Geometrical concepts in HOS

Polyspectra of signals are related to Fourier transforms of moments or cumulants called spectral moment functions. These functions have remarkable geometrical properties involving some specific surfaces in the frequency domain. Thus the stationary manifold characterizes the stationarity of a signal. Similarly normal manifolds and normal densities are related to normal signals. In this paper we analyze the connections between these manifolds and some properties of signals such as circularity spherical invariance, time reversibility or ergodicity. Geometry appears also in the time domain, for example when considering ordered signals. These signals are characterized by the fact that the mathematical expression of their moments requires that the instants are put in an increasing order. This ordering property is strongly related to a Markov property. It introduces some geometrical consequences in the frequency domain, and especially the apparition of principle value distributions in polyspectra. Some consequences are presented and analyzed