M.Sc Thesis

M.Sc StudentChen Ilan
SubjectThe Thershold Effect in the Estimation of Chaotic
DepartmentDepartment of Electrical and Computer Engineering
Supervisor PROF. Neri Merhav


Chaotic sequences and chaotic dynamical systems are attractive candidates for use in signal synthesis and analysis as well as in communications applications. Various methods for the estimation of chaotic sequences under noise were developed. However, although the methods were different, their qualitive performance was the same: for high SNR the performance was good, but below some threshold SNR, a sharp degradation in performance occurred.

We derive the maximum likelihood estimator of chaotic sequences generated by the r-diadic map in the presence of additive white Gaussian noise. We perform Monte-Carlo simulations, showing that the Bayesian Cramer-Rao bound is achieved beyond some threshold SNR.

Using information-theoretic tools, we prove that for any ergodic chaotic system there is a certain threshold SNR level, below which the ratio between the mean square error obtained by any estimator of the system's initial state and the Bayesian Cramer-Rao bound increases exponentially fast as the number of observations, N, grows without bound. We derive lower bounds on SNRth, the value of the threshold SNR, as function of the system's Lyapunov exponent. As a direct consequence, any estimator of the initial state of a chaotic system is necessarily asymptotically inefficient for SNR's below SNRth. Our bounds have two versions, one, for finite number of observations, and one for the asymptotic regime as N ® ¥ . We explain the connection between the existence of threshold effect in the estimation process of chaotic sequences, as indicated by our information-theoretic analysis, and the converse to the joint-source channel coding theorem. We apply our results to the dynamical system governed by the r-diadic map.

We examine the effect of dimensionality on joint source-channel codes based on multidimensional chaotic dynamical systems. We introduce a class of multidimensional chaotic systems which generalize the one-dimensional system governed by the r-diadic map, and for which we derive the maximum likelihood estimator and the Bayesian Cramer-Rao. We show that within this class of multidimensional chaotic systems, increasing the system's dimension decreases the code distortion, however, at the same time, the decoding process computational complexity is severely increased.