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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[edit]

The squared coherency spectrum is given by

which expresses the amplitude spectrum in dimensionless units.

See also[edit]


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