A Novel Technique for Designing Multirate Filter Banks Exploiting Quasi-Newton Optimization

Authors: Rohit Kumar; Dr. Bajrang Lal
DOI
10.5281/zenodo.19551963
DIN
IJOER-JUN-2025-1
Abstract

This paper proposes a new technique for the design of multirate filter banks with linear phase in frequency domain. To match the ideal system response, low-pass analysis prototype filter response is optimized to minimize an objective function. The objective function is formulated as a weighted sum of pass-band error and stop-band residual energy of low-pass analysis filter, the square error of the overall transfer function at the quadrature frequency and amplitude distortion of the filter bank. Quasi-Newton optimization method is used to minimize the objective function by optimizing the filter tap weights of the prototype filter. Simulation results show that the proposed method is able to perform better than other existing methods.

Keywords
Quadrature mirror filter bank Quasi-Newton optimization method Peak reconstruction error Linear phase.
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Introduction

Quadrature mirror filter (QMF) banks are multirate filter bank and have been extensively used for sub-band coding, where the signal is split into two or more sub-bands in the frequency domain, so that each sub-band signal can be processed in an independent manner and sufficient compression may be achieved [1]. At the receiver end the sub-band signals are recombined such that the original signal is properly reconstructed [2]. QMF banks find applications in many areas, such as analog to digital conversion [3], design of wavelet bases [4,5], image compression [6,7], digital trans-multiplexers [8], discrete multi-tone modulation systems [9], 2-D short-time spectral analysis [10], antenna systems [11], digital audio industry [12], biomedical signal processing [13,14,15].

Alias free efficient design of two-channel QMF banks while keeping minimum dimensions is a tough task. Therefore, various constrained and unconstrained optimization based techniques [16−31] have been developed for the design of linear phase QMF banks. Iterative methods [22−27] and genetic algorithms [28−31] have been proposed for the design problem of QMF based on multi-objective or single objective nonlinear optimization.

Fig. 1(a) shows the analysis and synthesis section of a popular multirate filter bank known as two-channel QMF bank. The discrete input signal x(n) is divided into two sub-band signals having equal band width, using the low-pass and high-pass analysis filters H0(z) and H1(z), respectively. Typical frequency responses of these filters are depicted in Fig. 1(b). The outputs of the synthesis filters are combined to obtain the reconstructed signal x̂(n). The reconstructed signal x̂(n) suffers from three types of errors: aliasing distortion (ALD), phase distortion (PHD), and amplitude distortion (AMD), due to the fact that the filters H0(z), H1(z), F0(z), and F1(z) are not ideal [32]. Therefore, the main stress of most of the researchers while designing the prototype filter for two-channel QMF bank has been on the elimination or minimization of these three distortions to obtain a perfect reconstruction (PR) or nearly perfect reconstruction (NPR) system [2, 16-21].

The overall transfer function of such an alias and phase distortion free system turns out to be a function of the filter tap coefficients of the low-pass analysis filter only [23]. Then, the AMD can only be minimized by optimizing the filter tap weights of the low-pass analysis filter using computer assistance techniques [2].

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Conclusion

A new iterative method for the design of multirate filter banks has been developed by formulating the perfect reconstruction condition in the frequency domain. The quadratic objective function is minimized without any matrix inversion which generally affects the effectiveness of some methods. Design example shows that the proposed technique is very effective in designing the quadrature mirror filters. The peak reconstruction error is minimum by the proposed method that makes it suitable for real time applications. Further, it is possible to extend this approach for the design of QMF banks with more than two bands.

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