603 ECE - Probability and Random Processes, Fall 2010
Homework #8
Due: 12/06/10, in class
I#F
30HG& 5 E
5 D
75
CB A@9864
)#)
3210 ( '&$"
%# !
and
(a) Find the probability density function
(b) Find the mean function
, and
and
are independent.
.
.
(c) Find the autocorrelation function
.
wide-sense stationary (WSS)?
2. In a simple quadrature amplitude modulation (QAM) communication system,
Q
P
(d) Is
, where
1. Let
is dened as:
d
S
d
S
b a % e b a % X V U S
H 14 1`CAfX d cH 14 1`Y'WTRQ
where is the carrier frequency, and and are independent random variables with mean zero
and variance . Assume all of the moments of and , except the rst moment (mean), are nonzero.
Q
.
b
g4
h
(b) Find the autocorrelation function of
wide-sense stationary?
strict-sense stationary? (Hint: To show that it is not, try showing that
is a discrete random variable.)
cB 4
.
)
for all .
(b) Find the mean function and autocorrelation function for
is a Gaussian random A@98 process.
Q
7
CB 4
H
x
(a) Find the probability density function
and variance . I observe the random process
t v # pD E
t pv D
be a Gaussian random variable with mean
.
Q
(c) Determine whether
for all
.
(c) Find the autocorrelation function
4. Let
(Note that
t pv
v
(b) Find the mean function
)
(Ct pv
(a) Find the probability density function
for all . If tails appears,
v
3. A fair coin is tossed. If heads appears,
.
depends
) x
3y0wCt pv
(d) Is
on .).
.
t r p
uHsQ qi
H
H
(c) Is
Q
I
(a) Find the mean function of
.
be a Gaussian random process with zero-mean and autocorrelation function
HP
.
. I run this sample through a square-law
3) I
)
30P FI "P FI I
!
FI
%
IF w
(b) Suppose that I form the linear combination
density function (pdf) of
, and calculate
.
. Find
CB 4
'"
!
; call it
.
!
"P F
(a) Suppose I take a sample at time
device to yield
. Find
5 E
5. Let
, the probability
I