# Cross-spectrum

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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.

## Contents

## DefinitionEdit

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]}

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

## Squared coherency spectrumEdit

The squared coherency spectrum is given by

which expresses the amplitude spectrum in dimensionless units.

## See alsoEdit

## ReferencesEdit

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