1 0 n 0 Error between 64257lter output and a desired signal Change the 64257lter parameters according to 1 57525u 1 Normalized LMS Algorithm Modify at time the parameter vector from to 1 ful64257lling the constraint 1 with the least modi6425 ID: 28975
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Lecture5 2ni))u(ni)d(n1i=0wi(n)u(ni 2M1i=0(u(ni))2=(n) M1i=0wi(n)u(ni)) M1i=0(u(ni))2=2e(n) Thus,theminimumofthecriterion +1))willbeobtainedusingtheadaptationequation+1)= Inordertoaddanextrafreedomdegreetotheadaptationstrategy,oneconstant, ,controllingthestepsizewillbeintroduced:+1)=)+ M1i=0(u(ni))2e(n)u(njj(nµ u Toovercomethepossiblenumericaldicultieswhen isveryclosetozero,aconstant0isused: +1)= a+u (n)2e(n)u(nj) ThisistheupdatingequationusedintheNormalizedLMSalgorithm. Lecture5ComparisonofLMSandNLMSwithintheexamplefromLecture4(channelequalization) 500 1000 1500 2000 250 0 103 102 101 100 Learning curve Ee2(n) for LMS algorithmtime step n =0.075 µ=0.025 µ=0.0075 500 1000 1500 2000 250 0 103 102 101 100 101 Learning curve Ee2(n) for Normalized LMS algorithmtime step n =1.5 µ=1.0 µ=0.5 µ=0.1 Lecture52.LMSAlgorithmwithTimeVariableAdaptationStepHeuristicsofthemethod:Wecombinethebenetsoftwodierentsituations:TheconvergencetimeconstantissmallforlargeThemean-squareerrorinsteadystateislowforsmallTherefore,intheinitialadaptationstagesiskeptlarge,thenitismonotonicallyreduced,suchthatinthenaladaptationstageitisverysmall.Therearemanyreceiptsofcoolingdownanadaptationprocess.Monotonicallydecreasingthestepsize Disadvantagefornon-stationarydata:thealgorithmwillnotreactanymoretochangesintheoptimumsolution,forlargevaluesofVariableStepalgorithm: +1)= (n(n)u (n)e(n)whereM(n µ0(n00µ1(n).001(n) Lecture5ComparisonofLMSandvariablesizeLMS( )withintheexamplefromLecture4(channelequalization)=[10;20;50] 500 1000 1500 2000 2500 103 102 101 100 2(n) for LMS algorithmtime step n =0.075 µ=0.025 µ=0.0075 500 1000 1500 2000 2500 103 102 101 100 101 2(n) for variable size LMS algorithmtime step n c=20 LMS µ=0.0075 Lecture5Veryfastcomputation:ifisconstrainedtotheform,onlyshiftingandadditionoperationsareDrawback:theupdatemechanismisdegraded,comparedtoLMSalgorithm,bythecrudequantizationofgradientestimates.*Thesteadystateerrorwillincrease*TheconvergenceratedecreasesThefastestofthem,Sign-Sign,isusedintheCCITTADPCMstandardfor32000bpssystem. Lecture54.LinearsmoothingofLMSgradientestimatesLowpasslteringthenoisygradientLetusrenamethenoisygradient (nw Jg (nw J=2u Passingthesignals)throughlowpasslterswillpreventthelargeuctuationsofdirectionduringadaptationprocess.LPFwhereLPFdenotesalowpasslteringoperation.Theupdatingprocesswillusethelterednoisygradient +1)= (n)µb Thefollowingversionsarewellknown:AveragedLMSalgorithmWhenLPFisthelterwithimpulseresponse(0)= 1)= +1)=weobtainsimplytheaverageofgradientcomponents: +1)= (n Nnj=nN+1e(j)u (j) Lecture55.NonlinearsmoothingofLMSgradientestimatesIfthereisanimpulsiveinterferenceineither)or),theperformancesofLMSalgorithmwilldrasticallydegrade(sometimesevenleadingtoinstabilty).Smoothingthenoisygradientcomponentsusinganonlinearlterprovidesapotentialsolution.TheMedianLMSAlgorithmComputingthemedianofwindowsize+1,foreachcomponentofthegradientvector,willsmoothouttheeectofimpulsivenoise.Theadaptationequationcanbeimplementedas+1)=*Experimentalevidenceshowsthatthesmoothingeectinimpulsivenoiseenvironmentisverystrong.*Iftheenvironmentisnotimpulsive,theperformancesofMedianLMSarecomparablewiththoseofLMS,thustheextracomputationalcostofMedianLMSisnotworth.*However,theconvergenceratemustbeslowerthaninLMS,andtherearereportsofinstabiltyoccurringwhithMedianLMS. Lecture57.LMSVolterraalgorithmWewillgeneralizetheLMSalgorithm,fromlinearcombinationofinputs(FIRlters)toquadraticcombina-tionsoftheinputs. (n)Tu )Linearcombinerbinerk(n)u(nk)+M1i=0M1j=iw[2]i,j(n)u(ni)u(nj)(3)QuadraticcombinerWewillintroducetheinputvectorwithdimension+1) ...u+1)...u+1)andtheparametervector 0(n)w[1]1(n)...wwM1(n)w[2]0,0(n)w[2]0,1(n)...wwM1,M1(n)TNowtheoutputofthequadraticlter(3)canbewritten (n)T andthereforetheerror (n)T isalinearfunctionofthelterparameters(i.e.theentriesof (n)) Lecture5andtheadaptationproblemdecouplesintwoproblems,i.e.theWieneroptimallterswillbesolutions n)u n)R4w n)u (n)andsimilarly,theLMSsolutionscanbeanalyzedseparately,forlinear, ,andquadratic, ,coe-cients,accordingtotheeigenvaluesof