**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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:Theconvolutionof two signals and in is denoted '''' and defined by

Note that this iscyclicor ''circular'' convolution.^{8.2}The importance of convolution inlinear systems theoryis discussed in §8.7Convolution is

commutative, i.e.,

Proof:

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

Subsections