  Massive multi-input multi-output (MIMO) can support high spectral efficiency
(SE) with simple linear transceivers, and is expected to provide high energy
efficiency (EE). In this paper, we analyze the EE of downlink multi-cell
massive MIMO systems under spatially correlated channel model, where both
transmit and circuit power consumptions, training overhead, channel estimation
and pilot contamination (PC) are taken into account. We obtain the maximal EE
for the systems with maximum-ratio transmission (MRT) and zero-forcing
beamforming (ZFBF) for given number of antennas and users by optimizing the
transmit power. The closed-form expressions of approximated optimal transmit
power and maximal EE, and their scaling laws with the number of antennas M are
derived for the systems with MRT and ZFBF. Our analysis shows that the maximal
EE decreases with M for both systems with and without PC, but with different
descending speeds. For the system without PC, the optimal transmit power should
be configured to increases with M, while for the system with PC, the optimal
transmit power should be configured as a constant independent from M. The
analytical results are validated by simulations under a more realistic
three-dimensional channel model.
