Civil Engineering Reference
In-Depth Information
Its mean value and variance are then given by
T
⎫
1
x
lim
∫
Xdt
=
⎪
⎪
⎬
⎪
T
T
→∞
0
(2.9)
T
1
2
2
()
∫
σ
=
lim
⎡ ⎤
xt
dt
⎣ ⎦
⎪
x
T
T
→∞
⎭
0
It will in the following also be assumed that the individual observations of the
fluctuating part
()
x t
within the time window between 0 and
T
may with sufficient
accuracy be fitted to a Gaussian probability distribution, as illustrated on the right hand
side of Fig. 2.2.
Fig. 2.2
Time domain statistics
Example 2.1
Given a zero mean variable
()
(
)
x t
a
cos
t
2
T
=⋅
ω
, where
ωπ
=
. Its variance is then given by
0
0
0
T
T
1
1
2
2
2
()
(
)
lim
∫
x
tdt
lim
∫
a
cos
t
dt
σ
=
=
⎡
ω
⎤
⎣
⎦
x
0
T
T
T
T
→∞
→∞
0
0
TnT
, where
n
is an integer, then
Substituting
=⋅
0
⎡
2
⎤
T
0
2
1
⎡
⎛ ⎞
2
⎤
a
T
π
2
⎢
⎥
lim
na
∫
cos
t
t
σ
=
⋅
⋅
=
⎢
⎜ ⎟
⎥
x
nT
⎢
⎥
2
⋅
n
→∞
⎢
⎥
⎣
⎝ ⎠
⎦
0
0
0
⎣
⎦
