s
(6.15)
rw = a An important benefit of this formulation is that the numerical condition number (ratio of largest to smallest eigenvalue [10]) of r is the square root of the sample covariance. Additionally, the QR decomposition itself can be efficiently implemented via a sequence of Givens rotations [10]. Diagonal loading can easily be accommodated in this framework by augmenting the data matrix Y with a diagonal matrix whose entries are the square root of the desired loading factor; that is, Y DL = 冋Y  √␦ e 1 … = 冋Y  √␦ I 册
√␦ e NM 册
(6.16)
where e 1 , . . . , e NM are the NM Euclidean basis vectors, and I is the NM dimensional identity matrix. The corresponding diagonally loaded sample covariance matrix is thus given by Y DL Y D′L = YY ′ + ␦ I
(6.17)
where ␦ is the desired loading factor. From (6.17) we see that the inclusion of diagonal loading in the data domain [10] has the effect of creating more data samples—albeit artificially. This is a reasonable result since diagonal loading is a form of structuredcovariance estimation and has been shown to produce results (under conditions that agree with the model) equivalent to those associated with far greater sample supports (see Chapters 2 and 5).
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175
Interestingly, this dataaugmentation approach can also be extended to include CMTs. This is of important practical value since a diagonally loaded SMI augmented with a CMT can closely approximate the highly effective PCCMT algorithm—without the need for a difficulttoimplement eigendecomposition. To see how this is accomplished, we first need to introduce an interesting Hadamard matrix multiplication identity discovered by Khatri and Rao [11]. Theorem
Let A and B be matrices of size n × n with corresponding factorizations A = ⌿′⌿ and B = ⍀′⍀, where ⌿ is of order q × n and ⍀ is of order p × n . Then the Hadamard product of A and B expressed in terms of its factors is given by A 䊊 B = (⌿ 䉺 ⍀)′ (⌿ 䉺 ⍀)
(6.18)
where 䉺 is the KhatriRao product, which consists of the pairwise Kronecker products of the columns of ⌿ and ⍀; that is, ⌿ 䉺 ⍀ = [ 1 ⊗ 1 2 ⊗ 2 … n ⊗ n ]
(6.19)
where i and i are the i th columns of ⌿ and ⍀, respectively. Proof
Although algebraically straightforward, the interested reader is referred to the book by Rao for a proof [11]. Now consider the application of a CMT to the SCM Rˆ 䊊 T = (YY ′ ) 䊊 T
(6.20)
As discussed in Chapter 5, the structure of T is quite simple in practice and can be essentially fixed for a given operating environment (channel mismatch and more). Being positivesemidefinite [12], a Gram or squareroot factorization of T is guaranteed [13]. Thus, (6.20) can be rewritten as (YY ′ ) 䊊 T = (YY ′ ) 䊊 (⌫′ ⌫ ) = ( Z ′Z ) 䊊 (⌫′ ⌫ ) = ( Z 䉺 ⌫ )′ ( Z 䉺 ⌫ )
(6.21)
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where T = ⌫′ ⌫ and, by definition, Z = Y ′. The advantage of (6.21) is that the same QR factorization and implementation architecture can be applied to the augmented data matrix ( Z 䉺 ⌫ )′. A simple example will illustrate the basic procedure. Consider a simple twochannel adaptive beamformer implementing a diagonally loaded SMICMT (obviously this is of no practical interest, but it is useful for illustrating the process). Assume that there are four i.i.d. samples available for estimating the SCM; that is, Y=
冋
2 + j 2.5
0.9 + j 4.5
2.3 − j 3.6
−0.6 − j 1.7
−1.2 + j 2.6
−1.5 − j 4.7
−3 + j 2.9
2.3 + j 1.9
册 (6.22)
from which YY ′ =
冋
52.8
−40.4 + j 9.4
−40.4 + j 9.4
58.9
册
(6.23)
To add 0dB of diagonal loading, we augment Y with a simple 2 × 2 diagonal matrix whose diagonal entry is 2 [the 1/4 factor omitted from (6.23) is absorbed into the loading factor]; that is, Y DL =
冋
册
2 + j 2.5
0.9 + j 4.5
2.3 − j 3.6
−0.6 − j 1.7
2
0
−1.2 + j 2.6
−1.5 − j 4.7
−3 + j 2.9
2.3 + j 1.9
0
2 (6.24)
Assuming a CMT of the form T=
冋
1
0.9
0.9
1
册
(6.25)
the corresponding diagonally loaded SMICMT covariance matrix is given by Rˆ = (Y DL Y D′L ) 䊊 T =
冋
56.8
−36.3 − j 8.4
−36.3 + j 8.4
62.9
册
(6.26)
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177
In factored form, (Y DL Y D′L ) 䊊 T can be rewritten as (Y DL Y D′L ) 䊊 T = ( Z ′Z ) 䊊 (⌫′ ⌫ )
(6.27)
where Z = Y D′L and ⌫=
冋
0.85
0.53
0.53
0.85
册
(6.28)
The corresponding 12 × 2 KhatriRao factor, Z 䉺 ⌫, is given by Z 䉺 ⌫ = [z 1 ⊗ ␥ 1 z 2 ⊗ ␥ 2 ]
(6.29)
where
z1 =
冤 冥冤 冥 2 − j 2.5
−1.2 − j 2.6
0.9 − j 4.5
−1.5 + j 4.7
2.3 + j 3.6
−0.6 + j 1.7
, z2 =
−3 − j 2.9
−2.3 − j 1.9
2
0
0
2
(6.30)
and ␥ 1 and ␥ 2 are the first and second columns of ⌫. The diagonally loaded SMICMT covariance matrix corresponding to the KhatriRao factored form is given by Rˆ = ( Z 䉺 ⌫ )′ ( Z 䉺 ⌫ ) =
冋
56.8
−36.3 − j 8.4
−36.3 + j 8.4
62.9
册
(6.31)
which is identical to (6.26). Although highly contrived and simple, the foregoing example nonetheless illustrates that the final STAP algorithm, which includes datadomain formulations and techniques for improving its robustness (e.g., diagonal loading and/or CMTs), can be more computationally demanding then one might at first expect. This is certainly true for the far more complex structuredcovariance estimation methods described in Chapter 5 and elsewhere. For
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further details on realtime implementation, the reader is referred to the literature ([9, 10]).
6.4 KnowledgeAided STAP Much of the emphasis in this book has been on developing STAP techniques that can work in realworld environments. A moment’s reflection will reveal that realworld ground clutter can be exquisitely complex—especially when manmade objects are factored in (buildings, power lines, roadways, and so forth). The presence of these large clutter discretes and abrupt discontinuities can have a significantly deleterious effect on STAP performance if not somehow accounted and/or compensated for [14–16]. One of the most exciting and farreaching areas of research beginning to address these problems is socalled knowledgeaided STAP (KASTAP) [14, 16]. The idea behind KASTAP is simple (its implementation, however, is very challenging [16]): Databases derived from a multitude of sources [e.g., digital terrain and elevation data (DTED) and SAR [16]] contain information about the environment that can be extremely useful in the STAP adaptation process. For example, as described by Melvin et al. [14], if one knows where the roadways are, resolution cells corresponding to potentially competing ground traffic can be excised from the training data, which can result in dramatically improved detection performance of targets with Doppler frequencies similar to the competing background traffic [14]. While certainly a new area of research, KASTAP is tantamount to the inclusion of priors in the statisticalestimation process [1], and it is thus a wellestablished statistical concept from a classical Bayesian perspective [1]. For example, if the underlying interference has a quiescent spacetime pdf R q , prior knowledge of the presence of a discrete, for example, would modify the total covariance R , resulting in R = R q + R d , where R d is the a priori covariance, associated with the clutter discrete only (that is, assumed to be independent of the nominal background clutter). If the clutter discrete is a singlepoint scatterer with associated spacetime steering vector d and power (variance) d2 , then for the Gaussian case, the composite total interference pdf is given by p (x) =
1
R  N
e −x′R
−1
x
(6.32)
where the total covariance R is the sum of the quiescent covariance R q and the factor due to the discrete; that is,
Other Topics
R = R q + d2 dd′
179
(6.33)
In this example, the discrete was treated as a rankone signal. In practice, due to the generally spacetimedistributed nature of the scatterer and uncertainties associated with channel mismatch, the actual covariance will generally be of full rank, that is, dd′ → (dd′ ) 䊊 T, where T is a CMT. The interested reader is referred to the literature for further details [14, 16].
6.5 Summary In this chapter, several important topics were surveyed. Section 6.2 introduced the statistical basis for STAP establishing the optimality of the max SINR beamformer for the additive Gaussian case and discussed current research trends in this area. Section 6.3 considered implementation issues with an emphasis on an important class of realtime QR factorization computing architectures. This popular implementation method was extended to the CMT approach introduced in Chapter 5, thereby expanding the class of robust STAP algorithms amenable to realtime implementation. Finally, Section 6.4 introduced some concepts in the newly emerging area of research known as KASTAP.
References [1]
Van Trees, H. L., Detection, Estimation, and Modulation Theory: Part I, New York: John Wiley & Sons, 1968.
[2]
DiFranco, J. V., and W. L. Rubin, Radar Detection, Dedham, MA: Artech House, 1980.
[3]
Anderson, T. W., An Introduction to Multivariate Statistical Analysis, 2nd ed., New York: John Wiley & Sons, 1984.
[4]
Wegman, E. J., S. C. Schwartz, and J. B. Thomas, (eds.), Topics in NonGaussian Signal Processing, New York: SpringerVerlag, 1989.
[5]
Fang, K. T., and Y. T. Zhang, Generalized Multivariate Analysis, New York: SpringerVerlag, 1990.
[6]
Richmond, C. D., ‘‘PDF’s, Confidence Regions, and Relevant Statistics for a Class of Sample CovarianceBased Array Processors,’’ IEEE Trans. on Signal Processing, Vol. 44, No. 7, July 1996, pp. 1779–1793.
[7]
Kelly, E. J., Adaptive Detection in Nonstationary Interference: Parts I and II, Technical Report 724, MIT Lincoln Laboratory, 1985.
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[8]
Nitzberg, R., Adaptive Signal Processing for Radar, Norwood, MA: Artech House, 1992.
[9]
Ward, C. R., P. J. Hargrave, and J. G. McWirther, ‘‘A Novel Algorithm and Architecture for Adaptive Digital Beamforming,’’ IEEE Trans. on Antennas and Propagation, Vol. 34, No. 3, March 1986, pp. 338–346.
[10]
Farina, A., and L. Timmoneri, ‘‘RealTime STAP Techniques,’’ Electronics and Communication Engineering Journal, Vol. 11, No. 1, February 1999, pp. 13–22.
[11]
Rao, C. R., and M. B. Rao, Matrix Algebra and Its Applications to Statistics and Econometrics, Singapore: World Scientific, 1998.
[12]
Guerci, J. R., ‘‘Theory and Application of Covariance Matrix Tapers for Robust Adaptive Beamforming,’’ IEEE Trans. on Signal Processing, Vol. 47, No. 4, April 1999, pp. 977–985.
[13]
Horn, R. A., and C. R. Johnson, Topics in Matrix Analysis, Cambridge, England: Cambridge University Press, 1991.
[14]
Melvin, W. L., ‘‘SpaceTime Adaptive Radar Performance in Heterogeneous Clutter,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 36, No. 2, April 2000, pp. 621–633.
[15]
Guerci, J. R., and J. S. Bergin, ‘‘Principal Components, Covariance Matrix Tapers, and the Subspace Leakage Problem,’’ IEEE Trans. on Aerospace and Electronic Systems, Vol. 38, No. 1, January 2002, pp. 152–162.
[16]
Guerci, J. R., ‘‘KnowledgeAided Sensor Signal Processing and Expert Reasoning,’’ Proc. of 2002 Workshop on KnowledgeAided Sensor Signal Processing and Expert Reasoning (KASSPER), Washington, D.C., April 3, 2002 (CD ROM).
About the Author J. R. Guerci is the deputy director of the Special Projects Office for the Defense Advanced Research Projects Agency (DARPA), where he is responsible for the development of next generation radar and sensor systems, including spacebased radar. He has more than 15 years of industrial and academic experience, and has published numerous technical papers including a book chapter he coauthored in Intelligent Signal Processing (IEEE Press, 2001). A member of the IEEE Radar Systems Panel, he received his Ph.D. in system engineering from Polytechnic University in New York.
181
Index Adaptive 1D processing, 31–39 Adaptive array processing, 11–42 1D, 31–39 introduction, 11 optimum spatial (angle) beamforming, 11–28 optimum temporal (Doppler/pulse), 29–30 summary, 42 Adaptive DPCA (ADPCA), 119 effectiveness reduction, 122 rankreduction mechanism, 121 signalindependent effective inverse, 125 theoretical performance, 122 threepulse, 121, 160, 161, 162 twopulse, 121 See also Displaced phase center array (DPCA) Adaptivity in nonstationary environments, 39–42 reduceddimension, 112 spatialonly, 40 Additive colorednoise, 23–28 Additive Gaussian noise (AGN), 9 Additive white noise, 19–23 Airborne MTI (AMTI) radar design, 54 with twoway antenna sidelobes, 51
See also Moving target indicator (MTI) Angledependent channel mismatch, 92–95 defined, 77 errors, 78 source, 93–94 See also Channel mismatch AngleDoppler airborne clutter structure, 54 contours, 101, 102 dependence for ULA radar, 2 steering vectors, 65 Angleindependent channel mismatch, 78–92 cause, 78 finite (nonzero) bandwidth case, 83–92 narrowband, impact, 79 narrowband array illustration, 78 narrowband case, 79–83 See also Channel mismatch Angle of arrival (AoA), 11, 92 Antenna array dispersion, 95 Antenna array misalignment, 99–103 clutter rank and, 100 velocity, 100 Arrays manifold effects, 92 nonlinear, 103, 104 UHF linear/circular, 104 183
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Beamformers 1D optimal, 66 max SINR, 69, 169 optimum, derivation of, 18–19 optimum ULA, 30 response, 14 singlebin postDoppler STAP, 123 spacetime, 4, 5, 66, 72 ULA adaptive, 12 Beamforming 1D, datadependent methods for, 147 adaptive, 18 optimal, 27 optimum spatial (angle), 11–28 Beamspace rank reduction, 127 Brennan’s rule, 99, 103, 133 true clutter rank and, 100 violation of, 103 Cancellation ratio (CR), 91–92 Capon estimator, 68, 69 Cell averaging constant falsealarm rate (CACFAR), 31 cellaveraging detector, 172 circuit insertion, 172 robust algorithm, 173 Channel mismatch, 77–95 analysis, 78 angledependent, 77, 92–95 angleindependent, 78–92 bandwidth, 77 categories, 76–77 decorrelating, 85, 86, 88 decorrelating random amplitude and phase, 85 defined, 77 i th jammer in absence of, 93 phaseonly, 82 spatial, 80 total interference covariance in presence of, 81 Chebyshev taper function, 17 weighting schemes, 16 Clutter airborne, angleDoppler structure, 54 cancellation, 63, 114
decorrelation, 98 Doppler power, 95, 96 eigenspectrum, 99 nonstationarity impact on SINR, 107 in perfect DPCA system, 117 point beams at (PBC) approach, 128 true rank, 133 Clutterplusnoise eigenspectrum, 103 Cluttertonoise ratio (CNR) defined, 58 nonstationarity, 77 specified, 58 Coherent processing interval (CPI), 3 DPCA filtered, 118 short, 99 Constant falsealarm rate (CFAR), 7 Covariance augmentation, 40 Covariance estimation error, 34 Covariance matrix clutter/jamming and, 63 clutteronly, 60 CMT application to, 40 diagonallyloaded SMICMT, 176–77 estimation/inversion STAP operations, 173 jammer, spatialonly, 63 NM dimensional spacetime clutter, 57–58 sample, 40, 46–49 total spacetime, 58, 63 unknown, 46 weight vector formulation, 173 Covariance matrix tapers (CMTs) application, to sample covariance matrix, 40 application example, 41 application to SCM, 40, 175 composite, 151 defined, 8 ICM model, 96 MaillouxZatman, 41–42 matrix, 40 rankone, 91 robustness to errors, 163 spatialonly, 40, 86, 87, 92 temporal, 99 Crab angle, 100
Index Crabbing, effect of, 102 Cross spectral metric (CSM), 137–43 defined, 142 effective covariance inverse for, 143 using, as rankordering metric, 142 See also Signaldependent methods Datadependent reducedrank STAP, 129–49 HungTurner projection, 134–36 methods comparison, 147–49 PC method, 129, 130–34 signaldependent methods, 136–47 signalindependent methods, 130–36 Dataindependent reducedrank STAP, 114–29 linear transformations, 125–29 postDoppler, 122–25 preDoppler, 114–22 Decorrelating channel mismatch, 85 clutter eigenspectrum impact, 90 errors, impact of, 89 impact on spacetime clutter covariance matrix, 88 random amplitude and phase, 85 SINR loss impact, 90 spatialonly, 88 See also Channel mismatch Defense Advanced Research Projects Agency (DARPA) radar, 6 photograph, 7 processed clutter returns, 8 specifications, 7 Degrees of freedom (DoFs) adaptive, 39 NM, 4 optimal spacetime processing, 115 spatiotemporal, 99 Diagonal loading, 39, 174 data matrix augmentation and, 174 limitations, 39 in PC method, 133 SMI, 76 SMICMT, 176 Digital terrain and elevation data (DTED), 178 Dirac delta function, 96
185
Discrete Fourier transform (DFT), 14 Displaced phase center array (DPCA) adaptive (ADPCA), 119, 120, 121, 122 case, 63 clutter, 117 concept illustration, 115 performance, 119 PRI requirement, 63 processing, 61 twopulse, performance, 119 twopulse canceller, 116, 118 twopulse system, 117 ULA, 115 Doppler aliasing, 59 filtering, 28, 114 processing, 30 Doppler shift distribution, 53 induced by moving targets, 52 normalized, 53, 55 spatially dependent, 54 Dopplershifted return, 29 Doppler steering vectors, 30, 66 Kronecker product, 55 M dimensional, 56 normalized 2D angle pattern, 57 Effective covariance inverse (ECI) mathematical derivation, 113 RIE method, 137 singular, 114 for twostage MWF, 147 Eigendecomposition, 37 Eigenvalues, 37, 49 distributions, 61 of estimated covariance, 38 exact and estimated comparison, 36 invariance property, 82 as noisefloor components, 36 total interference, 89 Elliptically contoured distributions (ECDs), 172 Euler’s Identity, 45 Fast Fourier transform (FFT), 14 Forwardbackward smoothing (FBS), 151, 157
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Fourier transform 2D discrete, 59 inverse, 96 Generalized likelihood ratio test (GLRT), 171–72 defined, 171 for finite sample case, 172 Geometric Sum Formula, 45 Global positioning systems (GPS), 3 GramSchmidt method, 37 Ground MTI (GMTI) radars, 52 Hadamard identity matrix, 86, 87 identity property, 153 identity row vector, 16 matrix multiplication identity, 80, 175 multiplicative property, 91 Hadamard products, 16, 22 identities, 87 performing, 16 HungTurner projection, 134–36 form, 134–35 PC method comparison, 135–36 See also Signalindependent methods Iceberg effect characteristics, 77 defined, 75 illustrated, 105 interference nonstationarity and, 103–6 Inertial navigation systems (INS), 3 Interference covariance matrix colorednoise, 82 estimating, 32 total, 59, 81 See also Covariance matrix Interference subspace leakage (ISL), 3 decorrelating, 76 defined, 95 effects, 95–99 mechanisms, 95 rangewalk source, 98 uncorrelated phenomenon, 76 Interferencetonoise ratio (INR), 75 Internal clutter motion (ICM), 1 CMT corresponding to, 96 defined, 95
impact on eigenspectrum, 96, 97 modeling, 151 SINR loss impact, 96, 97 for subspace leakage, 105 as temporal modulation effect, 96 Inverse displaced phase center array (IDPCA), 6, 7 Inverse Fourier transform, 96 Jammers impact on total interference eigenspectrum, 65 nulls, 40 power, 24 spatial only, covariance matrix, 63–64 steering vector, 24 uncorrelated, 25, 80 uncorrelated noise, 64 Jammertonoise ratio (JNR), 25, 149 KarhunenLoeve representation, 151 KarhunenLoeve transform (KLT), 130 analysis filter, 131 analysis tree, 131–32 KhatriRao factor, 177 product, 175 Knowledgeaided STAP (KASTAP), 9, 178–79 defined, 178 idea behind, 178 See also STAP Kronecker delta function, 152 identity, 88 products, 55, 56, 175 Lagrange multiplier, 69, 131 Likelihood function convexity of, 49 maximizing, 47 Linear frequency modulation (LFM), 94 Low Earth obit (LEO) SBR, 59 MaillouxZatman CMT, 41–42 defined, 41 illustrated, 41 jammer notch, 42 See also Covariance matrix tapers (CMTs)
Index Main lob, 14 Maximum likelihood estimate (MLE), 46–49 derivation, 46–49 of unknowncovariance matrix, 47 Meansquared error (MSE), 9 scalar, minimizing, 138 spacetime beamformer, 9 Minimum detectable velocity (MDV), 106 Minimum variance (MV), 70 Modelbased methods. See Structuredcovariance methods Moving target indicator (MTI) radar, 1 airborne (AMTI), 51, 54 ground (GMTI), 52 optimum spacetime processing for, 65–70 spaceborn, 52 STAP for, 3–7 twopulse canceller, 63 Multistage Wiener filter (MWF), 143–47, 162 advantage, 147–49 bias vector selection, 143 comparison, 147–49 defined, 143 design variables, 162–63 first stage, 144 optimal rankone weight vector, 144 shaping of quiescent steering vector, 149 twostage, 145–47 See also Signaldependent methods NeymanPearson criterion, 170 Noise additive colored, 23–28 additive Gaussian, 9 additive white, 19–23 total, 23 white, 19–23, 24 Noisefloor components, 36 Nonlinear arrays, 103, 104 Optimal beamforming, 27 Optimum spatial (angle) beamforming, 11–28
187
Optimum temporal (Doppler/pulse) processing, 29–30 Organization, this book, 7–9 Orthonormality, 82–83 Parametric multichannel signal modeling, 157–58 PCCMT method, 76, 154 PC method comparison, 154 performance, 155 Point beams at clutter (PBC) approach, 128 PostDoppler reducedrank STAP, 122–25 multibin, 127 singlebin, 126 threebin, 164–65 Power selected training (PST), 52 Power spectral densities (psd), 58 clutter Doppler, 96 illustrated, 60 Principal components (PC) method, 130–34 comparison, 147–49 defined, 129 performance, 133, 134 in practical applications, 133–34 rankk, 133 rank reduction, achieving, 132–33 straight SMI/diagonally loaded SMI and, 133, 134 See also Signalindependent methods Pulse repetition frequency (PRF), 3 Pulse repetition interval (PRI), 3 DPCA requirement, 63 staggered approach, performance, 129 Range walk, 98 Rank linear transformation (RLT), 113 datadependent, 114 dataindependent, 114 preDoppler dataindependent, 114 Rankordering metric (ROM), 130, 136 Rank reduction beamspace, 127 data dependent, 129–49 data independent, 114–29 postDoppler, 122–25 preDoppler, 114–22
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SpaceTime Adaptive Processing for Radar
Rank reduction (continued) principal components, 129, 130–34 signaldependent methods, 136–47 signalindependent methods, 130–36 Relative importance of eigenbeam (RIE) method, 136–37 comparison, 147–49 ECI, 137 Riemann sum approximation, 56, 88 Sample covariance matrix (SCM) CMT application to, 40, 175 estimating, 176 maximum likelihood, derivation of, 46–49 N × N, 47 Sample matrix inverse (SMI), 9 diagonalloading, 76 STAP weight vector formulation, 173 See also SMICMT Schur Product theorem, 91 Schwartz’s inequality, 13, 23, 144 Selfnulling, 33 Sidelobe canceller (SLC), 137 effective covariance inverse for, 139 spacetime, structure, 138 Sidelobes defined, 15 Doppler, 66 range, reduction, 94 Signaldependent methods, 136–47 cross spectral metric (CSM), 137–43 illustrative design example, 161–63 multistage Wiener filter (MWF), 143–47 relative importance of eigenbeam (RIE) method, 136–37 See also Datadependent reducedrank STAP Signalindependent methods, 130–36 HungTurner projection, 134–35 illustrative design example, 159–61 PC, 129, 130–34 rankordering metric (ROM), 130 See also Datadependent reducedrank STAP
Signalinterferencetonoise ratio (SINR) clutter nonstationarity impact on, 107 Doppler vs., 68 dropout, 67 loss, 28, 39, 67, 80, 163 loss curves, 159 loss relative to ideal threebin postDoppler, 165 max, 141 max beamformer, 69, 169 optimal, 68 performance impact, 91 for sixjammer example, 147, 148 spacetime beamformer, 72 unitary matrix and, 140 Signaltonoise ratio (SNR), 4 defined, 19 loss, 21, 83 loss ratio, 79 maximizing, 19 optimum, 20, 21–22, 79 output, 20–21 realized, 79 singlechannel, 21 tapered output, 21–22 whitenoise, 24 Six jammer case angular distribution, 25 covariance, 36 optimum beam pattern, 27 SINR performance for, 147, 148 SMICMT, 76, 155 covariance matrix, 176, 177 diagonallyloaded, 176, 177 Spacebased radar (SBR), 59 Spacetime adaptive processing. See STAP Spacetime beamformers illustrated, 66 optimal SINR, 72 response for sidelooking ULA, 5 Spacetime mismatch taper, 80 Spatialonly CMT, 87, 92 adaptivity, 40 elements, calculating, 86 See also Covariance matrix tapers (CMTs)
Index STAP, 51–72 beamformer, 3 bistatic, x, 77 circular array applications, 40 for clutter cancellation, 114 CPI dimension NM, 112 datadependent reducedrank, 129–49 dataindependent reducedrank, 114–29 efficiency, 9 fullDoF, 119 implementation, 173–78 introduction, 51–52 knowledgeaided (KASTAP), 9, 178–79 for MTI radar, 3–7 multibin postDoppler, 127 need in moving target indicator radar, 1–3 nonlinearity impact on, 105 optimum, for MTI radar, 65–70 performance factors, 75–107 postDoppler reducedrank, 122–25 preDoppler reducedrank, 114–22 realworld systems, 75 singlebin postDoppler, 126 statistical basis for, 169–73 taxonomy, 111 STAP algorithms comparison, 75 PCCMT, 76, 154, 155 SMICMT, 76, 155 taxonomy, 112 Steering vectors angleDoppler, 65 Doppler, 30, 55, 56, 57, 66 spacetime, 55, 117 spatial, 55 Vandermonde, 157 Structuredcovariance methods, 149–58 applicability, 149 CMT, 151–56 defined, 149 FBS, 157 parametric multichannel signal modeling, 157–58 TBT, 156–57
189
Synthetic aperture radar (SAR), x Threebin postDoppler rank reduction, 164–65 eigenspectrum for, 164 SINR loss relative to, 165 Threepulse ADPCA, 121, 160 eigenspectrum, 161 relative performance, 162 See also Adaptive DPCA (ADPCA) ToeplitzblockToeplitz (TBT), 151, 156–57 defined, 156 matrix, fitting, 157 structure, 113, 156 Trackwhilescan (TWS), 103 Twostage MWF, 145–47 ECI for, 147 illustrated, 145 NM dimensional spacetime weight vector for, 146 optimal SLC weight for, 145–46 See also Multistage Wiener filter (MWF) Uniform linear array (ULA) adaptive beamformer, 12 antenna array response, 44–45 DPCA, 115 N element, 3 optimum beamformer, 30 sidelooking, 60 whitenoise case, 20 Uniform linear array (ULA) radar, 1 angleDoppler dependence for, 2 clutter isorange ring for, 2 N element, 53 Vandermonde steering vectors, 157 Vandermonde structure, 62 White noise additive, 19–23 defined, 19 matched filter solution, 20 SNR, 24 ULA, 20 Wide sense stationary (wss) process, 57 ZeroDoppler filter, 52