Cancelling deterministic noise by constructing nonlinear inverses to linear filters

In this paper we propose a method of noise cancellation where the 'noise' is deterministic: that is, it is generated by some (possibly chaotic) dynamical system. The method exploits the fact that a time series measured from a nonlinear dynamical system can be used to reconstruct the phase space of the system, and that this is still true if the time series is first passed through a linear FIR filter. Since both filtered and unfiltered versions reconstruct the same system, there is an invertible (nonlinear) mapping from the filtered time series to the unfiltered one which undoes the effect of the filter. To perform signal separation we use a filter which stops the signal we wish to recover, so that the output is a filtered version of the noise alone. Using the nonlinear mapping the unfiltered noise time series can be recovered, and subtracted from the original (noisy) signal. Two examples of the recovery of simple signals from chaotic time series are given, and one in which one chaotic signal is separated from another.