How is time used in this autocorrelation expression?
The autocorrelation function of a random process $x(t)$ is defined as $$R(t,\tau) = {\mathbb E}\{x(t) x(t+\tau)\}.$$
For a stationary random process, this function does not depend on $t$, i.e., we have $R(t,\tau) = R(\tau) \; \forall t$. The correlation only depends on the time difference between two samples, not the &quot;absolute&quot; time $t$ when the correlation is measured.
For a stationary and ergodic process, ensemble averaging can be replaced by time averaging and we have $$R(\tau) = \lim_{T\rightarrow \infty} \int_T x(t) x(t+\tau) {\rm d}t,$$ which in the discrete case collapses to the sum you were showing.
The expression mentioned by Brown is a function of $t$ and $\tau$, no doubt about that. However, Brown argues (I haven't followed through the arguments, but they should be in the second paper he was citing) that the expression can be used to compute a &quot;narrowed autocorrelation function&quot; (what ever that is) and that if viewed as a function of $\tau$ it has peaks when $\tau$ is equal to multiples of the period $T$ of the periodic signal $f(t)$. This sounds like the particular choice of $t$ is irrelevant, which makes sense if we consider $f(t)$ to be stationary and ergodic. Then again, the notation if not really rigorous. It is worth reading some more though because a few equations later he introduces the bracket notation $\langle \cdot \rangle$ for an average over a period $T$. That would make a lot of sense for his definition of $S(\tau)$ too. In fact, if we do open Brown et al 1989 in equation (1b) he does include the $\langle \cdot \rangle$ notation. Maybe it was forgotten in the other paper?