Basic Wiener Filter Theory | Digital Signal Processing

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Many adaptive algorithms can be viewed as approximations of the discrete Wiener filter (Figure 10.5). Two signals, x_k and y_k are applied simultaneously to the filter. Typically, y_k consists of a component that is correlated with y_k and another that is not. The Wiener filter produces an optimal estimate of the part of y_k that is correlated with x_k which is then subtracted from y_k to yield e_k.
Assuming an FIR filter structure with N coefficients (or weights - the popular phrase in the literature), the error, e_k between the Wiener filter output and the primary signal, y_k is given by

The Wiener filter has a limited practical usefulness because

it requires the autocorrelation matrix, R, and the cross-correlation vector, P, both of which are not known a priori,
it involves matrix inversion, which is time consuming, and
if the signals are nonstationary, then both R and P will change with time and so W_opt will have to be computed repeatedly.

For real-time application, a way of obtaining W_opt on a sample-by-sample basis is required. Adaptive algorithms are used to achieve this without having to compute R and P explicitly or performing a matrix inversion.