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In this work, we study a multi-user wireless powered communication network(WPCN), where a multi-antenna base station (BS) sends an energy signal tomultiple single-antenna users equipped with non-linear energy harvesting (EH)circuits. The users, in turn, harvest energy from the received signal andutilize it for information transmission in the uplink. Furthermore, to jointlyoptimize the energy signal waveform and downlink beamforming, we assume thatthe BS broadcasts a pulse-modulated signal employing multiple energy signalvectors. We formulate an optimization problem for the joint design of thedownlink transmit energy signal vectors, their number, the durations of thetransmit pulses, and the time allocation policy for minimization of the averagetransmit power at the BS. We show that for single-user WPCNs, a single energysignal vector, which is collinear with the maximum ratio transmission (MRT)vector and drives the EH circuit at the user device into saturation, isoptimal. Next, for the general multi-user case, we show that the optimal signaldesign requires a maximum number of energy signal vectors that exceeds thenumber of users by one and propose an algorithm to obtain the optimal energysignal vectors. Since the complexity of the optimal design is high, we alsopropose two suboptimal schemes for WPCN design. First, for asymptotic massiveWPCNs, where the ratio of the number of users to the number of BS antennas,i.e., the system load, tends to zero, we show that the optimal downlinktransmit signal can be obtained in closed-form and comprises a sequence ofweighted sums of MRT vectors. Next, based on this result, for general WPCNswith finite system loads, we propose a suboptimal closed-form MRT-based designand a suboptimal semidefinite relaxation (SDR)-based scheme.

Optimal Energy Signal Design for Multi-user MISO WPCNs With Non-linear Energy Harvesting Circuits