Stability and overflow oscillations in 2-D state-space digital filters

Abstract

An important theorem relating to the Lyapunov stability of two-dimensional discrete systems is proven. Using this theorem it is shown that for any 2-D digital filter satisfying Shanks' criterion there exists a realization that cannot support overflow oscillations. In the process of proving the theorem some interesting results on the multi-dimensional bilinear transformation are developed. One of these results yields a simple test that can be used to check the stability of a 2-D discrete transfer function that has been obtained from the bilinear transform of a 2-D continuous transfer function with a 2-D Hurwitzian denominator polynomial. A technique is given for determining whether a normal realization exists for a given 2-D discrete system. Also, a theorem is presented that allows the determination of the norm of the minimum norm realization of a given transfer function. A noniterative technique for obtaining a low norm realization and an iterative technique for obtaining a minimum norm realization are developed.