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Blind Adaptive MMSE Equalization of Underwater Acoustic Channels Based on the Linear Prediction Method

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Blind Adaptive MMSE Equalization of Underwater Acoustic Channels Based on the Linear Prediction Method

    Blind Adaptive MMSE Equalization of Underwater Acoustic Channels Based on

    the Linear Prediction Method JMarineSci.Applf2011110:113120

    DOI:10.1007/sll804011.10509

    BlindAdaptiveMMSEEqualizationofUnderwaterAcoustic

    ChannelsBasedontheLinearPredictionMethod

    YinbingZhangr,JunweiZhao,YecaiGuoandJinmingLi

    .

    CollegeofMarine,NorthwesternPolytechnicalUniversity,Xian710072,China 2.CollegeofElectronicandInformationEngineering,NanjingUniversityoflnformationScience&Technology,Nanjing210044,China

    Abstract:Theproblemofblindadaptiveequalizationofunderwatersingleinputmultiple

    output(SIMO)

    acousticchannelswasanalyzedbyusingthe1inearpredictionmethod.Minimummeansquareerror(MMSE)

    blindequalizerswitharbitrarydelayweredescribedoffabasisofchannelidentification.Twomethodsfor

    calculating1inearMMSEequalizerswereproposed.Onewasbasedonfu11channe?

    dentificationandrealized

    usingRLSadaptivealgorithms.andtheotherwasbasedoffthezero

    delayMMSEequalizerandrealizedusing

    LMSandRLSadaptivealgorithms.respectively.Performanceofthethreeproposedalgorithmsand

    comparisonwithtwoexistingzero

    forcing(ZF)equalizationalgorithmswereinvestigatedbysimulations

utilizingtwounderwateracousticchannels.Theresultsshowthattheproposedalgorithmsare

    robustenoughto

    channe1ordermismatch.TheyhavealmostthesameperformanceasthecorrespondingZFal

    gorithmsundera

    highsignaltonoisefSNR1ratioandbetterperformanceunderalowSNR. Keywords:linearprediction;blindequalization;channe1identification;secondorderstatist

    ics;MMSE

    ArticleID:16719433(20?,0l0ll308

    1IntrOductiOn

    Time..varyingcharacteristicandmulti..pathfadingof underwateracousticchannelscaninducesevereinter symbolinterference(is1)iflhighdataratecommunication systems.Channelequalizationapplyingadaptivefiltersis oneofthetechniquestomitigatetheeffectsofISI. ConventiOnaIlvtheinitializationofanadaptivefilteris achievedbyaknowntrainingsequencefromatransmitter beforedatatransmission,sothatvaluablechannelcapacity isreduced.Recently,blindequalizationtechnique (Stojanovic,1996)hasattractedmoreandmoreattention. Comparedwithadaptiveequalizationtechnique,themajor advantageofsuchtechniqueisthatnotrainingsequenceis neededtostartuporrestartthesystemwheneverthe communicationbreaksdownunpredictably.

    Traditionally.symbolratesampledchanneloutputsequence isstationaryandhigherorderstatisticsareusedtoestimate thechanne1andtocalculatetheequalizer.Morerecently,it hasbeenshownthatthechanneloutputsequenceis

    cyclostationaryifthesamplingrateexceedsthesymbolrate, andthensecondorderstatistics(sos)containsufficient informationtoestimatemostcommunicationchannelsusing

    cyclostationarity(Tongeta1.,1994;Tongeta1.,1995; PapadiasandSlock,1999).Basedontheseminarworkof Tongela1.r1994),manyeffectiveblindmethodshavebeen Receiveddate:20090907.

    Foundationitern:SupportedbytheNationalNatura1ScienceFoundationof

    ChinaunderGrantNo.60372086andtheFoundationfortheAuthorof Nationa1ExcellentDoctoralDissertationofChinaunderGrantNo.200753.

    CorrespondingauthorEmail:ybzhangnwpu@I63.corn HarbinEngineeringUniversityandSpringer-VerlagBerlinHeidelberg2011

    proposedforestimatingthechannelfromtheoutputofonly secondorderstatistics.However.itturnsoutthatthese methodshavemuchcomputationalcomplexityortheyare verysensitivetochannelordermismatch(Moulineseta1.. 1995:Meraimeta1.,1997;Liueta1.1994;Albergeeta1. 2002),whicharemajorobstaclesfortheirrealtime

    implementations.Thepredictionerrormethodoffersan alternativetotheclassoftechniquesabove.Itwas introducedbySlock(1994),MeraimetaI.(1997),Ding (1997),GesberandDuhamei(1997),Tugnait(1999)and offeredgreatadvantagesoverotherS0Sbasedtechniques becauseofitsrobustnesstochannelordermismatchandlow computationa1complexity.Basedoffmultichannellinear predictionoftheobservationszeroforcing(ZF)and

    minimummeansquareerror(MMSE)equalizerswith arbitrarydelaywereinvestigatedinPapadiasandSlock f1999).Nevertheless,whencalculatingZFequalizers,not onlyther+1,step.aheadlinearpredictionofthenoisefree

    channeloutputshouldbeestimated,butalsothebackward linearpredictionofsomesufficientorderMofthe

predictionerrorofthepreviousf+11stepaheadlinear

    predictorneedtobecarriedout.WhencalculatingMMSE equalizers.ZFequalizersshouldbeworkedoutfirstand noisevariancemustbeestimatedcorrectly.Theseoperations makethetwokindsofequalizersverycomplicatedandhard torealize.AcomputationallyeffectiveblindZFequalization methodhasbeendiscussedinLiandFan(2000).Itisbased ontwolinearpredictionmodels.oneisusedtocalculatethe

    delayZFequalizerandtheotherisusedtocalculateZF zero

    equalizerswitharbitrarydelayonthebasisofthefirstone. However.onlyZFequalizersarepresented.InGiannakiand Ha1f0rdr1997).anapproachfordirectlyestimatingnonzero 114YinbingZhang,etalBlindAdaptiveMMSEEqualizationofUnderwaterAcousticChann

    elsBasedontheLinerPredicti0nMeth0d

    delayMMSEequalizerswasgiven.Nevertheless,thefirst coefficientofthechannelresponsemustbeknownapriori andnoisevarianceshouldbeestimatedcorrectly. Inordertoimprovetheperformanceofblindequalizers withouttheaforementionedlimitations,twomethodsfor findinglinearMMSEequalizerswitharbitrarydelayare presentedinthispaper

    identificationandrealized

    OBeiSbasedonfu11channel

    usingRLSadaptivealgorithm,

    theotherisbasedonthezerodelayMMSEequalizerand

    realizedusingLMSandRLSadaptivealgorithms. respectively.Simulationresultsshowthattheproposed methodsarerobusttochanne1ordermismatchandtheyhave betterperformancethanthecorrespondingZFalgorithms underlowSNR.Forthewholepapervectorsandmatrices

    areboldfacesmal1andcapital1etters.respectively.The notationstr(.),(.),(.)",and(.)standforthetrace, transpose,conjugatetranspose,andtheMoorePenrose

    pseudoinverse,respectively.I?istheLxLidentity

    matrixand0istheMxNzeromatrix.

    E1.]

    denotesthestatisticalexpectation.

    2Problemformulation

    Consideralineartimeinvariantcommunicationchannel

    Thereceivedbasebandsignaly(t)canbeexpressedas (f)=(f)+v(f)=?s,h(t-rz)+v(,)(1),)

    wheresldenotesthesymbolemittedbythedigitalsource attime,withbeingthesymbolduration;h(t)the overallcomplexbasebandequivalentimpulseresponseof thetransmitterfilter,unknownchannelandthereceiver filter;x(t)thechanneloutputwithoutnoise;andv(t)the channelnoisethatisassumedtobestationaryaswellas uncorrelatedwithsThefollowingassumptionsareheld throughoutthispaper:

    1)Thesymbolsequences,isstationarysubGaussian

    signalwithzeromeanandunitvariance.

    2)Thenoisev(t)isGaussianwithvariance.

    3)iscausalandhasfinitesupport[0].

    4)Thesubchannelshavenocommonzeros.

    TheoversamplingfactorisassumedtobeLandtheinitial samplingtimeinstantis,0.Theoversampledreceived signalcannowberepresentedas

    ,)+.+Ts/L)(2) 1+/=slh(to+,zT.IL

    y(n)=y(to+nT~/L.rs/C)nT~,/L)(3)()=(,.+,v(,z)=v.+) thenEq.(2)becomes

y(n)=?slh(nlL)+v(n)=x(n)+v()(4),=

    Define

    xi()=x(nL+i),Yi()=y(nL+i)

    hi()=(+),v):v(nL+i)

    wherei0,...,L1.Thenthesingleinputsingleoutput

    (SISO)systemofEq.(4)hasanequivalentS1MOdescription

    asfollows,

    Yi()=?sth(一『)+v()()+lJ()(6)J)

    Definethefollowingsymbolratevector, ?=)…一()lr,)=?…(,z)J

    )=()hL.()()=()VL-1()]

    TheEq.r41canberepresentedinavectorform J,()=?,("/)+():())(8)

    Furthermore,itcanberepresentedasthefollowingmatrix

    form,

    YN()=()+PN()=XN(n)+t'N()(9)

    ereHisa?whz×(?+^)blockToeplitzmatrix, s(n)isa(N+Lh)xlvectorandxN("),vN("),Yu()

    are?E×1vectorsasfoilowR

    H=

    (0).1l(L)

    (o)()

    x(n)=1().

    n):jY().

    ?()=l',T(,2)..

    ()=lss

    h(O)h(L)

    ..

    xy(JV+1)]

    yTn^,+1)]'

    .

    ',n-?十1)]

    

    L+11

    3Theproposedmethods

    3.1ZFequalizersandMMSEequalizers ConsiderthefractionallyspacedFIRlinearequalizershown

    inFig.1,whereg(,zJfori=0,...,?一1istheequalizer

    withorderLoftheithsubchanne1.Intheabsenceof noise,onenaturalchoiceistorequire)=w(,zd)for

    someintegerdelaydwithd?[0,L^+L].Thistypeof

    equalizerisknownaszeroforcing.Moreprecisely,aZF equalizerisdescribedby

    ?一1

    ??(1)glma(m)(12)

    JournalofMarineScienceandApplicationr201l11O:113120

    wheresuperscript(referstothedelayd.Choose N=L+1inEq.(10),thenEq.(12)canbewrittenas Hgd.zF=ed+1(13)

    where,

    =

    (0)T(L)]isa+1)×1

    vectoroftheequalizertapscorrespondingtodelay. dand

    g(,):(f)g(f)r.Pisa+?+1)xl

    vectorwithan1asthe(d+1)thelementandzeros elsewhere.TheexistenceofZFequalizersgd. hasbeen

    proven(GiannakiandHalford,1997;SlockandPapadias, 1995)ifthesubchannelshavenocommonzerosand LgL^1.Itcanbewritteninthefollowingexpression, gT:H"(:,+1)cRr(14)

    whereR:E{x(,z)x)]=HH"and

    the(d+1)thcolumnofthematrixH.

    Vo(n)

    H(:,+1)

    Fig.1FractionallyspacedFIRlinearequalizer AsZFequalizersdonotaddressnoisesuppression,another kindofequalizercalledblindMMSEequalizershasbeen proposedtofind():i=0,1,...,L1)suchthat

    Jd=E)(d)l)isminimized,where

    ;)=??g),)(15)i=o/=o

    Takethecomplexderivativewithrespecttotheunknown equalizertapsandsetthemtozero,MMSEequalizerwith arbitrarydelaydisobtained,

    T

    .MMs=H"(:,d+1))+(16)

    where=E{(")J,1H()).RLSandcyclicLMS

    algorithms[]havebeenproposedtorecursivelycalculate theequalizertaps.However,theycanonlybeusedto calculatezerodelayMMSEequalizerbasedonthe

    assumptionoftheknowledgeofh(0).Ifitismodifiedto getMMSEequalizerswitharbitrarydelay,noisevarianceof thereceiveddatamustbeestimatedcorrectly,whichmakes itimpracticalforrealization.Fortunately,accordingto Eq.(16),ifthecorrectestimatesoff():i=0,1,...,L1)

    115

    areobtained,thenH(:,d+1)willbeavailablesothatRLS

    algorithmcanbeusedtorecursivelycalculategd.MMsE. Basedonthisidea,channelidentificationbecomesamajor issuehere.

    FortheexistingSOSbasedchannelidentificationmethods, mostofthemaresensitivetochannelordermismatchor computationallycomplicated.Thepredictionerrormethod offersanalternativetothechannelidentification.Inthe followingsections.1inearpredictionbasedchannel identificationandequalizationmethodswillbepresented. s):)一一Y一一1)-,一一)(17)

    mintr{E(g(n)g"(n"

    )(18r{[一一][—一】"j

    ,,一一.=(0)0oo](19)一一-=0JJ()

    E=mi

    

    nE(e(())=h(O)h"(o)(20)

    knownapriori.SupposethesmallestchannelorderisL^ P?一:喇…(U-1)](21)

    wherelfori=1,,N1isaLxLmatrix.

    AccordingtothespecialstructureofH,itdemonstrates (1)=JIl(0)

    |Il(2)=h0)+|lI(0)f221

    |II()=(1)+..."h(Lh+l一?)

    Eq.(20)showsthatthepredictionerrorcovariancei'sa rank?onematrix.Anycolumnofthismatrixcanbeusedas 116YinbingZhang,eta1.BlindAdaptiveMMSEEqualizationofUnderwaterAcousticChan

    nelsBasedontheLinearPredictionMethod

    theestimateofh(0).ThenaccordingtoEq.(22),thewhole channelresponsecanbecalculatedrecursivelysothatthe

estimateofH(:,d+1)isobtained.NoticethatN?L^

    shouldbesatisfiedtoensuregoodestimationresult.

    ThroughtheanalysisaboveandcombiningEq.(16),Eq.(17),

    Eq.(20)andEq.(21),thefollowinglinearpredictionbased

    RLSalgorithm,namelyMMSERLS1forsimplicity,is givenforcomputingblindMMSEequalizerwitharbitrary

    delay.

    step1:InitializationQ1(0)=1J),where isasmallpositiveconstant.

    

    (.)=0L~c0v_0.E(.)=,Q^(.)=lI

    y=0_llJ,(0)=0,y?=0

    whereisasmallpositiveconstant. Foreachtimeinstantt7:1,2.,performStep2toStep5 Step2:Gettheoptimallinearpredictor

    1)y(F/1)

    

    1)Q1)y)1)

    1)y1)

    

    ()=.(1)+s(lH()

    Q():[Q1)K.()H1)e1)/L

    Step3:Estimateh(01

    ThecolumnofE(n)

    estimateofh(0).

    ,e(n):E1)+s")

    withthelargestnormistakenasthe Step4:Calculatetheestimatesof(1),h(2).,h(L7est1

    usingEq.(22).

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