Decomposition of 2-D Separable-Denominator Systems: Existence, Uniqueness, and Applications

This paper proves that any single-input/single-output (SISO) 2-D system with a separable denominator can be decomposed into a pair of 1-D systems having dynamics in different directions and that the minimal decomposition pair is unique modulo an invertible constant transformation. One of the 1-D systems is a single-input/multi-output system and the other is a multi-input/single-output system. On the basis of the reduced-dimensional decomposition, which directly connects a 2-D separable-denominator system to two 1-D systems, the paper studies the state-space realizations of 2-D separable-denominator systems from 2-D input-output maps. It is shown that the state-space realization problems of 2-D separable-denominator systems can be reduced into corresponding 1-D realization problems. Therefore, any 1-D state-space realization technique can be directly applied to the 2-D case.