In time series analysis, the cross-spectrum is used as part of a frequency domain analysis of the cross-correlation or cross-covariance between two time series.
Let
(
X
t
,
Y
t
)
represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions
γ
x
x
and
γ
y
y
and cross-covariance function
γ
x
y
. Then the cross-spectrum
Γ
x
y
is defined as the Fourier transform of
γ
x
y
Γ
x
y
(
f
)
=
F
{
γ
x
y
}
(
f
)
=
∑
τ
=
−
∞
∞
γ
x
y
(
τ
)
e
−
2
π
i
τ
f
,
where
γ
x
y
(
τ
)
=
E
[
(
x
t
−
μ
x
)
(
y
t
+
τ
−
μ
y
)
]
.
The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and its imaginary part (quadrature spectrum)
Γ
x
y
(
f
)
=
Λ
x
y
(
f
)
+
i
Ψ
x
y
(
f
)
,
and (ii) in polar coordinates
Γ
x
y
(
f
)
=
A
x
y
(
f
)
e
i
ϕ
x
y
(
f
)
.
Here, the amplitude spectrum
A
x
y
is given by
A
x
y
(
f
)
=
(
Λ
x
y
(
f
)
2
+
Ψ
x
y
(
f
)
2
)
1
2
,
and the phase spectrum
Φ
x
y
is given by
{
tan
−
1
(
Ψ
x
y
(
f
)
/
Λ
x
y
(
f
)
)
if
Ψ
x
y
(
f
)
≠
0
and
Λ
x
y
(
f
)
≠
0
0
if
Ψ
x
y
(
f
)
=
0
and
Λ
x
y
(
f
)
>
0
±
π
if
Ψ
x
y
(
f
)
=
0
and
Λ
x
y
(
f
)
<
0
π
/
2
if
Ψ
x
y
(
f
)
>
0
and
Λ
x
y
(
f
)
=
0
−
π
/
2
if
Ψ
x
y
(
f
)
<
0
and
Λ
x
y
(
f
)
=
0
The squared coherency spectrum is given by
κ
x
y
(
f
)
=
A
x
y
2
Γ
x
x
(
f
)
Γ
y
y
(
f
)
,
which expresses the amplitude spectrum in dimensionless units.
