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

A recursive solution is presented for the probability density function of the sum of
N
independent, random phase vectors. The recursion parameter is
N
, the number of vectors in the sum. This approach allows one to rapidly compute a complete set of these density functions for values of
N = 2, 3, ..., N_{max}
, where N
max
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.