基于LMS算法的自适应组合滤波器中英文翻译

1、百度文库-好好学习，天天向上百度文库-好好学习，天天向上(3)- (3)- # CombinedAdaptiveFilterwithLMS-BasedAlgorithmsAbstract:Acombinedadaptivefilterisproposed.ItconsistsofparallelLMS-basedadaptiveFIRfiltersandanalgorithmforchoosingthebetteramongthem.Asacriterionforcomparisonoftheconsideredalgorithmsintheproposedfilter,wetakethera

2、tiobetweenbiasandvarianceoftheweightingcoefficients.Simulationsresultsconfirmtheadvantagesoftheproposedadaptivefilter.Keywords:Adaptivefilter,LMSalgorithm,Combinedalgorithm,Biasandvariancetrade-off1IntroductionAdaptivefiltershavebeenappliedinsignalprocessingandcontrol,aswellasinmanypracticalproblems

3、,1,2.Performanceofanadaptivefilterdependsmainlyonthealgorithmusedforupdatingthefilterweightingcoefficients.ThemostcommonlyusedadaptivesystemsarethosebasedontheLeastMeanSquare(LMS)adaptivealgorithmanditsmodifications(LMS-basedalgorithms).TheLMSissimpleforimplementationandrobustinanumberofapplications

4、13.However,sinceitdoesnotalwaysconvergeinanacceptablemanner,therehavebeenmanyattemptstoimproveitsperformancebytheappropriatemodifications:signalgorithm(SA)8,geometricmeanLMS(GLMS)5,variablestep-sizeLMS(VSLMS)6,7.EachoftheLMS-basedalgorithmshasatleastoneparameterthatshouldbedefinedpriortotheadaptatio

5、nprocedure(stepforLMSandSA;stepandsmoothingcoefficientsforGLMS;variousparametersaffectingthestepforVSLMS).Theseparameterscruciallyinfluencethefilteroutputduringtwoadaptationphases:transientandsteadystate.Choiceoftheseparametersismostlybasedonsomekindoftrade-offbetweenthequalityofalgorithmperformance

6、inthementionedadaptationphases.WeproposeapossibleapproachfortheLMS-basedadaptivefilterperformanceimprovement.Namely,wemakeacombinationofseveralLMS-basedFIRfilterswithdifferentparameters,andprovidethecriterionforchoosingthemostsuitablealgorithmfordifferentadaptationphases.Thismethodmaybeappliedtoallt

7、heLMS-basedalgorithms,althoughwehereconsideronlyseveralofthem.Thepaperisorganizedasfollows.AnoverviewoftheconsideredLMS-basedalgorithmsisgiveninSection3proposesthecriterionforevaluationandcombinationofadaptivealgorithms.SimulationresultsarepresentedinSection4.2.LMSbasedalgorithmsLetusdefinetheinputs

8、ignalvectorX=x(k)x(k-1)x(k-N+1)tandvectorofkweightingcoefficientsasW=W(k)W(k)-W(k)t.Theweightingcoefficientsk01N-11)vectorshouldbecalculatedaccordingto:W二W+2卩EeXk+1kkkwherepisthealgorithmstep,Eistheestimateoftheexpectedvalueande=dWtXistheerroratthein-stantk,anddkisareferencesignal.kkkkDependingonthe

9、estimationofexpectedvaluein(1),onedefinesvariousformsofadaptivealgorithms:theLMS(EIXLeX),theGLMSEIXL迄k(1-a)eX,0a1/kkkkkki_qk-ik-iandtheSA(EIX了=Xsign(e),1,2,5,8.TheVSlMshasthesameformasthekkkkLMS,butintheadaptationthestepp(k)ischanged6,7.Theconsideredadaptivefilteringproblemconsistsintryingtoadjustas

10、etofweightingcoefficientssothatthesystemoutput,y_WtX,tracksareferencekkksignal,assumedasd=W*TX+n,wherenisazeromeanGaussiannoisewithkkkkkthevariancec2,andW*istheoptimalweightvector(Wienervector).Twocasesnkwillbeconsidered:W*_Wisaconstant(stationarycase)andW*istime-varyingkkisthezero-meanrandomperturb

11、ation,independenton=b21.NotethatanalysisZ(nonstationarycase).Innonstationarycasetheunknownsystemparameters(.theoptimalvectorW*)aretimevariant.ItisoftenassumedthatvariationofW*maybekkmodeledasW*_W*+Z“k+1kK.XandnwiththeautocorrelationmatrixG_EZZTkkkkforthestationarycasedirectlyfollowsforc2_0.Theweight

12、ingcoefficientvectorZconvergestotheWienerone,iftheconditionfrom1,2issatisfied.Definetheweightingcoefficientsmisalignment,13,V_W一W*.ItisduetobothTOC o 1-5 h zkkktheeffectsofgradientnoise(weightingcoefficientsvariationsaroundtheaveragevalue)andtheweightingvectorlag(differencebetweentheaverageandtheopt

13、imalvalue),3.Itcanbeexpressedas:(2)V_W-EW)+(EW)-W*),kkkkkAccordingto(2),theithelementofVis:kV(k)_(E(W(k)-W*(k)+(W(k)-E(W(k)_bias(W(+p()11ii百度文库-好好学习，天天向上百度文库-好好学习，天天向上 - wherebias(W(k)iistheweightingcoefficientbiasandP(k)isazero-meanirandomvariablewiththevariancec2.ThevariancedependsonthetypeofLMS-b

14、asedalgorithm,aswellasontheexternalnoisevariancec2.Thus,讦thenoisenvarianceisconstantorslowly-varying,c2istimeinvariantforaparticularLMS-basedalgorithm.Inthatsense,intheanalysisthatfollowswewillassumethatc2dependsonlyonthealgorithmtype,.onitsparameters.Animportantperformancemeasureforanadaptivefilter

15、isitsmeansquaredeviation(MSD)ofweightingcoefficients.Fortheadaptivefilters,itisgivenby,3:MSD=limeVtV13CombinedadaptivelterThebasicideaofthecombinedadaptivefilterliesinparallelimplementationoftwoormoreadaptiveLMS-basedalgorithms,withthechoiceofthebestamongthemineachiteration9.Choiceofthemostappropria

16、tealgorithm,ineachiteration,reducestothechoiceofthebestvaluefortheweightingcoefficients.Thebestweightingcoefficientistheonethatis,atagiveninstant,theclosesttothecorrespondingvalueoftheWienervector.LetW(k,q)bethei-thweightingcoefficientforLMS-basedalgorithmwiththeichosenparameterqataninstantk.Notetha

17、tonemaynowtreatallthealgorithmsinaunifiedway(LMS:q三卩,GLMS:q三a,SA:q三卩).LMS-basedalgorithmbehavioriscruciallydependentonq.Ineachiterationthereisanoptimalvalueqopt,producingthebestperformanceoftheadaptivealgorithm.Analyzenowacombinedadaptivefilter,withseveralLMS-basedalgorithmsofthesametype,butwithdiff

18、erentparameterq.TheweightingcoefficientsarerandomvariablesdistributedaroundtheW*(k),withibias(W(k,q)andthevariancec2,relatedby4,9:(4)W(,q)-W*(k)-biasW(k,q)KC,iiiqwhere(4)holdswiththeprobabilityP(K),dependentonK.Forexample,forK=2andaGaussiandistribution,P(K)=(twosigmarule).Definetheconfidenceinterval

19、sforW(k,q),4,9:(5)D(k)=W(k,q)-2kc,W(k,q)+2kc1iiqiqThen,from(4)and(5)weconcludethat,aslongasbiasW(k,q)iKQqW*(k)eD(k),iiindependentlyonq.Thismeansthat,forsmallbias,theconfidenceintervals,fordifferentqsofthesameLMS-basedalgorithm,ofthesameLMS-basedalgorithm,intersect.When,ontheotherhand,thebiasbecomesl

20、arge,thenthecentralpositionsoftheintervalsfordifferentqsarefarapart,andtheydonotintersect.Sincewedonothaveaprioriwewilluseainformationaboutthebias(W(k,q),ispecificstatisticalapproachtogetthecriterionforthechoiceofadaptivealgorithm,.forthevaluesofq.Thecriterionfollowsfromthetrade-offconditionthatbias

21、andvarianceareofthesameorderofmagnitude,.|biasW(k,q)i,4.qTheproposedcombinedalgorithm(CA)cannowbesummarizedinthefollowingsteps:Step1.CalculateW(k,q)forthealgorithmswithdifferentqsfromthepredefinedisetQ=%,q,i2Step2.Estimatethevarianceb2foreachconsideredalgorithm.qStep3.CheckifD(k)intersectfortheconsi

22、deredalgorithms.Startfromanialgorithmwithlargestvalueofvariance,andgotowardtheoneswithsmallervaluesofvariances.Accordingto(4),(5)andthetrade-offcriterion,thischeckreducestothecheckifvariance.Firsttwointervalsthatdonotintersectmeanthattheproposedtrade-offcriterionisachieved,andchoosethealgorithmwithl

23、argevariance.issatisfied,W.G,q)W(k,q)b2,nq纟Q.qlh+b)qmql(6)Vq:b2b2hqmqhIfnoD(k)iIftheD(k)intersect,thebiasisalreadysmall.So,checkanewpairofweightingicoefficientsor,ifthatisthelastpair,justchoosethealgorithmwiththesmallestintersect(largebias)choosethealgorithmwithlargestvalueofvariance.Step4.Gotothene

24、xtinstantoftime.ThesmallestnumberofelementsofthesetQisL=2.Inthatcase,oneoftheqsshouldprovidegoodtrackingofrapidvariations(thelargestvariance),whiletheothershouldprovidesmallvarianceinthesteadystate.Observethatbyaddingfewmoreqsbetweenthesetwoextremes,onemayslightlyimprovethetransientbehaviorofthealgo

25、rithm.Notethattheonlyunknownvaluesin(6)arethevariances.Inoursimulationsweestimatec2asin:q(k)-WQ-1)0.675、2,ii(7)c二medianqfork=1,2,.,Landc2c2.ZqThealternativewayistoestimatec2as:1nc2工Te2,forx(i)=0.nTi=1iExpressionsrelatingc2andc2insteadystate,fordifferenttypesofLMS-basednqalgorithms,areknownfromlitera

26、ture.ForthestandardLMSalgorithminsteadystate,8)c2andc2arerelatedc2=qc2,3.Notethatanyotherestimationofc2isnqqnqvalidfortheproposedfilter.ComplexityoftheCAdependsontheconstituentalgorithms(Step1),andonthedecisionalgorithm(Step3).Calculationofweightingcoefficientsforparallelalgorithmsdoesnotincreasethe

27、calculationtime,sinceitisperformedbyaparallelhardwarerealization,thusincreasingthehardwarerequirements.Thevarianceestimations(Step2),negligiblycontributetotheincreaseofalgorithmcomplexity,becausetheyareperformedattheverybeginningofadaptationandtheyareusingseparatehardwarerealizations.Simpleanalysiss

28、howsthattheCAincreasesthenumberofoperationsfor,atmost,N(L-1)additionsandN(L-1)IFdecisions,andneedssomeadditionalhardwarewithrespecttotheconstituentalgorithms.ofcombinedadaptivefilterConsiderasystemidentificationbythecombinationoftwoLMSalgorithmswithdifferentsteps.Here,theparameterqisy,.Q=q,q=，卩/1。.1

29、2Theunknownsystemhasfourtime-invariantcoefficients,andtheFIRfiltersarewithN=4.Wegivetheaveragemeansquaredeviation(AMSD)forbothindividualalgorithms,aswellasfortheircombination,Fig.1(a).Resultsareobtainedbyaveragingover100independentruns(theMonteCarlomethod),with卩=.Thereferencedkiscorruptedbyazero-mea

30、nuncorrelatedGaussiannoisewithc2=nandSNR=15dB,andkis.Inthefirst30iterationsthevariancewasestimatedaccordingto(7),andtheCApickedtheweightingcoefficientscalculatedbytheLMSwith百度文库-好好学习，天天向上百度文库-好好学习，天天向上百度文库-好好学习，天天向上- - AspresentedinFig.1(a),theCAfirstusestheLMSwith卩andthen,inthesteadystate,theLMSwit

31、h“/1O.Notetheregion,betweenthe200thand400thiteration,wherethealgorithmcantaketheLMSwitheitherstepsize,indifferentrealizations.Here,performanceoftheCAwouldbeimprovedbyincreasingthenumberofparallelLMSalgorithmswithstepsbetweenthesetwoalsothat,insteadystate,theCAdoesnotideallypickuptheLMSwithsmallerste

32、p.Thereasonisinthestatisticalnatureoftheapproach.Combinedadaptivefilterachievesevenbetterperformanceiftheindividualalgorithms,insteadofstartinganiterationwiththecoefficientvaluestakenfromtheirpreviousiteration,taketheoneschosenbytheCA.Namely,iftheCAchooses,inthek-thiteration,theweightingcoefficientv

33、ectorW,theneachindividualalgorithmcalculatesitsweightingcoefficientsinthe(k+1)-thiterationaccordingto:旷=旷+2卩Ek+1pkk0-1海*區:CdLMSaawnwaierumtFBr1&0CSWOD0i|*煜泊耐审材Fig.1.AverageMSDforconsideredalgorithms.Fig.2.AverageMSDforconsideredalgorithms.ss(9)lalrflfftf*I1Fig.1(b)showsthisimprovement,appliedonthepr

34、eviousexample.Inordertoclearlycomparetheobtainedresults,foreachsimulationwecalculatedtheAMSD.ForthefirstLMS()itwasAMSD=,forthesecondLMS(“/10)itwasAMSD=,fortheCA(CoLMS)itwasAMSD=andfortheCAwithmodification(9)itwasAMSD=.SimulationresultsTheproposedcombinedadaptivefilterwithvarioustypesofLMS-basedalgor

35、ithmsisimplementedforstationaryandnonstationarycasesinasystemidentificationofthecombinedfilteriscomparedwiththeindividualones,thatcomposetheparticularcombination.Inallsimulationspresentedhere,thereferencedkiscorruptedbyazero-meanuncorrelatedGaussiannoisewithc2=0.1andSNR=15dB.Resultsareobtainedbynave

36、ragingover100independentruns,withN=4,asintheprevioussection.Timevaryingoptimalweightingvector:TheproposedideamaybeappliedtotheSAalgorithmsinanonstationarycase.Inthesimulation,thecombinedfilteriscomposedoutofthreeSAadaptivefilterswithdifferentsteps,.Q=“，“/2,“/8;“=.Theoptimalvectorsisgeneratedaccordin

37、gtothepresentedmodelwithc2=0.001,andwithk=2.Inthefirst30iterationsthevariancewasestimatedZaccordingto(7),andCAtakesthecoefficientsofSAwith“(SA1).Figure2(a)showstheAMSDcharacteristicsforeachalgorithm.InsteadystatetheCAdoesnotideallyfollowtheSA3with“/8,becauseofthenonstationaryproblemnatureandarelativ

38、elysmalldifferencebetweenthecoefficientvariancesoftheSA2andSA3.However,thisdoesnotaffecttheoverallperformanceoftheproposedalgorithm.AMSDforeachconsideredalgorithmwas:AMSD=(SA1,“),AMSD=(SA2,“/2),AMSD=(SA3,“/8)andAMSD=(Comb).ComparisonwithVSLMSalgorithm6:InthissimulationwetaketheimprovedCA(9)from,andc

39、ompareitsperformancewiththeVSLMSalgorithm6,inthecaseofabruptchangesofoptimalvector.SincetheconsideredVSLMSalgorithm6updatesitsstepsizeforeachweightingcoefficientindividually,thecomparisonofthesetwoalgorithmsismeaningful.AlltheparametersfortheimprovedCAarethesameasin.FortheVSLMSalgorithm6,therelevant

40、parametervaluesarethecounterofsignchangem0=11,andthecounterofsigncontinuitym1=7.Figure2(b)showstheAMSDforthecomparedalgorithms,whereonecanobservethefavorablepropertiesoftheCA,especiallyaftertheabruptchanges.Notethatabruptchangesaregeneratedbymultiplyingallthesystemcoefficientsby1atthe2000-thiteratio

41、n(Fig.2(b).TheAMSDfortheVSLMSwasAMSD=,whileitsvaluefortheCA(CoLMS)wasAMSD=.Foracompletecomparisonofthesealgorithmsweconsidernowtheircalculationcomplexity,expressedbytherespectiveincreaseinnumberofoperationswithrespecttotheLMSalgorithm.TheCAincreasesthenumberofrequresoperationsforNadditionsandNIFtheV

42、SLMSalgorithm,therespectiveincreaseis:3Nmultiplications,Nadditions,andatleast2NIFdecisions.ThesevaluesshowtheadvantageoftheCAwithrespecttothecalculationcomplexity.ConclusionCombinationoftheLMSbasedalgorithms,whichresultsinanadaptivesystemthattakesthefavorablepropertiesofthesealgorithmsintrackingpara

43、metervariations,isthecourseofadaptationprocedureitchoosesbetteralgorithms,allthewaytothesteadystatewhenittakesthealgorithmwiththesmallestvarianceoftheweightingcoefficientdeviationsfromtheoptimalvalue.Acknowledgement.ThisworkissupportedbytheVolkswagenStiftung,FederalRepublicofGermany.基于LMS算法的自适应组合滤波器

44、摘要：提出了一种自适应组合滤波器。它由并行LMS的自适应FIR滤波器和一个具有更好的选择性的算法组成。作为正在研究中的滤波器算法比较标准，我们采取偏差和加权系数之间的方差比。仿真结果证实了提出的自适应滤波器的优点。关键词：自适应滤波器；LMS算法；组合算法；偏差和方差权衡1、绪论自适应滤波器已在信号处理和控制，以及许多实际问题1,2的解决当中得到了广泛的应用自适应滤波器的性能主要取决于滤波器所使用的算法的加权系数的更新。最常用的自适应系统对那些基于最小均方（LMS）自适应算法及其改进（基于LMS的算法）。LMS算法是非常简便，易于实施，具有广泛的用途1-3。但是，因为它并不总是收敛在一个可接受

45、的方式，所以有很多的尝试，以对其性能做适当改进:符号算法（SA）的8,几何平均LMS算法（GLMS）5,变步长LMS（最小均方比）算法6,7。每一种基于LMS的算法都至少有一个参数在适应过程（LMS算法和符号算法，加强和GLMS平滑系数，各种参数对变步长LMS算法的影响）中被预先定义。这些参数的影响关键在两个适应阶段：瞬态和稳态滤波器的输出。这些参数的选择主要是基于一种算法质量的权衡中所提到的适应性能。我们提出了一个自适应滤波器的性能改善的方法。也就是说，我们提出了几个基于LMS算法的不同参数的FIR滤波器，并提供不同的适应阶段选择最合适的算法标准。这种方法可以适用于所有的LMS的算法，虽然我

46、们在这里只考虑其中几个。本文的结构如下，作者认为的LMS的算法概述载于第2节，第3节提出了自适应算法的改进和组合标准，仿真结果在第4节。2、基于LMS的算法让我们定义输入信号向量X二x（k）x（k-1）x（k-N+1）T和矢量加权系数k为W二W（k）w（k）（k）t权重系数向量计算应根据：(1)k01N-1W二W+2卩EeX其中M为算法步长，E是预期值的估计。在e二d-WtX中，常数Kkkkk表式误差，dk是一个参考信号。根据（1）中不同的预期值估计在，我们可以得出k+1kkkdbk疋种各种形式的自适应算法的定义：LMS（X=eX），kkkkGLMSeX,0a1丿,SAiX丄Xsign（e）丿

47、,1,2,5,TOC o 1-5 h zkk/_ok-ik-ikkkk8.变步长LMS算法和基本LMS算法具有相同的形式，但在适应过程中步长M（k）是变化的6,7。正在研究中的自适应滤波问题在于尝试调整权重系数，使系统的输出y二WtX跟踪参考信号，d二W叫X+n中n是一个零均值与方差q2的高斯kkkkkkkkn噪声，W*是最佳权向量（维纳向量）。我们考虑两种情况：W*=W是一个常数kk（固定的情况下），W*随时间变化（非平稳的情况下）。在非平稳情况下，未知k系统参数（即W*最佳载体）是随时间变化的。我们假设变量W*可以建立模型kk为W*二W*+Z，它是随机独立的零均值，依赖于和n自相关矩阵k+

48、kKkkG=EIzZt=q21。注意：分析直接服从q2二0，如果1，2的条件是满足的，kkZZ那么加权系数向量收敛于维纳解。定义加权错位系数，1-,V二W-W*。是因为这两个梯度噪声（加权系kkk数的平均值左右的变化)和加权矢量滞后(平均及最佳值的差额)的影响，3。它可以表示为：V=W-EW+0W)-W*)TOC o 1-5 h zkkkkk根据(2),V是：(3)V(k)=G(W(k)-W*(k)+(W(k)E(W(k)匚bias(W(+p()11iibias(W()是加权系数的偏差，p()与方差a2是零均值的随机变量差，它ii取决于LMS的算法类型，以及外部噪声方差a2。因此，如果噪声方差

49、为常数或n是缓慢变化的，a2为某一特定的基于LMS时间不变的算法。在这个意义上说,在后面的分析中我们将假定a2只依赖算法类型，及其参数。自适应滤波器的一个重要性能衡量标准是其均方差(MSD)的加权系数。对于自适应滤波器，它被赋值，3：MSD二limeVtVkkks3、组合自适应滤波器合并后的自适应滤波器的基本思想是在两个或两个以上自适应LMS算法并行实现与每个迭代之间的最佳选择，9。在每次迭代中选择最合适的算法，选择最佳的加权系数值。最好的加权系数是1,即在给定的时刻，向相应的维纳矢量值最接近。让W(,q)是以基本LMS算法为基础的第i个加权系数，在瞬间选i择参数q和系数k。注意，现在我们可以

50、在一个统一的处理方式(LMS:q三p,GLMS:q三a,SA:q三p)下。基于LMS算法的行为主要依赖于q，在每个迭代中有一个最佳值qopt，生产的最佳表现的自适应算法。现在分析最小均方与一些基于相同类型的算法相结合的自适应滤波器，但参数q是不同的。加权系数周围分布随机变量W*()和bias(W(k,q)和方差a2，相关4，9：。TOC o 1-5 h ziiqW(,q)-W*()-biasV(,q)Ka(4)iiiq(4)中的概率P(K)依赖K的值.例如K=2勺高斯分布，P(K)=(两个。规则)。置信区间的定义W(,q),4,9(5)，q、KaW*(k)eD(k)iqiiDQ)=WQ,q)-

51、2ka,W(k,q)+2Kaiiqiq接着，从(4)式到(5)式我们认为只要iasW(k,q)关于独立q,这意味着，对于小偏差，置信区间对同一的LMS的算法是不同的,而对同一的LMS的算法则相交。另一方面，当偏置变大，然后中央位置的不同间隔距离很大，而且他们不相交。由于我们对有关信息bias(W(k,q)没有先验知识，我们将使用一种特定的统i计学方法得到的标准，即自适应算法选择的q值问题。这个标准的平衡状态，从或同一个数量级的，即biasW(k,q)=kc,I4。提出的联合算法(CA)现在可以被总结为下面的步骤：第1步：从不同预定义设置Q=,q,中为算法计算W(k,q)。i2i第2步：估计每个

52、算法的方差c2。q第3步：检查D(k)是否相交对于算法。从一个最大的差异值算法走向与差i异较小的值。根据(4),(5)和取舍的标准，如果下式成立那么将会减少这个检查：(6)WQ,q)一W(k,q)c2c2,nq纟Qhqmqhqlh如果没有D(k)相交(大偏差)选择具有最大的方差的值算法。如果相交，i偏差已经很小。因此，检查了一对新的加权系数，或者，如果DG)是最后一对,i只选择具有最小方差的算法。首先两个区间不相交意味着实现了取舍标准，并选择最大方差算法。第4步：转到下一个瞬间。元素的集合Q中最小的数L=2。在这种情况下，应提供良好的跟踪快速变化(最大的差异)，而其他应提供小的方差的稳定状态。通过增加更多的观察，这两个极端之间,我们可以稍微改进算法的瞬态行为。需要注意的是，只有未知值(6)的差异。在仿真中我们估计c24式：TOC o 1-5 h zc=median(WG)-W(-1)0.675“(7)qii当k=1，2，.，L和c2c2Zq替代的方法是估计C2为n1c2沁Xte2,forx(i)=0nT心1(8)有关表达式c2和c2在稳定状态为LMS算法的不同类型，从已知文献中可nq以看出。对于标准的LMS算法在稳定状态，c2和c2是相关的c2=qc2,3.需nqqn要注意的是，任何其他估计C2对于滤波器来说是有效的。q