# Area Spectral Efficiency of Ultradense Networks

**DOI:**https://doi.org/10.1007/978-3-319-32903-1_44-1

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## Synonyms

## Definition

Area spectral efficiency refers to the data rate that can be achieved per unit bandwidth and in a unit area of the wireless network. It has the unit of b/s/Hz/m^{2} or b/s/Hz/km^{2}.

## Historical Background

Network densification has been the main driver of wireless network capacity increase in the past and will play an even more crucial role in the development of the next generation mobile communication systems (5G). Network densification refers to the deployment of more base stations (BSs) and wireless access points per unit area and the associated technological advances to support such densification. There are three primary ways of increasing the wireless network capacity: (1) adding more spectrum, (2) enhancing spectral efficiency through advanced communication techniques, and (3) enhancing spatial reuse of frequency spectrum through network densification. The area spectral efficiency (ASE) is a major metric to measure the efficiency in the spatial reuse of frequency spectrum. According to a study by Webb (Henderson, 2007), among the 1-million-fold increase in wireless network capacity achieved in 50 years from 1950 to 2000, 15× improvement was achieved from a wider spectrum, 5× improvement from better Medium Access Control (MAC) and modulation schemes, 5× improvement by designing better coding techniques, and an astounding 2700× gain through network densification and reduced cell sizes. As we move to the next generation mobile communication systems and seek further capacity increases, network densification, manifested through heterogeneous and ultradense networks, will play an even more important role. First, the combined amount of spectrum available for licensed mobile broadband and unlicensed Wi-Fi is scarce. Spectrum approaching millimeter wavelengths is more abundant but is yet to be proven for cellular and Wi-Fi use due to difficulty in penetration and supporting significant mobility (Andrews et al., 2016a; Kutty and Sen, 2016). Second, the scope for enhanced spectral efficiency may be limited as many current wireless systems are already running at a spectral efficiency close to the performance limit prescribed by the Shannon. Therefore, mobile operators have increasingly turned to network densification and small cell technology to meet the 1000× capacity increase expected on 5G. Small cells are now deployed in massive numbers (Small Cell Forum, 2016), which drives the paradigm shift into ultradense networks. Ultradense networks feature a very dense deployment of BSs and wireless access points.

## Foundations

The ASE and wireless network capacity can be used interchangeably in the sense that the total capacity achieved by a network deployed in a given geographical area and using a specific amount of bandwidth is equal to the ASE times the size of the area and the amount of bandwidth. It is of crucial interest to investigate how the ASE varies as more and more BSs are deployed and the network becomes denser and denser.

### The SINR Invariance Principle for Low-to-Medium BS Density

*k*-th nearest BS has a distance of

*l*

_{ k }. Further assume that all BSs transmit at the same fixed power

*P*

_{ t }and the wireless signal experiences standard power-law attenuation such that the received power at a distance

*d*from the transmitter is \(P_{r}\left (d\right )=P_{t}Ld^{-\alpha }\), where

*L*is a reference path loss at unit distance and

*α*is the path loss exponent. For a “typical” user located at the origin and associated with its nearest BS, its SINR can be expressed as

*σ*

^{2}represents the noise power. Now consider scaling the distances between all BSs by a factor of

*t*. The density of BSs will increase by a factor of \(\frac {1}{t^{2}}\). Further assume that noise power is negligible, which is valid in an interference-limited regime where most wireless networks now operate in. The SINR then becomes

The SINR invariance principle implies that as the network densifies and the distances between transmitters and receivers reduce, the increase in interference will be counterbalanced by the increase in the desired signal. Consequently, the SINR will stay approximately the same. The principle suggests that other things being equal, the spectral efficiency, or equivalently the capacity, per BS cell is invariant as network densifies. Therefore, from the network perspective, the ASE, or equivalently the overall network capacity, will linearly increase with the number of cells per unit area; from the user perspective, as each user maintains the same SINR but shares its BS with an ever-smaller number of other users, each user can achieve approximately linear growth in its achievable data rate as BSs are added, until the limit of one user per cell is reached. The SINR invariance principle is not affected by the BS layout, transmit power, shadowing and fading distributions, and path loss exponent (Andrews et al., 2016b).

### Area Spectral Efficiency for Ultradense BS Regime

Recent research suggested however that the SINR invariance principle may no longer apply when the BS density becomes very large and that there may exist a limit in network densification, beyond which further densification will not bring the expected linear increase in capacity and may even reduce the capacity (Ding et al., 2015, 2016a,b; Ge et al., 2016; Liu et al., 2016a,b; Andrews et al., 2016b; Zhang and Andrews, 2015). Specifically, by incorporating those effects that have negligible impacts on the ASE when the BS density is small or moderate but become dominant factors determining the ASE when the BS density is very large, it was shown that the ASE may either exhibit a sublinear increase with the BS density, or reduce at certain region of the BS density, or even monotonically reduce to zero beyond a certain BS density threshold.

*x*experience LoS transmission, \(1-\mathrm{Pr}^{\mathrm{L}}\left (x\right )\) is the probability that the same pair of transmitter and receiver experiences NLoS transmission, and

*d*

_{1}is a parameter determining the decreasing slope of the linear function \(\mathrm{Pr}^{\mathrm{L}}\left (r\right )\). Ding et al. (2016a) investigated the variation of the ASE with the BS density, shown in Fig. 1. The ASE is determined using the following equation

*λ*is the BS density,

*γ*

_{0}is the SINR threshold required for establishing a connection, \(f_{\text{SINR}}\left (x\right )\) is the probability density function of the SINR, and the term \(\log _{2}\left (1+x\right )\) comes from the Shannon capacity formula. In the literature, the ASE has also been calculated using the following formula \(ASE=\lambda \Pr \left (\text{SINR}\geq \gamma _{0}\right )\log _{2}\left (1+\gamma _{0}\right )\), which reflects the fact that some wireless systems may not be able to explore the extra SINR above the SINR threshold

*γ*

_{0}to boost the data rate.

- 1.
when the BS density is small, the ASE quickly increases with the BS density because the network is generally noise-limited and adding more BSs immensely benefits the ASE by reducing the transmission distance between the BS and the mobile user (MU). Most transmissions in this regime are NLoS transmissions; however the transmission between a MU and its desired BS has a higher probability of being LoS transmission, and as the BS density further increases, this probability increases to a non-negligible value. Therefore, in this region, the ASE considering LoS/NLoS transmissions is higher than that without considering LoS/NLoS transmissions, but their trend is the same;

- 2.
when the network is dense enough, i.e., greater than 20 BS/km

^{2}in Fig. 1, the growth of the ASE with the BS density, when considering LoS/NLoS transmissions, becomes flat or even exhibits a decrease (for*γ*_{0}= 3 dB and*γ*_{0}= 6 dB), which is distinctly different from that predicted without considering LoS/NLoS transmissions. This can be explained by the so-called NLoS-LoS transition effect. Particularly, in this region, the transmissions between MSs and their desired BS are already dominated by LoS transmissions but in the beginning signals from interfering BSs, are mainly NLoS transmissions suffering higher loss. As BS density further increases and the distances between a MU and BSs, including both the desired BS and interfering BSs, further reduce, signals from interfering BSs start to transit from NLoS transmissions to LoS transmissions. Consequently, as the BS density increases, interference experiences a larger increase compared with the desired signal, causing the SINR to reduce sharply in this region. Therefore, the ASE remains largely flat with the increase in BS density or may even reduce; - 3.
when the network becomes very dense, i.e., greater than 100 BS/km

^{2}in Fig. 1, the dominant interfering BSs have completed their NLoS-LoS transitions. Both the desired signal and the dominant interference are now LoS transmissions. However, non-dominant interfering BSs further away from the MS may still experience the NLoS-LoS transition effect. Therefore, in this region, the SINR may still reduce modestly with the increase in BS density. This modest decrease in the SINR combined with the increase in the BS density causes the ASE to increase but at a rate below that predicted without considering LoS/NLoS transmissions.

^{2}. As we move into the regime of ultradense networks, we are bound to pass through the aforementioned second region. Therefore, it is important to consider the effect of LoS and NLoS transmissions when analyzing the ASE of ultradense networks.

*x*denotes the distance from the transmitter to the receiver,

*R*

_{ n }separates the attenuation function into several distinct regions, and

*α*

_{ n }denotes the path loss exponent for

*R*

_{ n }<

*x*≤

*R*

_{ n+1},

*η*

_{0}= 1, and \(\eta _{n}=\prod _{i=1}^{n}\frac {1+R_{i}^{\alpha _{1}}}{1+R_{i}^{\alpha _{i-1}}}\). As signals are attenuated faster at larger distances,

*α*

_{ n }<

*α*

_{ n+1}holds. Using the model, they predicted that the ASE may monotonically decrease to zero when the BS density increases beyond a certain critical threshold. When the SINR threshold

*γ*

_{0}= 20 dB, this critical BS density lies between 5000 −−25, 000 BS/km

^{2}depending on the values of

*α*

_{ i }s; when

*γ*

_{0}= 0 dB, this critical BS density lies between 2 × 10

^{5}− 3 × 10

^{5}BS/km

^{2}. In a separate work (Zhang and Andrews, 2015), Zhang and Andrews used a dual-slope (power-law) path loss function to model the different radio propagations in a region close to the transmitter and in a region far away:

*R*

_{ c }> 0 is the critical distance, and

*α*

_{0}and

*α*

_{1}are the near- and far-field path loss exponents with 0 ≤

*α*

_{0}≤

*α*

_{1}. Using the model, they predicted that the ASE grows linearly with the BS density

*λ*if

*α*

_{0}> 2, scales sublinearly with rate \(\lambda ^{2-\frac {2}{\alpha _{0}}}\) if 1 <

*α*

_{0}< 2, and decays to zero if

*α*

_{0}< 1. It is worth noting that compared with extensive studies on far-field propagation models, surprisingly little is known about near-field radio propagation. Therefore, the models in (5) and (6) are plausible. However, a common insight revealed in their work (Zhang and Andrews, 2015; Liu et al., 2016b) is that the ASE exhibits quantitatively and qualitatively different trends with the BS density

*λ*when the different propagation conditions in the near-field and in the far-field are considered.

^{2}and then starts to monotonically decrease to zero as the BS density further increases. In comparison, when the impact of this height is ignored, the ASE may increase monotonically toward infinity. Their result can be intuitively explained using Fig. 2, which shows that while the impact of antenna height may be small at small BS density, its impact may play a dominant impact on the ASE in an ultradense network with a large BS density. In 2016, Gruber considered the geometric constraints limiting the BS deployment locations. Specifically, by considering that BSs can only be deployed along the two sides of the street, some distinctly different results on the ASE were obtained compared with that assuming BSs can be deployed randomly and anywhere in the street. It is worth noting that while some assumptions used in these studies are quite plausible, e.g., as the density of BS increases, one would expect that the BS size and height also reduce instead of being constant; the insight revealed in these studies holds: as we move into the ultradense regime, such geometric restrictions as physical dimensions of transmitters and receivers and deployment restrictions on BSs that previously have small or negligible impact on the ASE may now become dominant factors in the consideration.

In summary, distinct from conventional low-to-medium density networks, in ultradense networks, impacts of LoS/NLoS transmissions, the different radio propagation conditions in the near-field and far-field of BSs, and network geometric constraints may play an important role in determining the ASE.

## Key Applications

The ASE is a fundamental performance metric of ultradense networks and more general wireless networks. The ASE determines the capacity that can be achieved by a wireless network and measures the efficiency in the spatial reuse of frequency spectrum. Knowledge of the ASE helps to guide the design and deployment of wireless networks.

## Cross-References

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