Civil Engineering Reference
In-Depth Information
Fig. 2.12
Cross spectrum decomposition into co-, quad-and phase spectra
2.7 The connection between spectra and covariance
()
()
Auto-spectra
may also be calculated from the auto covariance function
,
S
ω
Cov
τ
x
x
()
x t
is a stationary and zero mean stochastic variable, the
see Eq. 2.15. Assuming that
following applies:
⎡
T
T
⎤
⎛
⎞ ⎛
⎞
1
1
it
ω
−
it
ω
()
()
E
⎢
⎜
∫
xte dt
⎟ ⎜
∫
xte
dt
⎟
⎥
⋅
*
⎡
⎤
⎜
T
⎟ ⎜
T
⎟
EXX
⎢
⎥
kk
⎝
⎠ ⎝
⎠
⎣
⎦
⎣
0
0
⎦
()
S
lim
lim
ω
=
=
x
2/
T
Δ
ω
π
T
T
→∞
→∞
TT
1
(
)
it t
−
ω
−
() ()
21
∫∫
=
lim
E x t
⎡
⋅
x t
⎤
⋅
e
dt dt
⎣
1
2
⎦
1
2
2
T
π
T
→∞
00
TT
1
(
)
it t
−
ω
−
()
(
)
21
S
lim
∫∫
Cov
t
t
e
dt dt
⇒
ω
=
−
⋅
(2.80)
x
x
21
12
2
π
T
T
→∞
00
