Bispectrum
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2008-07-04T21:51:52Z
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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''<sub>3</sub>(''t''<sub>1</sub>, ''t''<sub>2</sub>) (third-order [[cumulant]]-generating function) is called the bispectrum or '''bispectral density'''. Applying the [[convolution theorem]] allows fast calculation of the bispectrum <math> B(f_1,f_2)=X^*(f_1+f_2).X(f_1).X(f_2)</math>.
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 [[electroencephalography|EEG]] [[signals (biology)|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.
==See 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
[[Category:Complex analysis]]
[[Category:Integral transforms]]
[[Category:Fourier analysis]]
[[Category:Image processing]]
[[Category:Time series analysis]]
[[Category:Nonlinear time series analysis]]
[[fr:Bispectre]]