  In this paper, the problem of outer beamformer design based only on channel
statistic information is considered for two-stage beamforming for multi-user
massive MIMO downlink, and the problem is approached based on
signal-to-leakage-plus-noise ratio (SLNR). To eliminate the dependence on the
instantaneous channel state information, a lower bound on the average SLNR is
derived by assuming zero-forcing (ZF) inner beamforming, and an outer
beamformer design method that maximizes the lower bound on the average SLNR is
proposed. It is shown that the proposed SLNR-based outer beamformer design
problem reduces to a trace quotient problem (TQP), which is often encountered
in the field of machine learning. An iterative algorithm is presented to obtain
an optimal solution to the proposed TQP. The proposed method has the capability
of optimally controlling the weighting factor between the signal power to the
desired user and the interference leakage power to undesired users according to
different channel statistics. Numerical results show that the proposed outer
beamformer design method yields significant performance gain over existing
methods.
