Symbolic dynamics for processing chaotic signals. I. Noise reduction of chaotic sequences

Chaotic signals attracted the attention among researchers because of their rich dynamics and their random-like behavior. What has been missing so far is an appropriate characterization of chaotic systems from a signal-processing point of view. This paper demonstrates that the framework of symbolic dynamics gives the possibility to partition the infinite number of finite-length trajectories of the piecewise-linear chaotic system into a countable number of trajectory-sets with common statistical properties. It turns out that this partitioning allows to derive noise-reduction schemes directly from the maximum-likelihood criteria. For the two proposed noise-reduction methods, the upper performance limits are given in an analytical form and the results are verified by applying the schemes to different types of one dimensional piecewise-linear Markov maps.