常用小波函数及Matlab常用指令

1、常用小波函数及Ma t lab常用指令一、常用小波函数与标准傅立叶变换相比，小波分析中用到的小 波函数没有唯一性，小波函数0）具有多样性。 由此而带来的问题是使用不同的小波基分析同一 个问题会产生不同的结果，没有一个选择最优小 波基的统一方法。目前主要是通过用小波分析方 法处理信号的结果与理论分析结果的误差莱判定 小波基的好坏，并由此选定小波基。常用的指导性选择标准有：(1) 0、“、0、/ 的支撑长度。即当时间或频 率趋于无穷大时，上述各量从有限值收敛到0的速 度;(2) 对称型。它在图象处理中对于避免移相非常有用;(3) ©和0 (若存在)的消失矩阶数。对于压缩非常 有用；正则性

2、。对信号或图象的重构获得较好的平滑效 果非常有用。 1、Haar 小波10<x<l/2屮h = -1/2 < 兀 W10 其他waveinfo(,haar,)HAARINFO Information on Haar waveletHaar WaveletGeneral characteristics: Compactly supported wavelet, the oldest and the simplest wavelet.scaling function phi = 1 on 0 1 and 0 otherwisewavelet function psi = 1 on

3、 0 0.5, = -1 on 0.5 1 and 0 otherwise.Family Short name ExamplesHaarhaarhaar is the same as db1Orthogo nalyesBiorthogo nalyesCompact supportyesDWTpossibleCWTpossibleSupport width Filters length Regularity Symmetry12haar is not continuousyesNumber of vanishing moments for psi图:10.500.20.40 60.311.21.

4、40危浏小彼的小彼函数210200.20.40 60.811.21.4在命令窗口输入2、db系列小波i DBINFO Information on Daubechies wavelets. Daubechies WaveletsGeneral characteristics: Compactly supported wavelets with extremal phase and highest number of vanishing moments for a given support width. Associated scaling filters arei minimum-phase

5、 filters.Daubechies db Family Short name Order N1 ExamplesN strictly positive integer db1 or haar, db4, db15OrthogonalyesBiorthogo nalyesCompact supportyesDWTpossibleCWTpossibleSupport width2N-1Filters length2NRegularityabout 0.2 N for large NSymmetryfar fromNumber of vanishing moments for psiNdb少卜波

6、的尺度函数I 3、Biorthogonal(biorNr.Nd)小波系主要特点体现在具有线性相位型，主要应用于信 号和图象的重构中。通常表示为biorNr. Nd形式。Nr=1Nd=1,3,5;Nr=2Nd=2,456,8Nr=3Nd=1,3,57,9;Nr=4Nd=4Nr=5Nd=5;Nr=6Nd=8FamilyShort nameOrder Nr,Nd r for rec on struction d for decompositi on General characteristics: Compactly supported biorthogonal spline wavelets fo

7、r which symmetry and exact reconstruction are possible with FIR filters (in orthogonal case it is impossible except for Haar)BiorthogonalbiorNr= 1 , Nd = 1,3, 5Nr = 2,Nd = 2, 4,6, 8Nr = 3,Nd = 1,3, 5,7, 9Nr = 4 , Nd = 4Nr = 5, Nd = 5Nr = 6 5 Nd = 8Examplesbior3.15 bior5.5Orthogonal （正交）noBiorthogona

8、l（双正交的）yesCompact supportDWTCWTSupport width 2Nd+1 for dec.yes possible possible 2Nr+1 for rec.5max（2Nr52Nd）+2 butFilters length essentiallybior Nr.NdIdIreffective lengtheffective lengthof Lo_Dof Hi_Dbior 1.122bior 1.362bior 1.5102bior 2.253bior 2.493bior 2.6133bior 2.8173bior 3.144bior 3.384bior 3.

9、5124bior 3.7164bior 3.9204bior 4.497bior 5.5911bior 6.817111 Regularity for psi rec. Nr-1 and Nr-2 at the knotsi Symmetryyesi Number of vanishing moments for psi dec. Nri Remark: bior 4.4,5.5 and 6.8 are such that rec on struction andi decomposition functions and filters are close in value.图:bior2.4

10、-解小液的尺度函数510bior2.4构小液的尺度函数bior2.4-解小波的小波函数 4、Coiflet(coifN)小波系由Daubechies构造,N=1,2,3,4,5.具有比dbN更好的 对称性。从支撑长度看，具有和db3N及sym3N具有 相同的支撑长度，从消失矩的数目看，具有和db2N 和symN相同的消失矩数目。图:1.5sifa卜波的尺度函数024681012141618Coiflets coifN = 1,2, .,5 co if 2, co if 4 yes yes yes possible possible General characteristics: Compac

11、tly supported wavelets with highest number of vanishing moments for both phi and psi for a given support width. Family Short name Order NExamplesOrthog onal Biorthogonal Compact support DWTCWTSupport width Filters length Regularity Symmetry6N-16Nnear fromNumber of vanishing moments for psi 2NNumber

12、of vanishing moments for phi 2N-1 5、SymletsA(symN)小波系Symlets函数系由Daub echi es提出的近似对称的小波 函数，是对db函数的改进，N=2，3,，8osym牛、波的尺度函数511'11>01234567 General characteristics: Compactly supported wavelets with least asymmetry and highest number of vanishing moments for a given support width. Associated scal

13、ing filters are nearlinear-phase filters. i Family iShort nameSymlets symN = 2,3, .Order N Examplessym25 sym8Orthog onalBiorthog onal Compact support DWTCWTSupport widthFilters lengthRegularity SymmetryNumber of vanishingyes yes yes possible possible 2N-1 2Nnear from moments for psi NDefiniti on:Fam

14、ilyShort name Orthogo nal Biorthogo nalCompact supportDWTCWT 6、Molet(morl)小波小波函数为：0(兀)=Ce_x2/2cos5x 尺度函数不存在，不具有正交性。morl(x) = exp(-xA2/2) * cos(5x)Morlet morl no no no no possibleSupport widthinfiniteEffective support -4 4 Symmetryyes 7、Mexican Hat (mexh)小波由Gauss函数的二阶导数构成。20(Q =厶/气1 F)2/2V3具有很好的时频局部化

15、能力，尺度函数不存在，不 具有正交性。1 Definition: second derivative of the Gaussian probability density function1 mexh(x) = c * exp(-xA2/2) * (1-xA2) where c = 2/(sqrt (3)*piA1/4)FamilyShort name Orthog onal Biorthogonal Compact support DWTCWTMexican hat mexhnonono no possible infinite -5 5 yesSupport width Effectiv

16、e support Symmetrymexihat小波的小液函数I 8> Meyer小波其小波函数和尺度函 数在频率域定义，为 具有紧支撑的正交小 波。二、小波分析工具箱常用函数介绍1、Cwt功能：一维连续小波变换 格式：(1)coefs=cwt(s,scales,5wname5)(2)coefs=cwt(sJscales/wname7plot,)S为待分析信号；scales为尺度向量：可以为离散值，表示为 31,32,33,.;也可以为连续值，表示为 amin:step:amax；还可以是混合情况，需要将离散 值写前面，连续值写后面a1 ,a2,a3 ,amin:step:amax返回

17、值为小波变换系数矩阵，矩阵的行数为尺度个 数，每一行的值为该尺度小波变换系数在命令窗口输入help cwt,可得指令的功能解释。 help cwt CWT Real or Complex Continuous 1-D wavelet coefficients. COEFS = CWT(S,SCALES,'wname') computes the continuous wavelet coefficients of the vector S at real, positive SCALES, using wavelet whose name is fwname The signa

18、l S is real, the wavelet can be real or complex. COEFS = CWT(S5SCALES；wname7plot,) computes and, in addition, plots the continuous wavelettransform coefficients. COEFS =CWT(S5SCALES；wnamePLOTMODE) computes and，plots the continuous wavelet transform coefficients.i Coefficients are colored using PLOTM

19、ODE.i PLOTMODE = Ivl* (By scale) ori PLOTMODE = 'gib1 (All scales) ori PLOTMODE = 'abslvl' or 'Ivlabs' (Absolute value and By scale) ori PLOTMODE = 'absglb* or 'glbabs' (Absolute value and All scales)I %维连续小波变换I load noissin;I s=noissin(1:100);i ls=length(s);1 w=cwt(s

20、312.12,10.24,15.48,1.2,2:2:10；db 37plot); xlabel('时间')ylabel('变换尺度)Absolute Values of C弟b Coefficients for a = 12.12 10.24 15.48 1.2 2 .时间2、单尺度一维离散小波变换(2) 格式：(1) ca, cd=dwt (x, wname?)_ca, cd=dwt (x, Lo-D, Hi-D) 方式(1)直接对信号在指定的小波形式下进行 分解，CQ为低频系数，cd为高频系数；方式(2)先利用小波滤波器指令wf订ters求取 分解用的低通和高通滤

21、波器，然后将信号通过滤 波器进行分解，可以达到同样的效果。 %单尺度一维离散小波变换； load noissin; s=noissin(1:1000); subplot(411);plot(s) cal ,cd1 =dwt(s,'haar'); subplot(423);plot(ca1) ylabelChaacal)*); subplot(424);plot(cd1); ylabel('haar(cd1)'); Io_d5hi_d=wfilters('haar7d,);i ca2, cd2=dwt(s, lo_d, h i_d);i subplot(4

22、,2,5);plot(ca2)i ylabel('haar(ca2)'); subplot(4,2,6);plot(cd2)ylabel('haar(cd2)');10020030040050060070080090010000(方县泪匸(邑u)脣 一|oo60oeo妣002C3单尺度一维离散小波逆变换idwt功能：单尺度一维离散小波逆变换X = idwt(CA,CD,'wname');X = idwt(CA,CD5Lo_R5Hi_R);X = idwt(CA3CD5'wname',L);X = idwt(CA,CD 丄 o_R,

23、Hi_R 丄)后两种对信号中间长度为L的部分进行重构 %单尺度一维离散小波逆变 换 load noissin; s=noissin(1:1000); subplot(6,2,1); plot(s) title('原始信号) cal Jcd1=dwt(sJ'db2'); x1 =idwt(ca1 ,cd1 ,'db2'); subplot(6,2,5) plot(x1) title('小波重构') errxl max=max(abs(s-x1); errxl =s-x1;subplot(626) plot(errxl) title('

24、;小波重构误差') axis(0,1000,-2e-11,2e-11); lo_d,hi_d,lo_r,hi_r=wfilters Cdb2!);ca,cd=dwt(s4o_d,hi_d);x2=id wt(ca,cd ,lo_r,hi_r); subplot(6,2,9);plot(x2);titleC滤波器重构)errx2max=max(abs(s-x2)errx2=s-x2;subplot(6,2,10);plot(errx2) titled滤波器重构误差 axis(0,1000<2e-11,2e-11);原始信号05001000小波重构ilvwvN05001000滤波器重

25、构05001000賀10小小波重构误差X1O1波器重构误差050010004、小波滤波器wfilters格式：LoD,HiD 丄 o-R,Hi-R=wfilters(twname,)(2) f 1 ,f2=wfilters(twname,type,)i LO_D,HI_D,L0_R,HI_R = WFILTERSfwname') computes four filters associated with the orthogonal or biorthogonal wavelet named in the string 'wname'.LO_D, the decompo

26、sition low-pass filterHI_D5 the decomposition high-pass filter LO_R, the reconstruct!on lowpass filter HI_R5 the rec on structi on high-pass filter F1,F2 = WFILTERS(,wname7type,) returns the following filters:LO_D and HI_D if 'type* = 'd* (Decomposition filters)LO_R and HI_R if 'type'

27、; = 'r' (Reconstruction filters)LO_D and LO_R if 'type' = T (Low-pass filters)HI_D and HI_R if 'type' = (High-pass filters)'type'd'分解滤波器'type'二R重构滤波器'type'=7低通滤波器 'type'二万高通滤 波器举例i lo_d,hi_d,lo_r,hi_r=wfilters(,haar'); figure(1)； subplo

28、t(221)；i stem(lo_d)； title(1o-d of haar1)；i subplot(222)i stem(hi_d)； title('hi-d of haar') subplot(223)； stem(lo)；i title('lo-r of haar*)i subplot(224)i stem(hi_r)i title(*hi-r of haar*)0.8lo-d of db11.520.40.20hi-dof db1hi-r of db11.&0 0.5(>"1 111.521.55、dwtmode功能：离散小波变换拓展模

29、式格式:(1) dwtmode(2) dwtmode('mode')说明：当对信号或图像的边缘进行处理时,需要 信号的边缘进行拓展。拓展模式有三种。该指令 在进行离散小波变换或小波包变换时，进行模式 拓展设定。类型说明补零模式，缺省设定对称延拓模式，即把边缘值进行复制平滑模式，对信号边 缘进行某种平滑处理6、wavedec功能：多尺度一维小波分解(一维多分辨分析函 数)格式:(1) c, 1二wavedec (x, n, ' wnaine?)(2) c, 1二wavedec(x, n, Lo-D, Hi-D)用小波或分解滤波器对信号X进行一维多尺度分解, n为尺度和正整

30、数。输出参数c是由冋,叫皿戸,cdj组成，L是由叫的长度叫的长度,cd/的长度,的长度组成。s：c：L:举例I %多尺度一维离散小波变换; load sumsin;| s=sumsin;I subplot(611) plot(s); title('原始信号') c)l=wavedec(s,3,'db1'); subplot(613)I plot(c);title(信号S3尺度分解');信号阳尺度分解L= 12512525050010007、appcoef功能：提取一维小波变换低频系数 格式:(1) A=appcoef(c,l,wname,N)(2) A=

31、appcoef(c,l,'wname')(3) A=appcoef(c,l,Lo-R5Hi-R)(4) A=appcoef(c5l5Lo-R5Hi-R ,N)说明：该函数是一个一维小波分解函数，用于从小波分解结构C,L中提取一维信号的低频 系数。格式（1）计算尺度N时的低频系数，格式（2）用于提取最后一个尺度的低频系数,格式（3）和（4）用滤波器提取低频系数。举例%提取一维小波变换低频 系数；load leleccum;s=leleccum(1:2000) subplot(421)plot(s);title('原始信号') c,l=wavedec(s535

32、9;db1'); cal =appcoef(c,l,'db1 ',1);i subplot(445)i plot(cal) ylabelfcal*)；i ca2=appcoef(c,l5'd b12);i subplot(4,8,17)i plot(ca2);ylabel('ca2');原始信号enn8、Detcoef功能：提取一维信号小波变换高频系数格式：(1) d=detcoef(c,l,N)提取N尺度的高频系 数。(2) d=detcoef(c,l),提取最后一尺度的 高频系数。i subplot(445)i plot(cdl)i ylab

33、el('ccH');i cd2=detcoef(c5l,2);i subplot(4,8,17)i plot(cd2); iylabelCcd2');举例%提取一维小波变换高 频系数；load leleccum;s=leleccum(1:2000) subplot(421) plot(s);title('原始信号') c,l=wavedec(s535,dbr); cd1=detcoef(c,l51);原始信号0500九、Waverec功能：多尺度一维小波重构 格式：(1) x=waverec(c, I, JwnameJ)(2) x=waverec(c5l

34、5Lo-R5Hi-R)(3) x= waverec (wavedec (x,N/wavename5)/ wavename5)】说明：该函数用指定的小波函数或重构滤波器对 小波分解结构(C,L)进行多尺度一维小波重构。举例:«subplot(312) plot(a) title('重构信号) err=s-a; subplot(313) plot(err) titlef 误差')1 %多尺度一维小波重 构；I load leleccum;I s=leleccum(1:3920)i subplot(311) plot(s); titlef原始信号')i c5l=wa

35、vedec(s53,'d b5J;i a=waverec(c5l,'db5')原始信号0JO050010001500 重嚮啓 T 2500300035004000I十、upwlev功能：单尺度一维小波分解的重构格式：(1) nc, nl, ca=upwlev (c, 1, ? wname?)(2) nc, nl, ca=upwlev (c, 1, Lo-R, Hi-R)说明：该函数用于对小波分解结构C,L进行单尺 度重构，返回上一尺度的分解结构并提取最后一 尺度的低频分量。%单尺度一维小波分解的重构；load sumsin;s=sumsi n;subplot(611)p

36、lot(s);title(源始信号')c,l=wavedec(s,3,'db1');subplot(613)plot(c)title('尺度3的小波分解结构)等效于质J©阳vedec(s越触站)xlabel('尺度3的低频系数和尺度3、2、1的高频系数')nc,nl=upwlev(c,l,'db1'); subplot(615);plot(nc);title('尺度2的小波分解结构Jxlabel('尺度2的低频系数和尺度2、1的高频系数')原始信号50-5IIIIIIIII010020030040

37、06006007008009001000g |iiiiiii_ '01002003004005006007008009001000尺度了的低频系数和尺度弘N 1的高频系数尺度3的小波分解结构50-50AAAA/VWVWWWWVWVW尺度2的小波分解结构2003004005006007008009001000尺度2的低频系数和尺度N 1的高频系数L= 1251252505001000十一、Wrcoef功能:对一维小波系数进行单支重构I 格式：x=wrcoef(，type，,c,wname；N) x=wrcoef(£type,c,l,Lo-R,Hi-R,N)(3) x=wrcoe

38、f(ttype,c,l,wname,) x=wrcoef(ttype,c,l,Lo-R,Hi-R)说明：对一维信号的分解结构C,L用指定的小波 函数或重构滤波器进行重构。当'type二a，时，对 信号的低频部分进行重构，此时N可以为0；当 'type二cT时，对信号的高频部分进行重构，此时 N为正整数。原始信号%对一维小波系数进行单支重构; load sumsin;s=sumsi n;subplot(611) plot(s);title。原始信号) c5l=wavedec(s5 5,'sym4); a5=wrcoef(,a,c,l/sym4,5);subplot(613

39、) plot(a5) title('低频部分重构信号') a51 =wrcoef(,dc,l/sym4,J5);subplot(615) plot(a51) title('高频部分重构信号')5 !jjjjjjjj0仍 YYYY 单杯巧Iiiiiiiii01002003004005006007008009001000低频部分重构信号22I|iiiiiii'01002003004005006007008009001000高频部分重构信号十二、upcoef功能:格式:一维系数的直接小波重构(1) y=upcoef('0',x,'wna

40、me',N)(2) y=iipcoef('0',x,'wname',N丄)(3) y=upcoefCO,x,Lo-R,Hi-R,N)(4) y=upcoef(<0,x,Lo-R,Hi-R,N,L)(5) y=upcoef(<0,x,wname,)(6) y=upcoefCO,x,Lo-R,Hi-R)说明：该函数用于一维小波分析，它用来计算向 量X （信号系数）向上N步的重构小波系数，N为正 整数。如果0 = 2,对低频系数进行重构；如果0 = d,对高频系数进行重构；对于（2）和（4）,则 是对向量X中间长度为L部分进行重构。原始信号尺度1的

41、低频系数宛1向上一步重构信号cd1=detcoef(c,l,1);scd1 =upcoef('d',cd1 ,'db6',1); subplot(626);plot(scd1); title('尺度1的高频系数ccM 向上一步重构信号')； axis(0,2000,-20,20); f1,f2=wfilters('db6','r'); ca2=appcoef(c,l/db6',2)； 10Qlsca2=upcoef('a',ca2,f1 ,f2,2); 'subplot(629) ;

42、plot(sca2);title('尺窿2的低频系数ca2 向上2步重构信号，)；axis(0,2000,200,600); Load leleccum; s= leleccum(1:2000); Plot(s) titlefM 始信号)； c,l=wavedec(s,3,'db6'); cal =appcoef(c,l,'db6',1); seal =upcoef('a',ca1 ,'db6',1); subplot(622) ;plot(sca1); title('尺度1的低频系数cal 向上一步重构信号

43、9;)； axis(0,2000,200,600); seal I=upcoef('a',ca1 ,'db6',1I subplot(625) ;plot(sca11); title(lca1 向上一步只取 1000 点重构信号')；axis(0,2000,200,600);原始信号尺度1的低频系数宛1向上一步重构信号600向上一步只取1000点重构信号尺度1的高频系数旳1向上一步重构信号400600100016002000尺度2的低频系数向上2步重构信号原始信号尺度1的低频系数宛1向上一步重构信号十三、wpdec功能：一维小波包的分解格式：(1) T=

44、wpdec(X, NJwname； E, P)说明：wpdec是一个一维小波包分解函数。I它根据小波函数'wname?(参见wfilters)、炳标准E和参数P对信 号X进行N层小波包分解，并返回小波包分解结构T, T为树结构。1 E is a string containing the type of entropy (see WENTROPY):E = 'shannon； 'threshold1,五orm； log energy*, 'sure "user'.I P is an optional parameter: 'shanno

45、n* or log energy1: P is not usedI 'threshold* or 'sure': P is the threshold (0 <= P) ,norm,: P is a power (1 <= P)'user':P is a string containing the name of an user-defined function. load noisdopp; x=noisdopp; t=wpdec(x,35'db1 ','shannon'); plot(t)0.9 -0.8

46、-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -01110.81Tree Decompositiondata for node: or (0,0).108 -6 -4 -20-2-4 -6 -8data for node: or (0,0).data for node: or (0,0).1°2004006008001000Tree DecompositionTree Decompositiondata for node: (2) or (1,1).3r2 -1 -0-1-2-3 -4 -100 200300400500data for

47、node: or (2,0).20-15 -'105 -0-5 -10-15 I11150100150200250Tree Decompositiondata for node: (5) or (2,2).3 I 1>210-1-2-341111150100150200250data for node: or (2,3).4111132 -1 -0 -1-2-3iiii50100150200#data for node: or (3,0).20 1 1 1 1 1Tree Decomposition#data for node: or (3,0).20 1 1 1 1 1#dat

48、a for node: or (3,0).20 1 1 1 1 1(0,0)15 -10 -5 -0-5 -10 -15 -20 -25111120406080100121十四、wprec功格举能：一维小波分解的重构x=wprec(t)load noisdopp; x=noisdopp; figure(1)； subplot(211)； plot(x) titleCJM 始信号') t=wpdec(x,3,'db1'shannon'); x1=wprec(t) subplot(212)plot(x1)title(重新信号')原始信号重构信号十五、wpco

49、ef功能：计算小波系数格式：(1)x=wpcoef (t, n)I (2) x=wpcoef (t)说明：wpcoef是一个一维或二维的小波包 分析函数。格式(1)返回与节点n对应的 系数。如果n不存在，x = ；1 x=wpcoef (t)等效于x= wpcoef (t, 0)load noisdopp; x=no isdopp;figure(1) subplot(311) plot(x) title(源始信号) t=wpdec(x,3,'db1 ','sha nnon'); cfs21 =wpcoef(t,2,1 ); cfs22=wpcoef(t,2,2)

50、;cfs31 =wpcoef(t,3,1 ); cfs32=wpcoef(t,3,2);subplot(323); plot(cfs21); title('小波包2,1的系数J; subplot(324); plot(cfs22);title('小波包2,2的系数J; subplot(325); plot(cfs31);title('小波包3,1的系数*);subplot(326); plot(cfs32);title('小波包3,2的系数J;原始信号10小波包即的系数十六、wprcoef功能：小波包分解系数的重构;格式：x= wprcoef (t, n)说明：

51、wprcoef是一个一维或二维的小波包 分析函数，计算节点n的小波包分解系数的 董初信号。1 X= wprcoef (t) = wprcoef (t, 0)该函数一次只能对一个节点进行重构，不能 同时对多点进行重构，可以通过多次调用 实现。 load noisdopp;x=noisdopp(1:1000); figure(1) subplot(311) plot(x) title('原始信号')i t=wpdec(x53,'db1 Vshannon*);i refs=wprcoef(t,2,0);i cfs21 =wpcoef(t,2,0); subplot(312) plot(cfs21) titled小波包节点(2, 0)系数') subplot(313) plot(rcfs)titled重构小波包节点(2, 0)信号)