Derivation of N-TH Order Cumulant Spectra

Higher-order spectra (HOS) provide the frequency-domain decomposition of higher-order moments by cross-frequency correlation. They establish the frequency-domain equivalent to correlation functions and form powerful representations for assessing nonlinear, non-Gaussian, or non-stationary systems and processes. HOS are subdivided into moment and cumulant spectra. While the latter provide a clear assessment of statistical dependence and favorable mathematical properties, cumulant spectra cannot be estimated directly. Their concept requires the identification and removal of spectra of lower order, analogously to their scalar-valued counterparts. So far, HOS applications have been based on third and fourth order and so has the derivation of cumulant spectra. Computational power, advanced methods, and new estimators put forward the interest in expanding HOS analysis to orders above four. This paper presents the combinatorial framework to define n
^th
-order cumulant spectra in the frequency domain. On this basis, sixth-order spectral estimates are employed to differentiate two processes of same PSD and trispectrum.