Exact Amplitude Distributions of Sums of Stochastic Sinusoidals

I consider a model applicable in many communication systems where the sum of n stochastic sinusoidal signals of the same frequency, but with random amplitudes as well as phase angles is present. The exact probability distribution of the resulting signal’s amplitude is of particular interest in many important applications, however the general problem has remained open. I derive new general formulae for the resultant amplitude’s distribution, in terms of any given general joint density of the random variables involved. New exact probability densities of cases of interest follow from the general formulae, which are plotted and shown to be consistent with simulations. Index Terms Stochastic sinusoidals, sums of sinusoidals, amplitude distribution, exact distribution.