On the Capacity of the Static Half-Duplex Non-Orthogonal AF Relay Channel

In this paper, we analyze the capacity of the static half-duplex single-relay amplify-and-forward (AF) system under both per-node and joint power constraints. Different from multiple-input multiple-output systems, the channel matrix of the cooperative AF system is a function of the parameters to be optimized and hence water-filling over the square of the singular values is no longer optimal. Furthermore, given that the mutual information of the AF system is not a concave function, conventional optimization methods cannot be applied. Instead, by deriving and comparing all local solutions, we characterize the optimal input covariance matrix at the source and the optimal power allocation scheme at the relay that maximize the achievable rate. First, for the individual power constraint scenario, it is shown that the capacity of the AF system is achieved by either a direct transmission (DT) scheme, a non-orthogonal AF (NAF) beamforming (BF) protocol with a unit-rank covariance matrix, or a NAF system using a specific full-rank covariance matrix. Then, for the global power constraint scenario, it is shown that only a DT or a NAF-BF protocol can achieve the capacity. In both cases, orthogonal transmission is strictly suboptimal. The capacity of the AF system is finally analyzed for some concrete examples, such as under asymptotically high and low transmission powers and for several network models.