Distributed Wireless Power Transfer with Energy Feedback
  Energy beamforming (EB) is a key technique for achieving efficient radio-frequency (RF) transmission enabled wireless energy transfer (WET). By optimally designing the waveforms from multiple energy transmitters (ETs) over the wireless channels, they can be constructively combined at the energy receiver (ER) to achieve an EB gain that scales with the number of ETs. However, the optimal design of EB waveforms requires accurate channel state information (CSI) at the ETs, which is challenging to obtain practically, especially in a distributed system with ETs at separate locations. In this paper, we study practical and efficient channel training methods to achieve optimal EB in a distributed WET system. We propose two protocols with and without centralized coordination, respectively, where distributed ETs either sequentially or in parallel adapt their transmit phases based on a low-complexity energy feedback from the ER. The energy feedback only depends on the received power level at the ER, where each feedback indicates one particular transmit phase that results in the maximum harvested power over a set of previously used phases. Simulation results show that the two proposed training protocols converge very fast in practical WET systems even with a large number of distributed ETs, while the protocol with sequential ET phase adaptation is also analytically shown to converge to the optimal EB design with perfect CSI by increasing the training time. Numerical results are also provided to evaluate the performance of the proposed distributed EB and training designs as compared to other benchmark schemes. 
Seunghyun Lee, Rui Zhang
cs.IT math.IT