Convolution

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Convolution

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Convolution



Definition: The convolution of two signals $x$ and $y$ in ${\ is denoted ''$x\'' and defined by

\

Note that this is cyclic or ''circular'' convolution.8.2 The importance of convolution in linear systems theory is discussed in §8.7

Convolution is commutative, i.e.,

\

Proof:

\

where in the first step we made the change of summation variable $l\, and in the second step, we made use of the fact that any sum over all $N$ terms is equivalent to a sum from $0$ to $N-1$.


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