Mixed‐Integer Linear Programming Design of Economical Decimation Filters via Self‐Reciprocal Polynomials With Unimodular Roots

This paper presents a novel method for designing low‐complexity, multiplier‐free filters tailored to multistage decimation architectures. The proposed decimation filters are constructed from building blocks P m , r ( z ), which are self‐reciprocal polynomials of degree 2 m + 1, where and r are design parameters. When r is selected within the admissible ranges identified in this work, these polynomials exhibit all their zeros on the unit circle (i.e., unimodular roots), making them well suited for multiplier‐free, intrinsically stable, and highly selective decimation filtering. After establishing key properties of the proposed building blocks that enable computationally efficient implementations, a mixed‐integer linear programming (MILP) optimization framework is introduced to determine a minimal‐complexity set of such blocks whose cascade satisfies prescribed filter specifications. Practical guidelines for implementing the resulting multistage decimation filters are provided. Finally, the proposed approach is evaluated through comparisons with existing techniques in the literature.