On maximal ratio combining in correlated Nakagami channels with unequal fading parameters and SNRs among branches: an analytical framework

We develop an analytical framework to study the performance of wireless systems using maximal ratio combining (MRC) with an arbitrary number of diversity branches in correlated multipath fading. We consider the coherent detection of digital signals received over correlated Nakagami (1960) fading channels, where the instantaneous signal-to-noise ratios (SNRs) of the diversity branches are not necessarily independent or identically distributed. Specifically: (1) these SNRs can be arbitrarily correlated; (2) the SNR distributions can be from different Nakagami families, i.e., fading parameters (ms) are not necessarily equal; and (3) the average SNRs (averaged over the fading) of the branches are not necessarily equal. We derive closed-form expressions for three performance measures of a MRC diversity system: (1) probability density function (p.d.f.) of the combiner output SNR; (2) symbol error probability (SEP) for coherent detection; and (3) outage probability. We obtain a canonical structure for these performance measures as a weighted sum of the corresponding expressions for a non-diversity (single-branch) system with appropriately-defined parameters. This result is fundamental: the canonical structure depends only on the properties of the channel and diversity combiner, and not on the specific modulation technique. Calculations of the SEP for specific modulation techniques are illustrated through examples.