Relay selection for SWIPT: Performance analysis of optimization problems and the trade-off between ergodic capacity and energy harvesting

In this paper, we focus on the optimal relay selection (RS) and ergodic capacity of an energy harvesting (EH) amplify-and-forward (AF) multi-relay network, where multiple relays cooperate with the source node and the destination node. We begin the work with the optimization of a single relay. In the first step, after the average end-to-end signal-to-noise ratio (SNR) is obtained, the high end-to-end SNR is approximated to achieve the PDF and CDF, and we jointly optimize time switching (TS) and power splitting (PS) ratios by using Alternate Convex Search algorithm, second derivatives and Hessian matrices. Following that, closed-form expressions for the outage probability and throughput in both delay-restricted and delay-tolerant transmission mode in the high SNR regime are derived. Most importantly, we achieve not only ergodic capacity in the considered EH multi-relay network but also the trade-off between it and the average EH. Our theoretical results are proved by numerical simulations, in which delay-tolerant throughput outperforms delay-restricted throughput thanks to the change of number of relays. In addition, the data rate can be significantly enhanced thanks to the achievement of optimal TS and PS ratios.