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62
Signals and Spectra Chap. 2
t
w 3 (t) = 1 e +(l-t)/T dl = T(1 - e -t/T )
L
If t 7 T, then Eq. (2–62b) becomes
T
w 3 (t) = 1 e +(l-t)/T dl = T(e - 1)e -t/T
L
0
0
Thus,
0, t 6 0
w 3 (t) = c T(1 - e -t/T ), 0 6 t 6 T
T(e - 1)e -t/T , t 6 T
See Example2_08.m. This result is plotted in Fig. 2–7.
Example 2–9 SPECTRUM OF A TRIANGULAR PULSE BY CONVOLUTION
In Example 2–7, the spectrum of a triangular pulse was evaluated by using the integral theorem.
The same result can be obtained by using the convolution theorem of Table 2–1. If the rectangular
pulse of Fig. 2–6a is convolved with itself and then scaled (i.e., multiplied) by the constant
1/T, the resulting time waveform is the triangular pulse of Fig. 2–6c. Applying the convolution
theorem, we obtain the spectrum for the triangular pulse by multiplying the spectrum of the rectangular
pulse (of Fig. 2–6a) with itself and scaling with a constant 1/T. As expected, the result is
the spectrum shown in Fig. 2–6c.
Example 2–10 SPECTRUM OF A SWITCHED SINUSOID
In Example 2–5, a continuous sinusoid was found to have a line spectrum with the lines located at
f = ;f 0 . In this example, we will see how the spectrum changes when the sinusoid is switched
on and off. The switched sinusoid is shown in Fig. 2–8a and can be represented by
w(t) =ßa t T bA sin v 0t =ßa t T bA cos av 0t - p 2 b
Using the FT of the rectangular pulse from Table 2–2 and the real-signal translation theorem of
Table 2–1, we see that the spectrum of this switched sinusoid is
W(f) = j A 2 T[Sa (pT(f + f 0)) - Sa (pT(f - f 0 ))]
(2–63)
See Example2_10.m for spectral plots of Eq. (2–63). This spectrum is continuous and imaginary.
The magnitude spectrum is shown in Fig. 2–8. Compare the continuous spectrum of Fig. 2–8 with
the discrete spectrum obtained for the continuous sine wave, as shown in Fig. 2–4a. In addition,
note that if the duration of the switched sinusoid is allowed to become very large (i.e., T : q),
the continuous spectrum of Fig. 2–8 becomes the discrete spectrum of Fig. 2–4a with delta
functions at f 0 and -f 0 .