  This work studies a beamforming design for downlink transmission of a
multi-user multiple-input single-output (MISO) system where each legitimate
user employs a power splitting (PS) based simultaneous wireless information and
power transfer (SWIPT) technique. The transmitter intends to send confidential
information to its legitimate users in the presence of purely unknown
eavesdroppers. Since the transmitter does not have any knowledge of the
eavesdroppers' channel state information (CSI), we consider an artificial noise
(AN) approach to establishing secure communication. This beamforming design is
developed by maximizing the AN power to interfere with the eavesdropper as much
as possible. Based on the assumption of imperfect CSI of legitimate users at
the transmitter, two robust design approaches for the joint beamforming and PS
ratio have been studied to maximize the AN power under both energy harvesting
(EH) and signal-to-interference-plus-noise ratio (SINR) requirements at each
legitimate user. In the first robust design, we consider the bounded channel
uncertainties, and employ semidefinite relaxation (SDR) and a linear matrix
inequality (LMI) representation to transform the original problem into a
semidefinite program (SDP). In the second robust design, we consider the
statistical channel uncertainties, and show that the proposed problem can be
reformulated into another form of SDP through both SDR and Bernstein-type
inequality. In addition, the non-linear energy harvesting (EH) model is
incorporated in this work as it could reflect the characteristics of practical
radio frequency(RF)-EH conversion circuit. Simulation results have been
provided to demonstrate the performance of our proposed robust designs.
