I Introduction
Due to the flying nature of unmanned aerial vehicles (UAVs), base stations (BSs) can be mounted on the UAV to support wireless communications and improve the performance of cellular networks. For example, UAVmounted base stations (UAVBSs) are introduced when a natural disaster interrupts communications or ground base stations are overloaded [1]. Compared with ground BSs, the flexibility of UAVBSs allows them to adapt their locations to the demand of users.
Most of the literature on the UAVBS focuses on its deployment. The work in [2] proposed that fixedwing UAVs at a constant height are more applicable for aerial networks due to less power consumption. Positions of UAVBSs were modeled as a 3D Poisson Point Process (3DPPP) distribution with a limited height in [1], but the analysis in [3] showed that the flexible height of UAV is not as helpful as a wellchosen fixed altitude. In [4], UAVmounted mobile base stations were deployed in a fixed altitude and placed along an optimal trajectory to cover as much as user equipment (UE) whose locations are already known in a given area.
Beyond the UAV deployment, the performance of 3D networks also attracts much attention in the existing literature. The work in [3] analyzed the average downlink spectral efficiency without considering the environment noise, while the authors of [5] evaluated the performance of UAV at a low altitude platform in terms of the coverage area and transmit power. Similarly, the optimal deployment model in [6] led to the analysis of coverage and transmit power. Furthermore, the analysis in [7] introduced a tractable analytical framework for the coverage and the rate in UAV based network with the coexistence of devicetodevice (D2D) network.
Although the path loss model has been considered as a key factor in the performance analysis for UAV networks, there is no consensus on this issue yet. For example, the work in [1] and [4] only considered the UAV hovering in a LoS dominated network for simplicity. To conduct a practical analysis for UAV, the authors of [8] proposed a general path loss model which considers both LoS and NLoS connections and their occurrence probabilities, depending on the elevation angle between a UAV and a user. Despite that this model has been widely adopted as the highaltitude model (a typical height is around 1000 meters), the network performance has not been investigated due to the complexity of the proposed model. On the other hand, the work in [9] provided a network analysis of the terrestrial cellular network where the antenna height between BSs and users is around 10m30m, together with 3GPP LoS and NLoS models. Considering that the height of UAVs is comparable with that of ground base stations in future UAV networks, the curent macrocelltoUE model (a typical height is around 32 meters) and picocelltoUE model (a typical height is around 10 meters) proposed for terrestrial communication in 3GPP standard can also be applied to the UAVbased network. Such macrocelltoUE model and picocelltoUE model are referred to as the lowaltitude model and the ultralowaltitude model hereafter. To our best knowledge, the path loss model for UAVBSs has not been adequately explored in the literature, so the intriguing question arises: which is the most appropriate path loss model when UAVs fly at a practical height, e.g., 50m100m? The reasons for such mediumaltitude deployment of UAVs are: i) UAVs should not fly too high (e.g., larger than 100 meters) because of the recently discovered network capacity crash [10], and ii) UAVs should not fly too low (e.g., lower than 10 meters) due to obvious safety reasons.
Motivated by the above theoretical gap and to answer such fundamental question, in this paper, we analyze the performance of UAVenabled networks on the condition of different path loss models. An interesting finding in our study is that although the 3GPP path loss model is developed for terrestrial communications, the network performance based on the lowaltitude model (a typical height is around 32 meters) is similar to that based on the highaltitude model, when a practical UAV height of 50 meters is considered. Moreover, the optimal UAV density to maximize the coverage probability and the area spectral efficiency (ASE) performance can be found from numerical results.
The rest of this paper is structured as follows. In Section II we introduce the system model of UAVenabled networks. In section III, three path loss models are investigated and compared. In section IV, we derive the analytical expressions of the coverage probability and the ASE for a 3D network with PPPdistributed UAVs, and then we propose an ideal case with teleportation UAVs to find the upper bound of the network performance. Numerical results and discussions are provided in Section V and the conclusions are drawn in Section VI.
Ii System Model
We consider a UAV network, where UAV aerial base stations follow a 3DPPP distribution with a density in an infinite 3D space , and the UAV height is set to , that is . Here, we consider practical values for around 50100 meters. We consider such mediumaltitude deployment of UAVs because UAVs should not fly too high (e.g., larger than 100 meters) since the recently discovered network capacity crash [9]
, and UAVs should not fly too low (e.g., lower than 10 meters) due to obvious safety reasons. User equipments (UEs) are Poisson distributed in the considered network with a density of
. Here, is assumed to be sufficiently larger than so that each UAV has at least one associated UE in its coverage area. The 3D distance between an arbitrary UAV and an arbitary UE is denoted by in km. Considering practical LoS and NLoS transmissions, we propose to model the path loss associated with distanceas a path loss function
. Such is segmented into 2 pieces, where is the path loss function for LoS transmission, is the path loss function for NLoS transmission and is the LoS probability function. In more detail,
is modeled as
(1) with and being the path losses at a reference distance and and being the path loss exponents for the LoS and the NLoS cases in , respectively. In practice, , , and are constants obtained from field tests [10].

is the probability function that a transmitter and a receiver have LoS connections. Also, the probability of NLoS is .
As a common practice in the field, each UE is assumed to be associated with the UAV that provides the strongest signal strength, and the multipath fading between an arbitrary UAVBS and an arbitrary UE is modeled as independently identical distributed (i.i.d.) Rayleigh fading. Thus, the channel gain denoted by
can be modeled as an i.i.d. exponential random variable (RV). The transmit power of each UAV and the additive white Gaussian noise (AWGN) power at each UE are denoted by
and , respectively.Iii Discussion and Analysis of path loss models
Since there is no consensus on proper path loss model for UAVenabled networks, we choose three widely adopted path loss models and apply them to the considered UAV networks.
Iiia Highaltitude model
The highaltitude model based on the elevation angle has been widely used in the satellite communication model, e.g., thousands of meters. The probability function that a transmitter and a receiver have a LoS connection at an elevation angle of can be expressed as [7]:
(2) 
where and are constant values that depend on the environment (rural, urban, dense urban, etc.). Furthermore, the elevation angle can be written as , so the LoS probability function for this highaltitude model can be reformulated as a new function with respect to :
(3) 
IiiB Lowaltitude model
Provided that the practical height of UAVBSs is usually limited to a medium altitude, like 50m and 100m. Such height is comparable to the antenna height of terrestrial base stations, we further analyze the path loss model proposed for 3GPP terrestrial communications and apply it to the considered UAV networks.
In particular, the 3GPP macrocelltoUE path loss model has been proposed for connection between a UE and its associated macrocell BS. Considering that the height of a macrocell base station is usually around 32m, which is slightly lower than the considered altitude of UAV around 50100m, it is reasonable to use this model to study the UAV network. In this case, the LoS probability function for this lowaltitude model can be expressed as [11]
(4) 
IiiC Ultralowaltitude model
To obtain a comprehensive insight of the proper path loss model for UAVs, we also introduce the 3GPP picocelltoUE model as the ultralowaltitude model, since the typical height of picocell base station is about 10m. In this case, the LoS probability function is defined as [11]
(5) 
IiiD The Comparison of the Three Path Loss Models
Fig. 1 compares the LoS probability functions for different path loss models. It can be seen from this figure that the LoS probability for the ultralowaltitude model drops very quickly with respect to the distance, followed by the lowaltitude model. Moreover, it should be noted that the highaltitude model generates different LoS probability functions for different altitudes.
Iv Analysis for the Proposed UAV Networks
To analyze the performance of UAVBSs based on the interested path loss models, we investigate the coverage probability and the ASE of the network in this section. The coverage probability represents the probability that the typical user is covered by the associated UAVBS and defined as the probability that the received signaltointerferencenoiseratio (SINR) is larger than a preset threshold , which can be expressed as
(6) 
where SINR is expressed as
(7) 
where and denote the transmission power of the UAVBS and additive white Gauss noise (AWGN) power, respectively. Moreover, is the sum of interference from other UAVBSs, and
is the channel gain of Rayleigh fading and can be modeled as a RV which follows a exponential distribution with the mean value of one. It can be further written as
(8) 
Obviously, when UAVBSs are HPPP distributed and randomly hovering in the network, the network performance reaches a lower bound because the mobility of UAVs is completed ignored. Such lowerbound performance is characterized in the following Theorem 1.
Theorem 1.
Considering the path loss of the LoS and the NLoS connection, the lower bound of the coverage probability can be expressed as
(9) 
where and .
The and are expressed as
(10) 
and
(11) 
where and are the solutions of and , respectively.
Moreover, and are expressed by
(12) 
and
(13) 
where is the Laplace transform of in the computation of interference.
Proof.
See Appendix A. ∎
In Theorem 1, we assume UAVs are randomly hovering in the network. On the other hand, if we consider the mobility of UAVs, the system performance can surely be improved. However, the analysis of such mobile UAVs is difficult because we need to further consider UAV mobility control management. Fortunately, we can instead consider a UAV teleportation model, where UAVs can instantaneously move to the positions directly overhead the users to show the upperbound performance of a UAV network. In this case, each user will be associated with its UAVBS overhead. Such upperbound performance is characterized in the following Corollary 1, which is derived from Theorem 1.
Corollary 1.
The coverage probability of teleporting UAVs can be expressed as
(14) 
In this case, the associated UAV is set at the positions overhead the users, so the space distance from users to their associated UAVs is rather than . In comparison with the case of HPPP distributed UAVs, the case of teleporting UAVs can provide the user with the strongest received signal power due to the minimized distance between them and the highest probability of having LoS connection. As a result, this teleporting model gives the upper bound of network performance. The detailed discussion on such upperbound performance will be shown in Sec.VC.
According to [12], the ASE can be expressed as
(15) 
where is the minimum SINR threshold for UE to work normally.
V Simulation Results
To find the appropriate path loss model when UAVs fly at a medium altitude, we use simulation results to demonstrate the coverage probability and the ASE of three LoS probability models and make a comparison. Parameters adopted in simulation are: dBm, dBm [11], dB, , [7]. To obtain the numerical results at the medium height, we choose to analyze UAVs at the height of 50m and 100m, which are the most practical cases in reality. For the highaltitude model, the relative parameters are: , , , [4], [13]. For the lowaltitude model, path loss parameters are: , , , [11]. For the ultralowaltitude model, path loss parameters are: , , , [11].
Va The coverage probability for hovering UAVs
Fig. 2 and Fig. 3 show the comparison of the coverage probability for UAVs hovering at 50m and 100m based on the investigated three models of path loss, i.e., the highaltitude model, the lowaltitude model and the ultralowaltitude model.
It can be seen from Fig. 2 that with the increase of the UAV density, the coverage probability of the highaltitude model first rises to the peak and then decreases. The optimal UAV density for this model is about 10 BSs/km. As for the lowaltitude model, the performance trend is similar to that of the highaltitude one, with a slightly different optimal density around 6 BSs/km. The explanations of these phenomena are:

For a sparse UAVBSs density, the distance from associated UAVBS to UE decreases with the increasing UAVBS density and the associated UAVBS is more likely to have a LoS transmission with UE, so the coverage probability grows as the UAVBS density increases.

For a dense UAVBSs density, although the associated UAV have a higher probability to transmit data via a LoS channel, other UAVs also produce strong interference through LoS paths, thus, the coverage probability decreasing after reaching the optimal point.
For the ultralowaltitude model, the performance is significantly different with the other two models. The reason is that the ultralowaltitude model is designed for a scenario where UAVs fly at a relatively low altitude and the transmission distance is quite limited. Furthermore, even when the UAV is hovering over user’s location, the probability of having LoS connection is still low since the minimum distance from UAVBS to UE is the height of UAV. As a result, the ultralowaltitude model is not suitable for the practical UAV scenario with a height around 50100 meters.
In Fig. 3, we can see that the performance of the lowaltitude model and highaltitude model is very similar when the UAVBS density is less than 2 BSs/km. When the density is between 2 BSs/km and 20 BSs/km, the coverage probability of the highaltitude model is higher than that of the lowaltitude model, but the lowaltitude model performs better than the highaltitude model when density is beyond 20 BSs/km.
VB The ASE for hovering UAVs
Fig. 4 shows the ASE performance of different path loss models for a height of 50m. As can be seen from this figure, the ASE of the highaltitude model and the lowaltitude model keep growing due to the increasing coverage probability, but the growing rate slows down when the density of UAVs is more than 10 BSs/km. This is because the declining coverage probability shown in Figs. 3 and 4 outweighs the increase of the UAV density. In this figure we can also find that the ASE for the ultralowaltitude model differs from the other two. As a result, when the height of UAV is around 50m, the highaltitude model and the lowaltitude model are equally good for the performance analysis of the UAVbased network.
Fig. 5 shows the comparison of the ASE in different models when the height of UAV is 100m. When the UAVBS density is lower than 20 BSs/km, the ASE of the highaltitude model and that of the lowaltitude model leave the similar trail. However, after reaching the density of 10 BSs/km, their ASE performance diverges. The drop of the ASE for the highaltitude model indicates that the deceasing coverage dominates the ASE performance compared with the growing UAV density. Considering that the lowaltitude model was developed for a height around 32 meters, it might not be suitable for the UAVs flying at 100 meters studied here. Hence, the highaltitude model might be more appropriate here. However, we may need to conduct reallife channel measurement to confirm this conjecture.
VC The performance for Teleporting UAVs
It can be seen from the previous simulation that when UAVs fly at the height of 50m, the coverage probability and the ASE performance of the highaltitude model and the lowaltitude model are very similar. However, when the height of UAV is at 100m, the performance of these two models deviate in dense networks. To verify whether these two models are still equally good for teleporting UAVs at 50m, we investigate and compare their coverage probability and ASE performance in this subsection.
From Fig. 6 and Fig. 7, we can see that the highaltitude model and the lowaltitude model generate similar results. In Fig. 6, we can see that the optimal UAV density for these two models can be found at around 6 BSs/km. Fig. 7 shows that the ASE of both increases linearly at first, and then grows slowly.
VD Comparison of the Upper and Lower bounds of Performance
From Fig. 2 and Fig. 4, we can see that the highaltitude model and the lowaltitude model are equally good for network performance analysis. Hence, we choose the highaltitude model to show the difference between the upper bound of ASE and the lower bound of ASE when the UAVs fly at the height of 50m. Such comparison is displayed in Fig. 8. It can be seen that when the density is lower than 10 BSs/km, the gap between the upper bound and the lower bound is large, which shows great promise for optimization of UAV mobility in UAVenabled networks. However, as UAV density increases, the ASE gain due to the UAV mobility becomes marginal, e.g., at a UAV density of 100 UAVBSs/km.
Vi Conclusion
In this paper, we have studied the performance of UAVenabled wireless networks. In order to identify the proper path loss models for UAVs flying at practical heights, such as 50m and 100m, we first analyze the performance when adopting the conventional highaltitude model based on the elevation angle. Then we further investigate the coverage probability and the ASE by using path loss models which have been widely applied to terrestrial communications, including the lowaltitude model and the ultralowaltitude model. From simulation results, we find that performance for networks with highaltitude model and the lowaltitude model are equally good when UAVs fly at the height of 50m, while the performance trend of the ultralowaltitude model is quite different. We also found that the number of the UAVs should be optimized for the benefits of the networks, which sheds new light on the design of the future UAVenabled networks.
Appendix A Proof of Theorem 1
When transmission between UAVBS and UE is LoS, two conditions should be satisfied: 1) The distance between UE and UAVBS is , and there is no UAVBS of LoS path within . 2) There is no UAVBS of NLoS path within . Based on these conditions, is computed as [10]
(16) 
When transmission between UAVBS and UE is NLoS, two conditions should be satisfied: 1) The distance between UE and UAVBS is , and there is no UAVBS of NLoS path within . 2) There is no UAVBS of LoS path within . So can be derived as
(17) 
conditioned on can be expressed as
(18) 
Then can be derived in the similar way.
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