Wiener-Filter

The Wiener filter or Wiener- Kolmogorov filter is a filter for signal processing, which was developed independently in the 1940s by Norbert Wiener and Andrei Nikolaevich Kolmogorov and 1949 published by Norbert Wiener. It performs an optimal noise reduction.

Properties

The Wiener filter is described by the following characteristics:

Model Properties

The input signal of the Wiener filter, a signal is disturbed by an additive noise assumed.

The output signal is given by the convolution of the input signal with the filtering function:

Error and squared error resulting from the deviation of the output signal from the delayed input signal. Depending on the value of the time offset d different problems can be considered:

  • For: prediction
  • For: filtering
  • For: Smoothing

If, as a convolution integral is:

We obtain the expected value of the squared error to:

In which

  • The auto-correlation function
  • The auto-correlation function
  • The cross-correlation functions and are

If the signal and noise are uncorrelated (and thus the cross-correlation is zero ), the following simplifications

The goal now is to minimize an optimal by determining.

Stationary solutions

The Wiener filter has a solution for each of the causal and non - causal case.

Non- causal solution

Under the condition that is optimal, the equation describing the minimum of the mean squared error ( Minimum Mean Square Error, MMSE) simplifies to

The solution is the inverse Laplace transform of both sides.

Causal solution

In which

  • The positive solution of the inverse Laplace transform,
  • The positive solution of the inverse Laplace transform and
  • The negative solution of the inverse Laplace transform of is.
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