2030 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 8, APRIL 15… · 2021. 1. 29. · 2030...

2030 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 8, APRIL 15, 2013

Maximally Robust Capon BeamformerMichael Rübsamen, Member, IEEE, and Marius Pesavento, Member, IEEE

Abstract—The standard Capon beamformer (SCB) achievesthe maximum output signal-to-interference-plus-noise ratio in theerror-free case. However, estimation errors of the signal steeringvector and the array covariance matrix can result in severeperformance deteriorations of the SCB, especially if the trainingdata contains the desired signal component. A popular techniqueto improve the robustness against model errors is to compute theCapon beamformer with the maximum output power, consid-ering an uncertainty set for the signal steering vector. However,maximizing the total beamformer output power may result in aninsufficient suppression of interferers and noise. As an alternativeapproach to mitigate the detrimental effect of model errors, wepropose to compute the Capon beamformer with the minimumsensitivity, considering the uncertainty set for the signal steeringvector. The proposed maximally robust Capon beamformer(MRCB) is at least as robust as the maximum output power Caponbeamformer with the same uncertainty set for the signal steeringvector. We show that the MRCB can be implemented efficientlyusing Lagrange duality. Simulation results demonstrate that theMRCB outperforms state-of-the-art robust adaptive beamformersin many scenarios.

Index Terms—Robust adaptive beamforming.

I. INTRODUCTION

B EAMFORMERS are spatial filters, which are used in anumber of application areas to receive the signal with aspecific spatial signature while interferers and noise are sup-pressed [1]–[4]. The performance of narrowband beamformersis commonly measured in terms of the signal-to-interference-plus-noise ratio (SINR) at the output of the beamformer. Theoutput SINR can be maximized by minimizing the total beam-former output power subject to a distortionless constraint forthe desired signal. This leads to the standard Capon beamformer(SCB) of [5]. The SCB is known to provide an excellent perfor-mance and a fast convergence rate if the beamformer trainingdata does not contain the desired signal component [6]. Thelatter condition is satisfied in some active radar and sonar sys-tems. However, in other application areas such as wireless com-munications and passive radar and sonar systems, the training

Manuscript received May 23, 2012; revised September 11, 2012 and De-cember 26, 2012; accepted December 28, 2012. Date of publication January 23,2013; date of current version March 22, 2013. The associate editor coordinatingthe review of this manuscript and approving it for publication was Dr. KainamThomas Wong. This work was supported in part by the European ResearchCouncil (ERC) Advanced Investigator Grants Program by Grant 227477-ROSEand the German Research Foundation (DFG) by Grant GE 1881/1-4.M. Rübsamen is with the Mobile and Communications Group, Intel, Neu-

biberg 85579, Germany (e-mail: [email protected]).M. Pesavento is with the Department of Electrical Engineering and Infor-

mation Technology, Technische Universität Darmstadt, Darmstadt 64289, Ger-many (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2013.2242067

data is “contaminated” by the desired signal component. Then,even small estimation errors of the signal steering vector or thearray covariance matrix can lead to a severe performance de-terioration of the SCB. Such estimation errors can be causedby environmental nonstationarities, look direction errors, arraycalibration errors, signal wavefront distortions, array geometrydistortions, near-far mismatches, local scattering, etc. If the pre-sumed signal steering vector deviates from the actual signalsteering vector, then the SCB tends to erroneously suppress thedesired signal component. This effect is commonly referred toas signal self-nulling [4]. A similar effect occurs in the case ofestimation errors of the array covariance matrix.A number of techniques have been proposed to improve

the robustness of the SCB against model errors [4], [7]. Forexample, the array response towards the desired signal can bestabilized by enforcing multiple distortionless or derivativeconstraints. However, this approach is only applicable if thearray manifold is known and each distortionless and derivativeconstraint reduces the beamformer degrees of freedom by one.Moreover, these constraints do not provide robustness againstarbitrary-type signal steering vector estimation errors, whichcan occur, e.g., due to signal wavefront distortions.An alternative approach to improve the robustness of the SCB

is to project the estimate of the signal steering vector onto theestimate of the signal-plus-interference subspace [8], [9]. Thecorresponding eigenspace beamformer provides a good perfor-mance for moderate values of the input signal-to-noise ratio(SNR) especially if the number of sensors is significantly largerthan the number of sources. However, the eigenspace beam-former is highly sensitive to estimation errors in the number ofsources, and for low SNRs, signal subspace swaps can result insevere performance losses [10], [11].The beamformer sensitivity to model errors can be limited

by enforcing a distortionless constraint for the presumed signalsteering vector and an upper bound on the norm of the weightvector [12], [13]. The corresponding norm-bounded Caponbeamformer (NBCB) is equivalent to a regularization of thesample covariance matrix, where the regularization parameterdepends on the upper bound on the norm of the beamformerweight vector. The choice of this upper bound is critical for thebeamformer performance.More recently, several set-based worst-case beamformers

have been developed [14]–[22]. These beamformers are basedon an uncertainty set for the signal steering vector. For allvectors within the presumed uncertainty set, the magnitude ofthe array response is constrained to be larger than or equal toone. In the case of spherical or ellipsoidal uncertainty sets, theinfinite number of constraints can be simplified analyticallyusing the worst-case principle. This allows to compute the

1053-587X/$31.00 © 2013 IEEE

RÜBSAMEN AND PESAVENTO: MAXIMALLY ROBUST CAPON BEAMFORMER 2031

beamformer weight vector by solving a convex second-ordercone programming (SOCP) problem [23]. Similar to theNBCB, the set-based worst-case beamformer with a sphericaluncertainty set is equivalent to a regularization of the samplecovariance matrix [14].It has been shown in [24] that the constraints of the set-based

worst-case beamformer of [14] limit the sensitivity to model er-rors. Additionally, if the signal steering vector lies within the in-terior of the presumed uncertainty set, then the constraints of thisbeamformer result in an incentive for a low sensitivity, whichbecomes stronger if the power of the desired signal increases.Such an incentive is desirable since signal self-nulling effectsare most severe in the high SNR regime. A similar incentive fora low sensitivity, which becomes stronger with the power of thedesired signal, does not exist for the NBCB [25].In [26], it has been shown that the set-based worst-case beam-

formers of [14] and [17] are equivalent to the Capon beam-former with the maximum output power, considering the un-certainty set for the signal steering vector. In other words, theset-based worst-case beamformers of [14] and [17] can be ob-tained by replacing the estimate of the signal steering vector inthe SCB formula by the vector from the presumed uncertaintyset, which results in the maximum output power. Therefore,we refer to this beamformer as maximum output power Caponbeamformer (MOPCB).It can be expected that maximizing the Capon beamformer

output power results in an insufficient suppression of interferersand noise. As an alternative approach, we propose to computethe Capon beamformer with the minimum sensitivity to modelerrors, considering the uncertainty set for the signal steeringvector. This beamformer is referred to as maximally robustCapon beamformer (MRCB). By definition, the sensitivity ofthe MRCB is less than or equal to that of the MOPCB with thesame uncertainty set.The proposed approach leads to a quadratically constrained

quadratic programming (QCQP) problem [23]. We show thatthis problem can be solved efficiently using Lagrange duality.Our simulation results suggest that the asymptotic growth rateof the computational complexity of the proposed algorithm isdominated by the eigendecomposition of the sample covariancematrix.We address the relation of theMRCB to the maximally robust

beamformer (MRB), that is the delay-and-sum beamformer. Weprove that the MRCB is equivalent to the MRB if the uncer-tainty set for the signal steering vector is sufficiently large. Con-sidering the exact array covariance matrix, we also show thatthe MRCB is equivalent to the MRB if the true signal steeringvector lies within the interior of the presumed uncertainty set,and if the power of the desired signal is sufficiently high.The SINR performance of the MRCB is compared numer-

ically with that of the SCB, MRB, NBCB, MOPCB, and theeigenspace beamformer. Our simulation results demonstratethat the MRCB frequently outperforms the other beamformerstested. It is also shown that the MOPCB often achieves itsbest SINR performance if the uncertainty set for the signalsteering vector is significantly larger as compared to the actual

error in the presumed signal steering vector, even if the exactarray covariance matrix is used. The MRCB typically achievesits best SINR performance for smaller signal steering vectoruncertainty sets than the MOPCB. This property of the MRCBis desirable, since small signal steering vector uncertainty setsresult in a high spatial resolution capability.

II. BACKGROUNDThe narrowband beamformer output at the th time instant is

(1)

where and are the array snapshot andbeamformer weight vectors, respectively, is the number ofsensors, denotes the set of complex numbers, and standsfor the Hermitian transpose. The snapshot vectors are modeledas

(2)

where is the number of sources, is the steeringvector (spatial signature) of the th source, is the base-band waveform of the th source at the th time instant,

is the noise vector, and stands for the transpose.Without loss of generality, we assume that the source with index

is the source-of-interest, while the sources with indicesare interferers. Hence, the received snapshot

vector can be expressed as

(3)

where andare the desired signal and interferer components, respectively.In the sequel, we assume that the desired signal, interferer, andnoise waveforms are quasi-stationary zero-mean random pro-cesses and that the desired signal waveform is uncorrelated withthe interferer and noise waveforms. Then, the array covariancematrix can be expressed as

(4)

where is the power of the desired signal,denotes the statistical expectation, and

(5)

is the interference-plus-noise covariance matrix. The beam-former performance is commonly measured in terms of theoutput SINR, defined as [4]

(6)

The output SINR can be maximized by minimizing the outputinterference-plus-noise power subject to a distortionless con-straint for the desired signal. This can be formulated as

(7)


The optimum weight vector of (7) is [4]

(8)

where we assumed that is non-singular. In practice, thisassumption is usually satisfied due to the presence of a whitenoise component. Substituting (8) in (6) yields the optimumSINR

(9)

Due to (4) and the distortionless constraint in (7), replacingby in the objective function of (7) leads to an addi-

tional constant term. Thus, the optimum weight vector of (7)does not change if is replaced by . The array covari-ance matrix can be estimated as

(10)

where is the number of training snapshot vectors. Replacingin (8) by the sample covariance matrix and by the

presumed (estimated) signal steering vector leads to the SCB[4]

(11)

where is assumed to be nonsingular. It is well-known thatestimation errors in and can lead to a severe performancedegradation of the SCB, especially if the SNR is high.

A. Beamformer Sensitivity

Let denote the presumed upper bound on the norm of thesignal steering vector estimation error. That is, we consider theuncertainty set for the signal steering vector

(12)

where is the Euclidean vector norm. The array responsefor the presumed (estimated) signal steering vector is .The maximum deviation of the array response from forall steering vectors in is

(13)

Clearly, the larger the uncertainty in the signal steering vector,the larger can be the deviation of the array response from .To obtain a measure for the beamformer sensitivity, we nor-malize (13) by . Moreover, we divide (13) by , sincewe are interested in the relative size of the deviation in the arrayresponse. This leads to the definition of the beamformer sensi-tivity [4], [27], [28]

(14)

where the square has been taken for convenience. If is small,then the beamformer is robust against signal steering vector esti-mation errors. Note that the beamformer sensitivity is defined asthe squared norm of the scaled weight vector ,which satisfies . The normalization with respect to

in (14) is important, since otherwise an arbitrarily lowsensitivity could be achieved by scaling the beamformer weightvector without changing the SINR performance.The definition of the beamformer sensitivity in (14) can also

be motivated by considering the effect of estimation errors in thesample covariance matrix on the beamformer output power. Letdenote the presumed upper bound on the Frobenius norm of

the estimation error in the sample covariance matrix. Then, thecorresponding maximum variation of the beamformer outputpower is

(15)

where denotes the Frobenius norm. Similar to (13), (15) in-creases with the norm of the beamformer weight vector. Hence,if is small, then the beamformer is robust against estimationerrors in and .The sensitivity (14) is minimized by the MRB

(16)

and all non-zero scalings of this weight vector. Since the MRBis non-adaptive, it does not suffer from signal self-nulling. Atthe same time, it does not adaptively suppress interferers. Forthis reason, the performance of the MRB is usually significantlyworse than that of the SCB in the low SNR regime.Over the last decades, a number of techniques have been pro-

posed to improve the robustness of the SCB against model er-rors. Two popular robust adaptive beamformers, the NBCB andthe set-based worst-case beamformer, are reviewed next.

B. Norm-Bounded Capon Beamformer

The beamformer sensitivity can be limited by enforcing a dis-tortionless constraint and an upper bound on the norm of thebeamformer weight vector. This leads to the NBCB [13], [28]

(17)

where is a user-defined parameter. Applying theCauchy-Schwarz inequality to the equality constraint of(17) yields . Hence, (17) is feasible if and onlyif . Numerical results show that choosingmoderately (1–3 dB) above this lower bound often leads to agood performance [24], [28].The problem (17) belongs to the class of SOCP problems

[23]. An efficient algorithm to solve this problem has been pre-sented in [26]. Simulation results suggest that the asymptoticgrowth rate of the complexity of this algorithm is dominated bythe eigendecomposition of , which is [29].


C. Set-Based Worst-Case Beamformer

The set-based worst-case beamformer of [14] minimizes theexpected output power subject to the constraint that the magni-tude of the array response is larger than or equal to one for allsteering vectors in . This can be formulated as

(18)

The constraints in (18) are satisfied if and only if they are sat-isfied for the worst-case steering vector, which minimizes themagnitude of the array response. In the sequel, we assume that

. Then, [14]

(19)

Thus, (18) can be written as

(20)

In the literature, the beamformer (20) is often referred toas “worst-case performance optimization based beamformer.”However, it has been explained in [24] that, even ifand , the optimum weight vector of (20) does not max-imize the worst-case output SINR, defined as

(21)

Since the beamformer (20) does not maximize the worst-caseSINR, should not necessarily be chosen according to the ex-pected size of the signal steering vector estimation errors. Infact, numerical results show that, even if , the bestSINR performance of the beamformer (20) is often attained ifis chosen significantly larger than the actual norm of the signalsteering vector estimation error.The robustness of the set-based worst-case beamformer (20)

can be explained as follows [24]. Any weight vector, which sat-isfies the constraint of (20) with strict inequality, can be scaledsuch that the constraint is still satisfied, but a lower value ofthe objective function is attained. Hence, the optimum weightvector has to satisfy the constraint of (20) with equality. Thus,the sensitivity of the beamformer (20) is

(22)

That means, the constraint in (20) limits the beamformer sen-sitivity. Moreover, if denotes the signal steeringvector estimation error, then the constraint in (20) yields

(23)

The minimization of the total beamformer output power leadsto an incentive for a small value of . The strength ofthis incentive increases with the power of the desired signal.The inequality (23) results in an incentive for a small value of

if the signal steering vector lies within the interior of ,

i.e., if . The sensitivity (22) is strictly increasing in. Thus, if , then the constraint in (20) leads

to an incentive for a low sensitivity, which becomes strongerwith the power of the desired signal. A similar signal-dependentincentive for a low sensitivity does not exist for the NBCB [25].Due to (23), the strength of the incentive for a low sensitivity

of the set-based worst-case beamformer increases also with ,assuming that . That means, the larger the uncertaintyset for the signal steering vector, the stronger is the incentive fora low sensitivity.The beamformer (18) is based on the spherical uncertainty

set . The generalization to ellipsoidal uncertainty sets canbe solved in a similar way [17]. For the sake of simplicity, wedo not further consider ellipsoidal signal steering vector un-certainty sets, even though the generalization to these sets isstraightforward.It has been shown in [26] that the set-based worst-case beam-

former (18) can be obtained by solving

(24)

Hence, the beamformer (18) is equivalent to the MOPCB, i.e.,the Capon beamformer with the maximum output power, con-sidering the uncertainty set for the signal steering vector. Anefficient technique to solve (24) based on the eigendecomposi-tion of the sample covariance matrix has been presented in [26].Simulation results suggest that the complexity of this techniqueis .

III. MAXIMALLY ROBUST CAPON BEAMFORMER

In general, the beamformer output power consists of a desiredsignal, an interference, and a noise component. The maximiza-tion of the Capon beamformer output power in (24) diminishesthe erroneous suppression of the desired signal component. Atthe same time, it may lead to a poor suppression of interferersand noise. As an alternative approach, we propose to computethe Capon beamformer with the minimum sensitivity to modelerrors, considering the uncertainty set for the signal steeringvector. The proposed beamforming problem can be formulatedas

(25)

Obviously, the sensitivity of the proposed beamformer is lessthan or equal to that of the MOPCB with the same uncertaintyset. In other words, the proposed beamformer is at least as robustagainst model errors as the MOPCB with the same uncertaintyset.


Substituting the equality constraint of (25) in the objectivefunction yields

(26)

Since any scaling of has no impact on the objective functionof (26), the constraint of this optimization problem can be re-placed by

(27)

The minimum in (27) is attained for . Usingthis result, (27) can be written as

(28)

where is the orthogonal projection ma-trix onto the range space of . Straightforward transformationsallow to write (28) as

(29)

Thus, (26) can be formulated as

(30)

where

(31)

The objective function and the constraint in the latter optimiza-tion problem are invariant with respect to the scaling of . Wechoose to scale this vector such that . This leadsto the optimization problem

(32)

Proposition 1: The optimization problem (32) and its relax-ation

(33)

where denotes the real part operator, have the same op-timum point. Moreover, strong duality holds for (33).

Proof: See Appendix A.

Due to the strong duality of (33), this optimization problemcan be solved using Lagrange duality ([23], Section 5.5.5). TheLagrangian function of (33), taking the squares of the left- andright-hand sides of the inequality constraint, can be written as

(34)

where and are real-valued Lagrange multipliers,

(35)

and stands for the identity matrix. The dual functionis defined as [23]

(36)

The latter function is finite if and only if and, where denotes the column-space operator. To

determine the dual function, we take the derivative of the La-grangian function [30]

(37)

where denotes the complex conjugate. Setting the deriva-tive to zero yields

(38)

where is the Moore-Penrose pseudo-inverse. Substituting(38) in (34) gives

(39)

Hence, the dual problem can be formulated as

(40)

If is an optimum point of (40), then is an op-timum point of (33).The optimization problem (40) can be simplified by maxi-

mizing over analytically. Thereto, we set the derivative of theobjective function with respect to to zero. This yields

(41)

Substituting (41) in (38) gives

(42)


Moreover, (40) can be reformulated as

(43)

Since the objective function of (43) is positive, this optimizationproblem can be written equivalently as

(44)

The semidefinite constraint is convex in and it issatisfied for . For ,

(45)

is positive definite, so we can write

(46)

Thus, for if and only if

(47)

The function can be rewritten using the eigendecomposi-tion of , that is

(48)

where the unitary matrix containsthe orthonormal eigenvectors, and the diagonal matrix

contains the eigenvalues. In thesequel, we assume without loss of generality that the eigen-values are sorted in decreasing order, i.e., .Using (48), we obtain

(49)

where is the th component of . It can be easilyshown that and that is strictly monotonicallyincreasing for positive values of . Moreover,

(50)

Consequently, there is a unique positive value such that

(51)

Then, for and for .The optimization problem (44) can therefore be written as

(52)

where can be determined, e.g., by means of the secantmethod. This method involves the evaluation of at a largenumber of points. Once and have been computed, (49) al-lows to efficiently determine using only operations.From (51), we obtain

(53)

(54)

(55)

These upper and lower bounds for may be used as initialvalues for the secant method.Proposition 2: The constraint

(56)

is satisfied for , but not for .Proof: See Appendix B.

The latter proposition allows to formulate (52) as

(57)

The dual function is always concave ([23], Section 5.1.2).The objective function of (43) has been obtained by maximizingthe dual function over . Hence, the objective functionof (43) is concave for ([23], Section 3.2.5).Due to the positiveness of this function for , itsreciprocal

(58)

is convex for ([23], Section 3.2.4). Thus, (57) isa convex optimization problem.The convexity of (57) implies that the optimum point is

if and only if . For matrix-valued functions,we have [4]

(59)

Hence,

(60)

Consequently, the optimum point of (57) is if and onlyif

(61)

Based on Proposition 2, it can be shown that is unboundedabove for , where means convergence from below.Therefore, if (61) is not satisfied, then the optimum point of (57)can be determined by searching for the root of using, e.g.,


the secant method [31]. This method requires the evaluation ofat a large number of points. We show in Appendix C thatand can be computed efficiently based on the eigen-

decomposition of . In particular, if and are available,then the evaluation of and requires only op-erations.In summary, the MRCB can be obtained as follows:

1: Compute the eigendecomposition of and .2: Search for , which satisfies (51) and (53)–(55).3: Search for the optimum point of (57).4: Compute the beamformer weight vector

(62)

where is given by (42).

Remark 1: The steps 2 and 3 of the proposed algorithmhave comparable computational complexities if they are imple-mented using the secant method, because and canboth be evaluated using operations, assuming that andare known. Moreover, step 4 can be implemented efficiently

based on the eigendecomposition of . Our simulation resultssuggest that the computational complexity of the proposedalgorithm is asymptotically (with ) dominated by theeigendecomposition in step 1. Hence, it can be expected that thecomputational complexity of the proposed algorithm is .Remark 2: Substituting (42) in (62) yields

(63)

where the scalar has no effect on the SINR performance.Consequently, the proposed beamformer is not equivalent toa regularization of . This is in contrast to the NBCB andMOPCB, see [13] and [14], respectively.Remark 3: If (61) holds, then the optimum steering vector of

(33) follows from (42) as

(64)

and the corresponding beamformer weight vector is

(65)

Hence, if (61) holds, then the MRCB provides the same SINRperformance as the MRB. Using (31), it can be shown that (61)is satisfied if

(66)

Consequently, the MRCB is equivalent to the MRB if the un-certainty set for the signal steering vector is sufficiently large,so that (66) holds.

Replacing in (61) by gives

(67)

Using (4), the latter inequality can be written as

(68)

where

(69)

(70)

(71)

If the actual signal steering vector lies in the interior of , then(see (29))

(72)

so . The quadratic term in (68) dominates over the linearand constant terms if is large. Consequently, if the true signalsteering vector lies in the interior of , and if the power ofthe desired signal is sufficiently high, then (68) is satisfied. Inthis case, the MRCB using the exact array covariance matrixprovides the same SINR performance as the MRB. It can beexpected that the MRCB shows a similar behavior if the samplecovariance matrix is used. This will be evaluated numerically inthe subsequent section.

IV. SIMULATION RESULTSLet us assume three far-field sources that impinge on a uni-

form linear array (ULA) of identical isotropic sen-sors. The displacement between adjacent sensors is half a wave-length. The directions-of-arrival (DOAs) of the two interferersare and relative to the broadside direction ofthe ULA. The presumed (estimated) DOA of the desired signalis . The actual steering vector of the desired signalis generated randomly as

(73)

where corresponds to , and is drawn in eachMonte-Carlo run independently from a zero-mean circu-larly symmetric complex Gaussian distribution with covariance

. The parameter is chosen such that the average norm ofis one. The desired signal, interferer, and noise waveforms

are generated as uncorrelated white standard complex Gaussianrandom processes. The noise waveforms are assumed to havethe same power in all sensors. The input interference-to-noiseratios (INRs) of the two interferers are set to 20 dB. The arraycovariance matrix is estimated using independentsnapshot vectors.We compare the performance of the MRCB with that of the

SCB, the MRB, the NBCB, the MOPCB, and the eigenspacebeamformer of [8], [9]. For the NBCB, we set ,where . For the MOPCB and MRCB, we use

such that in 95% of the Monte-Carlo runs.Note that the MOPCB is equivalent to the set-based worst-case


Fig. 1. Beamformer performance versus the input SNR (for snapshotvectors, , and ).

beamformer of [14]. As a reference, we also plot the optimumoutput SINR (9).First, we study the beamformer output SINR versus the input

SNR (averaged over the sensors). Fig. 1 shows that the perfor-mance of the SCB deteriorates severely for high input SNRs dueto the signal self-nulling effect. The other beamformers testedprovide a significantly improved robustness against signal self-nulling as compared to the SCB. TheMRB outperforms the con-ventional adaptive beamformers in the high SNR regime as itdoes not suffer from signal self-nulling. Fig. 1 demonstrates thatthe performance of the MRCB is similar to that of the MRB forhigh input SNRs. However, the MRCB outperforms the MRBfor low input SNRs. Overall, it can be observed that the MRCBis an attractive alternative to the other beamformers tested forthe whole range of input SNRs.In the next simulation example, we study the beamformer

performance versus the number of training snapshot vectors thatare used to estimate the array covariance matrix. The input SNRis set to dB. All other parameters are chosen as before.The performance of the MRB is not depicted as it is below theSINR range plotted. Fig. 2 shows that the MRCB outperformsthe other beamformers tested for snapshot vectors.For snapshot vectors, the eigenspace beamformerprovides a better performance as compared to the MRCB.Next, we analyze the effect of the angular distance between

the desired signal and the interferers. Thereto, we vary the pre-sumed (estimated) DOA of the desired signal. The number oftraining snapshot vectors is . All other parameters arechosen as before. Fig. 3 shows that the performance of all beam-formers deteriorates if the distance between and(the DOA of one of the interferers) decreases. The MRCB out-performs the other beamformers tested for the entire range ofpresumed signal DOAs considered.In Fig. 1, the MRCB provides a similar performance as the

MRB for high input SNRs, but for low input SNRs, the MRCBsignificantly outperforms the MRB. In Section III, we haveshown that the MRCB provides the same SINR performance asthe MRB if is sufficiently large so that (66) holds. Next, weanalyze the effect of on the performance of the MRCB and

Fig. 2. Beamformer performance versus the number of training snapshot vec-tors (for dB, , and ).

Fig. 3. Beamformer performance versus the presumed DOA of the desiredsignal (for dB, snapshot vectors, and ).

MOPCB. Thereto, we change the presumed upper bound onthe norm of the signal steering vector estimation errors. At thesame time, we do not change the statistics of the actual signalsteering vector estimation errors. Hence, we still draw the signalsteering vector estimation errors from a Gaussian distributionsuch that the average norm of is one. The presumed DOA ofthe desired signal is set to . Figs. 4, 5, and 6 depictthe beamformer performance for input SNRs of 10, 10, and30 dB, respectively. All other parameters are chosen as before.In Fig. 4, the MRCB outperforms the other beamformers

tested if . Clearly, if , then the optimumsteering vectors of (24) and (25) converge to . Therefore,the performance of the MRCB and MOPCB converges tothe performance of the SCB if . The performance ofthe MRB is not depicted in Fig. 4 as it is below the SINRrange plotted. Hence, the performance of the MRCB degradesto that of the MRB in the narrow interval ,where . In fact, (66) was satisfied in none of theMonte-Carlo runs of Fig. 4.


Fig. 4. Beamformer performance versus the presumed upper bound on thenorm of the signal steering vector estimation errors (for dB,

snapshot vectors, and ).

Fig. 5. Beamformer performance versus the presumed upper bound on thenorm of the signal steering vector estimation errors (for dB,snapshot vectors, and ).

Fig. 5 shows that, for an input SNR of 10 dB, the MRCBachieves its maximum output SINR at . If increasesbeyond this value, then the performance of theMRCB degrades,and it eventually converges to the performance of the MRB forlarge values of .For an input SNR of 30 dB, Fig. 6 shows that the performance

of the MRCB is close to that of the MRB if . Further-more, the MOPCB provides a similar performance as the MRBfor large values of . If the true signal steering vector lies in theinterior of , then the MOPCB has an incentive for a low sen-sitivity (see Section II-C). This incentive becomes stronger ifthe power of the desired signal or increases. Hence, it comesat no surprise that the MOPCB and MRB provide a similar per-formance if and are large.In Figs. 4–6, the MOPCB achieves an excellent performance

for large values of . Since this may be partially due to the esti-mation errors in the sample covariancematrix, we show in Fig. 7

Fig. 6. Beamformer performance versus the presumed upper bound on thenorm of the signal steering vector estimation errors (for dB,snapshot vectors, and ).

Fig. 7. Beamformer performance versus the presumed upper bound on thenorm of the signal steering vector estimation errors (for dB,

, and the exact array covariance matrix).

the beamformer performance using the exact array covariancematrix. All other parameters are chosen as before. ComparingFigs. 6 and 7 shows that the beamformers tested provide a sim-ilar performance with and without finite sample errors. In allMonte-Carlo runs of Fig. 7, the norms of the randomly gen-erated signal steering vector estimation errors were less than1.8. Therefore, Fig. 7 also shows that, even if there are no finitesample errors, the MOPCB achieves its best SINR performancefor values of that are significantly larger than the actual normsof the signal steering vector estimation errors.The latter simulation results may suggest that a large value

of should be chosen for the MOPCB irrespectively of the pre-sumed size of the signal steering vector estimation errors. Fig. 8depicts the beamformer performance versus the presumed DOAof the desired signal using . All other parameters arechosen as for Fig. 3. In particular, the input SNR is set todB. Comparing Figs. 3 and 8 shows that the MOPCB provides


Fig. 8. Beamformer performance versus the presumed DOA of the desiredsignal (for dB, snapshot vectors, and ).

a significantly better performance when if the desiredsignal is far from the interferers. At the same time, it can be ob-served that the larger value of results in a performance degra-dation of the MOPCB and MRCB if the desired signal is closeto the interferer (compare the performance for inFigs. 3 and 8). Hence, increasing the value of results in a degra-dation of the spatial resolution capability. Figs. 4–7 show thatthe MRCB achieves its best SINR performance for significantlysmaller values of than the MOPCB, and these smaller valuesof result in a better spatial resolution capability.Finally, Fig. 9 depicts beampatterns of the MRCB and

MOPCB. The beampatterns have been computed using theexact array covariance matrix. The actual and presumed DOAsof the desired signal are and , respec-tively. The presumed upper bound on the norm of the signalsteering vector estimation errors has been set to , andthe input SNR is 10 dB. All other parameters are chosen asbefore. The beamformer weight vectors have been scaled suchthat . The vertical dashed lines in Fig. 9 show thedirections of the desired signal and the two interferers. Fig. 9demonstrates that the MRCB leads to a wider main beam andlower sidelobes as compared to the MOPCB. The wider mainbeam results in an improved robustness against look directionerrors. Due to its lower sidelobes, it can be expected that theMRCB has a better noise suppression than the MOPCB. InFig. 9, the MRCB provides a worse interference suppression ascompared to the MOPCB. However, both beamformers suffi-ciently suppress the interferers so that the output interferencepowers are significantly lower than the output noise powers.If the beamformer weight vectors are scaled such that

, then the beamformer sensitivity (14) is propor-tional to the beamformer output power for spatially white noise.Hence, if , then the output power of the MRCB forspatially white input noise is always less than or equal to thatof the MOPCB with the same uncertainty set for the signalsteering vector. Further simulation results have shown that themagnitude of the array response towards the desired signal isusually higher for the MRCB as compared to the MOPCB. Inother words, the MRCB usually provides a higher output powerfor the desired signal as compared to the MOPCB. However,

Fig. 9. Array beampattern (for dB,, and the exact array covariance matrix; the beamformer weight vectors

have been scaled such that ).

it depends on the scenario whether the MRCB or the MOPCBprovides a better interference suppression. Consequently, themain advantage of the MRCB as compared to the MOPCB isits improved robustness against model errors, which results inan improved protection of the desired signal component and abetter noise suppression.

V. CONCLUSIONWe proposed the Capon beamformer with the minimum

sensitivity to model errors, considering an uncertainty set forthe signal steering vector. An efficient technique to computethe MRCB has been developed. Simulation results suggest thatthe asymptotic growth rate of the computational complexity ofthis technique is dominated by the eigendecomposition of thesample covariance matrix.If the uncertainty set for the signal steering vector is suffi-

ciently large, then the MRCB provides the same SINR perfor-mance as the MRB. Moreover, considering the exact array co-variance matrix, we showed that the MRCB provides the sameSINR performance as the MRB if the actual signal steeringvector lies within the interior of the presumed uncertainty set,and if the power of the desired signal is sufficiently high.Simulation results demonstrate that theMRCB is an attractive

alternative to state-of-the-art robust adaptive beamformers. Inparticular, theMRCB typically provides an improved protectionof the desired signal component and a better noise suppressionas compared to the MOPCB with the same uncertainty set forthe signal steering vector.

APPENDIX APROOF OF PROPOSITION 1

To show that (32) and (33) have the same optimum point, weshow that, if denotes the optimum point of (33), then

(74)

where denotes the imaginary part operator. For the proof,we assume that (74) is not satisfied and show that this leadsto a contradiction. Hence, we assume that ,


where denotes the non-zero imaginary part of , and. Then, it can be easily verified that is a

feasible point of (33), which leads to a lower value of the objec-tive function than . However, this contradicts the optimalityof . Consequently, (32) and (33) are equivalent.To show that strong duality holds for (33), let us consider its

relaxation

(75)

Any vector , which satisfies the first inequality constraint of(75) with strict inequality, cannot be optimum since a lowervalue of the objective function can be achieved by scaling thisvector. Hence, the optimum point of (75) has to satisfy the firstinequality constraint of this problem with equality. Therefore,(33) and (75) have the same solution. It has been shown in [32]that strong duality generally holds for optimization problemsover the complex plane with a quadratic objective function andup to two quadratic inequality constraints. Thus, strong dualityholds for (75). The Lagrange multiplier corresponding to thefirst constraint of this problem is restricted to be nonnegative.This restriction does not exist for the Lagrange multiplier cor-responding to the equality constraint of (33). For this reason, thesolution of the dual problem of (33) is larger than or equal to thesolution of the dual problem of (75). Therefore, strong dualityholds also for (33).

APPENDIX BPROOF OF PROPOSITION 2

Equation (46) and the strict monotony of imply thatfor . Consequently, (56) is satisfied

for .For , the bracketed matrix in (46) is positive

semidefinite with a single eigenvalue equal to zero. Theeigenvector corresponding to the zero eigenvalue is .Hence, also has exactly one eigenvalue, which is equalto zero, and the corresponding eigenvector is . Theconstraint (56) is satisfied for if and only ifis orthogonal to this eigenvector, i.e.,

(76)

However,

(77)

is positive definite. Thus, (56) cannot be satisfied for .

APPENDIX CIMPLEMENTATION OF (57)

The objective function of (57) and its derivative can be ex-pressed as

(78)

(79)

where

(80)

Using the matrix inversion lemma, we obtain

(81)

The eigendecomposition of allows to express as

(82)

where

(83)

is a diagonal matrix. Substituting (82) in (78) and using theeigendecomposition of gives

(84)

It can be easily verified that, if and are available, then (84)can be evaluated using operations. Moreover, since

(85)

and

(86)

can be evaluated with the same complexity.

REFERENCES[1] R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays.

New York, NY, USA: Wiley, 1980.[2] D. H. Johnson and D. E. Dudgeon, Array Signal Processing: Concepts

and Techniques, A. V. Oppenheim, Ed. Englewood Cliffs, NJ, USA:Prentice-Hall, 1993.

[3] L. C. Godara, “Application of antenna arrays to mobile communica-tions, Part II: Beam-forming and direction-of-arrival considerations,”Proc. IEEE, vol. 85, pp. 1195–1245, Aug. 1997.

[4] H. L. van Trees, Optimum Array Processing. New York, NY, USA:Wiley, 2002.

[5] J. Capon, “High-resolution frequency-wavenumber spectrum anal-ysis,” Proc. IEEE, vol. 57, pp. 1408–1418, Aug. 1969.

[6] I. S. Reed, J. D. Mallett, and L. E. Brennan, “Rapid convergence ratein adaptive arrays,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-10,pp. 853–863, Nov. 1974.

[7] A. B. Gershman, “Robust adaptive beamforming in sensor arrays,” Int.J. Electron. Commun., vol. 53, pp. 305–314, Dec. 1999.

[8] D. D. Feldman and L. J. Griffiths, “A constraint projection approachfor robust adaptive beamforming,” in Proc. IEEE Int. Conf. Acoust.,Speech, Signal Process. (ICASSP), Toronto, ON, Canada, May 1991,pp. 1381–1384.

[9] L. Chang and C.-C. Yeh, “Performance of DMI and eigenspace-basedbeamformers,” IEEE Trans. Antennas Propag., vol. 40, pp. 1336–1347,Nov. 1992.

[10] J. K. Thomas, L. L. Scharf, and D. W. Tufts, “The probability of asubspace swap in the SVD,” IEEE Trans. Signal Process., vol. 43, pp.730–736, Mar. 1995.

[11] M. Hawkes, A. Nehorai, and P. Stoica, “Performance breakdown ofsubspace-based methods: Prediction and cure,” in Proc. IEEE Int.Conf. Acoustics, Speech, Signal Process. (ICASSP), Salt Lake City,UT, USA, May 2001, pp. 4005–4008.


[12] Y. I. Abramovich, “Controlled method for adaptive optimization of fil-ters using the criterion of maximum SNR,” Radio Eng. Electron. Phys.,vol. 26, pp. 87–95, Mar. 1981.

[13] J. E. Hudson, Adaptive Array Principles, P. J. B. Clarricoats, G.Millington, E. D. R. Shearman, and J. R. Wait, Eds. New York, NY,USA: Peregrinus, 1981.

[14] S. A. Vorobyov, A. B. Gershman, and Z.-Q. Luo, “Robust adaptivebeamforming using worst-case performance optimization: A solutionto the signal mismatch problem,” IEEE Trans. Signal Process., vol. 51,pp. 313–324, Feb. 2003.

[15] S. Shahbazpanahi, A. B. Gershman, Z.-Q. Luo, and K. M. Wong,“Robust adaptive beamforming for general-rank signal models,” IEEETrans. Signal Process., vol. 51, pp. 2257–2269, Sep. 2003.

[16] S. A. Vorobyov, A. B. Gershman, Z.-Q. Luo, and N. Ma, “Adaptivebeamforming with joint robustness against mismatched signal steeringvector and interference nonstationarity,” IEEE Signal Process. Lett.,vol. 11, pp. 108–111, Feb. 2004.

[17] R. G. Lorenz and S. P. Boyd, “Robust minimum variance beam-forming,” IEEE Trans. Signal Process., vol. 53, pp. 1684–1696, May2005.

[18] Z. L. Yu and M. H. Er, “A robust minimum variance beamformer withnew constraint on uncertainty of steering vector,” Signal Process., vol.86, pp. 2243–2254, Sep. 2006.

[19] A. Hassanien, S. A. Vorobyov, and K. M. Wong, “Robust adaptivebeamforming using sequential quadratic programming: An iterative so-lution to the mismatch problem,” IEEE Signal Process. Lett., vol. 15,pp. 733–736, 2008.

[20] A. B. Gershman, N. D. Sidiropoulos, S. Shahbazpanahi, M. Bengtsson,and B. Ottersten, “Convex optimization-based beamforming,” IEEESignal Process. Mag., vol. 27, no. 3, pp. 62–75, May 2010.

[21] J. P. Lie, W. Ser, and C. M. S. See, “Adaptive uncertainty based itera-tive robust Capon beamformer using steering vector mismatch estima-tion,” IEEE Trans. Signal Process., vol. 59, pp. 4483–4488, Sep. 2011.

[22] S. E. Nai, W. Ser, Z. L. Yu, and H. Chen, “Iterative robust minimumvariance beamforming,” IEEE Trans. Signal Process., vol. 59, pp.1601–1611, Apr. 2011.

[23] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge,U.K.: Cambridge Univ. Press, 2004.

[24] M. Rübsamen and A. B. Gershman, “Robust adaptive beamformingusing multidimensional covariance fitting,” IEEE Trans. SignalProcess., vol. 60, pp. 740–753, Feb. 2012.

[25] M. Rübsamen, “Advanced direction-of-arrival estimation and beam-forming techniques for multiple antenna systems” Ph.D. dissertation,Technische Universität Darmstadt, Darmstadt, Germany, May 2011[Online]. Available: http://tuprints.ulb.tu-darmstadt.de/2635/

[26] J. Li, P. Stoica, and Z. Wang, “On robust Capon beamforming and di-agonal loading,” IEEE Trans. Signal Process., vol. 51, pp. 1702–1715,July 2003.

[27] E. N. Gilbert and S. P. Morgan, “Optimum design of directive antennaarrays subject to random variations,” Bell Syst. Tech. J., vol. 34, pp.637–663, May 1955.

[28] H. Cox, R. Zeskind, and M. Owen, “Robust adaptive beamforming,”IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, pp.1365–1376, Oct. 1987.

[29] G. H. Golub and C. H. van Loan, Matrix Computations. Baltimore,MD, USA: The Johns Hopkins Univ. Press, 1996.

[30] D. H. Brandwood, “A complex gradient operator and its application inadaptive array theory,” in Inst. Electr. Eng. Proc. F—Commun., RadarSignal Process., Feb. 1983, vol. 130, no. 1, pp. 11–16.

[31] J. Nocedal and S. J. Wright, Numerical Optimization, P. Glynn and S.M. Robinson, Eds. New York, NY, USA: Springer, 1999.

[32] A. Beck and Y. C. Eldar, “Strong duality in nonconvex quadratic opti-mization with two quadratic constraints,” SIAM J. Optim., vol. 17, no.3, pp. 844–860, 2006.

Michael Rübsamen (S’05–M’12) received theDipl.-Ing. degree from RWTH Aachen University,Germany, in 2006, and the Dr.-Ing. degree fromTechnische Universität Darmstadt, Germany, in2011, both in electrical engineering.During 2004–2005, he held a research scholarship

from Bell-Labs, Crawford Hill, NJ, where he workedon signal processing techniques for fiber optic com-munication systems. From 2006 to 2012, he was aResearch Scientist with the Communication SystemsGroup, Technische Universität Darmstadt. In 2009,

he visited Temasek Labs, Nanyang Technological University (NTU), Singapore.In June 2012, he joined the Mobile and Communications Group at Intel, Neu-biberg, Germany, where he is currently working as a Wireless System Engineer.His research interests are focused on statistical and array signal processing, andcommunication systems.Dr. Rübsamen received the Student Best Paper Award of the IEEE Sensor

Array and Multichannel Signal Processing (SAM) Workshop in 2010. He wasstudent contest finalist at the IEEE International Conference on Acoustics,Speech, and Signal Processing (ICASSP) in 2008. He was awarded a grantfrom the German National Academic Foundation in 2003. In 2002 and 2003,he received the Philips Student Award and the Henry-Ford-II Student Award,respectively. He served as Local Organization Co-Chair for the IEEE SAMWorkshop, Darmstadt, in 2008.

Marius Pesavento (M’00) received the Dipl.-Ing.and M.Eng. degree from Ruhr-Universität Bochum,Germany, and McMaster University, Hamilton,ON, Canada, in 1999 and 2000, respectively, andthe Dr.-Ing. degree in electrical engineering fromRuhr-Universität Bochum, Germany, in 2005.Between 2005 and 2007, he was a Research Engi-

neer at FAG Industrial Services GmbH, Aachen, Ger-many. From 2007 to 2009, he was the Director of theSignal Processing Section at mimoOn GmbH, Duis-burg, Germany. In 2010, he became a Professor for

Robust Signal Processing with the Department of Electrical Engineering andInformation Technology, Technische Universität Darmstadt, Darmstadt, Ger-many, where he is currently the head of the Communication Systems Group.His research interests are in the area of robust signal processing and adaptivebeamforming, high-resolution sensor array processing, transceiver design forcognitive radio systems, cooperative communications in relay networks, MIMOand multiantenna communications, space-time coding, multiuser and multicar-rier wireless communication systems ( 3G), convex optimization for signalprocessing and communications, statistical signal processing, spectral analysis,parameter estimation, and detection theory.Dr. Pesavento was a recipient of the 2003 ITG/VDE Best Paper Award, the

2005 Young Author Best Paper Award of the IEEE TRANSACTIONS ON SIGNALPROCESSING, and the 2010 Best Paper Award of the CROWNCOM confer-ence. He is a member of the Editorial board of the EURASIP Signal ProcessingJournal, Associate Editor for the IEEETRANSACTIONS ON SIGNAL PROCESSING,and a member of the Sensor Array and Multichannel (SAM) Technical Com-mittee of the IEEE Signal Processing Society (SPS).

2030 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 8, APRIL 15… · 2021. 1. 29. · 2030...

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Transcript of 2030 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 61, NO. 8, APRIL 15… · 2021. 1. 29. · 2030...