This report, presents an analytical framework for analyzing two-branch diversity systems for a Rayleigh fading channel. In many cases the fading received at both branches (i.e. a two-antenna element system) is correlated because of the proximity of the antenna elements to each other. It is also not uncommon for a diversity system to use antennas with different patterns or polarizations, this usually results in differences in average signal-to-noise ratios at both branches depending on which element is better matched to the signal environment. As will be shown, the performance of a diversity system depends greatly on the envelope correlation, average power imbalance and the combining scheme used on both branches.

An analytical expression for the probability density function of the signal-to-noise ratio at the output of a two-branch maximal ratio and selection diversity system is developed in this report. The two branches are assumed to be Rayleigh fading, correlated, as well as of unequal signal-to-noise ratios. Measurements were made in Rayleigh fading channels and compared to the analytical results. The analytical cumulative distribution functions (derived using probability distributions) were found to be within 1 dB of the measured results (statistics obtained from time combining) for both maximal ratio and selection diversity attesting to the validity of the analytic results. Also developed in this report are the exact analytical average probabilities of symbol error for coherent BPSK and coherent QPSK before and after maximal ratio combining for this environment. The diversity gain for selection, maximal ratio, and equal gain combining for the 10% probability level is presented as a function of power imbalance and correlation between branches for a two-branch Rayleigh diversity system