<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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.</p></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.

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

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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...

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

Published Online: Sep 28, 2019

Page range: 527 - 539

Received: Nov 29, 2018

Accepted: Mar 31, 2019

DOI: https://doi.org/10.2478/amcs-2019-0039

Keywords
periodic 2D systems, convolutional codes, realizations

© 2019 Diego Napp et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Published Online: Sep 28, 2019

Page range: 527 - 539

Received: Nov 29, 2018

Accepted: Mar 31, 2019

DOI: https://doi.org/10.2478/amcs-2019-0039

Keywordsperiodic 2D systems, convolutional codes, realizations

© 2019 Diego Napp et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
periodic 2D systems, convolutional codes, realizations