Autocorrelation

The autocorrelation of a signal $x \in \mathbb{C}^N$ is simply the cross-correlation of $x$ with itself:

$$(x \star x)(n) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\sum_{m=0}^{N-1}\overline{x}(m)x(m+n), \quad x \in \mathbb{C}^N$$

From the correlation theorem, we have

$$\zbox{(x \star x) \leftrightarrow \vert X(\omega_k)\vert^2}$$