On the statistics of a sum of harmonic waveforms

In this paper, we address certain aspects of the problem of statistically characterizing the electromagnetic field inside an enclosure. The field that we are interested in describing is time-harmonic and a three-dimensional spatial vector; therefore, two random variables are required for each vector component at each location in the enclosure. We can describe either the magnitude (or intensity) and phase, or the real (in-phase) and imaginary (quadrature) parts, of each spatial component. It is the relationship between these two modes of description that is addressed in this paper. We show that this relationship is given by the Blanc-Lapierre transform and when there is a sum of more than one time-harmonic field, by equations first derived by Kluyver. The relationships are derived for any form of distribution taken on by any of the random variable. We also address issues related to the approximation of the probability density function (pdf) of the amplitude of an electromagnetic field given a known pdf of the intensity of this field. The work presented herein fills in some of the gaps which were left in some literature wherein the independence of the variables to each other was assumed, that is, the independence of the in-phase to the quadrature variables.