Capacity and Power Allocation of Dual-Hop AF Relaying over Rayleigh Fading Channels

In this paper, we investigate the capacity and optimal power allocation (PA) scheme between the source and relay for a dual-hop amplify-and-forward (AF) system over non-symmetric Rayleigh fading channels with channel information available at the relay. At first, a closed-form expression of the mutual information (MI) between the input and output of the considered channel is obtained. Since only the exponential integral is involved, the derived expression is useful in finding the optimal PA to achieve the capacity. By further considering high and low signal-to-noise ratio (SNR) regimes, we present tight yet simple approximations to this MI, which can be used to show the advantage of knowing channel information at the relay. Then, focusing on the problem of optimal PA, we first derive a closed- form derivative of the MI. A simple bisection method is then proposed to find the optimal PA scheme. While uniform PA is shown to achieve the capacity at any SNR over the symmetric channel, its optimality can only be observed at low SNRs over a non- symmetric channel. In other SNR regimes, numerical results reveal that uniform PA experiences a significant loss. A comparison between the dual-hop and direct transmission scheme is also made, where we show that the dual-hop scheme using the optimal PA can provide impressive rate increases in medium SNR ranges in various network configurations.