信号处理07b

17§7-4.SignalRepresentation:AMultiresolutionAnalysis7.4.1.SpacesVnandPropertiesVn=thesetoffunctionwhoareconstanton)21,2[nnkk,kZ)317(),2(22ZkktspannnPropertiesVinbasislorthonormaanisZkktBasisVMaximalityRLVDensityneachforVVInclusionVtfVtfScalingZnnZnnnnnn),(:}0{:)(::)2()(:2117.4.2.SpacesWnandPropertiesnnnnnnWinbasislorthonormaanisneachforZkktwhereZkktspanW),2(2),2(222PropertiesnZnnnnnnnnnnnnn1221121limlimlim)(不仅是简单的叠加，而是重组，合成向量Daubechies小波产生的Wn比Haar的要好Note:VnandVmarenotorthonormal;ButnnmnWVAndWW187.4.3.WaveletDecompositionofSignalsFromthesecondpropertyonlastsection,oneconcludesthatZnZkktnn,),2(22isanorthonormalbasisinL2(R).Henceforanyf(t)L2(R),)327()(ˆ)(ˆ),()(,,nkknkntttftfwhere)2(2)(ˆ2,kttnnkn(Haarbasis)Formanypracticalsignals,thereoftenaremanysmallwaveletcoefficients.SothatthesignalcanberepresentedwithtrancatedwaveletserieswithfewertermsthanitsFourierseriesrequires.[seetheaboveexampleon7.2.2s(t)=exp(-10t)sin(100t)]HaarBasisforL2[0,1)TheHaarbasisn,kgivenon7.4.3isforL2(R).Theycanbeusedtodefinedperiodicwavelets:)337())(2(2)(~2,ZjnnknkjttTheoremFunctions1andkn,~withn0,k=0,1,...,2n-1formanorthonormalbasisforL2[0,1).Similarly,periodicscalingfunctionscanbegeneratedas)347())(2(2)(~2,Zjnnknkjtkforn0,andk=0,1,...,2n-1.7.4.4.FastHaarTransformConsiderasignal11)(nnnWVVtf.Wewanttodecomposef(t)intoitemsinVn-1andWn-1.Thisistosay:GivenZkknnkttf)(ˆ)(,)(,wewanttocompute)1(nkand)1(nksuchthat)357()(ˆ)(ˆ)(,1)1(,1)1(ZkknnkZkknnktttf19where)2(2ˆ),2(2ˆ2,2,ktktnnknnnknNoteZkZkknkn,ˆ,ˆ,1,1formsanorthonormalbasisforVn.Therefore,)2(2),(,)2(2),(12)1()1(12)1()1(ktttfkttfnnnknnnkTheDEandWEleadtoabeautifulsetofrelationsbetweenthesecoefficients.Recall)()12()2()(DEttt)()12()2()(WEttt(平移)))12(2()22()(ktktkt))12(2()22()(ktktkt(伸缩)))12(2()22()2(1ktktktnnn))12(2()22()2(1ktktktnnn(长度归一化)))12(2(2)22(221)2(22212)1(ktktktnnnnnn))12(2(2)22(221)2(22212)1(ktktktnnnnnn12,212,21,1ˆˆˆknknkn12,212,21,1ˆˆˆknknkn20故有如下迭代公式)(1221)(22112,212,21,1)1(ˆ,ˆ,ˆ,nknkknknknnkfff)(1221)(221)1(nknknkNowweobtain)367()(1221)(221)1(anknknk)367()(1221)(221)1(bnknknk此即S.Mallat1989年给出的多分辨率分析的方程(MRA)或为)377()(12)(221212121)1()1(nknknknkW此处的W为正交矩阵，即(MRA)矩阵形式，它阐明了正交双方的可逆计算。AFilter-BankImplementationofMRAIfwereferindex2k+1as“thepresenttimeinstant”,then2kisanindexrepresentingthe“immediatepast”.So,theequations(MRAseethelastsection)canbeimplementedasfollows:21TheMRAcanbecarriedoutwithnlevelsforsignalsoflengthN=2n.Thisdecompositionprocessis“reversible”:Onecanuse)1()0()0(,,,ntoperfectlyreconstructtheoriginalsignal)(n.Again,weusethe(DE)and(WE)toaccomplishthereconstruction:)ˆ21ˆ21()ˆ21ˆ21(ˆˆˆ12,2,)1(12,2,)1(,1)1(,1)1(,)(knknknkknknknkknknkknknkknknkIfk=even:(比较相同基函数的系数应相等))387(]0[]0[21)1(2/2121)1(2/21)1(2/21)1(2/21)(anknknknknkIfk=odd:)387](0[]0[)1(2/)1(2121)1(2/)1(2121)(bnknknk此处相当于补零形成向量，再通过高、低通滤波器得更精确的系数。22Thissuggestsafilter-bankimplementationforthereconstruction:first)1(nand)1(nareup-sampledbypaddingazerobetweeneachpairofsamplestheup-sampled)1(nand)1(narethenfedintoalowpassandahighpassFIRfilters,respectively,andthefilteredsequencesarecombinedtogenerate)(n.★TheimpulseresponseofH0(z)andH1(z)matchthecoefficientsif(DE)and(WE),respectively;★ThefiltersH0(z)andF0(z),andthefiltersH1(z)andF1(z)aremirror-imagesymmetric.Then-levelMRAisnowcompletewithadecompositionphaseandareconstructionphase:23§7-5.ImprovedWavelets:AFilter-BankApproach7.5.1.WaveletsviaDilationEquationsTodiscovernewandimprovedwavelets,letussupposethat)(tisabetterscalingfunction.UsingMRA,werelate)(ttothespacesVn:)397(),2(22ZkktspanVnnnSince10VV,)(tcanbeexpressedas:kkDEktct)()2(2)(As10VW,weshouldhavekkWEktdt)()2(2)(Hereweconsiderthecasewhenonlyfinitenumberofsck'andsdk'arenonzero:)()2(2)(0DEktctKkk)()2(2)(0WEktdtKkk24Ifwerequire)}({ktorthonormal,then1,0,)]2()][2([2)()()(2mccdtltcktcdtmttmkmkkllkkInaddition,weimposethe“alternatingflip”assumptiononsdk':KkcdkKkk,,1,0,)1(Since0)(dttWeobtainKkkKkKkkKkcdtktcdtt00)1(21)2()1(2)(011,0),(2mmcckmkk)2(0)1(0KkkKkcExample-1(K=1case)Inthiscaseonehasonlytwononzerocoefficients:c0andc1satisfying25)(210101012120WaveletHaarccccccExample-2(K=3case)Inthiscase,nonzerocoefficientsare:c0c1c2andc3satisfying)(0)(0)(10123312023222120cccccbccccaccccOnecanimposeonemoreconditiononsck'inordertomakethewaveletmore“regular”(smooth):0)(dtttwhichleadsto)'(00dkdKkk)(032012dcccHereishow(d’):26kkkkkkkkkkkkkkkkkkkddkdddttdtttkddtktkktddtkttddtkttd