  In this paper, we propose a new beam allocation strategy aiming to maximize
the average successful tracking probability (ASTP) of time-varying
millimeter-wave MIMO systems. In contrast to most existing works that employ
one transmitting-receiving (Tx-Rx) beam pair once only in each training period,
we investigate a more general framework, where the Tx-Rx beam pairs are allowed
to be used repeatedly to improve the received signal powers in specific
directions. In the case of orthogonal Tx-Rx beam pairs, a power-based estimator
is employed to track the time-varying AoA and AoD of the channel, and the
resulting training beam pair sequence design problem is formulated as an
integer nonlinear programming (I-NLP) problem. By dividing the feasible region
into a set of subregions, the formulated I-NLP is decomposed into a series of
concave sub I-NLPs, which can be solved by recursively invoking a nonlinear
branch-and-bound algorithm. To reduce the computational cost, we relax the
integer constraints of each sub I-NLP and obtain a low-complexity solution via
solving the Karush-Kuhn-Tucker conditions of their relaxed problems. For the
case when the Tx-Rx beam pairs are overlapped in the angular space, we estimate
the updated AoA and AoD via an orthogonal matching pursuit (OMP) algorithm.
Moreover, since no explicit expression for the ASTP exists for the OMP-based
estimator, we derive a closed-form lower bound of the ASTP, based on which a
favorable beam pair allocation strategy can be obtained. Numerical results
demonstrate the superiority of the proposed beam allocation strategy over
existing benchmarks.
