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Linear Minimum Mean-Square Error Filtering

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Background  ::  Filtering

Causal Wiener Filtering

The examples we have seen so far have produced noncausal filters, i.e., practically nonimplementable in many cases. We will see what can be done now to make causal filters.

Let us take $a =-\infty$ , with $\{X_t\}$ and $\{Y_t\}$ jointly and individually W.S.S., and take $b=t$ (filtering). Furthermore, assume that the filter is time-invariant and causal . Then the Wiener-Hopf equations can be written

\begin{displaymath}
\boxed{R_{XY}(s) = \int_{-\infty}^t h(s-\nu) R_Y(\nu) d\nu.}
\end{displaymath}

The question is, how can this be solved? Because the limit does not proceed to $\infty$ , we can't use conventional transform techniques.

Here are some facts to help. Suppose $S_Y(\omega)$ satisfies the following condition:

\begin{displaymath}\int_{-\infty}^\infty \frac{\log\vert S_Y(\omega)\vert}{1 +
\omega^2} d\omega < \infty
\end{displaymath}

(This is known as the Paley-Wiener condition.) Then it turns out that we can write

\begin{displaymath}S_Y(\omega) = S_Y^+(\omega) S_Y^(\omega)
\end{displaymath}

where

\begin{displaymath}\vert S_Y^+(\omega)\vert^2 = \vert S_Y^-(\omega)\vert^2 = S_Y(\omega)
\end{displaymath}

and $\Fc^{-1}S_Y^+(\omega)$ is zero for negative times (that is, it is causal) and $\Fc^{-1}S_Y^-(\omega)$ is zero for positive times (that is, it is anticausal). Moreover,

\begin{displaymath}\frac{1}{S_Y^+(\omega)}
\end{displaymath}

is also causal and

\begin{displaymath}\frac{1}{S_Y^-(\omega)}
\end{displaymath}

is also anticausal. The proof in general is rather difficult (we will skip it, but give some examples). This factorization is known as the spectral factorization .
Copyright 2008, by the Contributing Authors. Cite/attribute Resource . admin. (2006, June 07). Linear Minimum Mean-Square Error Filtering. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/lecture9_2.htm. This work is licensed under a Creative Commons License Creative Commons License