Convolution Example 1: Smoothing a Rectangular Pulse

Filter
input signal
.
Filter impulse response
.
Filter output signal
. |

Figure 7.3 illustrates convolution of

with

to get

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

(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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University