Connections between Linear Systems and Convolutional Codes

TL;DR

This paper explores the algebraic and duality relationships between different definitions of convolutional codes, establishing conditions for controllability and observability, and demonstrating their equivalence under certain restrictions.

Contribution

It clarifies the algebraic differences between convolutional code definitions and shows their equivalence when focusing on controllable and observable codes.

Findings

Bi-infinite support systems are dual to finite-support systems under Pontryagin duality.

Controllability and observability are characterized by the presence of finite or bi-infinite support trajectories.

Different definitions of convolutional codes are equivalent for controllable and observable 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.