Civil Engineering Reference
In-Depth Information
it
⎡ ⎤
⎡ ⎤⎡
ω
e
1
i
cos
ω
ω
t
⎤
(2.57)
⎢ ⎥
=
⎢ ⎥⎢
⎥
it
1
−
i
sin
t
−
ω
⎢ ⎥
⎣ ⎦⎣
e
⎦
⎣ ⎦
i
=−
1
(where
) and defining the complex Fourier amplitude
1
2
(
)
d
a
i b
(2.58)
=
−
⋅
k
k
k
∞
∞
i
k
e
ω
⋅
()
∑
(
)
∑
(
)
xt
X
,
t
d
then:
=
ω
=
ω
⋅
(2.59)
kk
kk
−∞
−∞
Taking the variance of the complex Fourier components in Eq. 2.59 and dividing by
Δ
,
(
)(
)
*
it
it
*
−
ω
ω
⎡
⎤
k
k
EX X
de
de
⋅
T
*
dd
kk
1
k
k
⎣
⎦
k
k
∫
dt
=
=
(2.60)
T
Δ
ω
Δ
ω
Δ
ω
0
which may be further developed into
*
⎡
⎤
EX X
(
) (
)
⋅
2
ai b ai b
1
4
+⋅
⋅
−⋅
c
⎣
kk
⎦
k
k
k
k
k
⇒
=
=
(2.61)
4
Δ
ω
Δ
ω
Δ
ω
It is seen (see Eq. 2.53) that this is half the auto spectral value associated with
ω
k
. Thus,
a symmetric double-sided auto spectrum associated with
+
k
may be
defined with a value that is half the corresponding value of the single sided auto-
spectrum. Extending the frequency axis from minus infinity to plus infinity and using the
complex Fourier components
−
k
as well as
X
given in Eq. 2.59 above, this double sided auto
k
spectrum is then defined by
⎡
*
⎤
EX X
⋅
*
2
dd
c
⎣
kk
⎦
(
)
k
k
k
S
(2.62)
±=
ω
=
=
x
k
Δ
ω
Δ
ω
4
Δ
ω
(
)
and
TN
S
which, in the limit of
, and
from which the variance of the process may be obtained by integration over the entire
positive as well as negative (imaginary) frequency range
→∞
, becomes the continuous function
±
x
