Free Books
   Free Books    Mathematics of the DFT  

Convolution Example 1: Smoothing a Rectangular Pulse

Figure 7.3: Illustration of the convolution of a rectangular pulse $ x=[0 ,0 ,0 ,0,1,1,1,1,1,1,0,0,0,0]$ and the impulse response of an ``averaging filter'' $ h=[1/3,1/3,1/3,0,0,0,0,0,0,0,0,0,0,0]$ ($ N=14$).
\includegraphics{eps/smoother-x}
Filter input signal $ x(n)$.

\includegraphics{eps/smoother-h}
Filter impulse response $ h(n)$.

\includegraphics{eps/smoother-y}
Filter output signal $ y(n)$.

Figure 7.3 illustrates convolution of

$\displaystyle x = [ 0, 0, 0,0,1,1,1,1,1,1,0,0,0,0] $

with

$\displaystyle h = \left[\frac{1}{3},\frac{1}{3},\frac{1}{3},0,0,0,0,0,0,0,0,0,0,0\right] $

to get

$\displaystyle y = x\circledast h = \left[0,0,0,0,\frac{1}{3},\frac{2}{3},1,1,1,1,\frac{2}{3},\frac{1}{3},0,0\right] \protect$ (7.2)

as graphed in Fig.7.3(c). In this case, $ h$ 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

$\displaystyle h=\left[\frac{1}{3},\frac{1}{3},0,0,0,0,0,0,0,0,0,0,\frac{1}{3}\right] $

(in which case $ \hbox{\sc Flip}(h)=h$), we obtain a noncausal filter $ h$ 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).

Next Section:
Convolution Example 2: ADSR Envelope
Previous Section:
Convolution as a Filtering Operation