Civil Engineering Reference
In-Depth Information
and where
, one for each harmonic
component. Alternatively, Eq. A.2 may be replaced by the exponential format (often
encountered in the literature)
ψ
are arbitrary phase angles between zero and
2
π
k
N
-
½
°
°
()
¦
(
)
xt
Re
c
exp
i t
=
⋅
ª
ωψ
+
º
(A.4)
®
¾
¬
¼
k
k
k
°
°
¯
¿
k
=
1
2
N
c
()
¦
, which in the limit of
k
The variance of
x t
is
Δω →
0
and
N
→∞
,
1
2
k
=
2
∞
N
c
2
¦
³
k
()
lim
S
d
σ
=
ω ω
(A.5)
x
x
2
0
Δω
→
k
1
=
0
N
→∞
I.e., if the discretization is sufficiently fine, then the variance of the simulated
representative,
()
x t
, is equal to or close enough to the variance of the parent variable.
The procedure is further illustrated in Example A.1 and Fig. A.2. Any number of such
representatives may be simulated simply by changing the choice of phase angles.
Obviously, the accuracy of such a simulation depends on the discretization fineness, but
there is also the unfavourable possibility of aliasing. Let
ω be the upper cut-off
frequency, beyond which there is none or only negligible spectral information about the
process. Assuming constant frequency segments
c
c
N
Δω ω
=
(A.6)
then each simulated time series will be periodic with period
T
2/
=
πΔω
(A.7)
Thus, time series without aliasing will be obtained if they are generated with a time step
( )
t
Δπω
2/2
c
≤
(A.8)
