1426 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 5, MAY 1998 Correspondence An Efficient Algorithm for the Design of Weighted Minimax -Channel Cosine-Modulated Filter Banks Chee-Kiang Goh and Yong-Ching Lim Abstract—This correspondence presents an efficient algorithm for the design of -channel cosine-modulated filter banks. Our algorithm needs only a few iterations to obtain a weighted minimax solution and provides flexible control of the ripples in the filter’s stopband, the overall filter bank transfer function, and the aliasing components. Index Terms—Multirate filter banks, signal processing, weighted least squares method. I. INTRODUCTION Multirate digital filter banks are used extensively in many appli- cations, especially subband coding of audio, images, and video [2], [3]. These filter banks decompose an input signal into its subband components (analysis phase) and reconstruct the signal from the downsampled version of these components (synthesis phase) with little or no distortion. The modulated filter bank [4], [5] has recently emerged as an attractive choice due to its ease of design and implementation. All the analysis and synthesis filters can be obtained by modu- lating the coefficient values of one prototype filter, and efficient polyphase structures in cascade with fast transforms are used for their implementation. A variety of approaches are available for the design of the prototype filters [6]–[9]. General nonlinear optimization techniques were typically employed, especially for the design of perfect reconstruction (PR) filter banks, which may lead to further difficulties such as nonconvergence and step-size selection. For practical applications with lossy channel coding and quantization, the PR property is desirable but not necessary. In addition, the PR filter bank with high stopband attenuation is generally difficult to achieve. Recently, several iterative methods yielding good near PR designs have been reported [7], [10], [11]. In this correspondence, we present an efficient algorithm for the design of -channel cosine-modulated near PR filter banks. Similar to the approach in [1], the weighted minimax design of the filter bank is formulated as an unconstrained quadratic programming problem with respect to the prototype linear-phase lowpass FIR filter. Typically, our algorithm needs only a few iterations to obtain a solution that is optimal in the weighted minimax sense. The algorithm also provides flexible control of the ripples in the prototype filter’s stopband, the overall filter bank transfer function, and the aliasing components. Good -channel cosine-modulated filter banks with stopband attenuation and signal-to-reconstruction-noise ratio exceeding 100 dB can be easily designed using our algorithm. II. -CHANNEL COSINE-MODULATED FILTER BANKS The impulse response , of a prototype linear-phase lowpass FIR filter is symmetrical. It can be shown that Manuscript received March 27, 1997; revised October 14, 1997. The associate editor coordinating the review of this paper and approving it for publication was Dr. Sergios Theodoridis. The authors are with the Department of Electrical Engineering, National University of Singapore, Singapore (e-mail: gck@vlsi.ee.nus.sg). Publisher Item Identifier S 1053-587X(98)03259-0. the analysis and synthesis filters of an -channel cosine-modulated filter bank are written as (1a) (1b) where and , , represent the impulse responses of the th channel analysis and synthesis filters, respectively. It is noted that other lossless modulation [8], [12] exists, but the underlying design of the prototype filters are similar to the Type-IV cosine modulation [8] used in (1). The reconstructed signal can be written in terms of these filters in the domain as (2) where (3) and denotes the overall filter bank transfer function, whereas , represents the aliasing components. Note that and that . The weighted minimax optimization problem for the design of the cosine-modulated filter bank is formulated as follows: Minimize subject to (4) where is a ripple ratio function specifying the tolerance in (4), and . The error function is defined as (5) The Fourier transform of the prototype filter is , and is the stopband edge of . In (5), the ripples in the overall filter bank transfer function are evaluated on grid of points in the region , and the ripples in the stopband of the prototype filter are evaluated on grid of points in the region . A weighted least squares objective function as shown in (6) can be formulated for the design of the prototype filter coefficients to solve the nonlinearly constrained problem in (4). (6) In (6), is a cost weighting function, and . 1053–587X/98$10.00  1998 IEEE