Definition
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Let represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions and and cross-covariance function . Then the cross-spectrum is defined as the Fourier transform of [1]
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where
- .
The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and (ii) its imaginary part (quadrature spectrum)
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and (ii) in polar coordinates
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Here, the amplitude spectrum is given by
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and the phase spectrum is given by
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Squared coherency spectrum
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The squared coherency spectrum is given by
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which expresses the amplitude spectrum in dimensionless units.
See also
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References
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- ^ von Storch, H.; F. W Zwiers (2001). Statistical analysis in climate research. Cambridge Univ Pr. ISBN 0-521-01230-9.