A test of the stability of the two-dimensional recursive digital filter

Abstract

A test for the stability of a two-dimensional recursive digital filter is proposed. Bilinear transformation reduces the stability test outside the unit circle in the complex plane to a test in the right half plane (i.e., to a test of a two-variable Hurwitz polynomial for a continuous system). Conditions for a two-variable polynomial to be a Hurwitz polynomial are given. The necessary and sufficient conditions for a two-variable rational function to be a strict-sense positive real function are given in a theorem. It is shown that a strict-sense two-variable positive real function can be found for each two-variable Hurwitz polynomial. From the theorem it follows that the test of a two-variable Hurwitz polynomial reduces to a test of a Hurwitz polynomial of a lower order with respect to one variable and to a sign test of a real two-variable polynomial. The test of the two-variable Hurwitz polynomial reduces to tests for signs of a set of real two-variable polynomials. This is an extension of Sturm's test for the roots of a one-variable algebraic equation.