Optimal Energy Beamforming for Distributed Wireless Power Transfer Over Frequency-Selective Channels

This paper analyzes the optimal transmission strategy and power allocation for a distributed wireless power transfer (WPT) system operating over frequency-selective fading channels. We consider K coordinated energy transmitters (CETs) coherently transmitting energy to a single user over N > K subchannels with individual power constraints. To maximize the total harvested energy, we derive the optimal co-phasing power allocation rule which has the following properties: 1) For any given subchannel, if the optimal power allocation of one CET is zero, then the power allocated by all the other K - 1 CETs to that subchannel is also zero (i.e., the subchannel is inactive); 2) For the non-zero power subchannels, the optimal power allocation obeys a proportionality principle that establishes a relationship between the powers allocated by all K CETs to all active subchannels. Based on this property, we prove that the optimal distributed WPT strategy is for all CETs to select no more than K subchannels. This is in sharp contrast to wireless information transmission where more than K subchannels may be used for capacity maximization. Numerical examples verify our theoretical results and show the performance gains of our proposed scheme compared to two benchmark schemes.