On Achievable Rate and Ergodic Capacity of NAF Multi-Relay Networks with CSI

This paper investigates the achievable rate and ergodic capacity of a non-orthogonal amplify-and-forward (NAF) half-duplex multi-relay network where multiple relays exploit channel state information (CSI) to cooperate with a pair of source and destination. In the first step, for a given input covariance matrix at the source, we derive an optimal power allocation scheme among the relays via optimal instantaneous power amplification coefficients to maximize the achievable rate. Given the nature of broadcasting and receiving collisions in NAF, the considered problem in this step is non-convex. To overcome this drawback, we propose a novel method by evaluating the achievable rate in different sub-domains of the vector channels. It is then demonstrated that the globally optimal solution can be derived in closed-form. In the next step, we establish the ergodic channel capacity by jointly optimizing the input covariance matrix at the source and the power allocation among the relays. We show that this is a bi-level non-convex problem and solve it using Tammer decomposition method. This approach allows us to transform the original optimization problem into an equivalent master problem and a set of sub-problems having closed-form solutions derived in the first step. The channel capacity is then obtained using an iterative water-filling-based algorithm. Finally, we analyze the capacity-achieving input covariance matrix at the source in high and low signal-to-noise ratio (SNR) regimes. At sufficiently high SNRs, it is shown that the transmit power at the source should be equally distributed in all broadcasting and cooperative phases. On the other hand, in low SNR regions, the source should spend all its power in the broadcasting phase associated with a relay having the strongest cascaded source-relay and relay-destination channels.