Optimal Frequency-Selective Energy Beamforming with Joint Total and Individual Power Constraints

This paper analyzes the optimal energy beamforming solution for a multiple-input single-output wireless power transfer (WPT) system over frequency-selective fading channels with joint total and individual antenna power constraints. To maximize the total harvested energy, we derive the optimal co-phasing power allocation rule which reveals that all K antennas will participate in energy beamforming with T < K antennas transmitting with their maximum individual powers due to the total power constraint. We prove that optimally no more than T + 1 subchannels are selected for power allocation. We highlight that the optimal power allocation solution can be efficiently obtained based on the corresponding low-complexity dual problem. Our proposed power allocation algorithm generalizes previous solutions considering only total or individual power constraints. Numerical examples verify our theoretical results and show the impact of the joint total and individual power constraints on the average harvested power.