The problem of extracting a signal xₙ from a noise-corrupted time series yₙ = xₙ+eₙ is considered. The signal xₙ is assumed to be generated by a discrete-time, deterministic, chaotic dynamical system F, in particular, xₙ = Fⁿ(x₀), where the initial point x₀ 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 xₙ , 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 xₙ provided δ < 5Δ, where Δ is a separation threshold for F. A filtering algorithm for the latter situation is presented.
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Steven P. Lalley (1999) studied this question.
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