  In millimeter wave cellular communication, fast and reliable beam alignment
via beam training is crucial to harvest sufficient beamforming gain for the
subsequent data transmission. In this paper, we establish fundamental limits in
beam-alignment performance under both the exhaustive search and the
hierarchical search that adopts multi-resolution beamforming codebooks,
accounting for time-domain training overhead. Specifically, we derive lower and
upper bounds on the probability of misalignment for an arbitrary level in the
hierarchical search, based on a single-path channel model. Using the method of
large deviations, we characterize the decay rate functions of both bounds and
show that the bounds coincide as the training sequence length goes large. We go
on to characterize the asymptotic misalignment probability of both the
hierarchical and exhaustive search, and show that the latter asymptotically
outperforms the former, subject to the same training overhead and codebook
resolution. We show via numerical results that this relative performance
behavior holds in the non-asymptotic regime. Moreover, the exhaustive search is
shown to achieve significantly higher worst-case spectrum efficiency than the
hierarchical search, when the pre-beamforming signal-to-noise ratio (SNR) is
relatively low. This study hence implies that the exhaustive search is more
effective for users situated further from base stations, as they tend to have
low SNR.
