Signal Processing 88 (2008

erpea

na,

ity L

et, Ca

Received 14 March 2007; received in revised form 31 October 2007; accepted 9 January 2008

Available online 15 January 2008

Keywords: Adaptive beamforming; Coherent processing; WINGS; Waveeld modeling; Robust

tems, wideband signals will be used to supporthigher data rate services [5].

ARTICLE IN PRESS

fax: +972 4 8795315.

E-mail addresses: mirid@rafael.co.il (M.A. Doron),

amir.nevet@RemonMedical.com (A. Nevet).

Tapped-delay line lters with adaptive coef-

cients are often used for wideband adaptive arrays

0165-1684/$ - see front matter r 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.sigpro.2008.01.007

1This author was with RAFAEL Armament Development

Authority Ltd., when this work was performed.1. Introduction

In the eld of wireless communication, radar,acoustics and seismic sensing, wideband adaptivebeamforming is an effective technique for rejectinginterference signals whose incident direction ofarrival (DOA) on a sensor arrays differs from that

of the desired signals [1]. The potential for usingadaptive beamforming to improve the performanceof sensor arrays was already recognized during theearly 1960s in the elds of sonar, radar and seismicsignal processing. Recently, smart antennas havebeen proposed as a promising solution that cansignicantly increase the data rate and improve thequality of wireless transmission [24]. Moreover, infuture generations of wireless communication sys-

Corresponding author. Tel.: +972 528890359;Abstract

In this paper we consider the application of interpolation-based focusing for wideband array processing and direction

nding using waveeld interpolation matrices. The focusing approach exploits the wideband characteristic of the signals

for coherent subspace processing, achieving improved performance and reduced computational complexity. We investigate

the use of waveeld modeling-based frequency transformations of the array manifold, which are data independent and do

not require any initial direction of arrival (DOA) estimates. We apply these transformations to the array wideband data,

constructing a virtual waveeld-interpolated narrowband-generated subspace (WINGS), on which any narrowband

adaptive beamforming scheme may be implemented. WINGS transformations are also shown to be optimal for frequency-

invariant beamforming (FIB). We treat the important issue of robust WINGS processing, i.e. reducing the

transformations sensitivity to errors in the array manifold, such as sensor gain, phase and location errors. We develop

two novel robust forms of the WINGS transformations designed to ensure robustness by controlling the noise gain of the

transformation, and study their performance numerically. Finally, we demonstrate via simulation the effectiveness of

WINGS processing for minimum variance distortionless response (MVDR) beamforming.

r 2008 Elsevier B.V. All rights reserved.Robust waveeld intwideband b

Miriam A. DoroaRAFAEL Armament Development Author

bRemon Medical Technologies, 7 Halamish Stre) 15791594

olation for adaptivemforming

, Amir Nevetb,1

td., P.O.B 2250 (23), Haifa 31021, Israel

esarea Industrial Park, Caesarea 38900, Israel

www.elsevier.com/locate/sigpro

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 157915941580[6]. The FIR implementation using the well-knownLMS algorithm is one of the more widely usedimplementations of time-domain broadband adap-tive beamformers. However, the FIR implementa-tion requires many adaptive coefcients and theconvergence of the LMS algorithm tends to be slow.An alternative implementation, utilizing latticestructures, achieves improved convergence rates,e.g. the QR-multi-channel lattice least-squares(QR-MLSL) scheme [7,8]. A different approach toovercome these problems utilizes pole-zero adaptivelters, e.g. [9]. The well-known sample matrixinverse (SMI) method also offers improved conver-gence [1,10], and is typically implemented for thewideband case in the frequency domain, i.e. it isimplemented as a narrowband beamformer ateach frequency bin. All of these methods arecomputationally expensive, have slow convergencerate and are prone to signal cancellation in coherentsource scenarios that may occur in multi-pathenvironments.The wideband focusing approach for adaptive

beamforming based on the concept of signal-subspace alignment was originally proposed in[11]. The focusing approach involved a pre-proces-sor that focuses the signal subspaces at differentfrequencies to a single frequency, followed by atime-domain narrowband beamformer. The meritsof the focusing approach are low computationalcomplexity, the ability to combat signal cancellationin coherent source scenarios and improved conver-gence properties. Focusing matrices were originallyproposed for wideband DOA estimation in thepioneering work of Wang and Kaveh [12,13]. Therewere many improvements to the design of thefocusing matrices, such as the use of rotationalsignal-subspace (RSS) focusing matrices proposedin [14], and later extended in [15]. The focusingapproach, while providing improved performance,typically required preliminary estimates of theDOA, which was a drawback. Focusing proceduresdesigned to achieve robustness against DOA esti-mation errors were developed recently, see e.g. [16].A different focusing approach based on spatial

interpolation, which does not require initial DOAestimates, was proposed and investigated in [17,18].The waveeld interpolation focusing approach [17]is based on a separable representation of the arraymanifold, which linearly separated the array geo-metry and frequency from the plane wave direc-tions, yielding closed form expressions for the

frequency transformations. The closely relatedinterpolated array approach [18] is based on theidea that the array manifold of a virtual array canbe obtained by linear interpolation of the arraymanifold of the real array within a limited sector.The interpolation matrix is computed numericallyby performing a least-squares (LS) t over the arraymanifold in the desired sector. Further interestingworks dealing with related wideband processingschemes can be found in [1924].In this paper we investigate the focusing approach

for array processing applications. We considerwaveeld modeling based [25] frequency transfor-mations of the array manifold, which do not requireinitial DOA estimates and may be applied to anyarray with a known arbitrary geometry. Applyingthese fixed data independent transformations to thearray wideband data, we construct a virtual wave-eld-interpolated narrowband-generated subspace(WINGS) array. The virtual WINGS data have anarrowband array manifold while preserving thewideband spectral content of the wideband signals.Alternatively, WINGS data can be treated as avirtual wideband array having a frequency-invariantbeam pattern. One may apply to the WINGS dataany adaptive beamforming algorithm such as SMIminimum variance distortionless response(MVDR), Frost, LMS, QR-LMSL, QR-GSC, etc.The adaptive WINGS-based wideband beamfor-mers use a single narrowband set of adaptive weights,thus achieving superior performance, fast conver-gence rate and low complexity. The application ofWINGS transformations to frequency-invariantbeamforming (FIB) is examined, showing theoptimality of WINGS transformations for FIB.Special attention is given to reducing the WINGStransformations sensitivity to errors in the arraymanifold, such as sensor gain, phase and locationerrors. We derive two novel forms of the WINGStransformations designed to ensure robustnessby controlling the noise gain of the transforma-tion, and show analytically that the robustWINGS transformations for direction nding en-sure a bounded variance degradation of the DOAestimates.The paper is organized as follows: we formulate

the problem of interest and dene the WINGSfocusing process in Section 2. In Section 3, webriey review the waveeld modeling formalism,and describe a separable representation of the arraymanifold, which linearly separated the array geo-metry and frequency from the plane wave direc-

tions. In Section 4, we use the waveeld modeling

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 15791594 1581formalism to formulate closed form expressions forthe WINGS transformations, and for their trans-formation error. We include useful expressions forthe beamspace WINGS transformations. We thendiscuss several applications of WINGS processing.First we consider the FIB problem, and showoptimality of WINGS transformations for FIB.We then turn to adaptive beamforming applicationsand describe a WINGS framework example forimplementing the wideband MVDR adaptive beam-former. In Section 5, we propose two approachesdesigned to ensure the robustness of the WINGStransformation. The rst employs a transformationbased on a rank-truncated pseudo-inverse, and thelatter approach directly constrains the noise powergain of the transformation. In Section 6, we relatethe variance of DOA estimates to the WINGSinterpolation transformations via the CRB, andshow that robust WINGS focusing for directionnding ensures a limited variance degradation. InSection 7, we numerically investigate the interpola-tion accuracy and the noise gain of WINGStransformations and their robust forms, and illus-trate the performance of WINGS MVDR proces-sing via a simulation example. Finally, Section 8contains the conclusion.

2. Problem formulation and WINGStransformations

Consider an arbitrary array of N sensors,sampling a waveeld generated by P widebandsources, in the presence of additive noise. Tosimplify the exposition, we conne our discussionto the free-eld model; one can nd more generalmodels for reverberant elds in [21,31]. The signalmeasured at the output of the nth sensor can bewritten in this case as

xnt XPp1

spt tnp nnt for T=2ptpT=2;

n 1; . . . ;N, (1)where fsptgPp1 and fnntgNn1 denote the radiatedwideband signals and the additive noise processes,respectively, and T is the observation interval.The parameters {tnp} are the delays associated withthe signal propagation time from the pth source tothe nth sensor. Let fr1; r2; . . . ; rNg denote thecoordinates of the array sensors, where r r;jin 2-D and r r; y;j in 3-D, and let g1; g2; . . . ; gP

be the DOAs of the sources, where g j in 2-D andg y;j in 3-D. The sources are far enough fromthe observing array so that the signal wavefronts areeffectively planar over the array. The observationtime interval is sectioned into K subintervals ofduration Td each, and a discrete Fourier transform(DFT) is applied to each subinterval. The frequencydomain array measurements at the kth subintervalmay be written as

xkoj Acojskoj nkojfor j 1; . . . ; J; k 1; . . . ;K , (2)

where xkoj; skoj and nkoj denote vectorswhose elements are the discrete Fourier coefcientsof the measurements, of the unknown source signalsand of the noise, respectively, at the kth subintervaland frequency oj. Agoj is the NP directionmatrix

Ago ag1 o; ag2o; . . . ; agPo: (3)The vector ago, referred to as the array manifold

vector, is the response of the array to an incidentplane wave at frequency o and DOA g. For an arraycomprised of identical omni-directional uncoupledsensors, the array manifold vector is

agom expfikrm g^g, (4)where g^ denotes a unit vector pointed in thedirection of g, and k o/c is the wave numberassociated with the frequency o. To simplify theexposition, our discussion is conned to spatiallywhite noise. We assume that the noise vectors nkojare independent samples of stationary, zero meancircular complex Gaussian random process, withcovariance s2j I, where the noise variance s

2j is

unknown. The signal vectors skoj are independentsamples of stationary, zero mean circular complexGaussian random process with unknown variancePj. The noise process is assumed uncorrelated withthe signal process.Our goal is to estimate the desired wideband

signals, radiated or transmitted by the sources, viaimplementing adaptive narrowband beamformingmethods on a virtual frequency-invariant narrowbandarray at a single frequency o0. Let Tj denote atransformation that maps the wideband arrayoutput from frequency oj to frequency o0, so thatthe signal subspaces are aligned across the fre-quency bandwidth

TjAgoj Ago0, (5)where oj are the frequency bins within the

bandwidth of the signals and o0 is the focused

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 157915941582frequency, i.e. Tj focuses the signal spaces Ag(oj) atfrequencies {oj} onto the signal subspace Ag(o0) atfrequency o0. Following the approach outlined in[11], we now construct the virtual time-domainvector yk(n) as

ykn XJj1

Tjxkoj eiojnT s

Ago0sn ~nn; (6)where s(n) is the temporal vector of widebandunknown source signals within the focused fre-quency band [o1:oj], Ts is the sampling time intervaland

~nn XJj1

Tjnoj eiojnT s (7)

is the transformed noise. We refer to yk(n) asthe WINGS vector, and to the interpolationfocusing transformations as WINGS transforma-tions. One can see that the temporal WINGS vectoryk(n) has a narrowband array manifold while preser-ving the wideband spectral content of the wideband

signals. Alternatively, WINGS data can be treatedas a virtual wideband array having a frequency-invariant beam pattern, allowing one to implementon it any narrowband adaptive beamformingscheme.In the following we consider data-independent

WINGS transformations that interpolate the wide-band array output from frequency oj to frequencyo0 using the waveeld modeling approach [25].

3. Waveeld modeling review

The WINGS transformation is derived from theseparable representation of the array output, whichlinearly separates the array geometry and frequencyfrom the plane wave directions. This representationis based on the waveeld modeling approach whichwas introduced and examined thoroughly in rela-tionship to array processing in [2528]. In thissection, we review the key results of waveeldmodeling.The waveeld modeling approach is based on the

idea that the output of almost any array of arbitrarygeometry can be written as the result of a vector oflinear functionals, which is called the array samplingoperator, operating on the impinging waveeld. Itwas shown in [25] that assuming this operator to be

linear and continuous, assumptions that are almostalways satised, one can write the signal-dependentpart of the array output as a product of an arraysampling matrix and a waveeld coefcient vector

xo Gowo, (8)where the sampling matrixGo . . . ; g1; g0; g1; . . .is independent of the waveeld and the coefcientvector w is independent of the array. The samplingmatrix G is composed of the coefcients gn of theorthogonal decomposition of the array manifold ag:

gno ZGagof ngdg; (9a)

ago Xn

gnof ng; (9b)

where G is the manifold of possible directions ofarrival:

G j s:t: j 2 p;p for 2D;y;j s:t: y 2 0;p;j 2 p;p for 3D:

((10)

The functions ff ngg comprise a complete andorthogonal basis set in L2(G), and are chosen to be

f ng 12p

p einj for 2D;Ylmy;j for 3D;

l 0; 1; . . . ; m l; . . . ; l;

8>:(11)

where the functions Ylmy;j denote the sphericalharmonics. Note, that for the 2-D case the index nbecomes a double index (l,m), where l 0,1,y,m l,y,l. For convenience of notation weexpress the orthogonal expansion (9) in a singleindex; this may be done by using the mappingn ll 1 m. Let us now rewrite (9b) in thematrix form

ago Gobg; bgn f ng. (12)We see that the array geometry and the frequency

appear only in matrix G, which is our samplingmatrix, whereas the plane wave direction appearsonly in vector b.Let us write the explicit expressions for the

sampling matrix G, for the simple common case ofan array comprising omni-directional and un-coupled sensors. Other more general cases such asarrays mounted on reectors or arrays comprisedof directional sensors are treated in [25]. Inserting

(4) into (9a) yields for omni-directional and

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 15791594 1583uncoupled sensors

Gmno ZGexpfikrm g^gf ngdg hno; rm: (13)

The functions fhno; rg comprise an orthogonalbasis spanning the Hilbert space of all possiblewaveelds generated by plane waves, and aregiven by

hno; r

2p

pinJnkreinj for 2D;

4pil jlkrYlmy;j for 3D;l 0; 1; . . . ; m l; . . . ; l;

8>:(14)

where Jnkr denote the Bessel functions of the rstkind, and jlx denote the spherical Bessel functionsof the rst kind.We should point out that although, hitherto,

vector bg and matrix G(o) have been innite, there isan effective cutoff for nbkmaxrmax, where kmax andrmax are the maximal wave number in the processingband and the maximal sensor distance from theorigin, respectively. This results from the fact thatfor nbkr, the Bessel function Jnkr in (14)decreases faster than exponentially to zero [29].Furthermore, the decay of the Bessel functionbegins when the order n of the Bessel function isequal to its argument. Let n be dened by

jJnkmaxrmaxjo for n4n (15)for some small e of our choice. For most applica-tions it is sufcient to consider n 1:5kmaxrmax. Aquantitative evaluation of the error caused by theBessel function truncation can be found in [25]. Itproves that by this choice, the truncation error isnegligible in comparison to other errors of ourtransformation, which are evaluated later. Hence-forth we shall regard G(o) and bg as the nite matrixand vector, respectively, obtained by truncating theoriginal G(o) and bg at jnj n. Then we can rewritethe separable array manifold representation (12) as

ago Gobg, (16)where G(o) is now an N 2n 1 matrix and bg avector of dimension 2n 1.

4. WINGS transformations

In this section, we use the waveeld modelingformalism to formulate closed form expressions forthe basic WINGS transformations satisfying rela-tion (5). We are interested in a data-independent

transformation that maps the array output fromfrequency oj to frequency o0, so that the signalsubspaces are aligned. Since the signal subspace isdetermined by the array manifold in the direction ofthe sources, such a transformation Tj should mapthe array manifold at one frequency to the arraymanifold in the same direction at another fre-quency:

ago0 Tjagoj ejg 8g, (17)where ejg denotes the transformation error vector.Applying (16) yields the following expression for thetransformation error for any Tj:

ejg G0bg TjGjbg G0 TjGjbg, (18)

where Gj Goj. Let ej be dened as the L2 normof the error

2j 1

N

ZGdgjjejgjj2; (19)

where jj jj is the Euclidian norm. Since the elementsof bg comprise a complete and orthogonal basis setin L2(G), we may consider G0 TjGj as anorthogonal decomposition of the error ejg; thus,we can use Parsevals identity to receive

2j 1

NjjG0 TjGjjj2F , (20)

where jj jjF denotes the matrix Frobenius norm.From (20) we see that the transformation minimiz-ing ej is given by the LS solution

Tj G0Gyj , (21)where Gy denotes the pseudo-inverse of G. TheWINGS transformation accuracy is

2WINGS MinTj

2j 1

NjjG0P?GHj jj

2F , (22)

where P?GHj

IGyjGj is the orthogonal projectionon the subspace spanned by GHj .Finally, let us discuss the general conditions

under which the WINGS transformation is numeri-cally stable and yields low interpolation errors. In[25], we showed that a frequency transformationo1 ! o2oo1 is equivalent to a spatial scaling ofthe array geometry by a factor of o2=o1. In orderfor this transformation to be numerically stable, thescaled virtual array must be contained in a manifoldM (not to be confused with the array manifold) thattightly bounds the unscaled array, M must bestarlike, i.e. one that can be represented by the form

r f(j). In addition, the array must sample the

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 157915941584boundary of M, qM, densely enough so that thewaveeld on qM can be reconstructed from thearray measurements. In [25], these two conditionsare termed the spectral sampling condition anddiscussed in detail, including a treatment of thespecial case of an array resonance.To give an example, a circular array (not at

resonance) will satisfy the two conditions if theorigin of the coordinate system is within the array(to ensure the shape is starlike), and the spacingbetween two adjacent elements is less than half awavelength (to satisfy the spatial sampling condi-tion). Setting the origin outside the array perimeterwill violate the starlike condition, thus forcing theWINGS transformations to effectively extrapolatethe waveeld outside of the circle. Since extrapola-tions are numerically unstable, this could cause thetransformation errors to increase rapidly. In prac-tice, a good generic choice is to place the origin atthe center of gravity of the array.

4.1. Beamspace WINGS transformations

In this paragraph we consider beamspace proces-sing and develop the equivalent sampling matrixrequired for WINGS processing implemented di-rectly in beamspace. Beamspace processing is acommon approach for reducing the complexity ofan adaptive beamformer via preliminary preproces-sing with a set of conventional beamformers, (seee.g. [1]). The space spanned by the output of thebeamformers is referred to as the beamspace.The transformation from element-space to beam-

space may be written as

xbs;koj BHbsojxkoj. (23)Let Gbso denote the beamspace sampling

matrix related to the beamspace processor outputxbso. It is straightforward to verify thatGbso BHbsoGo. (24)The appropriate WINGS transformation may

now be readily computed inserting the beamspacesampling matrix Gbso in (21).

4.2. WINGS transformations for frequency-invariant

beamforming (FIB)

Here we consider the FIB problem and provide aproof to the optimality of WINGS transformationsfor FIB. A FIB is a beamforming structure capable

of synthesizing far-eld beam patterns that exhibitminimal variation within the operating frequencyrange (see e.g. [3,4,30]). This technique provides asolution to the wideband source signal localizationand estimation by synthesizing a far-eld beampattern whose main beam-width, sidelobe levels(SLLs) and null directions are approximatelyfrequency invariant.The FIB beamforming matrices are chosen so

that their corresponding beam patterns are identicalfor all frequencies, where the narrowband far-eldbeam pattern BP(g,o) associated with the beam-forming weight vector wo is given byBPg;o wH oago. (25)The requirement of a narrow main beam-width

and low SLL can be achieved by a taperedbeamformer. The weight vector for linear uniformarrays is usually based on window functions such asDolphChebyshev, Kaiser, Hamming, etc. LetBPdg and wd denote the desired beam pattern ofthe array and its corresponding weight vector,respectively, both associated with a referencefrequency o0. The FIB weights wj minimize the L2distance of the actual beam pattern from the desiredbeam pattern [30]

wj ArgMinwj

ZGdg BPdg BPg;oj 2

ArgMinwj

ZGdg wHd ago0 wHj agoj 2: (26)

Inserting the orthogonal representation (16) ofthe array manifold, we get

wj ArgMinwj

ZGdg wHd G0 wHj Gjbg 2: (27)

Since the elements of bg are an orthogonal basis inL2G, applying Parsevals identity yields theequivalent LS minimization problem

wj ArgMinWj

wHd G0 wHj Gj 2: (28)

We see that the optimal FIB weights wj are givenby the LS solution of (28)

wHj wHd G0Gyj wHd Tj, (29)where Tj is the WINGS transformation given in (21).

4.3. WINGS for adaptive coherent wideband MVDR

beamforming

The benets of coherent wideband beamforming

are twofold. First, by coherently combining the

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 15791594 1585signal subspaces associated with different frequencybins, the effective source correlation matrix is fullrank, even in the extreme case of coherent sources[11]. Second, the computational complexity is fairlylow compared with the incoherent approach. Theformation of the J-focusing matrices is computationintensive, but since WINGS-focusing matrices aredata independent they can be prepared in advance.The third benet of coherent processing is theimproved convergence time of the adaptive algo-rithms.In this section, we describe a framework example

of the WINGS based focused coherent processingfor the MVDR adaptive beamformer using the SMIimplementation. To simplify the exposition, ourpresentation is conned to the well-known MVDR-SMI beamformer. However, in a similar mannerone can implement on the temporal WINGS vectorany desired adaptive beamforming scheme, such asthose based on the architecture of the cheaperand more stable generalized sidelobe cancellerGSC-SMI beamformer [1 Section 7.4].The MVDR-SMI method is implemented for

the wideband case in the frequency domain, i.e.it is implemented as a narrowband beamformer ateach frequency bin (see e.g. [1]). The DFT imple-mentation of the frequency domain MVDR-SMIbeamformer is based on estimating the narrow-band sample covariance matrix at each frequencybin

R^j 1

K

XKk1

xkojxHk oj. (30)

The narrowband MVDR-SMI adaptive weightvector is computed at each frequency bin as

w^goj R^1j agoj

aHg ojR^1j agoj

. (31)

The adaptive beam is then formed in the freq-uency domain.The WINGS MVDR-SMI adaptive beamformer

may be simply implemented as a narrowbandadaptive beamformer operating on the temporalWINGS data vector. The WINGS sample covar-iance matrix is estimated by

R^wings 1

KJ

Xk;n

yknyHk n, (32)

where ykn is the WINGS vector constructed by (6).

Now, the WINGS coherent adaptive MVDR weightvector is simply computed by

w^g;wings R^1wingsago0

aHg o0R^1wingsago0

, (33)

where o0 is the focusing frequency. The WINGSadaptive beam is then computed directly in the time-domain.Note that the adaptive WINGS beamformer uses

a single narrowband set of adaptive weights, thusachieving a substantial reduction in the computa-tional complexity. Furthermore, the WINGSsample covariance matrix (32) is averaged overKJ-independent data samples, which is considerablymore than the incoherent estimate (30) that employsonly K averages. This leads to a signicantlyimproved convergence rate as will be demonstratedin the numerical section.We note, that the simple coherent MVDR scheme

given above assumes that the noise and sourcesspectra are relatively at over the focused frequencyband. This assumption can be justied for acousticsonar applications, if the relative bandwidth is nottoo large. However, this is not the case for radarand RF communication applications. For thecase of spectrally colored sources, the use ofmaximum likelihood (ML)-based wideband beam-formers has been advocated by [21,31], which takethe spectral variation into account, thus achievingsuperior performance. Following [21,31] one mayimplement these ML methods in the virtual WINGSdomain.

5. Robust WINGS transformations

In this section we treat the important issue ofrobustness, i.e. reducing the transformations sensi-tivity to errors in the array manifold, such as gain,phase and location errors, and to numericallyproblematic array geometries such as resonant [22]or over-sampled arrays [27]. We propose twoapproaches designed to ensure the robustness ofthe WINGS transformation, by controlling thenoise gain of the transformation. The rst employsa rank-truncated pseudo-inverse scheme to limit thenorm of the transformation, thus controlling thewhite noise gain, and the latter is based on directlyconstraining the noise power gain of the transfor-mation for any noise of arbitrary but knowncovariance matrix. We note that the WINGSrobust schemes derived in this section can be

generalized in a straightforward manner to any

pwe argued that for ~5N, the effective rank depends

i1

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 157915941586logarithmically on ~. We dene the array to be over-sampled if roN.Using Parsevals identity one can show that theinterpolation-based processing, where the interpola-tion matrices are computed by performing a LS t,e.g. [18].

5.1. Robust rank-truncated least-squares WINGS

transformations

In this section, we follow a rank-truncatedpseudo-inverse approach to develop robust WINGSTransformations. The rank-truncation approachis based on examining and truncating the singularvalue decomposition (SVD) of G. Let G USVHbe the SVD of G, and let un; vn and sn denotethe nth left and right singular vectors and thenth singular value of G (arranged in a decreasingorder), respectively. Let us rewrite the WINGStransformation (21) in terms of the SVD compo-nents, fsnoj; vnoj; unojg of the samplingmatrix Gj

Tj G0Gyj G0XNi1

s1i ojviojuHi oj: (34)

One can see that difculties may arise due to thepseudo-inverse of Gj, if it has very small singularvalues. This condition of near-rank deciency of thesampling matrix occurs typically in spatially over-sampled arrays. It may also occur at resonancefrequencies of certain array geometries, such ascircular arrays (see [22,27]).In order to avoid reliance on small singular

components, we apply a rank-truncation procedurefollowing [32]. Let

Gp Xpi1

siojuiojvHi oj (35)

be the rank-p approximation of G, and let residualenergy function m2pG of the array be dened as theapproximation error

m2pG GGpk k2F XNnp1

s2n: (36)

We now introduce the concept of the effectiverank of the sampling matrix; r rankG is denedas the smallest integer r such that m2Gp~. In [25],error in the effective rank-truncated array manifoldThe solution above will be referred to as the rank-truncated least-squares (RTLS) WINGS transfor-mation. The transformation error of the RTLSWINGS transformation is given by

2r oj 1

N

ZGdgjjago0 Trjagojjj2,

1NjjG0

XNir1

viojvHi ojjj2F . (40)

The RTLS transformation can be shown to havea bounded norm

Trj

2Fp G0k k2F Gr

yj

2F VGN

Xri1

s2i oj, (41)

thus ensuring a limited white noise gain.In the following section, we discuss a more

sophisticated method for limiting the norm of theWINGS transformation, by directly constrainingthe noise power gain of the transformation.

5.2. Robust quadratically constrained least-squares

WINGS transformations

The second approach is based on directlyconstraining the noise power at the output of thetransformation. This approach is general in thesense that it can handle any additive noise witharbitrary but known spatial covariance Qj EfnojnH ojg. Let us minimize the transformationerror 2j dened in (20) subject to an inequalityconstraint on the output noise power

MinTj

jjG0 TjGjjj2F subject to

trTjQjTHj pd2. (42)argo Grobg is smaller than ~

2r ZGdgjjag argjj2 GGrk k2F m2Gp~.

(37)

The robust rank-r transformation Trj is theminimum 2-norm solution of the rank-decient LSproblem

MinTrj

jjG0 TrjGrj jj2F , (38)

given by

Trj G0Gryj G0

Xrs1i viu

Hi : (39)

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 15791594 1587The transformation satisfying (42) will be referredto as the quadratically constrained least-squares(QCLS) WINGS transformation. If the uncon-strained LS transformation (21) satises the con-straint trTjQjTHj pd2, we have a solution to (42).If not, then our solution occurs on the boundary ofthe constraint set. Thus, our goal is solving thequadratic equality constrained minimization

MinTj

jjG0 TjGjjj2F subject to trTjQjTHj d2.

(43)

Inserting (20) into (43) and using the method ofLagrange multipliers we minimize

MinTj

jjG0 TjGjjj2F l trTjQjTHj

trTjGjGHj lQjTHj TjGjGH0G0GHj THj , (44)

where l is a Lagrange multiplier. Taking thederivative with respect to THj and setting the resultequal to zero yields the solution

Tj G0GHj GjGHj lQj1. (45)Note that the Lagrange multiplier l may be

interpreted as a regularization parameter of the LSsolution of (20): taking l 0 yields the uncon-strained solution given in (21). The value of l isdetermined by solving to the equality constraint onthe noise power

hl trTjQjTHj G0 ~GH

j ~Gj ~GH

j lI1 2

F

d2, (46)where ~Gj Q1=2j Gj. Following [33] we nowrewrite (46) in terms of f ~Uj ; ~Vj ; ~Rjg the SVDcomponents of ~Gj, and recalling that the Frobeniusnorm is invariant with respect to orthogonaltransformations, we get

hl Z ~R2j lI1 2

F; (47)

where Z G0 ~GHj ~Uj. Inserting ~Rj diag ~s1; . . . ; ~sN we see that the relationship between l and theconstraint value d2 involves solving the secularequation

hl XNn1

XNk1

jZnkj2l ~s2n2

d2: (48)

Unfortunately, secular equations are not straight-

forward to solve. However, since h(l) is a mono-tonic decreasing function with respect to l for lX0and h04d2, a unique positive l* solving theconstraint hln4d2 must exist. Thus l* can befound through application of any standard root-nding technique, such as Newtons method. TheQCLS WINGS transformation can then be com-puted by evaluating (45) for l*.In practice we may wish to directly constraint the

transformations noise power gain Gn

MinTj

jjG0 TjGjjj2F subject to Gnpd2, (49)

Gn EjjTjnkjjj2Ejjnkjjj2

trTjQjTHj

trQj: (50)

We can write the required minimization in thesame form as (43)

MinTj

jjG0 TjGjjj2F subject to trTjQjTHj p~d2,

(51)

where ~d2 d2=trQj, and apply the QCLS

WINGS solution outlined above.In the sequel we consider the case Qj I and

refer to Gn as the white noise gain. The white noisegain constraint may be used to ensure robusttransformations, which are not highly sensitive tosmall amplitude, phase or position errors. Manyimportant sources of error, which occur in physicalsystems, are approximately uncorrelated fromsensor to sensor and can be modeled by adding acorresponding amount of spatially white noise toeach sensor. Thus, array gain against spatially whitenoise is a good measure of robustness, e.g. [34]presents a good application example for the robustadaptive beamforming problem.In the following section, we will show that robust

WINGS processing ensures a bounded CRB for theDOA error.

6. Robust WINGS focusing for direction nding withlimited variance degradation

Localizing multiple sources using an array ofsensors has been an active research area for manyyears. Several researchers advocated broadbandfocusing of the sensor data prior to applying ahigh-resolution direction-nding algorithm. Thecoherent focusing process reduces the dimensionof the problem by aligning the signal subspacefor the frequencies within the focusing band.

The benets cited include reduced computation,

improved performance in colored noise, in multi-

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 157915941588path scenarios, and improved resolution. However,coherent focusing can degrade the performanceof the direction-nding system if it is not per-formed appropriately. Several researches [20,22,35]linked the variance of DOA estimate to thefocusing transformations in special cases, derivingoptimal transformation under special considera-tions.In this section we address this problem; we show

that if the noise gain of the focusing matrices islarge, the variance degradation may be large as well.For example, interpolation-based transformationsuch as WINGS, when applied to a resonant array,may result in large errors. We consider the DOAestimation accuracy as given by the CRB for thesingle source case using the stochastic Gaussiansignal model. We will show that robust WINGStransformations having a bounded noise gain ensurea bounded CRB for the DOA error. The modelassumptions regarding the noise and signal are asspecied in Section 2. To simplify the exposition,our discussion is conned to 2-D azimuth-onlysystems, where y denotes the azimuth. The CRB forthe wideband DOA estimation using K snapshots isgiven by (see e.g. [1] Section 8.4.1.2)

CRB1y XJj1

2K

N

ASNRj21ASNRj

" #dHy ojI ayojaHy oj=jjayojjj2dyoj, (52)

where Pj and s2j are the unknown signal and noisevariance, respectively, ASNRj NPj=s2j is thearray signal-to-noise ratio (ASNR), and dyoj qayoj=qy is the derivative vector of ayoj withrespect to y. If the array manifold is conjugatesymmetric (a Ja) then dHy ay 0 and

CRB1y XJj1

2K

N

ASNRj21ASNRj

" #jjdyojjj2: (53)

The CRB above takes the azimuth y to be anunknown deterministic parameter. Let us nowexamine the hybrid2 CRB, which has beenshown to be useful in practical problems (e.g.[36,37]). Let us take y to be a random variablewhose a-priori probability density is uniformlydistributed over all possible look directions; for this

2The term hybrid refers to the fact that the variable vectorcontains both deterministic and random parameters.case we get [36]

HCRB1y EyfCRB1yg 1

2p

Z 2p0

dyCRB1y;

(54)

where HCRB denotes the hybrid CRB. Inserting theseparable orthogonal representation (16) of thearray manifold, and using (12), we generalize theresults given in [22] for the special geometry of amulti-ring circular array to any conjugate sym-metric 2-D arrays:

dyoj qayojqy

Gjqqby@y

GjDby;D diagfiN ; . . . ; 1; 0;1; . . . ;Ng. (55)

Inserting (55) into (53) and (54) and applyingParsevals Theorem, we get

HCRB1y XJj1

2K

N

ASNRj21ASNRj

" #1

2p

Z 2p0

jjdyojjj2

12p

XJj1

2K

N

ASNRj21ASNRj

" #jjGjDjj2F : (56)

Note that for small ASNR the hybrid CRB isproportional to ASNRj2 and for large ASNR, itis proportional to ASNRj1. We observe thatarray geometry is completely contained in thejjGjDjj2F term.Let us now examine the effect of the focusing

transformations on the hybrid CRB. Let HCRBFdenote the hybrid CRB of the virtual array datayk(n) given in (6) after the focusing process, and let~Rn denote the normalized spatial covariance matrixof the transformed noise process ~nn

~s2n ~Rn Ef~n ~nHg XJj1

s2j TjTHj ; (57)

where ~nn is dened in (7), and ~Rn is normalized sothat trfeRng N. Applying the spatial whiteninglter eR1=2n to the focused array data yk(n), we getthe following expression for the hybrid CRB afterfocusing:

HCRB1F ;y 1

2p2KJ

N

ASNRF 21ASNRF

jj ~R1=2n GjDjj2F ; (58a)

ASNRF NPJ

j1PjPJj1s

2j GnTj

; (58b)

whefocgainintema

mation diverges as the wave number approaches a

gain Gn as a function of the normalized wavenumber k0 in multiples of p/d. We can see thatthe interpolation error increases monotonicallywith a clear knee around k0 0:9 p=d, repre-senting the point where the array becomesunder-sampled for kj 1:1k0. In Fig. 1b, weshow how the robust RTLS and QCLS controlthe noise gain of the transformation at theexpense of mildly increasing the interpolationerror.

(2) Bandwidth: We now illustrate the effect of therelative bandwidth on the interpolation error.Again, we choose an equally spaced 20-elementlinear array, and set the focusing wave numberat k0 p/2d so the array is spatially over-

ARTICLE IN PRESS

Fig. 1. Performance of WINGS transformations for a 20-element

linear array: (a) e2 as a function of k0 and (b) Gn as a functionof k0.

M.A. Doron, A. Nevet / Signal Processing 88 (2008) 15791594 1589resonance frequency (see numerical example inFig. 4). This divergence of the noise gain results ina divergence of the HCRBF of the DOA. We alsosee from (58) that the robust WINGS transforma-tion, which has a bounded noise gain, ensures alimited variance degradation of DOA error. Ne-glecting the focusing errors, we can expect robustWINGS processing with near-unity noise gain toyield a relatively small degradation in the HCRB forthe case of constant SNR values over the frequencyband.

7. Numerical studies

7.1. Accuracy and noise gain of WINGS

transformations

In this section we conduct a numerical study ofthe various WINGS transformations. We evaluatetheir interpolation error, according to (20), andtheir white noise gain, dened by (50) with Qj I.We illustrate their dependence on various para-meters for several interesting cases. We compare theperformance of three WINGS transformations. Therst is the LS WINGS transformation given in (21),the second is the robust RTLS WINGS transforma-tion given in (39), where the residual energy for therank truncation (37) is taken to be ~ 0:001. Thethird transformation is the robust QCLS WINGStransformation dened by (45) and (48), where thenoise gain of the QCLS transformation is con-strained to unity.In our rst example we take a simple linear array,

consisting of omni-directional elements, equallyspaced with Dx d. In the following we refer tothe wave number k o=c associated with thefrequency o, and Gn denotes the white noise gain.

(1) Focusing wave number: First, we wish to examinethe inuence of the value of the focusing wavenumber k0 on the transformation error andnoise gain. We consider a 20-element array andinterpolate from kj 1.1k0 to k0, i.e. we reducethe frequency by 10%. In Fig. 1, we plot the

2resore ASNRF is the array signal-to-noise afterusing, and GnT trTTH =N is the white noiseof the transformation. From (58) we see how

rpolation-based transformation such as WINGSy result in large errors, when applied to anant array. The noise gain of the LS transfor-value of the interpolation error e , and the noise sampled by a factor of 2. In Fig. 2, we plot the

ARTICLE IN PRESS

Fig.

linea

of k

M.A. Doron, A. Nevet / Signal Processing 88 (2008) 157915941590value of the interpolation error e2, and the noisegain Gn as a function of the normalizedtransformed wave number kj in multiples of k0.We see again how the robust RTLS and QCLScontrol the noise gain of the transformation atthe expense of increasing the interpolation error.In Fig. 2b, we see a dramatic increase in thenoise gain of the LS transformation as kjdecreases below k0, due to the spatial extrapola-tion effect.Let us now examine the case where the focusingfrequency is higher than the processed fre-quency. Fig. 3 plots the squared interpolatedamplitudes of the array manifold elements for a20-element linear array for kj 0:7k0. We seethat the robust RTLS and QCLS transforma-

(3)

2. Performance of WINGS transformations for a 20-element

r array: (a) e2 as a function of kj and (b) Gn as a functionj.

Fig.

elemweighting to deemphasize the extrapolated zone.In this regime the transformations extrapolatethe array beyond its original physical bounds,which is an inherently noisy process.Note that one of the benets of using robusttransformations is that they are not highlysensitive to the choice of the focusing frequency,since they automatically suppress the magnitudeof the extrapolated values in the extrapolatedregime in order to maintain their low noise gain.Circular arrayresonance example: In this ex-ample we take a 2-D circular array andtions control the increase of the noise gain by

3. Squared interpolated amplitudes of the array manifold

ents for a 20-element linear array for kj 0.7k0.demonstrate how the transformations accuracyand noise gain may be affected by the existenceof resonances. In [25,27] it was shown that theresonance condition for a circular array ofradius R with omni-directional sensors is

JnkR 0; for some n. (59)If the resonance condition is satised then theresonant wave-functions

wr aJnkr einj (60)are unobservable by the array since they vanishat kR. The component of the array manifoldthat lies in the subspace of wr cannot beinterpolated, thus causing a sharp increase in theinterpolation error.The array we examine here is a circular equi-spaced array, with 14 omni-directional sensorsand unity radius, at an aperture range of3:5pkRp4:2. In this range, there is a resonance

ARTICLE IN PRESS

Fig.

circu

kjR

of k

M.A. Doron, A. Nevet / Signal Processing 88 (2008) 15791594 1591at kR 3:83171 . . ., for which J1kR 0. Weset the focusing wave number at kj 3.4. Fig. 4examines the performance of WINGS transfor-mations for a 14-element circular array; as thefrequency is swept through a resonance ofJ1kR at kjR 3:83171 we plot the value ofthe interpolation error, and of the noise gain asa function of the wave number.We see a divergence in the noise gain of theLS WINGS transformation as the wave numberapproaches the resonance frequency kjR 3:83171. The robust RTLS transformationdecreases the noise gain, consequently increas-ing the interpolation error only when theeffective rank drops by 2, which occurs quiteclose to the resonance frequency, while the

relative to the array, and the interference is locatedat abeaweby

4. Performance of WINGS transformations for a 14-element

lar array as the frequency is swept through a resonance at

3.83171: (a) e2 as a function of kj and (b) Gn as a functionj.n angle of 801. Fig. 5 shows the conventionalm pattern of the simulated array at 350Hz;see that a signal at 801 will be suppressed onlyQCLS transformation constrains unity noisegain, thus gradually increasing the interpolationerror over a relatively wide range of frequencies.

7.2. Discussion

Let us compare the two proposed approaches inperformance and complexity. QCLS is clearlyoptimal by denition since it minimizes the inter-polation error under the noise power gain cons-traint. In DF applications the QCLS transformationwill be expected to yield a lower MSE degradationof the DOA estimates, compared with that of theRTLS. The calculation of the QCLS transforma-tions is more complex than that of the RTLS, sinceit requires solving the secular Eq. (48) in order tond l the Lagrange multiplier, which can be solvedthrough application of any standard root-ndingtechnique, such as Newtons method. Considera-tions regarding computational costs are not relevanthere; since the WINGS-focusing matrices are dataindependent, they can be prepared in advance anddo not incur any real-time cost.To summarize, the real-time complexity is iden-

tical in both methods. The QCLS transformation isin principle superior in performance to the RTLS,though the performance difference observed in thenumerical examples does not seem to be verysignicant.

7.3. Simulation example

In this section we present a simulation exampledemonstrating the effectiveness and the reduction ofconvergence time achieved via WINGS processing.We will compare the frequency domain widebandSMI-MVDR to the focused WINGS-based SMI-MVDR procedure described in Section 4.3.In our example we take a uniform linear array of

12 microphones spaced half a wavelength apart at1 kHz. The array measures the acoustic waveeldgenerated by both narrowband and widebanddirectional signals representing two sources atdifferent locations. The signals of interest are spreadarbitrarily at a width of 40Hz around 350Hz. Thedesired signal is located in the direction of 9012 dB.

The inset in Fig. 6 shows the theoretical curves

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 157915941592The data includes the signals from the inter-ference and from the desired targetthe rstradiating 15 dB wideband power higher than thelast, white Gaussian noise is added yielding an SNRof 10 dB. After converting the data to beamspaceand selecting seven beams centered around thedirection of the target, it is being processed twice inparallelrst by implementing the SMI-MVDR atevery relevant frequency bin, and the second byimplementing narrowband SMI-MVDR onWINGS-transformed data, focused onto 330Hz.We chose to focus the data onto the lowestfrequency in band, since a high focusing frequency

Fig. 5. The conventional beam pattern of a 12-element array at

350Hz.requires the WINGS transformation to extrapolatethe array data beyond its physical boundaries.For each method we calculate the beamformer

array gain, which is dened by

AG SINRoutSINRin

, (61)

where SINRout and SINRin are the signal-to-interference plus noise ratios at the beamformeroutput and input, respectively.Fig. 6 plots the calculated AG averaged over 20

Monte Carlo runs, as a function of the number ofsnapshots averaged, each snapshot consisting 0.1 sof data. The signicant improvement in theconvergence time using WINGS processing resultsfrom averaging the sample WINGS covariancematrix (32) over KJ-independent data samples,which is considerably more than the incoherentestimate (30) that employs only K averages. Theapproximate theoretical expressions for the conver-gence behavior of the SMI-MVDR are taken fromSection 7.3.1 in [1]:

AGSMIK

K 2NK 1 AGSMI1 (62)

and for the coherent WINGS SMI-MVDR:

AGWINGS-SMIK

KJ 2NKJ 1 AGSMI1, (63)

where AGSMI and AGWINGS-SMI denote the arraygain of the non-coherent SMI-MVDR and that ofthe coherent WINGS SMI-MVDR, respectively.

Fig. 6. Array gain versus the number of snapshots for the

incoherent SMI-MVDR method and for the coherent WINGS-

based SMI-MVDR for a wideband 12-element linear array. The

inset shows the theoretical curves.computed for N 7 (the dimension of the beamspace), and J 4 (number of bins coherentlyprocessed). We see that the simulated results arecompatible with the theoretical curves.

8. Conclusion

In this paper we investigated the application ofinterpolation-based focusing for wideband adaptivearray processing. We considered the use of wave-eld modeling-based frequency transformations ofthe array manifold, which are data independent anddo not require any initial DOA estimates. They maybe applied to any array with a known arbitrarygeometry satisfying the spatial sampling condition.We applied these transformations to the arraywideband data, constructing the so-called WINGSarray, which can be associated with a virtualwideband array having a frequency-invariant beam

principle superior in performance to the RTLS,though the performance difference observed in the

ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 15791594 1593numerical examples does not seem to be verysignicant.Further benets of using robust transformations

were shown. In the application of focused proces-sing for direction nding, robust WINGS focusingwas shown to ensure a limited variance degradationof the DOA estimates. Furthermore, we demon-strated that robust WINGS processing is not likelyto be highly sensitive to the value chosen for thefocusing frequency, since the robust transforma-tions automatically suppress the extrapolated re-gime in order to maintain their low noise gain.Finally, we note that the robust schemes pre-

sented in this paper may be generalized in astraightforward manner to any interpolation-basedprocessing scheme, where the interpolation matricesare computed by means of a LS t.

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ARTICLE IN PRESSM.A. Doron, A. Nevet / Signal Processing 88 (2008) 157915941594

Robust wavefield interpolation for adaptive wideband beamformingIntroductionProblem formulation and WINGS transformationsWavefield modeling reviewWINGS transformationsBeamspace WINGS transformationsWINGS transformations for frequency-invariant beamforming (FIB)WINGS for adaptive coherent wideband MVDR beamforming

Robust WINGS transformationsRobust rank-truncated least-squares WINGS transformationsRobust quadratically constrained least-squares WINGS transformations

Robust WINGS focusing for direction finding with limited variance degradationNumerical studiesAccuracy and noise gain of WINGS transformationsDiscussionSimulation example

ConclusionReferences