On achievable rate and ergodic capacity of non-symmetric half-duplex NAF relay channels

In this paper, we investigate the achievable rate and ergodic capacity of a general non-symmetric half-duplex non-orthogonal amplify-and-forward (NAF) single-relay channel in Rayleigh fading environments, assuming that the channel state information is only available at the destination. The considered channel, which captures path loss and shadowing effects over the transmission links, includes the symmetric one as a special case. At first, for a given power allocation scheme, simple and closed-form expressions of the upper and lower bounds on the achievable rate are derived. As shown by various numerical examples, the gap between the upper and lower bounds is small in the entire range of SNRs, which makes them useful in finding the optimal power allocation solution to achieve the capacity. Focusing on the two extreme cases of low and high SNRs, we then provide relatively tight approximations of the achievable rate. Using these approximations, it is then revealed that at both high and low SNR regimes, the ergodic capacity is achieved when the relay is inactive. Equivalently, NAF relaying does not yield any advantage over direct transmission at low and high SNRs. The results and observations in this paper therefore provide some further important insights on NAF relaying.