Exact Analytical Solution for Dual-Hop and Opportunistic Dual-Hop AF Relaying Systems

Novel exact closed-form expressions are derived for the probability density function (PDF) and the cumulative distribution function (CDF) of the instantaneous received end-to-end signal-to-noise ratio (SNR) of dual-hop amplify-and-forward (AF) relaying systems operating over Rayleigh, Nakagami-m and Rician fading channels. New exact closed-form expressions are also obtained for opportunistic dual-hop AF relaying systems, with maximum relay-to- destination SNR relay selection. The average symbol error probability, ergodic capacity and outage probability are calculated using the derived PDF and CDF expressions. It is found that the opportunistic dual-hop AF system, with relay selection pool size M = 2, has at least 2.84 dB power advantage over the dual-hop AF system without relay selection for average error probabilities less than 10^-2 in the case of Rayleigh fading links. It is shown that although the performance of a dual-hop AF system with maximum relay-to-destination SNR relay selection is improved by increasing the selection pool size, the improvement has diminishing returns and a relay selection pool of more than 6 relays is not of practical benefit.