现代数字信号处理Advanced Digital Signal Processingch5 wavelet

1、advanced digital signal processing(modern digital signal processing)chapter 5 time-frequency analysis and wavelet transform general expression *( )( , )( ), ( , )ff t gt dtf tgt( , ): basis functiongt5.1 linear transforminner product wave & wavelet transformnwaves( ):,0,()0, ()0g ttgg waves are

2、non-compact (infinite) support functionsnon-compact support function: the functions extend to infinity in both directions.nwavelets( ), ( , )0, other g tta btwavelets are compact (finite) support functions. they vary with frequency as well as position compact support function: the functions are in a

3、 limit duration.nwave & wavelet transform (, ) , (, )0 is usually requirediigtgt dtwaveswaveletswave transform(wide-sense) wavelet transformnorthogonal transform northogonal basis functionnorthogonal (orthonormal) transform*0, (, ), (, )(, )(, ), ijijiijgtgtgt gt dtcij( , ): orthogonal (orthonor

4、mal) basis functiongtif c=1, then g(,t) is orthonormal basis function.nunderstanding of the (orthogonal) transform nintuitive interpretation of orthogonal transform for a given i, ()( ), (, )iiff tgt if f(t) is orthogonal with g(i,t), then f(i)=0, i.e. there is no component corresponding to the basi

5、s g(i,t) in f(t). otherwise, the components of f(t) corresponding to the basis g(i,t) will compose f(i) in space. on the other hand, the components of f(t) corresponding to the basis g(i,t) is orthogonal with any basis g(j,t), ij and will contribute nothing to f(j). decompositioneffectngeometric int

6、erpretation of orthogonal transformnnon-orthogonal transform the orthogonal transform of f(t) is a projection of f(t) into a orthogonal basis space formed by g(i,t), i=1,2,orthogonal projectionnon-orthogonal projection the same component of f(t) may project into different bases. redundancy will prob

7、ably exist in the transform results.nfourier transform (ft) ( )( ),( )(cossin)j tj tff t edtf t ef ttjt dtis the limitation of the orthonormal function set when j tjn tee 0.i.e. the ft is an orthonormal wave transform.lim0, lim0,iijtjttteenon-stationary (time-variant) signal nstationary (time-invari

8、ant) signal1234100 ;40 ;20 ;101234( )cos()cos()cos()cos()x tttttnnon-stationary (time-variant) signal1234100 ;40 ;20 ;1011223344( )cos(); ( )cos();( )cos(); ( )cos()x ttx ttx ttx tt x1(t) x2(t) x3(t) x4(t)nft of non-stationary (time-variant) signalsignals are different, but spectrums are similarndef

9、iciency of wave transform (e.g. ft) wave transforms are not suitable for time-variant signal since they dont include position (time) information in the transform results (e.g. ft analyzes the global frequency distribution of a signal, but it can not characterize the local behavior of the signal). ba

10、sic idea5.2 time-frequency analysisin ft, the local behavior of a signal is not represented in the signals frequency spectrumthe ft is not the most proper representation for the time-variant signals or the signals containing transient or localization componentstime-frequency analysis: characterizing

11、 the time and frequency information of a signal simultaneously in its spectrum examples of time-frequency analysis main tools of time-frequency analysisnshort time fourier transform (stft)nwavelet transformnwigner distribution (wd)nquadric transform (non-linear transform)ntime-frequency distribution

12、nwiger-ville distribution (non-stationary random signal) definition nstft of continuous time signal x(t)nstft of discrete time signal x(n),( ) ()jxstfttxw ted,( ) ()j mxmstftnx m w nm e5.3 short time fourier transformwhere w(t) is a real finite-width window function which slides along x(t) where w(n

13、) is a real finite-length window sequence which slides along x(n) the result of stft is a 2-d function which reflects the signal spectrum varied with time. nft of windowed x(t)is a compact support function (wavelet), and the stft isnwide-sense wavelet transform*( , , )()jw tw te *,( )( , , )( ), ( ,

14、 , )xstfttxw tdxw t the support width of the wavelet (i.e. the width of the window) is constant for all frequency components. the conflicting requirements between the frequency resolution & the time resolution in stftnfrequency resolution requirement the window width t should be wide enough to g

15、ive the desired frequency resolution. ntime resolution requirement the window width t should be narrow enough so as not to blur the time dependent events, i.e. the signal segment included in the window can be treated as stationary approximately. npartition of time-frequency plane in stftt1 21or, 4tf

16、t problems of stftnheisenberg uncertainty principlenstft is redundant representation not good for compressionnthe same and t throught the entire plane!t we cannot perfectly localize events in time and frequency simultaneously! multi-resolution analysis (mra)nbasic ideanthe high frequency components

17、vary rapidly in time. a relatively short signal segment can characterize them properly, hence a relatively narrow time window can be used (high time resolution and low frequency resolution). non contrary, the low frequency components vary slowly in time and a relatively wide time window should be us

18、ed (high frequency resolution and low time resolution). higher frequency more narrow time window lower frequency wider time window higher time resolution higher frequency resolution npartition of time-frequency planenmra different time and frequency resolutions are adopted to the different frequency

19、 (scale) components of signal at same time *,1,( )()( )( )( ),( )xaatwtax tdtaax tt dtx tt5.4 continuous wavelet transform (cwt) definitionncwt ( )0t dtwhere is mother (basis) wavelet which satisfies2( )( )tlr2( ) tdtenscaling & translation of mother wavelet where a is scaling (dilation) paramet

20、er and b is translation (shifting) parameter. is the basis function of cwt. it is called the analysis wavelet. ,( )at,1( )()attaa22,21( )()1( )attdtdtaatdteaanscaling (dilation)4a 2a 1a ntranslation (shifting)nscaling and translation4a 2a 1a 2a 16a 8a 2a 16a t representing cwt in frequency domain*1,

21、( )*()( )()xtft wtaftx taaaxa *11,( )()( )*()xttwtax tdtx taaaa*,( )()2jxawtaxaed if the central frequency of the ft of is 0, and its band width is b, then the central frequency and band width of the ft of are 0/a and b/a respectively, i.e.nproperties of waveletnfrequency spectrum analysis ability i

22、f the wavelet is a band-pass filter with relatively narrow passband, then the wavelet with different a can characterize the different frequency components of a signal.nconstant quality factor,( )at00/./aqconstbb a( ) t,( )ataninverse cwt (icwt)nadmissible conditionwhere is the ft ofc( ) 20( )cd 0( )

23、( )0j tt edt ( )0t dt( ) tthe satisfies the admissible condition is a admissible wavelet. ( ) ta basic restriction for constructing a mother wavelet nicwt,201( )( , )( )xadax twtat dca examples of mother wavelets properties of cwtnlinearityntime shifting 12,zxywtakwtak wta0,xwtat0()x tt,xwta( )x t12

24、( )( )( )z tk x tk y tnscalingnmoyal theorem (inner product theorem) nenergy of wt *1,2,1220( ),( )( ),( )( )( )aadax ttx ttdcx t x t dta2220( , )( )xdawtadcx tdta,xawt ( ),0tx,xwta( )x t12( )( )( )x tx tx t1212( , ),( , )( ),( )xxwtawtacx tx tnreproducing kernel equation0000*00,11(, , )( )( )( ),(

25、)aaaakaatt dtttcc0000*00,*,200020,( )( )1( , )( )( )( , )(, , )xaxaaxwtax tt dtdawtatt dtdacdawtakaada,201( )( , )( )xadax twtat dca icwtreproducing kernel: the dependence between 00,( ) and ( )aatt00, can be reproduced by ( , )xxwtawtaredundancy of cwtreproducing kernel equation0000,00in order to r

26、emove the redundency, an orthognal wavelet setis needed:0, or ( ),( ), and aaaattcaa5.5 discrete wavelet transform (dwt) definitionndiscretizing of the scaling & translation factor 00000, 0,1,2,; 1, 0,1,2,; 0jjaajak ak a 00jkmother (basis) waveletbasis with larger scale lower sampling rate 00000

27、0*000,( )( )( ),( ) , 0,1,;0,1,jjjjjjxakaakawtakax tt dtx ttjkndwt 0000020,0( )()( )jjjjj kjakatkatata: dwt or wavelet series,xwtj k*,( )( )( ),( ) , 0,1,;0,1,xj kj kwtj kx tt dtx ttjk22,2( )2()2(2)2jjjjj kjtkttkusually, are adopted, then 002 and 1a wavelet framenrequirements for discrete wavelet ba

28、sisncompleteness can characterize the x(t) completely? nreversibility can x(t) be restored from stably? nuniversality whether any x(t) can be represented by a linear combination of the wavelet basis000,jjxwtaka000,jjxwtaka000,( )?jjakat,( ),( )0,( )0j kx ttj kx t212,21,2,212( )( ),( )( ),( )( ),( )(

29、 )( ), 0j kj kj kj kj kx tx ttx ttx ttb x tx tb completeness uniqueness continuity1,2,12( ),( )( ),( ) ,( )( )j kj kx ttx ttj kx tx t22,( ),( )( ), 0j kj kx ttb x tb 121,2,( )( )( ),( )( ),( ) ,j kj kx tx tx ttx ttj k22,( )( ),( ), 0j kj ka x tx tta reversibility uniqueness continuitynframe let be a

30、 cluster of functions in hilbert space h, if for any function , it is held that then is a frame. moreover, if a=b, then is a tight frame and222( )( ),( )( ) , 0jja x tx ttb x tab( ),jth jz1( )( ),( )( )jjjx tx ttta( )x th( )jt( )jt if a=b=1, then ,222:22:2( ),( )( ),( )( ),( )( )( ),( )( )ijiiijjj j

31、 iiijj j iitttttttttthence is a set of orthogonal bases in h space. such a set of bases is orthonormal ifiz 2( )1, jtj2( ), ( ),( )0, iijtijttij( )jtndual frame ( ),( )jx ttwhere satisfies( )jt( )( ),( )( )jjjx tx ttt( )x trestoring 22211( )( ),( )( )jjbx tx ttax tand it is called the dual frame of(

32、 ).jt for convenience, when ab but , it is usually approximated asbabairibabawhere11ba02( )( )kjjktrtab2( )( )jjttab( )( )jjtt if a=b, thennwavelet frame if for any function x(t), the wavelet basis function satisfies then is a wavelet frame. its dual wavelet frame is 22211,( )( ),( )( )j kjkbx tx tt

33、ax twhich satisfies2,( )2(2)jjj kttk222,( )( ),( )( ), 0j kjka x tx ttb x tab ,( ),0,1,;j ktjkz2,( )2(2),0,1,;jjj kttkjkz if a=b, then,11( )( ),( )( )( , )( )j kj kxj kjkjkx tx tttwtj ktaaor,02( )( )lj kj kltrtab if ab, then,2( )( )j kj kttaband,( )( ),( )( )( , )( )j kj kxj kjkjkx tx tttwtj kt i.e.

34、 is an admissible wavelet. if is a wavelet frame, then it meets the three requirements for discrete wavelet basis proposed before, and20( )ln2ln222adb ,( )j kt,( ),0,1,;j ktjkz designing orthonormal wavelet basis with mranorthogonal wavelet basis: removing the information redundancy in the data afte

35、r transformation ,(,;,)( ),( )(,)mmnnmmnnjkjkmnmnkjkj kttcjj kkthen is orthogonal, and if c=a, thenis orthonormal. ,( )j ktif ,( )j ktreproducing kernel equation in dwt with tight frame:00001(,)( , ;,)( , )xxjkwtj kkj k j k wtj kanmra dissection of function space is a sequence of closed sub-spaces of2( )( )x tl r0111221,jjjvvw vvwvvw ,jvjz2( ),l r is a mra dissection of if the followings are held: (4) ( )();jjx tvx tkv1(1) for any ;jjvvjz jv2( )l r 2(2) ( ), ;jjj zj zvl rv 01(3) ( )(2 );jjx tvxtv00(5) there exi