Yarmouk University
Hijjawi Faculty of Engineering and Technology
Department of Communication Engineering
Random Processes CME 610
First Semester 2009/2010
Problem Set # 4
Instructor: Dr. Khaled Gharaibeh
Dec. 30, 2009
P
r
o
b
l
e
m
1
Given the complex process ()
()
tt
j
t
zx
y
where ()and ()
x
y
are zero mean independent processes with
RA
a
n
d
RB
xy
If
z(
t
)
is applied to a linear filter
L
whose impulse response
is
ct
ht
e Ut
such that ()
[ ()
]
tL
t
vz
a)
Find the autocorrelation function of ()
t
v
b)
Find the power spectral density of ()
t
v
c)
Find the power of
v
(t)
d)
Find the rms bandwidth of ()
t
v
Problem 2
Given the random process:
0
cos
(
)
Xt
A
t
where
A
and
0
are constants and
is a random variable uniformly distributed on the interval
]
2
,
0
[
.
(a)
Find the autocovariance functions of
X
(
t
).
(b)
Determine if
X
(
t
) is a widesense stationary process.
(c)
Define new random variables
)
(
)
(
)
(
)
(
)
(
3
2
1
t
X
t
X
t
Y
t
X
t
Y
where
is a constant (This is called GramSchmidt Orthogonalization). Find
such that the two processes
)
(
),
(
2
1
t
Y
t
Y
are orthogonal.
Problem 3
A real random process is defined by
x
(
t
) =
A
cos (
ω
o
t
) +
w
(
t
)
where
A
is a Gaussian random variable with mean zero and variance
σ
A
2
and w(t) is a white noise
process independent of
A
with variance
σ
w
2
(a)
What is the correlation function of
x
(t)?
(b)
Can the power spectral density of
x
(t) be defined? If so, what is the power spectral density
function?
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 Spring '10
 TTTTT
 Signal Processing, Autocorrelation, Stationary process, power spectral density, Yarmouk University Hijjawi, Yarmouk University Hijjawi Faculty of Engineering and Technology Department of Communication Engineering

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