  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.
