In the Name of God, the Compassionate, the Merciful
Stochastic Processes
Sharif University of Technology
Dr. Hamid Reza Rabiee
October 9, 2010
CE 40-695
Homework 2 (Introduction to Stochastic Processes)
Due Saturday, October 16th, 2010
1. Describe a random process that you are likely to encounter in the following situations:
a. listening to the daily weather forecast
b. paying the monthly telephone bill
c. leaving for work in the morning
Why is each process a random one?
2 ) and u[n] is White
2. A random process is defined as x[n] = Au[n] for all n, where A ∼ N (0, σA
2
Gaussian Noise with variance σu , and A is independent of u[n] for all n. Find the mean and
covariance sequences. What type of random process is x[n]?
3. The random processes x[n] and y[n] are both WSS. Every sample of x[n] is independent of every
sample of y[n]. Is z[n] = x[n] + y[n] WSS? If it is WSS, find its mean and autocorrelation
sequences. What about z[n] = x[n]y[n]?
4. Suppose that a random process x(t) is wide-sense stationary with autocorrelation Rxx (t, t + z) =
exp{−|z|/2}.
a. Find the second moment of x(5).
b. Find the second moment of x(5) − x(3).
5. Consider a random process x(t) defined by x(t) = U cos t + (V + 1) sin t, −∞ < t < ∞ where U
and V are independent R.Vs for which E(U ) = E(V ) = 0, E(U 2 ) = E(V 2 ) = 1
a. Find the autocovariance function Cxx (t, s).
b. Is x(t) WSS?
6. Consider a random process x(t) defined by x(t) = U cos(wt) + V sin(wt). − ∞ < t < ∞, where
w is constant and U and V are R.Vs.
a. Show that the condition E(U ) = E(V ) = 0 is necessary for x(t) to be stationary.
b. Show that x(t) is WSS if and only if U and V are uncorrelated with equal variances.
7. Show that if a normal process is WSS, then it is also strict-sense stationary.
8. Show that a random process which is stationary to order n is also stationary to all orders lower
than n.
9. Solve these problems from the Papoulis book :
9.4, 9.7, 9.8, 9.13, 9.14, 9.19, 9.20, 9.21, 9.22
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