Optimal Transmit Strategy for Multi-User MIMO WPT Systems With Non-Linear Energy Harvesters

In this paper, we study multi-user multi-antenna wireless power transfer (WPT) systems, where each antenna at the energy harvesting (EH) nodes is connected to a dedicated non-linear rectifier. We propose optimal transmit strategies which maximize a weighted sum of the average harvested powers at the EH nodes subject to a constraint on the power budget of the transmitter. First, for multiple-input single-output (MISO) WPT systems, we prove that the optimal strategy employs maximum ratio transmission (MRT) beamforming and scalar symbols with arbitrary phases and discrete amplitudes following a probability density function (pdf) with at most two mass points. Then, we prove that for single-input multiple-output (SIMO) WPT systems, the optimal transmit symbol amplitudes are discrete random variables, whose pdf also has no more than two mass points. For general multi-user MIMO WPT, we show that the optimal transmit strategy employs scalar unit-norm symbols with arbitrary phases and at most two beamforming vectors. To determine these vectors, we formulate a non-convex optimization problem and obtain an optimal solution based on monotonic optimization. Since the computational complexity of the optimal solution is high, we propose a low-complexity iterative algorithm to obtain a suboptimal solution, which achieves near-optimal performance. Our simulation results reveal that the proposed transmit strategy for multi-user MIMO WPT systems outperforms baseline schemes based on a linear EH model and a single beamforming vector. For a given transmit power budget, we show that the harvested power saturates when increasing the number of transmit antennas. Finally, we observe that the harvested power region spanned by multiple EH nodes is convex and the power harvested at one EH node can be traded for a higher harvested power at the other nodes.