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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 to multiple single-antenna users equipped with non-linear energy harvesting (EH) circuits. The users, in turn, harvest energy from the received signal and utilize it for information transmission in the uplink. Furthermore, to jointly optimize the energy signal waveform and downlink beamforming, we assume that the BS broadcasts a pulse-modulated signal employing multiple energy signal vectors. We formulate an optimization problem for the joint design of the downlink transmit energy signal vectors, their number, the durations of the transmit pulses, and the time allocation policy for minimization of the average transmit power at the BS. We show that for single-user WPCNs, a single energy signal vector, which is collinear with the maximum ratio transmission (MRT) vector and drives the EH circuit at the user device into saturation, is optimal. Next, for the general multi-user case, we show that the optimal signal design requires a maximum number of energy signal vectors that exceeds the number of users by one and propose an algorithm to obtain the optimal energy signal vectors. Since the complexity of the optimal design is high, we also propose two suboptimal schemes for WPCN design. First, for asymptotic massive WPCNs, 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 downlink transmit signal can be obtained in closed-form and comprises a sequence of weighted sums of MRT vectors. Next, based on this result, for general WPCNs with finite system loads, we propose a suboptimal closed-form MRT-based design and a suboptimal semidefinite relaxation (SDR)-based scheme.