2860 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 62, NO. 8, AUGUST 2014 A Mathematical Framework to the Computation of the Error Probability of Downlink MIMO Cellular Networks by Using Stochastic Geometry Marco Di Renzo, Senior Member, IEEE, and Peng Guan, Student Member, IEEE Abstract—In this paper, a mathematical framework to the computation of the error probability of downlink cellular net- works is introduced. It is based on the Poisson point process (PPP)-based abstraction for modeling the spatial locations of the base stations (BSs), and it exploits results from stochastic geometry for characterizing the distribution of the other-cell in- terference. The framework is applicable to spatial multiplexing multiple-input–multiple-output (MIMO) systems with an arbi- trary number of antennas at the transmitter (N t ) and at the receiver (N r ). If N t = N r =1, the mathematical approach can be used for arbitrary fading distributions on both useful and interfering links. If either N t > 1 or N r > 1, it can be applied to arbitrary fading distributions on the useful link and to Rayleigh fading on the interfering links. It is shown that the proposed approach leads to easy-to-compute integral expressions, which reduce to closed-form formulas in some asymptotic regimes. Furthermore, the framework is shown to provide insights for system design and optimization. The accuracy of the mathematical analysis is substantiated through extensive Monte Carlo simula- tions for various cellular network setups. Index Terms—Cellular networks, MIMO systems, network in- terference, stochastic geometry, error probability. I. I NTRODUCTION T HE mathematical modeling of cellular networks is usually performed through abstraction models, which rely upon simplified spatial models for the locations of the Base Stations (BSs). Common approaches include the Wyner model, the single-cell interfering model and the hexagonal grid model [1], [2]. However, these abstraction models are either inaccurate for many operating conditions or they require extensive nu- merical computations. As a result, the analysis and design of cellular networks is often conducted by resorting to network simulations for selected scenarios, which represent specific arrangements of BSs. Manuscript received December 13, 2013; revised April 15, 2014 and May 26, 2014; accepted June 24, 2014. Date of publication July 1, 2014; date of current version August 20, 2014. This work was supported in part by the European Commission through the FP7-PEOPLE MITN-CROSSFIRE Project under grant 317126. This paper was presented in part at the IEEE Interna- tional Conference on Computing, Networking and Communications (ICNC) Honolulu, Hawaii, USA, February 2014. The associate editor coordinating the review of this paper and approving it for publication was T. Tsiftsis. The authors are with the Laboratoire des Signaux et Systèmes, Unité Mixte de Recherche 8506, Centre National de la Recherche Scientifique-École Supérieure d’Électricité-Université Paris-Sud XI, Gif-sur-Yvette Cedex 91192, France (e-mail: marco.direnzo@lss.supelec.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCOMM.2014.2334293 To circumvent these problems, a new abstraction model to the mathematical analysis of cellular networks is emerging, which is referred to as Poisson Point Process (PPP)-based abstraction [1]–[5]. With the aid of this abstraction model, the locations of the BSs are modeled as points of a homogeneous PPP. Recent results have confirmed that the PPP-based ab- straction model is capable of accurately reproducing the main structural characteristics of operational cellular networks [6], [7]. The usefulness of the PPP-based abstraction model orig- inates from its analytical tractability and from the possibility of leveraging mathematical tools from applied probability, such as stochastic geometry, for mathematical performance analysis [8]–[10]. In particular, the authors of [7] have recently studied and compared different spatial stochastic models. They have shown that even though point processes different from the PPP may be more accurate in reproducing spatial structures of BSs, they are, in general, less mathematically tractable. To overcome this accuracy vs. mathematical complexity trade- off, the authors of [7] introduce the concept of “deployment gain”, which allows one to carry out the analysis based on the mathematically-tractable PPP-based abstraction model, subject to a correction factor (i.e., the deployment gain) that can be estimated and taken into account for a better and more accurate system design and optimization. In light of these considerations, the PPP-based abstraction model is now routinely used to the analysis and design of wireless networks in general and cellular networks in particular. Notable examples include [1], [2], [11]–[20]. For a compre- hensive literature survey, the interested reader is referred to [2] and [21]. More specifically, in [1] the coverage probability and the average rate of cellular networks for transmission over Rayleigh fading channels are computed in closed-form. In [13], the framework of [1] is extended to heterogeneous cellular networks, which are modeled as the superposition of many PPPs. In [12], the authors study heterogeneous cellular net- works based on the PPP-based abstraction model by assuming a cell association criterion based on the maximum Signal- to-Interference-Plus-Noise-Ratio (SINR). The analysis in [12] assumes that the SINR threshold is greater than 0 dB. This assumption is removed in [11], where arbitrary SINR thresholds are considered. In [15], the framework in [13] is generalized by studying heterogeneous cellular networks that employ a biased cell association criterion. Likewise, the framework in [11] is generalized in [14] for a biased cell association criterion. In [16], the authors incorporate the load characteristics of the 0090-6778 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.