DOAEstimationforCoherentSourcesinBeamspace UsingSpatialSmoothing

Transcription

1 DOAEstmatonorCoherentSouresneamspae UsngSpatalSmoothng YnYang,ChunruWan,ChaoSun,QngWang ShooloEletralandEletronEngneerng NanangehnologalUnverst,Sngapore, InsttuteoAoustEngneerng NorthwesternPoltehnalUnverst,X an,chna,77 Abstrat eamspae USIC (Ultple SIgnal Classaton) algorthmsotenusedtosolvethedoa(dretono Arrval) estmaton problem to take advantage o the benets o beamspae operatons, suh as redued omputaton omplet, redued senstvt to sstem errors, redued resoluton threshold, redued bas n the estmate,andsoon.owever,thsmethodwllaltowork properlwhenthesouresgnalsareoherentorstrongl orrelated,whhareotenenounterednsonarorradar envronment.hspaperusestheorward-bakwardspatal smoothng tehnque as the preproessor o beamspae USICorlneararras,aterwhhtheohereneothe soure sgnals an be removed n the ovarane matr. We show b omputer smulaton that the resoluton o DOA estmaton usng beamspae USIC ater spatal smoothngsbetterthanusngelementspaeusicater spatal smoothng. We also show that the beamspae SUIC usng spatal smoothng s more robust than elementspaeusicwhentherearesstemerrorsnthe lneararra..introduton he estmaton o the dreton o arrval (DOA) o multpleplanewavesmpngngonanarraosensorshas beenntensvelstudednreentearswthapplatons n derent areas suh as underwater aousts, radar or ommunatons.oneothemostpopularalgorthmsor perormngdoaestmatonstheusicalgorthm[]. Itsattratvenesssduetotheatthattprovdesgood resolutonwhlelmtngthesearhornomngsgnalsto a sngle dmenson. o urther redue the omputaton ompletousicalgorthm,beamspaeusicwas dsussednseveralreerenes,suhas[-6].eeptor the reduton o omputaton, there are several other advantagesoeamsapeusicomparedwthelement spaeusic,suhasreduedsenstvttosstemerrors, reduedresolutonthreshold,reduedbasntheestmate, andsoon. he man lmtaton o beamspae USIC s that t perormspoorlnthepreseneohghlorrelatedand oherent soure sgnals, same as that o element spae USIC.Unortunatel,stronglorrelatedsouresgnals areotenenounterednsonarorradarenvronment[7], wheresgnalorrelatonoursnmultpathorsomeother senaros.forelementspaeusic,orwardororwardbakward spatal smoothng tehnques have been used beoredoaestmatontooveromethsproblemwhenthe reevngarrasaunormlneararra[8-]. Inthspaper,weuseorward-bakwardspatalsmoothng tehnque as the preproessor o beamspae USIC to reoverthereduedrankotheovaranematrdueto the oherene o soure sgnals. Computer smulatons show that the ombnaton o spatal smoothng and beamspae USIC an aheve good resoluton perormane.oremportantl,theproposedmethodhas lowersenstvttosstemerrorsnpratalarrasthan that o the ombnaton o element spae USIC and spatalsmoothng..eamspaeusic Consderaunormlneararra(UA)o sensor elementswthsensorspangdreevngasetooherent orstronglorrelatedplanewavesemttedbp(p<) souresntheareldothearra.weassumeanarrowbandpropagatonmodel,.e.,thesgnalenvelopesdonot hangedurngthetmettakesorthewaverontstotravel throughthearra.supposethatthesgnalshaveaommon enterrequeno,then,theorrespondngwavelength s λ = /, where s the speed o propagaton. he reeved-vetor ( oarraattmets ( = A( Θ) s( + n( () where s( = [ s (,, s ( ] sthep-vetorothesoure P sgnals, A( Θ) = [ a(,, a( ] s the P steerng matr P nwhh a( πd sn θ λ = [, e,, e π ( ) d sn θ λ ] () stheresponseothelneararratothethsourearrvng romθ,andn( = [ n (,, n ( ] sanaddtve nose

2 proess,assumedtobeazero-meangaussannosevetor wthovaraneσ I,whereIdenotesan dentt matr. he element spae ovarane matr s gvenb = E{ ( ( } = A ( Θ) SA( Θ) + σ I (3) where S = E{ s( s ( } represents the soure ovarane matr. InthebeamspaeUSICalgorthm,elementoutputs ( are passed through a beamormng preproessor pror to applaton o the element spae USIC algorthm, as shownnfg..hepreproessortransormstheelement spaedatantobeamspae, ( = W ( (4) wherew s an beamormngmatrwth <. hs beamormng matr W onssts o beamormng vetors whh ormbeamstowardepeteddretons. he beamspaeovaranematrsgvenb = E{ ( UsngEqs.(3)and(4), wllbe θ θ P θ ( } (5) = W E{ ( ( } W = W W (6) he representaton o eamormng eamormng eamspae USIC Fg..eamspaeUSIC θ θ,,, θ P n terms o ts egenvalues µ µ and orrespondng egenvetors, t, t µ t, sasollows: = µ t t (7) = where t,, t orresponds to the beamspae sgnal P subspae and t,, t orresponds to the beamspae P+ nose subspae. he beamspae steerng vetors o the sgnalareorthogonaltothenosesubspae[4],thatsto sa, [ W a( ] t a ( Wt =, = P +,, (8) = hereore, we an orm the beamspae USIC spatal spetrumas a ( WW a( P ( = (9) USIC a ( W W a( where = [ t,, t ] n P + s the beamspae nose subspae egenvetormatr.wendtheloatonsothepeakso P ( asthedoaestmatesothesoures. USIC 3.eamspaeDOAEstmatonater SpatalSmoothng One assumpton made above states that the nomng sgnals are mutuall unorrelated over the tme o observaton. I soure sgnals orgnate rom derent transmttersoraremodulatedwthderentdatastreams, the wll onl be partall orrelated. owever, the resultrommultpathresponsesromthesametransmtter, the sgnals are oherent and the assumpton s nvald. When there are oherent (ompletel orrelated) soure sgnals, rank(s) wll be less than P. ene, the beamspaeusicalgorthmdesrbedaboveals.for unormlneararras,bapplngtheorward-bakward spatal smoothng method beore beamormng, a new rank-pmatrsobtanedwhhanbeusednplaeo nbeamspaeusictoremovethesgnaloherene. akward UAsensors Sub-arras n n Forward Equvalentlneararra Fg..Forward-bakwardspatalsmoothng Spatal smoothng starts b dvdng the -element UA nto K=-+ overlappng sub-arras o sze.he outputokthorwardsub-arrasdenotedb ( wth elements { ( t ),, ( t )}, and the output o the kth k k + bakward sub-arra s denoted b ( wth elements * * { (,, ( }.hen,aorwardandbakward k + k+ spatallsmoothedmatr salulatedas = N K b b { ( ( + ( k, k, k, k KN t= k = ( } (), he rank o s P there are at most /3 oherent soures.mustbeseletedsuhthat P + P / + ()

3 nwhhp sthenumberooherentsoures.itsalso possbletodospatalsmoothngbasedonlon ( or (,butnthsaseatmost/oherentsouresan behandled. husthebeamspaeusicalgorthmmabeappledto toomputethedoaestmatesotheoherentarrvals, asshownnfg.3.etterdoaestmatonperormanes epetedbusngtheombnatonoeamspaeusic and spatal smoothng. We wll show the perormane omparson results n the net seton va omputer smulaton. ( ( ) ( t Spatalsmoothng, DOAestmatonusng beamspaeusic eamormng, W W Fg3.Combnatonospatalsmoothngandbeamspae USICtoestmateDOA 4.PerormaneComparsonVa ComputerSmulaton he onventonal approah to deal wth the DOA estmatonproblemoroherentsouressombnngthe spatalsmoothngtehnqueandtheelementspaeusic algorthm(hereaterwereerttoas).we proposed to ombne beamspae USIC and spatal smoothng(hereaterwedenotetas)nthe prevous seton to aheve better DOA estmaton perormane. In ths seton, we wll use omputer smulatontoomparethesetwomethods. hearrausedorperormaneomparsonsaunorm lnear arra wth = sensors and hal wavelength spang. Assume there are two oherent sgnals havng equalpowerndentrom [,3 ], wth respet to the broadsdeothearra.hesgnaltonoseratosdened as log (var( s ( ) / σ ) p.henumberodatasample taken at eah sensor output s N=. o use the orward-bakwardspatalsmoothngtehnque,thelnear arrasdvdednto K = 6 sub-arras,andeahsub-arra has = 6 elements. he spatal smoothed ovarane matr anberegardedastheovaranematroa 6-element lnear arra. For ths equvalent arra, three beams were ormed usng onventonal beamormng method to steer to 3.6,.8 and 8., respetvel. Fg.4showsthebeampatternothesethreebeams. eampattern(d) DretonoArrval(Deg) Fg.4.eampatternothethreebeamsusedor beamspaeproessng 4..IdealnearArra Frst we onsder an deal lnear arra. ere the deal lneararrameansthattherearenosstemerrorsnthe arrasstemdesrbedpresent.hetpalspatalspetra oeusic,andwhensn= d are shown n Fg. 5. eause o the sgnal oherene,elementspaeusicannotresolvethetwo sgnals,asshownbdottedlnenfg.5.utnthesame ase,bothandanresolvethe twosgnalsorretl.headvantageothe spatalspetrumsthatthaslowerbakgroundandlower vallethanthato.asweknow,thswll helpwththedentatonopeaksorrespondngtothe DOAs. SpatalSpetrum(d) EUSIC -6-3 DretonoArrval(Deg) Fg.5.palspatalspetrumoEUSIC, andordeallneararra 3

4 In order to ompare the probablt o resoluton and SE (oot ean Square Error) o both algorthms at derentsns,5tmesotestsweredoneateahsn rom 3todwthdstep.heresultswereshown nfg.6.fromfg.6(a),weanseethattheresoluton thresholdoslowerthanthatoeusic- SS, and Fg. 6(b) shows that both methods have appromatelthesamese. 4..PratalnearArra In a pratal arra, there wll be some gan and phase unertant nevtabl. We onsder suh a lnear arra wthganandphaseerrorsspeedntable.hesegan andphaseerrorshavebeennormalzedtotherstelement. In ths ase, the tpal spatal spetra o EUSIC, and when SN = d are shown n Fg. 7. he element spae USIC wthout spatalsmoothngertanlannotresolvethetwosoures, and too als to resolve the two soures. Probabltoesoluton SE(Deg) (a)probabltoresoluton (b)se Fg.6.PerormaneomparsonooEUSIC,EUSIC- SSandordeallneararra able.ganandphaseerrorsoapratallneararra element gan phase element gan phase element gan phase element gan phase SpatalSpetrum(d) EUSIC - 3 DretonoArrval(Deg) Fg.7.palspatalspetrumoEUSIC, andorpratallneararra Onlstllanresolvethetwosoures.hs showsthatsmorerobustthan. Wealsodd5tmesotestsateahSNrom 3to dwthdsteptoalulatetheprobabltoresoluton andseoand.heresults wereshownnfg.8.weseethattheresolutonthreshold osstlllowerthanthato,but ts SE s a lttle bt larger than that o. Compared wth the deal arra ase, the resoluton thresholdonreasedduetotheganand phase unertant, but an mantan ts resolutonthreshold.itsstraghtorwardthatusic- SSsmorerobustthan,thoughtheresalttle sareotsestmatonaura. 5.Conluson ombnng the orward-bakward spatal smoothng tehnque and beamspae USIC, the DOA estmaton 4

5 Probabltoesoluton SE(Deg) (a)probabltoresoluton (b)se Fg.8.PerormaneomparsonoEUSIC,EUSIC- SSandorpratallneararra perormaneanbemproved,bothordeallneararras andorpratalarras.anothermportantadvantageo the method proposed n ths paper s ts redued omputatonomplet,whhwllsavealototmen the real-tme sstem. hereore, ths method s an eetve tehnque or real-tme DOA estmaton or oherentsouresnpratalsonarorradarsstems. eerenes []. O Shmdt, ultple emtter loaton and sgnal parameter estmaton, IEEE rans. Antennas and Propagaton,Vol.34,No.3,pp.76-8,986 []X..XuandK..ukle, Statstalperormane omparsonousicnelement-spaeandbeam-spae, Pro.oInt.Con.onAoust.,Speeh,SgnalProessng, pp.4-7,989 [3]X..XuandK..ukle, Aomparsono element and beam spae spatal-spetrum estmaton or multple soure lusters, Pro.oInt.Con.onAoust., Speeh,SgnalProessng,pp ,99 [4]..eeand.S.Wengrovtz, esolutonthreshold o beamspae USIC or two losel spaed emtters, IEEErans.Aoust.,Speeh,SgnalProessng,Vol.38, No.9,pp ,September99 [5]P.StoaandA.Nehora, Comparatveperormane stud o element-spae and beam-spae USIC estmators, Cruts,Sstems,SgnalProessng,Vol., No.3,pp.85-9,99 [6]Y.X.Yang,Studesonbeamormngandbeamspae hgh resoluton bearng estmaton tehnques on sonar sstem,x an:ph.d.thessnnorthwesternpoltehnal Unverst,June [7] S. akn, Arra sgnal proessng, New Jerse: Prente-all,985 [8] -J Shan,. Wa and. Kalath, On spatal smoothng or dreton-o-arrval estmaton o oherent sgnals, IEEE rans. Aoust., Speeh, and Sgnal Proessng,Vol33,No.4,pp.86-8,Aprl985 [9] S. U. Plla and.. Kwon, Forward/bakward spatal smoothng tehnques or oherent sgnal dentaton, IEEE rans. Aoust., Speeh, andsgnal Proessng,Vol.37,No.,pp.8-5,Januar989 [] J., Improvng angular resoluton or spatal smoothng tehnques, IEEE rans. Sgnal Proessng, Vol.4,No.,pp ,Deember99 []J.S.hompson,P..Grant,andulgrew, Perormaneospatalsmoothngalgorthmororrelated soures, IEEErans.SgnalProessng,Vol.44,No.4, pp.4-46,aprl996 []..Vanrees,Optmum Arra Proessng,New York:JohnWle&Sons,In., 5