This paper studies the asymptotic properties of average area spectral efficiency (ASE) of a downlink cellular network in the limit of very dense base station (BS) and user densities. This asymptotic analysis relies on three assumptions: 1) interference is treated as noise; 2) the BS locations are drawn from a Poisson point process; and 3) the path loss function is bounded above satisfying mild regularity conditions. We consider three possible definitions of the average ASE, all of which give units of bits per second per unit bandwidth per unit area. When there is no constraint on the minimum operational signal-to-interference-plus-noise ratio (SINR) and instantaneous full channel state information (CSI) is available at the transmitter, the average ASE is proven to saturate to a constant, which we derive in a closed form. For the other two ASE definitions, wherein either a minimum SINR is enforced or CSI is not available, the average ASE is instead shown to collapse to zero at high BS density. We provide several familiar case studies for the class of considered path loss models, and demonstrate that our results cover most previous models and results on ultradense networks as special cases.
However, other more recent works have shown that when adopting more realistic bounded path loss models, different and much more pessimistic trends can be derived or observed for average ASE under densification @cite_53 @cite_20 @cite_29 @cite_48 @cite_27 @cite_38 @cite_26 @cite_54 . The current situation can be summarized by saying that one finds in the literature the whole spectrum of asymptotic behaviors for ASE ranging from linear growth to collapse. The reasons behind having different asymptotic behaviors are mainly due to the used path loss model and the used metric to measure the average ASE. Different assumptions regarding the channel state information (CSI) availability and minimum operational SINR have been used in the literature, and changing these assumptions might lead to a totally different scaling law. Hence, we dedicate Section to rigorously defining the average ASE under different assumptions for the CSI availability and the minimum operational SINR required for a successful transmission.
@cite_38 @cite_26 @cite_48 @cite_53 @cite_29 @cite_54 @cite_27 @cite_20
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Existing cellular network analyses, and even simulations, typically use the standard path loss model where received power decays like @math over a distance @math . This standard path loss model is quite idealized, and in most scenarios the path loss exponent @math is itself a function of @math , typically an increasing one. Enforcing a single path loss exponent can lead to orders of magnitude differences in average received and interference powers versus the true values. In this paper, we study multi-slope path loss models, where different distance ranges are subject to different path loss exponents. We focus on the dual-slope path loss function, which is a piece-wise power law and continuous and accurately approximates many practical scenarios. We derive the distributions of SIR, SNR, and finally SINR before finding the potential throughput scaling, which provides insight on the observed cell-splitting rate gain. The exact mathematical results show that the SIR monotonically decreases with network density, while the converse is true for SNR, and thus the network coverage probability in terms of SINR is maximized at some finite density. With ultra-densification (network density goes to infinity), there exists a phase transition in the near-field path loss exponent @math : if @math unbounded potential throughput can be achieved asymptotically; if $ 0 , ultra-densification leads in the extreme case to zero throughput. We consider networks consisting of nodes with radios, and without any wired infrastructure, thus necessitating all communication to take place only over the shared wireless medium. The main focus of this paper is on the effect of fading in such wireless networks. We examine the attenuation regime where either the medium is absorptive, a situation which generally prevails, or the path loss exponent is greater than 3. We study the transport capacity, defined as the supremum over the set of feasible rate vectors of the distance weighted sum of rates. We consider two assumption sets. Under the first assumption set, which essentially requires only a mild time average type of bound on the fading process, we show that the transport capacity can grow no faster than O(n), where n denotes the number of nodes, even when the channel state information (CSI) is available noncausally at both the transmitters and the receivers. This assumption includes common models of stationary ergodic channels; constant, frequency-selective channels; flat, rapidly varying channels; and flat slowly varying channels. In the second assumption set, which essentially features an independence, time average of expectation, and nonzeroness condition on the fading process, we constructively show how to achieve transport capacity of spl Omega (n) even when the CSI is unknown to both the transmitters and the receivers, provided that every node has an appropriately nearby node. This assumption set includes common models of independent and identically distributed (i.i.d.) channels; constant, flat channels; and constant, frequency-selective channels. The transport capacity is achieved by nodes communicating only with neighbors, and using only point-to-point coding. The thrust of these results is that the multihop strategy, toward which much protocol development activity is currently targeted, is appropriate for fading environments. The low attenuation regime is open. We study in this paper the network spectral efficiency of a multiple-input multiple-output (MIMO) ad hoc network with K simultaneous communicating transmitter-receiver pairs. Assuming that each transmitter is equipped with t antennas and each receiver with r antennas and each receiver implements single-user detection, we show that in the absence of channel state information (CSI) at the transmitters, the asymptotic network spectral efficiency is limited by r nats s Hz as Krarrinfin and is independent of t and the transmit power. With CSI corresponding to the intended receiver available at the transmitter, we demonstrate that the asymptotic spectral efficiency is at least t+r+2radictr nats s Hz. Asymptotically optimum signaling is also derived under the same CSI assumption, i.e., each transmitter knows the channel corresponding to its desired receiver only. Further capacity improvement is possible with stronger CSI assumption; we demonstrate this using a heuristic interference suppression transmit beamforming approach. The conventional orthogonal transmission approach is also analyzed. In particular, we show that with idealized medium access control, the channelized transmission has unbounded asymptotic spectral efficiency under the constant per-user power constraint. The impact of different power constraints on the asymptotic spectral efficiency is also carefully examined. Finally, numerical examples are given that confirm our analysis We study the throughput limits of a MIMO (multiple-input multiple output) ad hoc network with K simultaneous communicating transceiver pairs. Assume that each transmitter is equipped with t antennas and the receivers with r antennas, we show that in the absence of channel state information (CSI) at the transmitters, the asymptotic network throughput is limited by r nats s Hz as K spl rarr spl infin . With CSI corresponding to the desired receiver available at the transmitter, we demonstrate that an asymptotic throughput of t+r+2 spl radic tr nats s Hz can be achieved using a simple beamforming approach. Further, we show that the asymptotically optimal transmission scheme with CSI amounts to a single-user waterfilling for a properly scaled channel.