Design of Sparse Filters by a Discrete Filled Function Technique

Abstract

In this paper, we consider the sparse filter design problem where some of the coefficients can be reduced to zeroes in order to lower implementation complexity. The objective is to choose the fewest number of nonzero filter coefficients to meet a given performance requirement. We formulate a discrete optimization problem to minimize the number of nonzero terms and develop a discrete search method to find the minimal nonzero terms. In each step, we need to consider a subproblem to design the filter coefficients with a given set of nonzero terms. We formulate this subproblem as a linear programming problem and apply an exchange algorithm to find the optimal coefficients. For illustration, we compare the proposed algorithm with existing methods and show that the proposed method gives better results in all our test cases. KeywordsSparse filter design • filled function • discrete search method 1 Introduction Digital finite-impulse response (FIR) filters design have been widely studied in the literature. There are many optimization methods available to deal with this class of problems. In particular, Parks-McClellan method is recognized to

Sparse FIR Filter Design Based on Signomial Programming

“…The problem in (14) is recognized as an SGP problem with 2K variables and 2(K + Q) inequality constraints.…”

“…We solve the problem in (14) by using the optimization method presented in [23]. The method referred to is able to solve a general SGP problem shown in (2).…”

“…For convenience, we describe here the optimization procedure from [23] tailored for SGP problem in (14). Let us first express the problem in a more compact form:…”

“…The functions fj(x), j = 1, 2, ..., 2(K + Q), are obtained by simple rearranging of the constraints in (14). Note that we left the constraints in a form fj(x)  0 instead of transforming them into a common SGP form fj(x)  1, because such a notation simplifies some of further expressions.…”

“…However, several approximate methods have been developed. Many of them utilize l0-norm and estimate the positions of zero coefficients by using greedy algorithms [12], [13], integer programming [2], [14], iterative second order cone programming [15], iterative shrinkage/thresholding [16], and genetic algorithms [17]. If l0-norm in (1) is replaced by l1-norm, the filter is obtained with many small coefficients [5].…”