Interpreting the Trispectrum as the Cross-Spectrum of the Wigner–Ville Distribution

The fourth-order time-invariant spectrum, or trispectrum, has a simple derivation as the cross-spectrum among frequency bands in the Wigner-Ville distribution (WVD). Viewed this way, the trispectrum gains intuitive meaning as a measure of the linear dependence of power across frequencies, which yields some insight into its structure and interpretation. We highlight, in particular, a two-dimensional subdomain as useful for identifying modulated oscillations when the modulating envelope is non-negative or lowpass. Spectral characteristics of the carrier and modulating signals are revealed along separate axes of a two-dimensional representation of this domain. The application of this framework, combined with a previously described additive decomposition technique for higher-order spectra, is demonstrated by the blind identification and separation of sleep spindles and beta bursts in EEG.