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.

Definition

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
where
The cross-spectrum has representations as a decomposition into its real part and its imaginary part
and 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.