Civil Engineering Reference
In-Depth Information
last section of the review by [PAN 01], which would be a
useful initial reference, presents an excellent overview of the
literature in this domain. We give a brief discussion of these
aspects in this section.
4.11.1.
Karhunen-Loève series
Here, we give a brief summary of Karhunen-Loève series,
drawing inspiration from [PAP 84] to initiate readers in the
discipline. Consider a stochastic process
xt
()
that can be
expressed as a series in the form
∞
()
()
∑
[4.61]
x t
=
c
ϕ
t
0< <
t
T
nn
n
=
1
is a set of complex and orthogonal
functions in the interval (0,
T
), which is therefore governed
by the relation
In this relation,
()
ϕ
n
t
T
() ()
(
)
[4.62]
∫
*
ϕϕ
t
t dt
=−
n
m
n
m
0
where
is the Dirac function. The coefficients
c
n
are
determined by the relation
δ
T
()
∫
[4.63]
c
=
x t
()
ϕ
*
t dt
n
n
0
Consider the proper functions of the integral equation
T
(
) (
)
( )
[4.64]
∫
Rt t
,
ϕ
t dt
=
λϕ
t
12
2
2
1
0
is the autocorrelation function of
xt
()
(
)
=
()
x
*
()
Rt
1
,
t
2
xt
1
t
2
which is not necessarily steady. In this expression, in the
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