Performance Comparison Analyses of the \(N\)th Best Relay Selection Schemes Over Independent and Non-identically Distributed Nakagami-\(m\) Fading Channels

In this work, we study the \(N\)th best relay selection schemes with the consideration that in some case the best relay is unavailable due to the restriction of practical implementation. With amplify-and-forward relaying protocols, the interested \(N\)th best relay schemes are investigated over independent and non-identically distributed (i.ni.d) Nakagami-\(m\) fading channels. For such opportunistic relaying schemes, we first obtain the closed-form expressions to the probability density function (PDF) and cumulative distribution function (CDF) of the instantaneous end-to-end signal-to-noise ratio with appropriate mathematical proof. Then, with the obtained CDF and PDF, three main measurements are investigated as well as the corresponding explicit solutions, \(i.e.\), outage probability, average symbol error ratio (SER), and ergodic capacity. At the same time, as a byproduct, the corresponding performance metrics over Rayleigh fading are also derived. Finally, the detailed performance comparison analyses are presented under different values of \(N\) and different Nakagami-\(m\) channel fading severity parameters. The numerical results show that the increase of \(N\) incurs the very severe loss in performance such outage probability, SER, and ergodic capacity. However, the loss in performance can be decreased greatly when the \(N\)th systems have bigger fading severity factors. The derivations are of significance because the Nakagami-\(m\) fading spans via the fading severity parameters a wide range of fading scenarios that are typical in realistic wireless relay networks.