  In this paper, we study the channel estimation problem in correlated massive
multiple-input-multiple-output (MIMO) systems with a reduced number of
radio-frequency (RF) chains. Importantly, other than the knowledge of channel
correlation matrices, we make no assumption as to the structure of the channel.
To address the limitation in the number of RF chains, we employ hybrid
beamforming, comprising a low dimensional digital beamformer followed by an
analog beamformer implemented using phase shifters. Since there is no dedicated
RF chain per transmitter/receiver antenna, the conventional channel estimation
techniques for fully-digital systems are impractical. By exploiting the fact
that the channel entries are uncorrelated in its eigen-domain, we seek to
estimate the channel entries in this domain. Due to the limited number of RF
chains, channel estimation is typically performed in multiple time slots. Under
a total energy budget, we aim to design the hybrid transmit beamformer
(precoder) and the receive beamformer (combiner) in each training time slot, in
order to estimate the channel using the minimum mean squared error criterion.
To this end, we choose the precoder and combiner in each time slot such that
they are aligned to transmitter and receiver eigen-directions, respectively.
Further, we derive a water-filling-type expression for the optimal energy
allocation at each time slot. This expression illustrates that, with a low
training energy budget, only significant components of the channel need to be
estimated. In contrast, with a large training energy budget, the energy is
almost equally distributed among all eigen-directions. Simulation results show
that the proposed channel estimation scheme can efficiently estimate correlated
massive MIMO channels within a few training time slots.
