Optimal Energy Beamforming under Per-Antenna Power Constraint

Energy beamforming (EB) is a key technique to significantly enhance the efficiency of wireless power transfer (WPT). In this paper, we study optimal EB under per-antenna power constraint (PAC) which is more practical than conventional sum-power constraint (SPC) at multi antenna energy transmitter (ET). We consider a broadcast network, where one multi antenna ET with PAC, transfers wireless energy for energy receivers (ER)s which are randomly placed within the cell area. First, we consider sum energy maximization problem with PAC without fairness and provide the optimal solution structure for general case. This optimal structure implies that similar to SPC, sending one energy beam is optimal with PAC which means that the rank of transmit covariance matrix is one. We also derive closed-form solutions for two special cases and propose two sub-optimal solutions for general case, which are very close to optimal numerical results. To consider the fairness among the ERs, we further propose max-min fair problem with PAC, and analyze it for the special case of two transmit antennas. Simulation results show its advantages in comparison to the recent works in the literature.