Why is Autocorrelation between a Zero-mean Random process and a finite deterministic sequence zero?

The Solution is given above:

The Question is, how did the $$\mathbb{E}{[x(k)f(l)]}$$ and $$\mathbb{E}{[x(l)f(k)]}$$ become zero? is there some rule that correlation between Random Process and Deterministic sequence is zero?

• In a sea of poorly asked "help with my homework" questions, this one is done well. Congrats, and thank you. Oct 17, 2022 at 19:41
• Note that $E\{x(k)f(l)\}$ is the plain old correlation between them -- autocorrelation in the correlation of a signal with itself. Oct 17, 2022 at 19:44

Since $$f(n)$$ is deterministic, we have

$$E\{x(m)f(n)\}=f(n)E\{x(m)\}\tag{1}$$

So the value of $$E\{x(m)f(n)\}$$ is simply $$f(n)$$ times the mean of $$x(n)$$, and since the mean of $$x(n)$$ is zero, so is $$E\{x(m)f(n)\}$$.

• Even though Tim’s answer is also valid, this looks like it should be the accepted answer.
– Jdip
Oct 17, 2022 at 21:00
• yes, this makes sense mathematically. Oct 17, 2022 at 21:06

$$E\{x(k)f(l)\} = 0$$ because $$x(n)$$ and $$f(n)$$ are uncorrelated.

Unless otherwise stated, in your coursework you can assume that any given random process is uncorrelated with another, or with a deterministic signal. The only reason that a signal and a random process may be correlated is because there's some causal link between them -- i.e., the deterministic signal is being generated in part by the random signal, or the random signal is being generated in part by the deterministic one.

• The only reason why $E\{x(m)f(n)\}=0$ is because $E\{x(n)\}=0$. If $E\{x(n)\}\neq 0$ then also $E\{x(m)f(n)\}\neq 0$ (unless $f(n)=0$). Oct 17, 2022 at 20:23
• Please explain how that is true in the case where $x(m) = \sin m$ and $f(m) = \sin m$. Oct 17, 2022 at 23:24
• $x(n)$ is a random signal, so it can't be $\sin(n)$. It could be $\sin(n+\phi)$ with some random phase $\phi$. If that random phase has a uniform distribution then you would have $E\{x(n)\}=0$ and, consequently, $E\{x(m)f(n)\}=0$. Oct 18, 2022 at 6:21