Capacity limit of static single-relay amplify-and-forward channels

In this paper, we establish in closed-form the capacity and characterize the optimal input covariance matrix at the source and the optimal power allocation scheme between source and relay for a half-duplex single-relay amplify-and-forward (AF) system with static channel gains. Different from multiple-input multiple-output (MIMO) systems, the channel matrix of the AF system is a function of the parameters to be optimized and hence water filling over the square of the singular values of this matrix is no longer optimal. Furthermore, given that the mutual information of the AF system is not a convex function, conventional optimization methods cannot be applied to find the optimal input covariance, the power allocation, and the capacity. Instead, by analyzing all local maximizers, it is shown that the capacity of the AF system is achieved by either the direct transmission (DT) scheme, the orthogonal AF (OAF) protocol, or the non-orthogonal AF (NAF) protocol using a non-diagonal covariance matrix. The choice of protocol depends on the signal-to-noise ratio (SNR) and network configuration. By further analyzing the asymptotic mutual information, it is shown that the DT scheme is dominant in low and high SNR regimes. The capacity of the AF system is also provided for comparison among several network models. Specifically, it is demonstrated that in a general linear network model, the NAF protocol can provide significant gains over the other schemes at medium SNRs.