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ECE 534 Midterm 2: Random Processes at UIUC, Fall 2006 - Prof. Bruce Hajek, Exams of Electrical and Electronics Engineering

University of Illinois - Urbana-ChampaignElectrical and Electronics Engineering
Prof. Bruce Hajek

The instructions and questions for the midterm exam of the ece 534: random processes course at the university of illinois at urbana-champaign, held in fall 2006. The exam covers topics such as l'hospital's rule, poisson random variables, independent increments processes, martingales, autocorrelation functions, and the distribution of random processes. Students are allowed to bring notes and must show their work.

Typology: Exams

Pre 2010

Uploaded on 03/11/2009

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University of Illinois at Urbana-Champaign
ECE 534: RANDOM PROCESSES
Fall 2006
Midterm 2
Monday, November 13, 2006
Name:
•This is a closed-book exam. You may consult both sides of two sheets
of notes, typed in font size 10 or equivalent handwriting size.
•Calculators, laptop computers, Palm Pilots, two-way email pagers, etc.
may not be used.
•Write your answers in the space provided.
•Please show all of your work. Answers without appropriate justification
will receive very little credit.
Score: (24 points)
1
pf3
pf4
pf5

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University of Illinois at Urbana-Champaign

ECE 534: RANDOM PROCESSES

Fall 2006

Midterm 2

Monday, November 13, 2006

Name:

  • This is a closed-book exam. You may consult both sides of two sheets

of notes, typed in font size 10 or equivalent handwriting size.

  • Calculators, laptop computers, Palm Pilots, two-way email pagers, etc.

may not be used.

  • Write your answers in the space provided.
  • Please show all of your work. Answers without appropriate justification

will receive very little credit.

Score: (24 points)

The following properties might be useful for you throughout the duration

of the exam:

  • L’Hospital’s rule: If lim t→∞

f (t) and lim t→∞

f (t) are both 0 or ∞, and

if

lim

t→∞

f

′ (t)

g

′ (t)

exists, then

lim

t→∞

f (t)

g(t)

= lim

t→∞

f

′ (t)

g

′ (t)

  • For a Poisson random variable Z λ

with parameter λ, its characteristic

function is given by

Zλ

(u) , E[exp (juZ λ

)] = exp

λ

e

ju

− 1

b) Is X λ

an independent increments process?

c) Is X λ

a martingale?

Let us now define the new random process

X

λ,∆

X

λ,∆

(t) : t ≥ ∆) with

parameter ∆, given by

X

λ,∆

(t) =

X

λ

(t, t − ∆).

d) Calculate the autocorrelation function of the process X λ

as well as for

the process

Xλ,∆. Is Xλ WSS? Is

Xλ,∆ WSS?

f) What kind of process X have we constructed in part e)? Explain.