Autocorrelation of cosine with Uniform random variable

Question

I have $Z$, which is a random variable with a Uniform(0,$2\pi$) distribution, and I need to find the autocorrelation function of $Y=cos(w_0t+Z)$, where $w_0$ is a constant. I understand that:

$R_Y(t,t+\tau)=E(cos(w_0t+Z)cos(w_0(t+\tau+Z)))=\frac{1}{2}(E(cos(w_0\tau)+cos(w_0(2t+\tau+2Z)))$

And I also know that I'm supposed to get:

$R_Y=\frac{1}{2}cos({w_0\tau})$

But I don't understand how this calculation goes. Can someone help me?

Answer 1

I think that the definition you want to apply also involves an average over a long time period. In other words, the meaning of E in your formulas is

$$ E(f(t,Z)) = \lim_{T \rightarrow \infty} \frac{1}{2 \pi T} \int_{0}^{2 \pi}\, dZ \int_{-T}^{T} dt \ f(t,Z)$$

from which you see that $E(\cos(w_0(2t+\tau+2Z)))=0$, so your expected result is recovered.