  In the paper, we make an investigation of robust beamforming for secure
directional modulation in the multi-user multiple-input and multiple output
(MU-MIMO) systems in the presence of direction angle measurement errors. When
statistical knowledge of direction angle measurement errors is unavailable, a
novel robust beamforming scheme of combining main-lobe-integration (MLI) and
leakage is proposed to simultaneously transmit multiple different independent
parallel confidential message streams to the associated multiple distinct
desired users. The proposed scheme includes two steps: designing the
beamforming vectors of the useful confidential messages and constructing
artificial noise (AN) projection matrix. Here, in its first step, the
beamforming vectors for the useful confidential messages of desired user k are
given by minimizing the useful signal power leakage from main-lobe of desired
user k to the sum of main-lobes of the remaining desired directions plus
main-lobes of all eavesdropper directions. In its second step, the AN
projection matrix is constructed by simultaneously maximizing the AN power
leakage to all eavesdropper directions such that all eavesdroppers are
disrupted seriously, where AN is viewed by the transmitter as a useful signal
for eavedroppers. Due to independent beamforming vectors for different desired
users, a more secure transmission is achieved. Compared with conventional
non-robust methods, the proposed method can provide a significant improvement
in bit error rate along the desired directions and secrecy-sum-rate towards
multiple desired users without needing statistical property or distribution of
angle measurement errors.
