Exact analytical solution for AF relaying systems with full selection diversity

New, exact closed-form expressions are derived for the probability density function (PDF) and the cumulative distribution function (CDF) of the instantaneous end-to-end signal-to-noise ratio (SNR) of amplify-and-forward (AF) relaying systems with “full” selection diversity. The direct path, from the source to the destination, as well as multiple dual-hop paths, through intermediate relays, are considered in the selection set. The selection process follows a maximum SNR policy, such that the path of the maximum end-to-end SNR is used as the communication link, whether this path is the direct path or one of the dual-hop paths. The derived expressions are used to obtain the first exact results for the average error probability and the outage probability of such AF relaying systems. The results are verified through simulations for identically distributed as well as non-identically distributed Nakagami-m fading links. The system performance is compared to that of conventional wireless systems which use only the direct link between the source and the destination for communication. The system performance is compared also to that of an opportunistic dual-hop AF system with maximum end-to-end SNR relay selection that excludes the direct path from the selection set. Results show that the full selection system performance is superior to those in the comparison. For example, in the case of Nakagami-m fading links with m = 4 and N = 2 candidate intermediate relays, an average error probability of 2×10^-4 in the case of AF relaying with full selection, occurs at 4.06 dB less than it occurs in the case of only direct transmission and at 0.95 dB less than it occurs in the case of maximum end-to-end SNR relay selection that excludes the direct path from selection. Results show also that the system provides diversity gain, proportional to the selection set size, N +1.