# Find the Autocorrelation and Cross Correlation of The Wiener Filter

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?

• 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 – Gideon Genadi Kogan Nov 17 '20 at 15:46