On the unit circle problem: The Schur-Cohn procedure revisited

Abstract

Abstract This paper presents several results concerned with finding the zero distribution of a polynomial with respect to the unit circle using variants of the Schur-Cohn procedure. First, new and simple proofs of the Schur-Cohn procedure in the regular and singular cases are presented. A new method for handling one type of singular case is also developed. Next, we consider several aspects of the ‘inverse problem’, in which one wishes to alter a given polynomial to have a prescribed zero distribution. Three methods for forcing all zeros of a polynomial to lie inside the unit circle are derived. Also, two algorithms for solving singular inverse problems are presented; specifically, one corrects a symmetric polynomial to ensure that all its zeros lie on the unit circle, and the other corrects a symmetric polynomial to ensure that none of its zeros lie on the unit circle. The inverse problems have applications in spectral estimation, signal processing and system identification.