**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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 abovegraphically.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 filterfor .^{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.