  We investigate energy efficiency (EE) optimization for single-cell massive
multiple-input multiple-output (MIMO) downlink transmission with only
statistical channel state information (CSI) available at the base station. We
first show that beam domain transmission is favorable for energy efficiency in
the massive MIMO downlink, by deriving a closed-form solution for the
eigenvectors of the optimal transmit covariance matrix. With this conclusion,
the EE optimization problem is reduced to a real-valued power allocation
problem, which is much easier to tackle than the original large-dimensional
complex matrix-valued precoding design problem. We further propose an iterative
water-filling-structured beam domain power allocation algorithm with low
complexity and guaranteed convergence, exploiting the techniques from
sequential optimization, fractional optimization, and random matrix theory.
Numerical results demonstrate the near-optimal performance of our proposed
statistical CSI aided EE optimization approach.
