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In signal processing and statistics , the cross-spectrum is a tool used to analyze the relationship between two time series in the frequency domain . It describes how the correlation between the two series is distributed over different frequencies. For example, if two microphones are recording audio in a room, the cross-spectrum can reveal the specific frequencies of sounds (like a hum from an appliance) that are prominent in both recordings, helping to identify common sources.
Technically, the cross-spectrum is the Fourier transform of the cross-covariance function . This means it takes the relationship between the two signals over time and represents it as a function of frequency.
Let
(
X
t
,
Y
t
)
{\displaystyle (X_{t},Y_{t})}
stochastic processes that are jointly wide sense stationary with autocovariance functions
γ
x
x
{\displaystyle \gamma _{xx}}
γ
y
y
{\displaystyle \gamma _{yy}}
cross-covariance function
γ
x
y
{\displaystyle \gamma _{xy}}
cross-spectrum
Γ
x
y
{\displaystyle \Gamma _{xy}}
Fourier transform of
γ
x
y
{\displaystyle \gamma _{xy}}
[ 1]
Γ
x
y
(
f
)
=
F
{
γ
x
y
}
(
f
)
=
∑
τ
=
−
∞
∞
γ
x
y
(
τ
)
e
−
2
π
i
τ
f
,
{\displaystyle \Gamma _{xy}(f)={\mathcal {F}}\{\gamma _{xy}\}(f)=\sum _{\tau =-\infty }^{\infty }\,\gamma _{xy}(\tau )\,e^{-2\,\pi \,i\,\tau \,f},}
where
γ
x
y
(
τ
)
=
E
[
(
x
t
−
μ
x
)
(
y
t
+
τ
−
μ
y
)
]
{\displaystyle \gamma _{xy}(\tau )=\operatorname {E} [(x_{t}-\mu _{x})(y_{t+\tau }-\mu _{y})]}
The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and (ii) its imaginary part (quadrature spectrum)
Γ
x
y
(
f
)
=
Λ
x
y
(
f
)
−
i
Ψ
x
y
(
f
)
,
{\displaystyle \Gamma _{xy}(f)=\Lambda _{xy}(f)-i\Psi _{xy}(f),}
and (ii) in polar coordinates
Γ
x
y
(
f
)
=
A
x
y
(
f
)
e
i
ϕ
x
y
(
f
)
.
{\displaystyle \Gamma _{xy}(f)=A_{xy}(f)\,e^{i\phi _{xy}(f)}.}
Here, the amplitude spectrum
A
x
y
{\displaystyle A_{xy}}
A
x
y
(
f
)
=
(
Λ
x
y
(
f
)
2
+
Ψ
x
y
(
f
)
2
)
1
2
,
{\displaystyle A_{xy}(f)=(\Lambda _{xy}(f)^{2}+\Psi _{xy}(f)^{2})^{\frac {1}{2}},}
and the phase spectrum
Φ
x
y
{\displaystyle \Phi _{xy}}
{
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
{\displaystyle {\begin{cases}\tan ^{-1}(\Psi _{xy}(f)/\Lambda _{xy}(f))&{\text{if }}\Psi _{xy}(f)\neq 0{\text{ and }}\Lambda _{xy}(f)\neq 0\\0&{\text{if }}\Psi _{xy}(f)=0{\text{ and }}\Lambda _{xy}(f)>0\\\pm \pi &{\text{if }}\Psi _{xy}(f)=0{\text{ and }}\Lambda _{xy}(f)<0\\\pi /2&{\text{if }}\Psi _{xy}(f)>0{\text{ and }}\Lambda _{xy}(f)=0\\-\pi /2&{\text{if }}\Psi _{xy}(f)<0{\text{ and }}\Lambda _{xy}(f)=0\\\end{cases}}}
Squared coherency spectrum [ edit ] The squared coherency spectrum is given by
κ
x
y
(
f
)
=
A
x
y
2
Γ
x
x
(
f
)
Γ
y
y
(
f
)
,
{\displaystyle \kappa _{xy}(f)={\frac {A_{xy}^{2}}{\Gamma _{xx}(f)\Gamma _{yy}(f)}},}
which expresses the amplitude spectrum in dimensionless units.
^ von Storch, H.; F. W Zwiers (2001). Statistical analysis in climate research . Cambridge Univ Pr. ISBN 0-521-01230-9 
![{\displaystyle \gamma _{xy}(\tau )=\operatorname {E} [(x_{t}-\mu _{x})(y_{t+\tau }-\mu _{y})]}](/media/api/rest_v1/media/math/render/svg/892f99784501ee709ba4657bc7c4102d2cb1de96)
