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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.

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  • $\begingroup$ The expression for Rww[n] results from directly applying the definition of autocorrelation to an LTI system. Please give a reference to how you obtained Rxx[n1,n2]. $\endgroup$ – Juancho Feb 4 '17 at 16:52
  • $\begingroup$ I went from $R_{WW}[n_{1},n_{2}]=E[W[n_{1}]W[n_{2}]]$ then used linearity of expectation to get that $R_{WW}[n_{1},n_{2}]=R_{UU}[n_{1},n_{2}]*h[n_{1}]*h[n_{2}]$ $\endgroup$ – Austin Feb 4 '17 at 16:57
  • $\begingroup$ You must have a wrong sign somewhere. Check for example this deduction (pg. 4). The operations for discrete time and continuous time are very similar. $\endgroup$ – Juancho Feb 4 '17 at 17:03
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It's basically just about using the definitions and doing the math:

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

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

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    $\begingroup$ too bad they didn't use the convention of lower case for "time-domain" functions leaving the upper case for "frequency-domain". using whatever notational convention remaining for the Fourier transform they might see that $$ \mathscr{F}\{R_{WW}[n]\} = |H(e^{j\omega})|^2 \mathscr{F}\{R_{UU}[n]\} $$ where $$ H(e^{j\omega}) = \mathscr{F}\{h[n]\} $$ $\endgroup$ – robert bristow-johnson Feb 4 '17 at 21:49
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    $\begingroup$ @robertbristow-johnson: It's an eternal dilemma: in many textbooks random processes have upper case letters to distinguish them from deterministic signals. $\endgroup$ – Matt L. Feb 4 '17 at 22:03
  • $\begingroup$ Thanks for your answer. I'm actually a bit confused about the starting point. Where does the complex conjugate come from? I know that $R_{WW}[n_{1},n_{2}]=E[W[n_{1}]W[n_{2}]]$, but haven't seen $R_{WW}[n]=E[W^{*}[k]W[k+n]]$ $\endgroup$ – Austin Feb 4 '17 at 22:21
  • $\begingroup$ i like the textbooks that use a boldface non-italics for the random process. problem is that notation can be confused with a vector. $\endgroup$ – robert bristow-johnson Feb 4 '17 at 22:57
  • $\begingroup$ @Jake: The definition with the complex conjugate is the general definition for complex-valued processes. You can ignore it for real-valued processes. $\endgroup$ – Matt L. Feb 5 '17 at 12:23

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