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Having the following system:

System

I'd like to know how to obtain the cross-correlation between$\ w(n) $ and$\ y(n) $ having the following conditions: $\ w(n) $ is a zero-mean white noise with power$\ P_w $ that is uncorrelated with$\ x(n) $, and $\ x(n) $ and $\ z(n) $ are Wide Sense Stationary random processes with $\ H(w) = |H(w)|exp[j \theta(w)] $

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    $\begingroup$ so, what's the problem you're facing that makes you ask this? I'm asking since $y(n) = w(n) + z(n)$, and you know what they say about linear operations! (they are linear :) ) $\endgroup$ – Marcus Müller Oct 9 '17 at 21:40
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\begin{align} R_{W,Y}[n] &= E\big[W[m]Y[m+n]\big]\\ &= E\big[W[m](Z[m+n] + W[m+n])\big]\\ &= R_{W,Z}[n] + R_{W,W}[n]\\ &= R_{W,Z}[n]+ P_w\delta[n]. \end{align} So the question is: can you figure out what $R_{W,Z}[n] = R_{W,h\star X}[n]$ is where you know that $\{X[n]\}$ and $\{W[n]\}$ are uncorrelated processes, that is, $R_{W,X}[n] = 0$ for all $n$. Hint: $Z[n]$ is a weighted linear combination of the $X[i]$, and, as Marcus Muller's comment points out, the curious fact about linear operations is that they are linear!

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  • $\begingroup$ Thanks for the answer! Really useful one! Anyway, I cannot find out by myself how to obtain $ R_{W,Z}[n] $. I mean, being $ x(n) $ and $ w(n) $ uncorrelated implies $ R_{W,X}[n] = 0 $, but how can be that related to $ R_{W,Z}[n] $? @dilip-sarwate $\endgroup$ – Ragnar Oct 10 '17 at 9:19
  • $\begingroup$ @Ragnar You have discrete-time input and output signals from a filter whose transfer function is specified for continuous time. So, you need to figure out what the corresponding discrete-time transfer function is, or, if you like, the $z$-transform of the impulse response. This will show that $Z[n]$ is a weighted sum of the $X[i]$'s as I pointed out in my comment and this will make $E\left[W[m]\sum_i h[i]X[m+n-i]\right]$ very easy to compute. $\endgroup$ – Dilip Sarwate Oct 10 '17 at 18:24

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