In this paper, an efficient method for designing perfect reconstruction (PR) two-channel finite impulse response (FIR) filter banks with low system delay is proposed. It is based on the use of nonlinear-phase FIR function in a structure previously proposed by Phoong et al. The design problem is formulated as a complex polynomial approximation problem and is solved effectively using the Remez exchange algorithm with very low design complexity. Design examples show that filter banks with flexible stopband attenuation and system delay can be readily obtained by the proposed algorithm.published_or_final_versio

Chan, SC

Ho, KL

Mao, JS

IEEE Signal Processing Letters

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Title Efficient method for designing two-channel PR FIR filter bankswith low system delayAuthor(s) Mao, JS; Chan, SC; Ho, KLCitation Ieee Signal Processing Letters, 2001, v. 8 n. 1, p. 16-19Issued Date 2001URL http://hdl.handle.net/10722/42830Rights Creative Commons: Attribution 3.0 Hong Kong License16 IEEE SIGNAL PROCESSING LETTERS, VOL. 8, NO. 1, JANUARY 2001An Efficient Method for Designing Two-Channel PRFIR Filter Banks with Low System DelayJ. S. Mao, S. C. Chan, and K. L. HoAbstract—In this paper, an efficient method for designing per-fect reconstruction (PR) two-channel finite impulse response (FIR)filter banks with low system delay is proposed. It is based on theuse of nonlinear-phase FIR function in a structure previously pro-posed by Phoong et al. [1]. The design problem is formulated asa complex polynomial approximation problem and is solved effec-tively using the Remez exchange algorithm with very low designcomplexity. Design examples show that filter banks with flexiblestopband attenuation and system delay can be readily obtained bythe proposed algorithm.Index Terms—Design method, filter banks, low system delay,perfect reconstruction, Remez exchange algorithm, two-channel.I. INTRODUCTIONB ECAUSE of the low delay requirement in subband coding,subband adaptive filtering, and other applications, thereis a growing interest in designing PR filter banks (FBs) withsuch properties. Conventional methods for designing low delaytwo-channel PR FB are usually based on numerical optimization[3], [7], which produces FB that is only pseudo-PR. More re-cently, several factorizations for PR low delay FB were reported[5], [6]. Unfortunately, they usually require unconstrained opti-mization of highly nonlinear functions, which is easily trappedin local minima and requires considerably high design com-plexity. Another approach based on linear programming has alsobeen proposed in [4]. The FB so obtained is PR. However, be-cause the FB does not have any structure, it is not robust to coef-ficient quantization and generally requires higher implementa-tion complexity. In this paper, a class of low delay two-channelFB, using the structure in [1], and [2], is proposed. In particular,we show that the functions and of the structure in [1]can be chosen as nonlinear-phase FIR functions to realize FBwith low system delay and good frequency characteristics. Wealso show that the design of the proposed FB can be formulatedas a complex polynomial approximation problem, which can besolved by the Remez exchange algorithm with very low designcomplexity. The paper is organized as follows: in Section II, abrief summary of the two-channel structural PR FB proposedin [1] is given together with the basic idea behind the proposedlow delay FIR FB and its design procedures. Several design ex-amples are given in Section III, and the conclusions are drawnin Section IV.Manuscript received January 21, 2000. The associate editor coordinating thereview of this manuscript and approving it for publication was Prof. A. M.Sayeed.The authors are with the Department of Electrical and Electronic Engineering,The University of Hong Kong, Hong Kong (e-mail: scchan@eee.hku.hk;jsmao@eee.hku.hk; klho@eee.hku.hk).Publisher Item Identifier S 1070-9908(01)01027-6.Fig. 1. Two-channel PR filter bank in [1].II. PROPOSED METHODA. Structurally Imposed FIR Biorthogonal FBsFig. 1 shows the structure of the two-channel PR FB pro-posed in [1]. The analysis lowpass filter and highpassfilter can be shown to beand(2.1)where and can be polynomials and rational func-tions to achieve PR. In [1], they are chosen to be identical,and the delay parameter is equal to 1. To design alinear-phase FIR FB with , they can be chosenas type-II linear-phase FIR functions with 1. Thelengths of and should be 2 and, respectively, with a system delay of 2 2 1. Forlow delay applications, the lengths of and are there-fore constrained by the system parameters and , and mightnot be able to provide the desired stopband attenuation. In thispaper, and are chosen as nonlinear-phase FIR func-tions. Under the same system delay 2 2 1, , andin low delay FBs are no longer constrained by the parametersand as their linear-phase counterparts. Therefore, the stop-band performance will significantly be improved by choosingand to be nonlinear-phase.B. Design of Analysis Lowpass FilterLet and be, respectively, the passband and stopbandcutoff frequencies of filter . Similarly, letand be the passband and stopband cutofffrequencies of . To design with nonlinear-phase, letus decompose it into its even and odd parts:. When the length of is even, and aretype-II and type-IV linear-phase FIR functions, respectively.Similarly, when the length of is odd, it can be decomposed1070–9908/00$10.00 © 2001 IEEEMAO et al.: EFFICIENT METHOD FOR DESIGNING TWO-CHANNEL PR FIR FILTER BANKS 17into a sum of type-I and type-III FIR functions [10]. For sim-plicity, only the even length case will be considered (details ofthe odd-length case will be given in [10]). and canbe written as follows:and(2.2)where and are polynomials with order2) 1. The desired response of , isequal to for 0 and zero otherwise. The errorfunction is then given by(2.3)whereand(2.4)The desired response of is seen to be, which is recognized as a complex polynomial approx-imation problem. They can be solved by a variety of methods[8], [9]. For simplicity, the complex polynomial is decomposedinto its real and imaginary parts and is solved by two indepen-dent real Chebyshev approximations [9]. The desired functionsof and are then given byand(2.5)Writing , the two Chebyshev approximation prob-lems can be written asand(2.6)where and are, respectively, the coefficients of thepolynomials and , which mini-mize the minimax problems in (2.6) in the intervaland(2.7)The interval (0 ] is an optional disjoint interval to controlthe values of and in the tran-sition band of . Equation (2.6) can readily be solved(a)(b)(c)Fig. 2. Example 1. (a) Magnitude responses of (z) and (z); (b) magnituderesponses of analysis filters H (z) and H (z): low delay (solid lines) andlinear-phase (dashed lines); and (c) Group delay responses of the proposed lowdelay analysis filters.using the function REMEZ in the signal processing Toolbox ofMATLAB.C. Design of Analysis Highpass FilterLet us assume that is properly designed so thatis a reasonably good lowpass filter. From (2.1), itcan be seen that the frequency response of dependson and . The error function of is, whereis the desired response of and is equal tofor and zero otherwise. The idealmagnitude response of suggests that its value should beclose to one, except around 2, where it is even smaller[see Fig. 2(a) for an example]. It then follows from (2.1) thatthe passband ripple of is approximately equal to thestopband error of . This allows us to minimize onlythe stopband attenuation of using , instead ofminimizing over the pass- and stop-bands and relies onthe high stopband attenuation of to achieve a smallpassband ripple. First of all, let us consider the case where18 IEEE SIGNAL PROCESSING LETTERS, VOL. 8, NO. 1, JANUARY 2001is an even-length FIR filter with length . The odd lengthcase can be derived similarly and will be described in [10].Using again the even and odd parts decomposition, canbe expressed as , whereand(2.8)where and are polynomials withorder 2) 1. Let the lowpass filter be writtenas , where is a complexfunction but is approximately equal to one in [0, ] for ,with sufficient high order. Substituting (2.8) into the errorfunction , one obtains(2.9)where , and. Therefore, the idealresponse of is , .Again, if and are used to approx-imate separately the real and imaginary parts of , theirideal responses areand(2.10). Writing , the problem can be solvedusing the Chebyshev approximation as in (2.6), with weightingfunctionsand(2.11)They again can be solved by the Remez exchange algorithm.Approximating the real and imaginary parts of [or] separately as two real Chebyshev approximationsdoes not necessarily produce an equiripple error function. Areweight technique is therefore used to modify the weightingfunction as , where, and are the error in approximatingthe real and imaginary parts of (or ). Fromexperimental results, is found to perform better than, and is used in this work. (2.6), (2.7) and (2.10), (2.11)are first solved using the Remez exchange algorithm. Afterthat, it is solved again using the new weighting function.Considerable improvement over the independent Chebyshevapproximations can be obtained.TABLE IOVERALL PERFORMANCE OF EXAMPLE 1 COMPARED TO OTHERCONVENTIONAL METHODSIII. DESIGN EXAMPLESExample 1: In this design example, and are non-linear-phase FIR functions with lengths 8 and 10.The delay parameters and are, respectively, 2 and 5. Theoverall system delay is 15 samples. Fig. 2 displays themagnitude responses of (solid line), (dashed line),and the frequency responses of the analysis filters (solid line)designed by the proposed method. To demonstrate the advan-tage of the proposed low delay FIR FB over the linear-phaseFB, a two-channel linear-phase FIR FB with the same systemdelay was also designed by the proposed Remez exchange al-gorithm. The delay parameters and are still 2 and 5, butthe length of and are shortened to 4 and8, respectively, due to the linear-phase constraint. The magni-tude responses of the linear-phase analysis filters are plotted asdashed lines in Fig. 2(b). It is observed that, under the samesystem delay ( 15) and cutoff frequencies ( 0.34and 0.66 ), the stopband attenuation of the low-delayanalysis filters are much higher than its linear-phase counter-part: 42 dB versus 26 dB for , and 40 dB versus 36 dBfor . This improvement of stopband attenuation, however,requires additional computations in implementing the FB (thenumber of variables in the low-delay and is 18, whilethat of the linear-phase is 6). It can also be seen from Fig. 2(c)that the proposed low delay FB is approximately linear-phasein their passbands. In order to compare the proposed methodto the conventional methods in [3], [4], the cutoff frequenciesand system delay are chosen to be the same as the design exam-ples in [3, Fig. 5] and [4, Fig. 3]. The stopband attenuations ofthe analysis filters in [3] are about 39 dB. The stopband atten-uations of and in [4] are, respectively, 45 dB and40 dB. Therefore, the proposed analysis filters have comparablepassband and stopband performance as those in [3] and [4]. Byadjusting the cutoff frequencies to ( 0.24 , 0.76 )and ( 0.4 , 0.6 ), the proposed method can stillprovide comparable stopband attenuation as that in [5, Fig. 1](55 dB versus 56 dB), and higher stopband attenuation thanthat in [7, Fig. 2] (30 dB versus 25 dB). It is also noted thatthe design and implementation complexities of the proposedmethod are very low and there is no reconstruction error, whichMAO et al.: EFFICIENT METHOD FOR DESIGNING TWO-CHANNEL PR FIR FILTER BANKS 19Fig. 3. Magnitude responses of H (z) and H (z) in Example 2.is always present in other methods based on constrained non-linear optimization [3], [7]. Furthermore, as mentioned earlier,the proposed filter bank structure is still PR, even under coef-ficient quantization, unlike the direct form in [4]. The designcomplexity of the proposed method is also much lower than theunconstrained nonlinear optimization methods in [5] and [6],thanks to the Remez Exchange algorithm. The overall perfor-mance comparison between the proposed method and the con-ventional low delay filter bank design methods [3]–[5] and [7]are summarized in Table I, where the arithmetic complexityis simply defined as the number of multiplications and addi-tions per sample in implementing the filter bank. This demon-strates the good performance, flexibility, low implementation,and design complexities of the proposed method as comparedwith conventional methods.Example 2: In this example, a two-channel low delay FIRfilter bank with higher order is designed. The lengths ofand are chosen to be odd numbers, 13 and 15.The overall system delay of the filter bank is 23 sampleswith 3 and 8. Fig. 3 plots the magnitude responsesof the analysis filters and . It can be seen that thestopband attenuation of and is about 39 dB, andtheir passband and stopband cutoff frequencies are, respectively,0.41 and 0.59 . It is also observed that due to thehigher system delay, the transition band is sharper than that ofExample 1.IV. CONCLUSIONAn efficient algorithm for designing two-channel low delayPR FIR filter banks is presented. It is based on the use of non-linear-phase FIR functions in the structure previously proposedby Phoong et al. [1]. The design problem is formulated as a poly-nomial approximation problem and is solved using the Remezexchange algorithm. Design examples show that filter bankswith flexible stopband attenuation and system delay can readilybe obtained by the proposed algorithm.REFERENCES[1] S. M. Phoong, C. W. Kim, P. P. Vaidyanathan, and R. Ansari, “A newclass of two-channel biorthogonal filter banks and wavelet bases,” IEEETrans. Signal Processing, vol. 43, pp. 649–664, Mar. 1995.[2] C. W. Kim and R. Ansari, “FIR/IIR exact reconstruction filter bank withapplications to subband coding of images,” Proc. 34th Midwest Symp.Circuits and Systems Symp., vol. 1, pp. 227–230, May 1991.[3] K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, “Low delay FIR filterbanks: Design and evaluation,” IEEE Trans. Signal Processing, vol. 42,pp. 24–31, Jan. 1994.[4] H. Ochi, M. Ohta, and S. Kinjo, “Linear programming de-sign of two-channel perfect-reconstruction biorthogonal filterbanks-linear-phase and low-delay,” Proc. Int. Symp. Circuits andSystems, vol. 4, pp. 2401–2404, June 1997.[5] G. Schuller and M. J. T. Smith, “A new algorithm for efficient low delayfilter bank design,” Proc. Int. Conf. Acoustics, Speech, and Signal Pro-cessing, vol. 2, pp. 1472–1475, May 1995.[6] R. Gandhi and S. K. Mitra, “Design of two-channel low delay perfectreconstruction filter banks,” Proc. 32nd Asilomar Conf. Signals, Systemsand Computers, vol. 2, pp. 1655–1659, Nov. 1998.[7] S. J. Yang, J. H. Lee, and B. C. Chieu, “Minimax design of two-channellow-delay perfect-reconstruction FIR filter banks,” Inst. Elect. Eng.Proc. Vis. Image Signal Process., vol. 146, pp. 15–24, Feb. 1999.[8] L. J. Karam and J. H. McClellan, “Complex Chebyshev approximationfor FIR filter design,” IEEE Trans. Circuits Syst. II, vol. 42, pp. 207–216,Mar. 1995.[9] S. C. Pei and J. J. Shyu, “Design of complex FIR filters with arbitrarycomplex frequency responses by two real Chebyshev approximations,”IEEE Trans. Circuits Syst. I, vol. 44, pp. 170–174, Feb. 1997.[10] J. S. Mao, S. C. Chan, and K. L. Ho, “Design and multiplier-less imple-mentation of a class of two-channel PR FIR filter banks and waveletswith low system delay,” IEEE Trans. Signal Processing.

Efficient method for designing two-channel PR FIR filter banks with low system delay

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Efficient method for designing two-channel PR FIR filter banks with low system delay

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