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 • LatticeImplementa- 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, hig her 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 effectiveprocessing. 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. IIRdesignscanoffercomputationaladvantages,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 thatismostappropriate. Uniformdecompositionsandoctave-banddecompositions areparticularly 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, t ransition bands, and stopband deviations. Alternately or in addition, time domain characteristics may be imposed, such as limits on the step response ripples, anddegree 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 theoverallbehaviorofthesystem. Inmostcases,oneviewshavingexactreconstructionasbeingideal. Occasionally,however,itmaybepossibletotradesomesmalllossinreconstructionqualityforsignif- icant gains in computation, speed, or cost. In addition to specifying the quality of reconstruction, it is generally possible to control the overall delayof the system from end toend. Insomeapplications, 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 parametersandoptimizingthesystemwith respecttothem. As is typical, the filterbankanalysisand 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 coverall 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 beimpossible in a single chapter. However, itispossibletogainsomeinsightandappreciationforgeneraldesignstrategiesgermanetothistopic. Inadditiontodiscussingdesig nmethodologiesforlinearanalysis-synthesissystems,wealsoconsider 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 linearfilter banks can berepresented 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,thereare only twoanalysisfilters: H 0 (z), alowpassfilter;andH 1 (z), ahighpass filter. Similarly, there are two synthesis filters: a lowpass G 0 (z), and a highpass G 1 (z). Let us consider this two-bandfilterbankfirst. Intheprocess,wewilldevelopadesignmethodology thatcanbeextended to the more complex problem of M-bandsystems. Examiningthetwo-bandfilterbankinFig.36.2, we seethattheinputx[n] islowpassandhighpass filtered,resultingin v 0 [n] and v 1 [n]. Thesesignals are then downsampled by a factorof 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 thez 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 . Inthesynthesissection,thesubbandsignalsy 0 [n] andy 1 [n] areupsampledtogives 0 [n] ands 1 [n]. Theyarethenfilteredbythelowpass andhighpassfilters,G 0 (z) andG 1 (z), respectively,beforebeing summed together. The upsampling operation (for an arbitrary positive integer R)canbedefined 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 oper ations, 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 reconstructedexactly 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 matr ix is to represent the equations in mat rix 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)     ACmatrix  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 y ields  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 ACmatrix 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) . Thisisthequadraturemirrorfilter(QMF)solution. Fromtheequationsin(36.3), it can beseenthat 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 ThequestionathandishowdowedetermineH 0 (z) andH 1 (z) suchthatT(z)isanintegerdelayz −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 showninFig.36.3. Oncedesigned,F 0 (z) canbefactoredintotwolowpassfilters,H 0 (z) andH 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 frequenciestobeω p = π −ω s ,andusingequalpassbandandstopbanderrorweightings. 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 p erfect 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 TABLE36.1 CQF (Smith-Barnwell) Filter Bank Coefficients with 40dB Attenuation 32-Tapfilter 16-Tapfilter 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-Tapfilter −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 show n 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) , wherex isthe input vector, g is the synthesis filter vector, and H is the ACmatrix. 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 introducedbyVaidyanathanin[30]. Itisbasedonalatticeimplementationstructureandisdiscussed 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 InadditiontothedirectformstructuresshowninFigs.36.1and36.2,filterbankscanbeimplemented 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 synthesisbankconsistsofthecascadeofinversesectionmatrices. Second,thestructurealsoprovides 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. Thisisequivalent 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 matr ix 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. Thematrix A is a matrix of filtercoefficientsandzerosthateffectively describe the decimated con- volution operations inherent in thefilter 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]      . ThezerosarerepresentedbyanM × M matrixofzeros,denotedO M . Withtheseterms,wecanwrite 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]      , wherei = 0, 1, ,L− 1 and N isassumed to be equal toLM. The synthesismatrix S is then given by S =           Q 0 Q M . . . Q iM . . . Q (L−1)M           . Finally, to achieveexactreconstruction we want the impulse responsesassociatedwith each of the M constituenttransferfunctionsintheperiodicallytime-varyingsystemtobeanimpulse. 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 centeredwithin 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 inplace, 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) equationsandMN unknowns. Clearly,thebestwecanhopefor is todetermineBinanapproximate 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 g ives the best synthesis filter set for a particular analysis set and system delayN − 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, inclusionof 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 cr iterion. Byproperlyweightingthereconstructionandfrequencyresponsetermsintheerrorfunction,exact reconstruction can be obtained, if such a solution exists. Ifanexactreconstruction solution doesnot exist, the design algorithm will find the locally optimal solution subject to the specified constraints. c  1999 by CRC Press LLC [...]... Acoustics, Speech, Signal Processing, Denver, CO, April 1980 [18] Croisier, A., Esteban, D and Galand, C., Perfect channel splitting by use of interpolation/decimation/tree decomposition techniques, Conf on Information Sciences and Systems, 1976 [19] Crochiere, R.E and Rabiner, L.R., Multirate Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1983 [20] Malvar, H.S., Signal Processing with... design theory, IEEE Trans on Signal Processing, 40(6), 1412–1429, June 1992 [4] Nayebi, K., Barnwell, T and Smith, M., Analysis-synthesis systems based on time varying filter banks, Intl Conf on Acoustics, Speech, and Signal Processing, IV, 617–620, March 1992 [5] Schuller, G and Smith, M.J.T., A new framework for modulated perfect reconstruction filter banks, IEEE Trans Signal Processing, August 1996 [6]... reconstruction, Proc IEEE Intl Conf Acoustics, Speech, Signal Processing, 1991 [22] Rothweiler, J., Polyphase quadrature mirror filters — a new sub-band coding technique, Proc IEEE Intl Conf Acoustics, Speech, Signal Processing, 1983 [23] Nussbaumer, H.J and Vetterli, M., Computationally efficient QMF filter banks, Proc IEEE Intl Conf Acoustics, Speech, Signal Processing, 1984 [24] Malvar, H., Modulated QMF... 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, Prentice-Hall, Englewood Cliffs, NJ, 1993 [28] Vetterli, M and Kovacevic, J., Wavelets and Subband Coding, Prentice-Hall, Englewood Cliffs, NJ, 1995 [29] Fleige, N.J., Multirate Digital Signal Processing, ... 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, John Wiley & Sons, New York, 1993 [15] Akansu, A and Smith, M., Eds., Subband and Wavelet Transforms: Design and Applications, Kluwer Academic Publishers, 1995 [16] Smith, M and Docef, A., A Study Guide to Digital Image Processing, Scientific Publishers,... the references References [1] Smith, M and Barnwell, T., 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,... the synthesis filters using the same wrap-around arithmetic within the same finite integer field The bands are then combined using modulo-N addition As it turns out, the resulting signal will not match the original However, the signal can be corrected by applying a mapping based on the gain of the filter banks, M, and the dynamic range, N Let us assume that the input is an image with N discrete levels,... framework for maximally decimated analysis/synthesis systems, Proc Intl Conf on Acoustics, Speech, and Signal Proc., 521–524, March 1985 [7] Smith, M and Barnwell, T., A procedure for designing exact reconstruction filter banks for treestructured subband coders, Proc Intl Conf on Acoustics, Speech, and Signal Proc., 27.1.1– 27.1.4, March 1984 [8] Nayebi, K., Barnwell, T and Smith, M., Time domain conditions... Signal Processing, John Wiley & Sons, New York, 1993 [30] Vaidyanathan, P.P., Quadrature mirror filter banks, M-band extensions and perfect reconstruction techniques, IEEE Trans on Acoustics, Speech, and Signal Processing, July 1987 [31] Egger, O and Li, W., Very low bit rate image coding using morphological operators and adaptive decompositions, ICIP ’94, 2, 326–330, Nov 1994 [32] Florencio, D.A.F and Schafer,... fij and gij to be nonlinear filters Thus, y0 [n] y1 [n] = = f00 (x0 [n]) + f01 (x1 [n]) f10 (x0 [n]) + f11 (x1 [n]) where fij (·) are the linear or nonlinear polyphase analysis filters To reconstruct the signal, the output can be expressed as a filtered combination of the channels, x0 = g00 (y0 ) + g01 (y1 ) x1 = g10 (y0 ) + g11 (y1 ) where gij (·) are the linear or nonlinear polyphase synthesis filters . and Rabiner, L.R., Multirate Digital Signal Processing, Prentice-Hall, Engle- wood Cliffs, NJ, 1983. [20] Malvar, H.S., Signal Processing with Lapped Transforms,. 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: