# Encyclopedia of Wireless Networks

Living Edition
| Editors: Xuemin (Sherman) Shen, Xiaodong Lin, Kuan Zhang

# Area Spectral Efficiency of Ultradense Networks

Living reference work entry
DOI: https://doi.org/10.1007/978-3-319-32903-1_44-1

## 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/m2 or b/s/Hz/km2.

## 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

Until recently, there was widely held belief that the ASE, or equivalently the network capacity, may increase indefinitely as the network densifies (Haenggi et al., 2009; Andrews et al., 2011). This view has been underpinned by the so-called SINR (signal-to-interference-plus-noise ratio) invariance principle, which is valid for a low-to-medium BS density. We use the following example to illustrate the SINR invariance principle. Consider a set of BSs deployed on an infinite plane, and number these BSs according to their distances to the origin such that the 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
\displaystyle \begin{aligned} \mathrm{SINR}=\frac{P_{t}Ll_{1}^{-\alpha}}{\sum_{k=2}^{\infty}P_{t}Ll_{k}^{-\alpha}+\sigma^{2}},{} \end{aligned}
(1)
where σ2 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
\displaystyle \begin{aligned} \mathrm{SINR} & =\frac{P_{t}L\left(tl_{1}\right)^{-\alpha}}{\sum_{k=2}^{\infty}P_{t}L\left(tl_{k}\right)^{-\alpha}+\sigma^{2}} \\ & \simeq\frac{P_{t}L\left(tl_{1}\right)^{-\alpha}}{\sum_{k=2}^{\infty}P_{t}L\left(tl_{k}\right)^{-\alpha}} \\ & =\frac{l_{1}^{-\alpha}}{\sum_{k=2}^{\infty}\left(l_{k}\right)^{-\alpha}},{} \end{aligned}
(2)
i.e., the SINR is invariant with the increase of BS density.

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.

In Ding et al. (2015, 2016a,b), and Ge et al. (2016), researchers considered the impact of non-line-of-sight (NLoS) and line-of-sight (LoS) transmissions on the ASE. NLoS and LoS transmissions are ubiquitous in wireless communications. Other things being equal, as the distance between a transmitter and a receiver increases, the probability that their direct LoS path gets blocked increases and the converse. NLoS transmissions will generally suffer much higher path loss than LoS transmissions. Specifically, by employing a 3GPP-endorsed LoS/NLoS probability model given below where $$\mathrm{Pr}^{\mathrm{L}}\left (x\right )$$ is the probability that a transmitter and a receiver separated by a distance 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 d1 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
\displaystyle \begin{aligned} ASE & =\lambda\Pr\left(\text{SINR}\geq\gamma_{0}\right)\\ &\quad\int_{\gamma_{0}}^{\infty}\log_{2}\left(1+x\right)f_{\text{SINR}}\left(x\right)dx \end{aligned}
(4)
where λ is the BS density, γ0 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 γ0 to boost the data rate. Fig. 1The variation of the ASE with the BS density at various SINR thresholds. Both the case considering the coexistence of NLoS and LoS transmissions and the case without considering the coexistence of NLoS and LoS transmissions are shown. The case without considering the coexistence is plotted using the result in Andrews et al. (2011) and assuming α = 3.75, P t  = 24 dBm, σ2 = −95 dBm, and L = 10−14.54. The case considering the coexistence of NLoS and LoS transmissions is obtained assuming d1 = 300 m, P t  = 24 dBm, σ2 = −95 dBm, α = 2.09, and L = 10−10.38 for LoS transmissions and α = 3.75 and L = 10−14.54 for NLoS transmissions. These parameters are chosen following 3GPP recommendations
Figure 1 shows that when LoS/NLoS transmissions are considered, the ASE exhibits not only quantitatively but also qualitatively different trends with the BS density, compared with the case that does not consider the distinction of LoS and NLoS transmissions. Specifically,
1. 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. 2.

when the network is dense enough, i.e., greater than 20 BS/km2 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 γ0 = 3 dB and γ0 = 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. 3.

when the network becomes very dense, i.e., greater than 100 BS/km2 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.

Open data at Ofcom, an independent regulator and competition authority for the UK (http://sitefinder.ofcom.org.uk/search), shows that in some dense urban regions in the UK, the BS density already exceeds 10 BS/km2. 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.
In 2016b, Liu et al. investigated the ASE from a different perspective by considering the transition from far-field to near-field radio propagation which may be triggered as the BS density increases to a very large value. Specifically, it is well known that wireless signal attenuates faster at larger distances. In a region close to the transmitter, the signal may suffer little attenuation, while this attenuation sharply increases as the distance from the transmitter becomes large. Based on some field measurements, they proposed the following multi-slope BPM model to better capture the signal attenuation
\displaystyle \begin{aligned} &g_{N}^{B}\left(\left\{ \alpha_{n}\right\} _{n=0}^{N-1};x\right)=\eta_{n}\left(1+x^{\alpha_{n}}\right)^{-1},\\ &R_{n}<x\leq R_{n+1},{} \end{aligned}
(5)
where 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, η0 = 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 γ0 = 20 dB, this critical BS density lies between 5000 −−25, 000 BS/km2 depending on the values of α i s; when γ0 = 0 dB, this critical BS density lies between 2 × 105 − 3 × 105 BS/km2. 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:
\displaystyle \begin{aligned} l_{2}\left(\alpha_{1},\alpha_{2};x\right)=\begin{cases} x^{-\alpha_{0}} & x\leq R_{c}\\ \eta x^{-\alpha_{0}} & x>R_{c} \end{cases},{} \end{aligned}
(6)
where $$\eta =R_{c}^{\alpha _{1}-\alpha _{0}}$$, R c  > 0 is the critical distance, and α0 and α1 are the near- and far-field path loss exponents with 0 ≤ α0 ≤ α1. Using the model, they predicted that the ASE grows linearly with the BS density λ if α0 > 2, scales sublinearly with rate $$\lambda ^{2-\frac {2}{\alpha _{0}}}$$ if 1 < α0 < 2, and decays to zero if α0 < 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.
In Ding and Perez (2016) and Gruber (2016), researchers considered the impact of some geometric conditions on the ASE. Specifically, in Ding and Perez (2016), Ding and David Lopez investigated the impact of antenna height on the ASE by considering a scenario where the BS antenna height is kept constant when its density increases. They showed that when the height of the BS relative to the MU is maintained at a constant value of 8.5 m, the ASE may peak when the BS density increases to ∼15 BS/km2 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. Fig. 2An illustration of the different impacts of the antenna height as the distance between MU and BS changes. When the distance d between the MU and BS is large, the impact of the relative height between BS antenna, h a , and MU, h p , on the propagation distance x is small and negligible. When the BS density increases to a very large value and hence the distance d reduces to a small value, the impact of h a  − h p on the propagation distance xis no longer negligible and may even be the dominant factor determining x

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.

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