Digital Signal Processing/Wiener Filters

A Wiener Filter is a filtering system that is an optimal solution to the statistical filtering problem.

Statistical Filtering

Statement of Problem

Note:
The operator E[] is the expectation operator, and is defined as:
${\displaystyle E[x]=\sum xf[n]}$
where fx[n] is the probability distribution function of x.

d[n] is the expected response value, or the value that we would like the input to approach.

${\displaystyle \sigma _{d}^{2}=E[d[n]d^{*}[n]]}$

e[n] is the estimation error, or the difference between the expected signal d[n] and the output of the FIR filter. We denote the FIR filter output with a hat:

${\displaystyle {\hat {d}}[n]=\sum _{k=1}^{M}w_{k}u[n-k+1]}$

Where the convolution operation applies the input signal, u[n], to the filter with impulse response w[n].

We can define a performance index J[w] which is a function of the tap weights of the FIR filter, w[n], and can be used to show how close the filter is to reaching the desired output. We define the performance index as:

${\displaystyle J[w]=E[e[n]e^{*}[n]]}$

J[w] is also known as the mean-squared error signal. The goal of a Wiener filter is to minimize J[w] so that the filter operates with the least error.

${\displaystyle Rw_{o}=p}$
${\displaystyle w_{o}=R^{-1}p}$