Convolution Example 3: Matched Filtering

Figure 7.5: Illustration of convolution of $y=[1,1,1,1,0,0,0,0]$ and ``matched filter'' $h=\protect\hbox{\sc Flip}(y)=[1,0,0,0,0,1,1,1]$ ($N=8$ ).

Figure 7.5 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

$$y\circledast h = [4,3,2,1,0,1,2,3]. \protect$$ (7.3)

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

$$\hbox{\sc Flip}(h) = [1,1,1,1,0,0,0,0]$$

which is the same as $y$ . When $h=\hbox{\sc Flip}(y)$ , we say that $h$ is a matched filter for $y$ .^7.7 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.(7.3) 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 in the presence of some amount of noise and/or interference from other signals. Thus, matched filtering is useful for detecting known signals in the presence of noise and/or interference [35].