# 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 :${\displaystyle B(f_{1},f_{2})=F^{*}(f_{1}+f_{2})\cdot F(f_{1})\cdot F(f_{2})}$, 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.[citation needed]

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.[citation needed]

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.[citation needed]