Estimating initial conditions of noisy chaotic signals generated by piecewise linear Markov maps using itineraries

Estimating a one-dimensional (1-D) chaotic signal in noise is an important problem in chaotic communications and information processing. This problem is theoretically equivalent to the estimation of the initial condition of a chaotic signal. A few studies on this initial condition estimation problem have been carried out for certain specific maps such as the tent map and the logistic map. This problem is investigated for the piecewise linear Markov maps as well as maps that are topologically conjugate to piecewise linear Markov maps. By using the one-to-one correspondence between the initial conditions of a chaotic map and its space of itineraries, several algorithms extending the halving method are developed to estimate the initial condition of a 1-D chaotic signal embedded in additive noise. Performance of these estimators is evaluated using Monte Carlo simulations. At high SNR, the variance of these estimators is found to approach the Cramer-Rao bound.