Wiener Filter Problem Setup
The input to the Wiener filter is assumed to be a signal, corrupted by additive noise, . The output, is calculated by means of a filter, using the following convolution:
where is the original signal (not exactly known; to be estimated), is the noise, is the estimated signal (the intention is to equal ), and is the Wiener filter's impulse response.
The error is defined as
where is the delay of the Wiener filter (since it is causal). In other words, the error is the difference between the estimated signal and the true signal shifted by .
The squared error is
where is the desired output of the filter and is the error. Depending on the value of, the problem can be described as follows:
- if then the problem is that of prediction (error is reduced when is similar to a later value of s),
- if then the problem is that of filtering (error is reduced when is similar to ), and
- if then the problem is that of smoothing (error is reduced when is similar to an earlier value of s).
Taking the expected value of the squared error results in
where is the observed signal, is the autocorrelation function of, is the autocorrelation function of, and is the cross-correlation function of and . If the signal and the noise are uncorrelated (i.e., the cross-correlation is zero), then this means that and . For many applications, the assumption of uncorrelated signal and noise is reasonable.
The goal is to minimize, the expected value of the squared error, by finding the optimal, the Wiener filter impulse response function. The minimum may be found by calculating the first order incremental change in the least square resulting from an incremental change in for positive time. This is
For a minimum, this must vanish identically for all which leads to the Wiener–Hopf equation:
This is the fundamental equation of the Wiener theory. The right-hand side resembles a convolution but is only over the semi-infinite range. The equation can be solved by a special technique due to Wiener and Hopf.
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