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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 and in is denoted '''' 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 , and in the second step, we made use of the fact that any sum over all terms is equivalent to a sum from to .

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Subsections

• Graphical Convolution
• Polynomial Multiplication
• Multiplication of Decimal Numbers

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