B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

502 RANDOM PROCESSES AND SPECTRAL ANALYSIS
Figure P.9.2-8
t ll 1 t t 1 tt1 1 11 t1 ill 1
t
--+-
9.2-9 Repeat Prob. 9 .2-8 if the impulses are equally likely to be positive and negative.
9.2-10 A sample function of a random process x(t) is shown in Fig. P9.2-10. The signal x(t) changes
abruptly in amplitude at random instants. There are an average of f3 amplitude changes (or
shifts) per second. The probability that there will be no amplitude shift in r seconds is given by
Po(r) = e - f3r. The amplitude after a shift is independent of the amplitude before the shift. The
amplitudes are randomly distributed, with a PDF Px (x). Show that
and
2{Jx 2
Sx if) = {32 + (2 :rr.f) 2
This process represents a model for thermal noise. 1
Figure
P.9.2-10
n
n r1
t--+-
9.3-1 Show that for jointly wide-sense stationary, real, random processes x(t) and y(t),
IRxy(r)I 'S [Rx(0)Ry (0)J 1 12
Hint: For any real number a, (ax - y) 2 2: 0.
9.3-2 If x(t) and y(t) are two incoherent random processes, and two new processes u(t) and v(t) are
formed as follows:
u(t) = 2x(t) - y(t)
v(t) = x(t) + 3y(t)
find Ru (r), Rv (r), Ruv (r), and R vu (r) in terms of R x (r) and Ry (r).
9.3-3 Two random processes x(t) and y(t) are
x(t) = A cos (wot + rp) and y (t) = B sin (ncvot + nrp + ifr)
where n = integer =/: 0 and A, B, ifr, and wo are constants and rp is an RV uniformly distributed
in the range (0, 2:rr). Show that the two processes are incoherent.
9.3-4 A sample signal is a periodic random process x(t) shown in Fig. P9.3-4. The initial delay b where
the first pulse begins is an RV uniformly distributed in the range (0, T b )-
(a) Show that the sample signal can be written as
x(t) = Co + L C n cos [nwo (t - b) + 0 n ]
n=l