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Sec. 2–2 Fourier Transform and Spectra 61
Example 2–8 CONVOLUTION OF A RECTANGLE WITH AN EXPONENTIAL
Let
t - 1 2 T
w 1 (t) =ßP Q and w 2 (t) = e -t/T u(t)
T
as shown in Fig. 2–7. Implementing step 3 with the help of the figure, the convolution of w 1 (t)
with w 2 (t) is 0 if t 6 0 because the product w 1 (l) w 2 (-(l - t)) is zero for all values of l. If
0 6 t 6 T, Eq. (2–62b) becomes
w 1 (Ò)
1
–T T 2T
Ò
w 2 (Ò)
–T T 2T
Ò
w 2 (–(Ò-t))
–T t T
2T
Ò
T
0.63T
w 3 (t)
–T T 2T
t
Figure 2–7
Convolution of a rectangle and an exponential.