Anybody give me an advice how to find the auto-covariance of the product of deterministic and wide-sense stationary signal. I couldn't find how to solve this, I have looked and searched the internet, I found just 1 similar topic here so I think someone in here may solve it.

The mean and autocorrelation of $X(t)$ are $m$ and $R$, respectively, and the $X(t)$ process is wide-sense stationary. $g(t)$ is a deterministic function, $Y(t)=X(t)\cdot g(t)$ is defined.

  • A. Find the mean, autocovariance and autocorrelation functions of the $Y(t)$ process.
  • B. Is the $Y(t)$ process wide-sense stationary ? Explain your answer.
  • 1
    $\begingroup$ Should we assume you already tried to apply the definition of mean and autocorrelation to Y? What did you get? $\endgroup$ – Juancho Jun 13 '17 at 20:08
  • $\begingroup$ I don't know how to relate deterministic and wide-sense signal, so I couldn't try with respect to any rule, just tried some of formulas but I did not sure about that because couldnt find even some example or theory about this. $\endgroup$ – Ofe Jun 13 '17 at 20:35
  • $\begingroup$ Look for the definition of a wide-sense stationary (WSS) process. Be careful that you do not confuse it with a strict-stationary process as they are different. You'll see that many deterministic processes also have WSS properties; constant mean across all time is one of them. $\endgroup$ – Envidia Jun 13 '17 at 21:35
  • $\begingroup$ Thank you, so I could use g(t) as an WSS ? $\endgroup$ – Ofe Jun 14 '17 at 3:48
  • $\begingroup$ The term "deterministic process" is peculiar. All moments in a WSS are constant. Your deterministic process would be a flat line. Not very interesting . I believe that you just confused the person who asked the question $\endgroup$ – user28715 Jun 14 '17 at 6:58

Let me give a clue $$ E\{ y \} = E\{ x g \} = E\{ x \} g $$

  • $\begingroup$ Is g(t) behave like constant in mean operator ? Could you be more spesific and please write also autocorrelation equation ? $\endgroup$ – Ofe Jun 14 '17 at 3:46
  • $\begingroup$ g(t) is deterministic. E{g(t)} is g(t). Expectations of anything deterministic is just itself. It's just like E{1} g(t). $\endgroup$ – user28715 Jun 14 '17 at 5:23
  • $\begingroup$ If I had any confidence that you would understand what I wrote, I would but since you don't understand the clue I have you, I don't. I think your confusion is rooted in not understanding the difference between an ensemble average and time average. $\endgroup$ – user28715 Jun 14 '17 at 6:45
  • $\begingroup$ Could you also tell me how to find autocovariance and autocorrelation of y(t) ? $\endgroup$ – Ofe Jun 14 '17 at 17:00

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