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.

Parallel concatenated convolutional codes from linear systems theory viewpoint

“…Following [31,32], we define a convolutional code as a submodule C of F n [z]. Since F[z] is a principal ideal domain, and C is a submodule of the free submodule F n [z], the convolutional code C is free and it has a well defined rank k (with k ≤ n).…”

Feedback equivalence of convolutional codes over finite rings

“…Two linear systems are feedback equivalent if one can transform one into the other by changes of coordinates together with feedback loops. Regarding equivalence relations in coding literature, due to the fact that a convolutional code can have many encoders, most works on the topic are focused on the study of equivalence relations between encoders of the same convolutional code: that is, the study of the conditions in which two encoding matrices are equivalent, and thus they generate the same code (see [4] for a general overview). In the theory of block codes over nite elds, MacWilliams stated that linear block codes are related by weight-preserving isomorphism (see [27]).…”

“…The connection between linear systems and convolutional codes over nite elds has been studied from di erent points of view depending on the approach to convolutional codes that is being used (see [4]). The available representations let us describe the dynamics of the encoders of the codes or constructing convolutional codes with certain good properties such as observability.…”

A State Space Approach to Periodic Convolutional Codes

“…Convolutional codes [10] are an important type of error correcting codes that can be represented as a time-invariant discrete linear system over a finite field [20]. They are used to achieve reliable data transfer, for instance, in mobile communications, digital video and satellite communications [10,23].…”

“…Since the sixties it has been widely known that convolutional codes and linear systems defined over a finite field are essentially the same objects [20]. More recently, there has been a new and increased interest in this connection and many advances have been derived from using the system theoretical framework when dealing with convolutional codes, see [12,17].…”