# 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:
$E[x] = \sum x f[n]$
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.

$\sigma_d^2 = E[d[n]d^*[n]]$

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:

$\hat{d}[n] = \sum_{k=1}^M w_k u[n-k+1]$

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:

$J[w] = E[e[n]e^*[n]]$

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.

$Rw_o = p$
$w_o = R^{-1}p$