357 1-4244-1164-5/07/$25.00 ©2007 IEEE. Genetic Algorithm and Its Application for the Design of QMF Banks with Canonical Signed Digit Coefficients: A Comparative Study and New Results P. Samadi, M. Ahmadi Electrical and Computer Engineering Department, University of Windsor, ON, Canada (samadi@ieee.org, ahmadi@uwindsor.ca) Abstract—In this paper Genetic Algorithm (GA) is used for designing Quadrature Mirror Filter (QMF) banks with Canonical Signed Digit (CSD) coefficients. The performance of Genetic Algorithm is investigated through a novel study on dependency of Population size and number of Generations to Mean Square Error (MSE). Two optimization techniques are also compared for retaining the CSD constraints of coefficients. The first method offers a restoration operation for each loop of Genetic Algorithm but the second one can easily keep the CSD constraints through a new chromosome coding scheme. It is shown that for both IIR QMF bank and FIR QMF bank; the second technique yields better performance. I. INTRODUCTION Quadrature Mirror Filter (QMF) banks have been subject of research for many years [1], [2]. They are applied for the situation where a discrete-time signal ] [n x is needed to be split into a number of sub-band signals ] [n x i so that each can be processed separately. Typical processing comprises down- sampling, coding for transmission and storage. Subsequently at some point, these signals are needed to be recombined together to have the original signal reconstructed. The most common applications are frequency domain speech samplers, subband coders for speech signals and digital trans-multiplexers used in FDM/TDM conversion [1], [2]. Based on the type of Synthesis and Analysis filter (FIR or IIR) and number of channels, there are different kinds of filter banks. In this research we have focused on 2-channel QMF bank both FIR and IIR. Fig. 1 shows the basic two-channel QMF bank. The analysis bank is composed of a low-pass filter ) ( 0 z H and a high pass filter ) ( 1 z H . These two filters split the incoming signal to two frequency bands. The sub-band signals are then down-sampled by a factor of two. Each down-sampled subband signal is encoded by exploiting the special spectral properties of the signal, such as energy level and perceptual importance. At the receiving end these signals are up-sampled by a factor of two and fed into the two-band synthesis filter bank ) ( 0 z G and ) ( 1 z G , and at the end the output is created by adding these two signals. Up-sampling of ] [ 0 n x and ] [ 1 n x result in images, which have to be eliminated by ) ( 0 z G and ) ( 1 z G . The output of H0(z) H1(z) 2  2  G0(z) G1(z) 2  x[n] + y[n] 2  x [n] x [n] 0 1 v [n] 0 v [n] 1 Fig. 1: The two-channel QMF Bank ) ( 0 z G and ) ( 1 z G are a good approximation of ] [ 0 n x and ] [ 1 n x , and the reconstructed signal ] [ n y closely resembles ] [n x . In QMF banks the response of ) ( 0 z H is the mirror image of the response of ) ( 1 z H . There are diverse techniques for design of QMF banks [3], [4], namely, non-optimization based and optimization based. In this work, we have exploited Genetic Algorithm as an optimization based method for computing the coefficients of the filter with CSD format. A Comparative study on two different optimization techniques and the performance of GA is also performed. Therefore, this paper is organized as follows. In next section we review the design of FIR and IIR QMF banks. In Section 3 we define GA based design technique with two different optimization techniques for CSD format. In section 4 comparisons between two techniques are given. Section 5 is the performance analysis of GA and the paper finishes with conclusion. II. DESIGN OF QMF BANKS Consider Fig. 1 which is a two-channel QMF bank. In this figure the relation between the input and output signal is as follows: ) ( )} ( ) ( ) ( ) ( { 2 1 ) ( )} ( ) ( ) ( ) ( { 2 1 ) ( 1 1 0 0 1 1 0 0 z X z G z H z G z H z X z G z H z G z H z Y - - + - + + = (1) Knowing that up-sampler and down-sampler are linear time- varying components, as a result the QMF bank structure is a Linear Time-Varying (LTV) system and there will be aliasing effect. It’s possible to choose the analysis and synthesis bank