Millimeter Wave Beam Alignment: Large Deviations Analysis and Design Insights

TL;DR

This paper analyzes the fundamental limits of beam alignment in millimeter wave communication, comparing exhaustive and hierarchical search methods, and finds that exhaustive search generally outperforms hierarchical search especially at low SNR.

Contribution

It provides the first large deviations analysis of beam alignment performance bounds, characterizes asymptotic misalignment probabilities, and compares the effectiveness of exhaustive versus hierarchical search methods.

Findings

Exhaustive search asymptotically outperforms hierarchical search in misalignment probability.

Exhaustive search achieves higher spectrum efficiency at low SNR.

Numerical results confirm the theoretical performance differences in practical regimes.

Abstract

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.