On the threshold effect in the estimation of chaotic sequences

Chaotic sequences and chaotic dynamic systems are attractive candidates for use in signal modelling, synthesis, and analysis as well as in communications applications. In most of the above applications, there is a frequent need to estimate the chaotic sequence from noisy observations. In previous works, various methods for the estimation of chaotic sequences under noise were developed. However, although the methods were different, their qualitative performance was the same: for high SNR the performance was good, but below some threshold SNR, a sharp degradation in performance occurred. Using information-theoretic tools, we quantify this threshold effect and obtain lower bounds on the value of the threshold SNR. We show that the lower bound depends on the system's Lyapunov exponent and the mutual information between the chaotic sequence and the noisy observations. This bound is further simplified to a bound depending on the system's Lyapunov exponent and the power spectrum of the chaotic sequence. Essentially, for SNRs below the threshold, the amount of information that the chaotic system produces about its initial state is larger than the maximum information that the channel can convey with low probability of error. Thus, degradation in distortion performance is unavoidable.