work in this area). Explicitly, following [I] we consider a sequence C,,..., C, of j single-parity-check codes with parameters (n,, n, - 1, 2), (n2.n2- 1, 2), ..., (n, ni - 1, 2), respectively, where i = 2,. . . , j 9li 2 ni-1 and their iterated direct product A = C,0 ... 0 C,.The parameters of the resulting (n,k , d) code A are the following: n = H , = , Jn , k = n,,] (ni - I), d = r I + , j 2 = 2,. Battail [I] conjectures that as j m the weight distribution of this iterated product code tends to a Gaussian distribution. This conjecture is indeed correct if n, is large enough, as can be proved by applying the Sidel'nikov theorem 141. This states the following: Let A(w) denote the cumulative distribution function of the weights of A, i.e. the function L A a + A(w) = 2 0.31 3 4 5 6 7 8 9 1 0 0.26 0.16 0.092 0.059 0.037 0.025 0.016 0.011 > 0.99 > 0.99 >O .99 > 0.99 > 0.99 > 0.99 > 0.99 0.99 0.98 a, 3>lr--Ow where n P =C j a , ;=0 n and a; = A,/2k with A, the number of codewords of A with weight j. Further, let 91)denote the Gaussian cumulative distribution function 1 ' F ( z )= e-tz/2dt / Jz;; -m If & denotes the minimum distance of the dual code , ' C then Use of the Sidel'nikov theorem allows the Battail conjecture to he proved by showing that the dual of the iterated code A has a minimum distance which increases without bound as j m. Consider the product C of the two codes C,,= (nh, nh - 1, 2) and C,= (n,,n, - 1, 2). The dual CLof C has the form - Ic:+uI&+uI "'Icn,-1+ulC:+c:+.. .+& 1 BATTA1L.G.: 'Construction explicite de bons codes longs', Ann. Tilicommun., 1989, 44, (729, pp, 3 9 2 4 2 CAIRE, G., TARICCO, G . , and BATTAIL, G.: In preparation LODGE. I., YOUNG,R., HOEHER, P., and HAGENAUER, I.: 'Sperable MAP 'filters' for the decoding of product and concatenated codes'. Roc. ICC '93, Geneva, Switzerland, pp. 1 7 6 1 7 4 5 , May 1993 4 MACWILLIAMS, F.J., and SLOANE, N.I.A.: 'The theory of errorcorrecting codes' (Amsterdam: North Holland, 1978) 5 MASSEY, F.1.: 'The Kolmogorov-Smirnov test for goodness of fit', American Stat. Ass. J., March 1951, pp. 68-78 3 + U / (2) where U is any binary n,-tuple, and e: are words of the dual code Chk= (n,,, 1, nh). The minimum Hamming weight of C,,0 C,is easily shown to be min{n,, n,}. Repeated use of this result proves that the minimum Hamming distance of A is n, . Thus, by choosing n, large enough the distance between the weight distribution of A and the Gaussian distribution can be made arbitrarily small. A different proof of this fact has been derived independently in [2]. In conclusion, by choosing long enough component codes the weight distribution of their iterated product is asymptotically Gaussian. Now the question is: what can we say of moderatelength codes, as those whose performance was evaluated in [3]? Closed-form calculation of their weight distribution is impractical, so we have resorted to a statistical goodness-of-fit test, which makes it possible to ascertain whether the code weights are distributed according to a Gaussian distribution. Among the goodnessof-fit tests, we have chosen that of Kolmogorov-Smirnov (KS) because it is based on the same distance measure (eqn. 1) and because it is applicable where the size of a statistical sample is small. The KS test runs as follows [5]. N words are drawn at random from code C, their weights computed, and their cumulative distribution function A d w ) evaluated. The deviation of AN(w) from e w ) is measured by A = SUP IAN(w)- F(ut)l w A is a random variable that depends on the sample of codewords drawn. If the codewords are drawn from a population whose distribution is actually Gaussian, then the quantity P { A < A,(N)} =a can be computed, and in fact it is extensively tabulated as a func- 924 References Design of low-delay FIR QMF banks using the Lagrange-multiplier approach E. Abdel-Raheem, El-Guibaly and A. Antoniou Indexing terms: Digital signal processing, Multirate signal processing A new approach for the design of two-channel perfect reconstructionFIR filter banks with short reconstructiondelays is presented. A low-order fdter is first designed and the objective function of the fdter bank is formulated as a quadratic programming problem with linear constraints. The Lagrange multiplier method is then used to design a higher-order fdter. The method is simple, efficient, and flexible and leads to a closed-form solution. A design example is included to illustrate the advantages of the method. Introduction: Quadrature mirror fdter (QMF) banks have received considerable attention in recent years and found applications in many areas such as sub-band coding and transmultiplexing [I]. A two-channel QMF bank is shown in Fig. 1. The reconstructed signal, in general, suffers from aliasing errors andlor amplitude and phase distortions. A perfect-reconstruction (F'R) system is one which is free from all errors and distortions, and the reconstructed signal is therefore just a delayed version of the input signal. Several approaches have been proposed for the design of two-channel linear-phase FIR QMF banks [I-41. The overall delay of such a system is determined by the lengths of the fdters. Two-channel QMF systems are widely used for tree-structured subband d i g systems. These systems usually suffer from relatively long recon- ELECTRONICS LETTERS 9th June 1994 Vol. 30 No. 12 ~ struction delays which is a serious problem in many communication systems. Thus, the design of two-channel QMF banks with low delays is desired. Such a design was presented in [4]; however, this approach involves calculating the pseudo-inverse of large matrices several times, which is time consuming. In this Letter, we propose an approach for designing two-channel PR FIR QMF banks with low delays. The method is based on the Lagrange-multiplier approach [3, 51, and a closed-form solution is obtained. LPF (ii) Form an error measure for the design of a highpass fdter of transfer function HI@)and length NI, and group delay k,. The error measure can be put in the form *i 1 = ZYTQiYi + P ~ Y+Idi where QI is an NI x NI matrix, PI is an NI x 1 vector, and d, is a constant [6]. (E) The optimisation problem for designing H,(z) becomes LPF subject to Cy, = m minimise (iv) Form the Lagrangian function 1 L(YI,A) = -2Y T Q m PTyl where + Fig. 1 Two-channel QMF system x = [A1 x z ' . + dl - XT(Cy, - m) (6) (7) A# ' Perfect-reconstruction system: The reconstructed signal in the two- channel QMF system of Fig. 1 is related to the input signal by + X(z) = T ( z ) X ( z ) A(z)X(-t) (1) (v) The necessary and sufficient conditions for the solution form the following set of linear equations: ~~ and A ( z ) = $Ho(-z)Fo(z) + Hi(-t)4(z)] are the channel and aliasing transfer functions, respectively. The aliasing term is cancelled by choosing F&) = 2H,(-z) and F,(z) = -2H&z). If we impose the following pure delay constraint T ( z )= 2-k ~ ~ from which a closed-form solution is obtained. 1 (2) where k is an integer, then f ( z ) = r"X(z) and a PR system is obtained. Let Hdz) and XI($ he the transfer functions of a lowpass filter of length No and a highpass fdter of length NI (NI>No), respectively. The frequency responses of these filters can be expressed as Ho (e3") = IHO ( e J Wle-Jwku ) (3) H1(eJW)= IH1(ej"')le-Jwkl where ko < (No - 1)/2 and k, < (NI - 1)/2 are the group delays of the lowpass fdter and the highpass filter, respectively. It is easy to verify that k = ko + k, is the total system delay which is assumed to be an odd integer. Let k(nT) and h,(nT) be the impulse responses of the lowpass fdter and the highpass fdter, respectively. Assuming a normalised sampling period T = 1, eqn. 2 can be expressed in the time domain as Design example: A low-delay QMF system with No = 20, NI = 24, a = 3, fi = 1, ko = 3, and k, = 6 was designed where a and f3 are the passband and stopband weights, respectively. The 20tap lowpass filter was designed using the approach reported in [6],assuming band-edge frequencies wpo = 0.47n, w,( = 0.57% and ko = 3. Then matrix C was formed. Matrix Q, was then calculated assuming band-edge frequencies opl= 0.6n, and wSI = 0.4n and k = 6. The coefficients of H,(z) were then obtained by solving eqn. 8. Applying a ramp input signal to the filter bank, the signal-to-noise ratio (SNR) as defmed in [4] was 233dB assuming infinite precision. This value is in the range of the SNRs for the systems designed in [3] and it is higher than those of the low-delay systems in [5]. The high SNR and the low error in the amplitude response and the delay characteristic of the channel depicted in Fig. 2 demonstrate the PR quality of the design. It is noted that the system delay is 9 samples, as opposed to 21 samples for the linear-phase case of [3]. 2i-1 r=O (-l)'ho(2i - 1- r ) h l ( r ) = AS(i - k') i = 1; 2 where k' = k o + k i + l ~ 2 R={ - 1 if NO+ N I is even if NO+ N I is odd Eqn. 4 can be expressed in matrix form as Cy, = m (5) where C is an R x NI matrix with its elements formed by Mn),n = 0, 1, ..., No - 1, and 0 a 0 1 0 2 0.3 0.4 0.5546,20 0 1 0 2 0-3 0.4 0-5 normalised frequency o b normalised frequency Fig. 2 Channel amplitude error and delay error for the QMF bank of the example a Channel amplitude error b Delay error with Lagrange-mdtiplier method: The PR QMF bank design procedure on be summarised as follows: (i) Design a lowpass filter of transfer function ITo@) and length No, and group delay ko using the approach reported in [6]. ELECTRONICS LE77ERS 9th June 7994 Vol. 30 Conclusions: The Lagrange-multiplier approach has been applied for the design of two-channel PR FIR QMF banks with low delays. The method is simple, efficient, and flexible and leads to a closed-form solution. A design example was given which demonstrates that the quality of the reconstruction is very good. We achieved low-delay designs using low-order filters while maintaining a high SNR, which is an advantage from the perspective of implementation. Also, we achieved the same low delays using higher-order filters. No. 72 925 Acknowledgment: The authors wish to thank Micronet, Networks of Centres of Excellence Program, and the Natural Sciences and Engineering Research Council of Canada for supporting this work. 8 IEE 1994 22 March 1994 Electronics Letters Online No: 19940629 E. Abdel-Raheem, F. El-Guibaly and A. Antoniou (Department of Electrical and Computer Engineering, University of Victoria, Victoria. BC V8 W 3P6, Cunudu) F. El-Guibaly: Corresponding author Refer1 VAIDYANAT”, P.P.:‘Multirate systems and filter banks’ (Prentice- Hall, 1993) 2 3 4 5 6 and ANTONIOU, A.: ‘QMF banks: Design and implementation perspectives’. Proc. IEEE Pac. Rim Conf. on Comm., Comp., Signal Processing, 1993, Victoria, BC, Canada, pp. 35&351 HORNG, B.R..and WILSON, A.N.:‘Lagrange multiplier approaches to the design of two-channel perfect-reconstruction linear-phase filter banks’, IEEE Trans., 1992, S P 4 , pp. 364-374 NAYEBI. K.. BARNWELL. T.P.. and SMITH. M.I.T.: ‘Time-domain filter bank analysis: A new design theory’, IEEE Trans., 1992, SP-40, pp. 1412-1430 FLETCHER, R : ‘Practical methods of optimizations (John Wdey & Sons, 1987.2nd Edn.) SUNDER. S., and RAMACHANDRAN, R.P: ‘A kaSt-SquareS design Of nonrecursive fdters satisfying prescribed magnitude and phase specifications,.IEEE ht. symp, Circuits and sys., 1993, Michigan, pp. 335-338 burst errors. The conditions of the situation are as follows. A conventional fmed block interleaver/deinterleaverhas a memory in matrix form. The input data to the deinterleaver are written row by row and the output data are read column by column. Let the number of the deinterleaver matrix columns and that of rows be M and N , respectively; if the deinterleaver contains multiple error bursts and the period of periodic burst errors is equal to nM (n is an integer), errors presumed to be separated gather to form new burst errors. If these new burst errors are decoded on a random error correction basis, the BER will be degraded. As long as the period of burst errors is constant, we can optimise the interleaving parameters to fit the error patterns. However, if the period of burst errors or data transmission rate is changeable, it becomes impossible to optimise the interleaving parameters to fit the changeable error patterns, and BER degradation is unavoidable. ABDEL-RAHEEM. E., EL-GUIBALY, F., Evaluation of pseudorandom interleaving: Radar pulses are gener- ated periodically and their duty cycle is usually long enough to cover multiple symbol periods. This brings about periodic burst errors. The period and duration of periodic burst errors therefore depend on the radar pulse repetition frequency, radar pulse duty cycle, and symbol transmission rate. Because these three parameters are changeable, the period and duration of periodic burst errors are also changeable. To compensate for such radar interference, pseudorandom interleaving seems to be effective because it does not meet the con&tions of BER degradation shown in the preceding =tion, Therefore, we investigated by computer simulation the effectiveness of pseudorandom interleaving. We evaluated the performance of three different systems: A: a system with fixed block interleaving on a bursty error channel Interleavingtechnique against periodic bursty errors caused by radar interference B: a system with pseudorandom interleaving on a bursty error channel N. Nakajima, T.K. Matsushima and J. Murakami C: a system without interleaving on an equivalent random error channel. Indexing term: R u h r interjerence, Error correction codes, Digital communicarion systems An ‘equivalent random error channel’ is defined as a channel in which errors occur uniformly and independently with a long-term BER eaual to that of a burstv error channel. System C is SUD- pseudorandom interleaving is not degraded. interleaver burst errOrS or AWGN Introduction: For certain kinds of radio communication system, + radar interference causes severe degradation in the system performance. Aikawa et al. [I] have reported that conventional interleaving in conjunction with random-error-correction coding is an interference is observed as periodic bursty errors. On the other hand, Mui [2] has suggested that certain patterns sequence * Viterbi decoder fixed Mock lpseudorandom deinterleaver

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