I have the following system:
$$S(t) = X(t) \cdot d(t)$$
where
$R_{XX}(\tau)=\frac{1}{2\pi}Sa(\frac{\tau}{2})cos(3\tau)$ and $d(t)=cos(2t + \theta)$ where $\theta \sim U[0,2\pi)$ and is independent of $X(t)$
I would like to compute $R_{SS}(\tau)$.
Normally, I would think to be given the autocorrelation of $d$, then I could apply the Winer-Khintchine theorem and using the fact that multiplication in time domain is convolution in frequency domain. But, I am stumped where to begin given that I am given the autocorrelation of X(t) but not d(t).
$R_{SS}(t,\tau)=E[S(t)S(t+\tau)]$
$=E[X(t)d(t)X(t+\tau)d(t+\tau)]$
$=E[X(t)X(t+\tau) d(t)d(t+\tau)]$
$=E[X(t)X(t+\tau)]E[d(t)d(t+\tau)]$ via independence
$=R_{SS}(\tau)E[d(t)d(t+\tau)]$
Attempting to solve for $R_{dd}=E[d(t)d(t+\tau)]$,
$E[cos(2t+\theta)cos(2(t+\tau)+\theta)]$
Using product identities, I get,
$\frac{1}{2}E[cos(4t + 2\theta + 2\tau)]$ which I believe is zero.