连续小波变换及其工程应用

139Daubechies{hk}{gk}MallatS.G.MallatAWaveletTourofSignalProcessing(SecondEdition)LaplaceHermitian6.16.1.1(harmonicwavelet)D.E.NewlandL2(R)FFTIFFTwe(t)wo(t),We(ω)=⎩⎨⎧04/1ππωπ4||2≤(6.1.1)Wo(ω)=⎪⎩⎪⎨⎧−04/4/ππiiπωππωπ4224≤−≤−(6.1.2)i=1−6.1.1a,b(a)We(ω)(b)Wo(ω)(c)W(ω)6.1.1We(ω),Wo(ω)W(ω)ω-4π-2π4π1/4π02πWe(ω)-2π0ω-4πi/4π4π2πWo(ω)ω4π02π1/2πW(ω)14004π16π32π8π1/2π,j=02π1/8π,j=21/16π,j=31/4π,j=1W(ω)=We(ω)+iWo(ω)W(ω)=⎩⎨⎧≤0422/1πωππ(6.1.3)6.1.1cW(ω)w(t)=we(t)+iwo(t)W(ω)w(t)=[exp(i4πt)–exp(i2πt)]/i2πt(6.1.4)(6.1.4)6.1.2(a)(b)6.1.2j,k∈Zw(2jt-k)=[exp(i4π(2jt-k))-exp(i2π(2jt-k))]⁄i2π(2jt-k)(6.1.5)(6.1.4)k1/2jw(t)v(t)v(t)=w(2jt–k)(j,k∈Z)6.1.3Re(w(t))ttIm(w(t))141v(t)V(ω)=∫∫+∞∞−−−+∞∞−−=dtektwdtetvtjjtj)2()(ωωp=2jt–kt=(p+k)/2j,dt=2-jdpV(ω)=1/2je–jωkW(ω/2j)j6.1.3w(t)w(2jt–k)(j,k∈Z)w(t)w(2jt–k)=∫+∞∞−−dtktwtwj)2()(_L2w(t)w(2jt–k)W(ω)V(ω)=ωωωdVW)()(∫+∞∞−j≠0W(ω)V(ω)w(t)w(2jt–k)W(ω)V(ω)=0w(t)w(t–k),(k≠0,k∈Z)w(t)w(t–k)=ωωωωdeWWkj−+∞∞−∫)()(=ωπωππdekj−∫42241=0w(t)6.1.56.1.31/26.1.2NewlandL2(R)x(t)∈L2(R)x(t)∑∑+∞−∞=+∞−∞=−=jkjkjktwa)2(,142aj,kx(t)),()2(),(,Ζ∈−=kjktwtxajkjNewlandNewlandFFTIFFTx(r)r=0,,N–1,N=n2as,s=0,,N–1Fs=FFT(x(r))asFsIFFTas=IFFTFss=]2[j]2[j+1]2[1+j1aN-s=sa(6.1.6)j=10log2(N/4)6.1.4166.1.4asasasj6.1.5jlog2(N/4)2jasaN/2|as|=|aN-s|a0,x0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15F0F1F8F15F2F3F4F5F6F7F9F10F11F12F13F14a0a1a8a15a2a3a4a5a6a7a9a10a11a12a13a14FFTIFFT143a1,a(N/2)-16.1.5asj6.1.3WaveletTime-FrequencyMap6.1.6jasjasas|as|2lg|as|26.1.6Parsevalxras22/20102||||||1NNrraaxN+=∑−=∑∑−=−−+−=++120222220))|||(|21(jjjkkNknjjaa(6.1.7)|as|26.1.7x(r)=sin(2π15tr)(r=0,…,511tr=r/320)Newland6.1.56.1.6a0,a1,a2,a3,a4,a5,a6,a7,…a(N/2)1,j=1,0j=1,j=2,j=log2(N/4)lg|as|2jt1446.1.7x(t)6.1.4XYDaubechiesMallatWickerhauseMallatbox-like⎪⎩⎪⎨⎧∈−=∧0],2[2)(21)(,nmmnnmππωπωψ(6.1.8)m,njn=2mm=0n=16.1.2t/s145Daubechies90DaubechiesLemare-MeyerDaubechiesDaubechiesMallatMallatXY5.3)(tψ)(tx(5.3.6)()()()dxWxtttτψτ+∞−∞=−∫(6.1.9))(txj)(,tnmψ∫+∞∞−−=dtttxnmWnmx)()(),,(,τψτ(6.1.10))()(),,(,ωψωωnmxxnmW∧∧∧⋅=(6.1.11)),,(ωnmWx∧Wx(m,n,τ))(ω∧x)(tx)(txFFT()xWω∧IFFTm,nj6.1.8146FFTIFFTx(r)Fs6.1.8jx(t)fsfN=fs/2m,n6.1.96.1.9m,n6.1.46.1.4166.1.100,Mallatx(t)∈L2(R)m=0n=fNmfN/2nfNnmfN/2fN/4n(m)mnfN/40FFTIFFT()tx(),,xWmnτ()ˆ,,xWmnω()ωxˆ(),ˆmnψω147Mallatj2jfN/2jm=i×fN/2jn=(i+1)×fN/2j6.1.11:6.1.116.1.56.1.6m=0n=fNn(m)fN/2nm0fNn(m)n(m)n(m)mnfN/20fN3fN/4fN/4···············148F()tβ()()rtrtHββ−=0,Hr(6.1.12)rH()tψ(){},tbaψ()⎟⎠⎞⎜⎝⎛−=−abtatbaψψ21,0,,≠∈aRba(6.1.13)ab(6.1.12)(6.1.13)ra6.1.7XXjx,)(⊂nnRnR∆∆N∆X⎟⎟⎠⎞⎜⎜⎝⎛−=→∆∆∆lglglim0NdB(6.1.14))(jxt∆0→∆∆∆k+∈Zk∆kN∆kX10.3.210.3.3(){}{}∑=++−+−+−++−+−+−−=kNjkjkjkjkkjkjkjkxxxxxxkP/11)1(2)1(1)1(1)1(2)1(1)1(0,,,min,,,max∆kNj/,2,10=0,,2,1NKKk=(6.1.15))(jx∆kN()∆∆=∆kkPNk+1(6.1.16)1491∆kN∆klg∆kNlg(6.1.14)()BdkkN−∆∆~21,kk∆klg∆kNlgbkdNBk+∆−=∆lglg21kkk≤≤(6.1.17))(jx()∑∑∑∑∑−+−−+−−=∆∆221212lglg)1(lglgloglg)1(kkkkNkNkkkdkkB21kkk≤≤(6.1.18)Bdli)(,nxililBd,12ilBd,6.1.86.1.12CO2XY6.1.12(a)X6.1.12(b)Y6530r/min2000Hz1024(a)X(b)Y150(c)0,0O(d)0,0Bd6.1.126.1.12(c)kjBd,jkOj,k6.1.12(d)0,01.384Bd=6.1.13(a)(b)XY2006.1.13(c)06.1.13(c)6.1.13(d)3536.10,2=Bd(a)X2logNr151(b)Y2(c)00,2O(d)00,2Bd6.1.13XY206.1.14(a)(b)XY301016.1.14(c)(e)6.1.14(c)5.4.3(c)2601.10,3=Bd6.1.14(d)3501.11,3=BdlogNr152(a)X3(b)Y3(c)00,3O(d)00,3Bd153(e)11,3O(f)11,3Bd6.1.14XY3016.2LaplaceHaarMorletMexico-HatDaubechiesLaplaceStrangG.LaplaceLawrenceC.FreudingerLaplaceLaplaceLaplaceLaplace154LaplaceLaplaceLaplaceLaplaceLaplaceLaplaceLaplace6.2.1LaplaceLaplace()()()()[]⎪⎩⎪⎨⎧+∈==−−−−−,0,,,,,21stjtWteAettττψτζωψτωτωζζγ(6.2.1){}τζωγ,,=τζω,,+∈Rω[)+⊂∈R1,0ζR∈τAsWfπω2=f{}τζγ,,f=LaplacefHzLaplaceζLaplaceLaplaceγψ{}0,08.0,2=γ5=sWsγψ6.2.1γψ()γψRe()γψIm()γψRe()γψIm155LaplaceLaplaceγψ()∫∞∞−∞=ωωωψγψdC2ˆ(6.2.2)()ωψγˆ()ωψγ{}0,1.0,1=γ()γψRe6.2.2(a)(b)()γψRe{}τζγ,,11f={}τζγ,,22f=0,21≠γγψψ(6.2.3)21,γγψψLaplace6.2.1Laplace(a)(b)6.2.2Laplace()γψRest()γψRestst1566.2.2LaplaceLaplaceLaplaceLaplaceLaplaceF,ZT{}{}[){}121212,,,,,,,0,1,,,,,mnpFfffRmZZRnZTRpZζζζτττ+++++⎧=⊂∈⎪⎪=⊂∈⎨⎪=⊂∈⎪⎩∩(6.2.4)FZTΓ=××LaplaceγψΨ{}(){}:,,,:,,ftfFTγΨψγΓψζτζΖτ=∈=∈∈∈(6.2.5)γψLaplaceΨ()txS()txLaplaceγψSγψ()()()θψψγγcos,22xtxt=(6.2.6)()txγψ0=θγκ()txγψ()()22,2xtxtγγγψψκ=(6.2.7)γΓ∈γκFTΓΖ=××τ()txγψτγκ()τκ157(){}max,,τζτγζκκτκfFf==Ζ∈∈(6.2.8)τγκτγκζ,fτγκ()τκLaplaceγψsW6.2.72()txγψ()1=τκ()[]1,0∈τκ()τκτ(){}Ζ∈∈=ζζκτγτ,:,FffP(6.2.9)τ()tx()()τκτ,()()τκτ′′,()txτ′sW()τκ′Laplaceγψfζ6.2.76.2.8()()τκτ,fζLaplace3()tx()()()()()()⎪⎩⎪⎨⎧≥+−=−−−000021,01.0,2sin00200tttntttnAttfetxnttfππζζ(6.2.10)()tn01.0=nA010Hzf=04.00=ζst00=[]5,5−200Hz()tx6.2.3.1(a)LaplaceΨΤ×Ζ×=ΓFMatlab{}20:5.0:5=F{}{}{}9.0:1.0:3.02.0:005.0:005.0=Ζ{}5:1.0:5−=ΤΖΨsW=4s200Hz6.2.3(b)(c)(d)(6.2.10)()txLaplace(b)τ()τκ(c)(d)()τκLaplacefζ1586.2.3()txLaplace6.2.3(b)()τκ0.5(b)()τκ0=τ6.2.3(a)6.2.3(c)00==tτ10Hzf=6.2.3(d)04.0=ζfζ()tx0f0ζ6.2.3(b)st6.3−=st1=f10Hzζ0ttζ0.040.6sζ6.2.8()τκ6.2.400==tτ()txγψτγκτP6.2.96.2.4()txγψτγκ0=τst()a()b()c()dHzfζ1596.2.410=fHz04.0=ζ6.2.9fζLaplaceγψ10=fHz04.0=ζ0=τ()tx()txγψ6.2.4fζ(6.2.10)()tx6.2.16.2.1()txnA*0.10.51.01.52.0()()τκmax0.9210.5000.2740.2080.150/Hzf10.0010.0010.0010.0010.00ζ