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ECE 603 - Probability and Random Processes, Fall 2009
Final Exam
December 15th, 10:30am-12:30pm, LGRT 0321
Overview
The exam consists of five problems for 110 points. The points for each part of each problem are given
in brackets - you should spend your two hours accordingly.
The exam is closed book, but you are allowed three page-sides of notes. Calculators are not allowed.
I will provide all necessary blank paper.
Testmanship
Full credit will be given only to fully justified answers.
Giving the steps along the way to the answer will not only earn full credit but also maximize the
partial credit should you stumble or get stuck. If you get stuck, attempt to neatly define your approach
to the problem and why you are stuck.
If part of a problem depends on a previous part that you are unable to solve, explain the method for
doing the current part, and, if possible, give the answer in terms of the quantities of the previous part
that you are unable to obtain.
Start each problem on a new page. Not only will this facilitate grading but also make it easier for you
to jump back and forth between problems.
If you get to the end of the problem and realize that your answer must be wrong, be sure to write “this
must be wrong because... ” so that I will know you recognized such a fact.
Academic dishonesty will be dealt with harshly - the minimum penalty will be an “F” for the course.
Some potentially useful information
Time Function Fourier Transform
C
D E9 F HGJI K
KLMIONP
Q
K
K RMIONP S
-T" F HUEVXW,Y
@ [Z
(^) sinc
-T"
O
P
\
T
]9
54_^
-T
T][
^
-T
T][ 76
P
\
T
]9
^
-T
T][
^
-T
T])
<; `a;>b % R
Q
ced Y
A@
[Z
c
8 E9e G I
K
K K
K0LfI Q K
K0RfI
-T"
sinc
-T"
Parseval’s Relation: If =
-T"
is the Fourier Transform of
E9
,
gih
h
K
E9e K
)j 9 F
g kh
h
K
-T"
K
[j T
Useful Facts:
- A random variable is jointly Gaussian with itself.
- If =
b
=
b
=ml^
b =on (^) are jointly Gaussian random variables, each with zero mean, then:
p 4 = = =ml=on^ 6 q^
p 4
p 4 (^) =ml)=on 6 (^)
p 4 = =ml 6
p 4 = =on (^6)
p 4 = =on 64 = =ml 6
- Let
E9e be a Gaussian random process with zero-mean and autocorrelation function
.
[10] (a) Suppose I take a sample at time
Q
; call it =
Q
. I run this sample through a
square-law device to yield =
P. Find S
R.
[10] (b) Suppose that I form the linear combination =
Q
(^) I. Find
T
$ c
, the proba-
bility density function (pdf) of =
, and calculate S
RMI.
- Consider zero-mean stationary Gaussian random processes =
E9e and
E9e , with respective power
spectral densities
$ -T" and
( -T"
. Suppose that
E9
and
E9e are independent.
[10] (a) Let
E9
E9
E9e
. Find the power spectral density of
E9
in terms of
$ -T" and
( -T"
. Show your work.
[10] (b) Let
E9 m
E9e
E9e
. Find the power spectral density of
E9e in terms of
$ -T" and
( -T"
. Show your work.
