Skip to main content
SpringerLink
Log in

Analysis of periodic linear systems over finite fields with and without Floquet Transform

  • Original Article
  • Published:
Mathematics of Control, Signals, and Systems Aims and scope Submit manuscript

Abstract

This paper develops the analysis of discrete-time periodically time-varying linear systems over finite fields. It is shown that the conditions for the existence of Floquet Transform for periodic linear systems over reals (or complex) do not carry forward for this case over finite fields. The existence of Floquet Transform is shown to be equivalent to the existence of an Nth root of the monodromy matrix for the class of non-singular periodic linear systems. As the verification of existence and computation of the Nth root is a computationally hard problem, an independent analysis of the solutions of such systems is carried without the use of Floquet Transform. It is proved that all initial conditions of such systems lie either in a periodic orbit or a chain leading to a periodic orbit. A subspace of the state space is also identified, containing all initial conditions lying on a periodic orbit. Further, whenever the Floquet Transform exists, more concrete results on the orbit lengths are established depending on the coprimeness of the orbit length with the system period.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The total number of points in the state space is \(q^n\).

  2. Any choice of invertible P(0) other than I, will result in another Floquet Transform where the LDSFF obtained would be similar to the LDSFF obtained by considering \(P(0) = I\).

  3. This means that if x(0) is on a orbit (or chain) of length L under (5), then \(\hat{x}(0)\) is on a orbit (or chain) of length L under (20)

References

  1. Albert R, Thakar J (2014) Boolean modeling: a logic-based dynamic approach for understanding signaling and regulatory networks and for making useful predictions. WIREs Syst Biol Med Wiley Period 6:353–369

    Article  Google Scholar 

  2. Anantharaman R, Sule V (2019) Analysis of periodic feedback shift registers. arXiv:1904.11794

  3. Arora S, Barak B (2009) Computational complexity: a modern approach. Cambridge University Press, Cambridge

    Book  Google Scholar 

  4. Bittanti S, Colaneri P (1996) Analysis of discrete-time linear periodic systems. Control Dyn Syst 78:313–339

    Article  Google Scholar 

  5. Bittanti S, Colaneri P (2000) Invariant representations of discrete time periodic systems. Automatica 36:1777–1793

    Article  MathSciNet  Google Scholar 

  6. Cook S (1971) The complexity of theorem proving procedures. In: Proceedings of the third annual ACM symposium on theory of computing, pp 151–158

  7. Cuk S (2016) Power electronics, vol 4. CreateSpace Independent Publishing Platform, Scotts Valley

    Google Scholar 

  8. Dooren PV, Sreedhar J (1994) When is a periodic discrete-time system equivalent to a time-invariant one? Linear Algebra Appl 212–213:131–151

    Article  MathSciNet  Google Scholar 

  9. Dubrova E (2012) A method for generating full cycles by a composition of NLFSRs. Des Codes Cryptogr 492:1–16

    Google Scholar 

  10. Felstrom AJ, Zigangirov KS (1999) Time-varying periodic convolutional codes with low-density parity-check matrix. IEEE Trans Inf Theory 45:2181–2191

    Article  MathSciNet  Google Scholar 

  11. Flamm DS (1991) A new shift-invariant representation for periodic linear systems. Syst Control Lett 17:9–14

    Article  MathSciNet  Google Scholar 

  12. Ghorpade SR (2018) A note on nullstellensatz over finite fields, pp 1–9. arxiv:1806.09489

  13. Gill AR (1962) Introduction to the theory of finite state machines. McGraw-Hill, New York

    MATH  Google Scholar 

  14. Gill AR (1966) Linear sequential circuits: analysis, synthesis and applications. McGraw-Hill, New York

    MATH  Google Scholar 

  15. Golomb SW (1982) Shift register sequences. Aegean Park Press, Laguna Hills

    MATH  Google Scholar 

  16. Golomb SW, Gong G (2005) Signal design for good correlation: for wireless communication, cryptography, and radar. Cambridge University Press, Cambridge

    Book  Google Scholar 

  17. Goresky M, Klapper A (2012) Algebraic shift register sequences. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  18. Hayakawa Y, Jimbo T (2008) Floquet transformations for discrete-time systems: equivalence between periodic systems and time-invariant ones. In: IEEE conference on decision and control

  19. Hilbert D (1978) Hilbert’s invariant theory papers, Translated from the German by Michael Ackerman, With comments by Robert Hermann, Lie Groups: History. Frontiers and Applications. Math Sci Press, Brookline, Massachusetts

  20. Klein A (2013) Stream ciphers. Springer, New York

    Book  Google Scholar 

  21. Mullen GL, Panario D (2013) Handbook of finite fields. CRC Press, Taylor and Francis Group, Boca Raton

    Book  Google Scholar 

  22. Napp D, Pereira R, Pinto R, Rocha P (2019) Periodic state-space representations of periodic convolutional codes. Cryptogr Commun 11:585–595

    Article  MathSciNet  Google Scholar 

  23. Otero DE (1990) Extraction of \(m^{th}\) roots in matrix rings over fields. Linear Algebra Appl 128:1–26

    Article  MathSciNet  Google Scholar 

  24. Rueppel RA (1986) Analysis and design of stream ciphers. Springer, Berlin

    Book  Google Scholar 

  25. von zur Gathen J, Gerhard J (2013) Modern computer algebra. Cambridge University Press, Cambridge

    Book  Google Scholar 

  26. Yakubovich VA, Starzhinskii V (1975) Linear differential equations with periodic coefficients, vol 1. Wiley, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai, India

    Ramachandran Anantharaman & Virendra Sule

Authors
  1. Ramachandran Anantharaman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Virendra Sule
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Virendra Sule.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Anantharaman, R., Sule, V. Analysis of periodic linear systems over finite fields with and without Floquet Transform. Math. Control Signals Syst. 34, 67–93 (2022). https://doi.org/10.1007/s00498-021-00304-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00498-021-00304-z

Keywords

Search

Navigation