This letter proposes a method for designing a class of M-channel, causal, stable, perfect reconstruction (PR) IIR cosine-modulated filter banks (CMFB). The proposed CMFB has the same denominator for all its polyphase components in the prototype filter. Therefore, the PR condition is considerably simplified, and it is relatively simple to satisfy the PR and the casual-stable requirements of the IIR CMFB. Design examples show that the proposed IIR CMFB has sharper cutoff, higher stopband attenuation, and passband flatness than its FIR counterparts, especially when the system delay is small.published_or_final_versio

Chan, SC

Ho, KL

Mao, JS

IEEE Signal Processing Letters

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Title Theory and design of a class of M-channel IIR cosine-modulatedfilter banksAuthor(s) Mao, JS; Chan, SC; Ho, KLCitation Ieee Signal Processing Letters, 2000, v. 7 n. 2, p. 38-40Issued Date 2000URL http://hdl.handle.net/10722/42838Rights©2000 IEEE. Personal use of this material is permitted. However,permission to reprint/republish this material for advertising orpromotional purposes or for creating new collective works forresale or redistribution to servers or lists, or to reuse anycopyrighted component of this work in other works must beobtained from the IEEE.38 IEEE SIGNAL PROCESSING LETTERS, VOL. 7, NO. 2, FEBRUARY 2000Theory and Design of a Class of M-Channel IIRCosine-Modulated Filter BanksJ. S. Mao, S. C. Chan, and K. L. HoAbstract—This letter proposes a method for designing a classof -channel, causal, stable, perfect reconstruction (PR) IIR co-sine-modulated filter banks (CMFB). The proposed CMFB has thesame denominator for all its polyphase components in the proto-type filter. Therefore, the PR condition is considerably simplified,and it is relatively simple to satisfy the PR and the casual-stable re-quirements of the IIR CMFB. Design examples show that the pro-posed IIR CMFB has sharper cutoff, higher stopband attenuation,and passband flatness than its FIR counterparts, especially whenthe system delay is small.Index Terms—Causal stable, cosine-modulated, filter banks, IIR,perfect reconstruction.I. INTRODUCTIONPERFECT reconstruction (PR) filter banks have importantapplications in signal analysis, coding, and the designof wavelet bases [1]. Fig. 1 shows the block diagram ofan -channel maximally decimated filter bank. The FIRcosine-modulated filter banks (CMFB) [2] or the extendedlapped transforms [8] have emerged as very good tradeoff be-tween filter quality, implementation, and design complexities.Recently, the theory and design of the more general arbitrarylength, biorthogonal CMFB has also been studied [3], [4]. Oneapplication of such biorthogonal CMFB is to realize filterbankwith low system delay. Although IIR filter banks can poten-tially offer low system delay and high stopband attenuation,their design is very involved. They are either limited to thetwo-channel case [5] or suffer from stability problems [7]. Infact, the difficulty in designing such IIR filter banks is to satisfythe complicated PR condition and the causal-stable requirementof the filters. In this paper, a novel method for designing aclass of -channel, causal-stable PR IIR CMFB is proposed.The proposed IIR CMFB has the same denominator for allthe polyphase components of its prototype filter. Therefore,the PR condition is considerably simplified, and it is simplerto satisfy the PR and the casual stable requirements. Designexamples show that the proposed IIR CMFB has sharpercutoff and higher stopband attenuation and passband flatnessthan its FIR counterparts, especially when the system delayis small. The layout of the letter is as follows. Section II isdevoted to the theory of the proposed -channel IIR CMFB.Their design procedure and several examples will be given inManuscript received September 28, 1999. The associate editor coordinatingthe review of this manuscript and approving it for publication was Prof. R.Shenoy.The authors are with the Department of Electrical and Electronic En-gineering, The University of Hong Kong, Kowloon, Hong Kong (e-mail:klho@eee.hku.hk).Publisher Item Identifier S 1070-9908(00)00923-8.Fig. 1. M -channel maximally decimated filter banks.Section III. Finally, the results are summarized in Section IV,the conclusion.II. PROPOSED -CHANNEL PR IIR CMFBThe analysis filters and the synthesis filters ina CMFB are obtained by modulating a prototype filter asfollows:(1)where is the number of channels. The modulation we shallbe using is the extended lapped transform (ELT) [8] given by(2)and is the time reverse of . The PR condition for thebiorthogonal CMFB is given by [3], [4](3)where is the type-I polyphase component of the proto-type filter . For simplicity, weassume that is even. Similar conditions can be derived whenis an odd number. ’s are usually chosen to be identicalfor all , and it determines the delay of the filter bank. In ourIIR CMFB, ’s are assumed to have the same denominator(i.e., for ). ThePR condition in (3) is then simplified to(4)To ensure that the analysis and synthesis filters be causal stable,all the roots of shall remain inside the unit circle. Forsimplicity, all zeros of are assumed to occur in complex1070–9908/00$10.00 © 2000 IEEEMAO et al.: THEORY AND DESIGN OF A CLASS OF FILTER BANKS 39conjugate pairs so that, where is a constant. Modifications to include afixed number of real zeros are easily made.III. DESIGN PROCEDURE AND DESIGN EXAMPLESDue to the cosine modulation in (2), the analysis filteris obtained by shifting to . For, these two copies will be widely sepa-rated, and the analysis filters look very like the original pro-totype filter. However, for and , the two copieswill overlap each other in the transition bands. In the case oforthogonal CMFB, the linear-phase property of ensuresthat the passband of and will be flat. However,in biorthogonal CMFB, is no longer symmetric, and therewill be bumping in the passband if the stopband attenuation ofalone is minimized. These bumps have been observed inthe design examples of other literatures [3], [4], [7]. This canbe suppressed by minimizing the passband ripples of( ) around ( ), at the expense of de-creased stopband attenuation. As current IIR CMFB is biorthog-onal, the following objective function is used:(5)wherestopband cutoff frequency whose value should be be-tween and ;passband cutoff frequency of ;constant less than used to suppress thebumping around ;constant between 0 and 1 used to control the relativeweighting of the stopband attenuation and the suppres-sion of the bump;desired response of at ;vector containing the coefficients of and the realand imaginary parts of ’s.The design problem can be formulated as the following con-strained nonlinear optimization problem:subjected to the PR condition in (4) and(6)This constrained nonlinear optimization can be solved by theNCONF/DCONF subroutine in the IMSL library, which isbased on sequential quadratic programming with an augmentedLagrangian line-search method. The initial guess to the op-timization program is obtained by designing an -channel,biorthogonal PR FIR CMFB with the desired delay. The initialvalues are chosen to be zero. The magnitude response ofthe prototype filter and analysis filters of a four-channel IIRCMFB designed by the proposed method are shown in Figs. 2and 3(a). The order of ( ) andare, respectively, 4 and 2. The magnitude responses ofthe numerator and denominator of the prototype filter are alsoshown in dotted and dashed lines, respectively. In this designexample, the DCONF program converged after about 150Fig. 2. Magnitude responses of four-channel IIR CMFB: prototype filterH(z)(solid line), numerator H (z) (dotted line), and denominator H (z) (dashedline).(a)(b)Fig. 3. Magnitude responses of four-channel biorthogonal CMFB: (a) IIRanalysis filters and (b) FIR analysis filters.iterations. Due to roundoff errors and the inherent limitation ofthe optimization program, the PR constraints can only be metapproximately. In this particular design example, the violationof PR constraints is of the order of 10 . In general, the numberof iterations used and the amount of constraint violation will de-pend on the difficulty of the problem and the initial guess used.To obtain a PR solution, the factorization technique recentlydeveloped in [9] is used to convert the nearly PR solution to astructurally PR system with a slight decrease in stopband atten-uation. Due to page limitation, details of this factorization areomitted here, and interested readers are referred to [9] for thederivations. Table I lists the polyphase components and40 IEEE SIGNAL PROCESSING LETTERS, VOL. 7, NO. 2, FEBRUARY 2000TABLE IPOLYPHASE COMPONENTS OF IIR CMFBIN EXAMPLE 1  = 3:130696027842296e   002,  = 2,k = 0; 1;    ; (M=2)   1the denominator of this PR IIR CMFB. For comparison,the magnitude response of a four-channel biorthogonal FIRCMFB with similar complexity (40-tap) is shown in Fig. 3(b).Both of them have the same system delay of 23 samples. Itcan be seen that the poles help to sharpen the transition bandand increase the stopband attenuation of the CMFB (about 7dB better than the FIR CMFB). The passband flatness of theproposed IIR CMFB is also significantly improved. Fig. 4shows the magnitude responses of another four-channel IIRCMFB designed using the proposed method. The orders ofthe numerator and denominator of the prototype filter areFig. 4. Magnitude responses of four-channel, IIR, cosine-modulated analysisfilters. System delay = 37.45 and 16, respectively. The system delay is increased to 37samples. It can be seen that CMFB with larger system delay hashigher stopband attenuation (40 versus 30) than its low-delaycounterparts. It is also observed that the bump of the low-delayFIR CMFB at and is very difficult to suppress withoutsignificantly decreasing its stopband attenuation.IV. CONCLUSIONA new method for designing a class of -channel causal-stable, PR IIR CMFB is presented. The proposed CMFB has thesame denominator for all its polyphase components in the pro-totype filter. Therefore, the PR condition is considerably sim-plified, and it is relatively simple to satisfy the PR and the ca-sual-stable requirements of the IIR CMFB. Design examplesshow that proposed IIR CMFB has sharper cutoff and higherstopband attenuation and passband flatness than its FIR coun-terparts, especially when the system delay is small.REFERENCES[1] P. P. Vaidyanathan, Multirate Systems and Filter Banks. EnglewoodCliffs, NJ: Prentice-Hall, 1992.[2] R. D. Koilpillai and P. P. Vaidyanathan, “Cosine-modulated FIR filterbanks satisfying perfect reconstruction,” IEEE Trans. Signal Processing,vol. 40, pp. 770–783, Apr. 1992.[3] T. Q. Nguyen and P. N. Heller, “Biorthogonal cosine-modulated filterbanks,” in IEEE Proc. ICASSP’96, vol. 3, May 1996, pp. 1471–1474.[4] S. C. Chan, L. Yi, and K. L. Ho, “M -channel compactly supportedbiorthogonal cosine-modulated wavelets bases,” IEEE Trans. SignalProcessing: Special Issue Theory Applicat. Filter Banks Wavelets, vol.46, pp. 1142–1151, Apr. 1998.[5] T. Q. Nguyen, T. I. Laakso, and T. E. Tuncer, “On perfect-reconstruc-tion allpass-based cosine-modulated IIR filter banks,” in Proc. IEEEISCAS-94, vol. 2, May 1994, pp. 33–36.[6] S. Znidar, “Cosine-modulated perfect reconstruction IIR filter bankswith good time-frequency resolution,” in Proc. 13th Int. Conf. DigitalSignal Processing, vol. 2, July 1997, pp. 1067–1070.[7] J. S. Mao, S. C. Chan, and K. L. Ho, “Theory and design of causal stableIIR PR cosine-modulated filter banks,” in Proc. IEEE ISCAS’99, vol. 3,June 1999, pp. 427–430.[8] H. S. Malvar, “Extended lapped transforms: Properties, applications,and fast algorithms,” IEEE Trans. Signal Processing, vol. 40, pp.2703–2714, Nov. 1992.[9] S. C. Chan, “Factorization of M -channel biorthogonal cosine-mod-ulated filter banks,” Dept. Elect. Electron. Eng., Univ. Hong Kong,Kowloon, internal rep., Dec. 1998.

Theory and design of a class of M-channel IIR cosine-modulated filter banks

Abstract—This letter proposes a method for designing a class of-channel, causal, stable, perfect reconstruction (PR) IIR co-sine-modulated filter banks (CMFB). The proposed CMFB has the same denominator for all its polyphase components in the proto-type filter. Therefore, the PR condition is considerably simplified, and it is relatively simple to satisfy the PR and the casual-stable re-quirements of the IIR CMFB. Design examples show that the pro-posed IIR CMFB has sharper cutoff, higher stopband attenuation, and passband flatness than its FIR counterparts, especially when the system delay is small. Index Terms—Causal stable, cosine-modulated, filter banks, IIR, perfect reconstruction

Theory and design of a class of M-channel IIR cosine-modulated filter banks

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