Graphical Convolution Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


Graphical Convolution

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

$\displaystyle y(n) \isdef (x\circledast h)_n \isdef \sum_{m=0}^{N-1}x(m)h(n-m) =
\left<x,\hbox{\sc Shift}_n(\hbox{\sc Flip}(h))\right>$   $\displaystyle \mbox{($h$\ real)}$

where $ x,y\in{\bf C}^N$ and $ h\in{\bf R}^N$. It is instructive to interpret the last expression above graphically, as depicted in Fig.7.5 above. The convolution result at time $ n=0$ is the inner product of $ x$ and $ \hbox{\sc Flip}(h)$, or $ y(0)=\left<x,\hbox{\sc Flip}(h)\right>$. For the next time instant, $ n=1$, we shift $ \hbox{\sc Flip}(h)$ one sample to the right and repeat the inner product operation to obtain $ y(1)=\left<x,\hbox{\sc Shift}_1(\hbox{\sc Flip}(h))\right>$, and so on. To capture the cyclic nature of the convolution, $ x$ and $ \hbox{\sc Shift}_n(\hbox{\sc Flip}(h))$ can be imagined plotted on a cylinder. Thus, Fig.7.5 shows the cylinder after being ``cut'' along the vertical line between $ n=N-1$ and $ n=0$ and ``unrolled'' to lay flat.


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[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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