Tài liệu 36 Filter Bank Design doc

Arrowood, J.; Randolph, T & Smith, M.J.T. “Filter Bank Design” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999 c  1999byCRCPressLLC 36 Filter Bank Design Joseph Arrowood Georgia Institute of Technology Tami Randolph Georgia Institute of Technology Mark J.T. Smith Georgia Institute of Technology 36.1FilterBankEquations TheACMatrix • Spectral Factorization • Lattice Implementa- tions • Time-Domain Design 36.2 Finite Field Filter Banks 36.3 Nonlinear Filter Banks References The interest in digital filter banks has grown dramatically over the last few years. Owing to the trend toward lower cost, higher speed microprocessors, digital solutions are becoming attractive for a wide variety of applications. Filter banks allow signals to be decomposed into subbands, often facilitating more efficient and effective processing. They are particularly visible in the areas of image compression, speech coding, and image analysis. The desired characteristics of a subband decomposition will naturally vary from application to application. Moreover,withinanygivenapplication, thereareamyriadofissuestoconsider. First, one mightconsiderwhethertouseFIRorIIRfilters. IIRdesigns canoffercomputationaladvantages, while FIR designs can offer greater flexibility in filter characteristics. In this chapter we focus exclusively on FIR design. Second, one might identify the time-frequency or space-frequency representation that is most appropriate. Uniform decompositions and octave-band decompositions are particularly popular at present. At the next level, characteristics of the analysis filters should be defined. This involves imposing specifications on the analysis filter passband deviations, transition bands, and stopband deviations. Alternately or in addition, time domain characteristics may be imposed, such as limits on the step response ripples, and degree of regularity. One can consider similar constraints for the synthesis filters. For coding applications, the charac- teristics of the synthesis filters often have a dominant effect on the subjective quality of the output. Finally, one should consider analysis-synthesis characteristics. That is, one has flexibility to specify the overall behavior of the system. In most cases, one views havingexact reconstructionas being ideal. Occasionally, however, it may be possible to trade some small loss in reconstruction quality for signif- icant gains in computation, speed, or cost. In addition to specifying the quality of reconstruction, it is generally possible to control the overall delay of the system from end to end. In some applications, such as two-way speech and video coding, latency represents a source of quality degradation. Thus, having explicit control over the analysis-synthesis delay can lead to improvement in quality. The intelligent design of applications-specific filter banks involves first identifying the relevant parameters and optimizing the system with respect to them. As is typical, the filter bank analysis and reconstruction equations lead to complex tradeoffs among complexity, system delay, filter quality, filter length, and quality of performance. This chapter is devoted to presenting an introduction to filter bank design. Filter bank design has reached a state of maturity in many regards. To cover all of c  1999 by CRC Press LLC FIGURE 36.1: Block diagram of an M-band analysis-synthesis filter bank. FIGURE 36.2: Two-band analysis-synthesis filter bank. the important contributions in any level of detail would be impossible in a single chapter. However, it is possible to gain some insight and appreciation for general design strategies germane to this topic. In addition to discussing design methodologies for linear analysis-synthesis systems, we also consider the design of a couple of new nonlinear classes of filter banks that are currently receiving attention in the literature. This discussion along with the referenced articles should provide a convenient introduction to the design of many useful filter banks. 36.1 Filter Bank Equations A broad class of linear filter banks can be represented by the block diagram shown in Fig. 36.1. This is a linear time-varying system that decomposes the input into M-subbands, each one of which is decimated by a factor of R. When R = M, the system is said to be critically sampled or maxi- mally decimated. Maximally decimated systems are generally the ones of choice because they can be information preserving, and are not data expansive. The simplest filter bank of this class is the two-band system, an example of which is shown in Fig. 36.2. Here, there are only two analysis filters: H 0 (z), a lowpass filter; and H 1 (z), a highpass filter. Similarly, there are two synthesis filters: a lowpass G 0 (z), and a highpass G 1 (z). Let us consider this two-band filter bank first. In the process, we will develop a design methodology that can be extended to the more complex problem of M-band systems. Examining the two-band filter bank in Fig. 36.2, we see that the input x[n] is lowpass and highpass filtered, resulting in v 0 [n] and v 1 [n]. These signals are then downsampled by a factor of two, leading to the analysis section outputs, y 0 [n] and y 1 [n]. The downsampling operation is time varying, which implies a non-trivial relationship between v k [n] and y k [n] (where k = 0, 1). In general, downsampling a signal v k [n] by an integer factor R is described in the time domain by the equation y k [n]=v k [Rn]. c  1999 by CRC Press LLC In the frequency domain, this relationship is given by Y k  e jω  = 1 R R−1  r=0 V k  e j  ω R + 2πr R   . The equivalent equation in the z domain is Y k (z) = 1 R R−1  r=0 V k  W r R z 1 R  where W r R = e −j 2πr R . In the synthesis section, the subband signals y 0 [n] and y 1 [n] are upsampled to give s 0 [n] and s 1 [n]. They are then filtered by the lowpass and highpass filters, G 0 (z) and G 1 (z), respectively, before being summed together. The upsampling operation (for an arbitrary positive integer R) can be defined by s k [n]=  y k [n/R] for n = 0, ±R, ±2R, ±3R, . 0 otherwise in the time domain, and S k  e jω  = Y k  e jRω  and S k (z) = Y k  z R  in the frequency and z domains, respectively. Using the expressions for the downsampling and upsampling operations, we can describe the two-band filter bank in terms of z-domain equations. The outputs after analysis filtering are V k (z) = H k (z)X(z), k = 0, 1. After decimation and recognizing that W 1 2 =−1, we obtain Y k (z) = 1 2  H k  z 1 2  X  z 1 2  + H k  −z 1 2  X  −z 1 2  ,k= 0, 1. (36.1) Thus, Eq. (36.1) defines completely the input-output relationship for the analysis section in the z domain. In the synthesis section, the subbands are upsampled giving S k (z) = Y k (z 2 ), k = 0, 1. This implies that S k (z) = 1 2 ( H k (z)X(z) + H k (−z)X(−z) ) ,k= 0, 1. Passing S k (z) through the synthesis filters and then summing yields the reconstructed output ˆ X(z) = 1 2 G 0 (z) [ H 0 (z)X(z) + H 0 (−z)X(−z) ] + 1 2 G 1 (z) [ H 1 (z)X(z) + H 1 (−z)X(−z) ] . (36.2) For virtually any application for which one can conceive, the synthesis filters should allow the input to be reconstructed exactly or with a minimal amount of distortion. In other words, ideally we want ˆ X(z) = z −n 0 X(z) , where n 0 is the integer system delay. An intuitive approach to handing this problem is to use the AC-matrix formulation, which we introduce next. c  1999 by CRC Press LLC 36.1.1 The AC Matrix The aliasing component matrix (or AC matrix) represents a simple and intuitive idea originally introducedin[6] for handling analysis and reconstruction. The analysis-synthesis equation (36.2) for the two-band case can be expressed as ˆ X(z) = 1 2 [ H 0 (z)G 0 (z) + H 1 (z)G 1 (z) ] X(z) + 1 2 [ H 0 (−z)G 0 (z) + H 1 (−z)G 1 (z) ] X(−z) . The idea of the AC matrix is to represent the equations in matrix form. For the two-band system, this results in ˆ X(z) = 1 2 [ X(z),X(−z) ]  H 0 (z) H 1 (z) H 0 (−z) H 1 (−z)     AC matrix  G 0 (z) G 1 (z)  , where the AC matrix is as shown above. The AC matrix is so designated because it contains the analysis filters and all the associated aliasing components. Exact reconstruction is then obtained when  H 0 (z) H 1 (z) H 0 (−z) H 1 (−z)  G 0 (z) G 1 (z)  =  T(z) 0  where T(z)is required to be the scaled integer delay 2z −n 0 .ThetermT(z)is the transfer function of the overall system. The zero term below T(z)determines the amount of aliasing present in the reconstructed signal. Because this term is zero, all aliasing is explicitly removed. With the equations expressed in matrix form, we can solve for the synthesis filters, which yields  G 0 (z) G 1 (z)  = 1 H 0 (z)H 1 (−z) − H 0 (−z)H 1 (z)  H 1 (−z) −H 1 (z) −H 0 (−z) H 0 (z)  T(z) 0  . (36.3) Often for a variety of reasons, we would like both the analysis and synthesis filters to be FIR. This means the determinant of the AC matrix should be a constant delay. The earliest solution to the FIR filter bank problem was presented by Croisier et al. in 1976 [18]. Their solution was to let H 1 (z) = H 0 (−z) and G 0 (z) = H 0 (z) G 1 (z) =−H 0 (−z) . This is the quadrature mirror filter (QMF) solution. From the equations in (36.3), it can be seen that this solution cancels all the aliasing and results in a system transfer function T(z)= H 0 (z)H 1 (−z) − H 0 (−z)H 1 (z) . As it turns out, with careful design T(z)can be made to be close to a constant delay. However, some amount of distortion will always be present. In 1980 Johnston designed a set of optimized QMFs which are now widely used. The coefficient values may be found in several sources [16, 17, 19]. Interestingly, the equations in (36.3) imply that exact reconstruction is possible by forcing the AC-matrix determinant to be a constant delay. The design of such exact reconstruction filters is discussed in the next section. c  1999 by CRC Press LLC FIGURE 36.3: Example of a zero-phase half-band lowpass filter. 36.1.2 Spectral Factorization The question at hand is how dowe determine H 0 (z) and H 1 (z) such that T(z)is an integer delay z −n 0 . A solution to this problem was introduced in 1984 [7], based on the observation that H 0 (z)H 1 (−z) is a lowpass filter [which we denote F 0 (z)] and H 0 (−z)H 1 (z) is its corresponding frequency shifted highpass filter. A unity transfer function can be constructed by forcing F 0 (z) and F 0 (−z) to be complementary half-band lowpass and highpass filters. Many fine techniques are available for the design of half-band lowpass filters, such as the Parks-McClellan algorithm, Kaiser window design, Hamming window design, the eigenfilter method, and others. Zero-phase half-band filters have the property that zeros occur in the impulse response at n =±2, ±4, ±6, ., etc. An illustration is shown in Fig. 36.3. Once designed, F 0 (z) can be factored into two lowpass filters, H 0 (z) and H 1 (−z). The design procedure can be summarized as follows. 1. First design a ( 2N − 1 ) -tap half-band lowpass filter, using the Parks-McClellan algo- rithm, for example. This can be done by constraining the passband and stopband cutoff frequencies to be ω p = π −ω s , and using equal passband and stopband error weightings. The resulting filter will have equal passband and stopband ripples, i.e., δ p = δ s = δ. 2. Add the value δ to the f [0] (center) tap value. This forces F(e jω ) ≥ 0 for all ω. 3. Spectrally factor F(z) into two lowpass filters, H 0 (z) and H 1 (−z). Generally the best waytofactorF(z)is such that H 1 (−z) = H 0 (z −1 ). Note that the factorization will not be unique and the roots should be split so that if a particular root is assigned to H 0 (z), its reciprocal should be given to H 0 (z −1 ). The result of the above procedure is that H 0 (z) will be a power complementary, even length, FIR filter that will form the basis for a perfect reconstruction filter bank. Note that since H 1 (z) is just a time-reversed, spectrally shifted version of H 0 (z),    H 0 (e jω )    =    H 1 (−e jω )    . Smith and Barnwell designed and published a set of optimal exact reconstruction filters [1]. The filter coefficients for H 0 (z) are given in Table 36.1. The analysis and synthesis filters are obtained from H 0 (z) by G 0 (z) = H 0  z −1  G 1 (z) = H 0 (−z) H 1 (z) = H 0  −z −1  . A complete discussion of this approach can be found in many references [1, 6, 7, 25, 27, 28]. c  1999 by CRC Press LLC TABLE 36.1 CQF (Smith-Barnwell) Filter Bank Coefficients with 40dB Attenuation 32-Tap filter 16-Tap filter 8.494372478233170D − 03 2.193598203004352D − 02 − 9.617816873474045D − 05 1.578616497663704D − 03 − 8.795047132402801D − 03 − 6.025449102875281D − 02 7.087795490845020D − 04 − 1.189065962053910D − 02 1.220420156035413D − 02 0.137537915636625D + 00 − 1.762639314795336D − 03 5.745450056390939D − 02 − 1.558455903573829D − 02 − 0.321670296165893D + 00 4.082855675060479D − 03 − 0.528720271545339D + 00 1.765222024089335D − 02 − 0.295779674500919D + 00 − 8.385219782884901D − 03 2.043110845170894D − 04 − 1.674761388473688D − 02 2.906699709446796D − 02 1.823906210869841D − 02 − 3.533486088708146D − 02 5.781735813341397D − 03 − 6.821045322743358D − 03 − 4.692674090907675D − 02 2.606678468264118D − 02 5.725005445073179D − 02 1.033363491944126D − 03 0.354522945953839D + 00 − 1.435930957477529D − 02 0.504811839124518D + 00 0.264955363281817D + 00 − 8.329095161140063D − 02 − 0.139108747584926D + 00 3.314036080659188D − 02 9.035938422033127D − 02 − 1.468791729134721D − 02 8-Tap filter − 6.103335886707139D − 02 6.606122638753900D − 03 3.489755821785150D − 02 4.051555088035685D − 02 − 1.098301946252854D − 02 − 2.631418173168537D − 03 − 6.286453934951963D − 02 − 2.592580476149722D − 02 0.223907720892568D + 00 9.319532350192227D − 04 0.556856993531445D + 00 1.535638959916169D − 02 0.357976304997285D + 00 − 1.196832693326184D − 04 − 2.390027056113145D − 02 − 1.057032258472372D − 02 − 7.594096379188282D − 02 For the M-channel case shown in Fig. 36.1, where the bands are assumed to be maximally deci- mated, the same AC-matrix approach can be employed, leading to the equations ˆ X(z) = 1 M  X(z), .,X(zW M−1 M )     x T      H 0 (z) ··· H M−1 (z) H 0 (zW 1 M ) ··· H M−1 (zW 1 M ) . . . . . . H 0 (zW M−1 M ) ··· H M−1 (zW M−1 M )         H      G 0 (z) G 1 (z) . . . G M−1 (z)         g , where W M = e −j 2π M . This can be rewritten compactly as ˆ X(z) 1 M x T (z)H(z)g(z) , where x is the input vector, g is the synthesis filter vector, and H is the AC matrix. However, the AC- matrix determinant for systems with M>2 is typically too intricate for the spectral factorization approach outlined above. An effective approach for handling the design of M-band systems was introduced by Vaidyanathanin [30]. It is based on a lattice implementation structure and is discussed next. c  1999 by CRC Press LLC FIGURE 36.4: Flow graph of a two-band lattice structure with three stages. 36.1.3 Lattice Implementations In addition to the directform structures shown in Figs. 36.1 and 36.2, filter banks can be implemented using lattice structures. For simplicity, consider the two-band case first. An example of a lattice structure for a two-band analysis system is shown in Fig. 36.4. It is composed of a cascade of criss- cross elements, each of which has a set of coefficients associated with it. Conveniently, each section, which we denote R m , can be described by a matrix. For the two-band lattice, these matrices have the form R m =  1 r m −r m 1  . Interspersed between the coefficient matrices are delay matrices, (z), having the form (z) =  10 0 z −1  . It can be shown [27] that lattice filters can represent a wide class of exact reconstruction filter banks. Two points regarding lattice filter banks are particularly noteworthy. First, the lattice structure provides an efficient form of implementation. Moreover, the synthesis filter bank is directly related to the analysis bank, since each matrix in the analysis cascade is invertible. Consequently, the synthesis bank consists of the cascade of inverse section matrices. Second, the structure also provides a convenient way to design the filter bank. Each lattice coefficient can be optimized using standard minimization routines to minimize a passband-stopband error cost function for the filters. This approach to design can be used for two-band as well as M-band filter banks [5, 27, 28]. 36.1.4 Time-Domain Design One of the most flexible design approaches is the time domain formulation proposed by Nayebi et al. [3, 8]. This formulation has enabled the discovery of previously unknown classes of filter banks, such as low and variable delay systems [12], time-varying filter banks [4], and block decimation systems [9]. It is attractive because it enables the design of virtually all linear filter banks. The idea underlying this approach is that the conditions for exact reconstruction can be expressed in the time domain in a convenient matrix form. Let us explore this approach in the context of an M-band filter bank. Because of the decimation operations, the overall M-band analysis-synthesis system is periodically time-varying. Thus, we can view an arbitrary maximally decimated M-band system as having M linear time invariant transfer functions associated with it. One can think of the problem as trying to devise M subsampled systems, each one of which exactly reconstructs. This is equivalent to saying that for each impulse input, δ[n − i], to the analysis-synthesis system, that impulse should appear at the system output at time n = i + n 0 ,wherei = 0, 1, 2, .,M − 1 and n 0 is the system delay. This amounts to setting up an overconstrained linear system AS = B, where the matrix A is created using the analysis filter coefficients, the matrix B is the desired response of zeros except at the appropriate delay points (i.e., δ[n − n 0 ]) and S is a matrix containing synthesis filter coefficients. c  1999 by CRC Press LLC Particular linear combinations of analysis and synthesis filter coefficients occur at different points in time for different input impulses. The idea is to make A, S, and B such that they describe completely all M transfer functions that comprise the periodically time-varying system. The matrix A is a matrix of filter coefficients and zeros that effectively describe the decimated con- volution operations inherent in the filter bank. For convenience, we express the analysis coefficients as a matrix h,where h =      h 0 [0] h 1 [0] ··· h M−1 [0] h 0 [1] h 1 [1] ··· h M−1 [1] . . . . . . . . . h 0 [N − 1] h 1 [N − 1] ··· h M−1 [N − 1]      . The zeros are represented by an M × M matrix of zeros, denoted O M . With these terms, we can write the (2N − M) × N matrix A, A =                h[n]     O M . . . O M O M     h[n]     . . . O M ··· ··· ··· ··· ··· O M . . . O M     h[n]                . The synthesis filters S can be expressed most conveniently in terms of the M × M matrix Q i =      g 0 [i] g 0 [i + 1] ··· g 0 [i + M − 1] g 1 [i] g 1 [i + 1] ··· g 1 [i + M − 1] . . . . . . . . . g M−1 [i] g M−1 [i + 1] ··· g M−1 [i + M − 1]      , where i = 0, 1, .,L−1 and N is assumed to be equal to LM. The synthesis matrix S is then given by S =           Q 0 Q M . . . Q iM . . . Q (L−1)M           . Finally, to achieve exact reconstruction we want the impulse responses associated with each of the M constituent transfer functions in the periodically time-varying system to be an impulse. Therefore, B is a matrix of zero-element column vectors, each with a single “one” at the location of the particular transfer function group delay. More specifically, the matrix has the form B =             O M O M . . . J M . . . O M O M             c  1999 by CRC Press LLC where J M is the M × M antidiagonal identity matrix J M =      0 ··· 01 0 ··· 10 . . . . . . . . . 1 ··· 00      . It is important to mention here that the location of J M within the matrix B is a system design issue. The case shown here, where it is centered within B, corresponds to an overall system delay of N − 1. This is the natural case for systems with N-tap filters. There are many fine points associated with these time domain conditions. For a complete discussion, the reader is referred to [3]. With the reconstruction equations in place, we now turn our attention to the design of the filters. The problem here is that this is an over-constrained system. The matrix A is of size (2N − M) × N. If we think of the synthesis filter coefficients as the parameters to be solved for, we find M(2N − M) equations and MN unknowns. Clearly, the best we can hope for is to determine B in an approximate sense. Using least-squares approximation, we let S =  A T A  −1 B . Here, it is assumed that  A T A  −1 exists. This is not automatically the case. However, if reasonable lowpass and highpass filters are used as an initial starting point, there is rarely a problem. This solution gives the best synthesis filter set for a particular analysis set and system delay N − 1. The resulting matrix AS = ˆ B will be close to B but not equal to it in general. The next step in the design is to allow the analysis filter coefficients to vary in an optimization routine to reduce the Frobenius matrix norm,       ˆ B − B       2 F . The locally optimal solution will be, S =  A T A  −1 B, such that       ˆ B − B       2 F is minimized . Any number of routines may be used to find this minimum. A simple gradient search that updates the analysis filter coefficients will suffice in most cases. Note that, as written, there are no constraints on the analysis filters other than that they provide an invertible A T A matrix. One can easily start imposing constraints relevant to system quality. Most often we find it appropriate to include constraints on the frequency domain characteristics of the individual analysis filters. This can be done conveniently by creating a cost function comprised of the passband and stopband filter errors. For example, in the two-band case, inclusion of such filter frequency constraints gives rise to the overall error function  =       ˆ B − B       2 F +  π p 0    1 − H 1 (e jω )    2 dω +  π π s    H 0 (e jω )    2 dω. This reduces the overall system error of the filter bank while at the same time reducing the stopband errors in analysis filters. Other options in constructing the error function can address control over the step response of the filters, the width of the transition bands, and whether an l 2 norm or an l ∞ norm is used as an optimality criterion. By properly weighting the reconstruction and frequencyresponse terms in the error function, exact reconstruction can be obtained, if such a solution exists. If an exact reconstruction solution does not exist, the design algorithm will find the locally optimal solution subject to the specified constraints. c  1999 by CRC Press LLC [...]... Thus, there is interest in designing filter banks that are less prone to producing annoying distortions in these cases Other nonlinear classes of filter banks can be considered that display different forms of distortion at low bit rates In the remainder of this chapter, we discuss the design of two nonlinear filter banks that are presently being studied 36. 2 Finite Field Filter Banks A new and interesting... must pay careful attention to the subband output size, filter length, and coefficient values during the design of the filter bank Nonetheless, it seems that finite field filter banks are potentially attractive in some applications 36. 3 Nonlinear Filter Banks One of the driving forces for research in filter banks is image coding for low bit rate applications Presently, subband image coders represent the best... This time domain formulation has been used to design an unprecedented variety of filter banks, including the first block decimation systems, the first time-varying systems, the first low delay systems, cosine modulated filter banks, nonuniform band filter banks, and many others [3, 4, 9, 10, 11] One of the most important in this list is cosine modulated filter banks because they can be implemented very efficiently... based on conventional linear filter banks suffer from ringing effects due to the Gibbs phenomenon These ringing effects occur around edges or high contrast regions One way to eliminate ringing is to use nonlinear filter banks There are pros and cons regarding the utility of nonlinear filter banks However, the design of the systems is rather new and interesting Nonlinear filter banks can be constructed within... possibilities for constructing nonlinear filter banks What is less obvious at this point is the impact of these systems in practical situations Given that development related to these filter banks is only in the formative stages, only time will tell Regardless of whether conventional or nonlinear filter banks are ultimately employed, the variety of design options and design techniques offer many useful solutions... filter banks, IEEE Intl Symposium on Circuits and Systems, San Diego, 947–950, May 1992 [10] Nayebi, K., Barnwell, T and Smith, M., Design and implementation of computationally efficient modulated filter banks, Proc Intl Symposium on Circuits and Systems, Singapore, 650–653, June 12-14, 1991 [11] Nayebi, K., Barnwell, T.P and Smith, M.J.T., Design of perfect reconstruction nonuniform band filter banks,... filter banks with perfect reconstruction, Electronics Lett., 26(13), 906–907, June 1990 [25] Mintzer, F., Filters for distortion-free two-band multirate filter banks, IEEE Trans on Acoustics, Speech, and Signal Processing, ASSP-33, 626–630, June 1985 [26] Akansu, Ali N and Haddad, R.A., Multiresolution Signal Decomposition, Academic Press, 1992 [27] Vaidyanathan, P.P., Multirate Systems and Filterbanks,... The design of digital filters for exact reconstruction in subband coding, Trans on Acoustics, Speech, and Signal Proc., ASSP-34(3), 434–441, June 1986 [2] Smith, M and Barnwell, T., A new filter bank theory for time-frequency representation, Trans on Acoustics, Speech, and Signal Proc., ASSP-35(3), 314–327, March 1987 [3] Nayebi, K., Barnwell, T and Smith, M., Time domain filter bank analysis: A new design. .. choose a system gain of 2 In this decomposition (Fig 36. 6), the lower band image is not what we are accustomed to observing in a conventional decomposition This case does have a lower first-order entropy than its conventional counterpart c 1999 by CRC Press LLC FIGURE 36. 6: A four-level octave band decomposition using finite field filter banks Finite field filter banks are still in their early phases of study... K., Barnwell, T.P and Smith, M.J.T., Design of low delay FIR analysis-synthesis filter bank systems, Proc Conf on Information Sciences and Systems, March 1991 [13] Nayebi, K., Barnwell, T and Smith, M., Time-domain view of filter banks and wavelets, Asilomar Conference on Signals, Systems and Computers, Nov 2-6, 1991 [14] Mersereau, R.M and Smith, M.J.T., Digital Filtering: A Computer Laboratory Textbook, . Technology 36. 1FilterBankEquations TheACMatrix • Spectral Factorization • Lattice Implementa- tions • Time-Domain Design 36. 2 Finite Field Filter Banks 36. 3. introduction to filter bank design. Filter bank design has reached a state of maturity in many regards. To cover all of c  1999 by CRC Press LLC FIGURE 36. 1: Block