  Intelligent reflecting surfaces (IRSs) constitute passive devices, which are
capable of adjusting the phase shifts of their reflected signals, and hence
they are suitable for passive beamforming. In this paper, we conceive their
design with the active beamforming action of multiple-input multipleoutput
(MIMO) systems used at the access points (APs) for improving the beamforming
gain, where both the APs and users are equipped with multiple antennas.
Firstly, we decouple the optimization problem and design the active beamforming
for a given IRS configuration. Then we transform the optimization problem of
the IRS-based passive beamforming design into a tractable non-convex
quadratically constrained quadratic program (QCQP). For solving the transformed
problem, we give an approximate solution based on the technique of widely used
semidefinite relaxation (SDR). We also propose a low-complexity iterative
solution. We further prove that it can converge to a locally optimal value.
Finally, considering the practical scenario of discrete phase shifts at the
IRS, we give the quantization design for IRS elements on basis of the two
solutions. Our simulation results demonstrate the superiority of the proposed
solutions over the relevant benchmarks.
