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Convolution Theorem for the DTFT

The convolution of discrete-time signals $ x$ and $ y$ is defined as

$\displaystyle (x \ast y)(n) \isdef \sum_{m=-\infty}^\infty x(m)y(n-m).
$

This is sometimes called acyclic convolution to distinguish it from the cyclic convolution used for length $ N$ sequences in the context of the DFT [248]. Convolution is cyclic in the time domain for the DFT and FS cases (i.e., whenever the time domain has a finite length), and acyclic for the DTFT and FT cases.3.6.

The convolution theorem is then

$\displaystyle \zbox {(x\ast y) \;\longleftrightarrow\;X \cdot Y}
$

That is, convolution in the time domain corresponds to multiplication in the frequency domain.



Proof: The result follows immediately from interchanging the order of summations associated with the convolution and DTFT:

\begin{eqnarray*}
\hbox{\sc DTFT}_\omega(x\ast y) &\isdef & \sum_{n=-\infty}^{\i...
...ad\mbox{(by the shift theorem)}\\
&\isdef & X(\omega)Y(\omega)
\end{eqnarray*}


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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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