信号处理071

1第七章小波与信号处理WAVELET&SIGNALPROCESSINGWITHAPPLICATIONS小波是应用数学的一个新领域80年代法国地质石油工程师J.Morlet提出Wavelet的概念，通过物理直观概念和信号处理的实际需要建立了经验的反演公式。但当时的数学家并未认可-1807年Fourier热力学工程师－FourierKingdom1986年数学家Mayer偶然构造出一个真正的小波基函数，并与Mallat合作建立了构造小波基函数的统一方法－多尺度分析方法及Mallat金字塔算法，小波才得到真正的发展。Daubechies-比利时女数学家－小波十讲－Daubechies小波近代科学400年来－从复杂事物中寻找简单规律－几乎忘记了非线性用局部线性化来取代非线性，用片面的美掩盖了整体的美。一叶障目不见泰山。自然世界的本性，丰富多彩、神奇多样化都来源于非线性－分形、Fractals-Choas。1963年气象学家洛伦兹用计算机仿真大气模型，仅忽略了一些看起来微不足道的参数，但结果却大相径庭。－科学别忘记了非线性小波经过近十年的研究，数学体系已经建立，理论基础扎实，应用广泛。小波分析，时间频率局部变换，多尺度分析，解决了传统的傅立叶变换难以解决的许多难题－数学显微镜，信号分析，数据压缩、模式识别、特征提取。AGeneralCommentonInformationRepresentation——APreludeinformationF(u)Today'scomputingdevicesarelargelylimitedtoprocessf(u)typeofinformation.Wherefrepresentstheinformation,anduisageneralindexatwhichtheinformationisdescribed.examplesifindexu:timet(speech,music,etc.)positionx(stillimage,etc.)(time,position)(t,x)(video,etc.)otherphysicalparameters...indexu:continuousversusdiscreteindexuversusinformationf:whichoneismoreimportantAcharacterofusefulinformation:redundancy.2TheNoteswerepreparedbasedonthefollowingreferences:1.J.S.walker,Fourieranalysisandwaveletanalysis.NoticesofAMS,vol.44,NO.6,pp.558-670,June/July1997.2.C.Mulcahy,plottingandschemingwithwavelets.Math.Magazine,b9,Dec.1996.3.G.strangandT.Nguyen,WaveletsandFilterBanks,Wellesley-CambridgePress,1996.4.W.-S.Lu,ELEC639ACourseNotesonWavelets,DeptofElec.&Comp.Engn.,Univ.ofVictoria,Jan1997.§7-1.什么是小波WhatAreWavelets?Awaveletisafunction(t)whoseaverageisZero,i.e.,)17(0)(dttExamples:WaveletsthatareusefulinDSPandotherapplicationsoftenpossessadditionalfeatures:Generate(viadilationandtranslation)orthonormalorbiorthonormal(basis)systemsinL2LocalbothintimeandfrequencydomainFastDecomposition/ReconstructionAlgorithmsVanishingmoments)27(1,...,1,00)(Kkdtttk3§7-2.SignalRepresentation:FourierSystemsdeFtfdtetfFtjtj)(21)()()(7.2.1SystemFeaturesAnOrthonormalSyteminL2(R)generatedwithone(Mother)function:ejtOffersFrequencyAnalysisofSignalsandDSPsystemFastAlgorithms(FFT,DFT,DCT,2D-DFT,……)InfiniteResolutioninFrequencydomainZeroResolutioninTimedomain不能表示突变信号，局部时间变化的信号7.2.2Application:SignalApproximation(LossyCommpression)Considerdiscretesignal)37()1024(1,,1,0MMkMkFfkwhere)47()(25)6.0(10tetFTheDFTof{fk}isgivenby)57(2,,0,,12,1ˆ10/2MMkefMfMlMkljlk1024M;1-M0,1,...,kMkFfk4Anapproximationof{fk}istheinverseDFTof)67}(2,,0,,12,ˆ{NNkfkCompressionRate=M/NThefigurebelowshowsaplotoftheoriginalsignalF(t),andthreeapproximatesofF(t)withN=512,256,and128.NoticetheGibbsoscillationsforN=256and128.7.2.3Application:SignalDenoisingConsideraModulatedSignal)77(]1,0[]eeee280t[ecos2s(t)222227/8)-(t640-6/8)-(t640-4/8)-(t-6403/8)-(t-6401/8)-(t-640trepresentingthebitsequence10110115Supposethesignalistransmittedthroughachannelthataddsacertainamountofnoisewhosefrequencyrangeisboundedby200HZ.Thenoise-contaminatedsignalcanthenberecoveredusingabandpassfilteringtechnique:STFT短时傅氏变换(Gabor变换)☆在频域和时域均有局部化功能;☆但是、其时频窗口的大小是固定的，窗口没有自适应性，只适合分析所有特征尺度大致相同的的各种过程。☆此外、其离散形式没有正交展开形式，难以实现高效算法。短时傅氏变换，只是向时频分析走出了重要的一步。dtetgtfGftj)()(),(Guass窗ataeatg4221)(参考书“子波变换与子波分析”赵松年窗口函数elsewheretttg,011,cos1)(ttgttgth8cos)6(4cos)4()(1ttgttgth6cos)6(4cos)4()(26易见、当两个频率分量之差为2时，STFT可以分辨出来而当两个频率分量之差为1时，STFT就不能分辨了。78§7-3SignalRepresentation:TheHaarWavelet7.3.1HaarScalingFunctionandHaarwaveletTheHaarscalingfunctionisdefinedby)87(0)1,0[1)(elsewherett尺度函数property:)97(1)(1)(2dttdttTheHaarwaveletisdefinedby)107(,0)1,[,1),0[,1)(2121elsewherettproperty:小波函数)117(1)(0)(2dttdtt7.3.2HaarScalingFunctionSeenFromSignalApproximationConsideracontinuous-timesignalf(t)anditsapprox.)127()()()(0kktkftfwhere(t)theHaarScalingfunctionWeseef0(t)asafinite-resolutionrepresentationoff(t)inthespaceV0spannedby{(t-k),kZ}Note:V0=thesetofallfunctionsthatarepiecewiseconstanton[k,k+1),kZ9and)137(,),(,0,1)()(lorthonormaisZkktlklkdtltktTwoproblems:(a)Whatiff0(t)asanapproximatelyisnotaccurateenough?(b)Whatiff0(t)istooaccurate?WecanmovefromV0uptospaceswithhigherresolutioninthecaseof(a).Orinthecaseof(b)wemovetospaceswithlowerresolution.Case(a)Anaturalwaytoobtainmoreaccuraterepresentationoff(t)istoincreasethenumberofsamples.Forexample,ifthesamplingrateisdoubled,thenweobtain)147()2()()(21kkktftfkkktftf)2()()(2110Where)157(,0),0[,1221elsewherettHereweseeaspaceV1:ZkktspanV),2(21isorthonormalCase(b)Wecanforinstanceusehalfofthesamplingratetorepresentthesignaliff0(t)is“tooaccurate”:)167()2(]2)12()2([)(1kktkfkftf11Where)177(,0)2,0[,12elsewherettNotethespaceV-1)187(),2(21ZkktspanVAcomparisionof:101,,VVV)197(101VVV§7-3HaarwaveletseenfromsignalreconstructionLetusconsiderasignaloffinitelengthinspaceV1:Weapproximatef(t)withasignal00)(Vspaceinta)217(2,,2,2)(12321000NNffffffata00)(Vta;lengthof20NaAstheaveragingisakindoflowpassfiltering,