LTI filtering for wide-sense stationary process Why is it that if $U[n]$ is wide-sense stationary and it is convolved with $h[n]$ to produce $W[n]$, the autocorrelation becomes $R_{WW}[n] = R_{UU}[n]*h[n]*h[-n]$?



I know that in general $R_{WW}[n_{1},n_{2}]=R_{UU}[n_{1},n_{2}]*h[n_{1}]*h[n_{2}]$ and that wide-sense stationary means $m_{U}[n] = m_{U}$ along with $R_{UU}[n_{1},n_{2}]=R_{UU}[n_{1}-n_{2},0]$, but I can't get to the above relation from these facts.
It's basically just about using the definitions and doing the math:

$$\begin{align}R_{WW}[n]&amp;=E\{W^*[k]W[k+n]\}\\
&amp;=E\left\{\sum_mU^*[k-m]h^*[m]\sum_lU[k+n-l]h[l]\right\}\\
&amp;=\sum_m\sum_lh^*[m]h[l]E\left\{U^*[k-m]U[k+n-l]\right\}\\
&amp;=\sum_m\sum_lh^*[m]h[l]R_{UU}[n+m-l]\\
&amp;=\sum_mh^*[m]\sum_lh[l]R_{UU}[n+m-l]\\
&amp;=\sum_mh^*[m]\left(h\star R_{UU}\right)[n+m]\\
&amp;=\sum_mh^*[-m]\left(h\star R_{UU}\right)[n-m]\\
&amp;=h^*[-n]\star h[n]\star R_{UU}[n]\end{align}$$

where $*$ denotes complex conjugation, and $\star$ denotes convolution.