Correlation

The cross-correlation of $x$ and $y$ in $\mathbb{C}^N$ is defined as:

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

Using this definition we have the correlation theorem:

$$\zbox{(x \star y) \leftrightarrow \overline{X(\omega_k)}Y(\omega_k)}$$

The correlation theorem is often used in the context of spectral analysis of filtered noise signals.