Connections between Linear Systems and Convolutional Codes
Abstract
The article reviews different definitions for a convolutional code which can be found in the literature. The algebraic differences between the definitions are worked out in detail. It is shown that bi-infinite support systems are dual to finite-support systems under Pontryagin duality. In this duality the dual of a controllable system is observable and vice versa. Uncontrollability can occur only if there are bi-infinite support trajectories in the behavior, so finite and half-infinite-support systems must be controllable. Unobservability can occur only if there are finite support trajectories in the behavior, so bi-infinite and half-infinite-support systems must be observable. It is shown that the different definitions for convolutional codes are equivalent if one restricts attention to controllable and observable codes.
Cite
@article{arxiv.math/0005281,
title = {Connections between Linear Systems and Convolutional Codes},
author = {Joachim Rosenthal},
journal= {arXiv preprint arXiv:math/0005281},
year = {2007}
}
Comments
28 pages, to appear in IMA volume on Codes, Systems and Graphical Models