APPLICATIONS OF WLS TECHNIQUES FOR DESIGNING DIGITAL FILTERS AND QMF BANKS

Abstract

In recent years, there has been an increasing interest in the weighted least squares (WLS) techniques for designing digital filters and quadrature mirror filter (QMF) banks, which play an important role in the field of digital signal processing. The key issues in digital filter design are the choice of suitable design criteria and optimization for the choice criteria. Many efforts have been made in designing digital filters, which are optimal in the minimax sense [4,25,27,39]. The design of digital filters, which are optimum in the minimax sense, requires the use of sophisticated optimization tools such as Remez exchange algorithm [7,79,100]. The WLS technique offers a promising alternative, which can be easily written in short computer codes. The optimum solution can be obtained analytically for any given least squares weighting function. But there are no known analytical methods for deriving the least squares weighting function that produces a minimax design. Hence, it is required to study various iterative techniques [18,113,130,137]. The primary reason for the choice of WLS techniques is that the best Chebyshev approximation is also the best WLS approximation, provided that the least squares updating weighting function should be suitably chosen [86,114]. The WLS approach is usually simpler compared with the weighted Chebyshev approach. The WLS techniques developed by Lawson and improved by others [130,113,137] produce an equiripple design if a suitable least squares weighting function is used. In digital filter bank context, many design methods have been proposed. Most of them are for two-channel QMF filter banks [66,72,94]. The design problems of QMF filter banks have received considerable attention since their introduction by Croisiere [102]. They have been widely used in one-dimensional and two-dimensional digital signal processing and have found applications in many areas such as subband coding and transmultiplexing [13]. Depending on the reconstruction at the output, QMF filter banks can be divided into two groups, perfect reconstruction (PR) QMF filter banks and near perfect reconstruction (NPR) QMF filter banks. The output ofa PR QMF filter banks is an exact delayed version ofthe input, whereas the output ofa NPR QMF filter banks has some small errors, which could be aliasing, magnitude, or phase distortions depending on the filter banks. In some applications, QMF filter bank with reconstruction delay less than AM (where Nis the order of the filter) is desired. The filter bank designed by Wu-Sheng [134] has reconstruction delay that is not constant in the transition band. This is relatively undesirable. From experiments, it was noticed that the undesirable artifacts could occur in the amplitude responses of the analysis and synthesis filters when the reconstruction delay (kj is significantly smaller than AM and if there is no constraint on its transition band. Such artifacts have been observed as well by other researchers when designing low-delay filter banks [17,54,56,89]. One of the solutions for this problem is to modify the objective function by including an additional term [75,134] in order to control these artifacts. This solution will force the pass band filter responses of the analysis and synthesis filters to be much larger than the transition responses [134]. Design of two-dimensional FIR filters has been an important research topic for 2-D digital signal processing. Many efforts have been spent on designing digital filters that are optimal in either the least square or the minimax sense [27,60,68]. By using the least square error criterion, one gets an overshoot offrequency response at the passband and the stopband edges caused by Gibbs' phenomenon [4]. The minimax design yields an equiripple solution and avoids the overshoot problem. However, it is complicated and requires more computational time [133]. The design of one or two-dimensional FIR filters can be performed with good computational efficiency using WLS design. To avoid the divergence of the Lim Lee Chen Yang (LLCY) algorithm [137] and to reduce the number of iterations required in the design process, a modified design of two-channel perfect reconstruction QMF filter banks has been carried out [142,146] using the iterative technique developed by Chen and Lee [18], along with certain modifications done on the weight updating technique described in [113]. It is preferable to use the WLS algorithm of Sunder and Ramachandran [113] to design a low pass filter, which is used in the construction of QMF filter banks. By using the WLS technique in [113], different QMF filter banks have been designed [142]. The resultants QMF filter banks have least square or equiripple reconstruction error with analysis and synthesis filters having equiripple or least square stopband errors. The results so obtained are superior as compared with other WLS techniques such as Lawson \Algazi [130] and LLCY [137]. The major drawback of FIR filter is the large number of arithmetic operations required in the implementation. Several techniques have been developed to improve FIR efficiency [69,79,120]. In [145], an investigation has been carried out on the iterative method proposed by Wu-Sheng [134] along with a modification based on improving the FIR efficiency. The technique in [134] depends on a self-convolution to reformulate a forth-order objective function as a quadratic function whose minimization leads to the design of the two-channel QMF filter banks. In our approach [145], an attempt has been in made to manipulate the filter impulse response to reduce its complexity while retaining the magnitude response of the filter within set specifications [11]. The filter design starts with the generation of a basis impulse response h. As a result, QMF filter banks with reduced computational efficiency are obtained. In [147,148,152], amodification on the QMF filter bank design given in [134] i made such that the transition-band constrains is added in the initial condition. This leads to have FIR QMF filter banks with constant group delay, improved peak reconstruction error and stop-band attenuation properties. But these improvements are obtained at the expense of computational efficiency and number ofiterations as compared with the other methods [53,134]. In [149,153], the WLS technique of [113] has been extended to design 2-D FIR digital filters. To achieve this, an intermediate desired frequency response Ghas been defined in order to obtain aclose-form solution in which all the parameters are kept in 2- D form. An extra updating desired frequency response is also introduced, which implicitly includes the weighting function such that the sum of weighted least square errors to be minimized are represented in a 2-D matrix form. The extra response is obtained by iteratively adjusting the designed frequency response Hwith the weight error. This is different from conventional WLS technique that rearranges the data of2-D form into their corresponding 1-D form. The computational complexity and the performance of the resultant 2-D digital filters are thus improved. It is confirmed through design examples that our approach is computationally efficient and leads to nearly optimal approximations. The thesis ends, as is customary, with reference to some problems, which could be taken up inthe future as an extension ofthis work.