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 C₃(t₁, t₂) (third-order cumulant-generating function) is called the bispectrum or bispectral density. Applying the convolution theorem allows fast calculation of the bispectrum \( B(f_1,f_2)=X(-f_1-f_2).X(f_1).X(f_2) \).

Bispectra 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 [1].

Bispectral measurements have been carried out for EEG signals monitoring[2].

In seismology, signals rarely have adequate duration for making sensible bispectral estimates from time averages.
See also

Trispectrum
References

^ Greb U, Rusbridge MG (1988). "The interpretation of the bispectrum and bicoherence for non-linear interactions of continuous spectra". Plasma Phys. Control. Fusion 30 (5): 537–49. doi:10.1088/0741-3335/30/5/005.
^ Johansen JW, Sebel PS (November 2000). "Development and clinical application of electroencephalographic bispectrum monitoring". Anesthesiology 93 (5): 1336–44. PMID 11046224.

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.
HOSA - Higher Order Spectral Analysis Toolbox: A MATLAB toolbox for spectral and polyspectral analysis, and time-frequency distributions. The documentation explains polyspectra in great detail.

Mathematics Encyclopedia