Wavelet Trasform ad its relatio to multirate filter bas Christia Walliger ASP Semiar th Jue 007 Graz Uiversity of Techology, Austria Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas
Outlie Short Time Fourier Trasformatio Iterpretatio usig Badpass Filters Uiform DFT Ba Decimatio Iverse STFT ad filter - ba iterpretatio Basis Fuctios ad Orthoormality Cotiuous Time STFT Wavelet Trasformatio Passig from STFT to Wavelets Geeral Defiitio of Wavelets Iversio ad filter - ba iterpretatio Orthoormal Basis Discrete Time Wavelet Trasf. Iverse Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas First, we will develop the short time Fourier trasform ( STFT ) ad its relatio to filter bas ad the the wavelet trasform ad its relatio to multirate filter bas. Therefore it is much easier to uderstad, if first the discret time STFT ad afterwards the cotiuous time STFT will be itroduced. Followed by cotiuous wavelet trasform ad discret wavelet trasform.
SHORT-Time FOURIER TRANSF. sdfgsdfg yxvyxvyxcv fgjfghj figure : STFT processig i time time frequecy plot = Spectogram figure : spectogram Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 3 I short time Fourier trasform, a sigal x() is multiplied with a widow v() ( typically fiite i duratio ). The Fourier trasform of the time domai product x()v() is computed, ad the the widow is shifted i time, ad the FT of the ew product computed agai. ( figure ) This operatio results i a separate FT for each locatio m of the ceter of the widow, which is typically a iteger multiple of some fixed iteger K ). (figure )
Defiitio: X STFT jω ( ) e, m = = x( ) v( m) e jω m... time shift variable ( typically a iteger multiple of some fixed iteger K) ω... frequecy variable π ω < π Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 4 From above discussio it is clear that the STFT ca be writte mathematically as show i the slide, where ω is cotiuous ad taes the usual rage betwee π ad + π.
Iterpretatio usig Badpass Filters Traditioal Fourier Trasform as a Filter Ba figure 3: Represetatio of FT i terms of a liear system 0 e jω. Modulator : performs a frequecy shift jω ( ) H e. LTI System : ideal lowpass filter Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas Before iterpretig the STFT i terms of filter bas, we will begi by represetig a filter ba iterpretatio for the traditioal Fourier Trasform. (figure 3) Figure 3 represets oly oe chael for oe specific frequecy ω 0. 5
jω ( ) H e Why is a ideal lowpass filter? Impulse Respose h() = for all jω jω H ( e ) h( ) e = πδ ( ω) = = a π ω < π oly zero - frequecy passes every other frequecy is completely supressed jω0 ( e ) y ( ) = X for all Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 6 h() = for all. This system is evidetly ustable, but let us igore these fie details for the momet. δ a (ω) is the Dirac delta fuctio. jω0 Summarizig, the process of evaluatig ( ) X ( e ) y = ca be looed upo as a liear system, which taes the iput x() ad produces a costat output y(). Therefore, the FT operator is a ba of modulators followed by filters. This system has a ucoutably ifiite umber of chaels.
Professor Horst Cerja, 9..005 7 Georg Holzma, Christia Walliger.06.07 Wavelet T. - Relatio to Filter Bas STFT as a Ba of Filters ( ) = = m j m j j STFT e m v x e m e X ) ( ) ( ) (, ω ω ω ) ( ) ( )) ( ( ) ( m j m j e m v m e v = ω ω Expasio of Defiito for further isight! with: Covolutio of x() with the impulse respose of the LTI System j e v ω ) (
figure 4: Represetatio of STFT i terms of a liear system I most applicatios, v() has a lowpass trasform V(e jω ). jω v( ) V ( e ) 0 v( jω j( 0 ) ) e V ( e ωω ) Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas Figure 4 shows the iterpretatio of the STFT i terms of a filter ba. ( Agai, oly oe chael ca be see). The first is a LTI filter followed by the modulator. 8
figure 5: Demostratio of how STFT wors Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas Figure 5 demostrates how the STFT wors. (a) FT of a arbitrary choose iput sigal x() (b) the widow trasform ad its shifted versio (c) output of LTI filter jω (d) traditioal Fourier trasform of X ( e 0 ) STFT, 9 Hece, the STFT ca be looed upo as a filter ba, with ifiite umber of filters ( oe per frequecy )!
I practice, we are iterested i computig the Fourier trasform at a discrete set of frequecies 0 ω 0 < ω < < ω M- < π Therefore the STFT reduces to a filter ba with M badpass filters H ( e jω ) = V ( e j( ωω ) ) figure 6: STFT viewed as a filter ba Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 0
Uiform DFT ba If the frequecies ω are uiformly spaced, the the system becomes the uiform DFT ba. The M filters are related as i the followig maer H z) H 0 ( zw ) ( = 0 M W = e j π M H ( e jω ) = H 0 e π j( ω ) M H 0( e ) = V ( e jω jω ) The uiform DFT ba is a device to compute the STFT at uiformely spaced frequecies. Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas jω jω The frequecy resposes H ( e ) are uiformly shifted versios of ( e ) H 0
Decimatio if passbad width of V(e jω ) is arrow output sigals y () are arrowbad lowpass sigals this meas, that y() varies slowly with time Accordig to this variyig ature, oe ca exploit that to decimate the output. Decimatio Ratio of M = movig the widow v() by M samples at a time if filters have equal badwidth = M maximally decimated aalyses ba figure 7: Aalysis ba with decimators Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas Figure 7 shows a decimated STFT system, where the modulators have bee moved past the decimators. I a more geeral system could be differet for differet, ad moreover ot be derived from oe prototype by modulatio. Such a system, however, does ot represet the STFT obtaiable by movig a sigle widow across the data x(). this systems will be admitted i the wavelet trasform. ( ) z H may
Time Frequecy Grid Uiform samplig of both, time ad frequecy ω figure 8: time frequecy grid Time spacig M correspods to movig the widow M uits ( = samples ) at a time. π frequecy spacig of adjacet filters = M Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 3
X Iversio of the STFT From traditioal Fourier viewpoit jω ( e m) STFT, is the FT. from the time domai product x( ) v( m) π jω ( e, ) jω x( ) v( m) = X STFT m e dω π 0 Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 4 For example, if we set = m we obtai the STFT iversio formula for x(m) as log as v(0) exists. If it does ot, we ca pic some other value of m.
Aother iversio formula is give by: π jω * ( e, m) v ( m) jω x( ) = X STFT e dω π m= 0 v = which is provided by ( m) m if m ( m) v but fiite divide right side of the formula by m v ( m) but if widow eergy is ifiite oe caot apply this formulatio Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 5
Filter Ba Iterpretatio of the Iverse With F (z) as sythesis - filter Recostructio ca be doe by the followig sythesis ba: figure 9: sythesis ba used to recostruct x() typically = M for all Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 6 SPSC Sigal Processig & Speech Commuicatio Lab The z Trasformatio of Xˆ M = 0 xˆ ( ) ( z) = X ( z ) F ( z) is give by xˆ M i time domai ( ) = x( m) f ( m) y = = 0 m= M y = 0 m= jω ( m) ( m) e f ( m) ( m) K STFT Coefficie ts Recostructio is stable, if the filters (z) F are stable! Perfect recostructio will be obtaied, if x ˆ( ) = x( ) Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 7
Basis Fuctios ad Orthoormality η m Fuctios of iterest ( ) =ˆ f ( m) Kbasis fuctios For these double idexed fuctios ( basis fuctios ), the orthoormality property meas that { ( )} η m ( m ) f ( m ) = δ ( ) ( m m ) * f δ = should be zero, except for those cases where = ad m = m Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas Remember:... filter umber m... time shift How should we desig the filters F ( z) i order to esure this orthoormality property? Therefore, the parauitary property of the polyphase matrix is sufficiet! 8
The Cotiuous - Time Case Mai poits: X STFT jω t ( jω,τ ) = x( t) v( t τ ) e dt ( STFT ) x x π j t ()( t v t τ ) = X ( jω, τ ) e Ω dω ( iv STFT ) STFT. * STFT. π jω t () t = X ( jω, τ ) v ( t τ ) dτ e dω ( iv STFT ) Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 9 Because of the close resemblace to the discrete time case, we oly summarize the mai poits for the cotiuous time case. Historically, the STFT was first developed for the cotiuous time case by Deis Gabor.
Choice of Best Widow R oot M ea S quare duratio of widow fuctio v(t) i time domai D t D t t v E () t = dt frequecy domai D f D f Ω V ( jω) πe = dω with: E... widow eergy E = v () t dt Ucertaity priciple: D t D f 0.5 Iff Gaussia widow, this iequality becomes a equality! Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas D t is the rms time domai duratio ad D f the rms frequecy domai duratio of the widow. 0
Filter Ba Iterpretatio figure 0: cotiuous STFT Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas Figure 0 shows agai the filterig iterpretatio for the cotiuous time STFT.
THE WAVELET TRANSFORM Disadvatage of STFT uiform time frequecy box ( D = cost., D cost. ) t f = The accuracy of the estimate of the Fourier trasform is poor at low frequecies, ad improves as the frequecy icreases. Expected properties for a ew fuctio: widow width should adjust itself with frequecy as the widow gets wider i time, also the step sizes for movig the widow should become wider. These goals are icely accomplished by the wavelet trasform. Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas
Passig from STFT to Wavelets Step : Givig up the STFT modulatio scheme ad obtai filters h () t a ( ) = h a t a > Kscalig factor, = it eger i the frequecy domai: ( jω) = a H ( ja Ω) H all reposes are obtaied by frequecy scalig of a prototype respose H( jω) Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas This is ulie the case of STFT, where all filters were obtaied by frequecy shift of a prototype. The scale factor a is meat to esure that the eergy () t h 3 dt is idepedet of.
Example: ( ) Assumig H jω is a badpass with cutoff frequecies α ad β. Also a =, β = α ad the ceter frequecy should be the geometrical mea of the two cutoff edges Ω = αβ = α figure : frequecy respose obtaied by scalig process Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 4 SPSC Sigal Processig & Speech Commuicatio Lab Ratio: badwidth ceter frequecy Ω = ( β α ) αβ = is idepedet of iteger I electrical filter theory such a system is ofte said to be a costat Q system! ceter frequecy ( Q... Quality factor Q = ) badwidth Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 5
filter ouputs ca be obtaied by: a e jω τ x ( ) () t h a ( τ t) dt Step : badwidth of H ( jω) Samplerate or i time domai widow legth step size Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 6 Sice the badwidth of H ( jω) is smaller for larger, we ca sample its output at a correspodigly lower rate. Viewed i time domai, the width of that we ca afford to move the widow by a larger step size! () t h is larger so
Therefore: τ = a T K it eger, a T Kstep size hece: h ( a ( a T t ) = h( T a t) Summarizig, we are computig: X X DWT DWT, (, ) a x() t h( T a t) = dt ( ) x( t) h ( a T t) = dt DWT...Discrete Wavelet Trasform figure : Aalysis ba of DWT Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 7 This ca be doe by replacig the cotiuous variable τ as show i the slide. The modulatio factor j τ e Ω has bee omitted. What we ca see is, that the above itegral represets the covolutio betwee x(t) ad h (), t evaluated at a discrete set of poits a T. I other words, the output of the covolutio is sampled with spacig a T. (figure is a schematic of this for a = ).
Time Frequecy Grid figure 3: time frequecy grid D D t f = cost. Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 8 Frequecy spacig is smaller at low frequecies, ad the correspodig time spacig is larger.
Geeral Defiitio of the Wavelet Trasform t q X CWT ( p, q) = x() t f dt p p p,q... real valued cotiuous variables Accordig to former defiitio: p = a q = a T f ( t) = h( t) X CWT ( p, q) ad X (, ) KKK wavelet coefficiets DWT Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 9
Iversio of Wavelet Trasform x where () t X (, ) ψ ( t) = ψ ( t) DWT are the basis fuctios Filter Ba Iterpretatio of Iversio Recostructio of x(t) as a desigig problem of the followig sythesis filter ba (, ) K sequece ( jω) K cotiuous itime X DWT F output of sythesis filter ba : () t = X ( ) f ( t a T ) x ˆ, DWT figure 4: sythesis ba Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas Figure 4 shows the sythesis filter ba. We have to be careful with the iterpretatio of this figure. Sice sequece, the sigal which is iput to the cotiuous time filter impulse trai. X DWT (, ) is a 30 F ( jω) is actually a
All sythesis filters are agai geerated from a fixed prototype sythesis filter f(t) ( mother wavelet ) f () t = a f ( a t) Substitutig this i the precedig equatio ad assumig perfect recostructio, we get with: () = ( ) ( t X, a f a t T ) x DWT () t = f () t ψ () t = a ψ ( a t T ) = a ψ [ a ( t a T )] Kset of basis fuctios ψ usig this, we ca express each basis fuctio i terms of the filter f () t ψ ( t) = f ( t a T ) Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 3
Orthoormal Basis Of particular iterest is the case where fuctios Therefore, we expect: ψ * { ( t) } ψ () t ψ () t dt = δ ( l) δ ( m) l m usig Parseval s theorem, this becomes π Ψ * ( jω) Ψ ( jω) dω = δ ( l) δ ( m) l m is a set of orthoormal ad get : X DWT * (, ) x() t ψ () t = dt Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 3 SPSC Sigal Processig & Speech Commuicatio Lab Comparig these results, we ca coclude: ψ Ad i particular for = 0 ad = 0: ( t) = h * ( a T t) * ψ () t = ( t) = h ( t) for the orthoormal case f ( t) = h * ( t) 00 ψ Discrete Time Wavelet Trasform Startig with the frequecy domai relatio ad a scalig factor a = H j ( e ) ( ) ω j ω = H e K is a oegative it eger ( ) jω for highpass H e ad =, = figure 5: Magitude resposes Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 33
Let G(z) be a lowpass with respose figure 6: Magitude respose of G(z) Usig QMF bas or its equivalet figure 7: 3 level biary tree-structured QMF figure 8: equivalet 4-chael system Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 34 SPSC Sigal Processig & Speech Commuicatio Lab 4 Resposes of the filters H ( z), G( z) H ( z ), G( z) G( z ) H ( z ),... figure 9: combiatios of H(z) ad G(z) Defiig the Discrete Time Wavelet Trasform M + ( ) = x( m) h ( m), m= y 0 M M ( ) = x( m) h ( m) ( D T WT ) m= y, M iscrete ime Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 35
Professor Horst Cerja, 9..005 36 Georg Holzma, Christia Walliger.06.07 Wavelet T. - Relatio to Filter Bas Iverse Trasform ( ) ( ) ( ) ( ) ( ) K,, 0 z G z H z F z H z F s s s = = figure 0: sythesis filters SPSC Sigal Processig & Speech Commuicatio Lab Professor Horst Cerja, 9..005 37 Georg Holzma, Christia Walliger.06.07 Wavelet T. - Relatio to Filter Bas For perfect recostructio ( ) ( ) x x = ˆ we ca express ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 0 0... + + + + = M M z Y z F z Y z F z Y z F z Y z F z X M M M M ad i time domai: ( ) ( ) ( ) ( ) ( ) = = = + + = 0 M m m M M M m f m y m f m y x
Mai Refereces Multirate Systems ad Filter Bas (Pretice Hall Sigal Processig Series) by P. P. Vaidyaatha Professor Georg Holzma, Horst Cerja, Christia 9..005 Walliger.06.07 Wavelet T. - Relatio to Filter Bas 38