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Encyclopedia > Wiener filter

The Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published [1]. In electronics and signal processing, a filter is a device or process that modifies a signal. ... Norbert Wiener (November 26, 1894 - March 18, 1964) was an American mathematician, known as the founder of cybernetics. ... // Events and trends The 1940s were dominated by World War II, the most destructive armed conflict in history. ...

1 Description
2 Model/problem setup
3 Stationary solution
- 3.1 Acausal solution
- 3.2 Causal solution
4 Non-stationary solution
5 See also
6 References

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]: Frequency response is the measure of any systems response to frequency, but is usually used in connection with electronic amplifiers and similar systems, particularly in relation to audio signals. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ...

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. In the mathematics of probability, a stochastic process can be thought of as a random function. ... Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ... In signal processing, cross-correlation is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. ... Minimum mean-square error (MMSE) relates to an estimator having estimates with the minimum mean square error possible. ... In general usage, noise can be considered data without meaning; that is, data that is not being used to transmit a signal, but is simply produced as an unwanted by-product of other activities. ... For Wikipedia statistics, see m:Statistics Statistics is the science and practice of developing human knowledge through the use of empirical data expressed in quantitative form. ...

Model/problem setup

The input to the Wiener filter is assumed to be a signal, $s (t)$ , corrupted by additive noise, $n (t)$ . The output, $x (t)$ is calculated by means of a filter, $g (τ)$ by means of the following convolution:

x (t) = g (τ) * (s (t) + n (t))

, where

$s (t)$ is the original signal (to be estimated)
$n (t)$ is the noise
$x (t)$ is the estimated signal (which we hope will equal $s (t)$

The error is $e (t) = s (t + d) - x (t)$ and the squared error is $e 2 (t) = s 2 (t + d) - 2 s (t + d) x (t) + x 2 (t)$ where

$s (t + d)$ is the desired output of the filter
$e (t)$ is the error

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

If $d > 0$ then the problem is that of prediction
If $d = 0$ then the problem is that of filtering
If $d < 0$ then the problem is that of smoothing

Writing $x (t)$ as a convolution integral: $x(t) = int_{-infty}^{infty}{g(tau)left[s(t - tau) + n(t - tau)right]dtau}$ . A prediction is a statement or claim that a particular event will come to pass in the future. ... In electronics and signal processing, a filter is a device or process that modifies a signal. ... This article is about the mathematical concept of convolution. ...

Taking the expectation of the squared error results in expectation in the context of probability theory and statistics, see expected value. ...

$E(e^2) = R_s(0) - 2int_{-infty}^{infty}{g(tau)R_{x,s}(tau + d)dtau} + int_{-infty}^{infty}{int_{-infty}^{infty}{g(tau)g(theta)R_x(tau - theta)dtau}dtheta}$

where

$,!R_s$ is the autocorrelation function of $s (t)$
$,!R_x$ is the autocorrelation function of $x (t)$
$R_{x,s}$ is the cross-correlation function of $x (t)$ and $s (t)$

If the signal $s (t)$ and the noise $n (t)$ are uncorrelated (i.e., the cross-correlation is zero) then note the following Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ... Autocorrelation is a mathematical tool used frequently in signal processing for analysing functions or series of values, such as time domain signals. ... In signal processing, cross-correlation is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. ...

$R_{x,s} = R_s$
$,!R_x = R_s + R_n$

The goal is to then minimize $E (e 2)$ by finding the optimal $g (t)$ .

Stationary solution

The Wiener filter has two solutions for two possible cases: causal and anticausal. A causal system is a system that depends only on the current and previous inputs. ... An acausal system is a system that depends on both the past and the future. ...

Acausal solution

$G(s) = frac{S_{x,s}(s)e^{alpha s}}{S_x(s)}$

Provided that $g (t)$ is optimal then the mmse equation reduces to $E(e^2) = R_s(0) - int_{-infty}^{infty}{g(tau)R_{x,s}(tau + d)dtau}$

And the solution, $g (t)$ is the inverse two-sided Laplace transform of $G (s)$ .

Causal solution

$G(s) = frac{H(s)}{S_x^{+}(s)}$

Where

$H (s)$ is the positive time solution of the inverse Laplace transform of $frac{S_{x,s}(s)e^{alpha s}}{S_x^{-}(s)}$
$S_x^{+}(s)$ is the positive time solution of the inverse Laplace transform of $S x (s)$
$S_x^{-}(s)$ is the negative time solution of the inverse Laplace transform of $S x (s)$

Non-stationary solution

References

[1]: Wiener, Norbert. Extrapolation, Interpolation, and Smoothing of Stationary Time Series. New York: 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

Categories: Linear filters | Estimation theory | Stochastic processes

Results from FactBites:

Wiener filtering (608 words)
	Wiener filtering is a method to recover the original signal as close as possible from the received signal.
	Filters are commenly used to extract a desired signal from a backgroud of random noise or deterministic interference.
	Thus, the performance of the wiener filter may be evaluated by listening to signals and noise.

Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (850 words)
	In signal processing, the Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published in 1949.
	The input to the Wiener filter is assumed to be a signal,
	The Wiener filter has solutions for two possible cases: the case where a causal filter is desired, and the one where a non-causal filter is acceptable.

More results at FactBites »

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