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Connections between linear systems and convolutional codes


Rosenthal, J (2001). Connections between linear systems and convolutional codes. In: Marcus, B; Rosenthal, J. Codes, systems, and graphical models (Minneapolis, MN 1999). New York: Springer, 39-66.

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.

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.

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Additional indexing

Item Type:Book Section, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2001
Deposited On:11 Mar 2010 14:13
Last Modified:19 Feb 2018 22:02
Publisher:Springer
Series Name:The IMA Volumes in Mathematics and its Applications
Number:123
ISBN:0-387-95173-3
OA Status:Green
Related URLs:http://www.springer.com/mathematics/analysis/book/978-0-387-95173-7 (Publisher)
http://www.ams.org/mathscinet-getitem?mr=1861952

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