# Bispectrum

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In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions.

## Definitions

The Fourier transform of the second-order cumulant, i.e., the autocorrelation function, is the traditional power spectrum.

The Fourier transform of C3(t1t2) (third-order cumulant-generating function) is called the bispectrum or bispectral density.

## Calculation

Applying the convolution theorem allows fast calculation of the bispectrum :${\displaystyle B(f_{1},f_{2})=F^{*}(f_{1}+f_{2})\cdot F(f_{1})\cdot F(f_{2})}$, where ${\displaystyle F}$ denotes the Fourier transform of the signal, and ${\displaystyle F^{*}}$ its conjugate.

## Generalizations

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.

## Applications

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] It was also shown that bispectra characterize differences between families of musical instruments.[3]

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

## References

1. ^ 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.
2. ^ Johansen JW, Sebel PS (November 2000). "Development and clinical application of electroencephalographic bispectrum monitoring". Anesthesiology. 93 (5): 1336–44. doi:10.1097/00000542-200011000-00029. PMID 11046224.
3. ^ Dubnov S, Tishby N and Cohen D. (1997). "Polyspectra as Measures of Sound Texture and Timbre". Journal of New Music Research. 26: 277–314. doi:10.1080/09298219708570732.
• 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. doi:10.1109/5.75086.
• 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.