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Correlation Theorem for the DTFT

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

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

The correlation theorem for DTFTs is then

$\displaystyle \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.


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