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.

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As a step towards decreasing power consumption in mobile communication networks, we investigate the energy efficiency (EE) of a scheduler that can be utilized at the transmitter in a broadcast channel: the random opportunistic beamforming algorithm. Using the ratio of rate to power as the EE metric and considering the constant circuit power, the EE at the transmitter decreases with the number of transmit antennas M . Two approaches of optimizing the power allocation when M = 1 are introduced. We show that the optimal EE scales double-logarithmically with the number of users, which has one antenna each. The appropriate expression for EE in communications is also discussed.

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