Design of Two-Channel Low-Delay FIR Filter Banks Using Constrained Optimization

Transcription

1 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. Deign of Two-Channel Low-Delay FIR Filter Bank Uing Contrained Otimization Abtract Robert Bregović and Taio Saramäki Signal Proceing Laboratory Tamere Univerity of Technology P. O. Box 553, FIN-33 Tamere, Finland and Thi aer how the efficiency of uing contrained otimization for deigning two-channel low-delay finite imule reone filter bank. The filter bank under conideration are quadrature mirror filter (QMF) bank and erfect recontruction (PR) biorthogonal filter bank. The deign roblem for both tye of bank are tated a contrained minimization roblem in form that enable u to minimize the maximum of the toband energie of the analyi filter() ubject to the given aband and tranition band contraint of the filter() a well a ubject to the given allowable recontruction error for QMF bank or the PR roerty for biorthogonal filter bank. For olving the given otimization roblem a modified Dutta-Vidyaagar otimization technique ha been ued. The efficiency of the rooed deign method i illutrated by mean of ome examle. Keyword: FIR filter bank, Low-delay, Two-channel, Otimization, QMF, Biorthogonal Introduction During the lat two decade, filter bank have found variou alication in many different area uch a eech coding, crambling, image comreion, and tranmiion of ignal through channel of different bandwidth [] [3]. The main idea of uing filter bank i the ability of the ytem to earate in the frequency domain the ignal under conideration into two or more ignal or to comoe two or more different ignal into a ingle ignal. Due to the filtering oeration that are erformed on the ignal aing through the filter bank, a delay between the outut and the inut ignal i introduced. In many alication, it i

2 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. deirable to kee thi delay a hort a oible. When uing linear-hae finite imule reone (FIR) filter in yntheizing filter bank, good filter roertie reult in many cae in high filter order and in an intolerably high delay. In thee alication, low-delay FIR filter bank have to be ued. Thee filter bank are characterized by the roerty that the order of the nonlinear-hae building-block FIR filter can be increaed to imrove the filter roertie without increaing the overall delay. For contructing low-delay FIR filter bank, quadrature mirror filter (QMF) bank and erfect recontruction (PR) filter bank have been ued. For the low-delay QMF bank [4] [6], the analyi filter are quadrature mirror filter and the PR roerty cannot be achieved, wherea for PR biorthogonal filter bank [6] [9], thee filter are related through the PR roerty and have different characteritic. For deigning low-delay QMF bank two iterative aroache have been reented by Xu, Lu, and Antoniou in [4] and [5]. The main drawback in thee iterative algorithm i that the tranition band rile of the amlitude reone i not automatically controlled and it i not traightforward to find roer value for the arameter ued in the rocedure to give a atifactory filter erformance in the tranition band. The larger i the difference between the filter order and the deired filter bank delay, the more difficult i the roblem. Low-delay PR biorthogonal filter bank have been firt introduced by Nayebi, Smith, and Barnwell in [7] and [8]. The filter bank reulting when uing their otimization technique are ubotimal, a ha been hown in ome later aer. However, they have made everal imortant obervation concerning the roertie of low-delay filter bank that aly to PR biorthogonal filter bank a well to QMF bank. Firt, it i not adviable to deign filter bank with a very mall delay comared to the filter order becaue after a certain filter order for the ame overall delay, larger order reult only in a negligible imrovement in the erformance of the filter bank. Second, additional contraint are uually neceary in the tranition band of the filter due to the oible artifact occurring in thee band. Abdel-Rahem, El-Guibaly, and Antoniou reented in [9] two aroache for deigning low-delay biorthogonal filter bank baed on the Lagrange-multilier method. Filter bank deigned uing their method have better roertie than thoe deigned by Nayebi, Smith, and Barnwell. The method ha, like for QMF bank, again roblem with the aband and tranition band rile. Thi aer how how the above-mentioned roblem can be olved in deigning twochannel low-delay FIR filter bank by uing contrained otimization. For both QMF bank and PR biorthogonal filter bank, the overall ynthei roblem i tated in the form to which 2

3 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. the algorithm of Dutta and Vidyaagar [] can be alied. Several examle are included illutrating the efficiency of the rooed ynthei cheme in imroving the filter bank erformance. More detail on how to aly the Dutta-Vidyaagar algorithm to deigning filter bank can be found in [] and [2]. A comrehenive review on two-channel FIR filter bank ha been given in [6]. 2 Two-channel filter bank Thi ection review ome baic relation for two-channel filter bank. The block diagram for a two-channel filter bank i hown in Fig. [2]. It conit of an analyi filter bank followed by downamler, uamler, and a ynthei bank. It i well known that the relation between the outut and inut of thi ytem i exreible a Y ( z) T( z) X ( z) + A( z) X ( z) =, () where the term T ( z) = [ H ( z) F ( z) + H( z) F ( z) ] (2a) 2 and A = z (2b) 2 ( z) [ H ( z) F ( z) + H ( z) F ( )] are the ditortion tranfer function and the aliaing tranfer function, reectively. The econd term can be made zero by electing the ynthei filter a F (z) = 2H ( z) and F (z) = 2H ( z). In thi cae, the reidual filter bank ditortion become T ( z) H ( z) H ( z) H ( z) H ( z) =. (3) To imlify the overall deign roblem to be decribed in the following ection, we ue, intead of H (z) and H (z), the following tranfer function [6]: N N n n G ( z) = g[ n] z H ( z) = h[ n] z (4a) n= n= N N n n n G ( z) = g[ n] z H( z) = ( ) h[ n] z. (4b) n= n= In the above, G (z) and H (z) are identical, wherea G (z) = H ( z). Therefore, G (e ) = H (e j(π ) o that the amlitude reone of G (z) i obtained from that of H (z) 3

4 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. by mean of the ubtitution π and vice vera. Fig. 2 exemlifie the above relation in addition to howing the contraint for G (z) and G (z) that are ued in the roblem to be tated in the following ection. The main advantage of uing the above tranfer function G (z) and G (z), intead of H (z) and H (z), lie in the fact that they are both lowa filter. Thi make their otimization more traightforward a they can be treated in the ame way and the ynthei formula can be exreed in a imlified manner. 3 Low-delay QMF bank In thi ection, the definition of low-delay QMF bank i given, an aroriate deign roblem i tated, and an otimization rocedure i uggeted for olving thi roblem. 3. Definition of low-delay QMF bank The low-delay QMF bank rooed by Xu, Lu, and Antoniou [4] are characterized by the following roertie: N. ( z) = G ( z) = g [ n] n G z, where N i an odd integer. n = 2. G (e ) aroximate zero on [, π] (toband) and unity on [, ] (aband). 3. T(z) = [G (z)] 2 +[G ( z)] 2 aroximate the delay z K with K i being an odd integer atifying K < N. Becaue of Proerty 3, the imule reone of G (z) cannot be ymmetric o that all the imule-reone value g [n] for n =,,, N are unknown. The recontruction error given by T jk 2 j( + ) ( ) [ ( )] [ ( )] π 2 jk e e = G e G e e (5) i deired to be made mall in the overall baeband [, π]. Due to the nonlinear-hae characteritic, the erformance of G (z) in the aband and in the tranition band mut alo be controlled. 3.2 Deign method for low-delay QMF bank For given N, ρ, ρ, δ, and δ a a well a K, the roblem i to find the N + unknown N imule-reone coefficient g [n] of G ( z) = g [ n] n n = z to minimize 4

5 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. ubject to ( e ) π G j 2 ε = d (6a) max G e [, ] ( ) δ, max G e (, ) ( ) δ, (6b) and [, π] ( e ) = [, π] 2 j( + π ) 2 jk [ G ( e )] [ G ( e )] e δ a max jk T e max (6c) where = ( ρ )π/2 and = (+ρ )π/2. Here the goal i to minimize the toband energy of the filter under conideration. The firt condition of Eq. (6b) force the maximum aband deviation of the amlitude reone from unity to be le than or equal to δ, the econd condition of Eq. (6b) force the tranition band maximum to be le than or equal to +δ, and the third condition of Eq. (6c) guarantee that the recontruction error i maller than or equal to δ a. The above otimization roblem can be olved by roerly alying the algorithm of Dutta and Vidyaagar [] (ee, e.g., [] or [2] for detail). 4 Low-delay PR biorthogonal filter bank In thi ection, the definition of low-delay PR biorthogonal bank i given, an aroriate deign roblem i tated, and an otimization rocedure i uggeted for olving thi roblem. 4. Definition of low-delay PR biorthogonal filter bank For low-delay PR biorthogonal filter bank G (z) and G (z), a given by Eq. (4a) and (4b), atify the following condition:. The imule reone of G (z) and G (z) are not ymmetric. + n= N N 2. The imule reone of E( z) = G ( z) G ( z) = e[ n] z n atifie / 2 for n = K e [ n] = (7) for n odd and n K, where K i an odd integer with K < (N + N )/2. An examle for an imule reone of E(z) i hown in Fig. 3. The econd condition imlie that the overall tranfer function between the outut and inut i T(z) = E(z) E( z) = z K 5

6 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. with K le than (N + N )/2 (comare Eq. (3) and (4) and Fig. 3). The high number of unknown (altogether N +N +2) and the PR condition with the delay le than half the um of the filter order make the ynthei of the overall ytem very nonlinear and comlicated. 4.2 Deign method for low-delay PR biorthogonal filter bank For given N, N, (k ) ρ and (k ) ρ for k =,, and δ a well a K, the roblem i to find the adjutable coefficient of G (z) and G (z), a given by Eq. (4a) and (4b), to minimize where ubject to π ( ε ε ) ε= max (8a) ( e ), ε k= Gk d for k=,, (8b) ( k ) 2 max ( k ) [, ] G k e ( ) δ, max G ( k ) ( k ) (, ) k ( e ) δ for k =,, (8c) and [, π] j( + π ) j( + π ) ( e ) G ( e ) G ( e ) G ( e ) max G e jk =. (8d) ( k ) ( k ) ( k ) ( k ) Here, = ( ρ ) π/2 and = ( ρ ) π/2 for k =, are the aband and + toband edge for G (z) and G (z) a hown in Fig. 2. Thi roblem i more comlicated than the one tated in Section 3.2. Here the goal i to minimize the maximum of the toband energie of the two filter under conideration. The condition of Eq. (8d) guarantee that a PR filter bank will be obtained. The above otimization roblem can alo be olved by roerly alying the algorithm of Dutta and Vidyaagar [] (ee, e.g., [] or [2] for detail). 5 Numerical examle Thi ection illutrate, by mean of examle, ome characteritic of two-channel low-delay QMF bank and low-delay PR biorthogonal filter bank. In addition, the filter bank reulting when alying the rooed otimization cheme are comared to thoe obtained uing other exiting technique. Examle. To how the flexibility of the rooed method, two low-delay QMF bank with N = 3, K = 5, ρ = ρ =.9, δ a =.38-4, and different aband rile δ =.87 and 6

7 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. δ = 4 have been deigned. Fig. 4 how the amlitude characteritic of the analyi filter a well a the recontruction error for both filter bank. A exected, a maller aband rile reult in a lower toband attenuation. In both cae, the achieved aband rile i very mall. Only in the tranition band, the amlitude characteritic exhibit the value of +δ. Moreover, the firt filter bank ha a erformance imilar to that of Examle 2 in [5]. Examle 2. It i deired to deign a QMF bank with K = 3, ρ =.72 ( =.586π), δ a = , and ρ = ρ =.6-3. Fig. 5 how the amlitude characteritic of the analyi filter a well a the recontruction error for a low-delay QMF bank of order N = 63 deigned uing the rooed method and for a linear-hae QMF bank of order N = 3 (Examle in [6]). The low-delay filter bank rovide ignificantly better toband attenuation. Examle 3. A low-delay PR biorthogonal filter bank with N = N = 3, K = 5, δ = 3.8-3, and ρ = ρ =.2 for k =, ha been deigned. The requirement are the ame a for the (k ) (k ) filter bank CLS32-5a in [9]. Fig. 6 how the amlitude characteritic of the analyi filter for both filter bank. Uing the rooed method, filter with coniderably better attenuation are obtained. Examle 4. It i deired to deign low-delay PR biorthogonal filter bank for K = 3, δ =., and ρ = ρ =.72 for k =,. The aband and toband edge for both G (z) (k ) (k ) and G (z) are thu located at =.44π and =.586π, reectively. Fig. 7 comare the otimized low-delay PR filter bank of order N = N = 33, N = N = 45, and N = N = 63, with a biorthogonal filter bank with linear-hae ubfilter of order N = N = 3. A can be exected, the toband attenuation of the analyi filter in the low-delay filter bank increae a the filter order are made higher. Reference [] H. S. Malvar, Signal Proceing with Laed Tranform. Norwood: Artec Houe, 992. [2] P. P. Vaidyanathan, Multirate Sytem and Filter Bank. Englewood Cliff, N.J.: Prentice Hall, 993. [3] N. J. Fliege, Multirate Digital Signal Proceing. Chiceter: John Wiley and Son,

8 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. [4] H. Xu, W.-S. Lu and A. Antoniou, An imroved method for the deign of FIR quadrature mirror-image filter bank, IEEE Tran. Signal Proceing, vol. 46, , May 998. [5] W.-S. Lu, H. Xu and A. Antoniou, A new method for the deign of FIR quadrature mirror-image Filter Bank, IEEE Tran. Circuit Syt. II, vol. 45, , July 998. [6] R. Bregović and T. Saramäki, Two-channel FIR filter bank A tutorial review and new reult, in Proc. Second Int. Workho on Tranform and Filter Bank, Brandenburg, Germany, March 999, TICSP #4, [7] K. Nayebi, T. P. Barnwell III and M. J. T. Smith, Time-domain filter bank analyi: A new deign theory, IEEE Tran. Signal Proceing, vol. 4, , June 992. [8] K. Nayebi, T. P. Barnwell III and M. J. T. Smith, Low delay FIR filter bank: Deign and evaluation, IEEE Tran. Signal Proceing, vol. 42,. 24 3, Jan [9] E. Abdel-Raheem, F. El-Guibaly and A. Antoniou, Deign of low-delay two-channel FIR filter bank uing contrained otimization, Signal Procing, vol. 48, , Feb [] S. R. K. Dutta and M. Vidyaagar, New algorithm for contrained minimax otimization, Mathematical rogramming, vol. 3,. 4 55, 977. [] R. Bregović and T. Saramäki, A general-uroe otimization technique for deigning two-channel FIR filter bank, in Proc. Euroean Signal Proceing Conference, EUSIPCO-2, Tamere, Finland, Set. 2, vol. I, [2] T. Saramäki, A generalized cla of coine-modulated filter bank, in Proc. Firt Int. Workho on Tranform and Filter Bank, Tamere, Finland, Feb. 998,

9 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. x[n] H (z) H (z) 2 2 F (z) 2 2 F (z) + y[n] Fig.. Two-channel filter bank. H (e ) H (e ) π/2 π +δ δ G (e ) H (e j(π ) ) G (e ) H (e ) π () () π/2 () () Fig. 2. Secification for G (z) and G (z) a well a the relation between H (z) and G (z) and H (z) and G (z). 9

10 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2..5 Imule reone for E(z): K=7, N +N = Imule reone for E( z) Imule reone for T(z)=E(z) E( z) n in amle Fig. 3. Imule reone for E(z), E( z), and T(z) for a PR filter bank with K < (N +N )/2. Recontruction error Amlitude in db x Normalized frequency (w/(2))

11 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. Fig. 4. Low-delay QMF bank with N = 3 and K = 5. The olid and dot-dahed line how the amlitude reone of analyi filter for δ =.87 and δ = 4, reectively. Recontruction error Amlitude in db x Normalized frequency (w/(2)) Fig. 5. QMF bank for K = 3. The olid and dot-dahed line how the amlitude reone of analyi filter for the low-delay QMF bank for N = 63 and the linear-hae QMF bank for N = 3, reectively. Amlitude in db Normalized frequency (w/(2)) Fig. 6. Low-delay PR biorthogonal filter bank with N = N = 3 and K = 5. The olid and dot-dahed line how the amlitude reone of analyi filter for the bank deigned with the rooed method and the bank CLS32-5a [9], reectively.

12 contrained otimization, CIT Journal of Comuting and Information Technology, vol. 8, no 4, , 2. a) b) c) Amlitude in db Amlitude in db Amlitude in db Normalized frequency (w/(2)) Fig. 7. Comarion for K = 3 between low-delay PR biorthogonal filter bank with (a) N = N = 33, (b) N = N = 45, (c) N = N = 63 and the linear-hae biorthogonal PR filter bank with N = N = 3. The olid and dot-dahed line give the amlitude reone of analyi filter for the low-delay bank and the linear-hae bank, reectively. 2