# Bispectrum

In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions. The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum. The Fourier transform of "C"3("t"1, "t"2) (third-order cumulant-generating function) is called the bispectrum or bispectral density. Applying the convolution theorem allows fast calculation of the bispectrum $B\left(f_1,f_2\right)=X^*\left(f_1+f_2\right).X\left(f_1\right).X\left(f_2\right)$.

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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