PapersNo Access

Kendall’s Tau Based Correlation Analysis of Chaotic Sequences Generated by Piecewise Linear Maps

Zouhair Ben Jemaa

Ecole Nationale d’Ingénieurs de Tunis, Laboratoire SysCom, Tunis, Tunisia

E-mail Address: zouhair.benjemaa@enit.rnu.tn

Search for more papers by this author

Daniele Fournier-Prunaret

LAAS-CNRS, INSA, Université de Toulouse, Toulouse, France

E-mail Address: daniele.fournier@insa-toulouse.fr

Search for more papers by this author

, and

Safya Belghith

Ecole Nationale d’Ingénieurs de Tunis, Laboratoire SysCom, Tunis, Tunisia

E-mail Address: safya.belghith@enit.rnu.tn

Search for more papers by this author

https://doi.org/10.1142/S0218127415501771Cited by:2 (Source: Crossref)

Abstract

In many applications, sequences generated by chaotic maps have been considered as pseudo-random sequences. This paper deals with the correlation between chaotic sequences generated by a given piecewise linear map; we have based the measure of the correlation on the statistics of the Kendall tau, which is usually used in the field of statistics. We considered three piecewise linear maps to generate chaotic sequences and computed the statistics of the Kendall tau of couples of sequences obtained from randomly chosen couples of initial conditions. We essentially found that the results depend on the considered chaotic map and that it is possible to approach the uncorrelated case.

Keywords:

References

1. Ben Jemaa, Z., Marcos, S. & Belghith, S. [2006] “ Performances of MMSE receiver when using super stable orbits based spreading sequences in DS-CDMA system,” Proc. European Signal Processing Conf. (EUSIPCO2006), Florence, Italy, 4–8 September 2006, pp. 1–4. Google Scholar
2. Ben Jemaa, Z., Fournier-Prunaret, D. & Belghith, S. [2011] “Independence test of chaotic sequences,” Nonlinear Maps and Their Applications (NOMA2011), Évora, Portugal, 15–16 September 2011; [2014] Springer Proceedings in Mathematics and Statistics, Vol. 57, pp. 95–108. Google Scholar
3. Ben Jemaa, Z., Fournier-Prunaret, D. & Belghith, S. [2013] “ Pearson correlation coefficient based independence analysis of chaotic sequences,” Nonlinear Maps and Their Applications (NOMA2013), Zaragoza, Spain, 3–4 September 2013. Google Scholar
4. Broomhead, D. S., Huke, J. P. & Muldoon, M. R. [1999] “ Codes for spread spectrum, applications generated using chaotic dynamical systems,” Dyn. Stab. Syst. 14, 95–105. Crossref, Google Scholar
5. Caroll, T. L. [2000] “ Spread spectrum sequences for unstable periodic orbits,” IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 47, 443–447. Crossref, Web of Science, Google Scholar
6. Chargé, P., Fournier-Prunaret, D. & Guglielmi, D. [2008] “ Features analysis of a parametric PWL chaotic map and its utilization for secure transmissions,” Chaos Solit. Fract. 38, 1411–1422. Crossref, Web of Science, Google Scholar
7. Fournier-Prunaret, D. & Taha, A. [2012] “ Chaotic sequences as pseudo-random sequences obtained from 2-dimensional maps,” Int. J. Appl. Math. Stat. Special Issue (SI) Statistical Chaos and Complexity 26, 92–105. Google Scholar
8. Furguson, T. S., Genest, C. & Hallin, M. [2000] “ Kendall’s tau for serial dependence,” Canadian J. Stat. 28, 587–604. Crossref, Web of Science, Google Scholar
9. Götz, M. & Schwarz, W. [1997] “ Spectral decomposition of the Frobenius–Perron operator and higher-order correlation functions of one-dimensional chaotic systems with fully stretching maps,” Proc. European Conf. Circuit Theory Design (ECCTD1997), Budapest, Hungary, 31 August–3 September 1997, pp. 1253–1258. Google Scholar
10. Hilliam, R. & Lawrance, A. [2001] “ A statistical perspective on the dynamics of bivariate chaotic maps for communications modelling,” Int. Symp. Nonlinear Theory and Its Applications (NOLTA2001), Miyagi, Japan, 28 October–1 November 2001. Google Scholar
11. Hilliam, M. R. & Lawrance, A. J. [2004] “ The dynamics and statistics of bivariate chaotic maps in communication modeling,” Int. J. Bifurcation and Chaos 14, 1177–1194. Link, Web of Science, Google Scholar
12. Kaddoum, G. [2008] “Contributions à l’amélioration des systèmes de communication multi-utilisateurs par chaos: Synchronisation et analyse des performances,” PhD thesis, INSA Toulouse. Google Scholar
13. Kohda, T. & Tsuneda, A. [1994] “ Explicit evaluations of correlation functions of Chebyshev binary and bit sequences based on Perron–Frobenius operator,” IEICE Trans. Fund. Electron. Commun. Comput. Sci. E77A, 1794–1800. Web of Science, Google Scholar
14. Kohda, T. & Fujisaki, H. [2000] “ The Jacobian elliptic Chebyshev rational maps can generate sequences of i.i.d. random variables,” Int. Symp. Circuits and Systems (ISCAS2000), Geneva, Switzerland, 28–31 May (IEEE), pp. 221–224. Google Scholar
15. Kohda, T., Fujisaki, H. & Ideue, S. [2000] “ On distributions of correlation values of spreading sequences based on Markov information sources,” Int. Symp. Circuits and Systems (ISCAS2000), Geneva, Switzerland, 28–31 May (IEEE), pp. 225–228. Google Scholar
16. Lawrance, A. J. & Balakrishna, N. [2008] “ Statistical dependency of chaos,” Int. J. Bifurcation and Chaos 18, 3207–3219. Link, Web of Science, Google Scholar
17. Lee, P. H., Chen, Y., Pei, S. C. & Chen, Y. Y. [2004] “ Evidence of the correlation between positive Lyapunov exponents and good chaotic random number sequences,” Comput. Phys. Commun. 160, 187–203. Crossref, Web of Science, Google Scholar
18. Lozi, R. [2012] “ Emergence of randomness from chaos,” Int. J. Bifurcation and Chaos 22, 1250021. Link, Web of Science, Google Scholar
19. Mazzini, G., Setti, G. & Rovatti, R. [1997] “ Chaotic complex spreading sequences for asynchronous DS-CDMA-Part I: System modeling and results,” IEEE Trans. Circuit Syst. 44, 937–947. Crossref, Web of Science, Google Scholar
20. Pareschi, F. [2006] “Chaos based random number generators: Monolithic implementation, testing and applications,” PhD thesis, Università di Bologna. Google Scholar
21. Rovatti, R., Setti, G. & Mazzini, G. [1998] “ Chaotic complex spreading sequences for asynchronous DS-CDMA-Part II: Some theoretical performance bounds,” IEEE Trans. Circuit Syst. 45, 496–506. Crossref, Web of Science, Google Scholar
22. Zheng, F., Tian, X., Song, L. & Li, X. [2008] “ Pseudo-random sequences generator based on the generalized henon map,” J. China Univ. Posts and Telecomm. 15, 64–68. Crossref, Google Scholar