Digital Filters Design for Signal and Image Processing

11.3. Algorithms used in stability tests

In this section we introduce a rank of stability tests for 2-D recursive filters that may be causal or semi-causal. We start with the Jury Table, which is the base for many tests discussed in the literature. We will also see other tests based on the rapid calculation of the Bezout resultant of a polynomial P(z) and its reciprocal polynomial P*(z). We close the section with a comparative discussion of rounding-off errors found in many algorithms implemented with floating point arithmetical operation.

11.3.1. The Jury Table

Here we introduce the Jury Table for a polynomial with one variable:

We can demonstrate [BEN 99] that the numbers p and q of its roots situated respectively in the open unit disk U and outside the closed unit disk are given by the signature of the Schur-Cohn hermitian quadratic form. The pioneering work of Hermite [HER 1856], Schur and Cohn [COH 22] produced this very useful result in the early 20^th century. Marden [MAR 66] later developed the algorithm.

There are different ways of presenting the results of Schur, Cohn and Marden in the form of tables. These presentations are the work of Jury [JUR 62]. These tables are variously called the Jury Table, the Marden-Jury Table or the Schur-Cohn Table. With certain singular polynomials P(z), it can happen that the construction rule does not apply; fortunately, these polynomials ...