Beneath the Noise, Chaos

The problem of extracting a signal x from a noise-corrupted time n series y x e is considered. The signal x is assumed to be genern n n n ated by a discrete-time, deterministic, chaotic dynamical system F, in nŽ . particular, x F x , where the initial point x is assumed to lie in a n 0 0 Ž . compact hyperbolic F-invariant set. It is shown that 1 if the noise sequence e is Gaussian then it is impossible to consistently recover the n Ž . signal x , but 2 if the noise sequence consists of i.i.d. random vectors n uniformly bounded by a constant 0, then it is possible to recover the signal x provided 5 , where is a separation threshold for F. A n filtering algorithm for the latter situation is presented.