Wiener filter equation

Example

Let $x(t)$ and $s(t)$, $t \in \R$, be real, jointly wide-sense stationary processes with mean $0$. Write $R_{xx}(\tau) = E[x(u + \tau)\, x(u)]$ and $R_{sx}(\tau) = E[s(u + \tau)\, x(u)]$; they do not depend on $u$. The task is to estimate $s(t)$ by a causal linear filter of the observations, $\hat s(t) = \int_0^\infty h(\tau)\, x(t - \tau)\, d\tau$, with the smallest mean square error $E\bigl[(s(t) - \hat s(t))^2\bigr]$. The optimal impulse response $h$ satisfies the Wiener filter equation $$R_{sx}(\sigma) = \int_0^\infty h(\tau)\, R_{xx}(\sigma - \tau)\, d\tau, \quad \sigma \ge 0.$$ This is a Wiener–Hopf equation of the first kind with kernel $R_{xx}$.

Check of the equation. The error $e(t) = s(t) - \hat s(t)$ of the best filter is orthogonal to every observation $x(t - \sigma)$ with $\sigma \ge 0$. Otherwise a small change of $h$ would lower the error. So $E[s(t)\, x(t - \sigma)] = \int_0^\infty h(\tau)\, E[x(t - \tau)\, x(t - \sigma)]\, d\tau$. The left side is $R_{sx}(\sigma)$ and the expectation under the integral is $R_{xx}(\sigma - \tau)$, which gives the equation. The condition holds only for $\sigma \ge 0$, because the filter may use only the past.