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.
