Directory
Encyclopedia

NodeWorks
                              ENCYCLOPEDIA

Link Checker
Home
Encyclopedia : W : WI : WIE :

Wiener filter

  

Wiener filter

The Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published [1].

Description


Unlike the typical filtering theory of designing a filter for a desired frequency response the Wiener filter approaches filtering from a different angle. By creating a filter that filters only on the frequency domain it is possible for the filter to pass noise. Wiener's solution was to require additional information regarding the spectral content of the original signal and the noise. Wiener filters are characterized by the following [2]:
  • Assumption: signal and (additive) noise are stochastic processes with known spectral characteristics or known autocorrelation and cross-correlation
  • Performance criteria: minimum mean-square error
  • An optimal filter can be found from a solution based on scalar methods

    The goal of the Wiener filter is to filter out noise that has corrupted a signal by statistical means.

    Model/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, by means of the following convolution:
    :, where
  • is the original signal (to be estimated)
  • is the noise
  • is the estimated signal (which we hope will equal

    The error is and the squared error is where

  • is the desired output of the filter
  • is the error

    Depending on the value of d the problem name can be changed:

  • If then the problem is that of prediction
  • If then the problem is that of filteringing
  • If then the problem is that of smoothing

    Writing as a convolution integral: .

    Taking the expectation of the squared error results in
    :
    where

  • is the autocorrelation function of
  • is the autocorrelation function of
  • is the cross-correlation function of and

    If the signal and the noise are uncorrelated (i.e., the cross-correlation is zero) then note the following

  • The goal is to then minimize by finding the optimal .

    Stationary solution


    The Wiener filter has two solutions for two possible cases: causal and anticausal.

    Acausal solution

    Provided that is optimal then the mmse equation reduces to

    And the solution, is the inverse two-sided Laplace transform of .

    Causal solution

    Where
  • is the positive time solution of the inverse Laplace transform of
  • is the positive time solution of the inverse Laplace transform of
  • is the negative time solution of the inverse Laplace transform of

    Non-stationary solution

    See also

  • Norbert Wiener
  • Kalman filter

    References

  • [1]: Wiener, Norbert. Extrapolation, Interpolation, and Smoothing of Stationary Time Series. New Yokr: Wiley, 1949.
  • [2]: Brown, Robert Grover and Patrick Y.C. Hwang. Introduction to Random Signals and Applied Kalman Filtering. 3 ed. New York: John Wiley & Sons. 1997


    		•
    NodeWorks boosts web surfing!
    Page Returned in 0.134 seconds - HTML Compressed 69.9%

    This article is from Wikipedia. All text is available
    under the terms of the GNU Free Documentation License.
     GNU Free Documentation License
    © 2008 Chamas Enterprises Inc.