Correlation Theorem for the DTFT
We define the correlation of discrete-time signals
and
by
![$\displaystyle \zbox {(x\star y)_n \isdefs \sum_m \overline{x(m)} y(m+n)}
$](http://www.dsprelated.com/josimages_new/sasp2/img169.png)
The correlation theorem for DTFTs is then
![$\displaystyle \zbox {x\star y \;\longleftrightarrow\;\overline{X}\cdot Y}
$](http://www.dsprelated.com/josimages_new/sasp2/img170.png)
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*}](http://www.dsprelated.com/josimages_new/sasp2/img171.png)
where the last step follows from the convolution theorem of
§2.3.5 and the symmetry result
of §2.3.2.
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Convolution Theorem for the DTFT