  In this paper, we investigate an energy efficiency (EE) maximization problem
in multi-user multiple input single output downlink channels. The optimization
problem in this system model is difficult to solve in general, since it is in
non-convex fractional form. Hence, conventional algorithms have addressed the
problem in an iterative manner for each channel realization, which leads to
high computational complexity. To tackle this complexity issue, we propose a
new simple method by utilizing the fact that the EE maximization is identical
to the spectral efficiency (SE) maximization for the region of the power below
a certain transmit power referred to as saturation power. In order to calculate
the saturation power, we first introduce upper and lower bounds of the EE
performance by adopting a maximal ratio transmission beamforming strategy.
Then, we propose an efficient way to compute the saturation power for the EE
maximization problem. Once we determine the saturation power corresponding to
the maximum EE in advance, we can solve the EE maximization problem with SE
maximization schemes with low complexity. The derived saturation power is
parameterized by employing random matrix theory, which relies only on the
second order channel statistics. Hence, this approach requires much lower
computational complexity compared to a conventional scheme which exploits
instantaneous channel state information, and provides insight on the saturation
power. Numerical results validate that the proposed algorithm achieves near
optimal EE performance with significantly reduced complexity.
