Efficient beam training is the key challenge in the codebook-based configuration of reconfigurable intelligent surfaces (RISs) because the beam training overhead can have a strong impact on the achievable system performance. In this paper, we study the performance tradeoff between overhead and achievable signal-to-noise ratio (SNR) in RIS beam training while taking into account the size of the targeted coverage area, the RIS response time, and the delay for feedback transmissions. Thereby, we consider three common beam training strategies: full search (FS), hierarchical search (HS), and tracking-based search (TS). Our analysis shows that the codebook-based illumination of a given coverage area can be realized with wide- or narrow-beam designs, which result in two different scaling laws for the achievable SNR. Similarly, there are two regimes for the overhead, where the number of pilot symbols required for reliable beam training is dependent on and independent of the SNR, respectively. Based on these insights, we investigate the impact of the beam training overhead on the effective rate and provide an upper bound on the user velocity for which the overhead is negligible. Moreover, when the overhead is not negligible, we show that TS beam training achieves higher effective rates than HS and FS beam training, while HS beam training may or may not outperform FS beam training, depending on the RIS response time, feedback delay, and codebook size. Finally, we present numerical simulation results that verify our theoretical analysis. In particular, our results confirm the existence of the proposed regimes, reveal that fast RISs can lead to negligible overhead for FS beam training, and show that large feedback delays can significantly reduce the performance for HS beam training.

翻译：高效波束训练是基于码本配置可重构智能表面（RIS）的关键挑战，因为波束训练开销对可达系统性能具有重要影响。本文研究了RIS波束训练中开销与可实现信噪比（SNR）之间的性能权衡，同时考虑了目标覆盖区域尺寸、RIS响应时间及反馈传输时延。为此，我们分析了三种常见波束训练策略：全搜索（FS）、分层搜索（HS）和基于跟踪的搜索（TS）。研究表明，对给定覆盖区域的码本照明可通过宽波束或窄波束设计实现，这两种设计导致可实现SNR存在两种不同标度律。类似地，开销存在两种状态：可靠波束训练所需导频符号数量分别依赖于SNR和独立于SNR。基于这些发现，我们探究了波束训练开销对有效速率的影响，并给出了使开销可忽略的用户速度上界。此外，当开销不可忽略时，我们证明TS波束训练相比HS和FS能获得更高有效速率，而HS波束训练是否优于FS则取决于RIS响应时间、反馈时延及码本尺寸。最后，我们给出数值仿真结果验证理论分析。特别地，仿真结果确认了所提状态的存在性，揭示快速RIS可使FS波束训练的开销可忽略，并表明大反馈时延会显著降低HS波束训练的性能。