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Convolution

The convolution of two signals $ x$ and $ y$ in $ {\bf C}^N$ may be denoted `` $ x\circledast y$ '' and defined by

$\displaystyle \zbox {(x\circledast y)_n \isdef \sum_{m=0}^{N-1}x(m) y(n-m)}
$

Note that this is circular convolution (or ``cyclic'' convolution).7.4 The importance of convolution in linear systems theory is discussed in §8.3.

Cyclic convolution can be expressed in terms of previously defined operators as

$\displaystyle y(n) \isdef (x\circledast h)_n \isdef \sum_{m=0}^{N-1}x(m)h(n-m) =
\left<x,\hbox{\sc Shift}_n(\hbox{\sc Flip}(h))\right>$   $\displaystyle \mbox{($h$\ real)}$

where $ x,y\in{\bf C}^N$ and $ h\in{\bf R}^N$ . This expression suggests graphical convolution, discussed below in §7.2.4.



Subsections
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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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