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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 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]



The cross-spectrum has representations as a decomposition into (i) its real part (co-spectrum) and (ii) its imaginary part (quadrature spectrum)

and (ii) in polar coordinates

Here, the amplitude spectrum is given by

and the phase spectrum is given by

Squared coherency spectrum

The squared coherency spectrum is given by

which expresses the amplitude spectrum in dimensionless units.

See also


  1. ^ von Storch, H.; F. W Zwiers (2001). Statistical analysis in climate research. Cambridge Univ Pr. ISBN 0-521-01230-9.

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