Filter Banks (779443), страница 4
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A simple parameterization for the matrices AI, thatguarantees the existence of A i l is t o use triangular matriceswith oneson the main diagonal. The inverses then are also triangular, so that theimplementation cost is somehow reduced. Examples are given in [146].Paraunitary FIR Filter Banks based on Rotations. Paraunitary filterbanks are easily derived from the above scheme by restricting thematrices AI,to be unitary. Interestingly, not all matrices have to be fully parameterizedrotation matrices in order to cover all possible unitary filter banks [41]. Thematrices Ao, . .
. , A K - only~ have t o belong to thesubset of all possible M X Munitary matrices which can be written asa sequence of M - 1 Givens rotationsperformed successively on the elements Ic, L 1 for Ic = 0,1,. . . , M - 2.+- cos&)-sin4p)- sin 4r) cos 4r)AI,=--1...11(k)-1--(k)4K-lsin 4K-l- sin 6 K(k)-1cos 6 K - 1 -COS(k)(6.96)The last matrix AK has to be a general rotation matrix.
Filter design canbe carried out by defining an objective function and optimizing the rotationangles.Paraunitary FIR Filter Banks based on Reflections. A secondwayof parameterizing paraunitary filter banks was proposed in [148]. Here, thepolyphase matrix is written as follows:E ( z ) = VK(Z)VK-l(Z)** V,(z)U.(6.97)The matrices V I , ( Zare) reflection-like matrices of the typeVI, = I - :.,IV+ z-1vI,v:,(6.98)170Banks.Chapter 6. FilterII@)Figure 6.20. M-channel filter bank with FIR filters; (a) analysis; (b) synthesis.where vk: is an M X 1 vector with llvkll = vTv = 1.It is easily proventhat V r ( ~ - ~ ) V k (=z )I , so thatthematricescanindeedbeusedforparameterization.
The matrix U has to be a general unitary matrix. Theparameterization (6.97) directly leads to an efficient implementation, whichis similar to the one discussed in Section 3.4.4 for the implementation ofHouseholder reflections: instead of multiplying an input vector z ( z ) with anentire matrix V k ( z ) ,one computes z ( z ) - vk[l - K'] [vTz(z)] in order toobtain Vk:(z)z(.z).In addition to the above parameterizations, which generally yield nonlinear phase filters, methods for designing linear-phase paraunitary filter bankshave also been developed.
For this special class the reader is referred to [137].6.5DFT Filter BanksDFT filter banks belong to the class of modulated filter banks, where allfilters are derived from prototypes via modulation. Modulated filter bankshave the great advantage that only suitable prototypes must be found, notthe complete set of analysis and synthesis filters. One prototype is requiredfor the analysis and one for the synthesis side, and in most cases the sameprototypes can be used for both sides. Due to the modulated structure veryefficient implementations are possible.In DFT banks, theanalysis and synthesis filters, Hk:(z) and Gk:(z),are1716.5.
DFT Filter BanksInorder to explain the efficient implementation of DFT banks, let usconsider the critically subsampled case. The analysis equation isL-l%(m) =pk:).(z ( m M - n)n=O1,-l(6.100)n=OWe now substitute n = i M + j , L = ML, and rewrite (6.100) asThus, the subband signals can be computed by filtering the polyphase components of the input signal with the polyphase components of the prototype,followedby an IDFT (without pre-factor l / M ) . On the synthesis side, thesame principle can be used. The complete analysis/synthesis system, whichrequires extremely low computation effort, is depicted in Figure 6.21.For critical subsampling, as shownin Figure 6.21, the PR condition iseasily found to be(6.102)Thismeans thatthe polyphasecomponents of P R FIRprototypesarerestricted to length one, and the filtering degenerates to pointwise scalarmultiplication.
Thus, critically subsampled DFT filter banks with P R mainlyreduce to the DFT.If oversampling by afactor p =E Z is considered, the PR conditionbecomes [33, 861(6.103)172BanksChapter 6. FilterClearly, if a filter bank provides P R in the critically subsampled case, it alsoprovides P R in the oversampled case, provided the outputsignal is downscaledby the oversampling factor. Thus, (6.102) is included in (6.103).
This is mosteasily seen from (6.103) for p = 2:In general, (6.103) means an increased design freedom compared to (6.102).This freedom can be exploited in order to design FIR prototypes P(,) andQ ( z ) with good filter properties.The prototypes are typically designed to be lowpass filters. A commondesign criterion is to minimize the stopband energy and the passband ripple:Sa!passband(IP(ej")l- 1)zdw +Sp IP(ej")12dwL min.(6.104)At this point it should be mentioned that all P R prototypes for M-channelcosine-modulated filter banks, which will be discussed in the next section, alsoserve as PRprototypes for oversampled 2M-channel DFT filter banks.
On theother hand, satisfying only (6.103) is not sufficient in the cosine-modulatedcase. Thus, oversampled DFT filter banks offermoredesignfreedomthancosine-modulated ones.MDFT Filter Bank. Figure 6.22 shows the MDFTfilter bank introducedbyFliege. Compared to the simple DFT filter banks described above, this filterbank is modified in such a way that PR is achieved with FIR filters [55], [82].The key t o P Ris subsampling the filter output signals by M/2, extracting thereal and imaginary parts, and using them to compose the complex subbandsignals yk:(rn), L = 0 , . . .
,M - 1. As can be seen in Figure 6.22, the extractionof the real and imaginary parts takes place in adjoining channels in reverseorder.1736.5. DFT Filter BanksX^@)Figure 6.22. Modified complex modulated filter bank with critical subsampling.DFT PolyphaseFilterBank with IIR Filters andPerfectReconstruction. We consider theDFT filter bank in Figure 6.21. Husmy andRamstad proposed to construct the polyphase components of the prototypeas first-order IIR allpass filters [74]:1ai + z - lm 1 + aiz-1’Pi(2)= -i = o , ... , A 4 - 1 .(6.105)Using the synthesis filtersthen ensures perfect reconstruction.
Unfortunately, this leads to a problemconcerningstability: if the analysis filters arestable,thesynthesis filtersdetermined according to (6.106) are not. This problem can be avoidedbyfiltering the subband signals “backwards”using the stable analysis filters.Then, the desired output signal is formed by another temporal reversal. Thisis not a feasible strategy if we work with one-dimensionalsignals, but in imageprocessing we a priori have finite-length signals so that this method can beapplied nevertheless.The quality of a filter bank is not only dependent on whetherit reconstructs perfectly or not. The actual purpose of the filter bank is to separatedifferent frequency bands, for example in order to provide a maximal coding174BanksChapter 6.
Filtergain. Thestopbandattenuationof theprototype P ( z ) composed of IIRallpasses is determined by the parameters ai, i = 0 , . . . , M - 1, so that theseare the design parameters. Husrrry and Ramstad statea stopband attenuationof 36.4 dB for the prototype P(,) of an eight-channel filter bank [74]. In viewof the extremely low computational cost this is an astonishing value.6.6Cosine-ModulatedFilterBanksCosine-modulated filter banks are very populardue to their real-valuednature and their efficient implementation via polyphase structure and fastDCT [116, 127, 94, 121, 87, 100, 110, 129, 681. Cosine-modulated filterbanks can be designed as pseudo QMF banks [127], paraunitary filter banks[94, 121, 87, 100, 1031, and also as biorthogonal filter banks allowing lowreconstructiondelay [110, 129, 68, 83, 861.
Perfect reconstruction is easilyachieved by choosing an appropriate prototype.For example, the MPEG audiostandard [l71 is based on cosine-modulated filter banks.In the following, we will consider biorthogonalcosine-modulated filterbanks where the analysis filters hk(n),Ic = 0, . . . , M - 1, are derived froman FIR prototypep ( n ) and the synthesis filters g k ( n ) , Ic = 0 , . . . ,M - 1, froman FIR prototype q(n) according tohk(lZ)= 2p(n)cos[G ( k + i )gk(n) = 2q(n)cos[$(v;)+qh],- 4 ,(k+i)n=O,..., L,-1n=O,..., L , - l .The length of the analysis prototype is L,, and the length of the synthesisprototype is L,. The variable D denotes the overall delay of the analysissynthesis system. A suitable choice for q5k is given by r$k = (-1)"/4[87,95].For the sake of brevity, we confine ourselves to even M , analysis andsynthesis prototypes with lengths L, = 2mM and L, = 2m'M, m,m' E IN,and an overall delay of D = 2sM 2M - 1 samples.
Note that the delaycan be chosen independently of the filter length, so that the design of lowdelay banks is included here. The most common case within this frameworkis the onewhere the sameprototype isusedforanalysis and synthesis.However, in order to demonstrate some of the design freedom, we start witha more general approach where different prototypes are used for analysis andsynthesis. Generalizations to all filter lengths and delays are given in [68].+Banks1756.6.FilterCosine-ModulatedIn order to derive the conditions that must be met by the prototypes P ( z )and Q(z) to yield perfect reconstruction, we first decompose them into 2Mpolyphase components. Note that in the case of DFT filter banks, only Mpolyphase components were used to describe an M-channel filter bank.
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