  Massive multiple-input multiple-output (MIMO) is promising for low earth
orbit (LEO) satellite communications due to the potential in enhancing the
spectral efficiency. However, the conventional fully digital precoding
architectures might lead to high implementation complexity and energy
consumption. In this paper, hybrid analog/digital precoding solutions are
developed for the downlink operation in LEO massive MIMO satellite
communications, by exploiting the slow-varying statistical channel state
information (CSI) at the transmitter. First, we formulate the hybrid precoder
design as an energy efficiency (EE) maximization problem by considering both
the continuous and discrete phase shift networks for implementing the analog
precoder. The cases of both the fully and the partially connected architectures
are considered. Since the EE optimization problem is nonconvex, it is in
general difficult to solve. To make the EE maximization problem tractable, we
apply a closed-form tight upper bound to approximate the ergodic rate. Then, we
develop an efficient algorithm to obtain the fully digital precoders. Based on
which, we further develop two different efficient algorithmic solutions to
compute the hybrid precoders for the fully and the partially connected
architectures, respectively. Simulation results show that the proposed
approaches achieve significant EE performance gains over the existing
baselines, especially when the discrete phase shift network is employed for
analog precoding.
