Polyspectra of ordered signals

Polyspectra are related to Fourier transforms of moment or cumulant functions of any order of random signals. They play an important role in many problems of signal analysis and processing. However, there are only a few statistical models giving explicitly the expression of polyspectra. Ordered signals are signals for which the explicit expression of the moment functions requires that the time instants appearing in these moments are put in an increasing order. There are many examples of such signals, the best known being the random telegraph signal constructed from a Poisson process. Some of these examples are presented and analyzed. The origin of the ordering structure is related with the point that real time is an oriented variable making a difference between past and future. This especially appears in Markov processes. The calculation of polyspectra is difficult because ordering is not adapted to Fourier analysis. By an appropriate grouping of various terms, the explicit expression of spectral moment functions is obtained. It shows in particular that many ordered signals present a normal density on the normal manifolds of the frequency domain and another contribution on the stationary manifold that is explicitly calculated. The analysis of the structure of this expression allows us to discuss some relationships with normal distribution, central limit theorem, and time reversibility.