robust adaptive beamforming using worst-case performance ...€¦ · array calibration, distorted...

IEEE TRANSACTIONSON SIGNAL PROCESSING,VOL. 51, NO. 2, FEBRUARY 2017 313

RobustAdaptiveBeamformingUsingWorst-CasePerformanceOptimization:A Solutionto theSignal

MismatchProblemSergiyA. Vorobyov, Member,IEEE, Alex B. Gershman, SeniorMember,IEEE, andZhi-QuanLuo, Member,IEEE

Abstract—Adaptive beamforming methods are known todegrade if someof underlying assumptionson the environment,sources,or sensor array become violated. In particular, if thedesired signal is present in training snapshots, the adaptivearray performance may be quite sensitive even to slight mis-matchesbetweenthe presumedand actual signal steeringvectors(spatial signatures). Such mismatches can occur as a result ofenvironmental nonstationarities, look direction errors, imperfectarray calibration, distorted antenna shape,as well as distortionscausedby medium inhomogeneities,near–far mismatch, sourcespreading, and local scattering. The similar type of performancedegradation can occur when the signal steering vector is knownexactly but the training samplesizeis small.

In this paper, we develop a new approach to robust adaptivebeamforming in the presenceof an arbitrary unknown signalsteering vector mismatch. Our approach is based on the opti-mization of worst-caseperformance. It turns out that the naturalformulation of this adaptive beamforming problem involvesminimization of a quadratic function subject to infinitely manynonconvex quadratic constraints. We show that this (originallyintractable) problem can be reformulated in a convexform astheso-calledsecond-ordercone(SOC) program and solvedefficiently(in polynomial time) using the well-established interior pointmethod. It is also shown that the proposed technique can be in-terpreted in terms of diagonal loading where the optimal value ofthe diagonal loading factor is computedbasedon the known levelof uncertainty of the signal steeringvector. Computer simulationswith severalfrequently encounteredtypesof signalsteeringvectormismatchesshow better performance of our robust beamformerascomparedwith existing adaptivebeamforming algorithms.

Index Terms—Optimal diagonal loading, robust adaptivebeamforming, second-orderconeprogramming, signal mismatchproblem, worst-caseperformance optimization.

I. INTRODUCTION

N recentdecades,adaptivebeamforminghasbeenwidelyusedin wirelesscommunications,microphonearrayspeech

processing,radar, sonar,medical imaging, radio astronomy,andotherareas.A traditionalapproachto thedesignof adaptivebeamformersassumesthat the desiredsignal componentsare

Manuscript receivedOctober 17, 2015; revised October 10, 2016. Thiswork wassupportedin partby theNaturalSciencesandEngineeringResearchCouncil (NSERC)of Canada,Communicationsand Information TechnologyOntario (CITO), Premier’s ResearchExcellence Award Program of theMinistry of Energy, Science,and Technology(MEST) of Ontario, CanadaResearchChairsProgram,andWolfgangPaulAwardProgramof theAlexandervonHumboldtFoundation.Theassociateeditorcoordinatingthereviewof thispaperandapprovingit for publicationwasDr. Rick S.Blum.

Theauthorsarewith theDepartmentof ElectricalandComputerEngineering,McMasterUniversity,Hamilton,ON, L8S 4K1 Canada.

Digital ObjectIdentifier 10.1109/TSP.2016.806865

1053-587X/03$17.00© 2017 IEEE

I

not presentin training data [1]. In this case,severalrapidlyconvergingtechniqueshavebeendeveloped[1]–[5] thatareap-plicableto problemswith small trainingsamplesize.Althoughthe assumptionof signal-freetraining snapshotsmay be truein someareas(suchasradar),therearenumerousapplicationswheretheobservationsarealways“contaminated”by thesignalcomponent.Such applications,for example,include mobilecommunications,passivesource location, microphonearrayspeechprocessing,medicalimaging,andradioastronomy.It iswell knownthatevenin theidealcasewherethesignalsteeringvector is exactlyknown, the presenceof the signalof interestin training datacell may dramaticallyreducethe convergenceratesof adaptivebeamformingalgorithmsascomparedwith thesignal-freetrainingdatacase[6]. This maycausea substantialdegradationof the performanceof adaptive beamformingtechniquesin situationsof small trainingsamplesize.

Whenadaptivearraysareappliedto practicalproblems,theperformancedegradationof adaptivebeamformingtechniquesmaybecomeevenmorepronouncedthanin theaforementionedidealcasebecausesomeof underlyingassumptionsontheenvi-ronment,sources,or sensorarraycanbeviolatedandthis maycausea mismatchbetweenthenominal(presumed)andactualsignalsteeringvectors.Adaptivearraytechniquesareknowntobeverysensitiveevento slightmismatchesof suchtypethatcaneasilyoccurin practicalsituationsasaconsequenceof look di-rectionandsignalpointingerrors[7], [8] or imperfectarraycal-ibrationanddistortedantennashape[9]. Othercommoncausesof modelmismatchincludearraymanifoldmismodelingduetosourcewavefrontdistortionsresultingfromenvironmentalinho-mogeneities[10], [11], near–farproblem[12], sourcespreadingandlocal scattering[13]–[16], aswell asothereffects[17]. Insuchcases,robustapproachesto adaptivebeamformingarere-quired[17]–[19].

There are severalexisting approachesto robust adaptivebeamforming.The most common is the so-called linearlyconstrainedminimum variance(LCMV) beamformer,whichprovides robustnessagainst uncertainty in the signal lookdirection. Recently,severalother techniquesaddressingthistypeof mismatchhavebeendeveloped(see[19] andreferencestherein). However, the applicability of these techniquesislimited by scenarioswith look direction mismatchesonly. Ifanyothertypesof steeringvectormismatchbecomedominant(e.g., mismatchesdue to array perturbations,array manifoldmismodeling,wavefrontdistortions,or sourcelocalscattering),thesemethodscannotbeexpectedto providesufficientrobust-nessimprovements[17].

314 IEEE TRANSACTIONSON SIGNAL PROCESSING,VOL. 51, NO. 2, FEBRUARY 2017

Severalotherapproachesareknownto beableto partlyover-cometheproblemof arbitrarysteeringvectormismatches.Themostpopularof themarethe quadraticallyconstrainedbeam-former (whoseimplementationis basedon the so-calleddiag-onal loading of the samplecovariancematrix [4], [18], [20])andthe eigenspace-basedbeamformer[6], [21]. However,themainshortcomingof the formerapproachis that it is not clearhow to obtaintheoptimalvalueof thediagonalloadingfactorbasedon the known level of uncertaintyof the signalsteeringvector,whereasthe latterapproachis essentiallyineffectiveatlow signal-to-noiseratios (SNRs)andwhenthe dimensionof

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nately,makesit difficult to apply the eigenspace-basedbeam-formerto wirelesscommunicationswherethedimensionof thesignal-plus-interferencesubspacemay be uncertainand rela-tively highdueto theeffectsof signallocalscattering[13]–[16].

In thispaper(alsosee[22]–[24]),wedevelopanewpowerfulapproachto robustadaptivebeamformingin the presenceofanarbitraryunknownsteeringvectormismatch.Our approachis basedon the optimization of worst-caseperformance.Itturnsout that the naturalformulationof this probleminvolvesminimizationof a quadraticfunctionsubjectto infinitely many

2

solve.However,weshowthatthisrobustadaptivebeamformingproblemcan be reformulatedas a convexsecond-ordercone(SOC) program and solved efficiently (in polynomial time)via the well-establishedinterior point method(see[27]–[29]).This result is somewhatsurprising from the optimizationtheorystandpointandis basedon a procedurethat transformsa semi-infinite nonconvexquadratically constrainedhomo-geneousquadraticminimization problem to a convex SOCprogram.Weshowthatourbeamformercanbeinterpretedasa

3

diagonalloadingfactor is computedbasedon theknownupperboundon thenormof thesignalsteeringvectormismatch.

Computersimulationswith severalfrequentlyencounteredtypesof signalsteeringvectormismatchesshowavisibleperfor-mancegainof theproposedbeamformeroverothertraditionalandrobustadaptivebeamformingtechniques.

Ourpaperisorganizedasfollows.Somebackgroundof adap-tive beamformingis presentedin SectionII, whereseveralpop-ularrobustadaptivebeamformingtechniquesareoverviewed.InSectionIII, we first describea newformulationof robustadap-tive beamformingbasedon theoptimizationof worst-caseper-formance.Then,we establishthediagonalloadingbasedinter-pretationof ourrobustadaptivebeamformingproblemandcon-vert it to a convexSOCproblemthatcanbeefficiently solvedusingthewell-establishedinteriorpointalgorithms.SectionIVpresentsour simulationresultswherethe performanceof theproposedmethodis comparedwith the existingalgorithmsinsituationswith differenttypesof thesignalsteeringvectormis-match.SectionV containsour concludingremarks.

1Additionally, this dimensionmustbeexactlyknownin this technique.2In optimizationtheory,NP-hardproblemsrepresentaclassof extremelydif-

ficult problemsthathaveno knownpolynomial-timesolutions[25].3Very recently,anotherrobust worst-caseoptimization-basedbeamformer

(which also belongsto the classof diagonal loading techniques)has beenindependentlyformulated;see[26].

the signal-plus-interferencesubspaceis high. This, unfortu-

nonconvexquadraticconstraintsandthereforeis NP-hard to

diagonalloadingapproach in which the optimal valueof the

II. BACKGROUND

Theoutputof a narrowbandbeamformeris givenby

where is the time index,is the complex vector of array observations,

is the complex vector ofbeamformerweights, is the numberof array sensors,and

and standfor thetransposeandHermitiantranspose,respectively.The observation(training snapshot)vector isgivenby

(1)

where , , and arethedesiredsignal,interference,and noisecomponents,respectively.Here, is the signalwaveform,and is thesignalsteeringvector.Theweightvectorcanbefoundfrom themaximumof thesignal-to-interference-plus-noiseratio (SINR) [5]

SINR (2)

where

(3)

is the interference-plus-noisecovariancematrix, andis the signal power. It is easyto find the solution for the

weight vectorby maintaininga distortionlessresponsetowardthedesiredsignalandminimizing theoutputinterference-plus-noisepower[5]. Hence,themaximizationof (2) is equivalentto[5]

subjectto (4)

From(4), the following well-knownsolutioncanbe found fortheoptimalweightvector[5]:

(5)

where is the normalizationconstantthatdoesnot affect the output SINR (2) and, therefore,will beomittedin theinterestof brevity.Thesolution(5) is commonlyreferredto as the minimum variancedistortionlessresponse(MVDR) beamformer[5], [30].

In practicalapplications,theexactinterference-plus-noiseco-variancematrix is unavailable.Therefore,thesampleco-variancematrix

(6)

isusedinsteadof (3) (see[1]). Here, is thenumberof trainingsnapshots(alsotermedthetrainingsamplesize).In thiscase,(4)shouldberewrittenas

subjectto (7)

VOROBYOV et al.: ROBUSTADAPTIVE BEAMFORMING USING WORST-CASEPERFORMANCEOPTIMIZATION 315

The solution to this problemis commonlyreferredto as thesamplematrix inversion(SMI) algorithm,whoseweightvector(after omitting the immaterial constant ) isgivenby [1]

(8)

Whenthesignalcomponentis presentin the trainingdatacell(cf. (1)), the useof the samplecovariancematrix (6) in placeof thetrueinterference-plus-noisecovariancematrix (3) affectstheperformanceof theSMI algorithmdramatically[6]. It iswellknownsincetheclassicpaper[1] thatin thecaseof signal-freetraining samples,the useof weight vector (8) providesrapidconvergenceof theoutputSINR to its optimalvalue

SINR (9)

so that the averageperformancelossesrelativeto (9) are lessthan3 dB if . However,this is no longertrue if thetrainingsnapshotsare“contaminated”by thesignalcomponent.It wasshownin [6] thatin thelattercasetheconvergenceto (9)becomesmuchslowerandgenerallyrequires .

Anotheressentialshortcomingof the SMI algorithmis thatit doesnotprovidesufficientrobustnessagainstamismatchbe-tweenthepresumedandactualsignalsteeringvectors and .Here, denotestheactualsteeringvectorthatcharacterizesthespatialsignatureof thesignal.In themismatchedcase

(10)

where isanunknowncomplexvectorthatdescribestheeffectof steeringvectordistortions.As a result,theSMI beamformertendsto “interpret” thesignalcomponentsin arrayobservationsas an interferenceand tries to suppressthesecomponentsbymeansof adaptivenulling insteadof maintainingdistortionlessresponsetoward (see[6] and[17]).

In themismatchedcase,(2) and(9) shouldberewrittenas

SINR (11)

and

SINR (12)

respectively.Severalrobustmodificationsof theSMI algorithmhavebeendevelopedto improveits performancein theabove-mentionedcaseswith signal steeringvector mismatchesandsmall trainingsamplesize.Oneof themostpopularrobustap-proachesis theso-calledloadedSMI (LSMI) algorithm,whichisbasedonthediagonalloadingof thesamplecovariancematrix[4], [18]. Theessenceof thisapproachis to replacetheconven-tional samplecovariancematrix by theso-calleddiagonallyloadedcovariancematrix

(13)

in theSMI algorithm(8). Here, is a diagonalloadingfactor,and is theidentitymatrix.Using(13),wecanwrite theLSMIweightvectorin thefollowing form [4]:

(14)

The main problemof the LSMI methodis how to chosethediagonalloadingfactor . Cox et al. [18] proposedto usetheso-calledwhitenoisegainconstrainttoobtainreasonablevaluesof thisparameter.Unfortunately,it is notclearhowto relatetheparametersof the white noisegain constraintandthe level ofuncertaintyof thesignalsteeringvector.Furthermore,therela-tionshipbetweenthediagonalloadingfactorandtheparametersof thewhitenoisegainconstraintisnotsimple,andtosatisfythisconstraint,a multistepiterativeprocedureis requiredto adjustthediagonalloadingfactor[18]. Eachstepof this iterativepro-cedureinvolvesanupdateof theinverseof thediagonallyloadedcovariancematrix,andasaresult,thetotalcomputationalcom-plexity of adaptivebeamformingwith thewhitenoisegaincon-straintmaybehigherthanthatof theSMI algorithm.Becauseof this, thediagonalloadingfactor is usuallychosenin a moread hocway, typically about10 , where is thenoisepowerin a singlesensor.

Another popularapproachto robustadaptivebeamformingin the generalcaseof an arbitrary mismatchis the so-calledeigenspace-basedbeamformer[6], [21] whosekeyideais touse,insteadof the presumedsteeringvector , the projectionofonto the samplesignal-plus-interferencesubspace.The eigen-decompositionof (6) yields

wherethe matrix containsthe signal-plus-interferencesubspaceeigenvectorsof , andthediagonalmatrix containsthe correspondingeigen-valuesof . Similarly, thematrix containsthe noise-subspaceeigenvectorsof , whereasthediagonalmatrix is built from thecor-respondingeigenvalues.Here, is the numberof interferingsources(or, mathematically,the rank of the interferencesub-space),whichis assumedto beknown.Theweightvectorof theeigenspace-basedbeamformeris givenby

(15)

where

(16)

aretheprojectedsteeringvectorandtheorthogonalprojectionmatrixontothesignal-plus-interferencesubspace,respectively.Inserting(16) into (15), thelatterequationcanberewrittenas

(17)

The eigenspace-basedbeamformeris known to be oneof themostpowerfulrobusttechniquesapplicableto arbitrarysteeringvectormismatchcase[21]. However,anessentialshortcomingof this approachis that it is limited to high SNR casesbe-causeat low SNRtheestimationof theprojectionmatrix ontothesignal-plus-interferencesubspacebreaksdownbecauseof ahigh probability of subspaceswaps[31], [32]. Moreover,theeigenspace-basedbeamformeris efficient only if the dimen-sionof thesignal-plus-interferencesubspaceis low andknownexactly.This makesit difficult to apply the eigenspace-basedbeamformerto wirelesscommunicationswherethedimension


of the signal-plus-interferencesubspacemay be uncertainandrelativelyhighdueto theeffectsof sourcescattering[13]–[16].

III. NEW APPROACHTO ROBUSTBEAMFORMING

In this section,we developa newadaptivebeamformerthatis robustagainstan arbitrarysignal steeringvectormismatchandsmall training samplesize.Our approachis basedon theworst-caseperformanceoptimization.Webeginwith theformu-lationof therobustadaptivebeamformingproblemandthende-velopaconvexoptimization-basedimplementationof ouradap-tive beamformerusingSOCprogramming.

A. Formulation

We assumethat in practical applications,the norm of thesteeringvector distortion can be bounded[33] by some4

knownconstant :

(18)

Then,theactualsignalsteeringvectorbelongsto theset

(19)

Indeed,if , then,accordingto (10), . Since canbeanyvectorin (19),weimposeaconstraintthatfor all vectorsthat belongto , the absolutevalue of the arrayresponse5

shouldnot besmallerthanone,i.e.,

for all (20)

Using(20), therobustformulationof adaptivebeamformercanbewrittenasthefollowing constrainedminimizationproblem:

subjectto for all (21)

Note that (21) representsa modified version of (7). Themain modification of (7) is that insteadof requiring fixeddistortionlessresponsetoward the singlesteeringvector , in(21), suchdistortionlessresponseis maintainedby meansof

6

vectorsgiven by the setinequalityconstraints for a continuumof all possiblesteering

. Hence,the constraintsin (21)guaranteethatthedistortionlessresponsewill bemaintainedintheworstcase, i.e., for theparticularvector thatcorrespondsto thesmallestvalueof . Therefore,suchadesignshouldimprove the beamformerrobustnessagainst signal steeringvector mismatchesthat satisfy (18) becausein this case,themismatchedvector belongsto theset .

For eachchoiceof , the conditionrepresentsa nonlinearand nonconvexconstrainton . Sincethereis an infinite numberof vectors in , thereis an in-finite numberof suchconstraints.Hence,(21) is asemi-infinite

4

consideredin [34], wherethe boundson the mismatchvector itself areusedratherthanthaton thenormof this vector.

5

thephaseof thebeamresponse.This issuemaybecritical when,for example,adaptivebeamforminghastobeperformedoverfrequencysubbandswhoseout-putsmustbecoherentlyintegrated.

6

straints) areusedin otheradaptivebeamformingtechniquesaswell [35]–[37].

Notethatthiscorrespondsto amuchmoregeneralclassof mismatchesthan

An importantissuethatisbeyondourpresentconsiderationis howtocontrol

Constraintsin the form of inequality (sometimesreferredto as soft con-

nonconvexquadraticprogram.It is well known that the gen-eral nonconvexquadraticallyconstrainedquadraticprogram-ming problemis NP-hardand, thus, intractable.However,aswe will shownext,dueto thespecialstructureof theobjectivefunctionandtheconstraints,theproblem(21) canbereformu-lated,surprisingly,asa convexSOCprogramand,thus,solvedefficiently (in polynomialtime)via thewell establishedinteriorpoint method.

Let usfirst convertthesemi-infinitenonconvexconstraintstoasingleconstraintthatcorrespondsto theworst-caseconstraintfrom (20). In particular,(21) canbeequivalentlydescribedas

subjectto (22)

Accordingto (19),we canrewritetheconstraintof (22) as

wheretheset is definedas

Applying the triangleandCauchy–Schwarzinequalitiesalongwith theinequality , we havethat

(23)

Moreover,it is easyto verify that

(24)

if is smallenough(i.e., if ) andif

where

angle

Note that we requirethat , as otherwise,thewhite noisegainof therobustbeamformermaybe insufficient[18].

Then,combining(23) and(24),we concludethat

and therefore,the semi-infinite nonconvexquadraticallycon-strainedproblem(22)canbewrittenasthefollowing quadraticminimizationproblemwith a singlenonlinearconstraint:

subjectto (25)

Thenonlinearconstraintin (25)is still nonconvexdueto theab-solutevalueoperationon the left-handside.An importantob-servationis thatthecostfunctionin (25) is unchangedwhenundergoesanarbitraryphaserotation.Therefore,if is anop-timal solutionto (25), we canalwaysrotate,without affectingtheobjectivefunctionvalue,thephaseof sothat is real.Thus,wecan,withoutanylossof generality,choose suchthat

Re (26)

Im (27)


Using this observationand employing(26) and (27) as addi-tional constraints,theconstraintin (25) canbewritten as

(28)

From(28)alongwith (27),it follows thatRe . Sincetheconstraint(26) is takeninto accountby (27) and(28), thereis no needto addthis constraintto the minimizationproblem(25).Therefore,this problemcanberewrittenas

subjectto

Im (29)

Note that theproblem(29) hasmuchsimplerformulationthan(21) andis convex.

B. Relationshipto theLSMI Beamformer

To clarify theproblem(29) more,notethat theconstraintin(22) is equivalentto

(30)

or, in other words, (29) correspondsto maximizationof theworst-caseoutputSINR. The equivalenceof the equalitycon-straint (30) andthe inequalityconstraintin (22) canbe easilyprovedby contradictionasfollows. If theyarenotequivalenttoeachother,thentheminimumof theobjectivefunction in (22)is achievedwhen . However,re-placing with , we candecreasetheobjectivefunction

by the factorof , whereastheconstraintin (22)will bestill satisfied.Thiscontradictstheoriginalstatementthattheobjectivefunctionis minimizedwhen . Therefore,theminimum of the objectivefunction is achievedat , andthisprovesthattheinequalityconstraintin (29) is equivalenttothe equalityconstraint . Therefore, isreal-valuedandpositive,andtheconstraintIm canbeignored.Usingthesefacts,we canrewrite theproblem(29)as

subjectto (31)

Thesolutionto (31) canbefoundby minimizing thefunction

where is a Lagrangemultiplier. Taking the gradient ofandequatingit to zerogives

Applying thematrix inversionlemmato thelatterequation,weobtain

(32)

which showsthat the proposedrobustbeamformerbelongstotheclassof diagonalloadingtechniques.

Note,however,thatit is noteasyto use(32)directly for com-putingtheoptimalweightvectorbecauseit is not clearhow to

obtaintheLagrangemultiplier in a closedform [26]. There-fore, to solve (29), an efficient SOC programming-basedap-proachis developedin thenextsection.

C. SOCImplementation

ThenextstepinvolvesdevelopingaSOCformulationof (29).Note thatalthoughwe haveshownin theprevioussectionthattheinequalityconstraintin (29)canbereplacedby equality,wewill usethis constraintin its original inequalityform, which issuitablefor theSOCimplementation.

Firstof all, weconvertthequadraticobjectivefunctionof (29)to a linearone.Let

(33)

betheCholeskyfactorizationof . Using(33),we canconverttheobjectivefunctionof (29) into

(34)

Apparently,minimizing isequivalenttominimizing(34).Hence,introducinga newscalarnon-negativevariable andanew constraint , we canconvert(29) into the fol-lowing problem:

subjectto

Im (35)

To facilitatethesolutionof (35),weneedto convertit to a real-valuedform. Introducing

Re Im

Re Im

Im Re

Re ImIm Re

werewrite(35) in termsof real-valuedvectorsandmatricesas

subjectto

(36)

Let usdefine

where is thevectorof zerosof aconformabledimension.Withthesenotations,(36)canbefurthertransformedto thefollowing


canonicaldual form of theSOCprogrammingproblem(which7

is equivalentto (8) in [29]):

subjectto

SOC SOC (37)

where is thevectorof variables,SOC is thesecond-orderconeof the dimension , which correspondsto the thinequalityconstraintin (36) ( ,2), and{0} is theso-calledzero conethat determinesthe hyperplanedue to the equalityconstraint . More specifically

SOC

where

Notethataftersolvingtheoptimizationproblem(37), theonlyparametersof interestin thevectorof variables aregivenby itssubvector . Theresultingweightvectorof our robustadaptivebeamformeris givenby

(38)

In summary,we convertedtherobustbeamformingproblem(21) to the canonicalSOC problem(37). Note that althoughthese problems are mathematicallyequivalent, the originalproblem(21) is computationallyintractable,whereasthe SOCproblem(37) can be easily solvedusing standardand highlyefficient interior point methodsoftwaretools, e.g., [29]. Forexample,using the primal-dual potential reduction method[27], thecomplexityof ourbeamformeris periteration[28], and the algorithm convergestypically in less than teniterations(a well-known andacceptedfact in the optimizationcommunity).Therefore,the overall complexity of our beam-former is . This is the sameorder of complexity asthat of the SMI algorithm.However,the SMI algorithmhasacomputationaladvantagein the on-line mode, wherethe RLSalgorithmcanbeusedto updatetheSMI beamformerweightswith the computationalcomplexity per updatingstep.Theweightvectorof our beamformercannotbeeasilyupdatedbut hasto berecomputedin eachstep.

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tively usedwhenapplyingtheSeDuMisoftwareof [29].Bothdualandprimal formsof SOCprogrammingproblemscanbealterna-

Fig. 1. OutputSINR versustrainingsamplesize ; first example.

IV. SIMULATIONS

In our simulations,we assumea uniform linear array withomnidirectionalsensorsspacedhalf a wavelength

apart.For eachscenario,200 simulationrunsareusedto ob-tain eachsimulatedpoint. In all examples,we assumetwo in-terfering sourceswith planewavefrontsand the directionsofarrival(DOAs)30 and50 , respectively.In all simulations,theinterference-to-noiseratio (INR) in a singlesensoris equalto30dB, andthesignalis alwayspresentin thetrainingdatacell.Fourmethodsarecomparedin termsof themeanoutputSINR:theproposedrobustbeamformer(38),theSMI beamformer(8),theLSMI beamformer(14) with ad hocchoiceof thediagonal

8

SINR(12) is alsoshownin all figures.TheSeDuMiconvexop-timization MATLAB toolbox [29] hasbeenusedto computethe weight vectorof our robustbeamformerthat employstheconstant

load, andtheeigenspace-basedbeamformer(17).Theoptimal

throughoutthe simulations(exceptonesimu-lation in the first example,where is varied;seeFig. 3), as-sumingthat the nominalsteeringvector is normalizedso that

( 10). The diagonalloadingfactor istakenin theLSMI beamformer.Furthermore,diagonalloadingwith the sameparameteris appliedto our robusttechniqueaswell but only in thecasewhenthesamplecovariancematrix islow rank(i.e., in thecasewhen ).

A. Example1: ExactlyKnownSignalSteeringVector

In this example,we simulatea scenariowhere the actualspatialsignatureof thesignalis knownexactly.Notethatevenin this ideal case,the presenceof the signalof interestin thetraining data cell may substantiallyreducethe convergenceratesof adaptivebeamformingalgorithmsas comparedwiththesignal-freetrainingdatacase[6].

In thisexample,theplane-wavesignalis assumedto impingeonthearrayfrom . Fig.1comparesfouraforementionedmethodsin termsof the meanoutputarraySINR (11) versusthenumberof trainingsnapshots for thefixed single-sensor

8Thisbeamformeris hereafterreferredto astheadhocLSMI technique.


Fig. 2. OutputSINR versusSNR;first example.

Fig. 3. OutputSINR versus ; first example.

SNR 10dB. Fig. 2 displaystheperformanceof thesetech-niquesversustheSNRfor thefixed trainingdatasize .Fig. 3 showstheperformanceof themethodstestedversus for

andSNR 10 dB. Additionally, thebeampatternsof our beamformerandthe ad hoc LSMI algorithmarecom-paredin Fig. 4 for andSNR 10 dB.

B. Example2: SignalLookDirection Mismatch

In thesecondexample,ascenariowith thesignallook direc-tion mismatchis considered.Weassumethatboththepresumedandactualsignalspatialsignaturesareplanewavesimpingingfrom theDOAs3 and5 , respectively.Thiscorrespondsto a2mismatchin thesignallook direction.

Fig.5showstheperformanceof themethodstestedversusthenumberof trainingsnapshots for the fixed SNR 10 dB.The performanceof thesealgorithmsversusthe SNR for thefixed trainingdatasize is shownin Fig. 6.

Fig. 4. Beampatternsof the proposedandad hoc LSMI beamformers;firstexample.

Fig. 5. OutputSINR versustrainingsamplesize ; secondexample.

Fig. 6. OutputSINR versusSNR;secondexample.


Fig. 7. OutputSINR versustrainingsamplesize ; third example.

C. Example3: SignalSpatialSignatureMismatchDuetoCoherentLocal Scattering

Ourthird examplecorrespondsto thescenariowherethespa-tial signatureof thedesiredsignalisdistortedby localscatteringeffects.In thisexample,thepresumedsignalspatialsignatureisaplanewaveimpingingonthearrayfrom3 , whereastheactualspatialsignatureis formedby five signalpathsandis givenby

(39)

where correspondsto thedirectpath,whereas ( 1, 2,3,4) correspondto thecoherentlyscatteredpaths.Wemodeltheth path asaaplanewaveimpingingonthearrayfrom the

direction . Theparameters , 1, 2, 3, 4 areindependentlydrawnin eachsimulationrun from auniformrandomgeneratorwith mean andstandarddeviation . Theparameters

, 1, 2, 3, 4 representpathphasesthatareindependentlyanduniformlydrawnfromtheinterval[0, 2 ] in eachsimulationrun.Notethat and ( 1, 2, 3, 4) changefrom run to runwhile remainingfrozen from snapshotto snapshot.This casecorrespondsto theso-calledcoherentscattering[14].

Fig. 7 displaystheperformanceof themethodstestedversusthenumberof trainingsnapshots for thefixed single-sensorSNR 10 dB. Note that theSNRin this exampleis definedby takinginto accountall signalpaths.

Theperformanceof thesamemethodsversustheSNRfor thefixed trainingdatasize is displayedin Fig. 8.

D. Example4: SignalSpatialSignatureMismatchDuetoIncoherentLocal Scattering

In thisexample,weassumeincoherentlocalscatteringof thedesiredsignal.The signal is assumedto havea time-varyingspatialsignaturethat is different for eachdatasnapshotandismodeledas

Fig. 8. OutputSINR versusSNR;third example.

where arei.i.d. zero-meancomplexGaussianrandomvari-ablesindependentlydrawnfrom a randomgenerator.As in thepreviousexample,the DOAs , 1, 2, 3, 4 are indepen-dently drawn in eachsimulationrun from a uniform randomgeneratorwith mean andstandarddeviation . NotethattheDOAs changefrom run to runwhile remainingfixedfrom snapshotto snapshot.At thesametime, therandomvari-ables changeboth from run to run and from snapshotto snapshot.This correspondsto the caseof incoherentlocalscattering[15], where the signal covariancematrix is nolongera rank-onematrix, and(11) for theoutputSINR shouldberewrittenin a moregeneralform [17]

SINR (40)

Theratio (40) is maximizedby [17]

(41)

where is the operatorthat computesthe principal eigen-vectorof a matrix.Notethatsolution(41) is of a little practicalusebecausein mostapplications,thematrix isunknown,andno reasonableestimateof it is available.

Fig. 9 displaystheperformanceof themethodstestedversusthe numberof training snapshots with the fixed SNR

10 dB. As in the previousexample,the SNR is definedby

The performanceof the samemethodsversusthe SNR fortakinginto accountall signalpaths.

thefixed trainingdatasize is displayedin Fig. 10.WestressthattheoptimalSINR for Figs.9 and10 is computedas

SINR

where is givenby (41).

E. Example5: Near–FarSignalSpatialSignatureMismatch

In thefifth example,we modeltheso-callednear-farspatialsignaturemismatchof the desiredsignal.In this example,the


Fig. 9. OutputSINR versustrainingsamplesize ; fourth example.

Fig. 10. OutputSINR versusSNR;fourth example.

presumedspatialsignatureof the signal is a planewave im-pingingon thearrayfrom thenormaldirection0 , whereastheactualspatialsignaturecorrespondsto thesourcelocatedin thenearfield of theantennaat a distancefrom the geometricalcenterof the array,where

is the lengthof arrayaperture. The sourceis assumed9

to be locatedon the line which is drawnfrom this geometricalcenterpoint in thenormaldirectionto thearrayaperture.

The performanceof the methodstestedversusthe numberof trainingsnapshots for thefixed SNR 10 dB is shownin Fig. 11. Fig. 12 showsthe performanceof thesetechniquesversustheSNRfor thefixed trainingdatasize .

F. Example6: SignalSpatialSignatureMismatchDuetoWavefrontDistortion

In ourlastexample,wesimulatethesituationwhenthesignalspatialsignatureis distortedby wavepropagationeffectsin aninhomogeneousmedium. We assumeindependent-increment

9

remainsmuchlargerthanFarfield conditionrequiresthatthedistancebetweenthesourceandantenna

[38].

Fig. 11. OutputSINR versustrainingsamplesize ; fifth example.

Fig. 12. OutputSINR versusSNR;fifth example.

phasedistortionsof thedesiredsignalwavefront[11], [39]. Ineachsimulation run, eachof thesephasedistortions(whichremainsfixed for all snapshots)is independentlydrawnfrom aGaussianrandomgeneratorwith varianceequalto 0.04.

Fig. 13 showstheperformanceof themethodstestedversusfor thefixed SNR 10dB.Theperformanceof thesetech-

niquesversustheSNRfor thefixed trainingdatasizeis shownin Fig. 14.

G. Discussion

Our simulation figures clearly demonstratethat in all ex-amples,the proposedrobustbeamformerconsistentlyenjoysthe best performanceamong the methods tested. Indeed,our new method outperformsthe SMI, ad hoc LSMI, andeigenspace-basedbeamformers,achievinga performancethatis consistentlycloseto theoptimalSINRfor all valuesof SNR.TheadhocLSMI algorithmis anotherwell-performingmethodasits performanceis comparablewith thatof our robustbeam-former in a partof theexamplestested.This canbeexplainedby thefact discoveredin SectionIII thatourbeamformer] be-


Fig. 13. OutputSINR versustrainingsamplesize ; sixth example.

Fig. 14. OutputSINR versusSNR;sixth example.

longsto theclassof diagonalloadingtechniques.However,ouralgorithmperformsbetterthanthead hocLSMI methodin thefourth exampleat all SNRs(Figs.9 and10) andin thesecond,third, fifth, andsixthexamplesathighSNRs(Figs.6, 8, 12and14). Note that in theseexamples,the aforementionedperfor-mancegainsof our beamformerascomparedwith the ad hocLSMI methodcanachieve2 dB. Clearly, this improvementinperformanceis due to a more properchoiceof the diagonalloadingfactor in our beamformerascomparedwith thead hocLSMI beamformer.

The performanceof the SMI and eigenspace-basedbeam-formersis muchworsethanthat of the proposedbeamformerandthe ad hoc LSMI beamformereitherat high SNRs(as inthecaseof SMI beamformer:Figs.2, 6, 8, 12,and14)or at lowSNRs(asin thecaseof eigenspace-basedbeamformer,thesamefigures).Moreover,in the fourth example,the SMI algorithmshowspoorperformanceat all valuesof theSNR(seeFig. 10).Note that theperformancebreakdownof theeigenspace-based

beamformerat low SNRsis causedby the subspaceswapef-fect [31], [32], whereastheaforementionedperformancelossesof theSMI algorithmat high SNRsaredueto the fact that in-creasingtheamountof thesignalcomponentin thetrainingdata

10

SINRof SMI-typebeamformers[6]. Furthermore,from Figs.1,5, 7, 9, 11, and13, it follows that the proposedalgorithmen-joysmuchfasterconvergenceratethantheSMI andeigenspace-basedalgorithms.It is worth noting that evenin the situationwithout anysteeringvectormismatch(Example1, Figs.1 and2), theproposedtechniquehassubstantiallybetterperformanceandfasterconvergenceratethantheSMI andeigenspace-basedbeamformers.

Fig. 3 demonstratesthattheproposedbeamformeris insensi-tive to thechoiceof theparameter

is known to lead to a substantialdegradation of the output

, whereasFig. 4 showsthatthe beampatternsof the proposedbeamformerand the LSMIbeamformerareverysimilartoeachother.Thiscanbeexplainedby the above-mentionedfact that the proposedbeamformerisequivalentto the LSMI beamformerwhosediagonalloadingfactor is optimally matchedto the known level of the steeringvectordistortion.

V. CONCLUSIONS

A new adaptivebeamformerwith an improvedrobustnessagainstan arbitraryunknownsignalsteeringvectormismatchhas beenproposed.Our techniqueoptimizes the worst-caseperformanceby minimizing the outputinterference-plus-noisepower while maintaining a distortionlessresponsefor theworst-case(mismatched)signal steering vector. A convexformulation for sucha robustadaptivebeamformingproblemis derivedusingsecond-orderconeprogramming.It is shownthat theproposedbeamformercanbeinterpretedasa diagonalloading approachwhose optimal diagonal loading factor ispreciselycomputedbasedon theknownlevel of uncertaintyinthe signal steeringvector.Computersimulationswith severalfrequently encounteredtypes of signal steeringvector mis-matchshowbetterperformanceof theproposedbeamformerascomparedwith severalpopular robustadaptivebeamformingalgorithms.

In additionto theofferedperformanceimprovementsrelativeto existing methods,the proposedbeamformerenjoyssimpleimplementation.The order of computationalcomplexity ofthis algorithm is comparablewith that of the SMI technique.It can be efficiently implementedusing currently availableconvexoptimizationsoftwaretoolboxesbasedon interiorpointalgorithms.

CKNOWLEDGMENT

The authorswish to thank the anonymousreviewersfortheir helpful commentsand suggestions,which led to theunderstandingof therelationshipbetweenourbeamformerandthediagonalloadingmethod.

10

trainingobservations(which in theidealcasemustcontaintheinterferenceandnoisecomponentsonly).

A

Thisdegradationoccursbecausethesignalcomponent“contaminates”the


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