Calculation of an autocorrelation function 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) &amp; = E\left\{x(t)x(t + \tau)\right\}\\
&amp; = E\left\{\left(A\cos(a) + Bw(t)\right)\left(A\cos(b) + Bw(t+\tau)\right)\right\}\\
&amp; = E\{A^2\cos(a)\cos(b) + AB\cos(a)w(t+\tau) + AB\cos(b)(wt)\\&amp;\quad + B^2w(t)w(t+\tau)\}\\
&amp; = 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\}\\&amp;\quad + E\left\{B^2w(t)w(t+\tau)\right\}\\
&amp; = E\left\{A^2\cos(a)\cos(b)\right\} + E\left\{B^2w(t)w(t+\tau)\right\}\\
&amp; = E\left\{A^2\cos(a)\cos(b)\right\} + B^2\left(R_w(\tau)\right)\\
&amp; = 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) &amp; = E\left\{A^2\cos(a)\cos(b)\right\} + B^2\left(\frac{N_0}{2}\right)(\delta(\tau))\\
&amp; = 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))\\
&amp; = \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.
I guess you've done almost everything right, but have a problem at the calculation of the expectation value regarding $t$. You should calculate the expectation value of the cosine function. Sadly, it does not simply "go away" as you wrote.

Have a look at the Wikipedia page. There you can find another, more explicit, formula for the auto-correlation function of a function $f(t)$:

$R_{\textrm{ff}}(\tau) = \int\limits_{-\infty}^\infty f(t+\tau)\bar{f}(t) \, \textrm{d}t$ .

(Note, that compared to the Wikipedia page, I have taken the liberty to use the variable $t$ in the integration instead of $u$, which would be the mathematically more accurate version.)

As you can see from this equation, you "integrate away" the dependency on t, and indeed you should be left with a function that is independent of $t$.

Note that there is also an version that does not go to infinite times, but is constrained to a period $T$. Maybe this version is more appropriate in your case. However, the same holds for this version: $t$ is integrated away and should not be a variable in the resulting formula.