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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

y&=&[1,1,1,1,0,0,0,0] \\

to get

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

$\displaystyle \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].

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University