- Cross-spectrum
-
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.
Contents
Definition
Let (Xt,Yt) represent a pair of stochastic processes that are jointly wide sense stationary with covariance functions γxx and γyy and cross-covariance function γxy. Then the cross spectrum Γxy is defined as the Fourier transform of γxy [1]
The cross-spectrum has representations as (i) a decomposition into its real part (co-spectrum) and its imaginary part (quadrature spectrum)
- Γxy(f) = Λxy(f) + iΨxy(f),
and (ii) in polar coordinates
Here, the amplitude spectrum Axy is given by
and the phase spectrum Φxy given by
- 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}" border="0">
Squared coherency spectrum
The squared coherency spectrum is given by
which expresses the amplitude spectrum in dimensionless units.
See also
References
Categories:- Frequency domain analysis
Wikimedia Foundation. 2010.