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. 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 (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.