Achievable Rate and Power Allocation for Single-Relay AF Systems Over Rayleigh Fading Channels at High and Low SNRs

In this paper, we propose a general method to analyze the achievable rate and to characterize the optimal power-allocation (PA) scheme for a wide range of half-duplex single-relay amplify-and-forward (AF) protocols over Rayleigh fading channels in high and low signal-to-noise-ratio (SNR) regimes. These include one-way (OW) dual-hop AF (DHAF) systems, OW nonorthogonal AF (NAF) systems, and two-phase two-way AF (2P TWAF) systems using either fixed-gain (FG) or variable-gain (VG) amplification coefficients. At high-SNR regimes, our main idea is to exploit the capacity of a two-branch maximal-ratio combining (MRC) system to obtain tight yet simple approximations to the achievable rates. In low-SNR regions, we use a simple approximation to the logarithm to compute the asymptotic achievable rates. For all considered AF protocols, the closed-form approximations are tight and easy to analyze since they involve only the exponential integral. Then, using the derived approximations, we analytically quantify the asymptotic PA schemes among the nodes to achieve the maximum rate and the sum rate for OW and TW schemes, respectively. The OW and TW protocols of interest are finally compared with the direct transmission (DT) scheme. Although OW relaying is inferior in terms of capacity in high- and low-SNR regimes, the 2P TW system might become advantageous in terms of sum-rate performance at high SNRs under the right channel conditions.