In this paper, we study an intelligent reflecting surface (IRS) assisted 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). The IRS phase shifts are adjusted so as to maximize the ideal SNR (i.e., without CSI error), based only on the estimated channels. First, we derive a closed-form expression of the worst-case SNR, and then formulate the robust (discrete) optimization problem. Moreover, we design and analyze 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. Finally, numerical simulations confirm the theoretical results. In particular, the proposed algorithm shows identical performance with the exhaustive search, and significantly outperforms a baseline scheme, namely, the activation of all IRS elements.

翻译：本文研究在非完美信道状态信息下，由智能反射表面辅助的单天线收发机通信系统。具体而言，我们针对有界信道不确定性，在满足最小信噪比约束的前提下，通过鲁棒选择IRS单元的二进制（开启/关闭）状态来最大化最差情况下的能量效率。IRS的相位调整基于仅估计的信道，以使理想信噪比（即无信道误差时）最大化。首先，我们推导出最差情况信噪比的闭式表达式，进而构建鲁棒（离散）优化问题。此外，我们设计并分析了一种动态规划算法，该算法理论上能够以多项式复杂度$O(L \log L)$（其中$L$为IRS单元数量）实现全局最优。最后，数值仿真验证了理论结果。特别地，所提算法性能与穷举搜索完全一致，且显著优于基线方案（即激活所有IRS单元）。