What would a cellular network designed for maximal energy efficiency look like? To answer this fundamental question, tools from stochastic geometry are used in this paper to model future cellular networks and obtain a new lower bound on the average uplink spectral efficiency. This enables us to formulate a tractable uplink energy efficiency (EE) maximization problem and solve it analytically with respect to the density of base stations (BSs), the transmit power levels, the number of BS antennas and users per cell, and the pilot reuse factor. The closedform expressions obtained from this general EE maximization framework provide valuable insights on the interplay between the optimization variables, hardware characteristics, and propagation environment. Small cells are proved to give high EE, but the EE improvement saturates quickly with the BS density. Interestingly, the maximal EE is achieved by also equipping the BSs with multiple antennas and operate in a "massive MIMO" fashion, where the array gain from coherent detection mitigates interference and the multiplexing of many users reduces the energy cost per user. The EE of cellular networks has been defined from different perspectives in the last decade [15] , [16] . One of the most common definitions is a benefit-cost ratio, where the service quality per area unit (a.u.) is compared with the associated energy costs. In this paper, the following general definition is used: spectral efficiency [bit/symbol/km 2 ] Transmit power + Circuit power per a.u. [J/symbol/km 2 ] which yields to an EE metric measured in bit/J. 1 Most of the early works that analyzed the EE, as defined above, have focused on the single-cell case wherein the interference from other cells is neglected. Examples in this context are [13], [17]-[23]. In particular, in [17] the authors consider the UL power allocation of multi-user MIMO systems and show that the EE is maximized when specific UEs are switched off. The UL was studied also in [18], where the EE was shown to be a concave function of the number of BS antennas, M, and the UE rates. The DL has been investigated in [13], [19], [20], whereof [13] and [19] show that the EE is a concave function of M, while a similar result is shown for the number of UEs, K , in [20] . Unfortunately, in these works, the system parameters were optimized by means of simulations that (although useful) do not provide a complete picture of how the EE is affected by M and K . Recently, [21] derived the optimal M and K for a given UL sum rate, while [23] derived the optimal M, K and UE rates to maximize the EE jointly for the UL and DL. These two works utilize a realistic power consumption model where the dissipation in circuits depends non-linearly on the above parameters. Simple closed-form expressions for the EE-maximizing parameter values and their scaling behaviors are derived for zero-forcing (ZF) processing with perfect channel state information (CSI) and verified by simulations for other processing schemes under imperfect CSI. The EE analysis of multi-cellular networks is much more involved than in the single-cell case due to the complicated network topology and the arising inter-cell interference. The simplest approach is to rely on heavy Monte Carlo simulations. Attempts in this direction can be found in [24] and [25]. Unfortunately, Monte Carlo simulated results are often anecdotal since one cannot separate fundamental properties from behaviors induced by parameter selection. Alternatively, simplified network topologies can be considered, such as the Wyner model [26] or the symmetric grid-based deployment [23] . While attractive for its analytical simplicity, the Wyner model does not well capture the essential characteristics of real and practical networks [27] . Similarly, the symmetric grid-based models cannot capture the irregular structure of small-cell network deployments. 1 Some prior works have considered erroneous EE metrics in bit/J/Hz, by forgetting to scale both the numerator and the denominator with the symbol rate (or the bandwidth) that was explicitly (or implicitly) assumed in order to compute the noise power and the dimensionless SNR value. These works are not mentioned in this section for obvious reasons.
In REF , the uplink energy efficiency maximization problem is formulated.
Deploying Dense Networks for Maximal Energy Efficiency: Small Cells Meet Massive MIMO
IEEE Journal on Selected Areas in Communications
IEEE Journal on Selected Areas in Communications
Mathematics Computer Science