An Application of Symbolic Computational Algebra in Signal Processing

The design of digital FIR filters and filter banks for wavelet systems are areas of signal processing where techniques from symbolic computational algebra are currently being applied. In this talk we will discuss the Selesnick-Burrus design equations for low-pass filters. This method specifies filters by imposing a chosen multiplicity K for the zero at omega=pi, a chosen number M of flatness constraints on the square magnitude response at omega=0, and a third number L of flatness constraints on the group delay response at omega=0. This leads to a system of polynomial equations in the filter coefficients and the problem becomes one of polynomial system solving. We will present some new results on solving the Selesnick-Burrus equations in the difficult cases where M is large and L is small relative to M ("Region II" in their terminology), and discuss how to determine the number of complex or real solutions for given (K,L,M).