Virtual branch analysis of symbol error probability for hybrid selection/maximal-ratio combining in Rayleigh fading

In this paper, we derive analytical expressions for the symbol error probability (SEP) for a hybrid selection/maximal-ratio combining (H-S/MRC) diversity system in multipath-fading wireless environments. With H-S/MRC, L out of N diversity branches are selected and combined using maximal-ratio combining (MRC). We consider coherent detection of M-ary phase-shift keying (MPSK) and quadrature amplitude modulation (MQAM) using H-S/MRC for the case of independent Rayleigh fading with equal signal-to-noise ratio averaged over the fading. The proposed problem is made analytically tractable by transforming the ordered physical diversity branches, which are correlated, into independent and identically distributed (i.i.d.) "virtual branches," which results in a simple derivation of the SEP for arbitrary L and N. We further obtain a canonical structure for the SEP of H-S/MRC as a weighted sum of the elementary SEP's, which are the SEP's using MRC with i.i.d. diversity branches in Rayleigh fading, or equivalently the SEP's of the nondiversity (single-branch) system in Nakagami fading, whose closed-form expressions are well-known. We present numerical examples illustrating that H-S/MRC, even with L much less than N, can achieve performance close to that of N-branch MRC.