  In massive MIMO systems, hybrid beamforming is an essential technique for
exploiting the potential array gain without using a dedicated RF chain for each
antenna. In this work, we consider the data phase in a massive MIMO
communication process, where the transmitter and receiver use fewer RF chains
than antennas. We examine several different fully- and partially connected
schemes and consider the design of hybrid beamformers that minimize the
estimation error in the data. For the hybrid precoder, we introduce a framework
for approximating the optimal fully-digital precoder with a feasible hybrid
one. We exploit the fact that the fully-digital precoder is unique only up to a
unitary matrix and optimize over this matrix and the hybrid precoder
alternately. Our alternating minimization of approximation gap (Alt-MaG)
framework improves the performance over state-of-the-art methods with no
substantial increase in complexity. In addition, we present a special case of
Alt-MaG, minimal gap iterative quantization (MaGiQ), that results in low
complexity and lower mean squared error (MSE) than other common methods, in the
case of very few RF chains. MaGiQ is shown to coincide with the optimal
fully-digital solution in some scenarios. For combiner design, we exploit the
structure of the MSE objective and develop a greedy ratio trace maximization
technique, that achieves low MSE under various settings. All of our algorithms
can be used with multiple hardware architectures.
