A new proof of the Jury test

The problem of determining the root distribution of a real polynomial with respect to the unit circle, in terms of the coefficients of the polynomial, was solved by Jury (1964). The calculations were presented in tabular form (Jury´s table) and were later simplified by Raible (1974). This result is now classical and is as important in the stability analysis of digital control systems as its continuous time counterpart, the Routh Hurwitz criterion. Most texts on digital control state the Jury test but avoid giving the proof. In this paper we give a simple, insightful and new proof of the Jury test. The proof is based on the behaviour of the root-loci of an associated family of polynomials that was introduced previously by the authors (1995). The proof reveals clearly the mechanism underlying the counting of the roots within and without the unit circle, and this is illustrated with an example