  The directionality of millimeter-wave (mmWave) communications creates a
significant challenge in serving fast-moving mobile terminals on, e.g.,
high-speed vehicles, trains, and UAVs. This challenge is exacerbated in mmWave
systems using analog antenna arrays, because of the inherent non-convexity in
the control of the phase shifters. In this paper, we develop a recursive beam
tracking algorithm which can simultaneously achieve fast tracking speed, high
tracking accuracy, low complexity, and low pilot overhead. In static scenarios,
this algorithm converges to the minimum Cram\'er-Rao lower bound (CRLB) of beam
tracking with high probability. In dynamic scenarios, even at SNRs as low as
0dB, our algorithm is capable of tracking a mobile moving randomly at an
absolute angular velocity of 10-20 degrees per second, using only 5 pilot
symbols per second. If combining with a simple TDMA pilot pattern, this
algorithm can track hundreds of high-speed mobiles in 5G configurations. Our
simulations show that the tracking performance of this algorithm is much better
than several state-of-the-art algorithms.
