Non-parametric Wiener filter for reducing noise on reproducible pure signals

This paper communicates a novel method of Wiener filtering signals from scientific instruments to reduce their noise content. The main prerequisite for the applicability of this technique is that the pure (i.e. noiseless) signal is reproducible, a condition that is satisfied by a wide range of experimental measurements. The benefits of this Wiener filter design approach are its simplicity, generality and practicality. In particular, signal-dependent or multiplicative noise are accommodated without complication. The non-parametric filter does not require models of pure signal or noise statistics, and is exactly optimal for the observed (i.e. noisy) signal ensemble. Implementation of this Wiener filter only requires measurement of observed signal correlation functions. After deriving the classical stationary signal Wiener filter, the analysis is extended to derive the reproducible stationary pure signal Wiener filter. A distinction is made between nonadaptive Wiener filters derived from correlations computed as ensemble averages, and adaptive Wiener filters derived from correlations computed as time averages. The analysis is extended again to encapsulate non-stationary signal ensembles and further extended to synthesize Wiener filters based on statistically good observed signal correlation estimates for arbitrarily many pure signal reproductions.