# 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(t1t2) (third-order cumulant-generating function) is called the bispectrum or bispectral density.

## Calculation

Applying the convolution theorem allows fast calculation of the bispectrum $B(f_1,f_2)=F^*(f_1+f_2).F(f_1).F(f_2)$, where $F$ denotes the Fourier transform of the signal, and $F^*$ 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.