Civil Engineering Reference
In-Depth Information
2.6 Cross-spectral density
The cross spectral density contains the frequency domain and coherence properties
between processes, i.e. it is the frequency domain counterpart to the concept of
covariance. Given two stationary time variable functions
()
()
x t
and
y t
, both with
()
()
length
T
and zero mean value (i.e.
⎣ ⎦ ⎣ ⎦
), and performing a Fourier
transformation (adopting a double-sided complex format) implies that
Ext
⎡ ⎤ ⎡ ⎤
Eyt
0
=
=
()
()
x t
and
y t
(
)
(
)
may be represented by sums of harmonic components
X
,
kk
t
and
k
Yt
,
, i.e.
ω
ω
()
()
(
)
x t
N
X
,
t
⎡ ⎤
⎡
ω
ω
⎤
∑
kk
lim
=
(2.69)
⎢ ⎥
⎢
⎥
(
)
y t
Y
,
t
N
→∞
−
⎣ ⎦
⎣
⎦
N
kk
where:
(
)
(
)
a
a
T
/2
(
)
⎡
ω
⎤
⎡
ω
ω
⎤
()
()
⎡
Xt
ω
ω
,
⎤
⎡ ⎤
xt
1
Xk
Xk
kk
k
i
⋅
ω
t
k
−⋅
i
ω
t
k
and
∫
k
=
⎢
⎥
⋅
e
⎢
⎥
=
lim
⋅
e
dt
⎢
⎥
⎢ ⎥
(
)
(
)
(
)
Yt
,
T
a
a
yt
ω
⎢
⎥
⎢
⎥
T
→∞
−
⎣
kk
⎦
⎣
⎣ ⎦
Yk
Yk
⎦
⎣
⎦
k
k
T
/2
k
2/
T
and where
ω
=⋅ Δ
ω
and
Δ=
ωπ
. The definition of the double-sided cross-
k
S
spectral density
associated with the frequency
ω
is then
xy
k
*
⎡
⎤
EX Y
⋅
1
⎣
kk
⎦
(
)
*
S
aa
±=
ω
=
⋅
(2.70)
xy
k
XY
k
2
T
k
Δ
ω
π
Since the Fourier components are orthogonal
(
)
S
,
t
hen
i
j
k
⎧
ω
⋅Δ
ω
=
=
⎪
(
)
(
)
xy
k
⎡
⎤
EX
,
t Y
,
t
ω
⋅
ω
=
(2.71)
⎨
⎣
i
i
j
j
⎦
0 when
ij
≠
⎪
⎩
()
x t
and
it follows from Eqs. 2.69 and 2.70 that an estimate of the covariance between
()
y t
are given by
⎡
⎤
⎛
N
⎞ ⎛
N
⎞
N
(
)
() ()
∑∑ ∑
[
]
Cov
E x t
y t
lim
E
X
Y
lim
E X
Y
=
⎡
⋅
⎤
=
⋅
=
⋅
⎢
⎜
⎟ ⎜
⎟
⎥
⎣
⎦
xy
i
j
k
k
N
→∞
N
→∞
⎢
⎥
⎝
⎠ ⎝
⎠
⎣
⎦
−
N
−
N
−
N
N
∑
(
)
Cov
lim
S
⇒
=
±
ωω
⋅ Δ
(2.72)
xy
xy
k
N
→∞
−
N
