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I am reading about the Wiener deconvolution in Wikipedia. In the expression of $G$ we have $$G=\frac{H^*S}{|H|^2S+N}$$ where $S = \mathbb{E}|X|^2$. Why do we have the $\mathbb{E}$ symbol? Isn't $X$ a deterministic signal? This would imply $S=|X|^2$. Am I right or am I missing some detail?

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If $x$ is a deterministic signal but isn't known to us, then it makes sense to treat it as a probabilistic signal.

Specifically in the case of the Wiener filter (and much of the other useful signal processing math) if you treat $x$ as a zero-mean Gaussian random variable, it doesn't just make the math easy, it makes a closed-form solution possible.

No real noise processes are truly Gaussian. No real noise processes are truly stationary. No real systems are truly linear. No real systems are truly time invariant. No real systems are actually "best" in the global sense when the MMSE is minimized. Yet the math gets so much easier for Gaussian stationary signals running through linear time-invariant systems where the cost goes as $x^2$ that if we can approximate our problem in those terms, then we can arrive at a solution quickly -- and the solution is just as good as the approximation, and very often plenty good enough.

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  • $\begingroup$ But what is $S$? Is it correct that $S=|X|^2$ if $X$ is deterministic? What am I supposed to put in place of $S$ in my algorithm to apply the filter? $\endgroup$
    – user171780
    Aug 17, 2021 at 17:11

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