Connections between Linear Systems and Convolutional Codes
نویسنده
چکیده
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.
منابع مشابه
Linear representations of convolutional codes over rings
In this paper we extend the relation between convolutional codes and linear systems over finite fields to certain commutative rings through first order representations . We introduce the definition of rings with representations as those for which these representations always exist, and we show that finite products of finite fields belong to this class. We develop the input/state/output represen...
متن کاملDecoding Algorithm for Convolutional Codes under Linear Systems Point of View
In this work the output-observability of convolutional codes is examined and a decoding algorithm for convolutional codes using linear algebra techniques is presented. Key–Words: Codes, linear systems, output-observability, generalized inverses.
متن کاملOn Behaviors and Convolutionail Codes
It is well known that a convolutional code is essentially a linear system defined over a finite field. In this paper we elaborate on this connection. We define a convolutional code as the dual of a complete linear behavior in the sense of Willems (1979). Using ideas from systems theory, we describe a set of generalized first-order descriptions for convolutional codes. As an application of these...
متن کاملMDP Periodically Time-Varying Convolutional Codes
In this paper we use some classical ideas from linear systems theory to analyse convolutional codes. In particular, we exploit input-state-output representations of periodic linear systems to study periodically time-varying convolutional codes. In this preliminary work we focus on the column distance of these codes and derive explicit necessary and sufficient conditions for an (n, 2, 1) periodi...
متن کامل