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120
w(t)
Signals and Spectra Chap. 2
4
3
2
1
–4 –3 –2 –1 1 2 3 4
t
Figure P2–19
(c) x(t) = 2 dw(t) .
dt
(d) x(t) = w(1 - t).
2–21 From Eq. (2–30), find w(t) for W(f) = Aß(f/2B), and verify your answer by using the duality
property.
2–22 Find the quadrature spectral functions X(f) and Y(f) for the damped sinusoidal waveform
w(t) = u(t)e -at sin v 0 t
where u(t) is a unit step function, a 7 0, and W(f) = X(f) + jY(f).
2–23 Derive the spectrum of w(t) = e -|t|/T .
★ 2–24 Find the Fourier transforms for the following waveforms. Plot the waveforrns and their magnitude
spectra. [Hint: Use Eq. (2–184).]
(a) ßa t - 3 b.
4
(b) 2.
(c) a t - 5 b.
5
2–25 By using Eq. (2–184), find the approximate Fourier transform for the following waveform:
sin(2pt/512) + sin(70pt512), 5 6 t 6 75
x(t) = e
0, t elsewhere
2–26 Evaluate the spectrum for the trapezoidal pulse shown in Fig. P2–26.
2–27 Show that
-1 {[w(t)]} = w(t)
[Hint: Use Eq. (2–33).]
2–28 Using the definition of the inverse Fourier transform, show that the value of w(t) at t = 0 is equal
to the area under W(f). That is, show that
q
w(0) = W(f) df
L
-q