#### 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

 (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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