Filter Banks (779443), страница 5
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Weuse the type-l decomposition given bym-lC p(2lM + j ) z-e ,P~(z)=j = 0 , . . . , 2 M - 1.(6.107)e=o6.6.1CriticallySubsampledCaseIn the critically subsampled case the analysis polyphase matrix can be writtenas [112, 681(6.108)where[Tl]k,j= 2cos[G( k + );(j -p) + 4 k ] ,k = O ,.", M - l ,j = o ,.", 2 M - 1 ,(6.109)andP o ( z 2 ) = diag [P0(-z2),Pl(-z2),. . . ,PM-l(-z2)],(6.110)P 1 ( z 2 ) = diag[ P M ( - ~ ~ ) , P M +. .~. ,P2M-l(-z2)](-~~), .Note that the matrices P 0 ( z 2 )and P1 ( z 2 ) contain upsampled and modulatedversions of the polyphase filters.For the synthesis polyphase matrix we getR ( z ) = [z-lQ1(z2),Qo(z2)]TT,(6.111)where[T2]k,j= 2cos[G(L + );(2M - 1 - j -k = O ,.", M - l ,p) - 4 4 ,j = o ,.", 2 M - 1 ,(6.112)andQo(z2) = diag [QM-I(-z~), .
. . ,Q1(-z2),Q~(-z2)] ,(6.113)Q1(z2) = diag [ Q ~ M - I ( - Z ~ ) , . . . , Q M + I ( - Z ~ ) ,Q M ( - ~ ~ ) .]176BanksChapter 6. FilterThe perfect reconstruction conditions are obtained by setting(6.114)Using the property [87]this yields the conditionswhich have to be met for k = 0 , . . . , % - 1. The relationship between qo andS isqo = 2s+ 1.(6.118))The condition (6.117) is satisfied for Q k ( z ) = az-PPk(z) and Q ~ + k ( z =az-8 P ~ + k ( z with)arbitrary a , p, which suggests the use of the same prototype for both analysis and synthesis.
Thus, with Q ( z ) = P ( z ) ,the remainingcondition isPZM-l-k(Z)Pk(Z)+ P M + k ( Z ) PM-l--k(Z)!z-'= 2"k=O,M...,--21.(6.119)The M/2equations in (6.119) may be understoodas P R conditions on M/2non-subsampled two-channel filter banks. The prototype can for instance bedesigned by using the quadratic-constrained least-squares (QCLS) approach,whichwas proposed byNguyen [lll].Here, we write all constraints givenby (6.119) in quadratic form and optimize the prototype using constrainednumerical optimization. The approach does notinherently guarantee PR, butthe PR constraints can be satisfied with arbitrary accuracy.Another approach, which guarantees P R and also leads to a very efficientimplementation of the filter bank, is to design the filters via lifting [129, 831.For this, we write the PR conditions asV ( z ) U ( z )=z-l(-z-z)'2MI,(6.120)1776.6.
Cosine-Modulated Filter Bankswhere2z - ~ Q ~ M - ~ - ~ () - z ( - l ) s - l Q z ~ - l - - k - ~ ( - z )(-1)Sz-1Qk+~(-z2)Qd-z2)l.(6.121)It is easily verified that (6.120) includes (6.117) and (6.116), but (6.120) canalso be derived straightforwardly from (6.114) by using the properties of thecosine functions [83]. The filter design is as follows.
We start withwhere the subscript 0 indicates that this is the 0th iteration. We haveV,(z)U,(z) =z-1I.2M~(6.123)Longer filters with the same delay are constructed by introducing matrices ofthe typeA i l ( z ) A i ( z )= I(6.124)Lin between the product(6.126)Note that Ui+l ( z ) and Vi+l( z )retain the structuregiven in (6.121). From thenew matrices thepolyphase componentsof the prototype areeasily extracted.The operation (6.126) can be repeated until thefilters contained in U i ( z )andV i ( z ) have the desired length. Since the overall delay remains constant, thisoperation is called zero-delay lifting.A second possibility is to introduce matrices178Chapter 6. Filter Banksand to construct the new filters asUi+l(Z)=Ci(Z)Ui(Z),(6.128)This type of lifting is known as maximum-delay lifting.
Again, Ui+l(z) andVi+l(z)have the structure given in (6.121), and since (6.120) is satisfied,PR is structurally guaranteed. Thus,filter optimization can becarried out byoptimizing the lifting coefficients in an unrestricted way.Also other lifting schemes can easily be found. The advantageof the aboveapproach is that only one lifting step with one lifting coefficient ai or ci isneeded in order to increase the length of two polyphase components of eachprototype.Implementation Issues. The straightforward polyphase implementationof(6.108) is depicted in Figure 6.23. On the analysis side, we see that alwaysthose two systems are fed with the same input signal which are connectedin (6.116).
In the synthesis bank, the output signals of the correspondingsynthesis polyphase filters are added. This already suggests the joint implementation of pairs of two filters. However, a more efficient structure can beobtained by exploiting the periodicies in the rectangular matrices 1'2 andT 2 and by replacing them with M X M cosine modulation matrices T 1 and-T--lT 2 = T , = T, [83]:2cos [$F ( k[TlIk,j+ +) ( j - g) + q5k] ,j = 0,. ..' M2 - 1=2 ~ 0 ~ [ $ F ( k + i ) ( M + j - g ) + 4 k j] ,= T , . . . , M - l(6.129)for k = 0,. . . , M - 1. This structure is depicted in Figure 6.24. Note that thefollowing signals are needed as input signals for the cosine transform:(6.130)Thus, all polyphase filtering operationscanbe carried out via the liftingscheme described above where four filters are realized jointly.Banks1796.6.FilterCosine-Modulated(b)Figure 6.23. Cosine-modulated filter bank with critical subsampling.(a) analysis;(b) synthesis.6.6.2Paraunitary CaseIn the paraunitary case with critical subsampling we haveB(z)E(z)= I M ,(6.131)which leads to the following constraints on the prototype:1.
The prototype has tobe linear-phase, that is, p ( L - 1- n) = p ( n ) .2. The same prototype is required for both analysis and synthesis.3. The prototype has tosatisfyPk (z)P/C(.l + h+ k (Z)PM+k=1z.(6.132)180BanksChapter 6. Filterli(-1)S-l-YM-1 (m)M-kM-lL;Figure 6.24.
Cosine-modulated filter bank with critical subsamplingimplementation structure. (a) analysis; (b) synthesis.dand efficientThe filter design may for instance be carried out by parameterizing thepolyphase components using the lattice structure shown in Figure 6.25 andchoosing the rotationangles so as to minimize an arbitraryobjective function.For this method a good starting point is required, becausewe have to optimizeangles in a cascade of lattices and the relationships between the angles andthe impulse response are highly nonlinear.
Alternatively, the QCLS approach[l111can be used, which typically is less sensitive to the starting point.As in the biorthogonal case, the polyphase filters can be realized jointly.Onecan use the structure in Figure 6.23 and implement two filters at atime via the lattice in Figure 6.25. However, the more efficient structure inFigure 6.24 can also be used, where four filters are realized via a commonlattice.
This was shown in [95] for special filter lengths. A generalization isgiven in [62].Banks1816.6.FilterCosine-Modulated-pk (z)/A\A/\In [l031 a method has been proposed that allows the design of discretecoefficient linear-phase prototypes for the paraunitary case. The design procedure is basedon a subspaceapproach that allowsus to perform linearcombinations of P R prototype filters in such a way that the resulting filter isalso a linear-phase P R prototype.
The filter design is carried out iteratively,while the PR property is guaranteed throughout the design process. In orderto give some design examples, Table 6.1 shows impulse responses of 8-bandprototypeswith integer coefficients and filter length L = 32. Because ofsymmetry, only the first 16 coefficients are listed. The frequency responsesof the filters #3 and #S are depicted in Figure 6.26.Closed Form Solutions. For filter length L = 2M and L = 4M closed formsolutions for PR prototypes are known.
The special case L = 2M is knownas the modulated lapped transform (MLT), which was introduced by Princenand Bradley [116]. In this case the PR condition (6.132) reduces to+PM+kPkwhich meansp 2 ( n )+ p 2 ( M=1,+ n) = 21 ~.(6.133)An example of an impulse response that satisfies (6.133) isP(n) =1msin [( n + -)-.(6.134)The case L = 4M is known as the extended lapped transform (ELT). TheELT was introduced by Malvar, who suggested the following prototype [95]:1p(n) =-GZ1cos (n+-2m[17r+ -)2 2M](6.135)182Chapter 6.
Filter Banks0.100.20.30.40.50.40.5Normalized Frequency(a)0.100.20.3Normalized Frequency(b)Figure 6.26. Frequency responses of 8-channel prototypes from Table3.1. (a) filter#3; (b) filter #6. For comparison the frequency response of the ELT prototype isdepicted with dotted lines.Table 6.1.Perfect reconstruction prototypes for 8-band filter banks with integercoefficients &(L - 1- n ) = p ( n ) ) .1r781011121314121112221--#3-#S--1-100002244667788-2190-1901-1681-426497254238026205967813197163591939822631247382639427421Banks1836.6.FilterCosine-Modulated6.6.3OversampledCosine-Modulated Filter BanksInthe oversampled case withoversampling by p =matrices may be written asEZ,the polyphase(6.136)andwithPe(z2’) = diag (pelv(-z2’),pe~+1(-z2’), .
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