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

500 RANDOM PROCESSES AND SPECTRAL ANALYSIS
3. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed., McGraw-Hill, New
York, 1984.
4. S. 0. Rice, "Mathematical Analysis of Random Noise," Bell. Syst. Te ch. J. , vol. 23, pp. 282-332, July
1944; vol. 24, pp. 46-156, Jan. 1945.
PROBLEMS
9.1-1 (a) Sketch the ensemble of the random process
x(t) = a cos (w e t+ 0)
where W e and 0 are constants and a is an RV uniformly distributed in the range (-A, A).
(b) Just by observing the ensemble, determine whether this is a stationary or a nonstationary
process. Give your reasons.
9.1-2 Repeat part (a) of Prob. 9.1-1 if a and 0 are constants but W e is an RV uniformly distributed in
the range (0, 100).
9.1-3 (a) Sketch the ensemble of the random process
x(t) = at + b
where b is a constant and a is an RV uniformly distributed in the range (-2, 2).
(b) Just by observing the ensemble, state whether this is a stationary or a nonstationary process.
9.1-4 Determine x(t) and R x (tJ, t 2 ) for the random process in Prob. 9.1-1, and determine whether
this is a wide-sense stationary process.
9.1-5 Repeat Prob. 9.1-4 for the process x(t) in Prob. 9.1-2.
9.1-6 Repeat Prob. 9.1-4 for the process x(t) in Prob. 9.1-3.
9.1-7 Given a random process x(t) = kt, where kis an RV uniformly distributed in the range (-1, 1).
(a) Sketch the ensemble of this process.
(b) Determine x(t).
(c) Determine Rx(t1 , t2).
(d) Is the process wide-sense stationary?
( e) Is the process ergodic?
(f) If the process is wide-sense stationary, what is its power Ps [that is, its mean square value
x 2 (t)?
9.1-8 Repeat Prob. 9.1-7 for the random process
x(t) = a cos (w e t+ 0)
where W e is a constant and a and 0 are independent RVs uniformly distributed in the ranges
(-1, I) and (0, 2n), respectively.