Problem RAS0103
1
1. Compute the mean value, correlation, and covariance functions of the following random pro
cesses
x
(
u,t
)
, u
∈ U
,t
∈ T
:
(a)
x
(
u,t
) =
a
i
(
u
)
∀
t
∈
(
i,i
+ 1]
, i
∈ {
0
,
±
1
,
±
2
,...
}
Here
T
is the real line with
x
(
u,t
) taking on a diﬀerent value in each unit
t
interval. As
sume that
{
a
i
(
u
)
}
is a sequence of identically distributed, uncorrelated random variables,
each with mean
m
and variance
σ
2
.
(b)
x
(
u,t
) =
N
X
n
=1
a
n
cos(
ω
n
t
+
φ
n
(
u
))
where
T
is the real line and
{
φ
n
(
u
)
}
is a set of independent real random variables, each
uniformly distributed on [

π,π
).
(c)
x
(
u,t
) =
t
X
n
=1
a
n
(
u
)
where
T
is the positive integers, and
{
a
i
(
u
)
}
is a sequence of identically distributed,
uncorrelated random variables, each with mean
m
and variance
σ
2
.
(d)
y
(
u,t
) =
A
(
u
) cos[
t
+
φ
(
u
)]
where
T
is the real line, and the random variables
A
(
u
) and
φ
(
u
) are independent real
random variables,
A
(
u
) has probability density function with mean
m
and variance
σ
2
,
and
φ
(
u
) has probability density
p
φ
(
u
)
(
θ
) =
1
π
, for 0
≤
θ
≤
π
and zero elsewhere.
Solution
1. Solution:
(a) Since
{
a
i
(
u
)
}
∞
i
=
∞
are uncorrelated with mean
m
and variance
σ
2
, then
Mean (expected) value
:
m
x
(
t
) =
E
{
x
(
u,t
)
}
=
E
{
a
i
(
u
)
}
=
m
for
t
∈
(
i,i
+ 1]. Hence
m
x
(
t
) =
m
x
=
m
for all
t
∈
(
∞
,
∞
).
Correlation
:
R
x
(
t
1
,t
2
) =
E
{
x
(
u,t
1
)
x
*
(
u,t
2
)
}
=
E
{
a
i
1
(
u
)
a
*
i
2
(
u
)
}
where
t
1
∈
(
i
1
,i
1
+
1]
,t
2
∈
(
i
2
,i
2
+ 1].