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I am dealing with a problem with Wiener filter. Assume the original function is $s(t)$ with observation $x(t) = s(t) + n(t)$. And $s(t)$ and $n(t)$ are WSS (wide sense stationary) and they are independent. Now I want to find the filter to estimate $h(t) = \frac{d s(t)}{dt}$.

The wiener filter involves finding the correlation functions. One step is to find $R_{x, h}$ which denotes the cross-correlation between the observations and the desired function.

Hence, $R_{x, h}(\tau) = E\left[\left(s(t+\tau) + n(t+\tau)\right)h(t) \right] = E\left[s(t+\tau) h(t) \right]$ (indept.)

Also, how to find the autocorrelation function $R_{h, h}(\tau) $?

It seems that this involves partial integral, and finally we would arrive the solution involved with $t$. So is there any mistake in my solution?

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    $\begingroup$ I suggest a convolution by a filter which is a frequency domain multiplication between the Wiener filter and the frequency representation of the time domain derivative $\endgroup$ – Gideon Genadi Kogan Nov 17 '20 at 15:46

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