Equal-gain combining with unequal energy constellations

To compute the optimal bias in equal-gain combining (EGC) receivers with unequal energy constellations such as M-ary pulse amplitude modulation (M-PAM) and M-ary square quadiature amplitude modulation (M-QAM), the estimation of the instantaneous fading gain magnitude in each diversity branch is needed, thus defeating the usefulness of EGC. In this paper, we present an exact analytical equation, which can be efficiently used to compute a suboptimal value for the bias, both for M-PAM and M-QAM constellations. The proposed suboptimal bias minimizes the symbol error probability, and depends on the statistics of the fading gain magnitudes and not on their instantaneous values. For the case of large number of branches, where the central limit theorem can be applied, the proposed equation for the computation of the suboptimal bias is derived in closed-form. Moreover. an approximate simple closed-form expression for the suboptimal bias is obtained, suitable for the high signal-to-noise ratio region. Numerical results verify the correctness and the accuracy of the proposed analytical formulation.