Graphical Convolution

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

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

where $x,y\in{\bf C}^N$ and $h\in{\bf R}^N$. We may interpret $h$ as the impulse response of a digital filter (hence the notation) [56]. It is instructive to interpret the last expression above graphically.

Figure: Illustration of convolution of $y=[1,1,1,1,0,0,0,0]$ and ``matched filter'' $h=[1,0,0,0,0,1,1,1]$ ($N=8$).

Figure 7.3 illustrates convolution of

$$\begin{eqnarray*} y&=&[1,1,1,1,0,0,0,0] \\ h&=&[1,0,0,0,0,1,1,1] \end{eqnarray*}$$

to get

$$y\ast h = [4,3,2,1,0,1,2,3] \protect$$ (7.2)

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

   FLIP$$(h) = [1,1,1,1,0,0,0,0]$$

which is the same as $y$. When $h=$FLIP$(y)$, we say that $h$ is matched filter for $y$.^7.3 In this case, $h$ is matched to look for a ``dc component,'' and also zero-padded by a factor of $2$. The zero-padding serves to simulate acyclic convolution using circular convolution. Note from Eq. (7.2) that the maximum is obtained in the convolution output at time 0. This peak (the largest possible if all input 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 various sinusoids at other frequencies and/or noise were added in.