Help with random variable linear estimation

[ashah99]

Homework Statement

Please see problem below

Relevant Equations

MSE = E((Xhat_k - X_k)^2)

Hi all, I have a problem on linear estimation that I would like help on. This is related to Wiener filtering.

Problem:

I attempted part (a), but not too sure on the answer. As for unconstrained case in part (b), I don't know how to find the autocorrelation function, I applied the definition, but how do I take it from here?

Here's the notes that I'm referencing:
https://mathhelpforum.com/attachments/1669130798979-png.45154/

A block diagram of what I think the scenario is:
https://mathhelpforum.com/attachments/1669130857779-png.45155/Attempt at solution:
https://mathhelpforum.com/attachments/1669130913277-png.45156/

 

[lippyka]

(a) The Wiener filter for this system is given by(z) = (Rxy(z))/(Rxx(z))where Rxx(z) and Rxy(z) are the autocorrelations of the input and output signals respectively. (b) For the unconstrained case, we can find the autocorrelation of the input signal as follows:Rxx(z) = E[x(n)x(n-k)] where E[.] is the expectation operator. To find Rxy(z), we must first compute the cross-correlation between the input and output signals, which is given by:Rxy(z) = E[x(n)y(n-k)]Using these two equations, we can then calculate the Wiener filter for the unconstrained case.