  This paper investigates the downlink (DL) transmit design for massive
multiple-input multiple-output (MIMO) low-earth-orbit (LEO) satellite
communication systems, where only the slow-varying statistical channel state
information is exploited at the transmitter. The channel model for the DL
massive MIMO LEO satellite system is established, in which both the satellite
and the user terminals (UTs) are equipped with uniform planar arrays. Observing
the rank-one property of the channel matrices, we show that the single-stream
precoding for each UT is the optimal choice that maximizes the ergodic sum
rate. This favorable result simplifies the complicated design of transmit
covariance matrices into that of precoding vectors without any loss of
optimality. Then, an efficient algorithm is devised to compute the precoding
vectors. Furthermore, we formulate an approximate transmit design based on the
upper bound on the ergodic sum rate, for which the optimality of single-stream
precoding still holds. We show that, in this case, the design of precoding
vectors can be simplified into that of scalar variables, for which an effective
algorithm is developed. In addition, a low-complexity learning framework is
proposed for optimizing the scalar variables. Simulation results demonstrate
that the proposed approaches can achieve significant performance gains over the
existing schemes.
