M.Sc Thesis | |

M.Sc Student | Chen Ilan |
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Subject | The Thershold Effect in the Estimation of Chaotic Sequences |

Department | Department 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 SNR_{th}, 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 SNR_{th}. 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.