Design of Low-Delay FIFt QMF zyx Banks Using the Lagrange-Multiplier Approach zyx Esam Abdel-Raheem, Fayez El-Guibaly, and Andreas Antoniou Department of Electrical and Computer Engineering University of Victoria, P.O. Box 3055, Victoria, B.C., Canada, V8W 3P6 email: fayez@sirius.uvic.ca Abstract- A new approach for the design of two-channel perfect reconstruction FIR filter banks with short recon- struction delays is presented. A low-order filter is first designed and the objective function of the filter bank is formulated as a quadratic programming problem with lin- ear constraints. Then the Lagrange-multiplier method is used to design a higher-order filter. The method is simple, efficient, and flexible and leads to a closed-form solution. Design examples are included to illustrate the advantages of the method. I. INTRODUCTION Quadrature mirror filter (QMF) banks have received considerable attention in recent years and found applica- tions in many areas such as subband coding and trans- multiplexing [l]. A two-channel QMF bank is shown in Fig. 1. The reconstructed signal, in general, suffers from aliasing error and/or amplitude and phase distortions. A perfect-reconstruction (PR) system is one which is free from all errors and distortions, and the reconstructed sig- nal is therefore just a delayed version of the input sig- nal. Several approaches have been proposed for the de- sign of two-channel linear-phase FIR QMF banks [1]-[4]. The overall delay of such a system is determined by the lengths of the filters. Two-cha.nne1 QMF banks are widely used for tree-structured subband coding systems. These systems usually suffer from relatively long reconstruction- delays, which is a serious problem in many communica- tion 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 inverse of large matrices several times, which is time consuming. In this paper, 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], [5] and a closed-form solution is obtained. 11. PERFECT-RECONSTRUCTION SYSTEM The reconstructed signal in the two-channel QMF sys- tem of Fig. 1 is related to the input signal by zyxwvutsrq X(Z) zyxwvutsrqp = zyxwvutsr T(z)X(z) + A(z)X(-z) (1) where T(r) = $[Ho(z)Fo(z) + H1(z)F1(z)] are the channel and aliasing transfer functions, respec- tively. The aliasing term is cancelled by choosing Fo(r) = 21Y1(-z) and zyxwv Fl(z) = -2Ho(-t). If we impose the pure delay constraint where zyxwvu IC is an integer, then X(t) = zyxw z-kX(z) and a PR system is obtained. Let Ho(z) and Hl(z) be the transfer functions of a low- pass filter of length NO and a highpass filter of length NI (NI > No), respectively. The desired (or ideal) frequency responses of these filters can be expressed as go(ej~T) = I jjo(ejwT) I e-jwkoT jwT) = I fil(ej~T) I e-jwkiT (3) 1 (e where li0 < (NO - 1)/2 and IC1 < (NI - 1)/2 are the desired passband group delays of the lowpass and the highpass filters, respectively [6], and T is the sampling period [7]. It is easy to verify that k = ko+tl is the desired total system delay which is assumed to be an odd integer. Let ho(nT) and hl(nT) be the impulse responses of the causal lowpass and highpass filters, respectively. Assuming a normalized sa.mpling period T = 1 s, equation (2) can be expressed in the time domain as 2-1 E ( - l ) r ho(2i - 1 - r)h,1(r) = fb(i - IC’), r=O i= 1,2, ... , R wlhere ko + kl + 1 2 k’ = - 1 if No + NI is even No+N1-l if No+N1 is odd R= { zyx 2 Equation (4) can be expressed in matrix form as (4) and Cy1 = m (5) 0-7803-2428-5195 $4.00 0 1995 IEEE 1057