On the Power Spectral Density of Chaotic Signals Generated by Skew Tent Maps

1. ISSCS’07 8-th International Symposium on Signals, Circuits and Systems July 12th, 2007 - Iasi, Romania On the Power Spectral Density of Chaotic Signals Generated by Skew Tent Maps Daniela Mitie Kato Marcio Eisencraft Escola de Engenharia – Universidade Presbiteriana Mackenzie São Paulo - Brazil

2. Topics • Chaotic signals • Skew tent family of maps • Power Spectral Density (PSD) of chaotic signals • Essential bandwidth x Lyapunov exponent • Conclusions

3. 1 Chaotic signals - Definitions • Chaotic signal: • Deterministic • Aperiodic • Sensitive dependence on initial conditions • One-dimensional discrete time dynamical systems • Lyapunov exponent h – divergence rate between orbits • h>0 implies chaos for aperiodic signals

4. 1 Chaotic signals - example

5. 1 Chaotic signals - Properties • Applications in Telecommunication Engineering • Broadband • Impulsive Autocorrelation Sequence (ACS) • Low values cross-correlation sequence • Interesting for Spread Spectrum systems • Problem: spectral characteristics – rarely studied deeply and fundamental for pratical implementation • Objective of this work: to answer questions like • Chaos  broadband? • How large is the bandwidth of a chaotic signal? • Can we control it?

6. Topics • Chaotic signals • Skew tent family of maps • Power Spectral Density (PSD) of chaotic signals • Essential bandwidth x Lyapunov exponent • Conclusions

7. 2. Skew tent maps • Definition: parameter: α • Lyapunov exponent

8. Topics • Chaotic signals • Skew tent family of maps • Power Spectral Density (PSD) of chaotic signals • Essential bandwidth x Lyapunov exponent • Conclusions

9. 3.1 PSD – Individual signals • Two forms of considering chaotic signals for PSD calculation • Deterministic individual signals • Sample functions of a stochastic process • Given a map and s0the generated signal is well defined • PSD is the DTFT of the Autocorrelation Sequence

10. 3.1 PSD – Individual signals • Same map with different s0 similar PSD • Interpretation of chaotic signal as sample function of ergodic stochastic process

11. 3.2 PSD – Stochastic process • We define the PSD as • Expectation taken over initial conditions that generate chaotic orbits • Advantage: higlight properties of the entire set of signals

12. 3.2 PSD – Stochastic process • The higher |α| the narrower the bandwidth of the generated chaotic signals • Signal of αdefines low-pass or high-pass • PSDs present symmetry around f = 0.5 for opposite α • Chaos is far way from being synonym for broadband non-correlated signals.

13. 3.2 PSD – Stochastic process • α > 0  ACS monotonically decreasing • α < 0  ACSoscilates • In this case, for any s0 the signals of s(n) and their nearest-neighbors are opposites

14. Topics • Chaotic signals • Skew tent family of maps • Power Spectral Density (PSD) of chaotic signals • Essential bandwidth x Lyapunov exponent • Conclusions

15. 4Essential bandwidth x Lyapunov Exponent • Essential bandwidth – frequency range where 95% of the signal power is concentrated • Normalized version - if B = 1 PSD spreads over all frequencies.

16. Topics • Chaotic signals • Skew tent family of maps • Power Spectral Density (PSD) of chaotic signals • Essential bandwidth x Lyapunov exponent • Conclusions

17. 5Conclusions • Chaotic signals are not necessarily broadband • Given a bandwidth B it is possible to find a simple piecewise linear generator map • One-to-one relationship between B and |α| chaotic digital modulation systems • Generalization of ours results to other maps seems to be possible using conjugacy

18. Thank you!Mulţumesc!Contact: marcioft@mackenzie.br