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Contents lists available at ScienceDirect

Signal Processing

Signal Processing 90 (2010) 1573–1581

0165-16

doi:10.1

� Cor

E-m

edu.cn

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

Robust Capon beamformer under norm constraint

Liu Congfeng�, Liao Guisheng

National Lab of Radar Signal Processing, Xidian University, Xian, Shaan Xi 710071, China

a r t i c l e i n f o

Article history:

Received 30 July 2008

Received in revised form

19 October 2009

Accepted 30 October 2009Available online 10 November 2009

Keywords:

Robust adaptive beamforming

Capon beamformer

Norm constraint

Diagonal loading

Negative loading

84/$ - see front matter & 2009 Elsevier B.V. A

016/j.sigpro.2009.10.027

responding author.

ail addresses: cfliu@mail.xidian.edu.cn (C. L

(G. Liao).

a b s t r a c t

In order to improve the robustness against the array steering vector mismatch, the

norm constraint on the weight vector is used. By the complete investigation on the

Capon beamformer under norm inequality constraint (NICCB), the existence of its

solution is analyzed in detail, the choice of the norm inequality constraint parameter for

NICCB is analyzed and the selecting range is given. In this paper, the Capon beamformer

under norm equality constraint (NECCB) is also proposed and is solved effectively. At

the end, numerical examples attest the correctness of the theory, and show that when

the norm constraint parameter is selected in the allowable range, the performances of

the optimal NICCB and NECCB vary unobvious, but for the same given norm constraint

parameter, NECCB has the better performance than NICCB, namely the optimal negative

loading that has the preferable robustness.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Beamforming is a ubiquitous task in array signalprocessing with applications, among others, in radar,sonar, acoustics, astronomy, seismology, communications,and medical imaging. Without loss of generality, weconsider herein beamforming in array processing applica-tions. The introduction to beamforming can be found in[1–9] and references therein.

The traditional approach to the design of adaptivebeamformers assumes that the desired signal componentsare not present in training data, and the robustness ofbeamformer is known to depend essentially on theavailability of signal-free training data. However, in manyimportant applications such as mobile communications,passive location, microphone array speech processing,medical imaging, and radio astronomy, the signal-freetraining data cells are unavailable. In such scenarios, thedesired signal is always present in the training snapshots,and the adaptive beamforming methods become very

ll rights reserved.

iu), gsliao@xidian.

sensitive to any violation of underlying assumptions onthe environment, sources, or sensor array. In fact, theperformances of the existing adaptive array algorithmsare known to degrade substantially in the presence ofeven slight mismatches between the actual and presumedarray responses to the desired signal [10–12]. Similartypes of degradation can take place when the arrayresponse is known precisely but the training sample sizeis small, namely when there is a mismatch between theactual and the estimated covariance matrix [13–15].Therefore, robust approaches to adaptive beamformingappear to be of primary importance in these cases [16,17].

Many approaches have been proposed to improve therobustness of the adaptive beamformer during the pastthree decades. Indeed, the literature on the robustadaptive beamformer is quite extensive. We provide abrief review. For more detailed recent critical reviews, see[18–25].

1.1. Robust approaches for signal direction mismatch

For the specific case of the signal direction mismatch,several efficient methods have been developed. Repre-sentative examples of such techniques are the linearly

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C. Liu, G. Liao / Signal Processing 90 (2010) 1573–15811574

constrained minimum variance (LCMV) beamformer [26],which is also denoted as the linearly constrained mini-mum power (LCMP) beamformer in reference [27] andthis paper, signal blocking-based algorithms [10,28], andBayesian beamformer [29]. Although all these methodsprovide excellent robustness against the signal directionmismatch, they are not robust against other types ofmismatches caused by poor array calibration, unknownsensor mutual coupling, near–far wavefront mismodeling,signal wavefront distortions, source spreading, and co-herent/incoherent local scattering, as well as other effects[17].

Chen and Vaidyanathan [25] consider a simplifieduncertainty set which contains only the steering vectorswith a desired uncertainty range of direction of arrival(DOA) , although the closed-form solution is given and thediagonal loading level can be computed by the iterationmethod systematically. How to determine the DOAuncertainty range is the critical problem.

1.2. Robust approaches for general mismatch

Several other approaches are known to provide theimproved robustness against more general types ofmismatches, for example, the algorithms that use thediagonal loading of the sample covariance matrix [14,16],the eigenspace-based beamformer [11,30,31], and thecovariance matrix taper (CMT) approach [32–34]. Forthe diagonal loading method, a serious drawback is thatthere is no reliable way to choose the diagonal loadinglevel, Vincent and Besson [35] propose the method toselect the optimal loading level with a view to maximiz-ing the signal-to-noise ratio (SNR) in the presence ofsteering vector errors and it is shown that the loading isnegative, but they cannot give the exact solution, insteadof the approximate solution. Moreover, they cannot givethe expression of steering vector errors. The eigenspace-based approach is essentially restricted in its performanceat low SNR and when the dimension of the signal-plus-interference subspace is high, and the dimension must beknown in the latter technique [31]. The CMT approach isknown to provide an excellent robustness in scenarioswith nonstationary interferers, however, its robustnessagainst mismatches of the desired signal array responsemay be unsatisfactory. Furthermore, it can also beexplained as the diagonal loading [33].

1.3. Uncertainty set constraint approaches for general

mismatch

Very recently, many approaches have been proposedfor improving the robustness of the standard minimumvariance distortionless response (MVDR) beamformer.Their main ideas are based on the definition of theuncertainty set and the worst-case performance optimi-zation, but these algorithms are all classified to thediagonal loading technique.

Li et al. [20] propose the robust Capon beamformerunder the constraint of steering vector uncertainty set, thenthe constraint of steering vector norm is imposed and the

doubly constrained robust Capon beamformer is proposed[22]. For the two beamformers, although they give the exactweight vectors and methods of finding the optimal loadinglevel, their performance improvements are not obvious.Actually, the constraint of uncertainty set is the essence ofthe two robust beamformers, and the two beamformershave the same robustness characteristic. Besson andVincent [36] also analyze the performance of the beamfor-mer under the uncertainty set constraint approximatively,but they do not give the exact loading level.

Vorobyov et al. [19] propose a robust beamformer inthe presence of an arbitrary unknown signal steeringvector mismatch. Although they prove the proposedapproach to be equivalent to the loading sample matrixinversion (LSMI) algorithm, they do not give the directmethod to compute the optimal weight vector. Furtherthe second-order cone (SOC) programming-based ap-proach is used to solve the original problem. Elnasharet al. [24] make use of the diagonal loading technique toimplement the robust beamformer, but the optimal valueof diagonal loading level is not solved exactly. Alterna-tively, the diagonal loading technique is integrated intothe adaptive update schemes by means of optimumvariable loading technique. Robert and Stephen [23] alsosolve the similar beamformer by the Lagrange multipliertechniques, but they express the weight vector and thearray manifold as the direct sum of the corresponding realand imaginary components. Mutapcic et al. [37] show thatworst-case robust beamforming with multiplicative un-certainty in the weights can be cast as a tractable convexoptimization problem, but they do not give the method ofsolving. In fact, the proposed robust beamformer withuncertain weights can be converted to that in [19]equivalently.

Shahram et al. [21] consider the general-rank signalmodel, and the robust beamformer is proposed for thedistributed sources. An elegant closed-form solution isgiven, but its performance improvement depends on theconstraint parameter severely, and is not very optimal.

1.4. Weight norm constraint approaches for general

mismatch

Li et al. [22] propose a Capon beamforming approachwith the norm inequality constraint (NICCB) to improvethe robustness against array steering vector errors andnoise. Although the exact solution is given and optimalloading level can be computed via the proposed method.Analysis and simulation shows that its efficiency is not asgood as expected. Since the constraint parameter deter-mines its robustness, how to select the constraintparameter is not discussed.

Quadratic inequality constraints (QIC) on the weightvector of LCMP beamformer can improve robustness topointing errors and to random perturbations in sensorparameter [27]. The weights that minimize the outputpower subject to linear constraints and an inequalityconstraint on the norm of the weight vector have thesame form as that of the optimum LCMP beamformerwith diagonal loading of the data covariance matrix. But

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the optimal loading level cannot be directly expressed as afunction of the constraint in a closed form, and cannot besolved. Hence, its application is restricted by the optimalweight vector finding. So some numerical algorithms areproposed to implement the QICLCMP, such as least meansquares (LMS) or recursive least squares (RLS) [27]. Butthe application effect is not good as expected.

1.5. Motivation and outline of this paper

From the reviews of the robust beamforming, we canconclude to as follows. (1) The robust approaches for signaldirection mismatch do not have robustness to generalmismatch, but the robust approaches for general mismatchdo not have strong roots in theory and they also do not havean exact solutions. (2) Although the robust approachesbased on the uncertainty set constraint or on SOC program-ming method yield exact solution, their performancedepends on the constraint parameter, and the interferenceparameter cannot be imposed. (3) Although the robustapproaches based on the uncertainty set constraint can besolved exactly or by some SOC programming method, buttheir performance depend upon the constraint parameter,and the interference parameter doesn’t be imposed in it. (4)The differences between uncertainty set constraint andweights norm constraint are selection of the constraintparameter and solution of the optimal loading level. Theconstraint parameter of the uncertainty set should beselected according to the mismatch, but the constraintparameter of the weights norm should be selected accordingto calculating condition, i.e. it is independent of themismatch, but for the two approaches, the solutions of theoptimal loading level are all determined by the existingmismatch and the given constraint parameter.

We deal with the robust Capon beamformer withweight norm constraint.

For the current proposed NICCB, the main problem ishow to select the norm constraint parameter. Hence, viaanalysis of the solution existence of the optimal Lagrangemultiplier, the choice of the norm constraint parameter isanalyzed and the selecting range is given. By analysis andsimulation, we see that its efficiency is not as good asexpected, and in this paper, the Capon beamformer undernorm equality constraint (NECCB) is proposed and issolved effectively. Numerical examples attest the correct-ness and the validity of the proposed algorithm, and showthat the NECCB has the best performance to overcome thesignal direction mismatch, i.e. the optimal negativeloading has the preferable robustness.

This paper is organized as follows. First, NICCB isintroduced and analyze. Second, the choice of the normconstraint parameter and the selecting range is discussed.Third, NECCB is proposed and is solved effectively. Finally,the simulation analyses and the conclusion are given.

2. Capon beamformer under norm inequality constraint(NICCB)

The Capon beamformer can experience significantperformance degradation when there is a mismatch

between the presumed and the actual characteristics ofthe source or array. The goal of NICCB is to impose anadditional inequality constraint on the Euclidean norm ofw for the purpose of improving the robustness to pointingerrors and to random perturbations in sensor parameters.Here w denotes the array weight vector. This requiresincorporating a norm inequality constraint on w of theform

JwJ2rB ð1Þ

where B is the norm constraint parameter. Consequently,the NICCB problem is formulated as follows:

minw

wHRw

s:t: wHs ¼ 1

JwJ2rB

8>><>>: ð2Þ

where R is the data covariance matrix, s is the presumedsignal steering vector, and ð�ÞH denotes conjugate trans-position, J � J denotes the vector l2 norm. For convenienceof analysis and analyzing the choice of the normconstraint parameter, the solution to NICCB [22] isintroduced in the following.

2.1. Solution to NICCB

Let S be the set defined by the constraints in aboveoptimization problem, i.e.:

S¼ fwjwHs ¼ 1; JwJ2rBg ð3Þ

Define

f1ðw; l;mÞ ¼wHRwþlðJwJ2� BÞþmð�wHs � sHwþ2Þ ð4Þ

where l is the real-valued Lagrange multiplier, and lZ0satisfied RþlI40 so that f1ðw; l;mÞ can be minimizedwith respect to w. m is the arbitrary Lagrange multiplier.Then:

f1ðw; l;mÞrwHRw; w 2 S ð5Þ

with equality on the boundary of S. For the standardCapon beamformer

minw

wHRw

s:t: wHs ¼ 1

(ð6Þ

The optimal solution is

w¼R�1s

sHR�1sð7Þ

where R�1 is the inversion of R, i.e. ð�Þ�1 denotes thematrix inversion. Here, we can have

JwJ2¼wHw¼

R�1s

sHR�1s

!HR�1s

sHR�1s¼

sHR�2s

ðsHR�1sÞ2ð8Þ

where R�2¼ ðR�1

Þ2¼ R�1

� R�1. The above result uses theHermitian property of R.

Consider the condition:

sHR�2s

ðsHR�1sÞ2rB ð9Þ

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When the above condition is satisfied, the standardCapon beamformer solution (7) satisfies the norm con-straint of NICCB. Hence, it is also the solution to NICCB.For this case, l¼ 0 and the norm constraint in NICCB isinactive.

Otherwise, we have the condition

Bo sHR�2s

ðsHR�1sÞ2ð10Þ

which is an upper range B so that NICCB is different fromthe standard Capon beamformer. To deal with this case,we can rewrite f1ðw; l;mÞ as follows:

f1ðw; l;mÞ ¼ ½w� mðRþlIÞ�1s�HðRþlIÞ½w� mðRþlIÞ�1s�

�m2sHðRþlIÞ�1s � lBþ2m ð11Þ

Hence, the unconstrained minimizer of f1ðw;l;mÞ, forfixed l and m, is given by

wl;m ¼ mðRþlIÞ�1s ð12Þ

Clearly, we have

f2ðl;mÞ9f1ðwl;m; l;mÞ

¼ � m2sHðRþlIÞ�1s � lBþ2mrwHRw; w 2 S ð13Þ

The maximization of f2ðl;mÞ is with respect to m. Hence, mis given by

m ¼ 1

sHðRþlIÞ�1s

ð14Þ

Insert m into f2ðl;mÞ, And let

f3ðlÞ9f2ðl; mÞ ¼ � lBþ 1

sHðRþlIÞ�1s

ð15Þ

The maximization of the above function f3ðlÞ withrespect to l gives

sHðRþlIÞ�2s

½sHðRþlIÞ�1s�2

¼ B ð16Þ

Hence, the optimal Lagrange multiplier l can beobtained efficiently via, for example, a Newton’s methodfrom the above equation of l. Note that using m in wl;myields

w ¼ðRþlIÞ�1s

sHðRþlIÞ�1s

ð17Þ

which satisfies the constraints of NICCB, i.e.,

wHs ¼ 1 ð18Þ

and

JwJ2¼ B ð19Þ

Hence, w belongs to the boundary of S. Therefore, w is oursought solution to the NICCB optimization problem,which has the same form as the Capon beamformer witha diagonal loading term lI added to R, namely, NICCB alsobelongs to the class of diagonal loading approaches.

From the above analysis, we can see that if theLagrange multiplier l is obtained, the optimal weightvector for NICCB will be solved. In order to obtain theLagrange multiplier l, we must solve the following

equation via Newton’s method, and let

hðlÞ ¼sHðRþlIÞ�2s

½sHðRþlIÞ�1s�2

ð20Þ

Hence, the key problem of NICCB is finding the optimalLagrange multiplier by the above equation (20). In thispaper, we will give the complete investigation on NICCB,and the existence of its solution is analyzed as follows.

2.2. Solution to the optimal Lagrange multiplier

In order to solve Eq. (20), we perform an eigenvaluedecomposition (EVD) of the sample covariance matrix asfollows:

R¼U � C � UH¼XMi ¼ 1

liuiuHi ð21Þ

where C¼ diagðl1; l2; . . . ; lMÞ is diagonal matrix,U¼ ðu1;u2; . . . ;uMÞ is Hermitian, and li ði¼ 1;2; . . . ;MÞand uiði¼ 1;2; . . . ;MÞ are the eigenvalues and eigenvectorsof R, respectively, and M is the total number of degrees offreedom. For the convenience of analysis, we assume thatthe eigenvalues/eigenvectors of R are sorted in descend-ing order, i.e.,

l1Zl2Z � � �ZlM ð22Þ

Therefore, we have

hðlÞ ¼

PMi ¼ 1

sH

uiuHi

s

ðliþlÞ2PMi ¼ 1

sH

uiuHi

s

liþl

� �2¼

PMi ¼ 1

JsH

uiJ2

ðliþlÞ2PMi ¼ 1

JsH

uiJ2

liþl

h i2ð23Þ

Therefore, hðlÞ is monotonically increasing function oflZ0 [22]. Then

hðlÞ ¼

PMi ¼ 1

JsH

uiJ2

ðliþlÞ2PMi ¼ 1

JsH

uiJ2

liþl

h i2r

PMi ¼ 1

JsH

uiJ2

ðlM þlÞ2PMi ¼ 1

JsH

uiJ2

l1þl

h i2

¼l1þllMþl

� �2 1PMi ¼ 1 JsHuiJ

2ð24Þ

hðlÞ ¼

PMi ¼ 1

JsH

uiJ2

ðliþlÞ2PMi ¼ 1

JsH

uiJ2

liþl

h i2Z

PMi ¼ 1

JsH

uiJ2

ðl1þlÞ2PMi ¼ 1

JsH

uiJ2

lM þl

h i2

¼lMþll1þl

� �2 1PMi ¼ 1 JsHuiJ

2ð25Þ

Let

g¼XMi ¼ 1

JsHuiJ2

ð26Þ

Alternately, the above inequality relationship can beexpressed as

ffiffiffiffiffigBp

rl1þllMþl

ð27Þ

ffiffiffiffiffigBp

ZlMþll1þl

ð28Þ

Next, we analyze the range of the Lagrange multiplier land its existence.

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(1) IfffiffiffiffiffigBp 41, using (27) and (28), we have

ffiffiffiffiffigBp ðlMþlÞrl1þlffiffiffiffiffigBp ðl1þlÞZlMþl)

lrl1 �

ffiffiffiffiffigBp lMffiffiffiffiffigBp � 1

lZlM �

ffiffiffiffiffigBp l1ffiffiffiffiffigBp � 1

8>><>>:

8>><>>: ð29Þ

SinceffiffiffiffiffigBp 41, but lM �

ffiffiffiffiffigBp l1o0, and lZ0, we havel1 �

ffiffiffiffiffigBp lM 403ffiffiffiffiffigBp ol1=lM . Therefore, the range of

the Lagrange multiplier l under 1offiffiffiffiffigBp ol1=lM is given

by

lð1Þmin90rlrl1 �

ffiffiffiffiffigBp lMffiffiffiffiffigBp � 19lð1Þmax ð30Þ

Then, we have

hðlð1ÞminÞ ¼ hð0Þ ¼sHR�2s

½sHR�1s�24B ð31Þ

hðlð1ÞmaxÞ ¼ hðlÞjlð1Þmaxr

l1þllMþl

� �2 1

g lð1Þmax¼ B

��� ð32Þ

Hence, when 1offiffiffiffiffigBp ol1=lM , there is unique solution

l 2 ½lð1Þmin; lð1Þmax� satisfying hðlÞ ¼ B.

(2) IfffiffiffiffiffigBp o1, using (27) and (28), we can have:

ffiffiffiffiffigBp ðlMþlÞrl1þlffiffiffiffiffigBp ðl1þlÞZlMþl)

lZffiffiffiffiffigBp lM � l1

1�ffiffiffiffiffigBp

lrffiffiffiffiffigBp l1 � lM

1�ffiffiffiffiffigBp

8>><>>:

8>><>>: ð33Þ

SinceffiffiffiffiffigBp o1, but

ffiffiffiffiffigBp lM � l1o0, and lZ0, we havethe implication

ffiffiffiffiffigBp l1 � lM 403ffiffiffiffiffigBp 4lM=l1. Therefore,

the range of the Lagrange multiplier l underlM=l1o

ffiffiffiffiffigBp o1 is given by

lð2Þmin90rlrffiffiffiffiffigBp l1 � lM

1�ffiffiffiffiffigBp 9lð2Þmax ð34Þ

Then, we have

hðlð2ÞminÞ ¼ hð0Þ ¼sHR�2s

½sHR�1s�24B ð35Þ

hðlð2ÞmaxÞ ¼ hðlÞjlð2ÞmaxZ

lMþll1þl

� �2 1

g lð2Þmax¼ B

��� ð36Þ

Hence, when lM=l1offiffiffiffiffigBp o1, there is no solution l 2

½lð2Þmin; lð2Þmax� satisfying hðlÞ ¼ B.

In a word, we can conclude that when 1offiffiffiffiffigBp o

l1=lM , there is a unique solution l 2 ½lð1Þmin; lð1Þmax� satisfying

hðlÞ ¼ B.

3. Norm inequality constraint parameter selection

From the above analysis, we can see that it isimportant to select the norm inequality constraintparameter B for NICCB. If the norm inequality constraintparameter B is large, it is inactive. On the contrary, if thenorm inequality constraint parameter B is small, there isno solution to satisfy NICCB.

We have the result that when 1offiffiffiffiffigBp ol1=lM , there is

a unique solution l 2 ½lð1Þmin;lð1Þmax� satisfying hðlÞ ¼ B.

Hence, we can have the selecting range of the norm

inequality constraint parameter B as follows:

1offiffiffiffiffigBp

ol1

lMð37Þ

i.e.,

1

g oBo 1

gl1

lM

� �2

ð38Þ

Add the condition BosHR�2s=ðsHR�1sÞ29B0, and weobtain

Bmin91

g oBomin B0;1

gl1

lM

� �2( )

9Bmax ð39Þ

If the norm inequality constraint parameter B is out ofthe above range, there is no solution to NICCB. Hence, thenorm inequality constraint parameter B should be chosenin the interval defined by the above inequalities.

4. Capon beamformer under norm equality constraint(NECCB)

From above analysis, we see that the norm inequ-ality constraint can enhance the robustness of NICCB.Since the inequality relationship has a wide range, thenorm of the weight vector will vary in the relevant widerange. If the fluctuation of weight vector norm is acute,the performance improvement will be weakened greatly.Because the norm equality constraint (NEC) is strongerthan the norm inequality constraint (QIC), NECCB willhave more ascendant robust performance than NICCB.Hence, NECCB is proposed and is solved effectively in thispaper.

NECCB is to impose an additional equality constrainton the Euclidean norm of w. The NECCB problem isformulated as follows:

minw

wHRw

s:t: wHs ¼ 1

JwJ2¼ B

8>><>>: ð40Þ

Comparing NECCB and NICCB, we can educe thefollowing conclusions: (1) The solution to NICCB isobtained on the boundary of its constraint. Similarly, forNECCB, the solution is also obtained on its constraintboundary. (2) The solving methods of the two beamfor-mers (or the optimization problem) are different. Theforenamed solution to NICCB, the Lagrange multiplier ofNICCB, is taken as positive real value only, but for NECCB,the Lagrange multiplier is taken as arbitrary real value,i.e., it will not only be the positive real value, but can alsobe of negative real value. Hence, if we are analyzing fromthe point of view of the solving optimization problem,NECCB has two solutions to the optimal Lagrange multi-plier, one is positive, and the other is negative. Actually,the positive one is the solution to NICCB. For the sake ofdistinguishing the other, the negative solution is ofinterest to NECCB. In order to solve NECCB, we mustmake use of the discussed results of NICCB, since themanipulation of some inequality, such as the inequality

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lessening and enlarging, is right only for the positive realvalue when we solve NICCB.

Similar to NICCB, the solution to NECCB can also besolved by the Lagrange multiplier methodology. And theoptimal weight vector of NECCB has the same form asNICCB. The difference between NECCB and NICCB is onlythe Lagrange multiplier �l, for NICCB, lZ0, here �l is ofarbitrary real value.

Although the solution to NECCB has the same form asNICCB, the range of the Lagrange multiplier is different. Inorder to use the analyzed results of NICCB for NECCB,replacing the Lagrange multiplier by its absolute value,namely the range of the Lagrange multiplier �l for NECCB,gives

ffiffiffiffiffigBp

rl1þj

�ljlMþj

�ljð41Þ

ffiffiffiffiffigBp

ZlMþj

�ljl1þj

�ljð42Þ

(1) IfffiffiffiffiffigBp 41, then

lM �ffiffiffiffiffigBp l1ffiffiffiffiffigBp � 1

rj �ljrl1 �

ffiffiffiffiffigBp lMffiffiffiffiffigBp � 1ð43Þ

If l1 �ffiffiffiffiffigBp lM 40, then

ffiffiffiffiffigBp ol1=lM , since lM �ffiffiffiffiffigBp l1o 0, but j �lj40. Therefore, if 1offiffiffiffiffigBp ol1=lM , we

have

�lð1Þ

min9�l1 �

ffiffiffiffiffigBp lMffiffiffiffiffigBp � 1r �lr

l1 �ffiffiffiffiffigBp lMffiffiffiffiffigBp � 1

9 �lð1Þ

max ð44Þ

Since �lð1Þ

max40 and �lð1Þ

min ¼ ��lð1Þ

maxo0, when 1offiffiffiffiffigBp o

l1=lM , the solution to NECCB in the range ½0; �lð1Þ

max� is the

same as of NICCB, but the solution in the range ½ �lð1Þ

min;0� is

the true solution to NECCB.(2) If

ffiffiffiffiffigBp o1, thenffiffiffiffiffigBp lM � l1

1�ffiffiffiffiffigBp rj �ljr

ffiffiffiffiffigBp l1 � lM

1�ffiffiffiffiffigBp ð45Þ

IfffiffiffiffiffigBp l1 � lM 40, then

ffiffiffiffiffigBp 4lM=l1, sinceffiffiffiffiffigBp lM �

l1o 0, but j �lj40. Therefore, if lM=l1offiffiffiffiffigBp o1, we have

lð2Þmin9�ffiffiffiffiffigBp l1 � lM

1�ffiffiffiffiffigBp r �lr

ffiffiffiffiffigBp l1 � lM

1�ffiffiffiffiffigBp 9lð2Þmax ð46Þ

Since �lð2Þ

max40 and �lð2Þ

min ¼ ��lð2Þ

maxo0, with the above

analysis of NICCB, we obtain that when lM=l1offiffiffiffiffigBp o1

there is no solution in the range ½0; �lð2Þ

max� to NECCB, but the

solution in the range ½ �lð2Þ

max;0� is the true solution to

NECCB.From above analysis, we conclude:

(I)

When 1offiffiffiffiffigBp ol1=lM , the solution in the range

½ �lð1Þ

min;0� is the true solution to NECCN, and the

norm equality constraint parameter B should bechosen in the interval defined by 1=goBominfðl1=lMÞ

2=g; B0g.

(II)

Fig. 1. Capon beamformer pattern comparison.

When lM=l1offiffiffiffiffigBp o1, the solution in the range

½ �lð2Þ

min;0� is the true solution to NECCB, and the normequality constraint parameter B should be chosen inthe range ðlM=l1Þ

2=goBominf1=g; B0g.

(III)

NECCB has the form of diagonal loading withnegative loading level, but NICCB has the form ofdiagonal loading with positive loading level.

5. Simulation analysis

In the following we take steps to validate thecorrectness and the efficiency of the proposed algorithms.In our simulations, we assume a uniform linear array withN=10 omnidirectional sensors spaced half a wavelengthapart. In all the examples, we assume that there is onedesired source, namely, there is a signal from direction 01,the signal noise ratio (SNR) is �5 dB. The presumed signaldirection is equal to 51 (i.e., there is a 51 directionmismatch).

For comparison, the benchmark standard Caponbeamforming algorithm corresponds to the ideal casewhen the covariance matrix is estimated by the maximumlikelihood estimator (MLE) and the actual steering vectoris used. This algorithm does not correspond to any realsituation but is included in our simulations for the sake ofcomparison only, and is denoted by Ideal-SCB in thefigure. The other algorithms include standard Caponbeamformer (SCB), NICCB, NECCB. For NICCB and NECCB,the constraint parameter is selected as the median of theallowable range.

5.1. Effectivity analysing

In order to show the effectivity of the proposedalgorithms, we first compare the pattern of the mentionedCapon beamforming algorithms. The Capon beamformerpattern is given in Fig. 1. Since the signal directionmismatches, the mainlobe of SCB departs from the signaldirection. The performance of NICCB is slightly better thanSCB, and NECCB is the best of all. The direction mismatchovercame commendably and NECCB also has lowersidelobe level. Here, NICCB uses the positive optimalloading level and NECCB uses the negative optimalloading level. From the comparison, we can see thatNECCB has better performance than NICCB.

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Variation of the beamformer SNR versus samplesnumber is given in Fig. 2. We can see that with changein the samples number, the SNRs vary correspondingly.The SNR of NICCB is almost closed to the SNR of SCB, andis lower than the SNR of Ideal-SCB, but NECCB is the bestof all, especially for the small number, it has preferableperformance. Hence, the norm constraint can improve theSNR, and NECCB has the highest SNR among the listedalgorithms.

The variation of the Capon beamformer output signalnoise ratio (SNR) versus signal direction mismatch isgiven in Fig. 3. We can see that with change in the signaldirection mismatch, the SNR varies correspondingly,when the angle error is in the range [�71, 71] NECCB hashigher SNR than SCB, NICCB, and Ideal-SCB. The NECCBhas higher SNR and it can be explained by Fig. 1 of thebeam pattern comparison. NECCB not only has goodpointing performance, but also has the lower sidelobelevel. For the same desired signal output, the output noiseof NECCB is less. The simulation results can also beexplained as follows: for the used scene, the signal noiseratio is �5 dB, and for NECCB, the optimal Lagrange is

Fig. 2. Output SNR versus samples number.

Fig. 3. Output SNR versus angle mismatch.

negative, i.e., the optimal loading level is negative, but forothers, the loading level is zero or positive. Therefore, forthe NECCB beamformer, the output noise powerdecreases, but for other beamformer, the output noisepower increases. Hence, the NECCB has higher output SNRthan others. For the sake of saving space, thecorresponding beam pattern comparison is not given,but in the simulation, NECCB pattern also points to theactual signal direction exactly. Hence, NECCB has thebetter robustness in the signal direction mismatch case.

From above analysis, we can see that NECCB has thebest robustness against the signal direction mismatch.

5.2. Correctness analysing

NICCB and NECCB have the same form as that of SCBwith diagonal loading. But their key problems are how tofind their own optimal loading level or Lagrange multi-plier. In order to show the impact of loading level on theCapon beamformer under norm constraint (NCCB) andattest the correctness of the proposed algorithms, thesimulation results are given as follows.

The variation of the output SNR versus diagonalloading level is given in Fig. 4. We can see that with the

change of the loading level in the range ½ �lð1Þ

min;�lð1Þ

max�, the

SNR of NCCB varies correspondingly. When the loadinglevel is positive, NCCB is NICCB, whereas, when theloading level is negative, NCCB is NECCB. By comparison,we can see that NECCB has higher SNR than NICCB, but forthe optimal loading, i.e., when the loading level is equal to�6.09, NECCB has the best pointing performance, and itsSNR is the highest one, where the optimal loading level

�6.09 is calculated using the equation hðlÞ ¼ B with

1offiffiffiffiffigBp o l1=lM in the range ½ �l

ð1Þ

min;0�. Hence, the

loading level has a great impact on the SNR of theCapon beamformer, and it determines the performanceimprovement.

The variation of the weight vector norm versus dia-gonal loading level is given in Fig. 5. We can see that with

Fig. 4. Output SNR versus loading level.

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Fig. 6. Output SNR versus constraint parameter.

Fig. 7. Weight norm versus constraint parameter.

C. Liu, G. Liao / Signal Processing 90 (2010) 1573–15811580

the change of the loading level in the range ½ �lð1Þ

min;�lð1Þ

max�,

the weight vector norm of NCCB varies accordingly. Formost loading level, the weight vector norm varies slightly,but when the loading level is small in negative domain,the weight vector norm is a little high, and the highestpoint is corresponding to the lowest point in Fig. 4.Therefore, the loading level has a great impact on theweight vector norm, especially for the negative loading.

From the above simulation results, we can see that theloading level has a great impact on the performance of theCapon beamformer, and NECCB has the best pointingperformance, i.e., optimal negative loading is the best.This is also consistent to the theory analysis. For therobust beamformer with diagonal loading, the improve-ment is determined by the optimal loading level. Whenthe loading level is optimal, the performance improve-ment will be optimal, but for other values, the improve-ment will be little, or even worse.

5.3. Constraint parameter selection analysis

For NCCB, there are two key problems: One is how tofind the optimal loading level, and the other is how to sel-ect the norm constraint parameter. Although we have sol-ved the two problems in theory, there is another keyproblem, and it is, how to select the optimal norm const-raint parameter. Therefore, the impact of norm constraintparameter on NCCB is analyzed here particularly.

The variation of the output SNR versus norm constraintparameter is given in Fig. 6. We can see that with changeof the norm constraint parameter in the allowable rangeðBmin; BmaxÞ, the SNR of the Capon beamformer varies.NICCB has a slightly higher SNR than SCB. NECCB has thehighest SNR. And with the norm constraint parameterincreasing, the SNR of NECCB increases correspondingly,but the SNR of NICCB is inclined to the SNR of SCB. Whenthe norm constraint parameter is equal to the maximum,the constraint is inactive, and the three SNRs tend to thesame value. Hence, the SNR is determined by the choice ofthe norm constraint parameter, especially for NECCB.

The variation of the weight vector norm versus normconstraint parameter is given in Fig. 7. When the norm

Fig. 5. Weight vector norm versus loading level.

constraint parameter is selected in the allowable rangeðBmin; BmaxÞ, the weight vector norms of NICCB and NECCBvary adaptively, and they are equal to the square rootof the constraint parameter approximately. This isconsistent with the theory, i.e., the solution is obtainedon the constraint boundary. The slight difference is causedby the approximations in computation.

From the above simulation results, we can see that ifthe norm constraint parameter is selected in the allow-able range, the norm constraint parameter has a greatimpact on the performance of NICCB and NECCB,especially for NECCB. But NECCB with the larger con-straint parameter has the better pointing performance;when the constraint parameter is selected as much largerin its allowable range, the optimal negative loading hasthe optimal improvement.

5.4. Simulation discussion

From the above analysis, we can conclude: (I) Theproposed algorithm is correct and effective. (II) The norm

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constraint can improve the robustness of the Caponbeamformer. Especially, the equality constraint has thepreferable improvement to overcome the steering vectormismatch. It also has good robustness for the samplesnumber. (III) When the norm constraint parameter isselected in the allowable range, NECCB has the bestperformance, i.e. the optimal negative loading has theoptimal improvement. This is because the norm equalityconstraint is stronger than the norm inequality constraint.

6. Conclusion

Norm constraints on the weight vector of the Caponbeamformer are used to improve robustness to pointingerrors and to random perturbations in sensor parameter.By a complete investigation on NICCB, the existence of itssolution is analyzed in detail, the choice of the norminequality constraint parameter for NICCB is analyzed andthe selecting range is given. Since the norm equalityconstraint is stronger than the norm inequality constraint,the performance of Capon beamformer under normequality constraint (NEC) is more robust than that ofNICCB. Hence, NECCB is proposed and is solved effectively.At last, numerical examples attest the correctness of thetheory, and show that when the norm constraint para-meter is selected in the allowable range, the performancesof the optimal NICCB and NECCB vary unobvious,respectively, but for the same given norm constraintparameter, NECCB has the better performance than NICCB,i.e. the optimal negative loading has the preferablerobustness.

Acknowledgement

This work was supported in part by the National Out-standing Young Foundation (60825104), the NationalNatural Science Foundation (60736009), the NationalPost-doctor Fundation (20090451251) and the Shaan XiIndustry Surmount Foundation (2009K08-31) of China.

References

[1] J. Capon, High resolution frequency–wavenumber spectrum analy-sis, Proc. IEEE 57 (1969) 1408–1418.

[2] J.S. Reed, J.D. Mallet, L.E. Brennan, Rapid convergence rate inadaptive arrays, IEEE Trans. Aerosp. Electron. Syst. AES-10 (6)(1974) 853–863.

[3] D.H. Johnson, D.E. Dudgeon, Array Signal Processing: Concepts andTechniques, Prentice-Hall, Englewood Cliffs, NJ, 1993.

[4] H. Krim, M. Viberg, Two decades of array signal processing research,IEEE Signal Process. Mag. 13 (4) (1996) 67–94.

[5] L.C. Godara, Application of antenna arrays to mobile communica-tions, Part II: Beam-forming and direction-of-arrival considerations,Proc. IEEE 85 (8) (1997) 1195–1245.

[6] H.L. Van Trees, Detection, Estimation, and Modulation Theory, PartIV, Optimum Array Processing, Wiley, New York, 2002.

[7] P.S. Naidu, Sensor Array Signal Processing, CRC Press, Boca Raton,FL, 2001.

[8] J.R. Guerci, Space–Time Adaptive Processing, Artech House, Nor-wood, MA, 2003.

[9] J. Li, P. Stotica, Robust Adaptive Beamforming, Wiley, New York,2006.

[10] H. Cox, Resolving power and sensitivity to mismatch of optimumarray processors, J. Acoust. Soc. Am. 54 (3) (1973) 771–785.

[11] D.D. Feldman, L.J. Griffiths, A projection approach to robustadaptive beamforming, IEEE Trans. Signal Process. 42 (1994)867–876.

[12] M. Wax, Y. Anu, Performance analysis of the minimum variancebeamformer in the presence of steering vector errors, IEEE Trans.Signal Process. 44 (1996) 938–947.

[13] E.K. Hung, R.M. Turner, A fast beamforming algorithm forlarge arrays, IEEE Trans. Aerosp. Electron. Syst. AES-19 (1983)598–607.

[14] B.D. Carlson, Covariance matrix estimation errors and diagonalloading in adaptive arrays, IEEE Trans. Aerosp. Electron. Syst. 24(1988) 397–401.

[15] M. Wax, Y. Anu, Performance analysis of the minimum variancebeamformer, IEEE Trans. Signal Process. 44 (1996) 928–937.

[16] H. Cox, R.M. Zeskind, M.H. Owen, Robust adaptive beamforming,IEEE Trans. Acoust., Speech, Signal Process. ASSP-35 (1987)1365–1376.

[17] A.B. Gershman, Robust adaptive beamforming in sensor arrays, Int.J. Electron. Commun. 53 (1999) 305–314.

[18] A.B. Gershman, Robust adaptive beamforming: an overview ofrecent trends and advances in the field, in: International Con-ference on Antenna Theory and Technique, 9–12 September 2003,Sewstopol, Ukraine, pp. 30–35.

[19] S.A. Vorobyov, A.B. Gershman, Z.-Q. Luo, Robust adaptive beam-forming using worst-case performance optimization: a solution tothe signal mismatch problem, IEEE Trans. Signal Process. 51 (2)(2003) 313–324.

[20] J. Li, P. Stotica, Z. Wang, On robust capon beamformer and diagonalloading, IEEE Trans. Signal Process. 51 (7) (2003) 1702–1715.

[21] S. Shahram, A.B. Gershman, Z. Luo, et al., Robust adaptivebeamforming for general-rank signal models, IEEE Trans. SignalProcess. 51 (9) (2003) 2257–2269.

[22] J. Li, P. Stotica, Z. Wang, Doubly constrained robust caponbeamformer, IEEE Trans. Signal Process. 52 (9) (2004) 2407–2423.

[23] G.L. Robert, P.B. Stephen, Robust minimum variance beamforming,IEEE Trans. Signal Process. 53 (5) (2005) 1684–1696.

[24] A. Elnashar, S.M. Elnoubi, H.A. El-Mikati, Further study on robustadaptive beamforming with optimum diagonal loading, IEEE Trans.Antennas Propag. AP-54 (12) (2006) 3647–3658.

[25] C.Y. Chen, P.P. Vaidyanathan, Quadratically constrained beamform-ing robust against direction-of- arrival mismatch, IEEE Trans. SignalProcess. 55 (8) (2007) 4139–4150.

[26] R.A. Monzingo, T.W. Miller, Introduction to Adaptive Arrays, Wiley,New York, 1980.

[27] Z. Tian, K.L. Bell, H.L. Van Trees, A recursive least squaresimplementation for LCMP beamforming under quadratic con-straint, IEEE Trans. Signal Process. 49 (6) (2001) 1365–1376.

[28] L.C. Godara, Error analysis of the optimal antenna array processors,IEEE Trans. Aerosp. Electron. Syst. AES-22 (1986) 395–409.

[29] K.L. Bell, Y. Ephraim, H.L. Van Trees, A Bayesian approach torobust adaptive beamforming, IEEE Trans. Signal Process. 48 (2000)386–398.

[30] L. Chang, C.C. Yeh, Performance of DMI and eigenspace-based beamformers, IEEE Trans. Antennas Propag. 40 (1992)1336–1347.

[31] C.-C. Lee, J.-H. Lee, Eigenspace-based adaptive array beamformingwith robust capabilities, IEEE Trans. Antennas Propag. 45 (1997)1711–1716.

[32] J. Riba, J. Goldberg, G. Vazquez, Robust beamforming for inter-ference rejection in mobile communications, IEEE Trans. SignalProcess. 45 (1997) 271–275.

[33] J.R. Guerci, Theory and application of covariance matrix tapers forrobust adaptive beamforming, IEEE Trans. Signal Process. 47 (1999)985–997.

[34] J.R. Guerci, J.S. Bergin, Principal components, covariance matrixtapers, and the subspace leakage problem, IEEE Trans. Aerosp.Electron. Syst. 38 (2002) 152–162.

[35] F. Vincent, O. Besson, Steering vector errors and diagonal loading,IEE Proc.-Radar Sonar Navig. 151 (6) (2004) 337–343.

[36] O. Besson, F. Vincent, Performance analysis of beamformers usinggeneralized loading of the covariance matrix in the presence ofrandom steering vector errors, IEEE Trans. Signal Process. 53 (2005)452–459.

[37] A. Mutapcic, S.-J. Kim, S. Boyd, Beamforming with uncertainweights, IEEE Signal Process. Lett. 14 (5) (2007) 348–351.