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Figure 7.5:
Illustration of convolution of
and ``matched filter''
(
).
![\includegraphics[width=2.5in]{eps/conv}](img1178.png) |
Figure 7.5 illustrates convolution of
to get
![$\displaystyle y\circledast h = [4,3,2,1,0,1,2,3]. \protect$](img1180.png) |
(7.3) |
For example,
could be a ``rectangularly windowed signal, zero-padded by
a factor of 2,'' where the signal happened to be dc (all
s).
For the
convolution, we need
which is the same as
. When
, we say that
is a
matched filter for
.7.7 In this case,
is matched to look for a
``dc component,'' and also zero-padded by a factor of
. 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
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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