  The dramatic increase of network infrastructure comes at the cost of rapidly
increasing energy consumption, which makes optimization of energy efficiency
(EE) an important topic. Since EE is often modeled as the ratio of rate to
power, we present a mathematical framework called fractional programming that
provides insight into this class of optimization problems, as well as
algorithms for computing the solution. The main idea is that the objective
function is transformed to a weighted sum of rate and power. A generic problem
formulation for systems dissipating transmit-independent circuit power in
addition to transmit-dependent power is presented. We show that a broad class
of EE maximization problems can be solved efficiently, provided the rate is a
concave function of the transmit power. We elaborate examples of various system
models including time-varying parallel channels. Rate functions with an
arbitrary discrete modulation scheme are also treated. The examples considered
lead to water-filling solutions, but these are different from the dual problems
of power minimization under rate constraints and rate maximization under power
constraints, respectively, because the constraints need not be active. We also
demonstrate that if the solution to a rate maximization problem is known, it
can be utilized to reduce the EE problem into a one-dimensional convex problem.
