Stability Testing of 2D Filters Based on Tschebyscheff Polynomials and Generalized Eigenvalues

For the stability of 2-dimensional (2D) filters it is required that the denominator polynomial of the transfer function (or the characteristic polynomial of the state-space model) have no zeros in the closed unit bidisc. A new algebraic test is presented to test this stability condition using 3 simple 1D conditions and a generalized eigenvalue problem. The approach is based on using the conjugate of the original polynomial together with Tschebyscheff polynomials to transform the problem of stability analysis of a polynomial with coefficients depending on a complex variable to a problem with a polynomial with coefficients depending on a real variable. It is shown that the latter condition can be tested using a generalized eigenvalue problem. The method is illustrated with an example.