# Bicoherence

In

mathematics , in the area ofstatistical analysis ,**bicoherence**is a squared normalised version of thebispectrum . The bicoherence takes values bounded between 0 and 1, which make it a convenient measure for quantifying the extent of phase coupling in a signal. It is also known as**bispectral coherency**. The prefix "bi-" in "bispectrum" and "bicoherence" refers not to two time series "x"_{"t"}, "y"_{"t"}but rather to two frequencies of a single signal.The "bispectrum" is a statistic used to search for nonlinear interactions. The

Fourier transform of the second-ordercumulant , i.e., theautocorrelation function, is the traditionalpower spectrum . The Fourier transform of C_{3}(t_{1},t_{2}) (third-ordercumulant ) is called bispectrum or**bispectral density**. They fall in the category of "Higher Order Spectra", or "Polyspectra" and provide supplementary information to the power spectrum. The third order polyspectrum (bispectrum) is the easiest to compute, and hence the most popular.The difference with measuring "coherence" (coherence analysis is an extensively used method to study the correlations in frequency domain, between two simultaneously measured signals) is the need for both input and output measurements by estimating two auto-spectra and one cross spectrum. On the other hand, bicoherence is an auto-quantity, i.e. it can be computed from a single signal. The coherence function provides a quantification of deviations from linearity in the system which lies between the input and output measurement sensors. The bicoherence measures the proportion of the signal energy at any bifrequency that is quadratically phase coupled.

Bispectrum and bicoherence may be applied to the case of non-linear interactions of a continuous spectrum of propagating waves in one dimension [

*http://www.iop.org/EJ/abstract/0741-3335/30/5/005*] .Bicoherence measurements have been carried out for EEG signals monitoring in

sleep , wakefulness andseizures [*http://cogprints.org/122/00/EEG_BICO.htm*] .**References*** Mendel JM. "Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications." "Proceedings of the IEEE",

**79**, 3, 278-305

* M J Hinich, "Testing for Gaussianity and linearity of a stationary time series", "Journal of Time Series Analysis"**3**(3), 1982 pp 169-176.

* [*http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=3013&objectType=file HOSA - Higher Order Spectral Analysis Toolbox*] . "(shareware for Microsoft Windows-type personal computers.)"

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