An efficient method for the design of low-delay two-channel, perfect reconstruction IIR filter banks is proposed. The design problem is formulated in terms of minimax designs of a general stable IIR filter that can be obtained using semidefinite programming and an FIR filter that can be obtained using the Remez exchange algorithm. A multiplierless implementation on this filter bank is also proposed and investigated.published_or_final_versio

Antoniou, A

Chan, SC

Lu, WS

Mao, JS

HKU Scholars Hub

Title Design and multiplierless implementation of two-channelbiorthogonal IIR filter banks with low system delayAuthor(s) Mao, JS; Lu, WS; Chan, SC; Antoniou, ACitation Proceedings - Ieee International Symposium On Circuits AndSystems, 2001, v. 2, p. II465-II468Issued Date 2001URL http://hdl.handle.net/10722/46254Rights Creative Commons: Attribution 3.0 Hong Kong LicenseDESIGN AND MULTIPLIERLESS IMPLEMENTATION OF TWO-CHANNEL BIORTHOGONAL IIR FILTER BANKS WITH LOW SYSTEM DELAY J. S. Mao*, W-S. Lu*, S. C. Chant, andA. Aiztoiziou* *Dept. of Electrical and Computer Engineering, University of Victoria, Victoria, B.C., Canada V8W 3P6 t Dept. of Electrical and Electronic Engineering, University of Hong Kong, Pokfulam Road, Hong Kong ABSTRACT An efficient method for the design of low-delay two-channel, per- fect reconstruction IIR filter banks is proposed. The design prob- lem is formulated in terms of minimax designs of a general stable IIR filter that can be obtained using semidefinite programming and an FIR filter that can be obtained using the Remez exchange algo- rithm. A multiplierless implementation on this filter bank is also proposed and investigated. 1. INTRODUCTION Low system delay is often desired for filter banks used in subband coding, subband adaptive filtering, and many other applications 11-31. An interesting problem formulation that can be used for the design of low-delay filter banks was proposed by Kim and Ansari in [4] and developed further by Phoong, Kim, Vaidyanathan, and Ansari in [5 ] .  This formulation entails two independent func- tions of :, a(:) and 3(:). that can be chosen to satisfy speci- fied frequency-domain requirements while maintaining the perfect reconstruction property (PR). More recent work on this type of filter banh includes the method in [6] where a(;) and a(;) are rational functions of : that represent IIR filters and the method in [7] where both a(-) and d(:) are polynomials in ;-l that rep- resent nonlinear-phase FIR filters. In this paper, we use the for- mulation in [4][5] to design low-delay filter banks but chose a ( z )  to be a polynomial in :-‘ that represents a linear-phase FIR fil- ter and 3(,) to be a rational function of z that represents an IIR filter. Thus a(;) can be obtained by designing the FIR filter us- ing the Remez exchange algorithm whereas 3(:) can be obtained by designing the corresponding IIR filter by using semidefinite programming (SDP). Computer simulations demonstrate that the proposed method yields satisfactory designs in terms of transition width, stopband attenuation, and implementation complexity com- pared with several existing methods. A multiplierless implemen- tation of such IIR filter bank is also proposed. 2. DESIGN OF IIR ANALYSIS FILTER BANK 2.1. Lowpass Analysis Filter Consider the 2-channel biorthogonal maximally decimated filter bank shown in Fig. 1 where the analysis lowpass filter H o  and high- pass filter HI are characterized by and HI(,)  = -cu(:2)Ho(z) + z--2Af-1 (Ib) respectively, function a(;) is a polynomial in and p(;) is a rational function in z .  The transfer functions of the synthesis filters are given by Go(:) = -HI( - : )  and GI(z) = Ho(-z ) .  A filter bank of this type is known to have the perfect reconstruction (PR) property regardless of the choices of a(:) and O ( z )  [4][5]. The formula of Ho ( z )  in ( la) suggests that if we design a lowpass filter represented by p ( z )  with passband [0,2w,] and passband group delay A -  f , then Ho will be a lowpass filter with passband [0, L+] and passband group delay 2N [5]. Let us assume that 8(;) is the transfer function of an IIR filter of the form n 1=0 By formulating an error function and minimizing it using SDP, a(=) can be determined. Assuming that the sampling frequency is 27r. an error function can be defined as where 3 d ( ~ )  is the desired frequency response of the IIR filter and is equal to e--3d(N-1’2) f o r d  E [0, 2wp] and zero elsewhere. An approximation for a(:) can be obtained by solving the minimiza- tion problem minimize 6 (32) for w E [0 ,2~) , ]  d ( z )  # 0 for I:/ 2 1 (3c) where 6 = m a z l E o ( ~ ) ) ) ,  denotes the denominator poly- nomial d ( ; )  obtained in the ( k  - 1)th iteration, IT=i(w) > 0 is a weighting function. and the stability of the IIR filter is assured by the constraint in (3c). Now if x = [6 b dIT with b = [bo . . . b,] and d = [cl0 . . . d,], then the constraint in (3b) is equivalent to 6 cll cl’ (4) 11-465 0-7803-6685-910 1/$10.000200 1 IEEE where the notation r ( x ,  w) 0 denotes that matrix r ( x ,  i.) is positive definite and a, for i = 1 , 2  are known functions of x and w [SI. Using the well-known Lyapunov stability theory [9], it can be shown that &-I(.) has all its zeros in the unit circle if and only if there exists a positive definite matrix P k - l  such that where D k - 1  is the canonical matrix obtained by using the coeffi- cients of &-I [8][9], and the notation + 0 denotes that the above matrix is positive definite. After the (k- 1)th iteration is complete, one can solve the Lyapunov equation Pk-1 -DT-lPk-~Dk--l = I for the positive definite solution of P k - l  and use it to construct the linear matrix inequality (LMI) in the kth iteration to assure the stability of the IIR filter. In the above equation, a parameter T > 0 is introduced to control the stability margin of the filter. The kth iteration of the design can now be formulated as the SDP problem minimize crx subject to: [ rg) Qk(x) 0 ] >- 0 wherec = [l O . . .  OITandr(x)  = diag[r(x,i.O),...,r(x,wm)] with ut, 1 5 i 5 m being a set of frequency points in the interval [O, 2w,]. The optimization problem in (5) can be efficiently solved using the LMI toolbox of MATLAB. With d ( s )  known, the anal- ysis lowpass filter Ho can be obtained. 2.2. Highpass Analysis Filter Now let us examine the design of the analysis highpass filter HI.  If function a(-) is assumed to be the transfer function of a linear- phase FIR filter of even length of the form a(:) = we can define the error function for the analysis highpass filter H1 as E l ( w )  = Hl(e.'") - Hld(d )  - -a(e'3u)Ho(e3") + e -J (2" '+1)d  - H l d ( d )  (6) where Hld(d )  is the desired frequency response of H1 and is equal to e-3u('d'+1) for w E [us, 7r ]  and zero elsewhere. Now let Ho(e3") = A(w)e-33"" , where A(u) is approximately equal to one in magnitude for w E [0, up]. and let the frequency response of the FIR filter be a(eJY) = e-JY(vck-1)'2 cos(ut/2)Pa(cosw), where P, (cos w) is a Chebyshev polynomial of order iV, / 2  - 1 that contains the coefficients of a(:). The error function in (6) can then be expressed as = e-~(2-q ' f l )d  [l - cos(w)P,(cos(2w))A(w)] for LJ E [O, x/2] By setting OlEl (i.)('/OP, to zero, we obtain the Chebyshev poly- nomial P,(cos(2w)) that minimizes IEl(w)J '  as P, (cos(2iJ)) = for i. # 7r/2 ( 7 )  lA(w)l- cosw Hence the design problem at hand can be formulated as the least pth minimization problem [IO1 minimizeLT ~~-~(r)lP,(z) - Pd(z)/~dx (8) where 5 = cos(2w). 2, = c0s(2dp), I ,  is the interval defined by I ,  = (-1, -x,) U (z.. l ) ,  and TT7a(w) is a weighting function given by TITa(z) = cos(0.5ar~~cosz)~~(0.5arccosz)~'. The op- timization problem in (8) with P = CO can be solved using the Remez exchange algorithm [I  I]. With a(2) known. the analysis highpass filter HI can be obtained. 2.3. Selection of Design Parameters We now describe some simple guidelines for the selection of pa- rameters N, n, h1, and N, for given design specifications. If filter Ho were a linear-phase half-band FIR filter. its length could be predicted by using a formula due to Herrmann et al. [ 1 I ]  as ( 9 )  where B = '"s2:-p'. F = 11.012, D = [O.O05309(log bso)2 + O.Oi1141ogb,, - 0.47611 logb:., -[O.O026G(log ~ 5 , ~ ) ~  + 0.5941(log6,,) + 0.42i81 In the above formulas, b,, is the stopband ripple, d p  and dS are the passband and stopband edgts of the analysis lowpass filter. re- spectively; [QJ denotes the largest integer smaller than Q. Now if HO is an IIR filter, then extensive simulations have shown that parameter N could be taken to be one eighth of L [IO]. i.e., L S N = [-1 where [Ql denotes the smallest integer larger than Q. A value of i\i smaller than this may still work. but the filter bank may exhibit undesirable artifacts in the transition band. Having chosen param- eter iV, the value n = N + 2 ur;ually leads to satisfactory designs. On the other hand, the values A f  = 3N - 1 and ]Ya = ahi lead to a highpass analysis filter HI whose frequency performance is comparable with that of the lowpass analysis filter Ho. In doing so, a good overall performance of the filter bank can be obtained. 3. A CASE STUDY 3.1. Design of a Filter Bank with Continuous Coefficients We have applied the design method proposed in Section 2 to de- sign an IIR filter bank with thc following specifications: b,, = 1.7iSe - 3, d p  = 0 . 4 5 ~ .  and w, = 0 . 5 5 ~ .  The length of a linear- phase half-band FIR filter was predicted to be L = GO: there- fore, parameter N for the low.delay IIR filter bank was chosen as IY = [L/S1 = 8; with N known. the other parameters can be deduced as follows: 11 = 10. 111 = 23. and A, = 32.  We first designed an FIR nonlinear-phase transfer function of order 35, represented by O l ( z ) ,  by using the method proposed in [7 ] .  A balanced approximation method [I31 was applied to the FIR transfer function & (:) to yield a stable rational function do(:) of order 10. This 1 3 ~ (  z ) was then used as the initial guess in the SDP optimization described in Section 2.  The SDP algorithm took 11-466 one iteration to converge and the result was a stable rational func- tion p ( ; )  that gave an IIR filter Ho with equiripple magnitude response. Next, the Remez exchange algorithm was applied to ob- tain a linear-phase FIR transfer function a(;) of order 31. The transfer functions Ho(;), HI( : ) ,  Go(;), and GI(;) were then constructed using (I). The system delay of the filter bank was 2 M  + 2N + 1 = 63 s. The performance of the filter bank de- signed was evaluated and compared with that of the designs ob- tained with several existing methods. These include the methods in [4][7] where the same filter bank formulation was applied but with different selections of Cl(;) and a(;). Based on the method proposed in [4], we obtained an allpass transfer function L?(z) of order 8 and a linear-phase FIR transfer function a(2) of order 31. By using the method proposed in [7], we obtained a nonlinear- phase FIR transfer function B(; )  of order 35 and a linear-phase FIR transfer function a(;) of order 31. The linear-phase FIR filter bank with 32 lattice stages proposed in [ 141 was also considered in our comparison study. All the designs have the same system delay of 63 s. The frequency responses of a(;), / 3 ( z ) ,  and analysis IIR fil- ters Ho and HI are shown in Figs. 2(a)-(c). The performance of the designs is illustrated in Table I, where Ad = dP - w, is the width of transition band for both Ho and HI,  and 6,". 6,, are the stopband attenuations of filters Ho and HI,  respectively. It is observed that the IIR filter bank designed by the proposed method outperforms the other methods in that it yields a narrower transi- tion width and improved stopband attenuation. With regard to the implementation complexity, the allpass function of order 8 is im- plemented with 8 multipliers and 24 adders [ 2 ] ,  the IIR function L?(z) is realized in terms of a 10-stage lattice and ladder structure [ 1 I], and the linear-phase FIR a(;) is implemented using the di- rect form [12]. It is noted that the proposed method requires less multiplications and additions (at the original sampling rate) com- pared to the method in [7] but is not as economical as the methods in [4] and [ 141. 3.2. Multiplierless Implementation of the Filter Bank It is advantageous to implement a filter bank without multiplica- tions. A commonly used approach is to employ individual filters with coefficients that can be expressed as sums of a small number of power-of-two terms. In this way the filters can be implemented in terms of additions of shifted versions of the input. In what fol- lows, we call such filter banks as filter banks with sum-of-power- of-two (SP2) coefficients. Optimal implementation of filter banks with SP2 coefficients can be achieved by using techniques such as simulated annealing or genetic algorithms but these methods re- quire a large amount of computation. The approach we take here c'an be described as follows. First we design a filter bank with con- tinuous coefficients. Next we assign a certain number of bits for each coefficient based on its magnitude, and determine its lower and upper SP2 bounds. For the filter bank designed in Section 3.1 there were 21 coefficients in a(:), five of which were very small in magnitude and were not sensitive to quantization; these were fixed to their nearest SP2 values. Then through an exhaustive search we selected either the upper bound or the lower bound for each of the I6 remaining coefficients such that the objective function is minimized. In (1 I )  Hqk(w) for k = 0 , l  are the frequency re- sponses with the coefficients quantized and Hk(4) for k = 0 , l  are the corresponding desired frequency responses. For the linear- phase FIR transfer function a(;), there were 16 coefficients that needed to be quantized to their SP2 values, and the exhaustive search was also performed as for transfer function b'(2). The search for the optimal SP2 coefficients for Cl(;) and a(;), which involved the selection of the optimal combination out of 2" com- binations in each case, was carried out on a Pentium I1 400 MHz PC. The resulting magnitude responses of the multiplierless 2- channel IIR filter bank are depicted in Fig. 2(d). The average number of power-of-two terms used was 2.9. It is observed that the filter bank with SP2 coefficients still offers more than 50 dB stopband attenuation. x (n) i Figure 1: A two-channel maximally decimated filter bank. 11-467 Group delay in passband 165’ nu S, (dB) S., tdBi Mult. Add. I I  0.1 0.1 0.172 0.1 45.5 55.2 42.5 55.5 45 52.7 42.5 52.8 11 26 17 23.5 28.5 34 49 31.5 Table 1 : Performance Comparisons I Allpnss [-I] I Non-LP FIR [7] I LP FIR FB [ 141 I Proposed W I Noimalizrd frequency Figure 2: Two-channel low-delay IIR filter bank. (a) Magnitude responses of IIR filters repre.sented by B(z )  (solid line) and the linear-phase FIR filter represented by a(=) (dashed line); (b) mag- nitude responses of the analysis IIR filters; (c) passband group de- lays of the analysis IIR filters; (d) magnitude responses of multi- plierless IIR analysis filter bank. (The heavy solid lines in (b) and (d) are the passband gains of filters HO and HI, multiplied by a factor of 200 in order to illustrate the passband ripple.) ACKNOWLEDGMENT The authors are grateful to PMC-Sierra, Micronet, NCE Program, and the Natural Sciences and Engineering Research Council of Canada for supporting this work. 4. REFERENCES [I]  T. A. Ramstad, “IIR filter banks for subband coding of im- ages,” Pi-oc. IEEE ISCAS’SS, vol. I ,  pp. 827-830. June 1988. [2] P. P. Vaidyanathan, Multirate Systeiiis arid Filter Banks, En- glewood Cliffs, NJ: Prentice Hall. 1992. [3] P. A. Naylor, Q. Tanrihdu and A. G. Constantinides. “Sub- band adaptive filtering for acoustic echo control using allpass polyphase IIR filterbanks,” IEEE T”. Speech arid audio Processirig. vol. 6, no. 2, pp. 143-155, March 1998. [4] C. W. Kim and R. Ansari. “IIR/IIR exact reconstructioh filter bank with applications to subband coding of images.” P roc. 34th Midwest CAS Sviiip., vol. 1, pp. 227-230, May 199 1. [5] S. M. Phoong, C .  W. Kim. P. P. Vaidyanathan. and R. Ansari, “A new class of two-channel biorthogonal filter b a d s  and wavelet bases.” IEEE Traris. Sigizcil Proccssirig. vol.43, no. [6] X. Zhang and T. Yoshikawa. “Design of two-channel stable IIR perfect reconstruction filter banks.” IEICE Traru. Firrida- rnerztals. vol. E8 I-A, no.8, pp. 1592-1597, August 1998. [7] J. S. Mao, S. C. Chan, and K. L. Ho. “Design of two-channel PR FIR filter banks with low system delay,” Proc. IEEE IS- 3. pp. 649-664, March 1995. CAS’2000. vol. I. pp. 627-630. May 2000. [8] W.4 .  Lu, “Design of stable minimax IIR digital filters using semidefinite programming,” Proc. IEEE ISCAS’2000, vol. I, pp. 355-358, May 2000. [9] T. Kailath, Liiiear Sysre~ns, Englewood Cliffs, NJ: Prentice- Hall, 1981. [IO] J. S. Mao, New Design arid Factori:atiorz Methods for- Per- fection Reconstruction Filter Batiks, Ph.D. Dissertation, The University of Hong Kong, April 2000. [ 1 I]  A. Antoniou, Digital Films: Aiialysis, Design, arid Applica- tions, 2nd ed., New Yorl..: McGraw-Hill, 1993. [I?] A. V. Oppenheim, R. W. Schafer and J. R. Buck. Discrcve- Tinze Signal Processing. Upper Saddle River, NJ: Prentice- Hall, 1999. [ 131 B. C. Moore, “Principal component analysis in linear system: Controllability, observalility. and model reduction,” IEEE Tnirzs. Airtornaric Colmvl, vol. AC-26, pp. 17-31, 1981. [ 141 T. Q. Nguyen and P. P. Vaidyanathan, “Two-channel perfect- reconstruction FIR QMF; structures which yield linear phase analysis and synthesis filters,” IEEE Trans. ASSP., vol. ASSP-37, pp. 676-690, May 1989. 11-468 

Design and multiplierless implementation of two-channel biorthogonal IIR filter banks with low system delay

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Design and multiplierless implementation of two-channel biorthogonal IIR filter banks with low system delay

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