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.
represent a pair of stochastic processes that are jointly wide sense stationary with autocovariance functions
and cross-covariance function
is defined as the Fourier transform of
[1] where 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 The squared coherency spectrum is given by which expresses the amplitude spectrum in dimensionless units.