University of Illinois at Urbana-Champaign
Department of Electrical and Computer Engineering
ECE 534 Random Processes
Spring 2005
Problem Set 6
Ergodicity, Karhunen-Loeve Expansions, Random Processes through Linear
Systems, Wiener Filtering
Issued: Wednesday, April 20th Due: Beginning of lecture on Wednesday, May 4
Reading from Hajek: Chapter 5, Sections 5.4–5.6; Chapter 6 (in particular, Sections 6.1–6.3);
Chapter 7 (in particular, Sections 7.1–7.4).
Reading from Stark and Woods: Chapter 8, Section 8.4–8.5; Chapter 7, Sections 7.3–7.5;
Chapter 9, Section 9.3.
Announcement: The Final Exam is scheduled for Thursday, May 12th, from 1:30-4:30pm
in room 1214 Siebel. The exam is closed-book but you can bring three 8.5×11-inch double-
sided sheets of handwritten notes. Calculators are allowed but will not be necessary. The final
exam covers everything from the beginning of the course up to (and including) the lecture on
Wednesday, May 4th and up to (and including) Problem Set 6.
Problem 6.1
This problem concerns ergodicity for random processes.
(a) Let X(t) be a wide-sense stationary Gaussian random process with zero-mean and corre-
lation function
RX(τ) = σ2e−α|τ|cos(2πf τ ),
where σ2,αand fare all constants. Show that X(t) is ergodic in the mean.
(b) Assume that a wide-sense stationary random process X(t) is ergodic in the mean and
that the limit of KX(τ) as τ→ ∞ exists. Show that
lim
|τ|→∞ KX(τ) = 0 .
Problem 6.2
From Hajek, Chapter 5: Problems 5 and 8.
Problem 6.3 (optional)
From Hajek, Chapter 5: Problems 7 and 9.
Problem 6.4
Let Nt,t≥0, be a Poisson random process with rate λ > 0.
