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

Abstract

AbstractA 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.