On a condition of higher order spectra factorization

This paper presents a necessary and sufficient condition that guarantees that a higher order cumulant spectrum is factorizable in an appropriate product form. This condition provides a way to check whether or not a given higher order spectrum is compatible with the stronger hypothesis of "white generation", where the analyzed signal is obtained by exciting a linear time invariant system with a white stationary non-Gaussian noise. The condition is based on symmetries of higher order spectra and on a generalization of a formula proposed by Marron et al. (1990) for third-order spectra phase unwrapping. It is expressed, at any order, as an identity relating six higher order spectrum values. This test can be applied directly on higher order spectral data and does not require transformations of the data or application of elaborate algorithms.