Modified version of Chen and Lee's algorithm for the design of QMF banks Sofija BOGDANOVA* Mitko KOSTOV* Momcilo BOGDANOV * * Faculty of Electrical Engineering, Ss. Cyril and Methodius University, Skopje, R. Macedonia Abstract A modification of the Chen and Lee’s algorithm for the design of quadrature mirror filter banks in the frequency domain is presented. It concerns updating the frequency response weighting function and results in significantly reduced number of iterations. 1 Introduction It is known that the design of QMF banks in the frequency domain can be accomplished by least- squares method and minimax method. The minimax design can be performed if an adequately updated weighting function is included in a least-squares objective function. The design is a typical uncon- strained and highly nonlinear optimization problem due to the fact that the objective function is a fourth- order function of the design parameters. In [1] Chen and Lee have proposed a linearization technique and derived an analytical design formula. Based on this analytical formula, the coefficients of the required low-pass filter with linear phase can be obtained by solving a set of linear equations at each iteration. By incorporating the proposed technique with a weighted least-squares algorithm they have obtained QMF banks having overall reconstruction error minimized in the minimax sense, in addition to the QMF filters having least-squares stopband error. The method of Chen and Lee leads to a very efficient algorithm for the considered design problem. An improved implementation of this algorithm, concerning the evaluation of two integrals involved in the computation of the objective function, has been reported in [2]. A recently developed a new iterative method [3] is also based on the same algorithm. There, the improvement results from the formulation of perfect reconstruction condition in the time domain. In this paper we propose an alternative imple- mentation of the algorithm of Chen and Lee, which leads to reduced number of iterations in the design process. The paper is organized as follows. Section 2 formulates the design problem. In Section 3, the algorithm of Chen and Lee is reviewed and exposed in a manner very close to that in [1]. Section 4 presents the proposed alternative realization of the algorithm. Two design examples for illustration and comparison are presented in Section 5. 2 Formulation of the design problem The following notation is assumed for a two-channel QMF bank: H 0 and H 1 are the lowpass and highpass filters, respectively, of the analysis section, and F 0 , F 1 are the corresponding filters of the synthesis section. Their impulse responses are h 0 (n), h 1 (n), f 0 (n) and f 1 (n). All filters are assumed to be linear phase and they are all of length N, where N is even. The conditions imposed on the analysis and synthe- sis filters for a perfect reconstruction system reduce the design problem to finding only the coefficients of the lowpass analysis filter. The frequency response of this prototype filter, H 0 (e jω ), satisfy so-called power complementary property ω ω ω ω all for 1 ) ( ) ( ) ( 2 ) ( 0 2 0 = + = π + j j j e H e H e T . The reconstruction error is defined as 1 ) ( ) ( - = ω ω j r e T e . The objective function E to be minimized in the weighted least-squares sense is given by [ ]   π π + = + = s d e H d e W E E E j r s r ω ω ω α ω ω ω α 2 0 0 2 ) ( ) ( ) ( where E r is the energy of the reconstruction error, E s is stopband energy related to the lowpass analysis filter H 0 , α is the relative weight between E r and E s , W(ω) is a weighting function, and ω s is the stopband frequen- cy. 3 Review of Chen and Lee's algorithm A basic idea in the algorithm of Chen and Lee [1] is that the conditions for a perfect reconstruction system are satisfied when the frequency response of the lowpass synthesis filter F 0 (e jω ) is very close to the frequency response of the lowpass analysis filter H 0 at k th iteration denoted as H 0 (k) (e jω ). Chen and Lee have reformulated the design problem as follows. The coefficients {f 0 (n), n = 0, 1, ... N/2-1} of the lowpass