Bispectrum

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(t₁, t₂) (third-order cumulant-generating function) is called the bispectrum or bispectral density.

Calculation

Applying the convolution theorem allows fast calculation of the bispectrum :${\displaystyle B(f_{1},f_{2})=F(f_{1})\cdot F(f_{2})\cdot F^{*}(f_{1}+f_{2})}$, 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.

Bispectral analysis describes observations made at two wavelengths. It is often used by scientists to analyze elemental makeup of a planetary atmosphere by analyzing the amount of light reflected and received through various color filters. By combining and removing two filters, much can be gleaned from only two filters. Through modern computerized interpolation, a third virtual filter can be created to recreate true color photographs that, while not particularly useful for scientific analysis, are popular for public display in textbooks and fund raising campaigns.

Bispectral analysis can also be used to analyze interactions between wave patterns and tides on Earth.[4]

A form of bispectral analysis called the bispectral index is applied to EEG waveforms to monitor depth of anesthesia.

See also

• Trispectrum

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. Bibcode:1988PPCF...30..537G. 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 (4): 277–314. doi:10.1080/09298219708570732.
4. Kamalabadi, F.; Forbes, J. M.; Makarov, N. M.; Portnyagin, Yu. I. (27 February 1997). "Evidence for nonlinear coupling of planetary waves and tides in the Antarctic mesopause". Journal of Geophysical Research: Atmospheres. 102 (D4): 4437–4446. Bibcode:1997JGR...102.4437K. doi:10.1029/96JD01996.

• Mendel JM (1991). "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.