On the SNR penalty of MPSK with hybrid selection/maximal ratio combining over i.i.d. Rayleigh fading channels

Closed-form expressions that lower and upper bound the penalty of hybrid selection/maximal ratio combining relative to maximal ratio combining (MRC) for -ary phase-shift keying (MPSK) modulations are proved. The bounds offer simple-to-evaluate explicit expressions, and are typically within 0.6 dB for hybrid systems with diversity order up to eight that use at least two branches, yet are independent of signal-to-noise ratio (SNR). Contrary to conclusions conjectured in a recently published paper, it is proved that the SNR penalty is not a constant, independent of SNR. It is also shown that previous estimates of the performance losses of selection diversity relative to MRC underestimate or lower bound the losses for MPSK modulation systems, and that the true loss can be significantly larger than previously believed. An upper bound to this loss is also obtained.