On the Probability Density Function of the Squared Envelope of a Sum of Random Phase Vectors

Abstract

A recursive solution is presented for the probability density function of the sum of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> independent, random phase vectors. The recursion parameter is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> , the number of vectors in the sum. This approach allows one to rapidly compute a complete set of these density functions for values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N = 2, 3, ..., N_{max}</tex> , where N <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</inf> typically corresponds to the total number of system users in a multiuser FHMA/MFSK application, or one plus the total number of jamming tones in an FH/MFSK spread-spectrum application. Such evaluations are necessary for exact calculation of the average error probability performance of such systems.