The dynamics of group codes: Dual abelian group codes and systems

Fundamental results concerning the dynamics of abelian group codes (behaviors) and their duals are developed. Duals of sequence spaces over locally compact abelian (LCA) groups may be defined via Pontryagin duality; dual group codes are orthogonal subgroups of dual sequence spaces. The dual of a com...

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Bibliographic Details
Published in:IEEE transactions on information theory 2004-12, Vol.50 (12), p.2935-2965
Main Authors: Forney, G.D., Trott, M.D.
Format: Article
Language:English
Subjects:
Applied sciences
Behavioral systems
Codes
Coding, codes
Computer science
control theory
systems
Conferences
Control theory. Systems
Controllability
duality
Exact sciences and technology
group codes
group systems
Information theory
Information, signal and communications theory
Laboratories
Legged locomotion
Linear code
Linear systems
Observability
Signal and communications theory
Space technology
State-space methods
System theory
Telecommunications and information theory
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Summary:Fundamental results concerning the dynamics of abelian group codes (behaviors) and their duals are developed. Duals of sequence spaces over locally compact abelian (LCA) groups may be defined via Pontryagin duality; dual group codes are orthogonal subgroups of dual sequence spaces. The dual of a complete code or system is finite, and the dual of a Laurent code or system is (anti-)Laurent. If C and C/sup /spl perp// are dual codes, then the state spaces of C act as the character groups of the state spaces of C/sup /spl perp//. The controllability properties of C are the observability properties of C/sup /spl perp//. In particular, C is (strongly) controllable if and only if C/sup /spl perp// is (strongly) observable, and the controller memory of C is the observer memory of C/sup /spl perp//. The controller granules of C act as the character groups of the observer granules of C/sup /spl perp//. Examples of minimal observer-form encoder and syndrome-former constructions are given. Finally, every observer granule of C is an "end-around" controller granule of C.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2004.838340