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-process ing point of view. This paper demonstrates that the framework of symbolic d ynamics gives the possibility to partition the infinite number of finite-le ngth trajectories of the piecewise-linear chaotic system into a countable n umber of trajectory-sets with common statistical properties. It turns out t hat this partitioning allows to derive noise-reduction schemes directly fro m the maximum-likelihood criteria. For the two proposed noise-reduction met hods, the upper performance limits are given in an analytical form and the results are verified by applying the schemes to different types of one dime nsional piecewise-linear Markov maps.