EE 805, Random Processes and Linear Systems
Jan. 27, 2012
OSU, Winter 2012
Due: Feb. 6, 2012
Problem Set 3
Problem 1
Consider the random process
X
(
t
) =
√
p
sin
‡
2
πf
0
t
+
B
[
n
]
π
2
·
,
nT
6
t <
(
n
+ 1)
T,
where
√
p
and
f
0
are known constants and
B
[
n
] is an iid Bernoulli random sequence taking on
values
±
1, each with probability 1/2, and
∞
< n <
∞
. Usually,
f
0
·
T
is an integer, and we
will assume so here.
(a) Sketch a sample path
X
(
t
).
(b) Find
m
X
(
t
) and
C
XX
(
s, t
).
(c) Is the process sss? wss? Explain.
Problem 2
Let
W
(
t
) (for
t
>
0) be a standard Wiener process with variance
σ
2
t
.
(a) Find two times
t
1
and
t
2
for which the two r.v.s
W
(
t
i
)
, W
(2) have a correlation coefficient
ρ
= 0
.
8 for both
i
= 1 and 2.
(b) For arbitrary
t
and
ρ >
0, can two such times
t
1
and
t
2
be found for which the correlation
coefficient between
W
(
t
) and
W
(
t
i
) is
ρ
, for both
i
= 1 and 2? Explain. What if
ρ <
0?
Problem 3
Let
A
and
B
be two uncorrelated r.v.s, each with zero mean and variance
σ
2
. Define (for fixed,
deterministic
ω
)
X
(
t
) =
A
cos(
ωt
) +
B
sin(
ωt
)
Y
(
t
) =

A
sin(
ωt
) +
B
cos(
ωt
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 Spring '12
 Eryilmaz
 Normal Distribution, Variance, Probability theory, Stochastic process, random process

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