Civil Engineering Reference
In-Depth Information
N
N
N
N
N
⎛ ⎞
(
)
(
)
⎜
⎝
∑ ∑
∑
⇒
Var
x
=
Cov x
⋅
x
=
ρ
x
⋅
x
⋅
σσ
(2.26)
i
i
j
i
j
i
j
i
1
i
1
j
1
i
1
j
1
=
=
=
=
=
()
i
x t
are independent (i.e. uncorrelated) then the variance of the sum of the processes
is the sum of the variances of the individual processes, i.e.
If
2
⎧
when
ij
N
N
σ
=
⎛ ⎞
⎪
(
)
x
i
⎝
∑ ∑
(2.27)
2
Cov x
x
Var
x
if
⋅
=
then
=
σ
⎨
⎜ ⎟
i
j
i
x
i
0 when
ij
⎪
⎩
≠
i
1
i
1
=
=
Example 2.3
Given an ensemble variable:
(
)
xa
sin
ωθ
=⋅
+
, where the probability density distribution of θ
1
⎧
for 0
2
≤≤
θπ
⎪
()
p
is:
θ
=
2
0 elsewhere
⎨
⎪
⎩
π
x
The ensemble covariance of
at a time lag τ
is then given by
k
2
π
( )
(
)
(
)
( )
(
)
(
)
Cov
E xt
,
xt
,
∫
p
xt
,
xt
,
d
τ
=
⎡
θ
⋅
+
τθ
⎤
=
θ
⋅
θ
⋅
+
τθ
θ
⎣
⎦
x
0
2
π
1
(
)
(
)
∫
a
sin
t
a
sin
t
d
=
⋅
ωθ
+
⋅
ωωτθθ
+
+
2
π
0
2
2
π
a
(
)
(
)
(
)
(
)
(
)
∫
sin
t
sin
t
cos
cos
t
sin
d
=
ωθ
+
⋅
⎡
ωθ
+
⋅
ωτ
+
ωθ
+
⋅
ωτ θ
⎤
⎣
⎦
2
π
0
(
)
2
⎡
2
π
2
π
⎤
sin
a
ωτ
(
)
2
(
)
(
)
∫
∫
=
cos
ωτ
sin
ω
t
+
θ
d
θ
+
sin 2
ω
t
+
θ
d
θ
⎢
⎥
2
π
2
⎢
⎥
⎣
⎦
0
0
which, after the substitution
ˆ
t
θω θ
=+, renders
(
)
2
ωπ
t
+
2
ωπ
t
+
2
sin
a
ωτ
ˆˆ
ˆˆ
()
(
)
2
Cov
cos
∫
sin
d
∫
sin 2
d
τ
=
ωτ
θ
θ
+
θ
θ
x
2
2
ω
t
ω
t
As shown in example 2.2, the first of these integrals is equal to
π
, while the second is zero, and
thus:
2
a
()
(
)
Cov
cos
τ
=
ωτ
x
2
