Exact distributions of envelopes of sums of stochastic sinusoids with general random amplitudes and phases

I consider a model used in many communication systems where the sum of n stochastic sinusoidal signals, each with general 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 use probabilistic methods to prove new formulae for the Exact General Envelope Density (EGED), in terms of any given general joint density of all the random variables involved. New exact probability densities of cases of interest follow from the general formulae. The resultant amplitude distribution is also obtained when a the signal comprises of a single sinusoidal sum corrupted by a Gaussian noise. Further, I derive a new general formula for the resultant signal’s Amplitude Characterisitc Function (ACF) and the Exact Envelope Characteristic function EECF. The characteristic function is transformed to obtain exact formulae for the pdf (probability density function). The Fourier transform of the ACF formula is further used to derive a general formula for the exact peak resultant amplitude in the presence of a Gaussian noise. In important cases interest it is shown that it is possible to represent the amplitude pdf in terms of the Fourier transform of Bessel functions and thus derive simpler exact expressions. I Consider two signals of sinusoidal sums corrupted by a Gaussian noise, for which previously the Ricean envelope density had only been verified for the very special case where n = 1 and the phase is uniform and independent of amplitude. Here a new formula for the exact density of the envelope of noisy stochastic sinusoids (EDENSS) is presented, which leads to the generalization of Ricean envelope density (GRED) formula under the most general conditions, namely n ≥ 1 signals, dependent amplitudes and phases having an arbitrary joint pdf. The power of our EECF and FT methods are further demonstrated by their ability to give an alternative derivation of the EGED formula.