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Periodic state-space representations of periodic convolutional codes

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Abstract

In this paper we study the representation of periodically time-varying convolutional codes by means of periodic input-state-output models. In particular, we focus on period two and investigate under which conditions a given two-periodic convolutional code (obtained by alternating two time-invariant encoders) can be represented by a periodic input-state-output system. We first show that one cannot expect, in general, to obtain a periodic input-state-output representation of a periodic convolutional code by means of the individual realizations of each of the associated time-invariant codes. We, however, provide sufficient conditions for this to hold in terms of the column degrees of the associated column reduced generator matrices. Moreover, we derive a sufficient condition to obtain a periodic state-space realization that is minimal. Finally, examples to illustrate the results are presented.

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Acknowledgments

This work was supported in part by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciêcia e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013 and also by Project POCI-01-0145-FEDER-006933 - SYSTEC - Research Center for Systems and Technologies - funded by FEDER funds through COMPETE2020 - Programa Operacional Competitividade e Internacionalização (POCI) - and by national funds through FCT - Fundação para a Ciência e a Tecnologia.

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Authors and Affiliations

  1. CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Diego Napp, Ricardo Pereira & Raquel Pinto

  2. SYSTEC, Faculty of Engineering, University of Porto, Porto, Portugal

    Paula Rocha

Authors
  1. Diego Napp
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  2. Ricardo Pereira
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  3. Raquel Pinto
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  4. Paula Rocha
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Corresponding author

Correspondence to Ricardo Pereira.

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This article is part of the Topical Collection on Special Issue on Coding Theory and Applications

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Napp, D., Pereira, R., Pinto, R. et al. Periodic state-space representations of periodic convolutional codes. Cryptogr. Commun. 11, 585–595 (2019). https://doi.org/10.1007/s12095-018-0313-6

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