Graphical Convolution

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Graphical Convolution

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Graphical Convolution

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

\

where $x,y\ and $h\. It is instructive to interpret the last expression above graphically.

Figure 8.3:Illustration of convolution of $y=[1,1,1,1,0,0,0,0]$ and its “matched filter” h=1,0,0,0,0,1,1,1.
\

Figure 8.3 illustrates convolution of

\



For example, $y$ could be a “rectangularly windowed signal, zero-padded by a factor of 2,” where the signal happened to be dc (all $1$s).
For the convolution, we need

\

which is the same as $y$. When $h=\, we say that $h$ ismatched filter for $y$.8.3 In this case, $h$ is matched to look for a “dc component,” and also zero-padded by a factor of $2$. The zero-paddingserves to simulate acyclic convolution using circular convolution. The figure illustrates the computation of the convolution of $y$ and $h$:

\

Note that a large peak is obtained in the convolution output at time $0$. This large peak (the largest possible if all signals are limited to$[-1,1]$ in magnitude), indicates the matched filter has “found” the dc signal starting at time $0$. This peak would persist even if varioussinusoids at other frequencies and/or noise were added in.

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