  This paper investigates a joint beamforming design in a multiuser
multiple-input single-output (MISO) communication network aided with an
intelligent reflecting surface (IRS) panel. The symbol-level precoding (SLP) is
adopted to enhance the system performance by exploiting the multiuser
interference (MUI) with consideration of bounded channel uncertainty. The joint
beamforming design is formulated into a nonconvex worst-case robust programming
to minimize the transmit power subject to single-to-noise ratio (SNR)
requirements. To address the challenges due to the constant modulus and the
coupling of the beamformers, we first study the single-user case. Specifically,
we propose and compare two algorithms based on the semidefinite relaxation
(SDR) and alternating optimization (AO) methods, respectively. It turns out
that the AO-based algorithm has much lower computational complexity but with
almost the same power to the SDR-based algorithm. Then, we apply the AO
technique to the multiuser case and thereby develop an algorithm based on the
proximal gradient descent (PGD) method. The algorithm can be generalized to the
case of finite-resolution IRS and the scenario with direct links from the
transmitter to the users. Numerical results show that the SLP can significantly
improve the system performance. Meanwhile, 3-bit phase shifters can achieve
near-optimal power performance.
