Filter bas Separately, the lowpass ad highpass filters are ot ivertible T removes the highest frequecy / ad removes the lowest frequecy Together these filters separate the sigal ito low-frequecy ad high-frequecy subsequeces ad T x is the complemet of T x It is said that these filters form a filter ba It ca be show that there exists a iverse filter ba or i other words the trasform which splits sigal ito lowfrequecy ad high-frequecy compoets is ivertible T
Decimatio The problem coected with filter bas is : The sigal legth has doubled sice each compoet has the legth equal to the legth of the iput sequece The solutio is to dowsample or decimate) Decimatio meas removig all odd-umbered compoets of the filtered sequece: y) y) y) y) y) y4) y3) y6)
Decimatio Decimatio is ot ivertible The odd-umbered compoets are lost Normally the eve-umbered compoets of both lowfrequecy ad high-frequecy ) decimated sigal parts will be eeded to recover all compoets of the origial vector For bad-limited sigals we ca recover the odd-umbered compoets from the eve-umbered compoets Two steps, filterig ad decimatio ca be doe with ew matrices ad B
Filter bas B Removig half of the rows leaves the matrices with a doubleshift To compesate losig half of the compoets we multiply the survivig compoets by
Filter bas Filterig exteds the legth of the output sigal compared to the legth of the iput sigal If the iput vector x has size N the the output vector y has size N l,where l deotes the legth of the impulse respose T T Thus the correspodig matrices, are matrices of size N N l ) To avoid such extesio we use the socalled circular extesio of the iput sigal I this case the covolutio y ) h ) x ) h ) x ) ca be cosidered as a circular covolutio ad become of size N N We will cosider matrices correspodig to the circular covolutio T, T
Filter bas The rectagular ad B fit ito a square matrix which represets the whole aalysis filter ba: B It executes the lowpass chael ad the highpass chael both decimated) The combied square matrix is ivertible
Filter bas The iverse is the traspose: T T B B The secod matrix is the sythesis filter ba T T B This is a orthogoal filter ba, because iverse=traspose
Filter bas B It is evidet that: T T T T B B B I The sythesis ba is the traspose of the aalysis ba Whe oe follows the other we have perfect recostructio The aalysis ba had two steps: filterig ad dowsamplig The sythesis ba also ca be orgaized to have two steps: upsamplig ad filterig The first step is to recostruct full-legth vectors The odd-umbered compoets are retured as zeros by upsamplig Upsamplig is deoted by )
Filter bas Applied to a half-legth vector it iserts zeros: y ) ) ) ) ) ) ~ y y y y y y y The secod step i the sythesis ba is filterig The lowpass filterig is equivalet to multiplyig the exteded vector by the matrix : S
Filter bas The highpass filterig is equivalet to multiplyig the exteded vector by matrix S The sythesis lowpass filter is: ) ) ) x x y ) ) ) x x y The sythesis highpass filter is: This is the Haar filter ba
y Example et x x), x), x), x3), x4), x5) be iput sequece of the the Haar filter ba Output of the lowpass filter is: x), x) x), x) x), x) x3), x3) x4), x4) x5), x Output of the highpass filter is: y x), x) x), x) x), x3) x), x4) - x3), x5) - x4), x v v Decimated output of the lowpass filter is: x), x) x), x3) x4), 5) x Decimated output of the highpass filter is: x), x) x), x4)- x3), 5) x 5) 5)
Example The exteded low-frequecy part is: ~ y x),, x) x),, x3) x4),, x 5) The exteded high-frequecy part is: ~ y x),, x) x),, x4) - x3),, x ~ y owpass filtered has the form w x), x), x) x), x) x), x3) x4), x3) x4) Highpass filtered w x), x), ~ y has the form x) x), x) 5) x), x3) x4), x4) - x Summig up w ) ad w after ormalizatio we get ~ x ), x), x), x), x3), x4) x ) 3)
Wavelet filterig The most importat feature of the wavelet filter bas is their hierarchical structure The recursive ature of wavelets leads to a tree structure of the correspodig filter ba The iput sequece is decomposed ito the so-called referece low-frequecy) subsequeces with dimiishig resolutios ad related with them the so-called detail highfrequecy) subsequeces At each level of decompositio the wavelet filterig is ivertible, that is, the referece sigal of this level together with the correspodig detail sigal provide perfect recostructio of the referece sigal of the ext level with higher resolutio)
Wavelet filterig x) Fig8 Oe level of wavelet decompositio followed by recostructio
Wavelet filterig x) h h r d h h r d Fig8 Multiresolutio wavelet decompositio
Wavelet filterig I the theory of wavelet filter bas such pairs of filters ad h g ) ad g iput sigal are foud that there exist pairs of the iverse filters providig the perfect recostructio of the The wavelet filterig provides perfect recostructio of the iput sigal, that is, the output sigal is determied as follows: y Ax ), where A is the gai factor, ad is the delay h ) At the th level of decompositio we obtai r ) with resolutio times scaled dow compared to the resolutio of the iput sigal ad d ), d ),, d with resolutio,,, sigal times scaled dow compared to the iput
Wavelet filterig At the level of decompositio the total legth of referece ad detail subsequeces is : th N N N N N ) ) N N N i i The sythesis wavelet filter bas always cosist of FIR liear phase filters that is very coveiet from the implemetatio poit of view
Wavelet ad scalig fuctio The remarable properties of wavelet filter bas follow from the dilatio equatio ad the wavelet equatio which coect pulse resposes of wavelet filters ad special fuctios called the wavelet ad the scalig fuctio: where ) A t h At ) S t g A t) h t S t ) S t g ) s t A ), ), ), ), At), S t) deote scalig fuctios ad 8) 8) 83) 84) At), S t) are the wavelets 8),8) are dilatio equatios ad 83),84) are wavelet equatios
Typical wavelet
Wavelets Wavelets are basis fuctios t) i cotiuous time A basis is a set of lieary idepedet fuctios that ca be used to produce all admissible fuctios f t) : b f t) b t), JK, f t) t) dt The wavelet trasform operates i cotiuous time o fuctios) f t) ad i discrete time o vectors) x) The output is the set of coefficiets b For ifiite sigals the basis is ecessarily ifiite For fiite legth vectors with compoets there will be basis vectors ad coefficiets The DWT is expressed by a N JK N N N N matrix
Wavelets The special feature of the wavelet basis is that all fuctios t) are costructed from a sigle mother wavelet t) This wavelet is a small wave a pulse) Normally it starts at time t ad eds at time t N The shifted wavelets start at time t ad ed at time t N The rescaled wavelets start at time t ad ed at time t N / A typical wavelet is compressed times ad shifted times t) t )
Scaled wavelet
Wavelet ad scalig fuctio et W be a subspace geerated by wavelets for a give the the followig equality holds where W, is the space cotaiig all fuctios for which / f t) dt is fiite ad that is, ay fuctio from of fuctios from W is the direct sum of subspaces, ca be represeted as a sum The scalig fuctios t ) with a give are a basis for the set of sigals, v a,, t) The chai of subspaces V V V V, V W V f t) b, t) w, w satisfies the coditios: f t) v J t) w t) J
Wavelet filterig a h a,, ) a h b,, ) t h t ) ) ),, t h t ) ) ),, t b t a t a t v ) ) ) ),,,,,, dt t t v a ) ),, dt t t v b ) ),,
Wavelet ad scalig fuctio Whe we perform the wavelet decompositio of the sigal it is divided ito differet scales of resolutio, rather tha differet frequecies Multiresolutio divides the frequecies ito octave bads with badwidth correspodig to the decompositio level, istead of uiform bads from to as the Fourier trasform does At each level of the wavelet decompositio time steps are reduced with a factor of ad the frequecy steps are icreased twice or i other words t ad
Haar example For lowpass filter with h ) ad h the ) dilatio equatio t) t) t ) It ca be show that the solutio is the box fuctio:, for t t), otherwise The coefficiets of the wavelet equatio are h ) ad The wavelet is a differece of half-boxes: h ) t) for for / t) t) t ) t / t is the Haar wavelet! If wavelet is a small wave the the Haar wavelet is a square wave
Haar example t) t) t )
Haar example et our iput sequece have legth the i terms of matrices the hierarchical filterig ca be writte as ) x 4 N y Tx, where, r r r r r r r r r r r r T is the matrix with rows equal to the Haar wavelets scaled by r
Haar example The wavelet trasform based o the Haar wavelets is orthoormal / sice the rescaled Haar wavelets form a t) t ) orthoormal basis ad the iverse matrix used for sythesis represets the trasposed matrix T We ca write V V W, where deotes the orthogoal sum V W all combiatios all combiatios a ) t of scalig fuctios at level, b ) t of wavelets at level We have two orthogoal bases for V, either the s at level or s ad s at level
Wavelet filterig For decompositio with three levels we get V 3 V W V W W V W W W The fuctios i W,, W W V are costat o,] The fuctios i are combiatios of wavelets f t) 3 The fuctio i V has a piece f t) i each wavelet subspace plus V ): W f t) ao t) b t) b t) b t)