Optimal ASK Levels for Channel Magnitude Based Diversity Reception in Rayleigh Fading

We consider two low-complexity receivers, namely, the energy detection (ED) receiver and the weighted envelope detection (WENVD) receiver, employing receive diversity, operating in flat Rayleigh fading with multi-level amplitude-shift keying, and having only channel magnitude information, that is, knowledge of the magnitudes of the fading gains of the receive diversity branches. A closed form expression for the approximate symbol error probability (SEP) of the WENVD receiver for high signal-to-noise ratio (SNR) is derived. Furthermore, analytical results on the optimization of the transmit symbol amplitude levels to minimize the SEP subject to a total energy constraint for high SNR and large number of receive diversity branches for both the ED and the WENVD receivers are presented. The performance improvement obtained by using these optimal levels over using symbol amplitude levels with equal spacing, that is, in arithmetic progression, for both the receivers is shown through numerical results.