ECE 534 Spring 2008
April 3, 2008
Homework Assignment 5
Due Date: Wednesday, April 16 (in class)
Reading: Chapter 5 of text, Sections 5.1-5.4. Also read the solutions to the even numbered problems
of Chapter 5 given at the end of the book.
Note: Exam 2 will held on April 16, from 7-8:30 PM in room 165 Everitt Lab. You will be allowed
two sheets of notes (8.5”x11”, both sides).
1. Problem 5.1.
2. Problem 5.5.
3. Problem 5.7. Hint: Exploit the continuity of RX(t,s) on T×Tto show that EYt+−Yt
−Xt2
goes to zero as →0.
4. Problem 5.11.
5. Problem 5.15
6. Problem 5.17. Hint: Look at Example 5.4.
7. (From Exam 2, Spring ’06) Let {Xt}be a continuous-time, zero-mean Gaussian random process
with autocorrelation function
RX(t, s) = 2ts +t2s2
(a) Does the random process defined by Yt=X0
texist in a mean square sense? Clearly explain
your answer.
(b) Find the mean and autocorrelation functions of {Yt}. Do these functions completely specify
all finite-order distributions of {Yt}? Clearly explain your answer.
8. (From Exam 2, Spring ’06): Let {Xk}be a discrete-time stationary zero-mean Gaussian process
with autocorrelation function RX(k) = α|k|, where αis a constant with |α|<1. Further, let A
be a Gaussian random variable with mean zero and variance 1 and independent of X. Determine
whether or not each of the following processes is mean ergodic in the mean square sense:
(a) Yk=Xk+A.
(b) Zk=AXk.
c
V. V. Veeravalli, 2008 1