Bispectrum

In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions. 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 C3(t1, t2) (third-order cumulant-generating function) is called the bispectrum or bispectral density. 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 ) {displaystyle B(f_{1},f_{2})=F^{*}(f_{1}+f_{2})cdot F(f_{1})cdot F(f_{2})} , where F {displaystyle F} denotes the Fourier transform of the signal, and F ∗ {displaystyle F^{*}} its conjugate. 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. Bispectrum and bicoherence may be applied to the case of non-linear interactions of a continuous spectrum of propagating waves in one dimension. Bispectral measurements have been carried out for EEG signals monitoring. It was also shown that bispectra characterize differences between families of musical instruments. In seismology, signals rarely have adequate duration for making sensible bispectral estimates from time averages.