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60
Signals and Spectra Chap. 2
Using Table 2–2, we find that the FT pair for the second derivative is
d 2 w(t)
dt 2
4 1 T ejvT - 2 T + 1 T e-jvT
which can be rewritten as
Referring to Table 2–1 and applying the integral theorem twice, we get the FT pair for the original
waveform:
Thus,
d 2 w(t)
dt 2 4 1 T 1 e jvT/2 - e -jvT/222 = -4 (sin pfT)2
T
w(t) 4 -4
T
(sin pfT) 2
(j2pf) 2
w(t) =a t T b 4 T Sa2 (pfT)
(2–61)
See Example2_07.m. This is illustrated in Fig. 2–6c.
Convolution
The convolution operation, as listed in Table 2–1, is very useful. Section 2–6 shows how the
convolution operation is used to evaluate the waveform at the output of a linear system.
DEFINITION. The convolution of a waveform w 1 (t) with a waveform w 2 (t) to produce
a third waveform w 3 (t) is
q
w 3 (t) = w 1 (t) * w 2 (t) ! w 1 (l) w 2 (t - l) dl
L
-q
(2–62a)
where w 1 (t)* w 2(t) is a shorthand notation for this integration operation and * is read
“convolved with.”
When the integral is examined, we realize that t is a parameter and l is the variable of integration.
If discontinuous waveshapes are to be convolved, it is usually easier to evaluate the
equivalent integral
q
w 3 (t) = w 1 (l) w 2 (-(l - t)) dl
L
-q
(2–62b)
Thus, the integrand for Eq. (2–62b) is obtained by
1. Time reversal of w 2 to obtain w 2 (-l),
2. Time shifting of w 2 by t seconds to obtain w 2 (-(l - t)), and
3. Multiplying this result by w 1 to form the integrand w 1 (l) w 2 (-(l - t)).
These three operations are illustrated in the examples that follow.