Wiener filter

The Wiener-Filter or Wiener-Kolmogoroff-Filter is a filter for signal processing , which was developed in the 1940s by Norbert Wiener and Andrei Nikolajewitsch Kolmogorow independently and published in 1949 by Norbert Wiener. Measured by the mean square deviation , it performs optimal noise suppression .

Use of the Wiener filter for noise suppression. (left: original, middle: noisy image, right: filtered image)

properties

The Wiener filter is described by the following properties:

1. Prerequisite: The signal and the additive noise are the same stochastic processes with known spectral distribution or known autocorrelation and cross-correlation
2. Error criterion: Minimum mean square deviation

Model properties

A signal disturbed by additive noise is assumed as the input signal of the Wiener filter . ${\displaystyle s\left(t\right)}$ ${\displaystyle n\left(t\right)}$

${\displaystyle y(t)=s(t)+n(t)}$

The output signal results from the convolution of the input signal with the filter function : ${\displaystyle x\left(t\right)}$ ${\displaystyle g\left(\tau \right)}$

${\displaystyle x(t)=g(\tau )*y(t)=g(\tau )*\left(s(t)+n(t)\right)}$

Error and square error result from the deviation of the output signal from the time-shifted input signal . Depending on the value d of the time offset, different problems can be considered: ${\displaystyle e(t)=s\left(t+d\right)-x(t)}$ ${\displaystyle e^{2}(t)=s^{2}\left(t+d\right)-2s(t+d)x(t)+x^{2}(t)}$ ${\displaystyle s\left(t+d\right)}$

• For  : prediction ${\displaystyle \left.d>0\right.}$
• For  : filtering ${\displaystyle \left.d=0\right.}$
• For  : smoothing ${\displaystyle \left.d<0\right.}$

If one represents as a convolution integral: ${\displaystyle x\left(t\right)}$

${\displaystyle x(t)=\int \limits _{-\infty }^{\infty }{g(\tau )\left[s(t-\tau )+n(t-\tau )\right]d\tau }}$,

the expected value of the quadratic error results in:

${\displaystyle E(e^{2})=R_{s}(0)-2\int \limits _{-\infty }^{\infty }{g(\tau )R_{y\,s}(\tau +d)d\tau }+\int \limits _{-\infty }^{\infty }{\int \limits _{-\infty }^{\infty }{g(\tau )g(\theta )R_{y}(\tau -\theta )d\tau }d\theta }}$

in which

• ${\displaystyle R_{s}}$ the autocorrelation of the function ${\displaystyle \left.s(t)\right.}$
• ${\displaystyle R_{y}}$ the autocorrelation of the function ${\displaystyle \left.y(t)\right.}$
• ${\displaystyle R_{y\,s}}$the cross-correlation of the functions and are ${\displaystyle \left.y(t)\right.}$ ${\displaystyle \left.s(t)\right.}$

If the signal and the noise are uncorrelated (and thus the cross-correlation is equal to zero), the following simplifications result ${\displaystyle s\left(t\right)}$ ${\displaystyle n\left(t\right)}$

• ${\displaystyle R_{y\,s}=R_{s}}$
• ${\displaystyle R_{y}=R_{s}+R_{n}}$

The goal now is to minimize by determining an optimal one. ${\displaystyle \left.E(e^{2})\right.}$ ${\displaystyle g\left(\tau \right)}$

Stationary solutions

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

Non-causal solution

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

With the proviso that is optimal simplifies the equation that the minimum of the mean square error ( Minimum Mean-Square Error describes MMSE) to ${\displaystyle g\left(t\right)}$

${\displaystyle E(e^{2})=R_{s}(0)-\int \limits _{-\infty }^{\infty }{g(\tau )R_{x,s}(\tau +d)d\tau }}$.

The solution is the inverse two-sided Laplace transformation of . ${\displaystyle g\left(t\right)}$ ${\displaystyle \left.G(s)\right.}$

Causal solution

${\displaystyle G(s)={\frac {H(s)}{S_{x}^{+}(s)}}}$

In which

• ${\displaystyle \left.H(s)\right.}$the positive solution of the inverse Laplace transform of , ${\displaystyle {\frac {S_{x,s}(s)e^{\alpha s}}{S_{x}^{-}(s)}}}$
• ${\displaystyle S_{x}^{+}(s)}$the positive solution of the inverse Laplace transform of and ${\displaystyle \left.S_{x}(s)\right.}$
• ${\displaystyle S_{x}^{-}(s)}$is the negative solution of the inverse Laplace transform of . ${\displaystyle \left.S_{x}(s)\right.}$

See also

• Kalman filter
• Viennese deconvolution

Individual evidence

1. ↑ Kristian Kroschel: Statistical Message Theory . Signal and pattern recognition, parameter and signal estimation. 3rd, revised and expanded edition. Springer, Berlin et al. 1996, ISBN 3-540-61306-4 .
2. ^ ^{A b} Norbert Wiener : Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Wiley, New York NY 1949.
3. ^ Robert Grover Brown, Patrick YC Hwang: Introduction to Random Signals and Applied Kalman Filtering. With MATLAB exercises and solutions. 3. Edition. Wiley et al., New York NY 1996, ISBN 0-471-12839-2 .