Noncoherent Reception of Multi-Level ASK in Rayleigh Fading with Receive Diversity

We consider a communication system with receive diversity in flat Rayleigh fading employing multi-level amplitude-shift keying (ASK) and noncoherent reception. Using the statistics of the decision variable of the quadratic receiver, a closed form expression for the symbol error probability (SEP) is derived. A formulation to determine the optimal signal amplitude levels of ASK constellation that minimize the SEP subject to a total signal energy constraint is presented. By approximating the derivative of the objective function, an approximate semi-analytical solution is obtained. We also present an asymptotic SEP expression that shows the variation of the SEP at high signal-to-noise ratio (SNR) and the diversity order of the system. An important result is that the optimum signal amplitude levels at high SNRs follow an approximately geometric progression in contrast to the more typical case of signal amplitude levels in arithmetic progression. Numerical results show the superiority of error performance of ASK with optimal signal amplitude levels as compared to that of ASK with signal amplitude levels in arithmetic progression.