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Intelligent Reflecting Surfaces (IRS) show a revolutionary potential for wireless communications. In this paper, a single IRS is used to achieve distributed multi-user beamforming and interference-free transmission. We first establish the IRS assisted multi-user system model and formulate an optimization problem called multi-user linearly constrained minimum variance (MU-LCMV) beamformer, under the criterion of minimizing the overall received signal power subject to a certain level of power response (e.g., unit power response) at desired signal directions and arbitrary low power response (e.g., zero power response) at the interference directions. A closed-form amplitudeunconstrained phase-continuous (AUPC) solution is derived first, then an amplitude-constrained phase-continuous (ACPC) solution is obtained by using sequential quadratic programming (SQP). Given the solutions, the IRS beam pattern shows that to achieve multi-user (𝑁 pairs of transceivers, 𝑁 > 1) transmission through a single surface, up to 𝑁 − 1 redundant beams are generated, significantly affecting power efficiency. The directions of the redundant beams are mathematically derived. The effect of mutual coupling on IRS is also analyzed to show the characteristic of side lobes. Simulation results verify the existence and accuracy of the redundant beam directions. This work can potentially enhance state-of-the-art wireless communication systems ranging from transceiver design, system and architecture design, network deployment and self-organizing-network operations.

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In this paper, we model, analyze and optimize the multi-user and multi-order-reflection (MUMOR) intelligent reflecting surface (IRS) networks. We first derive a complete MUMOR IRS network model applicable for the arbitrary times of reflections, size and number of IRSs/reflectors. The optimal condition for achieving sum rate upper bound with one IRS in a closed-form function and the analytical condition to achieve interference-free transmission are derived, respectively. Leveraging this optimal condition, we obtain the MUMOR sum rate upper bound of the IRS network with different network topologies, where the linear graph (LG), complete graph (CG) and null graph (NG) topologies are considered. Simulation results verify our theories and derivations and demonstrate that the sum rate upper bounds of different network topologies are under a 𝐾-fold improvement given 𝐾-piece IRS.