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

Problems 453
where M = ab - c 2 . Show that Px (x), py (y),Pxly (xly), and Pylx (y lx) are all Gaussian and that
x 2 = b, y 2 = a, and xy = c.
Hint: Use
8.2-10 The joint PDF of RVs x and y is given by
Pxy(x, y) = ke - < x2 +xy+y 2 )
Determine: (a) the constant k; (b) Px(x); (c) py (y) ; (d) Pxty(x, y); (e) Pylx (ylx). Are x and y
independent?
8.2-11 In the example on threshold detection (Example 8.16), it was assumed that the digits 1 and 0
were transmitted with equal probability. If P x (1) and P x (0), the probabilities of transmitting 1
and 0, respectively, are not equal, show that the optimum threshold is not O but is a, where
a =
a; Px (O) ln
2A p
P x (l)
Hint: Assume that the optimum threshold is a, and write P e in terms of the Q functions. For the
optimum case, dP e /da = 0. Use the fact that
and
I ix 2
Q(x) = 1---
e - y l 2 d y
../iii -00
--
dQ(x) I -x 2 /2
= ---e
dx
../iii
8.3-1 If an amplitude x of a Gaussian signal x(t) has a mean value of 2 and an RMS value of 3,
determine its PDF.
8.3-2 Determine the mean, the mean square, and the variance of the RV x in Prob. 8.2-3.
8.3-3 Determine the mean and the mean square value of RV x in Prob. 8.2-4.
8.3-4 Determine the mean and the mean square value of RV x in Prob. 8.2-6.
8.3-5 Find the mean, the mean square, and the variance of the RV x in Fig. P8.3-5.
Fi g ure P.8.3-5
Px( x )
-1 3