# Calculation of cross-correlation of a system with noise

Having the following 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 $\ H(w) = |H(w)|exp[j \theta(w)]$

• 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 :) ) – Marcus Müller Oct 9 '17 at 21:40

\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!
• 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 – Ragnar Oct 10 '17 at 9:19
• @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. – Dilip Sarwate Oct 10 '17 at 18:24