Filter Banks (779443), страница 3
Текст из файла (страница 3)
. . D‘(z)B:_,following(6.67)with D’(.) = J D ( z ) J , such that D ‘ ( z ) D ( z )= z P 1 1 .This means that allrotations areinverted and additional delay is introduced. The implementationis shown in Figure 6.11(b).6.2.BanksTwo-Channel FilterYom(K-)T7m...UN-1m-a1UN-lY 1 (m)159+--a0++(b)Figure 6.11.
Paraunitary filter bank in lattice structure; (a) analysis; (b) synthesis.6.2.7Linear-Phase Filter Banks in LatticeStructureLinear-phase PR two-channel filter banks can be designed and implementedin various ways. Since the filters do not have to be power-complementary, wehave much more design freedom than in the paraunitary case. For example,any factorization of a Nyquist filter into two linear-phase filters is possible.
ANyquist filter with P = 6 zeros can for instance be factored into two linearphase filters each of which has three zeros, or into one filter with four andone filter with two zeros. However, realizing the filters in lattice structure, aswill be discussed in the following, involves the restriction that the number ofcoefficients must be even and equal for all filters.The following factorization of E ( z ) is used [146]:E ( 2 ) = L N - l D ( 2 ) L N - 2 .
. . D(2)LO(6.68)withIt results in a linear-phase P R filter bank. The realization of the filter bankwith the decomposedpolyphase matrix is depicted in Figure 6.12. As inthe case of paraunitary filter banks in Section 6.2.6, we can achieve P R ifthe coefficients must be quantized because of finite-precision arithmetic. Inaddition, the structure is suitable for optimizing filter banks with respect togiven criteria while conditions such as linear-phase and PR are structurallyguaranteed. The number of filter coefficients is L = 2(N 1) and thus evenin any case.+160Chapter 6. Filter BanksgYo(m)-r-K...-aN-2Y 1 (m)-a0-a0x^(4+@)Figure 6.12. Linear-phase filter bank in latticestructure; (a) analysis; (b) synthesis.6.2.8Lifting StructuresLifting structures have been suggested in [71, 1411 for the design of biorthogonal wavelets. In orderto explain the discrete-time filter bank concept behindlifting, we consider the two-channel filter bank in Figure 6.13(a).
The structureobviously yields perfect reconstruction with a delay of one sample. Nowweincorporate a system A ( z ) and a delay z - ~ ,a 2 0 in the polyphase domainas shown in Figure 6.13(b).Clearly, the overall structure still gives PR, whilethe new subband signal yo(rn) is different from the one in Figure 6.13(a). Infact, the new yo(rn) results from filtering ).(Xwith the filterand subsampling.
The overall delay has increased by 2a. In the next step inFigure 6.13(c), we use a dual lifting step that allows us to construct a new(longer) filter HI (2) asH~(z)=z++- ~ ~ -z-~”B(z’)~z-~A(z’)B(z’).+ +Now the overall delay is 2a 2b 1 with a, b 2 0. Note that, although wemay already have relatively long filters Ho(z)and H l ( z ) , the delay may beunchanged ifwe have chosen a = b = 0. This technique allows us to designPR filter banks with high stopband attenuation and low overall delay. Suchfilters are for example very attractive for real-time communications systems,where the overall delay has to be kept below a given threshold.6.2.
Two-Channel Filter Banks161m(c)Figure 6.13. Two-channel filter banks in lifting structure.Figure 6.14. Lifting implementation of the 9-7 filters from [5] according to [37].The parameters are a = -1.586134342, p = -0.05298011854, y = 0.8829110762,6 = 0.4435068522, 6 = 1.149604398.In general, the filters constructed via lifting are non-linear phase. However,the lifting steps can easily be chosen t o yield linear-phase filters.Both lattice and lifting structures are very attractive for the implementation of filter banks on digitalsignal processors, because coefficient quantizationdoes not affect the PR property. Moreover, due to the joint realization ofHo(z)and H l ( z ) , the total number of operations is lower than for the directpolyphase implementation of the same filters.
To give an example, Figure 6.14shows the lifting implementation of the 9-7 filters from [ 5 ] , which are verypopular in image compression.162BanksChapter 6. FilterAn important result is that any two-channel filter bank can be factoredinto a finite number of lifting steps [37]. The proof is based on the Euclideanalgorithm [g].
The decomposition of a given filter bank into lifting steps is notunique, so that many implementations for the same filter bank can be found.Unfortunately, one cannot say a priori which implementation will performbest if the coefficients have to be quantized to a given number of bits.6.3Tree-Structured Filter BanksIn most applications one needs a signal decomposition into more than two,say M , frequency bands. A simple way of designing the required filters is tobuild cascades of two-channel filter banks. Figure 6.15 shows two examples,(a) a regular tree structure and (b) an octave-band tree structure. Furtherstructuresare easily found,and also signal-adaptive conceptshavebeendeveloped, wherethe treeis chosensuch that it is best matched to theproblem.In all cases, P R is easily obtained if the two-channel filter banks, which areused as the basic building blocks, provide PR.In orderto describe the system functions of cascaded filters with samplingrate changes, we consider the two systems in Figure 6.16.
It is easily seen thatboth systems are equivalent. Their system function isFor the system B2(z2)we haveWith this result, the system functions of arbitrary cascades of two-channelfilter banks are easily obtained.An example of the frequency responses of non-ideal octave-band filterbanks in tree structure is shown in Figure 6.17. An effect, which results fromthe overlap of the lowpass and highpass frequencyresponses, is the occurrenceof relatively large side lobes.6.3.
Tree-Structured Filter Banks1634hUU-I&-Figure 6.15. Tree-structuredfilterbanks;band tree structure.(a) regular tree structure; (b) octave-B2 (4Figure 6.16. Equivalent systems.164Chapter 6. Filter Banks1.o2g0.80.68g 0.42Fr,0.2n.o0.00.20.40.60.8m/Jc-1.0Figure 6.17. Frequency responses in tree-structured filter banks; (a) two-channelfilter bank; (b) octave-band filter bank.6.4UniformM-Channel Filter BanksThis section addresses uniform M-channel filter banks for which the samplingrate is reduced by N in all subbands. Figure 6.1 shows such a filter bank, andFigure 6.18 shows some frequencyresponses.
In orderto obtain general resultsfor uniform M-channel filter banks, we start by assuming N 5 M , where Mis the number of subbands.6.4.1Input-Output RelationsWe consider the multirate filter bank depicted in Figure 6.1. From equations(6.7) and (6.8) we obtain6.4. UniformM-Channel Filter Banks165W-a02.dMa@)Figure 6.18.
Frequency responses of the analysis filters in a uniform M-channelfilter bank; (a) cosine-modulated filter bank; (b) DFT filter bank.C1 N-lYj(z) = H k ( W h z k ) X(W&zk), k = 0 , . . . , MN i=O-1,(6.69)and_IX(Z)=M-lN-lC C Gk(z)Hk(W&z)X(W&z).k=OInorderGI,(z), k(6.70)i=oto achieve perfect reconstruction,suitablefilters H k ( z ) and= 0, . . . ,M - 1, and parameters N and M must be chosen. Weobtain the PR requirement by first changing the order of the summation in(6.70):X(Z)1 N-lM-]i=ok=O=-C X(W&Z)C G~(z)H~(W&Z). (6.71)Equation (6.71) shows that X ( z ) = 2 - 4X(z)holds if the filters satisfyM-lC Gk(z)Hk(W&z)= N z - ~&o, 0 < i < N - 1.k=O(6.72)166Chapter 6. Filter BanksUsing the notationH m ( z )=andz m ( z ) = [ X ( Z ) , X ( Z W N ) ,.
. . , X ( z W p ) ] T ,(6.75)the input-output relations may also be written asX(z) =1Hg T; ( z )z,(z).(6.76)[1,0,. . . ,O] .(6.77)Thus, PR requires that1N- g T ( z ) H Z ( z ) = 2-96.4.2ThePolyphaseRepresentationIn Section 6.2 we explained the polyphase representationof two-channel filterbanks. The generalization to M channels with subsampling by N is outlinedbelow.
The implementation of such a filter bank is depicted in Figure 6.19.Analysis. The analysis filter bank is described by!h(.)= E(z)Z P k ) ,(6.78)whereE ( z )=(6.81)M-Channel6.4. UniformBanks167FilterZ -1Z-1Figure 6.19. Filter bank in polyphasestructure.Synthesis. Synthesis may be described in a similar way:(6.82)withR ( z )=(6.83)Perfect Reconstruction. From (6.78) and (6.82) we conclude thePRrequirementR ( z ) E ( z ) = 2-40 I ,(6.84)+which results in an overall delay of Mqo M - 1 samples.
The generalizationto any arbitrary delay of Mqo r M - 1 samples is+ +(6.85)where 0< r < M - 1 [146].FIR Filter Banks. Let us write (6.84) as(6.86)and let us assume that all elements of E ( z ) are FIR.We see that theelementsof R ( z ) are also FIR if det{E(z)} is a monomial in z . The same arguments holdfor the more general P R condition (6.85). Thus, FIR solutions for both theanalysis and synthesis filters of a P R filter bank require that the determinantsof the polyphase matrices are just delays.168Chapter 6. Filter Banks6.4.3ParaunitaryFilter BanksThe paraunitarycase is characterized by the fact that the sumof the energiesof all subband signals is equal to the energy of the input signal.
This may beexpressed as llypll = l l z p l l V x p with llxpll < CO, where x p ( z )is the polyphasevector of a finite-energy input signal and yp(z) = E ( z ) xp(z)is the vector ofsubband signals. It can easily be verified that filter banks (oversampled andcritically sampled) are paraunitary if the following condition holds:k ( z )E ( z )= I .(6.87)This also implies thathk(n) = g;(-n)H I , ( z )= G k ( z ) ,t)k = 0 , . . . , M - 1.(6.88)Especially in the critically subsampled case where N = M , the impulseresponses hk(n-mM) andgk(n-mM), 5 = 0 , . . .
,M - l , m E Z,respectively,form orthonormal bases:(6.89)6.4.4Design of Critically Subsampled M-Channel FIRFilter BanksAnalogous to the lattice structures introducedin Sections 6.2.6 and 6.2.7, weconsider the following factorization of E ( z ) :where=[ IM-l .ol]0*(6.92)The matrices A k , k = 0,1,. . . , K are arbitrary non-singular matrices.
Theelements of these matrices are thefree design parameters, which can be chosenin order to obtain some desired filter properties. To achieve this, a usefulobjective function has to be defined and the free parameters have to be foundvia non-linear optimization. Typically, one tries to minimize the stopbandenergy of the filters.s169FilterM-Channel6.4.
UniformThe corresponding synthesis polyphase matrix can be designed asR ( z )= AilI'(z)ATII'(z)...I'(Z)A;',(6.93)(6.94)Clearly, both E ( z ) and R ( z ) contain FIR filters. The overall transfer matrixisR ( z ) E ( z )= C K I .(6.95)Figure 6.20 illustrates the implementation of the filter bank according to theabove factorizations.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
