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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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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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