Realization of 2D (2,2)-periodic encoders by means of 2D periodic separable Roesser models

It is well known that convolutional codes are linear systems when they are defined over a finite field. A fundamental issue in the implementation of convolutional codes is to obtain a minimal state representation of the code. Compared with the literature on one-dimensional (1D) time-invariant convolutional codes, there exist relatively few results on the realization problem for time-varying 1D convolutional codes and even fewer if the convolutional codes are two-dimensional (2D). In this paper we consider 2D periodic convolutional codes and address the minimal state space realization problem for this class of codes. This is, in general, a highly nontrivial problem. Here, we focus on separable Roesser models and show that in this case it is possible to derive, under weak conditions, concrete formulas for obtaining a 2D Roesser state space representation. Moreover, we study minimality and present necessary conditions for these representations to be minimal. Our results immediately lead to constructive algorithms to build these representations.