Unicast Multi-Antenna Relay Beamforming With Per-Antenna Power Control: Optimization and Duality

We consider amplify-and-forward multi-antenna relaying between a single pair of source and destination under relay per-antenna power constraints. We design the optimal relay processing matrix to minimize the maximum per-antenna power budget for a received SNR target. With given transmit and receive beamformers at the source and destination, respectively, we first focus on the equivalent system with single-antenna source and destination. Although non-convex, we show that the optimization satisfies strong Lagrange duality and can be solved in the Lagrangian dual domain. We reveal a prominent structure of this problem, by establishing its duality with direct SIMO beamforming system with an uncertain noise. This enables us to derive a semi-closed form expression for the optimal relay processing matrix that depends on a set of dual variables, which can be determined through numerical optimization with a significantly reduced problem space. We further show that the dual problem has a semi-definite programming form, which enables efficient numerical optimization methods to determine the dual variables with polynomial complexity. Using this result, the reverse problem of SNR maximization under a set of relay per-antenna power constraints is then addressed. We then consider the maximum relay beamforming achievable rate under different combinations of antenna setups at source and destination. In particular, we generalize the duality to MIMO relay beamforming vs. direct MIMO beamforming, and establish the dual relation of the two systems for different multi-antenna setups at source and destination.