Solving the Selesnick-Burrus Filter Design Equations Using Computational Algebra and Algebraic Geometry

Abstract

In a recent paper, I. Selesnick and C.S. Burrus developed a design method for maximally flat FIR low-pass digital filters with reduced group delay. Their approach leads to a system of polynomial equations depending on three integer design parameters $K,L,M$. In certain cases (their ``Region I''), Selesnick and Burrus were able to derive solutions using only linear algebra; for the remaining cases ("Region II''), they proposed using Gr\"obner bases. This paper introduces a different method, based on multipolynomial resultants, for analyzing and solving the Selesnick-Burrus design equations. The results of calculations are presented, and some patterns concerning the number of solutions as a function of the design parameters are proved.