Correlation Theorem for the DTFT

We define the correlation of discrete-time signals $x$ and $y$ by

$$\zbox {(x\star y)_n \isdefs \sum_m \overline{x(m)} y(m+n)}$$

The correlation theorem for DTFTs is then

$$\zbox {x\star y \;\longleftrightarrow\;\overline{X}\cdot Y}$$

Proof:

$$\begin{eqnarray*} (x\star y)_n &\isdef & \sum_m \overline{x(m)}y(n+m) \\ &=& \sum_m \overline{x(-m)}y(n-m) \qquad (m\leftarrow -m)\\ &=& \left(\hbox{\sc Flip}(\overline{x})\ast y\right)_n \\ &\;\longleftrightarrow\;& \overline{X} \cdot Y \end{eqnarray*}$$

where the last step follows from the convolution theorem of §2.3.5 and the symmetry result $\hbox{\sc Flip}(\overline{x}) \;\longleftrightarrow\; \overline{X}$ of §2.3.2.