1
EE 7615
Problem Set # 2
Due: 9,19,11
1) Please attempt each problem on a new page.
2) Show all your work clearly.
1) A system has input
X
which is random with mean
E
(
X
) =
m
. The output is
Y
. Suppose
the conditional density of
Y
given
X
=
x
is Gaussian with mean
x
, i.e.
f
Y

X
(
y

x
) =
1
√
2
πσ
exp[

(
y

x
)
2
2
σ
2
]
Find
E
(
Y
)
.
2)
{
X
(
t
)
}
is a WSS random process with
m
X
= 0
and
σ
2
X
= 5
. Can any of the following
functions be a candidate for the autocorrelation function of
{
X
(
t
)
}
? Justify your answers.
a)
R
X
(
τ
) = 6
u
(
τ
)
e

3
τ
where
u
(
τ
) = 1
for
τ
≥
0
and 0 otherwise.
b)
R
X
(
τ
) = 5 sin(5
τ
)
.
c)
R
X
(
τ
) = 5 cos(5
τ
)
.
d)
R
X
(
τ
) = 5(1 + 2
τ
2
)

1
+ 5 sin(
τ
)
.
3) A linear time invariant filter has transfer function
H
(
f
)
where
H
(
f
) =
(
1
1
≤ 
f
 ≤
2
0
otherwise.
The input to this filter is the Gaussian WSS random process
{
X
(
t
)
}
with mean zero and
autocorrelation function
R
X
(
τ
) =
e

τ

.
a) Find and sketch the power spectral density of the output process
{
Y
(
t
)
}
.
b) Find the pdf of the random variable
Y
t
.
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 Fall '11
 Naragipour
 Signal Processing, Stochastic process, Autocorrelation, Stationary process, power spectral density, Timeinvariant system

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