Realization of 2D (2,2)–Periodic Encoders by Means of 2D Periodic Separable Roesser Models

Abstract

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.

On the State Approach Representations of Convolutional Codes over Rings of Modular Integers

“…In the case of convolutional codes over finite rings, the minimality of the I/S/O representation is achieved by the reachability of the system. For instance, the necessary and sufficient conditions for the minimality of the state-space model for basic 2D convolutional codes over finite fields were found by a property of strongly modally reachability in [37], or by separable Roesser models for 2D periodic convolutional codes in [38].…”