  In this paper, we investigate the robust linear precoder design for three
dimensional (3D) massive multi-input multi-output (MIMO) downlink with uniform
planar array (UPA) and imperfect channel state information (CSI). In practical
massive MIMO with UPAs, the number of antennas in each column or row is usually
limited. The straightforward extension of the conventional DFT based beam
domain channel model widely used in massive MIMO with uniform linear arrays
(ULAs) can not apply. To overcome this issue, we establish a new beam domain
channel model by using sampled steering vectors. Then, a novel method to obtain
the beam domain channel power matrices and the instantaneous beam domain
channel coefficients is proposed, and an a posteriori beam domain channel model
which includes the channel aging and the spatial correlation is established. On
the basis of the a posteriori channel model, we consider the robust precoder
design with the expected weighted sum-rate maximization under a total power
constraint. By viewing the power constraint as a Riemannian manifold, we
transform the constrained optimization problem into an unconstrained
optimization problem on the Riemannian manifold. Then, we derive an iterative
algorithm to obtain the optimal precoders by setting the Riemannian gradient of
the objective function to zero. Furthermore, we propose a low complexity robust
precoder design by replacing the expected rates in the objective function with
their upper bounds. Simulation results show that the proposed precoders can
achieve significant performance gain than the widely used regularized zero
forcing (RZF) precoder and signal to leakage noise ratio (SLNR) precoder.
