563489578934

122
Signals and Spectra Chap. 2
2–36 Let x(t) =ßa t - 0.05 b. Plot the spectrum of x(t) using MATLAB with the help of
0.1
Eqs. (2–59) and (2–60). Check your results by using the FFT and Eq. (2–184).
★ 2–37 If w(t) = w 1 (t)w 2 (t), show that
where W(f) = [w(t)].
2–38 Show that
(a) 1 t
- q w(l)dl = w(t) * u(t).
(b) 1 t
- qw(l)dl 4 (j2pf) -1 W(f) + 1 W(0) d(f).
2
(c) w(t) * d(t - a) = w(t - a).
2–39 Show that
[Hint: Use Eq. (2–26) and integrate by parts. Assume that w(t) is absolutely integrable.]
2–40 As discussed in Example 2–9, show that
2–41 Given the waveform w(t) = Aß(t/T) sin v 0 t, find the spectrum of w(t) by using the multiplication
theorem as discussed in Example 2–10.
★ 2–42 Evaluate the following integrals.
q
sin 4l
(a)
d (t - l) dl.
L-q
4l
q
(b) (l 3 - 1) d (2 - l) dl.
L-q
2–43 Prove that
2–44 Evaluate y(t) = w 1 (t) * w 2(t), where
and
q
W(f) = W 1 (l)W 2 (f - l) dl
L
dw(t)
1
T ßa t T b * ßa t T b =a t T b
M(f) * d(f - f 0 ) = M(f - f 0 )
w 1 (t) = e 1, |t| < T 0
0, t elsewhere
w 2 (t) = e [1 - 2|t|], |t| < 1 T 2 0
0, t elsewhere
2–45 Given w(t) = 5 + 12 cos v 0 t, where f 0 = 10 Hz, find
(a) R w (t).
(b) P w (f).
2–46 Given the waveform
dt
-q
4 (j2pf)W(f)
w(t) = A 1 cos (v 1 t + u 1 ) + A 2 cos (v 2 t + u 2 )