Optimal Discrete Beamforming for Intelligent Reflecting Surface

This work pursues an optimal strategy of designing passive beamformer for intelligent reflecting surface (IRS) in order to maximize the overall channel strength. In particular, the choice of phase shift for each reflective element is restricted to $K \geq 2$ discrete values. Although the resulting discrete beamforming problem is believed to be NP-hard in some prior works, the paper shows that the global optimum of the binary case with $K=2$ can be achieved in linear time. For a general $K$ -ary beamforming problem with $K > 2$ , the state-of-the-art polynomial time algorithm is to greedily project the relaxed solution to the closest point in the constraint set. However, as shown in the paper, the performance of this greedy method cannot be guaranteed. In contrast, we propose a linear time algorithm that is capable of reaching a near-optimal solution with an approximation ratio of $(1+\cos(\pi/K))/2$ , i.e., its performance is at least 75% of the global optimum for $K\geq 3$ . Furthermore, inspired by the RFocus method in [1], we develop a statistic implementation of the above approximation algorithm in the absence of channel state information (CSI).