The amplitude modulation pattern of Gaussian noise is a fingerprint of Gaussianity

We introduce a new approach for Gaussianity testing using the envelope of a signal and its coefficient of variation. The envelope of a Gaussian signal follows the Rayleigh distribution, and given that the coefficient of variation of the Rayleigh distribution is invariant, the envelope of Gaussian noise's coefficient of variation is a universal constant and can be exploited as a discriminating statistic. The main application of this approach is to detect the Gaussianity of time series. However, the coefficient of variation of the envelope is also a measure of amplitude modulation patterns that captures the structure of the Fourier phase profile, making it a useful parameter to differentiate types of non-Gaussianity and for signal classification.