  We study two-user multiple-input single-output (MISO) wireless powered
communication networks (WPCNs), where the user devices are equipped with
non-linear energy harvesting (EH) circuits. We consider time-division duplex
(TDD) transmission, where the users harvest power from the signal received in
the downlink phase, and then, utilize this harvested power for information
transmission in the uplink phase. In contrast to existing works, we adopt a
non-linear model of the harvested power based on a precise analysis of the
employed EH circuit. We jointly optimize the beamforming vectors in the
downlink and the time allocated for downlink and uplink transmission to
minimize the average transmit power in the downlink under per-user data rate
constraints in the uplink. We provide conditions for the feasibility of the
resource allocation problem and the existence of a trivial solution,
respectively. For the case where the resource allocation has a non-trivial
solution, we show that it is optimal to employ no more than three beamforming
vectors for power transfer in the downlink. To determine these beamforming
vectors, we develop an iterative algorithm based on semi-definite relaxation
(SDR) and successive convex approximation (SCA). Our simulation results reveal
that the proposed resource allocation scheme outperforms two baseline schemes
based on linear and sigmoidal EH models, respectively.
