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In this paper, we study multi-user multi-antenna wireless power transfer(WPT) systems, where each antenna at the energy harvesting (EH) nodes isconnected to a dedicated non-linear rectifier. We propose an optimal transmitstrategy which maximizes a weighted sum of the average harvested powers at theEH nodes under a constraint on the power budget of the transmitter. First, formultiple-input single-output (MISO) WPT systems, we show that it is optimal totransmit scalar symbols with an arbitrary phase and an amplitude, whoseprobability density function (pdf) has at most two mass points, using maximumratio transmission (MRT) beamforming. Then, we prove that for single-inputmultiple-output (SIMO) WPT systems, the optimal transmit symbol amplitudes arediscrete random variables, whose pdf also has no more than two mass points. Forgeneral multi-user MIMO WPT systems, we show that the optimal transmit strategyinvolves scalar unit-norm symbols with arbitrary phase and at most twobeamforming vectors. In order to determine these vectors, we formulate anon-convex optimization problem and obtain an optimal solution based onmonotonic optimization. Since the computational complexity of the optimalsolution is high, we also propose a low-complexity iterative algorithm toobtain a suboptimal solution, which achieves nearly optimal performance. Oursimulation results reveal that the proposed transmit strategy for multi-userMIMO WPT systems outperforms two baseline schemes, which are based on a linearEH model and a single beamforming vector, respectively. For a given transmitpower budget, we show that the harvested power saturates when increasing thenumber of transmit antennas. Finally, we observe that the harvested powerregion spanned by multiple EH nodes is convex and the power harvested at one EHnode can be traded for a higher harvested power at the other nodes.

Optimal Transmit Strategy for Multi-user MIMO WPT Systems With Non-linear Energy Harvesters