In this paper, we study an intelligent reflecting surface (IRS)-aided communication system with single-antenna transmitter and receiver, under imperfect channel state information (CSI). More specifically, we deal with the robust selection of binary (on/off) states of the IRS elements in order to maximize the worst-case energy efficiency (EE), given a bounded CSI uncertainty, while satisfying a minimum signal-to-noise ratio (SNR). In addition, we consider not only continuous but also discrete IRS phase shifts. First, we derive closed-form expressions of the worst-case SNRs, and then formulate the robust (discrete) optimization problems for each case. In the case of continuous phase shifts, we design a dynamic programming (DP) algorithm that is theoretically guaranteed to achieve the global maximum with polynomial complexity $O(L\,{\log L})$, where $L$ is the number of IRS elements. In the case of discrete phase shifts, we develop a convex-relaxation-based method (CRBM) to obtain a feasible (sub-optimal) solution in polynomial time $O(L^{3.5})$, with a posteriori performance guarantee. Furthermore, numerical simulations provide useful insights and confirm the theoretical results. In particular, the proposed algorithms are several orders of magnitude faster than the exhaustive search when $L$ is large, thus being highly scalable and suitable for practical applications. Moreover, both algorithms outperform a baseline scheme, namely, the activation of all IRS elements.

翻译：本文研究了在不完美信道状态信息下，由单天线发射机和接收机构成的智能反射面辅助通信系统。具体而言，我们处理IRS单元二元（开/关）状态的鲁棒选择问题，旨在给定有界CSI不确定性的条件下，最大化最坏情况下的能效，同时满足最低信噪比要求。此外，我们不仅考虑了连续的IRS相移，也考虑了离散的IRS相移。首先，我们推导了最坏情况SNR的闭式表达式，然后针对每种情况分别构建了鲁棒（离散）优化问题。在连续相移情况下，我们设计了一种动态规划算法，该算法在理论上保证能以多项式复杂度$O(L\,{\log L})$达到全局最优，其中$L$是IRS单元数量。在离散相移情况下，我们开发了一种基于凸松弛的方法，以在多项式时间$O(L^{3.5})$内获得一个可行的（次优）解，并具有后验性能保证。此外，数值仿真提供了有益的见解并验证了理论结果。特别地，当$L$较大时，所提算法比穷举搜索快数个数量级，因而具有高度可扩展性并适用于实际应用。而且，两种算法的性能均优于基线方案，即激活所有IRS单元。