eigenvalue beamforming using a multirank mvdr ...978248/fulltext01.pdf1954 ieee transactions on...

1954 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 5, MAY 2008

Eigenvalue Beamforming Using a Multirank MVDRBeamformer and Subspace Selection

Ali Pezeshki, Member, IEEE, Barry D. Van Veen, Fellow, IEEE, Louis L. Scharf, Life Fellow, IEEE,Henry Cox, Life Fellow, IEEE, and Magnus Lundberg Nordenvaad, Member, IEEE

Abstract—We derive eigenvalue beamformers to resolve an un-known signal of interest whose spatial signature lies in a knownsubspace, but whose orientation in that subspace is otherwise un-known. The unknown orientation may be fixed, in which case thesignal covariance is rank-1, or it may be random, in which casethe signal covariance is multirank. We present a systematic treat-ment of such signal models and explain their relevance for mod-eling signal uncertainties. We then present a multirank general-ization of the MVDR beamformer. The idea is to minimize thepower at the output of a matrix beamformer, while enforcing adata dependent distortionless constraint in the signal subspace,which we design based on the type of signal we wish to resolve. Weshow that the eigenvalues of an error covariance matrix are funda-mental for resolving signals of interest. Signals with rank-1 covari-ances are resolved by the largest eigenvalues of the error covari-ance, while signals with multirank covariances are resolved by thesmallest eigenvalues. Thus, the beamformers we design are eigen-value beamformers, which extract signal information from eigen-modes of an error covariance. We address the tradeoff betweenangular resolution of eigenvalue beamformers and the fraction ofthe signal power they capture.

Index Terms—Eigenvalue beamforming, generalized sidelobecanceller, matched direction beamforming, matched subspacebeamforming, multirank MVDR beamformer.

I. INTRODUCTION

I N many applications of wireless communications, radar,sonar, and biomedical imaging, it is desired to separate

a signal of interest in the presence of interference and noiseusing measurements from sensor elements, e.g., see [1].

Manuscript received September 22, 2006; revised September 11, 2007. Theassociate editor coordinating the review of this manuscript and approvingit for publication was Prof. Alle-Jan van der Veen. This work is supportedin part under ONR contracts N00014-04-1-0084 and N0014-00-C-0145,DARPA contracts FA9550-04-1-0371 and FA8750-05-2-0285, and NIH award1R21EB005473.

A. Pezeshki is with the Program in Applied and ComputationalMathematics, Princeton University, Princeton, NJ 08544 USA (e-mail:[email protected]).

B. D. Van Veen is with the Department of Electrical and ComputerEngineering, University of Wisconsin, Madison, WI 53706 USA (e-mail:[email protected]).

L. L. Scharf is with the Department of Electrical and Computer Engineeringand the Department of Statistics, Colorado State University, Fort Collins, CO80523 USA (e-mail: [email protected]).

H. Cox is with Lockheed Martin-Orincon Defense, Arlington, VA 22203 USA(e-mail: [email protected]).

M. Lundberg Nordenvaad is with the Department of Computer Science andElectrical Engineering, Luleå University of Technology, Luleå, Sweden (e-mail:[email protected]).

Digital Object Identifier 10.1109/TSP.2007.912248

Typically, the problem is one of estimating a signal in themeasurement model

(1)

where is the spatial signature of interest, is the in-terference vector, and is broadband noise. When exactknowledge of the signature vector is available, adaptive beam-formers provide high spatial resolution and good interferencesuppression. However, in most situations the signature vectoris not perfectly known, due to factors such as multipath, localand random scattering, near field wavefront formation, randomfluctuations in the propagation medium, array flexing, arraycalibration errors, and movement of the source. Differencesbetween the presumed signature and the actual signature resultin signal suppression and poor interference rejection [2]–[5].

To date several methods have been reported to account for un-certainty in the signature vector. A few examples are: the robustadaptive beamformers of [6], which enforce a white noise gainconstraint; robust adaptive beamformers of [7], [8], which con-sider an ellipsoidal uncertainty for the signature vector; robustadaptive beamformers of [9] which optimize worst-case perfor-mance for a bounded norm distortion in the signature vector;robust adaptive beamformers of [10], which enforce a second-order distortionless constraint on a general rank signal covari-ance matrix; signal estimation methods of [11], [12], for the casewhere the unknown parameters of the signature vector are de-terministic; and Bayesian approaches of [13], [14], for the casewhere the unknown parameters of the signature vector are ran-domly drawn from a known probability density. Other examplesinclude the linearly constrained minimum variance (LCMV)beamformer in [15], Bayesian robust adaptive beamformer of[16], and signal blocking-based algorithms of [2], [17]. Thereader is referred to [18] for a comprehensive review of the rel-evant literature.

In this paper, we derive multirank generalizations of theMVDR beamformer to resolve an unknown signal of interest.We assume that the signature vector lies in a known linearsubspace and consider three signal models. In the first model,the signal is assumed to lie in a known one-dimensionalsubspace and has a known rank-1 covariance matrix. Thiscorresponds to the point source assumption in sensor arrayprocessing, where conventional adaptive beamformers havegood and predictable performance. In the second and thirdmodels, however, we assume that the signal lies in a knownmultidimensional subspace, but the signal orientation within thesubspace is otherwise unknown. The unknown orientation maybe fixed over a sequence of experimental realizations, in whichcase the signal covariance matrix is rank-one. Or the unknown

1053-587X/$25.00 © 2008 IEEE

PEZESHKI et al.: EIGENVALUE BEAMFORMING USING A MULTIRANK MVDR BEAMFORMER AND SUBSPACE SELECTION 1955

orientation may change from realization to realization, in whichcase the signal covariance is multirank. The former can be usedto model signals in the presence of slow-varying multipath,array calibration errors, and deterministic uncertainties aboutthe relative source/sensor array geometry. The latter can be usedto model signals in the presence of fast-varying multipath, localand random scattering, flexing arrays, and random uncertaintiesabout the relative source/sensor array geometry. Conventionalbeamformers suffer loss in detectability and resolution, due tomodel mismatch, for both of these models.

We begin by presenting a unified and systematic treatmentof our signal models and explain the rationale and relevance ofsuch models for problems in radar, sonar, wireless communi-cations, and biomedical imaging. This unified framework moti-vates the design and use of multirank beamformers as a generaltool for robust adaptive beamforming in many real-life applica-tions. We then present a multirank generalization of the MVDRbeamformer. The idea is to minimize the power at the output ofa matrix beamformer , while enforcing a data dependent dis-tortionless constraint in the signal subspace. More specifically,we enforce a constraint of the form , where isan orthonormal basis for the signal subspace and is a data de-pendent constraint matrix, which we design based on the typeof signal we wish to resolve. We note that minimizing the powerat the output of a matrix beamformer/filter under a constraint ina linear subspace has been considered before in [19] and [20].Such a problem has also been considered in [21], but in the con-text of spectrum estimation. What distinguishes the multirankbeamformers to be presented in this paper is that in [19]–[21] thedesired response is prespecified (namely ), whereasin our work the constraint matrix is a data dependent de-sign parameter. The design of has important and surprisingimplications for resolving signals in the presence of model un-certainties. Incidentally, when and are vectors the con-straint reduces in form to the constraint of a Frostbeamformer [22]. However, in the Frost beamformer the con-straint vector is again prespecified. Our multirank beamformersare also related to the minimum variance CDMA receivers of[23], [24], the biomagnetic spatial filters of [25], and the robustadaptive beamformers of [10].

The critical quantities for resolving signals that are drawnfrom a multidimensional subspace are eigenvalues of an errorcovariance matrix, , associated with the linear minimummean-squared error (LMMSE) in a generalized sidelobe can-celler (GSC). Although the GSC [26] is not necessary for im-plementing our beamformers, it is essential for understandingand interpreting multirank MVDR beamforming. We show thatsignals with rank-1 covariances are resolved by the dominanteigenvalues of , while signals with multirank covariancesare resolved by the subdominant eigenvalues of . In theformer case, the constraint matrix is selected to exploit thedominant eigenvectors of . In the latter case, is selectedto exploit the subdominant eigenvectors of . This is a fun-damental and surprising result. It shows that the dominant sub-space of is fundamental for resolving signals with rank-1covariances, or equivalently for beamforming in the presence ofdeterministic uncertainties. On the other hand, the subdominantsubspace of is fundamental for resolving multirank signals,or equivalently for beamforming in the presence of random un-certainties with known second-order statistics. We call the mul-tirank beamformers that resolve signals with rank-1 covariances

matched direction beamformers and the multirank beamformersthat resolve signals with multirank covariances matched sub-space beamformers. Our use of language, and view of matcheddirection and matched subspace scenarios, is consistent withthe use in [27]–[31], where matched direction detectors andmatched subspace detectors are developed.

More importantly, we show that matched direction andmatched subspace beamformers consist of a collection ofeigenvalue beamformers, where each eigenvalue beamformerextracts a fraction of the signal power from an eigenmodeof . A multirank beamformer diversity combines a set ofeigenvalue beamformers (dominant or subdominant) to capturea larger fraction of the signal power. We show that there existsa tradeoff between the fraction of the signal power captured atthe multirank beamformer output and the angular resolution ofthe beamformer. The eigenvalue beamformer associated withthe most dominant eigenvalue of offers the best angularresolution for resolving signals with rank-1 covariances. Onthe other hand, the best angular resolution for resolving signalswith multirank covariances is provided by the eigenvaluebeamformer corresponding to the most subdominant eigen-value of . Hence, eigenvalue beamformers are fundamentalfor resolving signals that are drawn from a multidimensionalsubspace, as each eigenvalue beamformer extracts signalinformation from an orthogonal subspace mode of the errorcovariance matrix and has an eigenvalue-dependent resolution.Numerical examples presented in Section VI demonstrate thekey role of eigenvalues of in resolving signals of interest.Finally, we note that preliminary versions of this paper werereported in [32]–[34].

II. SIGNAL MODELS

Consider the general -dimensional data model, consisting of a signal of interest plus interference

and noise . Assuming that , , and are un-correlated and have zero means, we may express the measure-ment covariance matrix as

(2)

where is the signal covariance,is the interference covariance, and isthe noise covariance. We assume that the signal lies in a linearsubspace and consider the three following signal models.

Model 1: Sequence of signals with a rank-1 covariance froma known one-dimensional subspace. The signal is modeled as

(3)

where is a known unit-norm vector, characterizing aknown one-dimensional signal subspace , and is a zero-mean random complex amplitude with variance .In this case, the signal has a rank-1 covariance matrix of theform

(4)

Realizations of are generated by randomly drawing realiza-tions of the complex amplitude , and a sequence of suchrealizations produces a rank-1 experimental(sample) covariance matrix. All realizations of lie in the known


Fig. 1. (a) Realizations of a signal from a known one-dimensional subspace;(b) realizations of a signal from an unknown one-dimensional subspace withina known �-dimensional subspace; and (c) realizations of a signal from a known�-dimensional subspace.

one-dimensional subspace (or on the line) , as illustrated inFig. 1(a). Each dot on the line depicts a realization

, .Model 2: Sequence of signals with a rank-1 covariance from

a known -dimensional subspace.The signal model is

(5)

where is a known matrix with orthonormal

columns , spanning a -dimensional subspace

. The vector is an unknown but fixed unit-norm com-plex vector that determines the orientation of the signal in ,and is a zero-mean random complex amplitude with variance

. Here, the signal is known to lie inside the -dimensionalsubspace but the coordinates of in are unknown. Thesignal has a rank-1 but unknown covariance matrix of the form

(6)

Realizations of are generated by randomly drawing realiza-tions of the complex amplitude , and a sequence of such re-alizations produces a rank-1 experimentalcovariance matrix. All realizations of lie on the unknown line

inside the known -dimensional subspace , as illus-trated in Fig. 1(b).

Model 3: Sequence of signals with a rank- covariance froma known -dimensional subspace. The signal is modeled as

(7)

where is a known matrix with orthonormalcolumns, spanning a -dimensional subspace . The vector

is a zero-mean complex random vector with aknown rank- covariance , normalized so that

, and is a random complex amplitude withvariance independent of . Here the signal has a knownrank- covariance matrix of the form

(8)

Realizations of are generated by randomly drawing realiza-tions of both the orientation vector and the complex amplitude, and a sequence of such realizations

produces a rank- experimental covariance matrix. Each real-ization of is built from a different linear combination of thecolumns of and produces a different point in the subspace

, as illustrated in Fig. 1(c).

Fig. 2. (a) Standing waves drawn from (3), (b) standing waves drawn from (5),and (c) fluctuating waves drawn from (7).

The three signal models introduced here capture a wealth ofeffects in radar, sonar, wireless communications, and biomed-ical imaging, as we describe in the remainder of this section.

A. Radar and Sonar

Let be the unit-norm bearing vector that car-ries the relative phases and amplitudes induced on the array el-ements due to a source radiating from angle at frequency .Depending on one’s assumptions about the mechanism that gen-erates , the wavefront may be modeled as(3), (5), or (7) as we now show.

Case 1: Standing waves from a known one-dimensional sub-space: If and , then may beexpressed as in (3) by setting . This is the typ-ical point source model in sensor array processing. The relativephasings and amplitudes induced by the signal on the array ele-ments do not vary from snapshot to snapshot, as all realizationsof are built from the fixed bearing vector . Thename standing waves indicates that all the wavefronts in the se-quence arrive from the same angle and have the same frequency.This is illustrated in Fig. 2(a) for the case of a uniform lineararray (ULA), where each realization of is a plane wave. In thisfigure, the straight line depicts the ULA and the oblique linesdepict a sequence of plane waves, which maintain their angle ofarrival and frequency over time. The only aspect that changesfrom one realization to another is the complex amplitude alongthe wavefronts.

Case 2: Standing waves from a known -dimensional sub-space: Several scenarios in radar and sonar give rise to thesignal model (5). In a multipath propagation scenario the signal

is a superposition of waves arriving from different angles(paths), all with the fixed frequency , where each pathhas a complex gain associated with a scattering object. Supposethe arriving angles are known, or are quantized, andthe corresponding path gains are unknown butfixed, as in the case of slow-varying multipath during an obser-vation period. Then, can be expressed as , where

, ,and is a zero-mean random complex amplitude. If we de-compose as , where is an orthonormalbasis for , we may then express as in (5) by defining

and .The relative phasings and amplitudes induced by the signal

on the array elements do not vary from snapshot to snapshot, asall realizations of are built from the same linear combinationof the columns of . The name standing waves indicates that all


wavefronts in the sequence have the same “shape”. This is illus-trated in Fig. 2(b) for the case of a ULA, where each realizationof is a “wrinkled” or nonplane wave, but the wrinkling doesnot change from realization to realization.

If the source is broadband with fixed but unknown discretespectrum on frequency band , but arrives from a fixed andknown angle , we may use the DTFT to expressas , where the ,

are the DTFT frequencies spanning , is the un-known but fixed source spectral content at frequency ,

, , and isa zero-mean random complex amplitude. Since columns of theDTFT matrix are orthonormal we may express in the formof (5) by defining , and .

The model (5) is also applicable when there is deterministicuncertainty about the bearing vector, due to array calibrationerrors. Let denote the uncertainty in the parameters of thebearing vector associated with the signal wavefront, so

, and assume that is fixed and known to lie in the in-terval . Let , be asampling of and define . If the sam-pling is sufficiently dense, then the true but unknown maybe represented as a linear combination of adjacent columns of

. Let be the matrix whose columns are the firstleft singular vectors of , where is the numerical rank of .Then, approximately lies in , so we may approximate

as , where is a unknown but fixed vectorof complex coefficients.

Case 3: Fluctuating waves from a known -dimensional sub-space: We now consider a few scenarios which give rise to thesignal model in (7). Consider a narrowband but spa-tially distributed source which radiates with angular power den-sity , from a continuum of angles .The covariance matrix for the source is given by

(9)

For sufficiently small , is numerically rank deficient.Thus, we may approximate with the rank- eigendecom-position , where is the numerical rank of

, contains the largest eigenvaluesof , with , and contains thecorresponding eigenvectors. Consequently, a distributed source

with known covariance may be modeled asin (7), provided that the orientation vector is randomly drawnfrom a distribution with known covariance

, and the complex amplitude is independently drawn from adistribution with variance .

The relative phasings and amplitudes induced by the signalon the array elements vary from snapshot to snapshot, as eachrealization of is built from a different linear combination ofthe columns of . The name fluctuating waves indicates thateach wavefront in the sequence will have a different shape. Thisis illustrated in Fig. 2(c) for the case of a ULA, where eachrealization of is a wrinkled wave, but the wrinkling changesfrom one realization to another.

Remark 1: If the array is an -element ULA with half-wave-length inter-element spacings and the angular power density

is uniform on the angular bandwidth , thenthe approximate numerical rank of is ,where is called the fractional wave-number bandwidth, and is approximated as . Thecolumns of are the first discrete prolate spheroidal wavefunctions and is the corresponding -dimensional Slepiansubspace [35]–[37].

Similarly, a broadband source at a fixed angle withpower spectral density on the frequency band

may be modeled as in (7). The signal covarianceis of the form (9), with playing the role of . The numer-

ical rank of is given by the time-bandwidth product ,where is the transmit time of the wavefront across the arrayand is thus a function of [38]. Hence, we may use the rank-eigendecomposition and model as in (7).

The model (7) is also applicable when the uncertainty inthe bearing vector changes randomly during the observationinterval, e.g., due to flexing of a towed array in sonar. Sup-pose is drawn randomly from a distribution over

. Then, the signal covariance matrixis1

(10)

If we take the numerical rank of to be , thenmay be approximated by a rank- eigendecomposition

. Therefore, we may again expressas , where the random change of from realization torealization is modeled by the -dimensional zero-mean randomvector with known covariance . Ingeneral, the numerical rank of increases as the support of

increases.

B. Communications

Signal models (3), (5), and (7) are applicable to communi-cation with a known code over a channel modeled as a -tapFIR filter. Let be an matrix whose columns are the

time shifts of the length code and let be thevector of channel coefficients. Then, the received signal is ofthe form , where denotes the transmitted bit oramplitude. If is assumed to be known from pilot symbols,then may be written as (3) by definingand . If is unknown but fixed, we may de-compose as , where is an orthonormal basis for

, and then write as in (5) by definingand . Finally, if is assumed to be randomwith known covariance , then we may ex-press as in (7) by defining and

.

1Provided that the variance of � is small, (10) may be approximated by acoherence loss model of the form � � � �� , where �denotes the Hadamard matrix product and �� is a real-valued symmetricToeplitz matrix, whose elements are Gaussian functions when � is Gaussianand sinc functions when � is uniform [10], [39]–[41].


C. Biomedical Imaging

Models (3), (5), and (7) are also relevant in biomedicalimaging. In electromagnetic brain imaging [42] an array ofelectric and/or magnetic sensors is used to measure the electricand/or magnetic fields produced due to the electrical activityof the brain. Focal sources in the brain are modeled as equiv-alent current dipoles, which are parameterized by location,orientation, and strength. For a given source location andbrain/sensor array geometry, the measured signal is describedas the weighted sum of the contributions due to unit strengthdipoles oriented in the three coordinate directions. Let be an

matrix, whose columns are the signals at the arraydue to unit strength sources in the -, -, and -coordinates.Then, the signal due to a source with orientation vector andamplitude may be expressed as . If is assumedto be known based on MRI and anatomical constraints, then

can be written as (3) by defining and. If is assumed to be unknown but fixed,

and we decompose as , where the matrixis an orthonormal basis for , then we may write as

in (5) by defining and .Finally, if is assumed to be random with known covari-ance , as in the rotating dipole model of[43], then the signal may be modeled as in (7) by defining

and .In Sections III–V, we will develop eigenvalue beamforming

methods to separate signals of the form (3), (5), and (7) frominterference and noise. We treat as a wavefront and use theterminology of sensor array processing for convenience.

III. MULTIRANK MVDR BEAMFORMING

Let be the output of a vector beamformer, applied to the array measurement vector .

In standard MVDR beamforming we design to minimizethe output power under theconstraint (u.c.) that the beamformer produces a unit-magnituderesponse to a waveform with a known signature vector . Thatis

(11)

where is the covariance of the array measurementvector . The constraint , , guarantees thatthe beamformer yields the desired response to any wavefrontwith the known signature vector . The MVDR beamformerperforms well and predictably when the signal model is of theform (3). However, when the signal of interest follows (5) or (7),the MVDR beamformer cannot match to the unknown signaturevector and its performance degrades unpredictably.

For signals of the form (5) and (7), where the -dimensionalsignal subspace is known, but the signal orientation is un-known, it is more natural to design a matrix beamformer

and control the beamformer response inthe subspace , which contains the signal of interest. Hence,we constrain our matrix beamformer to satisfy

(12)

where is a left-orthogonal matrix, i.e.,. This subspace constraint is equivalent to the constraint

where is anorthogonal matrix. This shows that in (12) images thelinear combinations (or vectors) as and the linearcombinations as zero. These are called distortionless andzero-forcing constraints, respectively. The problem of choosing

is deferred to Section IV. For now, we assume is an arbi-trary left-orthogonal matrix.

The matrix beamformer images the array measurementvector to the beamformer output vector . We wishto design to minimize the output power

, while forcing the subspace constraint in (12):

(13)

This may be viewed as a multirank generalization of the stan-dard MVDR beamformer. We note that multirank beamformingproblems of the form (13) have been considered in [19] and [20].In [19] and [20], the constraint matrix is prespecified or dataindependent. When and are vectors, the constraint in (13)is similar in form to the constraint of a Frost beamformer [22].Note that the desired response in the Frost beamformer is alsoprespecified and data independent.

The solution to (13) may be easily determined using themethod of Lagrange multipliers and completing the square toobtain the optimum rank- MVDR beamformer andthe minimum output power :

(14)

(15)

The multirank MVDR beamformer may also be formu-lated using a generalization of the GSC introduced in [26]. Al-though the GSC is not necessary for implementing , it isessential for understanding and interpreting multirank MVDRbeamforming. Let be a left-orthogonal matrixthat makes unitary and express the multirankbeamformer as

(16)

where . The beamformer outputis then written as

(17)

where , , and are the vectorsshown in the GSC diagram in Fig. 3.

In the GSC, the vector is decomposed into two sets of co-ordinates and , with the composite covari-ance matrix

(18)


Fig. 3. Generalized sidelobe canceller.

The top branch output is estimated from the bottom branchoutput , using the LMMSE filter . Hence,

is the error in estimating from . The errorcovariance is given by

(19)

which may also be expressed as

(20)

The output covariance matrix and the outputpower of the multirank MVDR beamformer areexpressed in terms of as

(21)

(22)

The top branch of the GSC passes the part of that lies inthe signal subspace and blocks the part of that lies inthe orthogonal subspace . Therefore, the signalis passed through the top branch along with the parts of inter-ference and noise that lie in . That is containsthe signal of interest and the resolutions of the interference andnoise onto . The bottom branch of the GSC passes the partsof interference and noise that lie in . However, due to the or-thogonality of and the signal does not leak through thebottom branch, and contains the resolutions of theinterference and noise onto , but no signal. In estimatingfrom , the interference that has leaked through the signal sub-space is estimated from the interference that lies inand subtracted out. Thus, the error vector contains the signal ofinterest, and reduced interference and noise. The output vector

is formed by transforming the error vector by the constraintmatrix . This is a subspace selection or coordinate selectionstep, which may be designed to extract the signal of interest from

. In beamforming applications the matrix may be steered toa particular angle , or more generally scanned to a particularvalue of a parameter of the uncertain signature vector, and ateach the output power is computed:

(23)

The plot of the output power versus is called a bearingresponse pattern.

Remark 2: From (9), it is easy to see that the basiscorresponding to a Slepian subspace steered to angle may beexpressed as

(24)

where corresponds torelative phasings on a ULA steered to electrical angle and

is the Slepian basis corresponding to angular bandwidth, centered at . The electrical angle is related

to the geometrical angle as .

IV. SUBSPACE SELECTION FOR EIGENVALUE BEAMFORMING:MATCHED DIRECTION BEAMFORMING VERSUS

MATCHED SUBSPACE BEAMFORMING

We now address the design of the constraint matrix to re-solve standing waves (signals of the form (5)) and fluctuatingwaves (signals of the form (7)). For ease of exposition, we as-sume that the interference covariance matrix is rank-1 ofthe form , so that .Inserting in (19), using and the matrix inversionlemma [44], we obtain the error covariance matrix

(25)

The first term in (25), , is the signal covariancematrix after it is passed through the top branch of the GSC.For standing waves (5) with given by (6) this term is

. For fluctuating waves (7) with the mul-tirank covariance given by (8) we have ,where is diagonal.2 The second term onthe right-hand side (RHS) of (25) is the covariance matrix for theinterfering wavefront, after it is suppressed by the GSC. The topbranch of the GSC reduces the interference power by a factorof . The bottom branch further suppresses the interfer-ence power by a factor of .

Remark 3: If the bottom branch of the GSC is switched off,the beamformer becomes a multirank Bartlett beamformer[36], [37], and will reduce to :

(26)

Thus, the factor represents the extra interference suppressionobtained with a multirank MVDR beamformer compared to amultirank Bartlett beamformer.

The output power of the multirank MVDR beamformeris related to the error covariance matrix via (22). We nowconsider choosing so that captures the signalpower for standing waves and for fluctuating waves.

A. Standing Waves and Matched Direction Beamforming

In the case of standing waves (5), the error covariance matrixis given by

(27)

2Without loss of generality, we assume � � �� ,since the known� can always be diagonalized and the eigenvector matrix of� can be absorbed in .


If the signal subspace and the interference subspaceare well separated, then the interference will be suppressed bythe GSC through and . In that case, the rank-1 signalterm contributes to only the largest eigenvalue of .Therefore, must carry the dominant eigenvector of . Thedominant eigenvector of identifies and the dominanteigenvalue of gives a noisy but interference free estimateof .

However, if the interference term is signifi-cant, the signal and interference terms together will determinethe two largest eigenvalues of . Thus, to prevent loss ofsignal power, must select the two most dominant eigenvec-tors of . The sum of the two largest eigenvalues of thencaptures all of the signal power, plus a fraction of the interfer-ence power and the noise power. In the more general case wherethe interference covariance matrix is not negligible and has rank

, or non-negligible interferer are present,the matrix must select the dominant eigenvectors of toavoid loss of signal power. The sum of the largest eigenvaluesof captures all of the signal power, and we may estimatethe signal power at the multirank beamformer output as

(28)

Each eigenvalue of captures an unknown fraction of thesignal plus interference power, plus .

The multirank beamformer , constructed using the dom-inant eigenvectors of , may be called a matched directionbeamformer, as it is designed to resolve signals that are drawnfrom an unknown direction inside the known subspace

.Remark 4: Matched direction beamforming may be posed as

a constrained max-min problem of the form

(29)

where is the set of all left-orthogonal matrices. Thus, ourmatched direction beamformers generalize the minimum vari-ance CDMA receivers of [23], [24] and the biomagnetic spatialfilters of [25], both of which are vector beamformersthat solve the max-min problem

(30)

where is the set of all unit-norm complex vectors.

B. Fluctuating Waves and Matched Subspace Beamforming

We now consider the case of fluctuating waves (7), wherethe signal covariance is . For expositionwe consider the Slepian case where . So the errorcovariance matrix in (25) is given by

(31)

When the interference term is negligible, allthe eigenvalues of are equal to ,

. Hence, may be selected as any collection ofof the eigenvectors of and may be estimated

as . When the interfer-ence term is significant, the largest eigenvalue of is contam-inated by the interference. However, the remaining eigenvaluesare equal to , . Thus, interfer-ence is eliminated if is selected from the subdominanteigenvectors of . In this case an estimate of the signal powerat the beamformer output is

(32)

We have deliberately written the index of the summation in (32)from to to emphasize that we start by including themost subdominant eigenvector of in and work towardmore dominant eigenvectors. In Section IV-C, we show that themost subdominant eigenvalue of gives the best angular res-olution for resolving fluctuating waves, and that there exists atradeoff between angular resolution and the fraction of signalpower captured at the beamformer output.

The beamformer , constructed from the subdominanteigenvectors of , resolves signals that are drawn from aknown multidimensional subspace, and is termed a matchedsubspace beamformer. Provided that the rank of the interferenceterm in is less than , the matched subspace beamformer

yields a noisy but interference free estimate of . Sinceeach eigenvalue of captures of the signal power ,we could normalize (32) by to make the signal term inequal to . However, this would scale the noise power by afactor of .

When the known is not proportional tothe identity matrix, then

. If the signal and interference subspaces are well sepa-rated the interference will be suppressed and the th eigenvalueof will capture of the signal power, and an estimateof the signal power at the beamformer output is given by (32).Since , the most subdominant eigenvalueof captures the smallest fraction of the signal power.However, this eigenvalue gives the best angular resolution forresolving fluctuating waves. Note that normalization of by

forces the signal power in to be ,but scales the noise power proportionally. When the interferenceterm is significant, but has rank , then the subdom-inant eigenvalues of provide interference free estimates offractions of the signal power.

Remark 5: Matched subspace beamforming may be posed asa min-min problem of the form

(33)

where is the set of all left orthogonal matrices. Themin-min problem in (33) is equivalent to minimizing the outputpower under the quadratic constraint

[45]. This establishes a connection betweenmatched subspace beamformers and robust adaptive beam-formers of [10], which enforce a quadratic constraint of theform .


C. Eigenvalue Beamforming

In both matched direction and matched subspace beam-forming, the th column of the multirank beamformer

is an eigenvalue beamformer ofthe form

(34)

where is the th column of . The outputpower of , given by , isan eigenvalue of . In matched directionbeamforming is the th most dominant eigenvector ofand is the th most dominant eigenvalue. In matched sub-space beamforming is the th most subdominant eigenvectorof and is its th most subdominant eigenvalue. Theeigenvalue beamformer extracts a fraction of the signalpower from an eigenmode of . When we steer aroundin the eigenvalue beamformers and their output powersvary with , by the virtue of their dependence on . Wecall the plot of versus an eigenvalue bearing responsepattern.

Matched direction and matched subspace beamforming arediversity combining techniques, in which per mode (per sub-space dimension) output powers extracted by eigenvalue beam-formers are summed up to produce an estimate of the signalpower. This is analogous to wireless communications where un-correlated paths are diversity combined. When the signal poweris distributed among multiple directions in the signal subspace

, either because the signal covariance is multirank or be-cause a signal with rank-1 covariance is in the presence of in-terference, each eigenvalue beamformer captures a fraction ofthe signal power, and the sum of the output powers is a diver-sity combination designed to capture all of the signal power.However, we shall show next that angular resolution decreasesas the fraction of the signal power captured by the multirankbeamformer increases.

Each eigenvalue beamformer extracts signal informationfrom an orthogonal subspace mode, at a different resolution.Hence, eigenvalue beamformers are fundamental for resolvingsignals that are drawn from a multidimensional subspace,and monitoring the output powers of eigenvalue beamformersindividually can be more insightful for resolving standing andfluctuating waves than monitoring their diversity sums.

D. Resolution Analysis

In the following resolution analysis, without loss of gener-ality, we assume that the signal subspace is a Slepiansubspace of dimension , which is steered around in .However, the argument easily extends to other signal subspaces,which in general are steered around in a parameter.

We first consider the matched subspace case. Suppose a fluc-tuating wave with angular bandwidth , centered at ,and with covariance matrix ,is incident on the array. If the interference is negligible, then

Fig. 4. Plots of the eigenvalues of �� versus �.

when is steered to angle , the error covariance matrixis given by

(35)

where . The th eigen-value of is equal to , and itdepends on the location of the source and the electrical angle

. The eigenvalues of are squares of the cosines of theprincipal angles [46] between and . When isfar from such that the angular bandwidthfor and for do notoverlap, all the principal cosines between andare zero. As moves closer to , so that and begin tooverlap, a few of the principal cosines (cosines of principal an-gles) become nonzero, while the rest stay close to zero. As theoverlap between and increases, the number of nonzeroprincipal cosines grows. Finally when and overlap com-pletely all principal cosines become one. This indicates that theangular bandwidth over which has only one nonzeroeigenvalue is wider than the angular bandwidth over which ithas nonzero eigenvalues.

Fig. 4 shows the evolution of the eigenvalues of versusfor the case where is a dimensional Slepian

subspace, centered at . We notice that as we move fromthe dominant eigenvalue towards the subdomi-nant one the eigenvalue bearing response patternsaround become sharper. This shows that the subdomi-nant eigenvalues of offer higher angular resolution forresolving fluctuating waves than the dominant ones, and pointsto a tradeoff between angular resolution and the fraction of thesignal power captured by the matched subspace beamformer.The eigenvalue beamformer associated with the most subdomi-nant eigenvalue of provides the highest resolution, but cap-tures only a small fraction of the signal power. We can cap-


ture a larger fraction of the signal power by diversity combiningthe eigenvalue beamformers associated with other subdominanteigenvalues of at the cost of reduced resolution. We presentnumerical examples to demonstrate this tradeoff in Section V.

Remark 6: The tradeoff between resolution and the fractionof signal power captured by the matched subspace beamformerextends to the case where ,

is diagonal, as shown in the Appendix. The mostsubdominant eigenvalue beamformer, which offers the highestangular resolution, captures the smallest per mode signal power

.The tradeoff between resolution and signal power also applies

in the matched direction case. If the interference is negligible themost dominant eigenvalue beamformer captures all of the signalpower and fixes the angular resolution. If a strong interferer ispresent at a nearby angle, then when is steered nearby,a fraction of the signal plus interference power will be cap-tured by the second most dominant eigenvalue of . Diversitycombining the two most dominant eigenvalue beamformers cap-tures all the signal power, but allows for more leakage of powerfrom the interferer and degrades the resolution of the source andinterference.

E. Comparison With Multirank Bartlett Beamformer

If the bottom branch of the GSC is switched off, the beam-former will reduce to a multirank Bartlett beamformer ofthe form , and the beamformer output covariancewill be . We can selectto retain either the dominant eigenvalues offor matched direction Bartlett beamforming or the subdomi-nant eigenvalues of for matched subspace Bartlett beam-forming. However, we note that in this case is different than

for matched direction MVDR and matched subspace MVDRbeamforming, because in general the eigenvectors of

and are different. They are thesame only when there is no interference. Matched direction andmatched subspace MVDR beamformers always provide betterinterference suppression than their Bartlett counterparts, due tothe extra interference reduction by .

The second matrix on the RHS of (19) is positive semidefinite(PSD), so the Bartlett matrix and theerror covariance matrix satisfythe inequality

(36)

where means that is PSD. This matrix in-equality holds at every electrical angle and is true even whenthere is no interference.3 The matrix inequality in (36) impliesthat at any given the th eigenvalue of is always greaterthan or equal to the th eigenvalue of . Thus, the bearingresponse pattern of the th Bartlett eigenvalue beamformer al-ways lies on or above the bearing response pattern for the thMVDR eigenvalue beamformer. Since both and

3Equality holds, if there is no interference and�� is steered to the electricalangle where the signal is located.

capture the same amount of noise power, and peakaround the same angle with nearly (within the factor ) thesame peak value, we conclude that the th MVDR eigenvaluebeamformer has a better angular resolution than the th Bartletteigenvalue beamformer. Numerical examples to be presentedvalidate this observation.

V. NUMERICAL EXAMPLES

We now demonstrate the role of the eigenvalue beamformersin resolving signals of the form (5) and (7), and study their per-formance in the presence of signal subspace mismatch usingsimple numerical examples. We consider a ULA withelements and half-wavelength inter-element spacing .Four narrowband sources of the rank-1 form (5) and four sourcesof the rank- form (7) are incident on the array. All sourcesare drawn from the dimensional Slepian sub-space, so is the four-dimensional Slepian subspace withfractional wavenumber bandwidth . Note that althoughwe discussed the Slepian subspaces in the context of fluctuatingwaves, they are also relevant for modeling standing waves. Anexample is the case where the unknown orientation vectoris drawn from a distribution with known covariance, but thenstays unchanged over a sequence of snapshots. The total signalpower for each source is and the noise powerat each sensor element is , resulting in an input SNRof 10 dB. The four rank-1 sources are centered at electrical an-gles 0, 1, 2.22, and 2.29 rad, and the four rank-4 sources arecentered at electrical angles , , , and rad.Going from negative to positive electrical angles, the subspaceorientation vectors for the rank-1 sources are, respectively,

,, and .

These orientation vectors are fixed, but they are unknown to thebeamformer. The orientation vectors for the multirank sourcesare randomly drawn from a distribution with covariance

.

A. Known Data Covariance

In the first example, we assume that the theoretical data co-variance is known, and that there is no mismatch betweenthe signal subspace used in beamforming and the ac-tual signal subspace. Fig. 5(a) and (b) shows the bearing re-sponse patterns for the conventional and standard MVDR beam-formers, respectively. These plots are obtained by steering the

bearing vector in electrical angle and computingthe output power (for conventional) and

(for MVDR) at each . As the plotsshow, the conventional beamformer produces false peaks nearrank-1 sources and the MVDR beamformer does not resolvethese sources at all. They also render wide peaks around themultirank sources and underestimate their powers by approxi-mately 6 dB.

The behavior of the Bartlett beamformer may be explained asfollows. Suppose there is no interference. Then, the power at theoutput of the Bartlett beamformer is

. When we sweep over , for a rank-1 source with covari-ance , there exists a narrow


Fig. 5. Bearing response patterns for (a) conventional (Bartlett) beamformerand (b) standard MVDR beamformer. Dashed vertical lines indicate true sourcelocations. Going from negative to positive electrical angles in each plot the firstfour sources generate fluctuating waves with rank-4 covariances and the nextfour sources generate standing waves with rank-1 covariances.

band of angles around inside which is notsmall. In this angular band the conventional beamformer passesa fraction of the source power and the bearing response patternpeaks. This also happens for the multirank sources with covari-ance . The power of the source inthis case is equally distributed among four dimensions in theSlepian subspace and when is close to ,is approximately one, and the power at the output of the beam-former is approximately 6 dB.

The behavior of the MVDR beamformer may be explainedusing the GSC in Fig. 3 by replacing with andchoosing such that is unitary. Thetop branch of the GSC is a conventional beamformer. Whenis close to this branch passes some of the signal. This is truefor both rank-1 and multirank sources. However, since is mis-matched with the actual signal subspace some of the signal leaksthrough the bottom branch, which after LMMSE estimation re-sults in signal suppression

Fig. 6(a)–(d) shows the eigenvalue bearing response patterns, . We notice that the most dominant

eigenvalue resolves the rank-1 sources. It captures allthe power of the rank-1 sources at 0 and 1 rad. It alsocaptures a large fraction of the powers of the rank-1 sources at

2.22 and 2.29 rad. However, since each source actsas an interference for the other a small fraction (-18 dB) of thesignal power is captured by the second most dominant eigen-value . The subdominant eigenvalues and

in Fig. 6(c) and (d) show no indication of presence ofthe rank-1 sources.

The dominant eigenvalue also gives interfer-ence free, but noisy, estimates of a fraction of the powerof the multirank sources, as the interference nearby issufficiently suppressed by the GSC. The peak values of

at locations of multirank sources are approxi-mately 6 dB, which is consistent with the signalcovariance model . However, note that

is wider around multirank sources than the sub-dominant eigenvalue bearing response patterns ,

, and thus offers worse angular resolution. In partic-ular the multirank sources at and are notwell-resolved by .

Fig. 6(b)–(d) shows that each of the subdominant eigenvalues, , captures a fraction ( in

Fig. 6. MVDR Eigenvalue bearing response patterns: (a) �� ,(b) �� , (c) �� , and (d) �� . Dashed verticallines indicate true source locations. Going from negative to positive electricalangles in each plot the first four sources generate fluctuating waves with rank-4covariances and the next four sources generate standing waves with rank-1covariances.

this example) of the power of the multirank sources. They alsoshow that the most subdominant eigenvalue offersthe highest angular resolution for resolving multirank sources.We can diversity combine the eigenvalue beamformers (multi-rank matched direction and matched subspace beamforming) tocapture a larger fraction of the signal power at the expense ofresolution.

Fig. 7(a)–(d) shows the Bartlett eigenvalue bearing responsepatterns , . Each of the subdominanteigenvalues of captures the same fraction of the signalpower as its MVDR counterpart. However, the most dominantBartlett eigenvalue beamformer no longer provides an inter-ference free estimate of the power of the multirank source at

0.049 rad, as it can not sufficiently suppress the rank-1 in-terferer at 0 rad. Furthermore, the resolution of each Bartletteigenvalue beamformer around a multirank source is less thanthe resolution of its MVDR counterpart. For example, whilethe most dominant Bartlett eigenvalue beamformer resolves therank-1 signals at 0 and 1 rad it fails to resolve the tworank-1 sources at 2.22 and 2.29 rad, which are closein angle. Comparison of and with

and around the rank-1 sources at2.22 and 2.29 rad clearly shows the superior interferencesuppression capability of MVDR eigenvalue beamformers.

B. Experimental Data Covariance

In most applications the data covariance matrix is notknown and has to be estimated from multiple array snapshots.Assuming copies of the array measurement vector , say

, are available, we may use the sample covari-ance matrix in place of .We consider the case where the number of data samplesis . The type of sources and their locations


Fig. 7. Bartlett eigenvalue bearing response patterns: (a) �� ,(b) �� , (c) �� , and (d) �� . Dashed verticallines indicate true source locations. Going from negative to positive electricalangles in each plot the first four sources generate fluctuating waves with rank-4covariances and the next four sources generate standing waves with rank-1covariances.

Fig. 8. Eigenvalue bearing response patterns: (a) �� ,(b) �� , (c) �� , and (d) �� . The datacovariance matrix � is estimated from � � �� snapshots. Dashedvertical lines indicate true source locations. Going from negative to positiveelectrical angles in each plot the first four sources generate fluctuating waveswith rank-4 covariances and the next four sources generate standing waveswith rank-1 covariances.

are identical to those of Section V-A. Fig. 8(a)–(d) shows theplots of the eigenvalues of versus

. Similar to the known data covariance case, the dominanteigenvalues of resolve the rank-1 sources and the sub-dominant ones resolve the multirank sources. However, dueto low sample support, loss of signal power is observed. Thissuggests that does not perfectly fit the four dimensional

Fig. 9. Eigenvalue bearing response patterns: (a) �� ,(b) �� , and (c) �� . The signal is modeled with a �� dimensional Slepian subspace and is mismatched to the actual �� dimensional Slepian signal subspace. Dashed vertical lines indicate true sourcelocations. Going from negative to positive electrical angles in each plot the firstfour sources generate fluctuating waves with rank-4 covariances and the nextfour sources generate standing waves with rank-1 covariances.

signal subspace model, and brings us to the question of signalsubspace mismatch.

C. Signal Subspace Mismatch

We now investigate the effect of signal subspace mis-match, where the Slepian subspace used in beamforming isdifferent from the actual Slepian subspace used in generatingthe data. Fig. 9(a)–(c) shows the plots of the eigenvalues of

versus , when is takento be the dimensional Slepian subspace instead ofthe four dimensional Slepian subspace used in generating thedata. In this case, has three eigenvalues. As can be seen,the dominant eigenvalue of captures a very small fraction(-15 dB) of the power of the rank-1 sources at ,2.22, and 2.29 rad. At first it may appear thatcaptures all the power of the rank-1 source at 0. However,the power at that location is due to leakage of power from thenearby multirank source, a result of signal subspace mismatch.

The subdominant eigenvalues of , say andin Fig. 9(b) and (c), still resolve the multirank

sources. This behavior may be explained using the GSC inFig. 3. Roughly speaking, since here the dimension of thesignal subspace is underestimated (three instead of four) thepart of the signal that lies in the subspace spanned by the fourthSlepian basis vector, associated with , leaks throughthe bottom branch of the GSC. The Slepian basis vectorsassociated with are different from those associatedwith . However, the three-dimensional Slepian sub-space with lies inside the four-dimensional Slepiansubspace with . Therefore, when estimating the topbranch from the bottom one and forming the error, the signal


Fig. 10. Eigenvalue bearing response patterns: (a) �� ,(b) �� , (c) �� , (d) �� , and (e) �� .The signal is modeled with a �� dimensional Slepian subspace andis mismatched to the actual �� dimensional Slepian signal subspace.Dashed vertical lines indicate true source locations. Going from negative topositive electrical angles in each plot the first four sources generate fluctuatingwaves with rank-4 covariances and the next four sources generate standingwaves with rank-1 covariances.

components in the top branch that are correlated with those inthe bottom branch are suppressed. The signal suppression canbe more extreme for resolving rank-1 sources than multiranksources, because the power of a rank-4 source is equally dis-tributed among four modes in the Slepian subspace. The powerof a rank-1 source on the other hand is concentrated along onedirection in the space and hence may be significantly reducedwhen the dimension of the signal subspace is underestimated.

Next we overestimate the signal subspace dimension bytaking the signal subspace to be the dimensionalSlepian subspace. Fig. 10(a)–(d) shows the eigenvalue bearingresponse patterns for this case, where has five eigenvalues.The multirank sources are again resolved by the subdominanteigenvalues of . The rank-1 sources at and areresolved at almost full power with the most dominant eigen-value . However, the rank-1 sources at 2.22 and

2.29 rad are not fully resolved by . Compared tothe previous mismatch scenario, here the dominant eigenvalueof provides better “detectability” (larger power level)around the location of the rank-1 sources. Roughly speaking,since here the dimension of the signal subspace is overestimatedthe signal does not leak through the bottom branch of the GSC.However, the interfering sources are not as strongly suppressed

as before, because less correlated interference passes throughthe bottom branch. This effect is evident in Fig. 10(b), where

peaks in the middle of the rank-1 sources locatedat 2.22 and 2.29 rad.

VI. CONCLUSION

In many practical imaging and beamforming problems thesignature vector of the signal of interest is not perfectly known,but the subspace in which the signature vector lies is knownor can be approximated. The unknown orientation of the signalin the subspace may stay fixed, in which case the signal has arank-one but unknown covariance matrix, or it may change ran-domly from one realization to another, in which case the signalhas a known multirank covariance matrix. We present a uni-fied and systematic treatment of such signal subspace and signalcovariance models and discuss their relevance in radar, sonar,wireless communications, and biomedical imaging.

We derive two multirank generalizations of the MVDR beam-former, namely matched direction and matched subspace beam-formers, by introducing a data dependent constraint matrix thatis designed to extract information from the error covariance ma-trix associated with a GSC. When the signal of interest is drawnfrom an unknown but fixed direction within a known multidi-mensional subspace, and has a rank-1 covariance matrix, it is thedominant eigenvalues of the error covariance matrix that resolvethe signal. When the signal is randomly drawn from a knownmultidimensional subspace, and has a known multirank covari-ance matrix, it is the subdominant eigenvalues that resolve thesignal. In the former case, the constraint matrix is chosen to se-lect the dominant eigenvectors of the error covariance, while inthe latter case it is chosen to select the subdominant eigenvec-tors. This leads to eigenvalue beamforming, where each eigen-value beamformer extracts signal information from an orthog-onal subspace mode, at a different resolution. Matched directionand matched subspace beamforming are diversity combiningtechniques, in which per mode (per subspace dimension) outputpowers of eigenvalue beamformers are diversity combined toproduce an estimate of the signal power. As the fraction of thesignal power captured by the multirank beamformer increasesthe angular resolution decreases.

APPENDIX

Let and be matrices with eigenvaluesand , then [47,

ch. 9]

(A.1)

Set and . Since, ,

and have equal eigenvalues and does notvary with , it is easy to see that the shape of therank- matched subspace bearing response pattern


, associated withthe diagonal but nonidentity , follows the shape of the rank-matched subspace bearing pattern associated with .For exposition, let and , corresponding to a rank-2matched subspace beamformer. Then, we have

(A.2)

Since we can write

(A.3)

Since does not vary with the inequality in (A.3), which holdsfor any , shows that the width ofaround has to be less than or equal to the width of

around .

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Ali Pezeshki (S’95–M’05) received the B.Sc. andM.Sc. degrees in electrical engineering from theUniversity of Tehran, Tehran, Iran, in 1999 and2001, respectively, and the Ph.D. degree in electricalengineering at Colorado State University, Ft. Collins,in 2004.

During 2005, he was a Postdoctoral Researcherwith the Signal Processing and CommunicationsLaboratory at Colorado State University. SinceJanuary 2006, he has been a Postdoctoral ResearchAssociate with The Program in Applied and Com-

putational Mathematics at Princeton University, Princeton, NJ. His researchinterests are in statistical signal processing and coding theory, as they apply tosensing, communications and data networking.

Barry D. Van Veen (S’81–M’86–SM’97–F’02) wasborn in Green Bay, WI. He received the B.S. degreefrom Michigan Technological University, Houghton,in 1983 and the Ph.D. degree from the University ofColorado, Boulder, in 1986, both in electrical engi-neering. He was an ONR Fellow while working onthe Ph.D. degree.

In spring 1987, he was with the Department ofElectrical and Computer Engineering at the Uni-versity of Colorado—Boulder. Since August 1987,he has been with the Department of Electrical and

Computer Engineering (ECE) at the University of Wisconsin–Madison, wherehe is currently a Professor. He coauthored, with S. Haykin, Signals and Systems(New York: Wiley, 1999 (1st ed.) and 2003 (2nd ed.)). His research interestsinclude signal processing for sensor arrays and biomedical applications ofsignal processing.

Dr. Van Veen was a recipient of a 1989 Presidential Young InvestigatorAward from the National Science Foundation and a 1990 IEEE Signal Pro-cessing Society Paper Award. He served as an Associate Editor for the IEEETRANSACTIONS ON SIGNAL PROCESSING and on the IEEE Signal ProcessingSociety’s Statistical Signal and Array Processing Technical Committee andthe Sensor Array and Multichannel Technical Committee. He received theHoldridge Teaching Excellence Award from the ECE Department at theUniversity of Wisconsin in 1997.

Louis L. Scharf (S’67–M’69–SM’77–F’86–LF’07)received the Ph.D. degree from the University ofWashington, Seattle.

From 1971 to 1982, he served as Professor ofelectrical engineering and statistics at ColoradoState University (CSU), Ft. Collins. From 1982 to1985, he was Professor and Chairman of electricaland computer engineering at the University ofRhode Island, Kingston. From 1985 to 2000, he wasProfessor of electrical and computer engineeringat the University of Colorado, Boulder. In January

2001, he rejoined CSU as Professor of electrical and computer engineeringand statistics. He has held several visiting positions here and abroad, includingthe Ecole Superieure d’Electricité, Gif-sur-Yvette, France; Ecole NationaleSuperieure des Télécommunications, Paris, France; EURECOM, Nice, Italy;the University of La Plata, La Plata, Argentina; Duke University, Durham,NC; the University of Wisconsin, Madison; and the University of Tromsø,

Tromsø, Norway. His interests are in statistical signal processing, as it appliesto adaptive radar, sonar, and wireless communication. His most importantcontributions to date are to invariance theories for detection and estimation;matched and adaptive subspace detectors and estimators for radar, sonar, anddata communication; and canonical decompositions for reduced dimensionalfiltering and quantizing. His current interests are in rapidly adaptive receiverdesign for space-time and frequency-time signal processing in the wirelesscommunication channel.

Prof. Scharf was Technical Program Chair for the 1980 IEEE InternationalConference on Acoustics, Speech, and Signal Processing (ICASSP), Denver,CO; Tutorials Chair for ICASSP 2001, Salt Lake City, UT; and Technical Pro-gram Chair for the Asilomar Conference on Signals, Systems, and Computers2002. He is past-Chair of the Fellow Committee for the IEEE Signal ProcessingSociety and serves on its Technical Committees for Theory and Methods and forSensor Arrays and Multichannel Signal Processing. He has received numerousawards for his research contributions to statistical signal processing, includingan IEEE Distinguished Lectureship, an IEEE Third Millennium Medal, and theTechnical Achievement and Society Awards from the IEEE Signal ProcessingSociety.

Henry Cox (S’61–M’64–SM’82–F’83–LF’01)received the B.S. degree in physics from the Collegeof the Holy Cross, Worcester, MA, in 1959 and theSc.D. degree in electrical engineering from Massa-chusetts Institute of Technology (MIT), Cambridge,in 1963.

He is currently a Senior Fellow at LockheedMartin DoD Systems, Arlington, VA. He was SeniorVice President and Chief Technology Officer (CTO)of Orincon before the merger with Lockheed Martinin June 2003. Previously, he was Divisional Vice

President of BBN Systems. During his carrier as a naval officer, he held anumber of important research and development positions, retiring as Captainof the U.S. Navy. He was the Project Manager for the Undersea SurveillanceProject, Division Director at Defense Advanced Research Projects Agency(DARPA), and Officer in Charge of the New London Laboratory of the NavalUnderwater Systems Center. He is the author of more than 60 technical papers.His recent contributions include development and application of robust adaptivebeamforming algorithms to large towed arrays, development of simplifiedtechniques for matched field processing, pioneering work in bistatic activesonar, new processing methods for Doppler exploitation in active sonar, newtechniques for passive ranging, and development of system approaches for pas-sive distributed systems, shallow-water active sonar and offboard sensors. Hisinterests include antisubmarine warfare, adaptive signal processing, antennaarrays, underwater acoustics, and sonar system design and analysis.

Dr. Cox is a Fellow of the Acoustical Society of America. He was awardedthe Gold Medal of the American Society of Naval Engineers. He received theDistinguished Technical Achievement Award of the Ocean Engineering Society.He was elected to the National Academy of Engineering in 2002.

Magnus Lundberg Nordenvaad (M’07) was bornin Luleå, Sweden, in 1973. He received the M.Sc.degree in computer science and engineering fromLuleå University of Technology (LTU), Luleå,Sweden, in 1998 and the Ph.D. degree in signal pro-cessing from the School of Electrical and ComputerEngineering, Chalmers University of Technology,Gothenburg, Sweden, in 2003.

He has held visiting positions at Purdue University,West Lafayette, IN, and Colorado State University,Ft. Collins. Currently, he is an Assistant Professor

with the Department of Computer Science and Electrical Engineering, LTU, andholds a senior research position at the Swedish Defence Research Agency. Hisresearch interests lie in statistical signal processing and how it applies to digitalcommunications, radar, sonar, process diagnostics and land-mine detection.

Dr. Lundberg has received several awards and grants for his research. Theseinclude the 1998 MD110 User-Group Award for the Best Master’s Thesis in thetelecommunications area in Sweden that year and a postdoctoral scholarshipaward from the Swedish Research Council.

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