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Embedded SystemsFPGAElectronics

Mathematics of the DFT
    Fourier Theorems for the DFT
       Signal Operators
          Convolution
             Graphical Convolution

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Graphical Convolution

As mentioned above, cyclic convolution can be written 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 this expression 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.

Previous: Convolution Example 3: Matched Filtering
Next: Polynomial Multiplication

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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