New algorithm for minimax design of sparse IIR filters

Abstract

Abstract-Sparse digital filters are of importance as they offer improved implementation efficiency relative to their nonsparse counterparts. This paper examines the design of minimax sparse IIR filters from a sparse representation point of view. The result is a new algorithm that accomplishes a design with three phasesidentification of the whereabouts of zero coefficients; optimal design subject to the sparsity constraint; and performance enhancement by further dimension reduction of the working subspace. A design example is presented for illustrating the new algorithm. To date, most of the work in this area has been focused on sparse FIR filters, except that of [9]. This paper presents a new algorithm for minimax design of stable sparse IIR filters. The method is different from that of [9] in several ways. First, the new algorithm is built on a linear sparse representation framework. This linear representation is made possible due to the fact that the poles of a sparse IIR filter are very close to those of a nonsparse IIR counterpart as long as both filters approximate the same frequency response, which is in a certain sense similar to an observation made in [16]. Second, here we adopt a constrained formulation to avoid the use of a regularization parameter as in [9]. In this way, the filter's sparsity is explicitly related to the minimax error of the filter, which in turn makes sparsity tuning easier and more intuitive. Third, the new algorithm includes an additional phase to enhance performance by further reduction of the working subspace's dimension and re-optimization. A design example is presented to illustrate the new algorithm. I. INTRODUCTION

Minimax Design of Sparse IIR Filters Using Sparse Linear Programming

“…as an evaluation value of the following algorithm. Hence, we attempt to obtain the combination of ν zero coefficients that minimized (34). Afterwards, we employ the successive thinning algorithm presented in [24] for sparse optimization of the numerator coefficients.…”

“…With respect to the design of sparse filters, researchers have focused on the design of FIR filters [21]- [32], the main reason being the highly complicated design of IIR filters compared with that of FIR filters, which is because of the non-linearity of the design problem caused by the presence of a denominator polynomial. Hence, there have been limited studies on the design of sparse IIR filters [33], [34]. For instance, a recent study [34] employed second-order cone programming (SOCP) [1] to design an IIR filter.…”

“…Hence, there have been limited studies on the design of sparse IIR filters [33], [34]. For instance, a recent study [34] employed second-order cone programming (SOCP) [1] to design an IIR filter. Because of the stability constraint, sparse optimization of denominator coefficients is difficult.…”

“…Because of the stability constraint, sparse optimization of denominator coefficients is difficult. Therefore, sparse optimization of the numerator coefficients was introduced in [34]. Additionally, the design examples are limited to the case of secondorder denominator polynomial.…”

“…This consists of multiple steps and includes sparse optimization (sparse linear programming). Since the denominator coefficients affect the stability constraint, the proposed method considers the sparse optimization of denominator coefficients, similar to [34]. Further, a method that can design Copyright c 2021 The Institute of Electronics, Information and Communication Engineers denominator coefficients of any order is required.…”