Distributed Multiantenna Frequency-Selective Energy Beamforming With Joint Total and Individual Power Constraints

We analyze distributed multi-antenna energy beamforming over frequency selective fading channels in wireless power transfer (WPT) systems with joint total and individual transmit power constraints. The constraints allow the WPT system to limit energy consumption for cost or environmental factors, and prevent individual coordinated energy transmitters (CETs) from overdriving their high-powered amplifiers due to hardware limitations. The harvested power at the energy receiver is modeled as an arbitrary non-linear, continuous, and non-decreasing function of the received radio frequency (RF) power. We show that the optimal frequency selective energy beamforming solution for <inline-formula> <tex-math notation="LaTeX">${K}$ </tex-math></inline-formula> CETs with <inline-formula> <tex-math notation="LaTeX">${M}$ </tex-math></inline-formula> antennas each is equivalent to maximizing the total RF power coherently combined at the energy receiver over <inline-formula> <tex-math notation="LaTeX">${N}$ </tex-math></inline-formula> shared subchannels. We obtained the optimal frequency selective energy beamforming strategy with the following insightful properties: First, for any subchannel, either all antennas of all <inline-formula> <tex-math notation="LaTeX">${K}$ </tex-math></inline-formula> CETs allocate zero power to this subchannel (i.e., inactive), or all antennas of all <inline-formula> <tex-math notation="LaTeX">${K}$ </tex-math></inline-formula> CETs allocate non-zero power to this subchannel (i.e., active). Second, for the active subchannels, we show that the optimal power allocation obeys a proportionality principle for the powers from all antennas of all <inline-formula> <tex-math notation="LaTeX">${K}$ </tex-math></inline-formula> CETs, which reveals that higher powers are not necessarily allocated to subchannels with higher channel gains. Third, we prove that optimally no more than <inline-formula> <tex-math notation="LaTeX">${T}\,\,+$ </tex-math></inline-formula> 1 subchannels are selected for energy beamforming, where <inline-formula> <tex-math notation="LaTeX">${T}$ </tex-math></inline-formula> (less than <inline-formula> <tex-math notation="LaTeX">${K}$ </tex-math></inline-formula>) is the number of CETs transmitting at their maximum individual powers due to the total power constraint. Based on the above properties, we show that the energy beamforming solution can be efficiently achieved via solving a low-complexity dual problem in simpler form with only <inline-formula> <tex-math notation="LaTeX">${K} +1$ </tex-math></inline-formula> variables compared to the original primal problem with NMK variables. Numerical examples verify our theoretical findings and show the impact of the joint total and individual power constraints on the average harvested power.