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Convolution

The cyclic convolution of $ x$ and $ y$ is defined as

$\displaystyle (x * y)(n) \mathrel{\stackrel{\Delta}{=}}\sum_{m=0}^{N-1}x(m)y(n-m), \quad x,y \in \mathbb{C}^N
$

Cyclic convolution is also called circular convolution, since $ y(n-m) \mathrel{\stackrel{\mathrm{\Delta}}{=}}y(n-m\left(\mbox{mod}\;N\right))$ .

Convolution is cyclic in the time domain for the DFT and FS cases, and acyclic for the DTFT and FT cases.

The Convolution Theorem is then

$\displaystyle \zbox{(x*y) \leftrightarrow X \cdot Y}
$


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``Review of the Discrete Fourier Transform (DFT)'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2020-06-27 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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