Civil Engineering Reference
In-Depth Information
+∞
2
x
(
)
∫
S
d
(2.63)
σ
=
±
ω
ω
x
−∞
Thus, the connection between double- and single-sided spectra is simply that
()
( )
=⋅ ±
. Assuming that the process is stationary and of infinite length, such
that the position of the time axis for integration purposes is arbitrary, then it is in the
literature of mathematics usually considered convenient to introduce a non-normalized
amplitude (which may be encountered in connection with the theory of generalised
Fourier series and identified as a Fourier constant)
S
ω
2
S
ω
x
x
T
−⋅
i
ω
k
(
)
()
∫
a
ω
=
x t
⋅
e
dt
=
T
⋅
d
(2.64)
kk
k
0
in which case the double-sided auto-spectral density associated with
±
is defined by
k
(
)
(
*
)
aT aT
/
/
*
⋅
dd
1
⋅
±=
k
k
(
)
kk
*
S
aa
ω
=
=
⋅
(2.65)
x
k
k
k
2/
T
2
T
Δ
ω
π
π
and
TN
→∞
In the limit of
this may be written on the following continuous form
1
(
)
*
() ()
S
lim lim
a
a
±=
ω
⋅
ω
⋅
ω
(2.66)
x
2
T
π
TN
→∞
→∞
and accordingly, the single sided version is given by
1
()
*
() ()
S
lim
a
a
ω
=
⋅
ω
⋅
ω
(2.67)
x
T
π
T
→∞
where it is taken for granted that
N
is sufficiently large. The auto-spectral density
()
S
ω
is defined by use of circular frequency
ω
as shown above. It may be replaced
x
()
1
Hz k
−
by a corresponding definition
S
f
using frequency
f
(with unit
=
). Since
x
()
()
and
f
,
they must give the same contribution to the total variance of the process, and thus
S
S
f
f
ωω
⋅Δ
and
⋅Δ
both represent the variance of the process at
ω
x
x
()
( )
( )(
)
S
f
⋅Δ
f
=
S
ωω
⋅Δ
=
S
ω π
⋅
2
⋅Δ
f
x
x
x
2
()
( )
*
() ()
S
f
2
S
lim lim
a
f
a f
⇒
=⋅
π
ω
→∞
=
⋅
⋅
(2.68)
x
x
T
TN
→∞
