Optimal Multilevel ASK With Noncoherent Diversity Reception in Uncorrelated Nonidentical and Correlated Rayleigh Fading

A noncoherent quadratic diversity reception system for multilevel amplitude-shift keying (ASK) signals in Rayleigh fading with different cases of the channel covariance matrix is considered. These include the important case of distributed antennas, which exhibit uncorrelated fading with distinctmean-squared channel gains (uncorrelated nonidentical fading) and weighted energy detectors, and the cases of exponentially correlated and uniformly correlated channels with equal mean-squared channel gains. For these cases, closed-form expressions for the symbol error probability (SEP) of the system are derived using the decision statistics obtained from the maximum likelihood receiver. The problem of determining the optimal amplitude levels that minimize the SEP under a total signal energy constraint is formulated and solved analytically for the high signal-to-noise ratio (SNR) configuration and numerically for the general configuration. For the high SNR configuration, it is shown that the optimal ASK levels should follow an approximately geometric progression. It is also observed that an equally spaced ASK constellation is not optimal in terms of achieving the minimum SEP. Numerical results are presented to highlight performance trends for channel gain imbalance and correlation.