  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.
