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Robust adaptive array processing

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Title:
Robust adaptive array processing theory and applications
Creator:
Wang, Zhisong ( Dissertant )
Li, Jian ( Thesis advisor )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
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2005
Language:
English
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vii, 108 leaves : ill. ; 29 cm.

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Subjects / Keywords:
Beamforming ( jstor )
Capons ( jstor )
Covariance ( jstor )
Imaging ( jstor )
Mathematical vectors ( jstor )
Matrices ( jstor )
Signal processing ( jstor )
Signals ( jstor )
Steering ( jstor )
Ultrasonography ( jstor )
Dissertations, Academic -- Electrical and Computer Engineering -- UF ( lcsh )
Electrical and Computer Engineering thesis, Ph. D ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Adaptive beamformers have better resolution and much better interference rejection capability than the data-independent beamformers, provided that the array steering vector corresponding to the signal of interest is accurately known. however, the adaptive beamforming techniques rely significantly on the assumption of the incident signals, the sensor array, and the surroundings; and are very sensitive to the inevitable mismatch between the assumptions and the actual characteristics. Consequently, adaptive beamformers often suffer from severe performance degradations in practical applications. We developed a novel robust Capon beamformer (RCB) that can maintain both the robustness of the data-independent beamformers and the adaptivity of the standard Capon beamformer (SCB) (a type of the adaptive beamformer). We showed that RCB belongs to the class of diagonal loading approaches. However, the amount of diagonal loading can be precisely calculated based on the uncertainty set of the array steering vector. We showed how the proposed robust Capon beamformer can be efficiently computed at a cost comparable to that of SCB. We also provided deep insights into the relationships among the recent three robust adaptive beamformers. Although the robust adaptive beamformers obtained from the two different formulations appear to be rather different we found that they are equivalent in terms fo the signal-to-interference-plus-noise ration (SINR), however, RCB also has some unique features. By comparing these three beamformers for degenerate vs non-degenerate ellipsoidal uncertainty sets of the steering vector, we showed the impressive advantages of RCB over the other two methods: simpler implementation, lower computational complexity, and more accurate power estimation. In addition, we applied RCB to various applications, and extended RCB to accommodate their specific goals and requirements. For the application of aero-acoustic imaging, we devised a constant-beamwidth RCB and a constant-powerwidth RCB for consistent acoustic imaging. This means that for an acoustic wideband monopole source with a flat spectrum, the acoustic image for each frequency bin stays approximately the same. For ultrasound imaging, we developed a time-delay based RCB and a time-reversal based RCB. We demonstrated the excellent performance of RCB and its various extensions by simulated and experimental examples.
Thesis:
Thesis (Ph. D.)--University of Florida, 2005.
Bibliography:
Includes bibliographical references.
General Note:
Printout.
General Note:
Vita.
Statement of Responsibility:
by Zhisong Wang.

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Copyright Zhisong Wang. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ROBUST ADAPTIVE ARRAY PROCESSING: THEORY AND APPLICATIONS


By
ZHISONG WANG















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA


2005


























To my parents and my wife.














ACKNOWLEDGMENTS

I would like to sincerely thank my advisor (Prof. Jian Li) for her support, encouragement, inspiration, and patience in this research. I am greatly indebted to her for introducing me to the area of robust adaptive array signal processing, and for her guidance throughout the development of this dissertation. My special appreciation is due to Prof. Toshikazu Nishida, Prof. Mark Sheplak, and Prof. Louis N. Cattafesta III for serving on my supervisory committee, and for their valuable guidance and constructive comments. I am deeply grateful to Prof. Petre Stoica for his advice, comments, and suggestions.

I am deeply thankful to my parents (Lin Wang and Yuyue Li), my sisters, and my brothers-in-law for their endless love, constant encouragement, and support. I sincerely thank my wife (Yu Chen) for her love, inspiration, patience, and care. I am especially thankful to my father who was unable to see the completion of my study.
I gratefully acknowledge Dr. Jianhua Liu, Dr. Guoqing Liu, Dr. Xi Li, and Dr. Renbiao Wu for their help during this work. I wish to thank other fellow graduate students in the SAL group with whom I had the great pleasure of interacting. Finally, I would like to thank all of the people who helped me during my Ph.D. study.















TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . iii

ABSTRACT . vi

CHAPTER

1 INTRODUCTION . 1

1.1 Background on Array Signal Processing . 1
1.1.1 D ata M odel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Uniform Linear Array . 4 1.1.3 Classification . 5 1.1.4 Beamforming . 6
1.2 Motivation for Robust Adaptive Beamforming . 7 1.3 Scope of the W ork . 9
2 ROBUST CAPON BEAMFORMING (RCB) . 10

2.1 Problem Formulation . 10 2.2 Standard Capon Beamforming . 13
2.2.1 Spatial Filtering SCB . 13 2.2.2 Covariance Fitting SCB . 14
2.3 Robust Capon Beamforming .15
2.3.1 Non-Degenerate Ellipsoidal Uncertainty Set . 15 2.3.2 Flat Ellipsoidal Uncertainty Set . 22
2.4 Diagonal Loading Interpretation of RCB . 25
2.4.1 Nondegenerate Ellipsoidal Uncertainty Set . 26 2.4.2 Flat Ellipsoidal Uncertainty Set . 26
2.5 Numerical Examples . . 27

3 APPLICATION OF RCB TO AEROACOUSTICS . 39

3.1 Introduction . . 39 3.2 Data Model and Problem Formulation of Acoustic Imaging . .41 3.3 Constant-Powerwidth RCB . 44 3.4 Constant-Beamwidth RCB . 45 3.5 Numerical Examples . . 46

4 APPLICATION OF RCB TO ULTRASONICS . 57

4.1 Introduction . 57 4.2 Problem Formulation . 60 4.3 Time-Delay Based RCB . . 64









4.4 Time-Reversal Based RCB . 68 4.5 Simulated and Experimental Examples . . 72 5 CONCLUSIONS AND FUTURE WORK . 90

5.1 Conclusions . . 90 5.2 Future Work . . 91 APPENDIX

A LINKING RCB TO VOROBYOV AND COLLEAGUES' APPROACH . 93 B CALCULATING THE STEERING VECTOR . 96 C LINK BETWEEN RCB AND LORENZ AND BOYD'S APPROACH . 97 REFERENCES . 100 BIOGRAPHICAL SKETCH . 108














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ROBUST ADAPTIVE ARRAY PROCESSING: THEORY AND APPLICATIONS By

Zhisong Wang

May 2005

Chair: Jian Li
Major Department: Electrical and Computer Engineering

Adaptive beamformers have better resolution and much better interference rejection capability than the data-independent beamformers, provided that the array steering vector corresponding to the signal of interest is accurately known. However, the adaptive beamforming techniques rely significantly on the assumptions of the incident signals, the sensor array, and the surroundings; and are very sensitive to the inevitable mismatch between the assumptions and the actual characteristics. Consequently, adaptive beamformers often suffer from severe performance degradations in practical applications.

We developed a novel robust Capon beamformer (RCB) that can maintain both the robustness of the data-independent beamformers and the adaptivity of the standard Capon beamformer (SCB) (a type of the adaptive beamformer). We showed that RCB belongs to the class of diagonal loading approaches. However, the amount of diagonal loading can be precisely calculated based on the uncertainty set of the array steering vector. We showed how the proposed robust Capon beamformer can be efficiently computed at a cost comparable to that of SCB. We also provided deep insights into the relationships among the recent three robust adaptive beamformers.








Although the robust adaptive beamformers obtained from the two different formulations appear to be rather different, we found that they are equivalent in terms of the signal-to-interference-plus-noise ratio (SINR). However, RCB also has some unique features. By comparing these three beamformers for degenerate vs non-degenerate ellipsoidal uncertainty sets of the steering vector, we showed the impressive advantages of RCB over the other two methods: simpler implementation, lower computational complexity, and more accurate power estimation. In addition, we applied RCB to various applications, and extended RCB to accommodate their specific goals and requirements. For the application of aero-acoustic imaging, we devised a constantbeamwidth RCB and a constant-powerwidth RCB for consistent acoustic imaging. This means that for an acoustic wideband monopole source with a flat spectrum, the acoustic image for each frequency bin stays approximately the same. For ultrasound imaging, we developed a time-delay based RCB and a time-reversal based RCB. We demonstrated the excellent performance of RCB and its various extensions by simulated and experimental examples.















CHAPTER 1
INTRODUCTION

Array signal processing [1]-[8] deals with the problem of extracting information (the number of sources, the signal waveforms, their spatial locations, etc.) from the measurements taken from an array of spatially distributed sensors in the propagating wavefield. The wavefield can be electromagnetic, acoustic, or seismic. Sensor arrays have been extensively used in many diverse applications including radar, sonar, acoustics, communications, speech processing, seismology, astronomy, and medical imaging.

The main fields of array signal processing include source localization, source waveform estimation, source intensity estimation, and channel estimation. The objectives for different applications are not the same. For example, in radar, the locations of the sources are the most important. In communications, the information-bearing waveforms are most critical. In acoustic measurements, people are concerned with the sound pressure levels or intensity levels of the sources. In seismic applications, the interest lies in the seismic structure, which can be determined by examining the the space-time propagation effects between the sources and the array. In ultrasound imaging, the reconstructed tissue characteristics and image contrast are crucial for diagnosing illnesses.

1.1 Background on Array Signal Processing
1.1.1 Data Model

Because of the different configurations, characteristics, and goals in the distinct applications, the associated array signal processing problems can be very different. We use a narrowband far-field data model to show some concepts of array signal









processing. Wideband signals in the near-field are discussed later. A signal is narrowband if the propagation time across the array is small compared to the inverse of its bandwidth. In [9], the following three conditions are given to determine whether a source is in the far-field.
2D 2
2D> (1.1)


r > D (1.2)

r > A (1.3)

where r is the range from the source, and D is the aperture of the array.

Consider an array comprising M sensors. Assume that there are J narrowband sources in the far-field. We assume that the data and weights are complexvalued, since in many applications a quadrature receiver is used to generate in-phase and quadrature signals, which are combined to yield the complex baseband signals. Assume that the output of each sensor contains the superposition of the J planar waveforms and additive sensor noise. Then the output of the ith sensor is given by

J
Xm(t) E- am(Oj)sj(t) + nm(t) (1.4)
j=1

where sj(t) denotes the jth emitter signal, Oj is the direction of arrival (DOA) of sj(t), am(Oj) is a complex scaler representing the propagating delay of sj(t) and the gain and phase adjusted by the mth sensor, and nm(t) is the additive noise. In matrix notation, we have

x(t) = [a(01) . a(9j)] s(t) + n(t) = A(O)s(t) + n(t) (1.5)

where x(t) = [x(t) . XM(t)]T is the data vector at the time t, s(t) = [s,(t) . sj(t)]T is the signal vector, n(t) = [nx (t) . nM(t)]T is the noise vector, 0 is a J-dimensional parameter vector corresponding to the true DOAs, and a(Oj) = [ai(Oj) . aM(Oj)]T is the array steering vector representing the array response to a unit wavefront from









the direction 9j. The collection of the array steering vectors over the parameter space of interest, {a(0j)10j E e}, is often called the array manifold.

The array steering vector a(0j) can be written as
a(03) = [Hi(Wc)eiwcrl . HM(wc)e-iwcrM]T (1.6)



where Hm(wc) is a complex scalar representing the gain and phase at the mth sensor, Tm denotes the time needed for the wave to travel from a reference point to the ith sensor, wc is the center or carrier frequency of the signal. If the sensors are identical and omnidirectional over the DOA range of interest, and the first sensor of the array is chosen as the reference point, we can obtain the following simplified form for the steering vector:

a(Oj) = [1 e-iwT2 . e-icTM]T (1.7)

The theoretical covariance matrix of the array output vector is defined as

R = E{x(t)x*(t)} (1.8)

where E{.} is the expectation operator. In practice, the theoretical covariance matrix R is not known. Hence a sample covariance matrix needs to be estimated based on the available data samples. Assume that the array output is sampled at N time instants, and these snapshots are collected in the columns of an M x N data matrix XN,

XN = [x(1) . x(N)] = A(O)SN + NN (1.9)

Then the sample covariance matrix can be obtained by R= NXNXv (1.10)
N XX

which is the maximum likelihood estimate of the theoretical covariance matrix. To make A nonsingular, at least M snapshots are required. The more snapshots, the better the sample covariance matrix approximates the theoretical covariance matrix.









The cost, however, is longer data acquisition time and more computation. Given only a finite number of snapshots, undesirable phenomena such as beampattern distortion and SNR loss may occur.

1.1.2 Uniform Linear Array

If an array consists of several identical sensors uniformly spaced on a line, it is called a uniform linear array (ULA). Let d denote the distance between two neighboring sensors, and let 0 denote the DOA of the signal counterclockwise from the broadside of the array. If we choose the first sensor of the ULA as the reference point, we have
(m dsinO (1.11)
c

where c is the wave propagating velocity. Let A be the signal wavelength. Then we have

A = f" (1.12)
~2

Define
fs = f dsinO (1.13)
c
w, = 2drfs = WC (1.14)
C

where w8 is referred to as spatial frequency. Inserting (1.14) into (1.11), and then into (1.7) gives

a(w,) = [1 e-i" . e-i(M-1)w']T (1.15)

Note that the steering vector in (1.15) is directly analogous to the tapped delay line vector used for an FIR filter. In fact, a beamformer based on the data collected at an array of sensors is a spatial filter analogous to a temporal FIR filter based on the data collected at one sensor. The vector a(wS) is uniquely defined if and only if the following condition is satisfied:


IWsI < 7r (.6


(1.16)






5

which is equivalent to

dl sinol _ A/2 (1.17)

which is satisfied if

d < A/2 (1.18)

For a ULA, if the intersensor spacing d is larger than A/2, spatial aliasing (grating lobes in the beampattern) will occur. On the other hand, the spatial resolution of a beamformer is inversely proportional to the array aperture. Hence, it is conventional to choose

d = A/2 (1.19)

For the sake of simplicity, we consider a ULA with half-wavelength intersensor spacing in the numerical examples in Chapter 3. The algorithms, however, can be applied to arbitrary arrays.

1.1.3 Classification

The array signal processing methods are of two kinds: nonparametric and parametric. The nonparametric method includes delay-and-sum beamforming [4], [5], standard Capon beamforming [10]-[12], robust adaptive beamforming (e.g., the robust Capon beamforming approaches [13]-[15]). The parametric method includes MUSIC [16]-[23], ESPRIT [24]-[28], maximum likelihood (ML) [29]-[32], and weighted subspace fitting (WSF) [33], [34]. The delay-and-sum beamforming, standard Capon beamforming, robust Capon beamforming, and MUSIC are discussed later.

By definition, a nonparametric method makes no assumption of the covariance structure of the data. It only uses the array covariance matrix and the array steering vector. In contrast, a parametric method requires knowledge of the model in (1.5). Furthermore, the noise n(t) is assumed to be spatially white, with its components having identical variances. Then we obtain


E{n(t)n"(t)} = cy.I


(1.20)









where u0 is the noise power (variance). In addition, the signal covariance matrix


S = E{s(t)s*(t)} (1.21)


is often assumed to be nonsingular, which means that the signals may be partially correlated but not coherent. Then the theoretical covariance matrix of the array output vector can be written as


R = E{x(t)x*(t)} = A(O)SA*(0) + a2I (1.22)


If the narrowband signals are uncorrelated, S is a diagonal matrix and
J
R= E a(j)a*(Oj) + aI (1.23)
j=1

where ({a3}J=) are the powers of the J signals impinging on the array. In the presence of coherent signal and interferences, S is a singular matrix.

1.1.4 Beamforming

A beamformer is a processor that applies a weight vector on the output of an array of sensors to estimate the signal of interest in the presence of noise and interferences. The goal of beamforming is to perform spatial filtering, passing the signal of interest from a given direction (or location), and at the same time, suppressing the background noise and directional interferences [35]. Beamforming is an essential task in array signal processing with ubiquitous applications. Beamforming can be used for either transmitting or receiving signals. We considered beamforming only for the purpose of reception.

As its name implies, a data-independent beamformer selects its weight vector regardless of the incoming data. On the other hand, an adaptive (data-dependent) beamformer takes advantage of the incoming data and adjusts its weight vector as a function of the data. Standard data-independent beamformers include the delayand-sum approach and methods based on various data-independent weight vectors for









sidelobe control [5], [8]. Adaptive beamformers include the standard Capon beamformer, linearly constrained minimum variance beamformer, and their extensions. Adaptive beamformers use the block mode or online mode to update the weight vectors. The beamformers that exploit a known theoretical covariance matrix are referred to as optimum beamformers.

1.2 Motivation for Robust Adaptive Beamforming

The data-dependent Capon beamformer adaptively selects the weight vector to minimize the array output power, subject to the linear constraint that the signal-ofinterest (SOI) does not suffer from any distortion [10], [11]. The Capon beamformer has better resolution and much better interference rejection capability than the dataindependent beamformer, provided that the array steering vector corresponding to the SOI is accurately known.

However, in practice, the knowledge of the SOI steering vector is often imprecise, because of differences between the assumed signal arrival angle and the true arrival angle; or between the assumed array response and the true array response (array calibration errors). Whenever this happens, the Capon beamformer may suppress the SOI as an interference, which results in significantly underestimated SOI power and drastically reduced array output signal-to-interference-plus-noise ratio (SINR). Then the performance of the Capon beamformer may become worse than that of the standard data-independent beamformers [36], [37].

The same happens when the number of snapshots is relatively small (i.e., about the same as, or smaller than, the number of sensors). In fact, a close relationship exists between the cases of steering vector errors and small-sample errors [38] in the sense that the difference between the sample covariance matrix R (estimated from a finite number of snapshots) and the corresponding theoretical covariance matrix R can be viewed as due to steering vector errors.









Many approaches have been proposed during the past 3 decades to improve the robustness of the Capon beamformer. The literature on robust adaptive beamforming is extensive [8], [13], [15], [35], [39]-[45].

To account for array steering vector errors, additional linear constraints (including point and derivative constraints) can be imposed to improve the robustness of the Capon beamformer [46]-[49]. However, these constraints are not explicitly related to the uncertainty of the array steering vector. Moreover, for every additional linear constraint imposed, the beamformer loses one degree of freedom (DOF) for interference suppression. These constraints belong to the class of covariance matrix tapering approaches [50].

Subspace based adaptive beamforming methods [38], [51] require knowledge of the noise covariance matrix. Hence they are sensitive to imprecise knowledge of the noise covariance matrix and also to the array steering vector errors. Making these methods robust against array steering vector errors will not cure them of being sensitive to imprecise knowledge of the noise covariance matrix.

Diagonal loading (including its extended versions) has been a popular and widely used approach to improve the robustness of the Capon beamformer [2], [51][69]. One representative of the diagonal loading based approaches is the norm constrained Capon beamformer (NCCB), which uses a norm constraint on the weight vector to improve robustness against array steering vector errors, and control white noise gain [2], [52]-[55]. However, for NCCB and most other diagonal loading methods, it is not clear how to choose the diagonal loading level based on information about the uncertainty of the array steering vector.

Recently some methods with a clear theoretical background [13]-[15], [44], [45], [61], [62] (unlike early methods) make explicit use of an uncertainty set of the array steering vector. In [61], a polyhedron is used to describe the uncertainty set. Spherical and ellipsoidal (including flat ellipsoidal) uncertainty sets were considered in









[13]-[15], [44], [45]. The robust Capon beamforming approaches presented in [44], [45] coupled the spatial filtering formulation of the standard Capon beamformer (SCB) in [10] with a spherical or ellipsoidal uncertainty set of the array steering vector. We coupled the covariance fitting formulation of SCB in [12] with an ellipsoidal or spherical uncertainty set, to obtain a robust Capon beamformer (RCB) [13], [14].

Interestingly, the methods in [13], [14], [44], [45] turn out to be equivalent and to belong to the extended class of diagonal loading approaches, but the corresponding amount of diagonal loading can be calculated precisely based on the ellipsoidal uncertainty set of the array steering vector. However, our RCB in [13] is simpler and computationally more efficient than its equivalent counterparts; and its computational complexity is comparable to that of SCB. Moreover, our RCB gives a simple way of eliminating scaling ambiguity, when estimating the power of the desired signal. The approaches in [44], [45] did not consider the scaling ambiguity problem.

1.3 Scope of the Work

Our study focused on theory and applications of the RCB algorithm, which is a natural extension of the covariance fitting formulation of SCB to the case of uncertain array steering vectors. We derived RCB for two cases: non-degenerate ellipsoidal constraints on the steering vector; and flat ellipsoidal constraints. We presented a diagonal loading interpretation of RCB, and showed that its diagonal loading level can be calculated precisely based on the uncertainty set of the steering vector. We also provided insights into relationships among the recent three robust adaptive beamformers. In addition, we address several extensions of RCB for aero-acoustic and ultrasonic applications. We offer numerous simulated and experimental examples to demonstrate the efficacy of RCB and also its extensions. More applications and analyses on RCB can be found in [63]-[66].















CHAPTER 2
ROBUST CAPON BEAMFORMING (RCB) In this chapter, we first formulate the problem of interest. Then we briefly review two formulations of the standard Capon beamformer, namely the spatial filtering SCB and the covariance fitting SCB, and demonstrate their equivalence. Next, we focus on the robust Capon beamformer by considering two cases( i.e., the case of non-degenerate ellipsoidal constraints on the steering vector and the case of flat ellipsoidal constraints). These two cases are treated separately because of the differences in their detailed computational steps and in the possible values of the associated Lagrange multipliers. We will also provide a diagonal loading interpretation of RCB.

2.1 Problem Formulation

Consider an array comprising M sensors. Let R denote the theoretical covariance matrix of the array output vector. We assume that R > 0 (positive definite) has the following form:
K
R 2 2
a0a0a0 + S Uaka* + Q (2.1)
k=1

where (W, {u2}=l) are the powers of the (K + 1) uncorrelated signals impinging on the array, (ao, {ak}K l) are the so-called steering vectors that are functions of the location parameters of the sources emitting the signals (e.g., their directions of arrival (DOAs)), (.)* denotes the conjugate transpose, and Q is the noise covariance matrix (the "noise" comprises nondirectional signals; and hence Q usually has full rank as opposed to the other terms in (2.1) whose rank is equal to one). We assume that the first term in (2.1) corresponds to the signal-of-interest (SOI), and that the remaining









rank-one terms correspond to K interferences. We assume that


[ao112 = M (2.2)

where 11 denotes the Euclidean norm. We note that the above expression for R holds for both narrowband and wideband signals. For narrowband signals, R is the covariance matrix at the center frequency. For wideband signals, R is the covariance matrix at the center of a given frequency bin. Let


R U U* (2.3)


where the columns of U contain the eigenvectors of R; and the diagonal elements of the diagonal matrix r, -yi _ y2 . _yM, are the corresponding eigenvalues. In practical applications, R is replaced by the sample covariance matrix A, where
N

-E EYnY* (2.4)
n=1
N is the number of snapshots and Yn is an M x 1 vector representing the nth snapshot with the form

Yn = aoso(n) + en, (2.5)

with so(n) denoting the waveform of the SOI, and en being the interference-plusnoise vector for the nth snapshot. In the radar application, the SOI-free data vectors are available, so the interference-plus-noise covariance matrix lin may be estimated and used for beamforming. However, in many other applications, the SOI is always present in the data. Hence we cannot exploit the interference-plus-noise covariance matrix. Instead, we design beamformers based on R. Throughout our study, we considered this more general circumstance.

The robust adaptive beamforming problem we will deal with in this chapter is as follows: extend the Capon beamformer so as to be able to accurately determine the power of SOI, even when knowledge of its steering vector, a0, is imprecise. More









specifically, we assume that our only knowledge about a0 is that it belongs to an uncertainty ellipsoid. The non-degenerate ellipsoidal uncertainty set has the following form

{a I (a- fa)* C (a- a) 1} (2.6)

where d (the assumed steering vector of SOI) and C (a positive definite matrix) are given. The ellipsoidal uncertainty set above is equivalent to {aIa Cu+d, IIull _ i} (2.7)


where C2 is the Hermitian square root matrix of C (i.e., C1(C1)* - C). If we assume that u is a circularly symmetric random vector, with a mean of zero and covariance matrix equal to identity matrix I, then we can prove the following results: E [a] = a (2.8)

and

E [(a - d)(a - d)] = C (2.9)

The ellipsoidal uncertainty set may be generated from the mean and covariance matrix of the array steering vector, calculated based on the measurements from repeated trials in the experiments. Even without the above statistical assumptions, we can still exploit semidefinite programming to obtain a tight ellipsoidal uncertainty set, the minimum volume ellipsoid covering all the samples of the array steering vectors [45]. The case of a flat ellipsoidal uncertainty set is considered in Section 2.3.2.

In particular, if C in (2.6) is a scaled identity matrix, (i.e., C = JI), we have the following uncertainty sphere:

{a I Ia- dll2 < 6} (2.10)

where c is a user parameter that can be calculated based on a priori knowledge of the array steering vector. For example, if we know the varying ranges of the gain, phase,









and position errors of each sensor, we can determine the corresponding uncertainty region, which is an annulus sector. To illustrate this point, we use a uniform linear array. Assume that at the mth sensor, the uncertainty is caused by gain error rm, phase error 7mi, and element position error AXm and Ay,m. Thus we have am = dm(1 + rm)eipm, m = 1,.,M (2.11)

where the total phase error Pm is Axm sin 0 + Aym cos 0 (2.12)
Pm = m � Wc (.2
C

with 0 being the signal direction of arrival, wc representing the carrier frequency, and c denoting the wave propagating velocity. Assume that the varying ranges for rm, 7m, Ax,, Aym are known. Then we can determine the uncertainty region Qm for am and the maximum total error em, which is given by em= max lam- mI, m= ,.,M (2.13)
amEnm

Hence we can choose c as follows:
M
E l 12 (2.14)
m=1
From an application standpoint, RCB makes it possible to avoid the time-consuming array calibration process, provided that the steering vector uncertainty set is known.

Our study focused on estimating the SOI power a0 from R (or more practically R) when the knowledge of a0 is imprecise. However, our beamforming approaches we present herein can also be used for other applications, such as signal waveform estimation [44], [45].

2.2 Standard Capon Beamforming
2.2.1 Spatial Filtering SCB

The common formulation of the beamforming problem that leads to the spatial filtering form of SCB [5], [10], [11] is as follows.









(a) Determine the M x 1 weight vector w0 that solves the following linearly constrained quadratic problem: minw*Rw subject to w*ao 1 (2.15)
w

(b) Use w*Rw0 as an estimate of o2.

The solution to (2.15) is easily derived:

w0 a=R-lao (2.16)

Using (2.16) in Step (b) yields the following estimate: &2 a a-=a0 (2.17)

Note that (2.15) can be interpreted as an adaptive spatial filtering problem: given R and ao, we wish to determine the weight vector w0 as a spatial filter that can pass SOI without distortion; and at the same time minimize undesirable interference and noise contributions in R.

2.2.2 Covariance Fitting SCB

The Capon beamforming problem can also be reformulated into a covariance fitting form. To describe the details of our approach, we first proved that a2 in (2.17) is the solution to the following problem [12], [14]: maxa2 subject to R- U2aoa > 0 (2.18)

where the notation A > 0 (for any Hermitian matrix A) means that A is positive semi-definite. The previous claim follows from the following readily verified equivalences (here R-1/2 is the Hermitian square root of R-1): R - 2a0a* > 0 �


I - a2R1/2aoa*R12 > 0 '::

1 - a2aR-lao > 0 <=>

or < a*RUIao = Oo (2.19)
0 a0









Hence a2 = &0 is indeed the largest value of a2 that satisfies the constraint in (2.18).

Note that (2.18) can be interpreted as a covariance fitting problem: given R and a0 we wish to determine the largest possible SOI term (o2aoa*) that can be a part of R, under the natural constraint that the residual covariance matrix be positive semi-definite.

2.3 Robust Capon Beamforming The robust Capon beamformer is derived by a natural extension of the covariance fitting SCB in Section 2.2.2 to the case of uncertain steering vector. In doing so we directly obtain a robust estimate of a, without any intermediate calculation of a vector w [13], [14].

2.3.1 Non-Degenerate Ellipsoidal Uncertainty Set

When the uncertainty set of the steering vector a is a non-degenerate ellipsoid as in (2.6), the RCB problem has the following form [13], [14]:

max a2 subject to R - a2aa* > 0

for any a satisfying (a - a)* C-1 (a - a) < 1 (2.20)

where a and C are given.

The RCB problem in (2.20) can be readily reformulated as a semi-definite program (SDP) [14]. Indeed, using a new variable X = 1/a2 along with the standard technique of Schur complements [5], [67] we can rewrite (2.20) as: min X subject to
xja

a*


[(a ) - a -Y ] > 0 (2.21)

The constraints in (2.21) are so-called linear matrix inequalities, and hence (2.21) is an SDP, which requires O(oM6) flops if the SeDuMi type of software [68]









is used to solve it, where o is the number of iterations, usually on the order of v/-. However, the approach we present below only requires O(M3) flops.

For any given a, the solution &0 to (2.20) is indeed given by the counterpart of (2.17) with a0 replaced by a, as shown in Section 2.2.2. Hence (2.20) can be reduced to the following problem

mina*R-la subject to (a - d)* C-1 (a - a) < 1 (2.22)
a

To exclude the trivial solution a = 0 to (2.20), we assume that


*C-la > 1 (2.23)

Note that we can decompose any matrix C > 0 in the form:


C-1 = 1D*D (2.24)
E

where for some E > 0,

D = v/C-1/2 (2.25)

Let

=Da, ai D = IR=DRD* (2.26)

Then (2.22) becomes

minA*Al%- subject to - a2 < (2.27)

Hence without loss of generality, we will consider solving (2.22) for C = EI, i.e., solving the following quadratic optimization problem under a spherical constraint: mina*R-la subject to Ila - i112 < E (2.28)
a

To exclude the trivial solution a = 0 to (2.28), we now need to assume that


Iall2 > (


(2.29)








Let S be the set defined by the constraints in (2.28). To determine the solution to (2.28) under (2.29), consider the function: hi(a, v) = a*R-la + v (la - 2- 2 (2.30)

where v > 0 is the real-valued Lagrange multiplier satisfying R + vI > 0 so that the above function can be minimized with respect to a. Evidently we have hi(a, v) < a*R-ia for any a E S with equality on the boundary of S. Equation (2.30) can be written as
hi(a, v)= a -(R-1-+-I) -1ic]*(R-1i+vI)[a -(R-1+I-V a (R-1 + vI)-a + va*a - vc (2.31)

Hence the unconstrained minimization of h, (a, v) w.r.t. a, for fixed v, is given by i = - + a (2.32)

= .-(I + vR)-i a (2.33)

where we have used the matrix inversion lemma [5] to obtain the second equality. Clearly, we have

h2(v) = h,(do, v) - v2 *(R1 � vI)-a+ va - (2.34)

< a*R-1a for any a E S (2.35)

Maximization of h2(V) with respect to v gives V + 1) (2.36)


which indeed satisfies

Ildo - d =1 (2.37)

Hence do belongs to the boundary of S. Therefore, do is the sought solution.









Using (2.33) in (2.37), the Lagrange multiplier v > 0 is then obtained as the solution to the constraint equation:

h2(V) A (I + vR)-l 2 = (2.38)

Making use of R = UPU* and z = U'a, (2.38) can be written as h2(V) Z i2 (2.39)


Note that h2 (v) is a monotonically decreasing function of v > 0. According to (2.29) and (2.38), h2(0) > E and hence v 0 0. From (2.39), it is clear that lim ,+ h2(V) = 0 < c. Hence there is a unique solution v > 0 to (2.39). By replacing the -ym in (2.39) with -M and -y, respectively, we can obtain the following tighter upper and lower bounds on the solution v > 0 to (2.39): Hall - < V < Hall - (2.40)


By dropping the 1 in the denominator of (2.39), we can obtain another upper bound on the solution v to (2.39):

< M Zm12 2(2.41) The upper bound in (2.41) is usually tighter than the upper bound in (2.40) but not always. In summary, the solution v > 0 to (2.39) is unique and it belongs to the following interval:

Hl-Ve < v < min{ ( M.IZ,122 Ha VI} (2.42)



Once the Lagrange multiplier v is determined, do is determined by using (2.33) and &0 is computed by using (2.17) with a0 replaced by Ao. Hence the major computational demand of our RCB comes from the eigen-decomposition of the Hermitian matrix R, which requires O(M3) flops. Therefore, the computational complexity of our RCB is comparable to that of the SCB.









Next observe that both the power and the steering vector of SOI are treated as unknowns in our robust Capon beamforming formulation (see (2.20)), and hence that there is a "scaling ambiguity" in the SOI covariance term in the sense that (a2, a) and (U2/a, a'/2a) (for any a > 0) give the same term U2aa*. To eliminate this ambiguity, we use the knowledge that Hla 112 = M (see (2.2)) and hence estimate o as [14]

UOo = &2llioll/M (2.43)

where &2 is obtained via replacing a0 in (2.17) by fo in (2.32). The numerical exam2 &
ples in [14] confirm that a0 is a (much) more accurate estimate of ao than &0. To summarize, our proposed RCB approach consists of the following steps. The Proposed RCB (spherical constraint) Step 1: Compute the eigen-decomposition of R (or more practically of fr). Step 2: Solve (2.39) for v, e.g., by a Newton's method, using the knowledge that

the solution is unique and it belongs to the interval in (2.42). Step 3: Use the v obtained in Step 2 to get f0 = a - U (I + vr)-1 u-i (2.44)


where the inverse of the diagonal matrix I + vr is easily computed. (Note that

(2.44) is obtained from (2.33).)

Step 4: Compute &2 by using

&0 = 1 2 (2.45)
i*Ur (V-2I + 2v-lr + r) -1u*a where the inverse of v-21 + 2v-IT + r2 is also easily computed. Note that a0 in (2.17) is replaced by do in (2.32) to obtain (2.45). Then use the &0 in (2.43)
2
to obtain uai0 as the estimate of o0r.









We remark that in all of the steps above, we do not need to have Y,, > 0 for all m = 1, 2,., M. Hence R or R can be singular, which means that we can allow N < M to compute Rt.

In other applications, such as communications, the focus is on SOI waveform estimation. Let so(n) denote the waveform of the SOI. Then once we have estimated the SOI steering vector with our RCB, so(n) can be estimated like in the SCB as follows:

90(n) = **x, (2.46)

where a0 in (2.32) is used to replace a0 in (2.16) to obtain o: v0 Rfi� (2.47)

(R+ -)1a (2.48)
d* (R + 4) -1 R (R + I -a

Note that our robust Capon weight vector has the form of diagonal loading except for the real-valued scaling factor in the denominator of (2.48). However, the scaling factor is not really important since the quality of the SOI waveform estimate is typically expressed by the signal-to-interference-plus-noise ratio (SINR) SINR = u~j'ra�o2 (2.49)
w (zk=1 Uaka* + Q wO

which is independent of the scaling of the weight vector.

When C is not a scaled identity matrix, the diagonal loading is added to the weighted matrix A defined in (2.26) and we refer to this case as extended diagonal loading. To exclude the trivial solution a = 0 to (2.20), we now need to assume, like in (2.29), that

115112 >(2.50)

which is equivalent to
5*c-lii > 1 (2.51)









The discussions above indicate that our robust Capon beamforming approach belongs to the class of (extended) diagonally loaded Capon beamforming approaches. However, unlike earlier approaches, our approach can be used to determine exactly the optimal amount of diagonal loading needed for a given ellipsoidal uncertainty set of the steering vector, at a very modest computational cost.

Our approach is different from the recent RCB approaches in [44], [45]. The latter approaches extended Step (a) of the spatial filtering SCB in Section 2.2.1 to take into account the fact that when there is uncertainty in a0, the constraint on w*ao in (2.15) should be replaced with a constraint on w*a for any vector a in the uncertainty set (the constraints on w*a used in [44] and [45] are different from one another); then the so-obtained w is used in w*Rw to derive an estimate of a., as in Step (b) of the spatial filtering SCB. Unlike our approach, the approaches of [44] and [45] do not provide any direct estimate A0. Hence they do not provide a simple way (such as (2.43)) to eliminate the scaling ambiguity of the SOI power estimation that is likely a problem for all robust beamforming approaches (this problem was in fact ignored in both [44] and [45]). Yet SOI power estimation is often the main goal in many applications including radar, sonar, acoustic and medical imaging.

Despite the apparent differences in formulation, we prove in Appendices A and C that our RCB gives the same weight vector as the approaches presented in [44], [45], yet our RCB is computationally more efficient [13]. The approach in [44] requires O(QM3) flops [69], where p is the number of iterations, usually on the order of VrM7A, whereas our RCB approach requires O(M3) flops. Moreover, our RCB can be readily modified for recursive implementation by adding a new snapshot to A and possibly deleting an old one. By using a recursive eigen-decomposition updating method [70], [71] with our RCB, we can update the power and waveform estimates in O(M2) flops. No results are available so far for efficiently updating the second-order cone program (SOCP) approach in [44]. The approach in [45] can be implemented









recursively by updating the eigen-decomposition similarly to our RCB. However, its total computational burden can be higher than for ours, as explained in the next subsection.

We also show in Appendix B that, although this aspect was ignored in [44], [45], the approaches presented in [44], [45] can also be modified to eliminate the scaling ambiguity problem that occurs when estimating the SOI power a0 [13].

2.3.2 Flat Ellipsoidal Uncertainty Set

When the uncertainty set of a is a flat ellipsoid, as is considered in [45] to make the uncertainty set as tight as possible (assuming that the available a priori information allows that), (2.20) becomes [13]

maxa2 subject to R- 2aa* > 0
a=Bu+ , Ilull <1 (2.52)


where B is an M x L matrix (L < M) with full column rank and u is an L x 1 vector. (When L = M, the second constraint in (2.52) becomes (2.6) with C = BB*.) Below we provide a separate treatment of the case of L < M due to the differences from the case of L = M in the possible values of the Lagrange multipliers and the detailed computational steps. The RCB optimization problem in (2.52) can be reduced to (see (2.22)):

min(Bu + d)*R-(Bu + d) subject to Ilull _< 1 (2.53)
U
Note that

(Bu + a)*R-l(Bu + a) = u*B*R-Bu + a*RlBu + u*B*R-a + a*R-& (2.54)

Let
1 = B*R-lB > 0 (2.55)

and


(2.56)









Using (2.54)-(2.56) in (2.53) gives
minu*Ru+a7*u+u*7 subject to lull 1 (2.57)
auul
U

To avoid the trivial solution a = 0 to the RCB problem in (2.52), we impose the following condition (assuming fi below exists, otherwise there is no trivial solution). Let fi be the solution to the equation Bfi+a= 0 (2.58)

Hence
i= -Bt (2.59)

Then we require that
d*Bt*Btfq > 1 (2.60)

where Bt denotes the Moore-Penrose pseudo-inverse of B.
The Lagrange multiplier methodology can again be used to solve (2.52) [72]. Let
hi (u, 0) = u*ltu + A*u + u* A + P(u*u - 1) (2.61)

where 0 > 0 is the Lagrange multiplier [73]. Differentiation of (2.61) with respect to u gives
lAfi + A7 + of, = 0 (2.62)

which yields

fi = -(A + I)-A (2.63)

If I1 -'A11 < 1, then the unique solution in (2.63) with 0 = 0, which is fi = -l-la, solves (2.57). If I1-'A11 > 1, then f > 0 is determined by solving h2 (0) I(f:t + PI) - 2 = 1 (2.64)

Note that h2(0) is a monotonically decreasing function of 0 > 0. Let


(2.65)


A = Ott*









where the columns of U contain the eigenvectors of R and the diagonal elements of the diagonal matrix f, 'i > 9Y > "'" _ 'L, are the corresponding eigenvalues. Let a *(2.66) and let it denote the lth element of 2. Then

1 1 ,12
Z I) 2 - 1 (2.67)


Note that lim h2(i ) = 0 and h2(0) = IRtIA7II > 1. Hence there is a unique solution to (2.67) between 0 and oo. By replacing the 5' in (2.67) with L and 1 respectively, we obtain tighter upper and lower bounds on the solution to (2.67):

11a11- a _< _I - (2.68)

Hence the solution to (2.67) can be efficiently determined by using, for example, the Newton's method, in the above interval. Then the solution 0 to (2.67) is used in (2.63) to obtain the fi that solves (2.57).
To summarize, our proposed RCB approach consists of the following steps. The Proposed RCB (flat ellipsoidal constraint) Step 1: Compute the inverse of R (or more practically of A) and calculate A and
a using (2.55) and (2.56), respectively.

Step 2: Compute the eigen-decomposition of R (see (2.65)). Step 3: If 111-14[I < 1, then set 0 = 0. If 111t-1A11 > 1, then solve (2.67) for 0, e.g.,
by a Newton's method, using the knowledge that the solution is unique and it
belongs to the interval in (2.68).

Step 4: Use the 0 obtained in Step 3 to get


f = -(6 () + I)-1*A


(2.69)









(which is obtained from (2.63)). Then use the fi to obtain the optimal solution

to (2.52) as
do = Bfi + d (2.70)
Step 5: Compute 82 by using (2.17) with a0 replaced by do and then use the &2 in


(2.43) to obtain the estimate of a02.

Hence, under the flat ellipsoidal constraint the complexity of our RCB is also O(M3) flops, which is on the same order as for SCB and is mainly due to computing R-1 and the eigen-decomposition of A. If L < M, then the complexity is mainly due to computing R-1. Note, however, that to compute P, we need O(L3) flops while the approach in [45] requires O(M3) flops (and L < M).

2.4 Diagonal Loading Interpretation of RCB

In many applications, such as in communications or the global positioning system, the focus is on SOI waveform estimation. The waveform of the SOI, s0(n), as in (2.5) can be estimated as follows: g0(n) = (2.71)

where *� is the corresponding weight vector. For RCB, we can substitute the estimated steering vector ao in lieu of a0 in (2.16) to obtain w.
Diagonal loading is a popular approach to mitigate the performance degradations of SCB in the presence of steering vector error or the small sample size problem. As the name implies, its weight vector has a diagonally loaded form: w = ,(R + RI)-1 (2.72)

where 6 denotes the diagonal loading level. Also, in (2.72) K is a scaling factor, which can be important for accurate power estimation; however, it is immaterial for waveform estimation since the quality of the SOI waveform estimate is typically









measured by the signal-to-interference-plus-noise ratio (SINR) SINR = o,0 r*ao12 (2.73)
(K 0.2
k=l aka + Q)

which is independent of ii.

As a matter of fact, RCB can be interpretted in the unified framework of diagonal loading based approaches.

2.4.1 Nondegenerate Ellipsoidal Uncertainty Set

Using do in (2.32) to replace a0 in (2.16), we can obtain the following RCB weight vector for the case of non-degenerate ellipsoidal uncertainty set: RRCB+-J- (2.74)
d" (R+ )I)-1R (R+ I)-( When C is not a scaled identity matrix, the diagonal loading is added to the weighted matrix 1 defined in (2.26) instead of R and we refer to this case as the extended diagonal loading.

2.4.2 Flat Ellipsoidal Uncertainty Set

The RCB weight vector for the case of flat ellipsoidal Uncertainty set has the form:

W RCBF = -(R + BB*) - 1 (
0 (2.75)
d* (Ra+ -IBB') -1R (R + -IBB*)To obtain (2.75) we have used the fact (also using (2.63) in (2.70)) that

R-, =o -R-1B(R + PI)-1 + R-1a

- _R-B(B*R-B + PI)-B*R-ld + R-ld R + (� BB* a (2.76)

where the last equality follows from the matrix inversion lemma. We see that in this case, the RCB weight vector again has an extended diagonally loaded form.









Despite the differences in the formulation of our RCB problem and that in [45], we can prove that the WRCB.F in (2.75) and the optimal weight in [45] are identical (Appendix C).

2.5 Numerical Examples

Next, we provide numerical examples to compare the performances of the SCB and RCB. In all of the examples considered below, we assume a uniform linear array with M = 10 sensors and half-wavelength sensor spacing, and a spatially white Gaussian noise whose covariance matrix is given by Q = I.

Example 2.1: Comparison of SCB and RCB for the case of finite number of snapshots without look direction errors. We consider the effect of the number of snapshots N on the SOI power estimate when the sample covariance matrix 1A in (2.4) is used in lieu of the theoretical array covariance matrix R in both the SCB and RCB. (Whenever Rt is used instead of R, the average power estimates from 100 Monte-Carlo simulations are given. However, the beampatterns shown are obtained using A from one Monte-Carlo realization only.) The power of SOI is 4 = 10 dB and the powers of the two (K = 2) interferences assumed to be present are o2 = a' = 20 dB. We assume that the steering vector uncertainty is due to the uncertainty in the SOI's direction of arrival 00, which we assume to be 00 � A. We assume that a(Oo) belongs to the uncertainty set

Ia(0o) - all2 < ; a = a(00 + A) (2.77)

where c is a user parameter. Let co = Ila(Oo) - i112. Then choosing e = co gives the smallest set that includes a(Oo). However, since A is unknown in practice, the E we choose may be greater or less than co. To show that the choice of c is not a critical issue for our RCB approach, we will present numerical results for several values of E. We assume that the SOI's direction of arrival is Oo = 0' and the directions of arrival of the interferences are 01 = 600 and 02 = 800.






28

12 I I I I I I I I I I I I I I

A 0 0 0 0
A
8) o 8) o
.E 2- 2W BU
o o
0 -2 . -2

-6- A RCB (Sample R) -6 A RCB (Sample R)
o SCB (Sample R) 0 SCB (Sample R)
-8 - ROB (Theoretical R) -8 - RCB (Theoretical R)
- - SCB ('heoretical RR j SOB corTaR
-10 I -110 I
10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100
Number of Snapshots Number of Snapshots
(a) (b)

Figure 2.1: Comparison of -0 (SCB using ft and R) and 0 (RCB using 1A and R) versus N. (a) c = 0.5. (b) e = 3.5. The true SOI power is 10 dB and Eo = 0 (i.e., no mismatch).
2 22
In Figure 2.1, we show uo6 and &o versus the number of snapshots N for the no mismatch case; hence A = 0 in (2.77) and consequently EO = 0. Note that the power estimates obtained by using R approach those computed via R as N increases, and that our RCB converges much faster than the SCB. The SCB requires that N is greater than or equal to the number of array sensors M = 10. However, our RCB works well even when N is as small as N = 2.

Figure 2.2 shows the beampatterns of the SCB and RCB using R as well as 1. with N = 10, 100, and 8000 for the same case as in Figure 2.1. Note that the weight vectors used to calculate the beampatterns of RCB in this example (as well as in the following examples) are obtained by using the scaled estimate of the array steering vector v/M60/f0iI in (2.16) instead of Ao. The vertical dotted lines in the figure denote the directions of arrival of the SOI and the interferences. The horizontal dotted lines in the figure correspond to 0 dB. Figure 2.2(a) shows that although the RCB beampatterns do not have nulls at the directions of arrival of the interferences as deep as those of the SCB, the interferences (whose powers are 20 dB) are sufficiently suppressed by the RCB to not disturb the SOI power estimation. Regarding the poor









29




















4 04 0I I I I

20 20- . . , .
,.- . . ., " . ." ' ,
0 - . . . . . . . . . . . . . . . . . . . . . . . i. . . . . . . . ,/ : , ,. .



=-20 - 8 -20E E
-40 -40
M C
<-60-1 <6


-80 -I -80 __________________________-10j I I I I I -10
-80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80
0 degree degree

(a) (b)
40 I I I I40


20- 20


M 0 . . . . . . M 0, . . . . ; .
E

E E
� -4 0- -4 0

-80 -80

-80 -80
-lG I I I I I I -100 I I I I I

-80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80
degree degree

(c) (d)


Figure 2.2: Comparison of the beampatterns of SCB and RCB when E = 3.5. (a) Using R. (b) Using R with N = 10. (c) Using R with N = 100. (d) Using A with N = 8000. The true SOI power is 10 dB and co = 0 (no mismatch).









,- . I I I . . I I . ,- . I I I I I I I .
A A
8 -AB
V
0 4 4(5a
EE
2-.
0 0 0 0 3: 0 0 0 0 0 0 0
0 0 00
a 2 , o -2 o
Co -4- -4-6 r RCB (Sample R) -6 A RCB (Sample R)
o SCB (Sample R) o SCB (Sample R)
-8 - RCB (Theoretical R) -8 - RCB (Theoretical R)
1 SCB ('heoretical A . SCB (Theoretical A
-I0 C p -10 C P I I I I
10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100
Number of Snapshots Number of Snapshots
(a) (b)

Figure 2.3: Comparison of &0 (SCB using R and R) and U0 (RCB using R and R) versus N. (a) c = 2.5. (b) c = 4.5. The true SOI power is 10 dB and fo = 3.2460 (corresponding to A = 2.00).


performance of SCB for small N, note that the error between R and R can be viewed

as due to a steering vector error [38].

Example 2.2: Comparison of SCB and RCB for the case of finite

number of snapshots in the presence of look direction errors. This example

is similar to Example 2.1 except that now the mismatch is A = 20 and accordingly

Eo = 3.2460. We note from Figure 2.3 that even a relatively small A can cause a

significant degradation of the SCB performance. As can be seen from Figure 2.4, the

SOI is considered to be an interference by SCB and hence it is suppressed. On the ,2
other hand, the SOI is preserved by our RCB and the performance of 60 obtained

via our approach is quite good for a wide range of values of c. Note that the RCB

also has a smaller "noise gain" than the SCB.

Example 2.3: Comparison of the RCB method and a fixed diagonal

loading level based approach. In Figure 2.5, we compare the performance of

our RCB with a fixed diagonal loading level based approach. Specifically, the fixed

loading level was chosen equal to 10 times the noise power (assuming the knowledge

of the noise power). Consider the same case as Figure 2.4(d) except that now we























20,
0 . . . . . , : . . . ,. . . . . . . . . . . . :. . . . . . . .
-201



-401

-60

-8o1
_10 C IB


-60 -30 0
0 degree

(b)


0 30 60 90 -'-0 -60 -30 0
0 degree 0 degree
(c) (d)


Figure 2.4: Comparison of the beampatterns of SCB and RCB when C = 1.0 when
(a) using R and (b) using A with N = 10, and when c = 4.5 for (c) using R and (d) using R with N = 10. The true SOI power is 10 dB and co = 3.2460 (corresponding to A = 2.00).


0 degree


30 60









assume that R is available and we vary the SNR by changing the SOI or noise power. For Figures 2.5(a), 2.5(c) and 2.5(e), we fix the noise power at 0 dB and vary the SOI power between -10 dB and 20 dB. For Figures 2.5(b), 2.5(d) and 2.5(f), we fix the SOI power at 10 dB and vary the noise power between -10 dB and 20 dB. Figures 2.5(a) and 2.5(b) show the diagonal loading levels of our RCB as functions of the SNR. Figures 2.5(c) and 2.5(d) show the SINRs of our RCB and the fixed diagonal loading level approach and Figures 2.5(e) and 2.5(f) show the corresponding SOI power estimates, all as functions of the SNR. Note from Figures 2.5(a) and 2.5(b) that our RCB adjusts the diagonal loading level adaptively as the SNR changes. It is obvious from Figure 2.5 that our RCB significantly outperforms the fixed diagonal loading level approach when the SNR is medium or high.

Example 2.4: Comparison of RCB, SCB and the delay-and-sum method in the presence of array calibration errors. We consider an imaging example, where we wish to determine the incident signal power as a function of the steering direction 0. We assume that there are five incident signals with powers 30, 15, 40, 35, and 20 dB from directions -35', -15', 0', 100, and 400, respectively. To simulate the array calibration error, each element of the steering vector for each incident signal is perturbed with a zero-mean circularly symmetric complex Gaussian random variable so that the squared Euclidean norm of the difference between the true steering vector and the assumed one is 0.05. The perturbing Gaussian random variables are independent of each other.

Figure 2.6 shows the power estimates of SCB and RCB, obtained using R, as a function of the direction angle, for several values of c. The small circles denote the true (direction of arrival, power)-coordinates of the five incident signals. Figure 2.6 also shows the power estimates obtained with the data-independent beamformer using the assumed array steering vector as the weight vector. This approach is referred to as the delay-and-sum beamformer. We note that SCB can still give good direction of













































SNR (dB) SNR

(a) (b)


-10 -5 0 5
SNR (dB)


5
SNR (dB)

(e)


10 15 20


A
15 A


0 0
CC 10 1 0


5
0


0



-0 -1 0 51 10 1 5 20
SNR (dB)

(d)

151 I I I

fA


0
0
n 0

E 0

w0
0-0
0
5 0
CI)
-5
Fixed diagonal loadn
A RCB


20 -10 -5 0 5
SNR (dB)


Mf


10 15 20


Figure 2.5: Comparison of a fixed diagonal loading level approach and our RCB when E = 4.5 and co = 3.2460 (corresponding to A = 2.00). (a) (c) (e) Signal power change.

(b) (d) (f) Noise power change.


0 ixddiagonal loaing RB


0Fxddiagona odn
A
A
A


A

0 0 0
00


'0


i i i i0


I I









-RCB -RCB
40 - . SCB 40- . . SCB
-Delay-and-sum - Delay-and-sumn
C - 305 30
S s * , 5 *

E 20 ,2 ti s , i , 55 5, 51
,,,' , , ',,, , .: ,,
Z55
10 - 3: 10
0 0
0 - V 0 - 555 5 5,

-10 -10
-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60
0 degree 0 degree
(a) (b)

Figure 2.6: Power estimates (using R) versus the steering direction 0. (a) e = 0.03.
(b) c = 0.1. The true powers of the incident signals from -350, -150, 00, 100, and 400 are denoted by circles, and c0 = 0.05. arrival estimates for the incident signals based on the peak power locations. However, the SCB estimates of the incident signal powers are way off. On the other hand, our RCB provides excellent power estimates of the incident sources and can also be used to determine their directions of arrival based on the peak locations. The delay-andsum beamformer, however, has much poorer resolution than both SCB and RCB. Moreover, the sidelobes of the former give false peaks.

Example 2.5: Comparison of SCB, RCB with spherical constraint and RCB with flat ellipsoidal constraint in the presence of look direction errors. We examine now the effects of the spherical and flat ellipsoidal constraints on SOI power estimation. We consider SO power estimation in the presence of several strong interferences. We will vary the number of interferences from K = 1 to K = 8. The power of SO is a2 = 20 dB and the interference powers are 2 2 =a 40

dB. The SOl and interference directions of arrival are 00 = 100, 1 = -75' 02 =

-60',03 = -45', 04 = -30', 05 = -10', 06 = 25',07 = 35',08 = 50'. We assume that there is a look direction mismatch corresponding to A = 20 and accordingly e0 = 3.1349.












-0 0
'615 - 15
1E E
(a a)
100- 3p10 w
o =
a.O

--- -- -- -- -cq 01 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
Number of Interferences Number of Interferences
(a) (b)

Figure 2.7: Comparison oy (SCB), UO (RCB with flat ellipsoidal constraint with L = 2), and setv (RCB with spherical constraint), based on R, versus the number of interferences K. (a) 6 = 1. 80. (b) 6 = 2.40. The true SOI power is 20 dB and c= 3.1349 (corresponding to A = 20).

Figure 2.7 shows the SOI power estimates, as a function of the number of interferences K, obtained by using SCB, RCB (with flat ellipsoidal constraint), and the more conservative RCB (with spherical constraint) all based on the theoretical array covariance matrix R. For RCB with flat ellipsoidal constraint, we let B contain two columns with the first column being a(Oo + A) - a(Oo + A - 3) and the second column being a(0o + A) - a(Oo + A + 3). Note that choosing 3 = A = 20 gives the smallest flat ellipsoid that this B can offer to include a(00). However, we do not know the exact look direction mismatch in practice. We choose 3 = 1.80 and 3 = 2.40 in Figures 2.7(a) and (b), respectively. For RCB with spherical constraint, we choose 6 to be the larger of Ila(Oo+A) -a(Oo+A-)12 and Ila(Oo+A) -a(Oo + A +3)112. Note that RCB with flat ellipsoidal constraint and RCB with spherical constraint perform similarly when K is small. However, the former is more accurate than the latter for large K.

Figure 2.8 gives the beampatterns of the SCB and RCBs using R as well as R with N = 10 for various K. For large K, the more conservative RCB with spherical constraint amplifies the SOI while attempting to suppress the interferences,


25 A' ROB (Flat ellipsoid)
-- RC (Sphere) --o- SCB - -a-----


25 R B (Flat elli psoid)
. . . . .(Sphere)
-0 C .-.









as shown in Figure 2.8. On the other hand, the RCB with flat ellipsoidal constraint maintains an approximate unity gain for the SOI and provides much deeper nulls for the interferences than the RCB with spherical constraint at a cost of worse noise gain. As compared to the RCBs, the SCB performs poorly as it attempts to suppress the SO. Comparing Figures 2.8(b) with 2.8(a), we note that for small K and N, RCB with spherical constraint has a much better noise gain than RCB with flat ellipsoidal constraint, which has a better noise gain than SCB. From Figure 2.8(d), we note that for large K and small N, RCB with flat ellipsoidal constraint places deeper nulls at the interference angles than the more conservative RCB with spherical constraint.

Figure 2.9 shows the SOI power estimates versus the number of snapshots N for K = 1 and K = 8 when the sample covariance matrix R is used in the beamformers. Note that for small K, RCB with spherical constraint converges faster than RCB with flat ellipsoidal constraint as N increases, while the latter converges faster than SCB. For large K, however, the convergence speeds of RCB with flat ellipsoidal constraint and RCB with spherical constraint are about the same as that of SCB; after convergence, the most accurate power estimate is provided by RCB with flat ellipsoidal constraint.




























=-2 - , -20







;1% 60 -30 0 30 60 90 O90o -60 -30 0 3 60 9
0 degree 0 degree
)(b)


-RC (Sphere) RCB (Sphere)
20 . scB 20 X 0 0 6 - 0


40 .40
-20- ' B 20'- SC


-4 0 . . . . . .
Es E

<-60 I6

-90 ' 8I 0-90 -60 30 0 30 60 90 -1 60 --1 0 30 60 90
0 degree 0 degree
(c) (d)

Figure 2.8: Comparison of the beampatterns of SCB, RCB (with flat ellipsoidal constraint) and RCB (with spherical constraint) when 6 = 2.40. (a) K = 1 and using R. (b) K = 1 and using ft with N = 10. (c) K = 8 and using R. (d) K = 8 and using ft with N = 10. The true SOI power is 20 dB and Eo = 3.1349 (corresponding to A = 20).










38
































25 T 1 1 T r r 1 1 25 1 T 1 r T 1 1 T


20 r- 20--S10 I 1 .
10

EE
Ti - 5- -------- -- -o 0 .- e---e---.---.o----o--- 0 0
.0 - ---0
00



-10, P , ROB (Flat ellipsoid) -to -A ROB (Flat ellipsoid)
-o- ROB (Sphere) -w ROB (Sphere)
-0- SOB -a- SOB
- , 0 -1 1
0 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100
Number of Snapshots Number of Snapshots

(a) (b)


Figure 2.9: Comparison of the SOI power estimates, versus N, obtained using SCB, RCB (with flat ellipsoidal constraint) and RCB (with spherical constraint), all with R when 6 = 2.40. (a) K = 1. (b) K = 8. The true SOI power is 20 dB and co = 3.1349 (corresponding to A = 2�).














CHAPTER 3
APPLICATION OF RCB TO AEROACOUSTICS
3.1 Introduction

The main motivation for our work in this chapter comes from an acoustic imaging application in which the goal is to consistently estimate the SOI in the presence of strong interferences as well as some uncertainty in the SOI direction of arrival. Due to its sensitivity to steering vector mismatch and small sample size, SCB has not been used very much in acoustic imaging despite its potential benefits. The various advantages of RCB, including robustness against array steering vector errors and small sample size, high resolution, and superb interference suppression capability make it a very promising approach to cure the problem of SCB.

Although RCB was devised under the narrowband assumption, it can also deal with wideband acoustic signals by first dividing the array outputs into many narrowband frequency bins using the fast Fourier transform (FFT) and then applying the narrowband RCB to each bin separately. However, it is well-known that as the frequency increases, the beamwidth of both data-independent and data-adaptive beamformers decreases. This beamwidth variation as a function of frequency will subject the signals incident on the outer portions of the main beam to lowpass filtering and lead to distorted signal spectra or inaccurate SOI power estimation [74]-[76]. Hence it is desirable that the beamwidth of a beamformer remains approximately constant over all frequency bins of interest. In fact, a constant-beamwidth beamformer is desirable in many applications including ultrasonics, underwater acoustics, acoustic imaging and communications, and speech acquisition [74], [76], [77]. This prevents future corrections for different frequencies, and contributes to consistent









sound pressure level (SPL) estimation, which means that for an acoustic wideband monopole source with a flat spectrum the acoustic image for each frequency bin stays the same.

In the past five decades, many approaches have been proposed to obtain constant-beamwidth beamformers including harmonic nesting [75], [77]-[79], multibeam [74], [80], [81], asymptotic theory based methods [82], and approximation of a continuously distributed sensor via a finite set of discrete sensors [83]-[85]. Among these approaches, harmonic nesting is commonly used for acoustic imaging via microphone arrays. For example, in [77]-[79] a shading scheme is used for a directional array consisting of a set of harmonically nested subarrays, each of which is designed for a particular frequency bin. For each array element, shading weights are devised as a function of the frequency. This shading scheme can provide a constant beamwidth for frequencies between 10 and 40 kHz when used with the delay-and-sum (DAS) beamformer. Hereafter, this approach will be referred to as the shaded DAS (SDAS).
In this chapter, we show that we can achieve a constant beamwidth across the frequency bins for an adaptive beamformer, by combining our RCB with the shading scheme devised in [77]-[79] provided that there are no strong interfering signals near the main beam of the array. We refer to this approach as the constant-beamwidth RCB (CBRCB) algorithm [86]. We also show that we can attain a constant powerwidth, and hence consistent power estimates across the frequency bins, by using RCB with a frequency-dependent uncertainty parameter for the array steering vector; we refer to the so-obtained beamformer as the constant-powerwidth RCB (CPRCB) [86]. CBRCB and CPRCB inherits the strength of RCB in the robustness against array steering vector errors and finite sample size problems, high resolution, and excellent interference suppression capability. Moreover, they both can be efficiently implemented at a comparable computational cost with that of SCB.









3.2 Data Model and Problem Formulation of Acoustic Imaging

We focus herein on forming acoustic images using a microphone array, which are obtained by determining the sound pressure estimates corresponding to the twoor three- dimensional coordinates of a grid of locations. The signal at each grid location of interest is referred to as the SOL.

First we introduce a wideband data model. Assume that a wideband SOI impinges on an array with M elements. We divide each sensor output into N nonoverlapping blocks with each block consisting of I samples. We then apply an I-point FFT to each block to obtain I narrowband frequency bins. The data vector, yi(n), for the ith frequency bin and the nth snapshot can be written as

yj(n)=aj(xo)sj(n)+ej(n), n=I,.,N; i=1,.,I (3.1)

where si(n) stands for the complex-valued waveform of the SOI that is present at the location coordinate x0 and the ith frequency bin, ei(n) represents a complex noise plus interference vector for the coordinate xo and the ith frequency bin, and aj(xO) is the SOI array steering vector, which depends on both x0 and the ith frequency bin. The nominal or assumed array steering vector ai (xo) has the form: rO [ _�ej27rfiTr1 e27rfir2 . ej27rfi-M ]T di (x0)- [ rj f7;'r r2 rM3.2)

where Tm and TO denote the propagation time from the source at x0 to the mth sensor and the array center, respectively. rm and r0 represent the distance from the steered location to the mth sensor and the array center, respectively. fi is the center of the ith frequency bin. Note that this steering vector corresponds to the measurement at the array center for the signal at the steered location. Due to the spherical spreading of acoustic waves, the actual sound pressure at the steered location should be the one at the array center multiplied by r0. If the steered location is in the far-field of the array, we can use


di(xo) = [ eii e 2fir2 . ej27fim]T (33


(3.3)









The covariance matrix for the ith frequency bin can be written as I = E[yi(n)y(n)], i=1.I(3.4)


where E [.] is the expectation operator, and (.)* denotes the conjugate transpose.

In aeroacoustic measurements using arrays, the sound pressure response is normally shown. The intensity of the sound pressure response is measured on a logarithmic scale referred to as the sound pressure level (SPL), which is defined as [87]

SPL = 20log1o(prmg/prer) dB, (3.5)

where ptm. denotes the root-mean-squared pressure in Pa and Pref stands for the reference pressure. For air, the reference pressure is 20 pPa corresponding to the hearing threshold of young people.

Next, we determine a scaling coefficient needed for calculating SPL estimates based on k-. Let y(l), 1 = 1, 2,. , I, represent a wideband time-domain sequence with I samples, and let Y(i), i = 1, 2, . , I, denote the frequency-domain sequence obtained by applying an I-point FFT to {y(/)}. Let yi(l),l = 1,2. ,I, be the narrowband time-domain sequence corresponding to the ith frequency bin of the above frequency-domain sequence, in other words, {yi(1)} is obtained by using an inverse FFT on {0, -. , 0, Y(i), 0, . , 0}. Making use of only Y(i), we can determine a root-mean-squared pressure estimate for the ith frequency bin, Prms, as follows:


Pr. "= lYi(1) 1= 2jIY(i)12 (3.6)


The second equality is due to the fact that J=1 Iy,(l)12 = 1 Y(i)12 according to the Parseval's theorem. Substituting (3.6) into (3.5), the SPL estimate for the ith narrowband frequency bin can be written as 1Olog1o(JY(i)J2/(p2efI2)) in dB. Therefore, a scaling coefficient, -, should be used when estimating the sound pressure from the data in the frequency domain. Let ai denote the root-mean-squared pressure









estimate for the ith narrowband frequency bin obtained using R,. Then, according to the previous discussion,

SPL = 1Ologlo (u/(p I2)) dB, (3.7)

In this chapter we distinguish between beampattern and powerpattern and make use of this distinction in the design of the robust constant-beamwidth and constant-powerwidth beamformers. We define the beampattern for the ith frequency bin as

BPi(x) = la*(x)w(xg)12 (3.8)

where w(xg) denotes the beamformer's weight vector corresponding to a given location, xg, and x is varied to cover each location of interest.

Next, we introduce the powerpattern, which at the ith frequency bin is defined as

PPi(x) = la!(xg)w(x)12 (3.9)

We remark that the beampattern shows how the beamformer will pass the SOI and interfering signals when it is steered to xg, whereas the powerpattern shows how the beamformer will pass the signal at xg when it is steered to x. PPi(x) can be used to measure approximately the normalized power responses corresponding to a series of locations, and hence it is named powerpattern.

To see this, we assume that the theoretical covariance matrix for the ith frequency bin has the following form:

R, = c2 ai(xo)ai.(xo) + Q, (3.10)

where au denotes the SOI power and Q stands for the interference-plus-noise covariance matrix. If xg = xo and the signal-to-interference-plus-noise-ratio (SINR) is high, then it follows that w*(x)Riw(x) - o~w*(x)a(xg)j2 cx Ppi(x).

We next use an imaging example to illustrate the concept of beampattern and powerpattern. Consider a function [a*(xl)w(x2)1 of two coordinate variables, x,









and x2. Then the beampattern is a slice of this function for x2 fixed, whereas the powerpattern is a slice for x, fixed. Note that the beampattern and powerpattern of the DAS beamformer are identical since its weight vector and steering vector have the same functional form. However, this is not the case for an adaptive beamformer due to the fact that its weight vector depends not only on the corresponding steering vector, but also on the data.

Without loss of generality, we consider herein two-dimensional (2-D) array imaging, in which the beamwidth or powerwidt is defined as the diameter of a circle having the same area as the 3-dB contour of the main lobe of a 2-D beampattern or powerpattern. The beamwidth shows how the nearby signals impact the estimation of SO. The powerwidth, on the other hand, shows how SOI impacts the estimation of the nearby signals. From now on, we will concentrate on the ith frequency bin.

3.3 Constant-Powerwidth RCB

The beamwidth of RCB decreases with the frequency but generally it does not depend on the choice of c if the SOI is far from the interferences. On the other hand, the powerwidth of RCB depends on the signal-to-noise-ratio (SNR), the frequency, and c. Since the steering vector and hence its uncertainty set are both functions of the frequency (see (3.2) and (2.10)), it is natural to consider altering the uncertainty parameter E with the frequency. Intuitively, a larger e will yield a larger powerwidth. By choosing a frequency-dependent parameter e for RCB, we obtain the constantpowerwidth robust Capon beamformer (CPRCB), which is able to provide consistent SPL estimates across the frequency bins for the source of interest [86]. However, the beamwidth of CPRCB changes with the frequency in the same way as that of RCB.

Although it is difficult to yield an analytical formula for choosing E as a function of frequency to guarantee a nearly constant powerwidth, such a choice can be readily made numerically via a contour plot of the powerwidths of RCB with respect









to the frequency and c. Given a, desired powerwidth, we can determine E as a function of frequency from the contour plot.

3.4 Constant-Beamwidth RCB

In [77]-[79] a shading scheme is used for a directional array consisting of a set of harmonically nested subarrays, each of which is designed for a particular frequency bin. For each array element, shading weights are devised as a function of the frequency. This shading scheme can provide a constant beamwidth for frequencies between 10 and 40 kHz when used with the DAS beamformer. We refer to this approach as the shaded DAS (SDAS). Our RCB can be readily combined with the shading scheme of [77], [78] to obtain a constant beamwidth for the desired frequency band, as explained below. We refer to this approach as the constant-beamwidth RCB (CBRCB) algorithm [86].

Let v denote the M x 1 vector containing the array element shading weights for the given frequency bin. The assumed array steering vector for CBRCB can now be written as:

di = v D d, (3.11)

where 0 denotes elementwise multiplication [5]. Accordingly, the covariance matrix for the CBRCB is tapered as follows:

11. G) (VVT). (3.12)


Since both R and vvT are positive semi-definite matrices, R is also positive semidefinite [5], [88]. Note that RCB can be viewed as a special case of CBRCB with all the elements of v being 1.
Similarly to (2.20) in Section 2.3.1, the CBRCB has the form: max 02 subject to or - 2aia* > 0
-2I(ai
11ai - ii112 < �,(3.13)









Table 3.1: Comparison of the features of CBRCB and CPRCB
Approach Constant Beamwidth Constant Powerwidth Loss of DOF
CBRCB Yes Yes Yes
CPRCB No Yes No


which can be solved like the RCB problem.

The ability of CBRCB to retain a constant beamwidth across the frequencies makes it suitable for many applications such as speech acquisition. Furthermore, CBRCB can also achieve constant powerwidth, which is essential for consistent imaging. However, CBRCB is less powerful and flexible than CPRCB for applications where constant powerwidth is demanded. First, CPRCB has more degrees of freedom for interference suppression than CBRCB since at each frequency the shading scheme involved in the latter tends to deactivate some elements of the full array. In addition, CPRCB can be used with arbitrary arrays, while a special underlying array structure is required for CBRCB due to the particular shading scheme employed. As mentioned earlier, SDAS can also be utilized to yield constant beamwidth. Nevertheless, SDAS is data-independent and hence has poorer resolution and worse interference suppression capability than CBRCB and CPRCB. The features of the CBRCB and CPRCB approaches are listed in Table 3.1 in terms of constant beamwidth, constant powerwidth and loss of degree of freedom (DOF). The computational costs of the DAS, RCB, SCB, CBRCB and CPRCB approaches are listed in Table 3.2 in flops.

3.5 Numerical Examples

We provide some simulated examples to compare the performances of the DAS, SDAS, SCB, RCB, CPRCB and CBRCB approaches for acoustic imaging. We use the Small Aperture Directional Array (SADA) [77], [78], which consists of 33 microphones arranged in four circles of eight microphones each and one microphone at the array center. The diameter of each circle is twice that of the closest circle











47













Table 3.2: Comparison of the computational cost of DAS, RCB, SCB, CBRCB and CPRCB

Approach DAS SCB RCB CBRCB CPRCB

Computation O(MZ) O(M3) O(M3) O(M3) O(M3)


o
3
0 0

2 0
a 0
1 0
0 0 000
%. 0 0 0 0 0 0 0 0
000 0 0
0
0 0
0



-3
0

4 -2 0 2 4
x (in)


(a)
4.,,


11 0
o o

01 ooooo
0o0 0 0
-110


-2


-3



-4 -2 0 2 4
x (In)


(b)


0

3
a 0

2
0 o



0-0 0 0 0 0-11
0 0
0
-2


0 0


-4 -2 0 2 4 -4 -2 0 2 - 4
x (In) x (in)


(c) (d)



Figure 3.1: Microphone layout of the Small Aperture Directional Array (SADA) and its three clusters. (a) SADA. (b) Cluster 1. (c) Cluster 2. (d) Cluster 3.


0
0 0
0
0 0

0 0 0 0 0
0 0
0
0 0
0


1,




















f (Hz) xl0

Figure 3.2: Cluster shading weights for SADA, as functions of the frequency, with w1, w2 and w3 corresponding to Cluster 1, Cluster 2 and Cluster 3, respectively. it encloses. The maximum radius of the array is 3.89 inches. Figure 3.1 shows the microphone layout of the SADA and its three subarrays used in the shading scheme (referred to as clusters in [77], [78]). Note that some microphones are shared by different clusters. Each cluster of SADA has the same directional characteristics for a given wavenumber-length product kD,,, where k is the wavenumber and D,, is the diagonal distance between the elements of the nth cluster. The wavenumber-length products at 10 kHz for Cluster 3, at 20 kHz for Cluster 2, and at 40 kHz for Cluster 1 are the same. According to the array coordinate frame, the array is located in the x-y plane, with center location at (0, 0, 0). Note that inch is used as the unit for the 3-D coordinates.

We assume that the distance between the array and the source is known and plot the 2-D images by scanning the locations on a plane parallel to the array and situated 5 feet above. We assume that a belongs to the uncertainty set


Ila- a2
where c is a user parameter chosen to account for the steering vector uncertainty. Note that this form of uncertainty set used in the CPRCB can cover many kinds of array errors, including calibration errors, look direction errors, or array covariance









estimation errors due to a small snapshot number (sample size). The uncertainty set for the CBRCB is the same as above except that 5 is used instead of 5. Figure 3.2 shows the SADA cluster shading weights as a function of the frequency bins, with wl, w2 and w3 corresponding to Cluster 1, Cluster 2 and Cluster 3, respectively. Since some array elements are shared by different clusters, the shading weights of those elements are the sum of the corresponding cluster shading weights.

In the simulated examples below, we consider an array that is identical to SADA, with a wideband monopole source (flat spectrum from 0 Hz to 70 kHz) located at (0, 0, 60) (except for Figure 3.9) in the array coordinate frame and a spatially white Gaussian noise with SNR equal to 20 dB and the SPL equal to 20 dB for each frequency. We use an 8192-point FFT on the non-overlapping blocks (each containing 8192 samples) of simulated data to convert the wideband signal into 8192 narrowband frequency bins.

In practice, the length for FFT operations is often chosen to be the nearest power of two with respect to the number of samples per block. The minimum number of frequency bins, which is required to decompose a wideband signal into narrowband bins, can be determined by the following constraint [8]: 1ax < 1 (3.15)


where Tmax denotes the maximum time delay across the array aperture, B is the bandwidth of each FFT bin, and < means much smaller than. It follows that A 1
< g (3.16)
cTf/I

where A denotes the array aperture, c is the sound speed, f, is the sampling frequency and I represents the number of FFT bins. Inserting the array parameters, we obtain I > 81. Hence, 8192 samples per block satisfies the above constraint. A smaller sample number can also be used, e.g., 1024 samples per block with similar imaging results.






50

20 M c.
18-- C
16- DAS



6
14



2
C� I .


1 1.5 2 2.5 3 3.5 4
f (Hz) X 10'

Figure 3.3: Comparison of the beamwidths for the CBRCB, RCB, SDAS and DAS methods with N = 64. We used E = 2.0 for RCB and CBRCB.

Example 3.1: Comparison of CBRCB, RCB, SDAS and DAS in terms of beamwidth, powerwidth, and consistency of acoustic imaging. Figure 3.3 compares the 3-dB beamwidths as functions of the frequency, corresponding to the CBRCB, RCB, SDAS and DAS methods when N = 64. Note that herein the beamwidth of RCB coincides with that of DAS and the beamwidth of CBRCB coincides with that of SDAS. CBRCB and SDAS achieve constant beamwidth over the frequency band from 10 to 40 kHz, whereas the beamwidths of RCB and DAS are frequency dependent and decrease appreciably with the frequency. We used f = 2.0 for RCB and CBRCB. Other choices of e for RCB and CBRCB yield the same results and hence they are not shown here. We remark that one can not achieve constant beamwidth for RCB by varying e. For example, it can be shown that the beampattern of RCB is independent of c if Q in (3.10) is white Gaussian noise, i.e., Q = oI, where a,, denotes the noise power and I is an identity matrix.

Figure 3.4 compares the 3-dB powerwidths of the CBRCB, RCB, SDAS and DAS methods, as functions of the frequency, when N = 64. Note that the powerwidths of the DAS and RCB methods decrease drastically as the frequency increases, while SDAS and CBRCB can both achieve approximately constant powerwidth. In addition, CBRCB has much smaller powerwidth than SDAS. As can be seen from the






51

20 2
18 -~RB18]-- C
-SDAS W~ AS
16 . S 16] DAS
14 ' 14 '.
12 . 212
,, � ~.- --b. . . . . . . . . . . . - A. . . .- A,, --A. ]. . . . . . . . . .-A --A. . . . . ., .

6 - 6S.-.-.6 *I"0 . - .o -. ._ __ OI I I I

1 1.5 2 2.5 3 3.5 4 1 1.5 2 2.5 3 3.5 4 f (Hz) X10 f (Hz) xl0
(a) (b)

Figure 3.4: Comparison of the powerwidths for the CBRCB, RCB, SDAS and DAS methods with N = 64 when (a) E = 1.0 for RCB and c = 0.5 for CBRCB and (b) c = 2.0 for RCB and e = 1.0 for CBRCB. figure, we can also adjust the powerwidth for CBRCB and RCB by choosing different values of e.

Figure 3.5 compares the acoustic imaging results or sound pressure level (SPL) estimates obtained via the DAS, SDAS, RCB and CBRCB methods for the narrowband frequency bins at 10 kHz and 40 kHz, with N = 64. The z axes show the SPL. We used c = 2.0 for RCB and c = 1.0 for CBRCB. Note that we choose f for CBRCB to be one half of that for RCB due to the fact that the squared norm of the steering vector for CBRCB is about one half of that of RCB. As can be seen, the DAS method has poor resolution and high sidelobes and its images vary considerably with the frequency. RCB cannot be used to obtain consistent imaging results over different frequency bins, either, though it has much better resolution than DAS. It is worth noting that both SDAS and CBRCB maintain approximately the same SPL estimates across the frequency bins, but the latter has much better resolution and lower sidelobes and hence better interference rejection capability than the former. It is obvious that CBRCB significantly outperforms the other methods. According to the previous discussions and the results shown in Figures 3.3 and 3.4, it is the














0 0 0







--0 -11-1
-20 -20,
-30> -302 11 2

-2 -2 x (i) y ft -2 -2 x(ft)
(a) (b)

30,n m 64.(a)DASwit f . 10. (D
20, kl . w . 20-, w


f 10k. -(0fw -20, -20,i
-30 > -30>
2 fF. 2
0 �1 0 1
Y f) -1 -1 0y (ft) - 0
y f -2 -2 x (ft) -2 -2 x (ft)
(c) (d)
Figure 3.5: Comparison of the acoustic imaging results obtained via the DAS, SDAS, RCB and CBRCB methods with N = 64. (a) DAS with f = 10 kHz. (b) DAS with f -- 40 kHz. (c) SDAS with f -10 kHz. (d) SDAS with f = 40 kHz. (e) RCB with f = 10 kHz. (f) RCB with f =40 kHz. (g) CBRCB with f = 10 kHz. (h) CBRCB with f = 40 kHz. For RCB, c = 2.0. For CBRCB, c = 1.0. The z axes show the SPL.

constant powerwidth rather than the constant beamwidth that contributes to the better performance of CBRCB as compared to SDAS.

Example 3.2: Comparison of CPRCB and CBRCB in terms of consistency of acoustic imaging. Figure 3.6 shows the contours of the 3-dB powerwidth of RCB as c and the frequency vary, when N = 64. As can be seen, the contours are almost linear with respect to the frequency and C. Therefore, given a desired powerwidth, we can readily determine c as a function of the frequency from the corresponding contour plot. Then CPRCB will have a constant powerwidth across the frequency bins.





































































y (ft) -2 -2


-1


30201 10



-10

-20 -'


0 y (It)


~-20


-30>
2K
2

0

ft)








30,

20, 10

CO 0,

-10

-20

-30 - 4


2 2

0








Figure 3.5: Continued.


1 1.5 2 2.5
f (Hz)


3 3.5 4
x 104


Figure 3.6: Contour plots of the powerwidth, versus E and the frequency, for the RCB method. The numbers on the contours are the 3-dB powerwidths in inch.


Wto 2

-2 -2 x(ift)


(h)


-2 x


30,

20, 100-0

-101

-20

-30
2


-. . .

-2 i i i










20, 20,
10- 10,

-10, W) -10,
-20, -20
-30_,. -30 -.
2 2�
1 2 1 "2
0
0 1 0
y(t) -2 -2 x(t) Y 0 -2 2 - x( )

(a) (b)

Figure 3.7: Acoustic imaging results obtained via the CPRCB method with N 64.
(a) f 10 kHz and c = 1.3. (b) f 40 kHz and c 13.

30 30" .




20 220-20 -20-30 -30-. -.
2 � ".2 . ,
-1 -I 1 2)y(ft) -2 -2 x (ft) -2 -2 x (ft)
(a) (b)

Figure 3.8: Acoustic imaging results obtained via the CBRCB method with N 64 and c = 0.65. (a) f = 10 kHz. (b) f = 40 kHz.

In Figure 3.7, we show the imaging results obtained via the CPRCB approach by choosing f = 1.3 when f = 10 kHz and c = 13 when f = 40 kHz from the 3 inch powerwidth contour in Figure 3.6. The similarity of the SOI SPL estimates obtained with CPRCB at these two frequencies, especially near the powerwidth area, verifies the consistency of CPRCB in powerpattern across the frequencies.

Figure 3.8 shows the imaging results obtained via the CBRCB approach with c = 0.65, for f = 10 kHz and f = 40 kHz. Again we note the consistency in the imaging results. Therefore, both CPRCB and CBRCB are suitable for applications where consistent SPL estimates are desirable. However, the sidelobes in Figure 3.7(b)









-U- PRCB-0-CPRCB
0-- SCB 35 -0- SCB
W- SDAS W- SDAS
DAS .-DAS
3 30
CO" 20 -n S25 - "
2- ,.A - -A -A- A . r - A'
03
15
"A", , - A - b . , ., - o ' 1
, .- A - "
" L, .4.-0'' ' . . e. '


5 10 15 2 0 5 101 5 20
K K
(a) (b)

Figure 3.9: Comparison of the SINR and SOI SPL estimates obtained via the CBRCB, CPRCB, SCB, SDAS and DAS methods, versus the number of interferences K, for the narrowband frequency bin at f = 20 kHz. (a) SINR. (b) SOI SPL estimate. For CPRCB, E = 2.0. For CBRCB, e = 1.0. We consider a look direction error case where the assumed source location is (0, 0, 60) but the actual point source is located at (0.2, 0.2, 60) with SNR equal to 20 dB. The INRs are equal to 40 dB.

are higher and rougher than those in Figure 3.8(b). Despite this fact, CPRCB does not perform worse than CBRCB, see the next example.

Example 3.3: Comparison of CPRCB, CBRCB, SCB, SDAS and DAS in the presence of look direction errors. We consider a look direction error case where the assumed source location is (0, 0, 60) but the actual source is located at (0.2, 0.2, 60) with SNR equal to 20 dB. Also we consider a varying number of interferences from K = 0 to K = 20, which are situated on a circle with a radius of 20 inches and have an INR equal to 40 dB. The circle is on a plane parallel to the array and situated 60 inches above. We assume that the theoretical covariance matrix R is known here.

Figure 3.9 compares the SINR and SOI SPL estimates obtained via the CBRCB, CPRCB, SCB, SDAS and DAS methods, versus the number of interferences K, for the narrowband frequency bin at f = 20 kHz. For CPRCB, 6- = 2.0. For CBRCB, c = 1.0. Note that SCB is very sensitive to the steering vector mismatch and suffers from severe performance degradation in SINR and SOI SPL estimates. Although DAS


4 . . . U









and SDAS are robust against array errors, they have poor capacity for interference suppression. Consequently, their SINRs and SOI SPL estimates are unsatisfactory. CBRCB and CPRCB outperform the other approaches due to their robustness to steering vector errors, better resolution and much better interference rejection capability than the data-independent beamformers. As can be seen, CPRCB has higher SINR than CBRCB. This is due to the fact that the former has more degrees of freedom (DOFs) for interference suppression than the latter. It might seem surprising that the performance of SCB improves as the number of interferences K increases. There is a simple explanation for this. When K is small, SCB has enough many DOFs and the SOI is suppressed as interference. As K increases, SCB focuses more on suppressing the interferences than the SOI since the INR is much higher than the SNR.















CHAPTER 4
APPLICATION OF RCB TO ULTRASONICS
4.1 Introduction

The data-independent delay-and-sum (DAS) beamformer is widely used for ultrasound imaging applications [89]-[91], although it has lower resolution and worse interference suppression capability than an adaptive beamformer such as the standard Capon beamformer (SCB) [10], provided that the array steering vector corresponding to the signal of interest (SOI) is accurately known. However, there are many factors in practice that can degrade the performance of SCB including pointing errors due to inaccurate knowledge of the source location, array calibration errors due to imprecise knowledge of the transducer responses and positions, and array covariance matrix estimation errors due to a small sample size. Owing to these problems, the performance of SCB may become worse than that of the standard data-independent beamformers such as the DAS beamformer. Consequently, SCB has not been used extensively despite its potential benefits.

Much work has been done to improve the robustness of SCB over the past three decades and the literature on robust adaptive beamforming is extensive. However, most of the early suggested methods do not directly address the problem of the uncertainty of the array steering vector [51], [53]-[55], [58], [60]. The robust Capon beamformer (RCB) we presented in [13], [14], on the other hand, couples the formulation of the covariance fitting based SCB in [12] with an ellipsoidal uncertainty set of the array steering vector. In addition, RCB comprises a simple way of eliminating the scaling ambiguity when estimating the power of the desired signal.









The aforementioned approaches are all devised based on the narrowband assumptions. Wideband signals are often encountered in ultrasound imaging applications such as pulse-echo detection. For a wideband signal, we can divide the array outputs into many narrowband frequency bins using the Fourier transform, apply the narrowband beamformer for each frequency bin and then combine the narrowband estimation results. However, such incoherent approaches have the disadvantage that their performances are limited by the threshold effects of the observation time and the signal-to-noise ratios of the individual narrowband beamformers [92], [93]. To make better use of the large bandwidth, several methods have been proposed for wideband far-field processing using passive arrays. For example, the coherent signal-subspace method (CSM) in [94], [95] utilizes focusing matrices to combine the signal subspaces at different frequencies into one focused covariance matrix. However, the estimation errors of the focusing matrices may result in the source location bias. Such bias can be avoided by the steered minimum variance (STMV) method in [96], which uses a steered covariance matrix in the space-time domain, obtained after inserting proper time delays to the data samples for each direction of arrival. Nevertheless, neither CSM nor STMV can be directly applied to ultrasound imaging, which often requires wideband near-field processing using active arrays.

To deal with the ultrasound imaging problem, we may explore two options. The first one borrows the idea from the STMV method in [96] and the observation that the number of transmitters in active arrays corresponds to the number of snapshots in passive arrays [97]. By aligning the received signals from a complete multistatic data set to the focal point based on the appropriate time delays, we can construct a pseudo-covariance matrix for beamforming. The second one comes from exploiting a covariance matrix interpretation of the time reversal operator as shown in [97]. In the past decade, time reversal has been a very active domain of research and has found many applications including medical diagnostics and therapeutics, nondestructive









evaluation, underwater acoustics, acoustic room dereverberation, communications [98]-[104]. Time reversal can be cast into two main categories as physical time reversal and synthetic time reversal. For the physical time reversal, the recorded signals at the array are time reversed and physically re-emitted into the medium to optimally focus on the targets, whereas for the synthetic time reversal, the backpropagation is done numerically to form images. The essential distinction between the physical time reversal and synthetic time reversal is that the former does not require knowledge of the medium, while the latter does. Specifically, the Green's function in the medium is needed for synthetic time reversal. In practice, the Green's function can be measured as used for acoustic room dereverberation [99], [105]. Also distorted Born iterative (DBI) method and the multiple frequency DBI method can be used to estimate the inhomogeneous Green's function in the medium [106], [107].

In this chapter, we show that RCB can be used for ultrasound imaging in two ways, one based on time delay and the other based on time reversal. The timedelay based RCB utilizes a pseudo-covariance matrix constructed in the space-time domain similarly to [96], while the time-reversal based RCB exploits a covariance matrix interpretation of the time reversal operator in the frequency domain as shown in [97]. Due to its inherent robustness, RCB can allow array calibration errors, caused by transducer magnitude and phase response mismatch as well as transducer position errors, and array covariance matrix estimation errors, as a result of small sample sizes. In addition, the time-delay based RCB can tolerate the misalignment of data samples and the time-reversal based RCB can withstand the uncertainty of the Green's function. Furthermore, RCB has the desirable features including high resolution and low sidelobes, both of which are very important for ultrasound imaging [91], [108], [109]. Due to its adaptivity, RCB has higher resolution than DAS. RCB belongs to the class of the (extended) diagonal loading based approaches and it can obtain the optimal diagonal loading level based on the uncertainty set of the array









steering vector [13]. Hence RCB has low sidelobes even in the presence of array errors and small sample sizes, which characterizes the diagonal loading based approaches [55]. RCB can also be efficiently computed at a comparable cost with that of SCB by using the Lagrange multiplier methodology [13]. Finally, simulated and experimental examples are presented to illustrate the effectiveness of RCB for ultrasound imaging.

4.2 Problem Formulation

Consider an active array of M transducers, which uses the pulse-echo mode to probe the unknown propagating medium, i.e., pulses are emitted and data sequences of the back-scattered echoes are stored. Depending on how data are acquired, there are two major methods, namely a monostatic approach and a multistatic approach. In a monostatic approach, a single array element is used and it acts both as a transmitter and a receiver, whereas in a multistatic approach, a transmitter is scanned across the array aperture with multiple receivers.

A complete data set, obtained by a multistatic approach for data acquisition, is considered in the chapter. This means that the resulting multistatic array response matrix P(t) with t denoting the time instant, includes the A-scan data for all possible transmitter and receiver pairs of the array. Specifically, the A-scan data sequence PiJ,(t) is obtained by recording the back-scattered echo at the jth transducer, after transmitting a pulse from the ith transducer. Assume that each A-scan data sequence contains N data samples. Then P(t) has a total of M x M x N data samples.

Let x and z represent the lateral and axial axes of a coordinate system, respectively, (xi, zi) denote the location of the ith transmitter, and (xj, zj) be the location of the jth receiver. Let p = (XF, ZF) represent the location of a focal point of interest in the imaging region. The time delay due to the ultrasonic wave propagation from the ith transmitter to the focal point p and then back to the jth receiver is
1 1
Ti,j(p) = [(Xi - XF) + 2 -[(X XF) (Z - ZF)2]2 (4.1)
C C









where c is the sound propagation speed in the medium.

Given a focal point p, we define yi,j (p) as the sample selected according to the time delay Ti,j(p) from the A-scan data sequence Pjj(t), i.e., yij(p) = Pij(TiAP)). Due to the acoustic impedance variations in the medium, either reflection or scattering occurs depending on the size of the target. We can write yij(p) as the sum of two terms:


yi,j(p) = i(p) + eij(p), i 1= , .,M; j =,.,M (4.2)

where O3(p) denotes the on-axis signal that is proportional to the ultrasonic reflectivity or scattering strength at the focal point p, and etj(p) denotes the residual term due to the contributions from the noise and interferences (off-axis signals) at other locations.

Let


Yi(P) = [ Yi,I(P) Yi,2(P) . YiM(P) IT, i= 1,.,M (4.3)

and

ei(p) = [ ejil(p) ei,2(p) . ei,M(p) IT, i= 1,.M (4.4)

It follows that

yi(p)=a~I(p)+ej(p), i=l,.,M (4.5)

where yi(p) represents the aligned array data vector due to the ith transmitter, a denotes the array steering vector and is approximately equal to 1M�I, which is an M by 1 vector with unit elements. In practice the elements of the steering vector a are also affected by data misalignment or array calibration errors due to the transducer magnitude and phase response mismatch as well as transducer position errors.

Let Y(p) be a matrix with yi(p) as its ith column, i.e.,


Y(P) = [ Yl(P) Y2(P) . YM(P) (


(4.6)









Then the pseudo-covariance matrix relating to the ultrasonic reflectivity or scattering strength at the focal point p can be written as R(p) = y(p)y(p) (4.7)

where (.)* denotes the conjugate transpose. With probability 1, R(p) has full rank. Note that R(p) is calculated in the same way as the sample covariance matrix for passive arrays. We refer to the resulting space-time matrix R(p) as a pseudo-covariance matrix due to the fact that different columns of Y(p) correspond to different transmitters and hence contain the contributions from the targets on distinct arcs, which coincide on the same focal point.

Both SCB and the DAS methods can be applied to the data model in (4.5) for ultrasound imaging. The goal of time-delay based SCB is to pass the on-axis signal (the signal on the focal point) without distortion, and at the same time minimize the contributions from the off-axis signals (other signals on the arcs). However, in practice the time-delay based SCB suffers from severe performance degradations due to the small sample size (note that the number of snapshots N is equal to the number of transmitters M) and array steering vector errors caused by data sample misalignment or array calibration errors. In fact, there is a close relationship between the cases of steering vector errors and small-sample errors [38] in the sense that the difference between the sample covariance matrix (estimated from a finite number of snapshots) and the corresponding theoretical covariance matrix can be viewed as due to steering vector errors. The conventional DAS method, on the other hand, simply adds up all the signals whether they are on-axis or off-axis. It cannot adapt to the incoming data and hence has poor resolution and off-axis signal suppression capability.

From the above discussions, we see that the time-delay based ultrasound imaging in the presence of steering vector mismatch and small sample size needs to be formulated as a robust adaptive beamforming problem. In particular, we will deal with the following problem in this chapter: extend the standard Capon beamformer so as









to be able to determine accurately the ultrasonic reflectivity or scattering strength for each focal point of interest, even when only an imprecise knowledge of its steering vector is available. More specifically, we assume that the only knowledge we have about a is that it belongs to the following uncertainty sphere: {a I Ia - all2 < El (4.8)

where & and c are given (a is the assumed steering vector, and f is a user parameter describing the uncertainty of the array steering vector). From now on, we will concentrate on the focal point p. For notational simplicity, whenever convenient, we will drop the dependence on p in the following sections.

Next, we present some preliminaries on time reversal. From the array response matrix P(t) in the time domain, we can readily obtain the array response matrix P(w) in the frequency domain, which is symmetric but not Hermitian. Assume that there are K isotropic point targets located at Y1, ., YK, without multiple scattering among them and assume that the transducers are all isotropic point transmitters and receivers. Then P(w) has the form [103]:
K
P(w) = f(w) ( (i(w)g(yi, w)gT(yi, w) (4.9)
i=1

where f(w) is the Fourier transform of the transmitted pulse f(t), (i(w) denotes the scattering coefficient of the ith target, (.)T denotes the transpose, and g(yi, w) is the illuminating Green's vector onto the array from the point yi, which can be written as [103]:

g(yi,w) = [ G(yj,x1,w) G(yi,x2,w) "" G(yi, xM, w) ]T (4.10)

where xj denotes the location of the jth transducer, G(yi, xj, w) is the Green's function in the medium at the frequency w. For the homogeneous background in twodimensional free space with a known sound speed c, the Green's function can be









written as

G(yi, xj, w) = --jHo (kjyi - xjj) (4.11)

where k denotes the wavenumber (k = w/c), Ho is the zeroth order Hankel function of the first kind. Note that in practice, P(w) in (4.9) should also contain an additive term due to the noise.

The time reversal operator is defined as

T(w) = P*(w)P(w) (4.12)

As stated in [103], the time reversal operator contains information obtained from probing the medium twice. The connection between the eigenvectors of the time reversal operator and the scatterers was investigated and exploited by the D.O.R.T method (French acronym for decomposition of the time reversal operator) [110]. Recently it was further shown in [97] that the time reversal operator for active arrays can be interpretted as a covariance matrix for passive arrays, and the illuminating Green's vector in the medium can be used as the assumed steering vector. Therefore, we can explore some algorithms originally developed for passive arrays for timereversal based imaging. The limitation, however, is that we need to have sufficient a priori knowledge of the medium to obtain the Green's vector. If the knowledge is imprecise, there will be steering vector mismatch and consequently, the performance of a time-reversal based adaptive beamformer may degrade severely. This motivates us to consider a robust adaptive beamformer. Similar to the time-delay based method, we can apply RCB by allowing the Green's vector to be within an uncertainty set as in (4.8).

4.3 Time-Delay Based RCB

In this section we focus on the time-delay based RCB, with additional discussions on time-delay based SCB and DAS. We show how to calculate the weight vectors of these beamformers and use coherent processing for ultrasound imaging.









Note that for the time-delay based methods, we use R in (4.7) as the covariance matrix and 1Mxl as the assumed steering vector a, i.e., d = 1Mxl.

The common formulation of the adaptive beamforming problem that leads to the standard Capon beamformer, when the array steering vector is assumed to be a, is as follows [10], [5].

(a) Determine the M x 1 vector w that is the solution to the following linearly constrained quadratic problem: minw*Rw subject to w* = 1 (4.13)
w

(b) Use w*Rw as an estimate of the sound pressure u2.

The solution to (4.13) is easily derived:

R-1 R-1Mxl
W = - = (4.14)
6"R-16 l*mxlR-11Mxl

Using (4.14) in Step (b) above yields the following estimate of a2:

-2 = 1 = 1 (4.15)
6"R-16 1'J�1R-l1Mx1

The problem with the standard Capon beamformer is that it is very sensitive to steering vector errors and small sample sizes. In our recent papers [13], [14], we have proposed a robust Capon beamformer (RCB), which extends the SCB so as to be able to accurately determine the power of the signal-of-interest (SOI) even when only an imprecise knowledge of its actual steering vector is available.

To derive the robust Capon beamforming approach, we use the reformulation of the Capon beamforming problem in [12], to which we append the uncertainty set in (4.8) [13], [14]. Proceeding in this way we directly obtain a robust estimate of a2, without any intermediate calculation of a vector w: maxo2 subject to R- a2aa* > 0
.2 ,a
Ila - d112 < E (4.16)









Note that the first line above can be interpreted as a covariance fitting problem: given R and a, we wish to determine the largest possible SOI term, 02aa*, that can be a part of R under the natural constraint that the residual covariance matrix is still positive semidefinite.

For any given a, the solution &2 to (4.16) is given by the counterpart of (4.15) with 5 replaced by a, as shown in [14]. Hence (4.16) can be reduced to the following problem

mina*R-la subject to Ila- d112 < 1 (4.17)
a
To exclude the trivial solution a = 0 to (4.17), we assume that

II&112 > E (4.18)

Because the solution to (4.17) (under (4.18)) will evidently occur on the boundary of the constraint set [13], we can re-formulate (4.17) as the following quadratic problem with a quadratic equality constraint: mina*R-la subject to Ila-112-- (4.19)
a

The solution to the above optimization problem can be determined by using the Lagrange multiplier methodology and it has the form [13], [14]: a = + I)' ) (4.20)

S - (I + AR)-1 d (4.21)

where we have used the matrix inversion lemma [5] to obtain the second equality. The Lagrange multiplier A > 0 can be obtained as the solution to the constraint equation:

(I + AR)-1 112 - = 0 (4.22)

the left-hand side of which is a monotonically decreasing function of A for A > 0. Hence A can be solved easily, e.g., by a Newton's method. We then use the soobtained A in (4.21) to obtain a. However, the norm of the resulting a tends to be








small due to the scaling ambiguity [13], [14]. Hence, we use the scaled estimate of the array steering vector as follows: d= V-Ma/JlaI (4.23)

The scaling above is based, without loss of generality, on the constant norm assumption hJiJ2 = M.

To obtain the weight vector for RCB, A in (4.23) is used to replace a in (4.14):

- ,R-1 (4.24)


We remark that the major computational demand of our RCB comes from the eigendecomposition of the Hermitian matrix R, which requires O(M3) flops (we can easily inverse R in (4.24) based on the eigen-decomposition). Therefore, the computational complexity of RCB is comparable to that of SCB.

When the pseudo-covariance matrix R in (4.13) is replaced with an identity matrix corresponding to the pseudo-covariance matrix for spatially white noise, it becomes a delay-and-sum beamformer as follows: minw*w subject to w*d = 1 (4.25)
w

The solution is
a = 1MX (4.26)
M M

From the weight vector in (4.26), we see that DAS is a data-independent approach.

Once the weight vector * of a beamformer is obtained, we can estimate 3/ as (we reinstate the dependence on p for clarity): 0i(P)=**(P)Yi(), i=1,.,M (4.27)

We can then use coherent processing to compute the final estimate /3(p) as follows:
M
ow(P) = Z' i(P) (4.28)
i=1









When {f (p)} are calculated for a grid of points in the imaging region of interest, we obtain an ultrasound image.

Using (4.26) into (4.27) and then into (4.28), we obtain the following wellknown form of the DAS estimate based on a complete multistatic data set:
1 M(M
/(p) = AI y,(P) (4.29)
i=1 j=1
If instead a monostatic data set is used for DAS, it is easy to show that


(p) Yii (P) (4.30)


Note from (4.29) and (4.30) that the DAS using a multistatic data set exploits more data samples than the DAS using a monostatic data set. Hence the image obtained by the former should have better quality than that obtained by the latter at the expense of extra data collection and more computational load [89]. This will also be verified by our examples later on.

4.4 Time-Reversal Based RCB

In this section we discuss the time-reversal based RCB and several other algorithms including the time-reversal based SCB, matched field (MF), and multiple signal classification (MUSIC).

We consider herein a coherent approach, which exploits the time-bandwidth product of the wideband signal. We use a scheme to construct the focusing matrices for the time reversal operators, similar to the rotational signal subspace (RSS) focusing approach in [95]. The difference is that the latter is used for passive arrays in the far field and requires the initial estimates of the source locations, while the former is used for active arrays in the near field and does not demand the initial estimates of the source locations. Specifically, in the former method we use focusing matrices to transform the time reversal operators at different frequency bins onto a









single reference frequency, and then sum the transformed time reversal operators to obtain the coherently focused time reversal operator.

Let

Y(w) = P*(w) (4.31)

Substituting (4.31) into (4.12) yields T(w) = Y(w)Y*(w) (4.32)

Assume that the reference frequency is equal to wo, and we wish to find the unitary focusing matrices for J frequency bins at wj, j = 1, 2, ., J. Unitary focusing matrices are desired due to the fact that they have better performances in preserving the shape of the noise covariance matrix and the array SNR than nonunitary focusing matrices. In [95], the following constrained minimization problem is used to determine the unitary focusing matrices F(wj):

min IIY(wo) - F(wj)Y(wj)IIF subject to F*(wj)F(wj) = I (4.33)
F(wj)

where 1H" IF denotes the Frobenius matrix norm. Let the singular value decomposition (SVD) of Y(wj)Y*(wo) be:

Y(wj)Y*(wo) = U(wj)E(wj)V*(wj) (4.34)

It can be proved that the solution to the problem in (4.33) is [95] F(wj) = V(wj)U*(wj) (4.35)

Then the coherently focused time reversal operator is given by
J
Tr(wo) = -F(wj)T(wj)F* (wj) (4.36)
j=1

Note that for the coherent approach, T(wo) is used as the covariance matrix and the illuminating Green's vector (wo,p) is used as the assumed steering vector for the focal point p.









Next, we briefly present several time-reversal based methods, whose formulations are analogous to the time-delay based counterparts in the previous section. To simplify the notation, we omit the argument w0 in T(wo) and g(p, wo).

The time-reversal based RCB has the form:

mina*T-la subject to Ila- (p) 12 < 6 (4.37)
a

Note that the norm square of g(p) depends on the location of the focal point and is not equal to M. Hence e should be chosen as a function of g(p). In addition, after the estimated steering vector a is obtained, we now need to scale it based on the constant norm assumption ILIllg(P)ll as follows: AL = IIg(p)lla/Ilall (4.38)

The remaining steps of the time-reversal based RCB are in direct analogy to those of the time-delay based RCB, and are hence omitted herein for brevity.

The time-reversal based SCB can be written as:

minw*Tw subject to w*g(p) = 1 (4.39)
w

The solution is

g*(p)T-l (p) (4.40)
The time-reversal based MF has the form:

minw*w subject to w*g(p) - 1 (4.41)
w

The solution is
w = (p) (4.42)
II (p)I I
As its name implies, the time-reversal based MF matches its weight vector to the field of a virtual medium with the Green's vector equal to g(p) at the focal point p. Note that the formulation of MF in (4.41) is similar to that of the time-delay based DAS in (4.25).









Once we have obtained the weight vector w for the time-reversal based SCB, RCB and MF, we can estimate the magnitude square of the on-axis signal as (we reinstate the dependence on p for clarity):


1'(p) = w*(p)tw(p) (4.43)

When {I/12(p)} are calculated for a grid of points in the imaging region of interest, we obtain an ultrasound image.

In [102], MUSIC was considered for time-reversal based imaging. MUSIC was a super-resolution approach originally proposed for passive arrays [17]. In the following, we briefly review the MUSIC algorithm. We will compare its performance with our proposed methods in Section 4.5. Let {Vm}mi be the eigenvectors of associated with the eigenvalues arranged in a decreasing order, and let S and G denote the following two matrices formed from {vm}.

S = [V1" . VK], G = [VK+I"". VM] (4.44)


where K denotes the dimension of the signal subspace, e.g., the number of point targets in the medium. Note that S and G satisfy S*S=I, G*G=I and S*G=0 (4.45)


The locations of the K targets are given by the positions of the K highest peaks of the function
1
h(p) =*(p)GG* (p) (4.46)

As is well-known, MUSIC is not an intensity estimator but a locator. It cannot cope with distributed targets. Also MUSIC demands the information of the dimension of the signal subspace K, which is unknown in practice and may be difficult to accurately estimate in the presence of colored noise.









4.5 Simulated and Experimental Examples

In this section, we provide several simulated and experimental examples to compare the performances of the time-reversal and time-delay based methods including DAS, SCB, RCB, MF and MUSIC.

The Field II software [111] is used as a simulation tool to generate a complete multistatic data set. In the simulations, the probing pulse is differential Gaussian and has the form:

f()= -2r2V2 (t - ) exp [ 2 V2 (t -)2] (4.47)


where v denotes the central frequency (v = 3 MHz). The sampling frequency is 100 MHz. The array comprises 21 transducers with inter-sensor spacing (referred to as pitch in ultrasound imaging) equal to A, where A is the central wavelength. The transducers are isotropic and located at
xj = (-5A + 1 1A, 0), j = 1,.,21. (4.48)

2

There are two point targets located at (-2A, 20A) and (3A, 19A) in a free-space homogeneous medium with a sound speed equal to 1540 m/s. In all the simulated images below, a 30 dB display dynamic range is used with the plus symbols denoting the true positions of the two targets. The unit for the axes is A. For the time-delay based methods, each A-scan data sequence is preprocessed by a matched filter based on the probing pulse. For the time-reversal based methods, we focus the time reversal operators at all of the frequency bins onto the central frequency (corresponding to the spectral peak of the transmitted pulse) to obtain a coherently focused time reversal operator. Unless specifically clarified, the complete multistatic data set is employed.

Figures 4.1 and 4.2 show the imaging results obtained via the time-reversal and time-delay based methods, respectively, for the case of no transducer position errors. We see from Figure 4.1 that the time-reversal based MF has poor resolution and high









sidelobes. Although the time-reversal based SCB image has peaks corresponding to the targets, it also contains some sidelobes like that of MF. Both MUSIC and the time-reversal based RCB have good cross-range resolution and low sidelobes. On the other hand, their range resolutions are not as good. This is due to the fact that the Green's vector is more sensitive in the cross range than in the range. We can verify this by imaging the norm of the difference of the Green's vector at a given point with those at its surrounding points. To improve the range resolution of the timereversal based methods, one method is to combine the direction of arrival estimation with the time delay estimation as in [103], [104] for imaging in the random media. However, this method requires very complicated procedures to estimate the time delay. Note from Figure 4.1 that the time-reversal based RCB outperforms the other three methods. From Figure 4.2, we see that the time-delay based SCB yields a very poor image. This is probably due to the high sidelobes caused by the small sample size. Comparing the DAS image obtained using monostatic data and that obtained using multistatic data, we observe that the targets are more localized in the latter with weaker trailing effects near the targets. A simple explanation is that averaging by using more samples lowers the noise contributions to the estimate at the focal point and hence higher SNR will result. This is actually why we concentrate on the multistatic data processing in this chapter. In terms of resolution and sidelobes, the time-delay based RCB has the best imaging result among all the time-delay based methods considered here. As can be seen from Figures 4.1 and 4.2, the time-delay based approaches have better range resolution but worse cross-range resolution than the time-reversal based counterparts.
Figures 4.3 and 4.4 are similar to Figures 4.1 and 4.2, except that now each transducer position is perturbed by a zero-mean Gaussian random variable with variance equal to 0.01A/2. The perturbing Gaussian random variables are independent of each other. This will lead to data misalignment for the time-delay based methods









and uncertainty in the Green's function for the time-reversal based methods. Hence there are array steering vector errors. As a result, degradations occur in the ultrasound images. Note that the time-reversal based RCB and the time-delay based RCB still achieve good performances, outperforming the other methods considered. This verifies the robustness of RCB against steering vector errors.

Next, we present some experimental results to compare the performances of the time-delay based DAS, SCB and RCB approaches. We did not consider the timereversal based methods here due to lack of information for the Green's function in the experimental media. The complete multistatic data sets were obtained from the Bioacoustics Research Laboratory of the University of Illinois at Urbana-Champaign (the data set for the wire phantom in a complex pattern and the data set for the rat mammary tumor) and the Biomedical Ultrasonics Laboratory of the University of Michigan at Ann Arbor (all the other data sets). To reconstruct an ultrasound image, we first use a digital bandpass filter on the data. Then we apply the aforementioned beamforming algorithms on the filtered data. Finally, we apply envelope detection, gain compensation, scan conversion and logarithmic compression. Except specifically stated otherwise, all the images below are shown over a 50 dB display dynamic range.

We first consider a wire phantom containing six wire targets. The data are collected using a 128-element linear array (M = 128). The transducer center frequency is 3.5 MHz and the sampling rate is 13.8889 MHz. The sound velocity is 1480 m/s. In Figure 4.5, we show the ultrasound images obtained via DAS, SCB and RCB using the entire array and only the central 64 elements of the array, respectively. The latter can be regarded as using a smaller array with M = 64. Note that SCB yields very poor images where the wire targets are lost. Although the wires are more recognizable in the DAS images than in the SCB images, the sidelobes of the DAS images are still very high. Clearly, RCB significantly outperforms DAS and SCB in terms of resolution and sidelobes.









To take a closer look at the resolution of DAS and RCB, in Figure 4.6 we compare the ultrasound images obtained via DAS and RCB using only the central 32 and 64 elements of the array over a 20 dB display dynamic range. This can be regarded as using smaller arrays with 32 and 64 elements, respectively. As can be seen from Figures 4.6(a) and 4.6(b), the resolution of the DAS improves if more transducers are used. In Figures 4.6(c)-4.6(f), we vary 6 from 0.2 to 20 for RCB and show the resulting images. Note that as E increases, the resolution of RCB becomes lower. This is not surprising since c is a parameter used to adjust the robustness of RCB. In fact, we can prove that the bigger the c, the smaller the Lagrange multiplier A and the bigger the loading level. Consequently, the less significant role the pseudocovariance matrix R plays and the closer RCB is to DAS. On the other hand, the smaller the c, the closer RCB is to SCB and the more sensitive RCB becomes to the steering vector errors. From Figure 4.5, we can easily observe that, for a large range of choices for E, RCB using an array with 32 transducers can achieve a similar resolution as that of DAS using an array with 64 transducers, outperforming that of DAS using an array with 32 transducers. This means that RCB can double the resolution of DAS without complicating the array hardware at all.

As a further illustration, we use a data set containing some wire targets in a more complicated pattern. The data are collected using a 64-element linear array (M = 64). The transducer center frequency is 2.6 MHz and the sampling rate is 25 MHz. The sound velocity is 1450 m/s. In Figure 4.7, we show the ultrasound images obtained via DAS, SCB and RCB. For DAS, we compare its images obtained using monostatic data and multistatic data. Note that by choosing the A-scans with each transducer acting both as a transmitter and the only corresponding receiver, the monostatic data can be easily obtained from the multistatic data. As can be seen from Figures 4.7(a) and 4.7(b), the image obtained based on multistatic data is much better than that obtained based on monostatic data. Due to its sensitivity









to steering vector errors and small sample sizes, SCB yields a very poor image as in Figure 4.7(c), in which the wires are hardly recognizable. The RCB image, on the other hand, has very good resolution and low sidelobes, surpassing both SCB and DAS in performance.

To investigate the contrast resolution, we consider now a cyst phantom containing four cyst targets. The data are collected using a 128-element linear array (M = 128). The transducer center frequency is 3.5 MHz and the sampling rate is 13.8889 MHz. The sound velocity is 1480 m/s. In Figure 4.8, we show the ultrasound images obtained via DAS, SCB and RCB using only the central 32 and 64 transducers of the array. This can be regarded as using smaller arrays with 32 and 64 elements, respectively. For RCB, we choose e = 2. By using a 50 dB display dynamic range in Figures 4.8(a)-4.8(f), we can discover the cysts from the images obtained via SCB and RCB. However, the cysts are invisible from the images obtained via DAS (we see instead some arcs). After checking carefully, we find out that, there are some samples with very high amplitudes close to the end of several A-scan data sequences, which results in very large DAS estimates at the arcs and make the cysts indistinguishable from the background. As expected, the more transducers used, the more arcs will appear in the DAS images. By increasing the display dynamic range to 80 dB for DAS in Figures 4.8(g)-4.8(h), the cysts become visible. SCB and RCB do not have such a problem since they can adjust their weight vectors and suppress the signals corresponding to the abnormal samples in the A-scan data. Comparing the RCB images and the SCB images, we notice that the cysts are much more discernible in the former than in the latter. Hence RCB has better contrast than SCB.

We examine next a heart phantom. The data are collected using a 64-element linear array (M = 64). The transducer center frequency is 3.333 MHz and the sampling rate is 17.76 MHz. The sound velocity is 1540 m/s. In Figure 4.9, we show the ultrasound images obtained via DAS, SCB and RCB using the entire array and









only the central 32 elements of the array, respectively. For RCB, we choose c = 2. Indeed we cannot find the heart phantom at all from the poor SCB images. However, the heart phantom is apparent in the DAS images and the RCB images since DAS and RCB are much more robust against steering vector errors and small sample sizes than SCB. It is clear that DAS has poor resolution and high sidelobes. Due to its inherent adaptiveness, RCB does not suffer from these problems.

Finally, we consider in vivo imaging of a rat mammary tumor. The data are collected using a 64-element linear array (M = 64). The transducer center frequency is 2.6 MHz and the sampling rate is 25 MHz. The sound velocity is 1500 m/s. In Figure 4.10, we show the ultrasound images obtained via DAS, SCB and RCB using the entire array and only the central 32 elements of the array, respectively. As expected, for each method applied to the same data, the more transducers used, the better resolution we have. For DAS, we compare its images obtained using monostatic data and multistatic data. Clearly, the images based on the multistatic data are much better than those based from the monostatic data, as expected. Comparing the images obtained using the same multistatic data, we see that the SCB images are worse than those of DAS and RCB due to the fact that SCB is vulnerable to the steering vector errors and small sample sizes. As can be seen, RCB has better imaging results than DAS in terms of resolution and sidelobes. In particular, the RCB image with M = 32 is very similar to the DAS image with M = 64, and it is superior to the DAS image with M = 32.














































S2 �

01



2--4



-6 --6
5 10 20 10 15 20
Depth Depth

(a) (b)



4 4







-2 0_2

-4 -4

6 -6
5 10 15 20 5 10 15 20
Depth Depth

(c) (d)


Figure 4.1: Simulated ultrasound imaging results obtained via time-reversal based methods. (a) MF, (b) SCB, (c) MUSIC, and (d) RCB with c = HgIg12/1OO. A 30 dB display dynamic range is used. There are no transducer position errors.









































j -2J510 15 20 5 10 Is 20
Depth Depth


(a) (b)




5 1
C 0


-.2



-66
510 15 20 5 10 15 20
Depth Depth

(c) (d)


Figure 4.2: Simulated ultrasound imaging results obtained via time-delay based methods. (a) DAS (monostatic data), (b) DAS (multistatic data), (c) SCB, and (d) RCB with E = 1. A 30 dB display dynamic range is used. There are no transducer position errors.









































.





-6
10 15 20 10 15 20
Depth Depth

(a) (b)





8 A2



-6

-4 -4

5 10 15 20 5 10 15 20
Depth Depth

(c) (d)


Figure 4.3: Simulated ultrasound imaging results obtained via time-reversal based methods. (a) MF, (b) SCB, (c) MUSIC, and (d) RCB with c = Ig12/10, A 30 dB display dynamic range is used. The transducer position errors are Gaussian distributed with variance equal to 0.01A/2.





































-4





Depth Depth
(a) (b)
0 2 0
= C
50 -2


-4
-6 -6







5 10 15 20 5 10 15 20
Depth Depth
(c) (d)


Figure 4.4: Simulated ultrasound imaging results obtained via time-delay based methods. (a) DAS (monostatic data), (b) DAS (multistatic data), (c) SCB, and (d) RCB with E = 1. A 30 dB display dynamic range is used. The transducer position errors are Gaussian distributed with variance equal to 0.01A/2.









82








30 30

40 _____________________________ 40

5~ 50
E 60 60
a� _______r-_____________________( 70 U 0
C C
80 5


110
It ' I ,-1120 2

-50 0 50 -50 0 50
Lateral distance (mm) Lateral distance (mm)

(a) (b)


30 30

40 40

50

E 60 E 6

70




1007


120
10


-50 0 50 -40 -30 -20 -10 0 10 20 30 40
Lateral distance (mm) Lateral distance (mm)

(c) (d)


30

40




5O5
E 7o


5 80 80

90O9

100 100

110 10
12120
0 500
-50 0 50 -50 0 5
Lateral distance (mm) Lateral distance (mm)

(e) (f)


Figure 4.5: Experimental ultrasound imaging results for the wire phantom. (a) DAS with M = 64, (b) DAS with M = 128, (c) SCB with M = 64, (d) SCB with M = 128,

(e) RCB with M = 64, and (f) RCB with M = 128. A 50 dB display dynamic range is used. For RCB, E = 10.









83









30 40 50
E 60 7 70



90 Xi 90

10o 100

110 110

120 120

-50 0 50 -50 0 50
Lateral distance (mm) Lateral distance (ram)

(a) (b)


303

40 50



E E0
70 70

80 8
90 90

100 O

110! 1

120 1

-50 0 50 -50 0 50
Lateral distance (mm) Lateral distance (mm)

(c) (d)


30 30

40 40

50 50
EB EO
is a
70 70

8a0 80

~90 ~90

100 100



1120

-50 0 50 -50 0 50
Lateral distance (mm) Lateral distance (mm)

(e) (f)


Figure 4.6: Experimental ultrasound imaging results for the wire phantom. (a) DAS with M = 32, (b) DAS with M = 64, (c) RCB with M = 32 and e = 0.2, (d) RCB with M = 32 and c = 1, (e) RCB with M = 32 and c = 5, and (f) RCB with M = 32 and c = 20. A 20 dB display dynamic range is used.








84
































20 120


-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60
Lateral distance (mam) Lateral distance (mm)
(a) (b)
E
(D (
1. 80 80



20 ~12














-60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60
Lateral distance (mm) Lateral distance (mm)
(c) (d)


Figure 4.7: Experimental ultrasound imaging results for the wire phantom with M = 64. (a) DAS (monostatic data), (b) DAS (multistatic data), (c) SCB, and (d) RCB with c = 4. A 50 dB display dynamic range is used.

































5050
E E6

S70 70

80 80

90 90

0100
1110

120 120

-50 0 50 -50 0 50
Lateral distance (mm) Lateral distance (mm)

(a) (b)


30 3

40 4

50 5

E E




90i9

O 1D


11C 1 1
120; t211 1 1 1

-50 0 50 -50 0 50
Lateral distance (mm) Lateral distance (mm)

(c) (d)


Figure 4.8: Experimental ultrasound imaging results for the cyst phantom. (a) DAS with M = 32, (b) DAS with M = 64, (c) SCB with M = 32, (d) SCB with M = 64,

(e) RCB with M = 32, (f) RCB with M = 64, (g) DAS with M = 32, and (h) DAS with M = 64. A 50 dB display dynamic range is used for (a)-(f), and a 80 dB display dynamic range is used for (g)-(h). For RCB, e = 2.






































Lateral distance (mm)
(e)


Lateral distance (mm)

(f)


0 50 -50 0
Lateral distance (mm) Lateral distance (mm)
(g) (h)


Figure 4.8: Continued.









87








20 2

40 4



E 60 60
E E


ca
0100




,40 I 4
160 I6

-80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80
Lateral distance (mm) Lateral distance (mm)

(a) (b)


20 20

40 40

0 60
EE
(D 80 CD 80
* 1oo a too"


.,120 120

140 140

160 160=


-80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80
Lateral distance (mm) Lateral distance (mm)

(c) (d)


20 2

40 4








160 6
E E

~80 8





140 140




-80 -60 -40 -20 0 20 40 s0 80 -80 -60 -40 -20 0 20 40 60 80
Lateral distance (mm) Lateral distance (mm)

(e) (f)


Figure 4.9: Experimental ultrasound imaging results for the heart phantom. (a) SCB with M = 32, (b) SCB with M = 64, (c) DAS with M = 32, (d) DAS with M = 64,

(e) RCB with M = 32, and (f) RCB with M = 64. A 50 dB display dynamic range is used. For RCB, c = 2.





























35 65
40

!5C







-20 -15 -10 -5 0 5 t0 15 20 -20 -15 -10 -5 0 5 10 15 20
Lateral distance (mm) Lateral distance (mm)
(a) (b)


25 25

35

E
EE
60 60

65 65









-20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20
Lateral distance (mm) Lateral distance (mm)
(c) (d)


Figure 4.10: Experimental ultrasound imaging results for the rat tumor. (a) DAS (monostatic data) with M =32, (b) DAS (monostatic data) with M =64, (c) DAS

(multistatic data) with M =32, (d) DAS (multistatic data) with M =64, (e) SCB

with M =32, (f) Sen with M =64, (g) RCB with M =32 and 2, and (h) RCB

with M =64 and c = 4. A 50 dB display dynamic range is used.









89



























252 303


-35
-E
EE
W-40 -;





25 25
5






30 g30












40 60

o C
45 2~5








55 5

60 60

65 65

-20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20
Lateral distance (mm) Lateral distance (mm)

(g) (h)



Figure 4.10: Continued














CHAPTER 5
CONCLUSIONS AND FUTURE WORK
5.1 Conclusions

We summarize the issues addressed in the dissertation and the contributions we made. We have developed a novel robust Capon beamformer (RCB) by combining the covariance fitting formulation of the standard Capon beamformer (SCB) with an ellipsoidal (including flat ellipsoidal) uncertainty set of the array steering vector. We have shown how to efficiently compute RCB at a comparable computational cost with that associated with SCB. The data-adaptive RCB is much less sensitive to steering vector mismatches than SCB and yet it can retain the appealing properties of SCB including better resolution and much better interference rejection capability than the standard data-independent beamformer. We have shown that the RCB belongs to the class of diagonal loading approaches but the amount of diagonal loading can be precisely calculated based on the uncertainty set of the steering vector. We have provided deep insights into the relationships among the recent three robust adaptive beamformers. We have shown that, although the robust adaptive beamformers obtained from the two different formulations appear to be rather different, they are equivalent in terms of the signal-to-interference-plus-noise ratio (SINR). Furthermore, we have demonstrated some unique features of RCB. Due to the fact that both the power and the steering vector of the signal of interest are treated as unknowns, there is a "scaling ambiguity" in the signal covariance term. This problem is easily overcome in the RCB framework since the estimated steering vector can be readily scaled. We have also compared these three beamformers for both the cases of degenerate and non-degenerate ellipsoidal uncertainty sets of the steering vector,









and demonstrated the impressive advantages of RCB over the other two methods: simpler implementation, lower computational complexity and more accurate power estimation.

In addition to the theoretical study, we have applied the RCB algorithms to various applications and extended RCB to accommodate the specific goals and requirements. For the application of aero-acoustic imaging, we have devised a constantbeamwidth RCB and a constant-powerwidth RCB for consistent acoustic imaging, which means that for an acoustic wideband monopole source with a flat spectrum the acoustic image for each frequency bin stays approximately the same. For the application of ultrasound imaging, we have developed a time-delay based RCB and a time-reversal based RCB, which have the desirable features including high resolution, low sidelobes and robustness against steering vector errors. The effectiveness of RCB and its various extensions has been demonstrated by simulated and experimental examples.

5.2 Future Work

Several possible directions for future work are as follows.

The dissertation focuses on the batch mode (block adaptation) implementation of RCB based on the stationary assumption. In the batch mode, a sample covariance matrix needs to be constructed from a temporal block of array data before calculating the RCB weight vector. In the nonstationary or time-varying situations such as the target tracking applications, it is more suitable to implement RCB in the online mode (continuous adaptation), which means that the weights are adjusted as the data are sampled.

Throughout this dissertation, we have assumed a rank-one signal covariance matrix. This assumption may be violated in sonar and wireless communications where scattering sources or fluctuating wavefronts may exist. Hence it is of interest






92

to extend RCB to the general case of a signal subspace with a dimension larger than one.

The performance of RCB will degrade in the presence of coherent signal and interferences. Preliminary work has been done to extend RCB for a simple case where a priori knowledge of the directions of arrival of the pair of coherent signal and interference is given. A more general case, where several coherent multipaths impinge on the array from unknown directions of arrival with respect to that of the signal, deserves attention for the future work.














APPENDIX A
LINKING RCB TO VOROBYOV AND COLLEAGUES' APPROACH

We repeat our optimization problem:

mina*R-la subject to Ia - all2 = e. (A.1)
a

Let a0 denote the optimal solution of (A.1). Let Wo- aR-la� (A.2)


We show below that the wo above is the optimal solution to the following SOCP considered in [44]:

minw*Rw subject to w*d> /I4w + 1, Im (w*a) = 0. (A.3)
W

First we show that if I1111 < Fc, then there is no w that satisfies w*d > VcIlwIl + 1. By using the Cauchy-Schwarz inequality, we have


Vx/IIwII + 1 <_ w* _ VCIlwil (A.4)

which is impossible. Hence the constraint in (2.29), which is needed for our RCB to avoid the trivial solution, must also be satisfied by the approach in [44].

Next let

w = w0 + y. (A.5)

We show below that the solution of (A.3) corresponds to y = 0.

Insertion of (A.5) in (A.3) gives:
2 1
min y*Ry + 2 Re (y*ao) + I (A.6)
yaR-lao a*R-lao




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xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0008233800001datestamp 2009-01-28setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Robust adaptive array processingdc:creator Wang, Zhisongdc:publisher Zhisong Wangdc:date 2005dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00082338&v=00001003320266 (alephbibnum)dc:source University of Florida