Circuits, Systems, and Signal Processing

, Volume 30, Issue 1, pp 185–208 | Cite as

Chaos Communication Performance: Theory and Computation

  • G. Kaddoum
  • Anthony J. Lawrance
  • P. Chargé
  • D. Roviras


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.


Bit energy Bit error Chaos shift keying communication Exact and Gaussian approximations Multi-user 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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) CrossRefGoogle Scholar
  2. 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) CrossRefGoogle Scholar
  3. 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) CrossRefGoogle Scholar
  4. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. Cheng, N.C. Beaulieu, Accurate DS-CDMA bit-error probability calculation in Raleigh fading. IEEE Trans. Wirel. Commun. 1, 3–15 (2002) CrossRefGoogle Scholar
  6. 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) CrossRefGoogle Scholar
  7. 7.
    R. Eposito, Error probabilities for the Nakagami channel. Proc. IEEE Trans. Inf. Theory 13, 145–148 (1967) CrossRefGoogle Scholar
  8. 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. 9.
    M.P. Kennedy, R. Rovatti, G. Setti, Chaotic Electronics in Telecommunications (CRC Press, London, 2000) Google Scholar
  10. 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. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 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) Google Scholar
  13. 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. 14.
    L.E. Larson, J.-M. Liu, L.S. Tsimring, Digital Communications Using Chaos and Nonlinear Dynamics (Springer, New York, 2006) CrossRefGoogle Scholar
  15. 15.
    F.C.M. Lau, C.K. Tse, Chaos-based Digital Communications Systems (Springer, Heidelberg, 2003) Google Scholar
  16. 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) zbMATHCrossRefMathSciNetGoogle Scholar
  17. 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) CrossRefMathSciNetGoogle Scholar
  18. 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. 19.
    A.J. Lawrance, J. Yao, Likelihood-based demodulation in multi-user chaos shift keying communication. Circuits Syst. Signal Process. 27, 847–864 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 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) CrossRefGoogle Scholar
  21. 21.
    J.G. Proakis, Digital Communication (McGraw–Hill, Boston, 2001) Google Scholar
  22. 22.
    T. Schimming, M. Hasler, Optimal detection of differential chaos shift keying. IEEE Trans. Circuits Syst. I 47, 1712–1719 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    P. Stavroulakis, Chaos Applications in Telecommunications (CRC Press, New York, 2006) Google Scholar
  24. 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) CrossRefGoogle Scholar
  25. 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) CrossRefGoogle Scholar
  26. 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) CrossRefGoogle Scholar
  27. 27.
    W.M. Tam, F.C.M. Lau, C.K. Tse, Digital Communications with Chaos (Elsevier, Oxford, 2007) Google Scholar
  28. 28.
    A. Uchida, S. Yoshimori, M. Shinozuka, T. Ogawa, F. Kannari, Chaotic on-off keying for secure communications. Opt. Lett. 26, 866–868 (2001) CrossRefGoogle Scholar
  29. 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. 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

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • G. Kaddoum
    • 1
  • Anthony J. Lawrance
    • 2
  • P. Chargé
    • 3
  • D. Roviras
    • 4
  1. 1.LACIME Laboratory, École de Technologie SupérieureUniversité du Québec, 1100MontréalCanada
  2. 2.Department of StatisticsUniversity of WarwickCoventryUK
  3. 3.LATTIS/INSAUniversity of ToulouseToulouse cedex 7France
  4. 4.CNAM ParisLAETITIA LaboratoryParisFrance

Personalised recommendations