Skip to main content
Springer Nature Link
Log in

Chaos Communication Performance: Theory and Computation

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

In this paper new and existing approaches are developed to compute the bit-error rate for chaos-based communication systems. The multi-user coherent antipodal chaos shift keying system is studied and evaluated in its coherent form, in the sense of perfect synchronisation between transmitted and received chaotic sequences. Transmission is through an additive white Gaussian noise channel. Four methods are interrelated in the paper, three approximate ones and an exact one. The least accurate but most well known is based on simple Gaussian approximation; this is generalised to better reveal its structure. Two accurate and computationally efficient approximate methods are based on conditional Gaussian approximation and the statistical distribution of the typically non-constant bit energy. The most insightful but computationally expensive one is based on exact theory and rests on explicit mathematical results for particular chaotic maps used to spread binary messages. Both upper and lower bounds to the bit-error rate are suggested. The relative advantages of the different approaches are illustrated with plots of bit-error rate against signal to noise ratio.

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
$34.99 /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 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

References

  1. A. Abdi, C. Tepedelenlioglu, M. Kaveh, G. Giannakis, On estimation of K parameter for Rice fading distribution. IEEE Commun. Lett. 5, 92–94 (2001)

    Article  Google Scholar 

  2. A.D. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fisher, J. Garcia-Ojlavo, C.R. Mirasso, L. Pesquera, A. Shore, Chaos-based communications at high bit rates using commercial fibre-optic links. Nature 437/17, 343–346 (2005)

    Article  Google Scholar 

  3. P. Chargé, D. Fournier-Prunaret, V. Guglielmi, Features analysis of a parametric PWL chaotic map and its utilization for secure transmissions. Chaos Solitons Fractals 38, 1411–1422 (2008)

    Article  Google Scholar 

  4. C.C. Chen, K. Yao, K. Umeno, E. Biglieri, Design of spread-spectrum sequences using chaotic dynamical systems and ergodic theory. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48, 1110–1114 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Cheng, N.C. Beaulieu, Accurate DS-CDMA bit-error probability calculation in Raleigh fading. IEEE Trans. Wirel. Commun. 1, 3–15 (2002)

    Article  Google Scholar 

  6. H. Dedieu, M.P. Kennedy, M. Hasler, Chaos shift keying: modulation and demodulation of a chaotic carrier using self-synchronizing Chua’s circuits. IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. 40, 634–642 (1993)

    Article  Google Scholar 

  7. R. Eposito, Error probabilities for the Nakagami channel. Proc. IEEE Trans. Inf. Theory 13, 145–148 (1967)

    Article  Google Scholar 

  8. G. Kaddoum, P. Charge, D. Roviras, D. Fournier-Prunaret, Comparison of chaotic sequences in a chaos based DS-CDMA system, in International Symposium on Nonlinear Theory and its Applications (NOLTA), Vankuver, Canada (2007)

    Google Scholar 

  9. M.P. Kennedy, R. Rovatti, G. Setti, Chaotic Electronics in Telecommunications (CRC Press, London, 2000)

    Google Scholar 

  10. T. Kohda, A. Tsuneda, Pseudonoise sequences by chaotic nonlinear mapsand their correlation properties. IEICE Trans. Commun. E76-B, 855–862 (1993)

    Google Scholar 

  11. G.K. Kolumban, Theoretical noise performance of correlator-based chaotic communications schemes. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 1692–1701 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Kolumban, B. Vizvari, W. Schwarz, A. Abel. Differential chaos shift keying: a robust coding for chaotic communication. Presented at Nonlinear Dynamics of Electronic Systems, Seville, Spain (1996)

  13. G. Kolumban, G. Kis, Z. Jako, M.P. Kennedy, FM-DCSK: a robust modulation scheme for chaos communication. IEEE Trans. Fundam. Electron. Commun. Comput. Sci. 81, 1798–1802 (1998)

    Google Scholar 

  14. L.E. Larson, J.-M. Liu, L.S. Tsimring, Digital Communications Using Chaos and Nonlinear Dynamics (Springer, New York, 2006)

    Book  Google Scholar 

  15. F.C.M. Lau, C.K. Tse, Chaos-based Digital Communications Systems (Springer, Heidelberg, 2003)

    Google Scholar 

  16. A.J. Lawrance, N. Balakrishna, Statistical aspects of chaotic maps with negative dependence in a communications setting. J. R. Stat. Soc., Ser. B 63, 843–853 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. A.J. Lawrance, G. Ohama, Exact calculation of bit error rates in communication systems with chaotic modulation. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50, 1391–1400 (2003)

    Article  MathSciNet  Google Scholar 

  18. A.J. Lawrance, T. Papamarkou, Higher order dependency of chaotic maps, In Nonlinear Theory and Its Applications 2006 (NOLTA2006), Bologna, Italy (2006)

    Google Scholar 

  19. A.J. Lawrance, J. Yao, Likelihood-based demodulation in multi-user chaos shift keying communication. Circuits Syst. Signal Process. 27, 847–864 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. W.C. Lindsey, Error probabilities for Rician fading multichannel reception of binary and N-ary signals. IEEE Trans. Inf. Theory 10, 339–350 (1964)

    Article  Google Scholar 

  21. J.G. Proakis, Digital Communication (McGraw–Hill, Boston, 2001)

    Google Scholar 

  22. T. Schimming, M. Hasler, Optimal detection of differential chaos shift keying. IEEE Trans. Circuits Syst. I 47, 1712–1719 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. P. Stavroulakis, Chaos Applications in Telecommunications (CRC Press, New York, 2006)

    Google Scholar 

  24. M. Sushchik, L.S. Tsimring, A.R. Volkovskii, Performance analysis of correlation-based communication schemes utilizing chaos. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47, 1684–1691 (2000)

    Article  Google Scholar 

  25. W.M. Tam, F.C.M. Lau, C.K. Tse, M. Yip, An approach to calculating the bit-error rate of a coherent chaos-shift-keying communication system under a noisy multiuser environment. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49, 210–233 (2002)

    Article  Google Scholar 

  26. W.M. Tam, F.C.M. Lau, C.K. Tse, A.J. Lawrance, Exact analytical of bit error rates for multiple access chaos-based communication systems. IEEE Trans. Circuits Syst. II 51, 473–481 (2004)

    Article  Google Scholar 

  27. W.M. Tam, F.C.M. Lau, C.K. Tse, Digital Communications with Chaos (Elsevier, Oxford, 2007)

    Google Scholar 

  28. A. Uchida, S. Yoshimori, M. Shinozuka, T. Ogawa, F. Kannari, Chaotic on-off keying for secure communications. Opt. Lett. 26, 866–868 (2001)

    Article  Google Scholar 

  29. J. Yao, Optimal chaos shift keying communications with correlation decoding, in IEEE International Symposium on Circuits and Systems (ISCAS2004), vol. IV, Vankuver, Canada (2004), pp. 593–596

    Google Scholar 

  30. J. Yao, A.J. Lawrance, Bit error rate calculation for multi-user coherent chaos-shift-keying communication systems. IEICE Trans. Fundam. E87-A, 2280–2291 (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LACIME Laboratory, École de Technologie Supérieure, Université du Québec, 1100, rue Notre-Dame Ouest, Montréal, QC, H3C 1K3, Canada

    G. Kaddoum

  2. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    Anthony J. Lawrance

  3. LATTIS/INSA, University of Toulouse, 135 Avenue de Rangueil, 31071, Toulouse cedex 7, France

    P. Chargé

  4. CNAM Paris, LAETITIA Laboratory, 292 rue Saint-Martin, 75003, Paris, France

    D. Roviras

Authors
  1. G. Kaddoum
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Anthony J. Lawrance
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. P. Chargé
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. D. Roviras
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Anthony J. Lawrance.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kaddoum, G., Lawrance, A.J., Chargé, P. et al. Chaos Communication Performance: Theory and Computation. Circuits Syst Signal Process 30, 185–208 (2011). https://doi.org/10.1007/s00034-010-9217-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-010-9217-1

Keywords