Graphical Convolution

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

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

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.