The Wiener filter as developed by Norbert Wiener in the early 1940 may be considered as a first application of stochastic signal models for filter optimization. Characteristic to such an approach is that some a priori knowledge about the signals has to be available. The filter is a solution to a classical communication problem: An additively disturbed signal has to be separated from noise.
Wiener assumed that the filter input signals can be modeled by stationary stochastic processes and that their power spectral densities are known. He found a linear filter that optimally reconstructs the input signal or a linear transformation thereof from an input that is additively disturbed by noise. The original Wiener solution was derived for time-continuous quantities. The filter is described by its (infinitely) long impulse response or the Fourier transform of it. Solutions are available for a noncausal and a causal filter. In both case, the observation intervals are infinitely long.
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