Convolution Example 1: Smoothing a Rectangular Pulse
Figure 7.3:
Illustration of the convolution of a rectangular pulse
and the impulse response
of an ``averaging filter''
(
).
Filter output signal  .
|
Figure 7.3 illustrates convolution of
with
to get
![$\displaystyle y = x\circledast h = \left[0,0,0,0,\frac{1}{3},\frac{2}{3},1,1,1,1,\frac{2}{3},\frac{1}{3},0,0\right] \protect$](http://www.dsprelated.com/josimages_new/mdft/img1178.png) |
(7.2) |
as graphed in Fig.
7.3(c).
In this case,

can be viewed as a ``moving three-point average''
filter. Note how the corners of the rectangular pulse are ``smoothed''
by the three-point filter. Also note that the pulse is smeared to the
``right'' (forward in time) because the filter impulse response starts
at time zero. Such a filter is said to be
causal (see
[
68] for details). By shifting the impulse response left one
sample to get
(in which case

), we obtain a noncausal filter

which is
symmetric about time zero so that the input signal is smoothed ``in
place'' with no added delay (imagine Fig.
7.3(c) shifted left
one sample, in which case the input pulse edges align with the
midpoint of the rise and fall in the output signal).
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