A sample of a random process is given as:
$$ x(t) = A\cos(2\pi f_0t) + Bw(t) $$
where $w(t)$ is a white noise process with $0$ mean and a power spectral density of $\frac{N_0}{2}$, and $f_0$, $A$ and $B$ are constants. Find the auto-correlation function.
Here's my attempt at a solution:
Let $a = 2\pi f_0t$, and $b = 2\pi f_0(t+\tau)$
\begin{align} \text{Autocorrelation of } x(t) & = E\left\{x(t)x(t + \tau)\right\}\\ & = E\left\{\left(A\cos(a) + Bw(t)\right)\left(A\cos(b) + Bw(t+\tau)\right)\right\}\\ & = E\{A^2\cos(a)\cos(b) + AB\cos(a)w(t+\tau) + AB\cos(b)(wt)\\&\quad + B^2w(t)w(t+\tau)\}\\ & = E\left\{A^2\cos(a)\cos(b)\right\} + E\left\{AB\cos(a)w(t+\tau)\right\} + E\left\{AB\cos(b)(wt)\right\}\\&\quad + E\left\{B^2w(t)w(t+\tau)\right\}\\ & = E\left\{A^2\cos(a)\cos(b)\right\} + E\left\{B^2w(t)w(t+\tau)\right\}\\ & = E\left\{A^2\cos(a)\cos(b)\right\} + B^2\left(R_w(\tau)\right)\\ & = E\left\{A^2\cos(a)\cos(b)\right\} + B^2\left(\frac{N_0}{2}\right)(\delta(\tau))\\ \end{align}
The expectation terms with the noise in them all equal $0$ (the last is just the auto correlation of white noise ... hence the simplification above. Using trigonometric identities: $$ \cos(a)\cos(b) = \frac 12\left[\cos(a + b) + \cos(a - b)\right] $$
we have:
\begin{align} \text{Autocorrelation of } x(t) & = E\left\{A^2\cos(a)\cos(b)\right\} + B^2\left(\frac{N_0}{2}\right)(\delta(\tau))\\ & = E\left\{\left(A^2\right)\frac 12\left[\cos(a+b)+\cos(a-b)\right]\right\} + B^2\left(\frac{N_0}{2}\right)(\delta(\tau))\\ & = \left(\frac{A^2}{2}\right)\left[E\{\cos(a+b)\} + E\{\cos(a-b)\}\right] + B^2\left(\frac{N_0}{2}\right)(\delta(\tau))\\ \end{align}
We're dealing with constant terms, so expectation term goes away and subbing in our initial conditions we get: $$ \frac {A^2}2 \left[\cos(2\pi f_o(2t + \tau) + \cos(2\pi f_o\tau)\right] + B^2\left(\frac{N_0}{2}\right)(\delta(\tau)) $$
For some reason I can't help but feel I did something incorrectly calculating that autocorrelation ... it's supposed to be a function of $\tau$, but has a $t$ is in there ... I would very much appreciate it if someone could point me in the right direction, or explain what I messed up. I don't know whether it matters, but in this class we're dealing with only wide sense stationary processes.