  In this paper, multiuser beam training based on hierarchical codebook for
millimeter wave massive multi-input multi-output is investigated, where the
base station (BS) simultaneously performs beam training with multiple user
equipments (UEs). For the UEs, an alternative minimization method with a
closed-form expression (AMCF) is proposed to design the hierarchical codebook
under the constant modulus constraint. To speed up the convergence of the AMCF,
an initialization method based on Zadoff-Chu sequence is proposed. For the BS,
a simultaneous multiuser beam training scheme based on an adaptively designed
hierarchical codebook is proposed, where the codewords in the current layer of
the codebook are designed according to the beam training results of the
previous layer. The codewords at the BS are designed with multiple mainlobes,
each covering a spatial region for one or more UEs. Simulation results verify
the effectiveness of the proposed hierarchical codebook design schemes and show
that the proposed multiuser beam training scheme can approach the performance
of the beam sweeping but with significantly reduced beam training overhead.
