autocorrelation

The autocorrelation of a function over time represents how much we know about the function at the current time plus some delay from the value at the current time.

The autocorrelation of a function \(f\), \(A_f\) is

\begin{equation} A_f(t, t + \Delta t) = \mathbb{E}\{f^*(t)f(t + \Delta t)\} \end{equation}

where the \(*\) represents complex conjugation.

For a wide-sense stationary stochastic process, we can do away with the dependence on current time, and focus only on time delay

\begin{equation} A_{f}(\Delta t) = \mathbb{E}\{f^*(t)f(t + \Delta t)\} \end{equation}

The fourier transform of the autocorrelation is the power spectral density by the Wiener-Khinchin theorem[STUB].

Autocorrelation is highest at 0.

So, a very highly varying signal may have an autocorrelation pulse only at 0 since you can't predict much about even points very close ahead in time.

The autocovariance is obtained from subtracting the product of the means at each time, or at the same time in the case of a WSS signal.