  The directionality of millimeter-wave (mmWave) communications introduces a
significant challenge in serving fast-rotating/moving terminals, e.g., mobile
AR/VR, high-speed vehicles, trains, UAVs.This challenge is exacerbated in
mmWave systems using analog beamforming, because of the inherent non-convexity
in the analog beam tracking problem. In this paper, we obtain the Cram\'er-Rao
lower bound (CRLB) of beam tracking and optimize the analog beamforming vectors
to get the minimum CRLB. Then, we develop a low complexity analog beam tracking
algorithm that simultaneously optimizes the analog beamforming vector and the
estimate of beam direction. Finally, by establishing a new basic theory, we
provide the theoretical convergence analysis of the proposed analog beam
tracking algorithm, which proves that the minimum CRLB of the MSE is achievable
with high probability. Our simulations show that this algorithm can achieve
faster tracking speed, higher tracking accuracy and higher data rate than
several state-of-the-art algorithms. The key analytical tools used in our
algorithm design are stochastic approximation and recursive estimation with a
control parameter.
