Beneath the noise, chaos

Abstract

The problem of extracting a signal cursive Greek chin from a noise-corrupted time series yn = cursive Greek chin + en is considered. The signal cursive Greek chin is assumed to be generated by a discrete-time, deterministic, chaotic dynamical system F, in particular, cursive Greek chin = Fn(cursive Greek chi0), where the initial point cursive Greek chi0 is assumed to lie in a compact hyperbolic F-invariant set. It is shown that (1) if the noise sequence en is Gaussian then it is impossible to consistently recover the signal cursive Greek chin, but (2) if the noise sequence consists of i.i.d. random vectors uniformly bounded by a constant δ > 0, then it is possible to recover the signal cursive Greek chin provided δ < 5Δ, where Δ is a separation threshold for F. A filtering algorithm for the latter situation is presented.