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In this paper, we analyze and optimize the energy efficiency of downlinkcellular networks. With the aid of tools from stochastic geometry, we introducea new closed-form analytical expression of the potential spectral efficiency(bit/sec/m$^2$). In the interference-limited regime for data transmission,unlike currently available mathematical frameworks, the proposed analyticalformulation depends on the transmit power and deployment density of the basestations. This is obtained by generalizing the definition of coverageprobability and by accounting for the sensitivity of the receiver not onlyduring the decoding of information data, but during the cell association phaseas well. Based on the new formulation of the potential spectral efficiency, theenergy efficiency (bit/Joule) is given in a tractable closed-form formula. Anoptimization problem is formulated and is comprehensively studied. It ismathematically proved, in particular, that the energy efficiency is a unimodaland strictly pseudo-concave function in the transmit power, given the densityof the base stations, and in the density of the base stations, given thetransmit power. Under these assumptions, therefore, a unique transmit power anddensity of the base stations exist, which maximize the energy efficiency.Numerical results are illustrated in order to confirm the obtained findings andto prove the usefulness of the proposed framework for optimizing the networkplanning and deployment of cellular networks from the energy efficiencystandpoint.

System-Level Modeling and Optimization of the Energy Efficiency in Cellular Networks -- A Stochastic Geometry Framework