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Signal Processing

Signal Processing 93 (2013) 804–810

0165-16

http://d

n Corr

Beijing

E-m

wangju

journal homepage: www.elsevier.com/locate/sigpro

Robust Capon beamforming against large DOA mismatch

Wei Zhang a,b,n, Ju Wang a, Siliang Wu a

a School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, Chinab Communications Research Group, Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, UK

a r t i c l e i n f o

Article history:

Received 4 April 2012

Received in revised form

17 July 2012

Accepted 1 October 2012Available online 11 October 2012

Keywords:

Capon beamformer

Robust adaptive beamformer

DOA mismatch

Robustness

84/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.sigpro.2012.10.002

esponding author at: School of Informatio

Institute of Technology, Beijing 100081, Chin

ail addresses: wei.zhang@sheffield.ac.uk (W. Z

@bit.edu.cn (J. Wang), siliangw@bit.edu.cn (S

a b s t r a c t

In the presence of significant direction-of-arrival (DOA) mismatch, existing robust Capon

beamformers based on the uncertainty set of the steering vector require a large size of

uncertainty set for providing sufficient robustness against the increased mismatch. Under

such circumstance, however, their output signal-to-interference-plus-noise ratios (SINRs)

degrade. In this paper, a new robust Capon beamformer is proposed to achieve robustness

against large DOA mismatch. The basic idea of the proposed method is to express the

estimate of the desired steering vector corresponding to the signal of interest (SOI) as a linear

combination of the basis vectors of an orthogonal subspace, then we can easily obtain the

estimate of the desired steering vector by rotating this subspace. Different from the

uncertainty set based methods, the proposed method does not make any assumptions

on the size of the uncertainty set. Thus, compared to the uncertainty set based robust

beamformers, the proposed method achieves a higher output SINR performance by

preserving its interference-plus-noise suppression abilities in the presence of large DOA

mismatch. In addition, computationally efficient online implementation of the proposed

method has also been developed. Computer simulations demonstrate the effectiveness and

validity of the proposed method.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

The Capon beamformer [1], also named the minimumvariance distortionless response (MVDR) beamformer,chooses the weight vector by minimizing the array outputpower subject to a look direction constraint [2,3]. Thestandard Capon beamformer (SCB) has high resolutionand good interference suppression ability if the steeringvector of the SOI is known accurately [4]. However, theknowledge of the steering vector corresponding to the SOImay be imprecise because of some factors, such as DOAmismatch, array calibration error, local scattering, near–far spatial signature mismatch and finite sample effect[4–13]. Whenever this happens, the output SINR of theSCB degrades dramatically [10]. In this paper, we focus

ll rights reserved.

n and Electronics,

a.

hang),

. Wu).

our study on improving the robustness against large DOAmismatch.

Many approaches such as the eigenspace-based beamfor-mer (ESB) [11], the diagonal loading (DL) method [14] andthe norm constrained Capon beamformer [12,15] were pro-posed for improving the robustness of the SCB over the pastdecade. Recently, based on the uncertainty set of the steeringvector, some robust beamformers were proposed [5,4]. Basedon worst-case performance optimization, a novel methodbased on the second-order cone programming (SOCP) pro-blem was proposed [5]. However, the SOCP problem needs touse some specific optimization toolboxes such as [16,17] toobtain its solution, which have a high computational cost andlimit its practical implementation. Moreover, it is difficult forthe online implementation of the SOCP problem. The basicidea behind the robust Capon beamformer (RCB) of [4] is toestimate the desired steering vector in an uncertainty set bymaximizing the array output power. It was proved in [4] thatthe beamformers of [5] and the RCB have the same weightvector and belong to the class of DL approaches.

W. Zhang et al. / Signal Processing 93 (2013) 804–810 805

In order to achieve sufficient robustness, we shouldchoose an optimal size of the uncertainty set for theuncertainty set based methods. However, due to theuncertainty of the actual DOA of the SOI, it is difficult tofind the optimal size. Obviously, a small uncertainty levelcannot provide sufficient robustness against large DOAmismatch. Nevertheless, in the presence of large DOAmismatch, the beamformer of [5] and the RCB require alarge size of uncertainty set for providing robustnessagainst the increased mismatch. In such a case, theirinterference-plus-noise suppression abilities weakened,which results in severe degradation of the output SINR[18,19]. To mitigate this problem, some advanced robustbeamformers were proposed [13,19,20]. An importantadvantage of the beamformer of [13] is that it can flexiblycontrol the beamwidth of the robust response region withconstraints on array magnitude response. However, thebeamformer of [13] is based on the SOCP problem, whichalso hits the wall of computational complexity. In addi-tion, some degrees of freedom (DOFs) of the beamformerof [13] for interference suppression are sacrificed. Aniterative robust Capon beamformer (IRCB) based on theRCB was proposed [21,19], which searches for the desiredsteering vector iteratively using a small uncertaintysphere. However, the IRCB may converge to an interfer-ence steering vector instead of the desired steering vector[22,23], provided that the interference power is higherthan the SOI power. A completely automatic diagonalloading (ADL) method for robust Capon beamforming wasproposed [20], which computes the DL level automaticallyfrom the received data without specifying any user para-meter. However, the ADL method is only useful for thecase of small sample sizes. This is because the samplecovariance matrix becomes very close to the actualcovariance matrix when the number of snapshots avail-able is sufficient to produce a well-conditioned samplecovariance matrix [24], resulting in the decrease of the DLlevel which, in turn, provides insufficient robustnessagainst DOA mismatch.

In this paper, we propose a novel method to achieverobustness against large DOA mismatch. The basic idea ofthe proposed method is first to form an orthogonal sub-space calculated with steering vectors for those DOAswithin an uncertainty region where the SOI arrives with ahigh probability; the estimate of the desired steeringvector can then be expressed as a linear combination ofthis orthogonal subspace. Finally, we can obtain theestimate of the desired steering vector by rotating thisorthogonal subspace. For the proposed method, it isshown that the problem of finding the desired steeringvector is an eigendecomposition problem that can beeasily solved without any specific optimization software,which is required by the SOCP based methods. Differentfrom the uncertainty set based methods, the proposedone does not need to know the size of the uncertainty set.Moreover, an online implementation of the proposedmethod is also developed for updating the weight vectorwhenever a new snapshot is received.

This paper is organized as follows. In Section 2, the signalmodel and some background on adaptive beamforming arepresented. In Section 3, we develop the proposed method

and its online implementation. Computer simulation resultsare presented in Section 4. Finally, conclusions are drawn inSection 5.

2. Background

Consider a uniform linear array (ULA) with M sensors,the received data of the ULA at the nth snapshot can beexpressed as

x½n� ¼ ½x1½n�,x2½n�, . . . ,xM ½n��T ¼

XP

i ¼ 1

aðyiÞsi½n�þn½n� ð1Þ

where xi½n� is the received data at the ith sensor, si½n� is theith source, ½��T denotes the transpose operation, n½n� is thenoise with a power s2, P is the number of signals, and

aðyiÞ ¼ ½1,e�j2pd sinðyiÞ=l, . . . ,e�j2pðM�1Þ d sinðyiÞ=l�T ð2Þ

is the M � 1 steering vector of the ith signal in direction yi,with d being the adjacent sensor spacing and l denotingthe signal wavelength.

Assume that all impinging signals and noise are un-correlated with each other. Then the covariance matrixcan be expressed as

Rxx ¼ E½x½n�x½n�H� ¼XP

i ¼ 1

s2i aðyiÞaðyiÞ

Hþs2I ð3Þ

where E½�� denotes the expectation, ½��H represents theHermitian transpose, s2

i is the ith source power and I isthe identity matrix. In practice, Rxx is replaced by thesample covariance matrix

R̂xx ¼1

N

XN

n ¼ 1

x½n�xH½n� ð4Þ

where N is the number of snapshots.Without loss of generality, we assume that the first

signal is the SOI. Then the Capon beamformer is obtainedby solving the following optimization problem:

minw

wHR̂xxw subject to wHaðy1Þ ¼ 1 ð5Þ

where w is the M � 1 complex weight vector, and y1 andaðy1Þ are the presumed steering direction and the presumedsteering vector, respectively.

The solution to (5) is given by

w¼R̂�1

xx aðy1Þ

aHðy1ÞR̂�1

xx aðy1Þ

ð6Þ

So the beamformer output power is given by

Po ¼1

aHðy1ÞR̂�1

xx aðy1Þ

ð7Þ

3. Proposed method and its online implementation

3.1. Proposed method

We assume that the SOI comes from a DOA uncertaintyregion with a high probability, while no interference comesfrom this region. It should be noted that this assumption is

W. Zhang et al. / Signal Processing 93 (2013) 804–810806

also implemented in the beamformers of [13,25,26]. Supposethat the uncertainty region is defined as F¼ ½y1�Dy,y1þDy�, where Dy is the DOA uncertainty range. Now weform a positive definite matrix Q [2,25,23]

Q ¼Z y1þDy

y1�DyaðyÞaHðyÞ dy: ð8Þ

As an approximation, we propose to use a sampled angle setby uniformly discretizing ½y1�Dy,y1þDy� into J grid pointsto calculate Q numerically. We then form a column ortho-gonal matrix U from the principal eigenvectors of Q corre-sponding to the K largest eigenvalues of Q. Since the K

eigenvalues contain most of the energy in the eigenvalues ofQ, we have Q CUXUH , where X is the K�K diagonal matrixwith its diagonal elements given by the principal eigenvaluesof Q [27]. That means any steering vector whose DOA comesfrom F can be expressed as a linear combination of thecolumns of U [23,27]. Therefore, the estimate of the desiredsteering vector can be reconstructed as a linear combinationof the columns of U. Hence, the proposed estimate of thedesired steering vector is given by

a¼Ur ð9Þ

where r is the rotating vector.To obtain the rotating vector r, we propose to max-

imize the output power Po or equivalently, minimize thedenominator of (7) with aðy1Þ replaced by Ur subject to aquadratic equation constraint rHr¼M, which maintainsthe norm of the proposed estimate of the desired steeringvector to avoid scaling ambiguity. Hence, the proposedoptimization problem can be formulated as

minr

rHR̂Ur subject to rHr¼M ð10Þ

where R̂U ¼UHR̂�1

xx U. The solution to (10) can be found bymeans of minimization of the function

Hðr,mÞ ¼ rHR̂UrþmðM�rHrÞ ð11Þ

where m is a Lagrange multiplier. Setting the gradient of(11) with respect to r equal to zero, we obtain the solutionfor r, given by the eigenvector of R̂U that corresponds tothe smallest eigenvalue of this matrix. So the rotatingvector is expressed as

r¼M½R̂U � ð12Þ

where M½�� is the operator that yields the eigenvectorcorresponding to the minimal eigenvalue. Substitutingthis solution into (9), we obtain the proposed estimateof the desired steering vector. Note that the calculatedrotating vector should be scaled so that its norm equals toM. Now the proposed estimate of the desired steeringvector is given by

a¼

ffiffiffiffiffiMp

JrJUr ð13Þ

Using a obtained in (13) to replace aðy1Þ in (6), we canthen obtain the weight vector of the proposed method.

It should be noted that the proposed method is notapplicable when NoM, because if NoM snapshots areused to form R̂xx, M�N eigenvalue estimates are zero [24].In such a case, R̂xx is rank deficient and not invertible. Toovercome this problem, we suggest using the DL method

[14] or covariance matrix estimation methods such as[20,28] to estimate an enhanced covariance matrix, thencalculating the estimate of the desired steering vectorusing the proposed method with sample covariancematrix replaced by the enhanced covariance matrix.

3.2. Online implementation of the proposed method

In practice, computationally efficient online implementa-tion is often required where the beamformer weight vector isupdated online. In this section, we develop an online imple-mentation for the proposed method with a low computa-tional cost.

Using the matrix inversion lemma [29], the matrixR̂�1

xx ½n� can be updated as

R̂�1

xx ½n� ¼ Z�1R̂

�1

xx ½n�1��Z�1R̂�1

xx ½n�1�x½n�

Z�1xH½n�R̂�1

xx ½n�1�

1þZ�1xH½n�R̂�1

xx ½n�1�x½n�ð14Þ

where Z is the forgetting factor. Then the optimizationproblem (10) can be reformulated as

minr

r½n�HR̂U ½n�r½n� ð15Þ

s:t: r½n�Hr½n� ¼M ð16Þ

where r½n� and R̂U ½n� are the estimates of the rotatingvector r and R̂U at the nth iteration, respectively. Utilizingthe conventional steepest descent (SD) algorithm, thetentative rotating vector r½n� to (15) is given by

r½n� ¼ r½n�1��uSD½n�g½n� ð17Þ

where uSD½n� is the step-size of the SD algorithm, given by

uSD½n� ¼1

tr½R̂U ½n��ð18Þ

where tr½�� denotes the trace operator, and

g½n� ¼ R̂U ½n�r½n�1� ð19Þ

is the conjugate derivative of (15) with respect to rn½n�.In order to satisfy the norm constraint (16), we project

the tentative rotating vector r½n� onto the norm constraintboundary by simply scaling r½n�, i.e.,

r½n� ¼

ffiffiffiffiffiMp

Jr½n�Jr½n� ð20Þ

To summarize, the online implementation of the pro-posed method consists of the following steps.

(1)

Update R̂�1

xx ½n� and R̂U ½n� when the nth data snapshotis received.

(2)

Define a tentative rotating vector using (17). (3) Project the tentative rotating vector onto the norm

constraint boundary using (20).

(4) Obtain the proposed estimate of the desired steering

vector by setting a½n� ¼Ur½n�.

(5) Use a[n] obtained in step 4 to calculate the weight

vector as

w½n� ¼R̂�1

xx ½n�a½n�

aH½n�R̂�1

xx ½n�a½n�ð21Þ

W. Zhang et al. / Signal Processing 93 (2013) 804–810 807

Computational complexity

Both the inverse matrix operation and the eigendecom-position operation need a computational complexity of

3.3.

OðM3Þ. Since the orthogonal subspace U can be calculated

offline, the computational complexity of the proposedmethod is OðM3

Þ, and its online implementation has acomplexity of OðM2

Þ per iteration. The RCB algorithm needsto perform eigendecomposition operation on the samplecovariance matrix, which has a complexity of OðM3

Þ. TheSOCP based methods in [5,25] have at least the complexityof OðM3:5

Þ, and the SOCP based methods in [13,26] have acomputational complexity of OðM3:5

ÞþOðJM2:5Þ, where J is

the number of sampled points in the DOA uncertaintyregion. Therefore, the proposed methods have much lowercomputational cost than the SOCP based methods. More-over, unlike the SOCP based methods, the proposed methodhas an important advantage for being easily implementedwithout any specific optimization software.

4. Simulations

In this section, simulations are carried out to investi-gate the performance of the proposed method comparedwith the SCB, the RCB, the ESB, the ADL and the beam-former of [13]. We consider a ULA with M¼10 sensorsand half-wavelength spacing between adjacent sensors.The SOI with signal-to-noise ratio (SNR) of 10 dB arrivesfrom direction y1 ¼ 01. Two interfering signals withinterference-to-noise ratio (INR) of 30 dB impinge on thearray from the directions �401 and 501, respectively. Thearray is steered toward the direction y1 ¼ y1þD1, whereD1 is the DOA mismatch. Two cases of RCB with uncer-tainty level b¼ 5 and b¼ 8 are considered. We alsoconsider the ESB of g¼ 3 and g¼ 4, where g is theestimated number of signals. The forgetting factor Z isset to 0.9998. The DOA uncertainty region for the beam-former of [13] and the proposed method is given by[�71, 71], and the number of grid points of the DOAuncertainty region is fixed to J¼100. Four principal eigen-vectors of Q corresponding to the four largest eigenvalueshave been used in the proposed method. All results areaveraged based on 100 independent simulation runs.

−80 −60 −40 −20−100

−80

−60

−40

−20

0

Arr

ay b

eam

patte

rn (d

B)

De

RCB (β=8)RCB (β=5)Proposed methodProposed method (online)ESB (γ=3)ESB (γ=4)Beamformer of [13]ADLSCB

Fig. 1. The resultant beampatt

4.1. Beampattern of beamformers

In the first example, we consider the resultant beam-pattern of the beamformers when the number of snap-shots is N¼1000. The DOA mismatch is D1 ¼ 61. We cansee from Fig. 1 that all these beamformers have deep nullsat the interferences DOAs. Note that the SOI is consideredto be an interference by the SCB, and hence, the SOI iscancelled by the SCB. On the other hand, the SOI ispreserved by these robust beamformers. Moreover, theESB of g¼ 3, the proposed method and its online imple-mentation point their mainlobes to the desired lookdirection rather than the presumed one. Due to theflexibility in setting the beamwidth of the beamformerof [13], it has formed a broad mainlobe according to theDOA uncertainty region. However, the two RCBs and theADL method have their mainlobes toward the presumedlook direction.

4.2. Output SINR versus the number of snapshots

In the second example, we consider the effect of thenumber of snapshots on the output SINR of the beam-formers. The parameters of the SOI remain the same asthe first example. Fig. 2 shows the output SINR of thebeamformers versus the number of snapshots. As shown,the output SINR of the SCB degrades significantly with aDOA mismatch of 61, and the RCB of b¼ 5 does not workwell. The performance of the ADL method is good whenthe snapshot number is small; however, its performancedegrades as the number of snapshots increases. Never-theless, it can be clearly seen that the proposed methodand its online implementation, the RCB of b¼ 8, the ESB ofg¼ 3, and the beamformer of [13] can provide sufficientrobustness and achieve high output SINR. Additionally,the ESB achieves the best performance when the numberof snapshots is small; however, it suffers from severelyperformance degradation if the number of signals isoverestimated. It should be noted from Fig. 2 that theproposed methods provide about 5 dB improvement inoutput SINR over the RCB of b¼ 8 and about 2 dBimprovement over the beamformer of [13].

0 20 40 60 80gree

ern of the beamformers.

−15 −10 −5 0 5 10 15

−80

−60

−40

−20

0

20

Out

put S

INR

(dB

)

SNR (dB)

Optimal beamformerRCB (β=8)RCB (β=5)Proposed methodProposed method (online)ESB (γ=3)ESB (γ=4)Beamformer of [13]ADLSCB

Fig. 3. Output SINR versus SNR.

200 400 600 800 1000 1200 1400 1600 1800 2000

−80

−60

−40

−20

0

20

Out

put S

INR

(dB

)

Number of snapshots

Optimal beamformerRCB (β=8)RCB (β=5)Proposed methodProposed method (online)ESB (γ=3)ESB (γ=4)Beamformer of [13]ADLSCB

Fig. 2. Output SINR versus the number of snapshots.

W. Zhang et al. / Signal Processing 93 (2013) 804–810808

4.3. Output SINR versus SNR

In the third example, we investigate the effect ofthe input SNR on the performance of the beamformersfor the same scenario as in the previous example, andthe number of snapshots is fixed at N¼500. Fig. 3 showsthe output SINR of the beamformers versus input SNR. Asshown, the performance of the proposed methods, thebeamformer of [13], the ESB of g¼ 3, the ADL and the RCBof b¼ 8 significantly outperform the SCB, the ESB of g¼ 4and the RCB of b¼ 5 in terms of output SINR.

4.4. Output SINR versus DOA mismatch

In the fourth example, the DOA mismatch is uniformlydistributed on [0,71] while the actual DOA of the SOI is 01.Other parameters remain the same as the second example.The result of output SINR versus the DOA mismatch is shownin Fig. 4. It can be observed that the performance of the RCBof b¼ 5 is better than that of the RCB of b¼ 8 when DOAmismatch is small. The reason for this is that a smalluncertainty level has sufficient capability to provide robust-ness against a small DOA mismatch. Nevertheless, a largeuncertainty level for the RCB algorithm will result in moreinterference and noise components in the beamformer

output. When the DOA mismatch increases, the performanceof the two RCBs degrade dramatically. However, the RCB ofb¼ 8 has a better ability of accommodating the increasedDOA mismatch and a higher output SINR than the RCB ofb¼ 5. We also see the performance of the ADL methoddegrades as the DOA mismatch increases. On the other hand,the performance of the proposed methods, the ESB of g¼ 3,and the beamformer of [13] remains satisfactory as the DOAmismatch varies.

4.5. Effect of moving interference

In the last example, the effect of moving interference isexamined. We consider a nonstationary scenario withonly one interference with INR¼30 dB, which changesits DOA rapidly, whereas the SOI is stationary and has thesame DOA mismatch as in the first example. The DOA ofinterference versus snapshot index n is given by

yjamming ¼ 501þn

100

� �1 ð22Þ

The parameters of the SOI remain the same as the firstexample. The instantaneous output SINR of the beamformersare shown in Fig. 5. It can be clearly seen that the proposedmethod and its online implementation, and the beamformer

0 1 2 3 4 5 6 7−50

−40

−30

−20

−10

0

10

20

Out

put S

INR

(dB

)

DOA mismatch (Degree)

RCB (β=8)RCB (β=5)Proposed methodProposed method (online)ESB (γ=3)ESB (γ=4)Beamformer of [13]ADLSCB

Fig. 4. Output SINR versus DOA mismatch.

100 200 300 400 500 600 700 800 900 1000

−80

−60

−40

−20

0

20

Out

put S

INR

(dB

)

Number of snapshots

Optimal beamformerRCB (β=8)RCB (β=5)Proposed methodProposed method (online)ESB (γ=3)ESB (γ=4)Beamformer of [13]ADLSCB

Fig. 5. Output SINR versus the number of snapshots.

W. Zhang et al. / Signal Processing 93 (2013) 804–810 809

of [13] can follow the environmental changes, whereas othermethods tested cannot follow nonstationary environment.However, this advantage of the proposed batch method andthe beamformer of [13] is achieved at the expense of amuch higher computational complexity. In practice, it maynot be possible to update the weight vector with such highcomputational complexity. Therefore, the online imple-mentation of the proposed method is a suitable choice fornonstationary environment.

5. Conclusions

A new robust Capon beamformer has been proposed forimproving the robustness of the SCB against large DOAmismatch. Unlike existing methods based on the uncertaintyset of the steering vector, the proposed method does not needto set a large size on the uncertainty set against large DOAmismatch, thereby providing a higher output SINR perfor-mance than the uncertainty set based methods by protectingthe interference-plus-noise suppression ability. Moreover, anefficient online implementation of the proposed method isalso provided. Simulation results have been presented todemonstrate the effectiveness of the proposed method.

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