Wiener filter

Wiener filtering allows us to estimate the value of one wide-sense stationary random process from measurements of another WSS random process that is jointly WSS.

Essentially, a Wiener filter is an LMMSE estimator that uses the values of one process to estimate the values of the other. This can be written in the form:

$$ \hat{y} [n] = \mu_y + \sum_{j = 0}^{L-1} h[j] \underbrace{(x[n-j] - \mu_x)}_{\tilde{x}[n-j]} $$

This is equivalent to $x$ convolved with $h$, an FIR filter:

$$ \hat{y}[n] = (h \ast x)[n] $$

This FIR filter $h[\dot]$ satisfies:

$$ (h \ast C_{xx})[m] = C_{yx}[m], \forall m $$

In the frequency domain, this can be rewritten as:

$$ H(e^{j\Omega}) = \frac{D_{yx}(e^{j\Omega})}{D_{xx}(e^{j\Omega}) $$

This is the frequency response of the unconstrainted Wiener filter - that is, $x[n]$ for all $n$ can be used.

$$ \frac{1}{2\pi} \int_{-\pi}^{\pi} (D_{yy}(e^{j\Omega}) - \frac{D_{yx}(e^{j\Omega})}{D_{xx}(e^{j\Omega})}) $$