# Graphical Convolution

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

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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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### Graphical Convolution

Note that the cyclic convolution operation can be expressed in terms of previously defined operators as

where and . It is instructive to interpret the last expression above graphically.

Figure 8.3 illustrates convolution of

For example, could be a ''rectangularly windowed signal, zero-padded by a factor of 2,'' where the signal happened to be dc (all s). For the convolution, we need

which is the same as . When , we say that ismatched filter for .8.3 In this case, is matched to look for a ''dc component,'' and also zero-padded by a factor of . The zero-paddingserves to simulate acyclic convolution using circular convolution. The figure illustrates the computation of the convolution of and :

Note that a large peak is obtained in the convolution output at time . This large peak (the largest possible if all signals are limited to in magnitude), indicates the matched filter has ''found'' the dc signal starting at time . This peak would persist even if varioussinusoids at other frequencies and/or noise were added in.

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