Beneath the noise, chaos

The problem of extracting a signal x n from a noise-corrupted time series y n = x n + e n is considered. The signal x n is assumed to be generated by a discrete-time, deterministic, chaotic dynamical system F, in particular, x n = F n (x 0 ), where the initial point x 0 is assumed to lie in a compact hyperbolic F-invariant set. It is shown that (1) if the noise sequence e n is Gaussian then it is impossible to consistently recover the signal x n , 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 x n provided δ < 5Δ, where Δ is a separation threshold for F. A filtering algorithm for the latter situation is presented.