[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Periodic state-space representations of periodic convolutional codes

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aleixo, J.C., Rocha, P.: Roesser Model Representation of 2D Periodic Behaviors: the (2,2)-Periodic Siso Case. In: 2017 10th International Workshop on Multidimensional (Nd) Systems (NDS), pp. 1–6. https://doi.org/10.1109/NDS.2017.8070617 (2017)

  2. Aleixo, J.C., Rocha, P., Willems, J.C.: State space representation of siso periodic behaviors. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 1545–1550. https://doi.org/10.1109/CDC.2011.6160552 (2011)

  3. Bocharova, I.E., Kudryashov, B.D.: Rational rate punctured convolutional codes for soft-decision viterbi decoding. IEEE Trans. Inf. Theory 43(4), 1305–1313 (1997). https://doi.org/10.1109/18.605600

    Article  MathSciNet  MATH  Google Scholar 

  4. Costello, D.J.: Free distance bounds for convolutional codes. IEEE Trans. Inf. Theory 20(3), 356–365 (1974). https://doi.org/10.1109/TIT.1974.1055223

    Article  MathSciNet  MATH  Google Scholar 

  5. Fekri, F., Sartipi, M., Mersereau, R.M., Schafer, R.W.: Convolutional codes using finite-field wavelets: time-varying codes and more. IEEE Trans. Signal Process. 53(5), 1881–1896 (2005). https://doi.org/10.1109/TSP.2005.845484

    Article  MathSciNet  MATH  Google Scholar 

  6. Fornasini, E., Pinto, R.: Matrix fraction descriptions in convolutional coding. Linear Algebra Appl. 392(Supplement C), 119–158 (2004). https://doi.org/10.1016/j.laa.2004.06.007. http://www.sciencedirect.com/science/article/pii/S0024379504002836

    Article  MathSciNet  MATH  Google Scholar 

  7. Gluesing-Luerssen, H., Schneider, G.: State space realizations and monomial equivalence for convolutional codes. Linear Algebra Appl. 425(2), 518–533 (2007). https://doi.org/10.1016/j.laa.2007.03.004. http://www.sciencedirect.com/science/article/pii/S002437950700122X. Special Issue in honor of Paul Fuhrmann

    Article  MathSciNet  MATH  Google Scholar 

  8. Johannesson, R., Zigangirov, K.S.: Fundamentals of Convolutional Coding. IEEE Press, New York (1999)

    Book  MATH  Google Scholar 

  9. Kailath, T.: Linear Systems. Prentice Hall Information and System Sciences Series. Prentice-Hall, Englewood Cliffs (1980)

    Google Scholar 

  10. Kuijper, M., Polderman, J.: Reed-Solomon list decoding from a system theoretic perspective. IEEE Trans. Inf. Th. IT-50, 259–271 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kuijper, M., Willems, J.C.: A behavioral framework for periodically time-varying systems. In: Proceedings of the 36th IEEE Conference on Decision and Control, vol. 3, pp. 2013–2016. https://doi.org/10.1109/CDC.1997.657060 (1997)

  12. Lee, P.J.: There are many good periodically time-varying convolutional codes. IEEE Trans. Inf. Theory 35(2), 460–463 (1989). https://doi.org/10.1109/18.32142

    Article  MathSciNet  Google Scholar 

  13. Mooser, M.: Some periodic convolutional codes better than any fixed code (corresp.) IEEE Trans. Inf. Theory 29(5), 750–751 (1983). https://doi.org/10.1109/TIT.1983.1056727

    Article  MATH  Google Scholar 

  14. Napp, D., Perea, C., Pinto, R.: Input-state-output representations and constructions of finite support 2D convolutional codes. Adv. Math. Commun. 4(4), 533–545 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Palazzo, R.: A time-varying convolutional encoder better than the best time-invariant encoder. IEEE Trans. Inf. Theory 39(3), 1109–1110 (1993). https://doi.org/10.1109/18.256526

    Article  MathSciNet  MATH  Google Scholar 

  16. Rosenthal, J.: Connections between linear systems and convolutional codes. In: Marcus, B., Rosenthal, J. (eds.) Codes, Systems and Graphical Models, IMA, vol. 123, pp. 39–66. Springer (2001)

  17. Rosenthal, J., York, E.V.: BCH convolutional codes. IEEE Trans. Automat. Control 45(6), 1833–1844 (1999)

    MathSciNet  MATH  Google Scholar 

  18. Truhachev, D., Zigangirov, K.S., Costello, D.J.: Distance bounds for periodically time-varying and tail-biting ldpc convolutional codes. IEEE Trans. Inf. Theory 56(9), 4301–4308 (2010). https://doi.org/10.1109/TIT.2010.2053873

    Article  MathSciNet  MATH  Google Scholar 

  19. Viterbi, A.: Convolutional codes and their performance in communication systems. IEEE Trans. Commun. Technol. 19(5), 751–772 (1971). https://doi.org/10.1109/TCOM.1971.1090700

    Article  MathSciNet  Google Scholar 

Download references

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.

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ricardo Pereira
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Raquel Pinto
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paula Rocha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricardo Pereira.

Additional information

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-018-0313-6

Keywords

Mathematics Subject Classification (2010)

Search

Navigation