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Random Processes and Spectral Analysis Chap. 6
★ 6–45 A USSB signal is
v(t) = 10 Re{[x(t) + jxN(t)]e j(v ct+u c ) }
where u c is a random variable that is uniformly distributed over (0, 2p). The PSD for x(t) is given
in Fig. P6–31. Find
(a) The PSD for v(t).
(b) The total power of v(t).
6–46 Show that R xN (t) = R x (t).
6–47 Show that R xxN (t) = RN x (t),
where the caret symbol denotes the Hilbert transform.
6–48 A bandpass random signal can be represented by
s(t) = x(t) cos(v c t + u c ) - y(t) sin(v c t + u c )
where the PSD of s(t) is shown in Fig. P6–48. u c is an independent random variable that is uniformly
distributed over (0, 2p). Assume that f 3 - f 2 = f 2 - f 1 . Find the PSD for x(t) and y(t) when
(a) f c = f 1 . This is USSB signaling, where y(t) = xN(t).
(b) f c = f 2 . This represents independent USSB and LSSB signaling with two different
modulations.
(c) f 1 6 f c 6 f 2 . This is vestigial sideband signaling.
(d) For which, if any, of these cases are x(t) and y(t) orthogonal?
p s (f)
A
–f 3 –f 2 –f 1 f 1 f 2 f 3
f
Figure P6–48
★ 6–49 Referring to Prob. 6–48(b), how are the two modulations m 1 (t) and m 2 (t) for the independent
sidebands related to x(t) and y(t)? Give the PSD for m 1 (t) and m 2 (t), where m 1 (t) is the modulation
on the USSB portion of the signal and m 2 (t) is the modulation on the LSSB portion
of s(t).
6–50 For the bandpass random process, show that Equation (6–133m) is valid (property 13).
6–51 For the bandpass random process, show that Equation (6–133i) is valid (property 9).
★ 6–52 Referring to Example 6–12, find the PSD for a BPSK signal with Manchester-encoded data. (See
Fig. 3–15.) Assume that the data have values of a n =;1 which are equally likely and that the data
are independent from bit to bit.
6–53 The input to an envelope detector is an ergodic bandpass Gaussian noise process. The RMS value
of the input is 2 V and the mean value is 0 V. The envelope detector has a voltage gain of 10. Find
(a) The DC value of the output voltage.
(b) The RMS value of the output voltage.