Connections between Linear Systems and Convolutional Codes
<|reference_start|>Connections between Linear Systems and Convolutional Codes: 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.<|reference_end|>
@article{rosenthal2000connections,
    title={Connections between Linear Systems and Convolutional Codes},
    author={Joachim Rosenthal},
    journal={arXiv preprint arXiv:math/0005281},
    year={2000},
    archivePrefix={arXiv},
    eprint={math/0005281},
    primaryClass={math.OC cs.IT math.IT}
}