Factorizability of complex signals higher (even) order spectra: a necessary and sufficient condition

This paper presents a necessary and sufficient condition for the factorizability of higher order spectra of complex signals. Such a factorizability condition can be used to test if a complex signal can model the output of a linear and time invariant system driven by a stationary non-Gaussian white input. The condition developed here is based on the symmetries of higher order spectra and on an extension of a formula proposed by Marron et al. (1990) to unwrap third order spectrum phases. It is an identity between the products of six higher order spectra values (which reduces to four values if only phases are considered). Our factorizability test requires no phase unwrapping, unlike existing methods developed in the cepstral domain. Moreover its extension to the N-th order case is direct. Simulations illustrate the deviation to this factorizability condition in a factorizable case (linear system) and a non-factorizable case (non-linear system)