# Bispectrum

In

mathematics , in the area ofstatistical analysis , the**bispectrum**is a statistic used to search for nonlinear interactions. TheFourier 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 -generating function) is called the bispectrum or**bispectral density**. Applying theconvolution theorem allows fast calculation of the bispectrum $B(f\_1,f\_2)=X^*(f\_1+f\_2).X(f\_1).X(f\_2)$.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. A statistic defined analogously is the "bispectral coherency" or "bicoherence".

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*] .Bispectral measurements have been carried out for EEG signals monitoring [

*http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=11046224&dopt=Abstract*] .In

seismology , signals rarely have adequate duration for making sensible bispectral estimates from time averages.**ee also**Trispectrum **References***Mendel JM. "Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications". "Proc. IEEE",

**79**, 3, 278-305

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