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Suppose ${X_n; n \in Z}$ and ${Z_n; n \in Z}$ are mutually independent, i.i.d. Gaussian random process with auto correlations $R_x(k) = {\sigma_x}^2\delta(k)$ and $R_z(k) = {\sigma_z}^2\delta(k)$ These processes are used to construct new processes as follows:

$Y_n = Z_n + rY_{n-1}$

$U_n = X_n + Y_n$

$W_n = U_n - rU_{n-1}$

Find the covariances and power spectral densities of ${U_n}$ and ${W_n}$. Find $E[(X_n - W_n)^2]$

The recursive definition is really confusing me. Any help on how this problem is to be tackled?

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Try to write out the $n$'th term of $U_n$ and $W_n$ and look for a pattern. For example: $U_n=X_n+Z_n+rY_{n-1}=X_n+Z_n+rZ_{n-1}+r^2Y_{n-2}$ and so on. This yields $U_n=X_n+\sum_{k=0}^{n-1} r^k Z_{n-k}$. –  Stefan Hansen Nov 2 '12 at 10:31
    
Any luck with @StefanHansen's hint? –  Did Dec 2 '12 at 14:27
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