MmWave UAV Networks with Multi-cell Association: Performance Limit and Optimization

05/15/2019
by   Chun-Hung Liu, et al.
Mississippi State University
0

This paper aims to exploit the fundamental limits on the downlink coverage and spatial throughput performances of a cellular network consisting a tier of unmanned aerial vehicle (UAV) base stations (BSs) using the millimeter wave (mmWave) band and a tier of ground BSs using the ultra high frequency (UHF) band. To reduce handover signaling overhead, the ground UAVs take charge of control signaling delivery whereas the UAVs are in charge of payload data transmission so that users need to simultaneously associate with a ground BS and a UAV in this network with a control-data plane-split architecture. We first propose a three-dimensional (3D) location distribution model of the UAVs using stochastic geometry which is able to generally characterize the positions of the UAVs in the sky. Using this 3D distribution model, we propose the multi-cell coverage probability and the volume spectral efficiency of the network, derive their explicit low-complexity expressions and find their upper limits when each of the UAVs and ground BSs is equipped with a massive antenna array. We further show that the multi-cell coverage probability and the volume spectral efficiency can be maximized by optimally deploying and positioning the UAVs in the sky and thereby their fundamental maximal limits are found. These important analytical findings are validated by numerical simulations.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 22

06/12/2018

Downlink Analysis in Unmanned Aerial Vehicle (UAV) Assisted Cellular Networks with Clustered Users

The use of unmanned aerial vehicles (UAVs) operating as aerial base stat...
10/26/2019

Stochastic Geometry Analysis of Hybrid Aerial Terrestrial Networks with mmWave Backhauling

To meet increasing data demands, service providers are considering the u...
01/23/2021

A 3D Modeling Approach to Tractable Analysis in UAV-Enabled Cellular Networks

This paper aims to propose a three-dimensional (3D) point process that c...
10/09/2018

Integrating UAVs into Existing Wireless Networks: A Stochastic Geometry Approach

The integration of unmanned aerial vehicles (UAVs) into wireless network...
07/20/2020

A 3D Tractable Model for UAV-Enabled Cellular Networks With Multiple Antennas

This paper aims to propose a three-dimensional (3D) point process model ...
03/22/2021

Stochastic Geometry Modeling and Analysis for THz-mmWave Hybrid IoT Networks

Terahertz (THz) band contains abundant spectrum resources that can offer...
02/12/2020

Towards Reliable UAV Swarm Communication in D2D-Enhanced Cellular Network

In the existing cellular networks, it remains a challenging problem to c...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

I Introduction

Insatiable mobile throughput demand is the main driver for mobile network operators to adopt millimeter wave (mmWave) spectra that support gigabit connections for data-intensive applications such as virtual reality, augmented reality and immersive gaming. Exploiting the use of the mmWave band may effectively alleviate the spectrum crunch problem in the next generation cellular networks. However, its effectiveness may be significantly weakened by environmental blockages thanks to the high path loss and low penetration characteristics of mmWave channels [1, 2, 3]. A very attractive means of enhancing the propagation performance of the mmWave channels is to use unmanned aerial vehicles (UAVs) equipped with multiple antennas as the “flying” base stations (BSs) using the mmWave band because UAVs are able to agilely position themselves to ameliorate their channel quality in accordance of environmental variations. Due to the mobility of UAVs, UAV-assisted communication techniques will play an important role in cellular networks to fulfill the goals of reliable coverage, high-speed, secure and public safety wireless communications [4][5].

Despite many of the advantages of UAV communications, to successfully employ UAVs to enhance the performances of cellular networks faces a few practical challenges. First, it should be noticed that UAVs have a limited communication capability owing to their limited size and power. As such, energy-efficient deployment, operation and management are particularly important issues for a cellular network using UAVs as access points or BSs (referred to as a UAV network). Moreover, a UAV network has a highly dynamic network topology due to the high mobility of UAVs so that new communication protocols need to be devised to mitigate the impact of intermittent network connectivity on the network performances. Another fairly challenging issue is how to make UAVs communicate each other and have good backhaul communication mechanisms so that they can be effectively coordinated and scheduled to do cooperative communications, position control, interference management, energy replenishment, etc. [6, 5]. How to tackle these practical networking problems and how the obtained solutions to these problems, if adopted, fundamentally influence the network performances in the network are two paramount questions for the success of UAV networks.

I-a Motivation and Prior Work

In light of these aforementioned issues in mmWave communications and UAV networks, a mmWave UAV network is suggested to adopt a plane-split architecture which has a design of splitting the data and control planes of the cellular networks. Such a plane-split architecture design may considerably reduce the networking complexity of a UAV network, and yet it may incur a more challenging context of network connectivity since users need to reliably and simultaneously connect to a ground terminal/BS taking charge in control signaling and a UAV taking charge in payload data delivery. To the best of our knowledge, the fundamental performances of mmWave UAV networks with a plane-split architecture, such as coverage and spatial network throughput, are not yet addressed and studied in the literature.

There are already a few prior works focusing on the performance analysis of UAV networks. The majority of them particularly studies the coverage performance of UAV networks with different constraints (typically see [7, 8, 9, 10, 11, 12, 13, 14]). Reference [7], for instance, studied how to optimize the coverage of a single cell by effectively deploying multiple UAVs over the cell. Under a minimal transmit power constraint, an optimal UAV placement algorithm was proposed in [8] to maximize the number of covered users by means of a UAV deployment decoupled in the vertical and horizontal dimensions. The coverage problem of a finite network of UAVs which were modeled as a uniform binomial point process serving a given region was investigated in [9] and the downlink coverage probability of a reference receiver was derived by assuming Nakagami-m fading for all wireless links. For the work in [10], the authors analyzed the coverage performance of UAV-assisted terrestrial cellular networks in which UAVs are randomly deployed in the 3D space with a height constraint, and they then proposed a cooperative UAV clustering scheme to offload traffic from ground BSs to cooperative UAV clusters. Reference [11] studied how to optimize the coverage of a UAV network by fast deploying UAVs in the sky. It considered two fast deployment optimization problems: one is to minimize the maximum deployment delay with fairness consideration and the other is to minimize the total deployment delay with efficiency consideration. The coverage performance of a reference user in a finite network of multiple UAVs was investigated in [13] and a mixed mobility model which characterizes the movement process of a UAV in the 3D cylindrical region was proposed. In addition to the prior works on the coverage problem of UAV networks, there are some other prior works that focused on some special performance metrics of UAV networks. For example, reference [15] proposed a framework for optimizing the performance of finite UAV networks in terms of the average number of bits transmitted to users as well as the flight time of UAVs. Another example is the work in [16] where the secrecy rate performance of a mmWave UAV network modeled by Matérn hardcore point processes was analyzed.

I-B Contributions

These aforementioned prior works can provide us with a good picture pertaining to how to use UAV as aerial BSs to improve the performances of cellular networks. Nonetheless, some crucial points in the prior works are not yet adequately in existing studies. For example, the majority of the prior works are built on a simple single cell model with a limited number of UAVs so that their analytical results may not be straightforwardly extended to their corresponding counterpart in a large-scale cellular network with inter-cell interference. Also, most of the prior works consider a fairly simple 3D distribution model of UAVs with a constant height that may not be able to generally evaluate the performance of UAV networks. Furthermore, the prior works on UAV networks using the mmWave band are still minimal and there seem no prior works focusing on the performance analysis of mmWave UAV networks with a plane-split architecture. Accordingly, the fundamental coverage and throughput performances of multi-cell mmWave UAV networks with a plane-split architecture are still unclear for the time being, which are the main focuses in this paper. The main contributions of this paper are summarized as follows:

  • We first propose a mmWave cellular network consisting of one tier of ground BSs which use the UHF band and are in charge of the control signaling plane and one tier of UAV (BSs) that adopt the mmWave band and take charge of the data plane. A 3D random distribution model of the UAVs is also proposed and it is able to generally characterize the positions of the UAVs in the sky.

  • Due to the plane-split architecture of the mmWave UAV network, a multi-cell cell association scheme is proposed to help users connect to a ground BS and a UAV at the same time. The multi-cell coverage (probability) and the volume spectral efficiency of the network in the downlink are accordingly proposed as the performance metrics of the mmWave UAV network.

  • The explicit and low-complexity expressions of the multi-cell coverage and volume spectral efficiency are derived when all BSs are equipped multiple antennas and their fundamental upper limits are found for the case of massive antenna array.

  • We particularly show that the multi-cell coverage and volume spectral efficiency can be maximized by optimizing the intensity and height of the UAVs when all UAVs are controlled at the same height and their fundamental maximal limits are characterized for massive antenna array.

  • We also particularly show that the multi-cell coverage does not depend on the intensity of the UAVs and, in addition, the volume spectral efficiency linearly increases with the intensity of the UAVs when all elevation angles from a user to all UAVs remain the same. Under the circumstances, the fundamental maximal limit on the multi-cell coverage can be only achieved by using massive antenna array whereas fundamental maximal limit on the volume spectral efficiency is infinity.

Moreover, some numerical simulation results are provided to validate our analytical findings and observations.

I-C Paper Organization

The rest of this paper is organized as follows. In Section II, we first describe the cellular network model consisting of one tier of mmWave UAVs and one tier of UHF ground BSs and how the UAVs and the ground BSs are distributed in the sky and on the ground, respectively. Section III studies the limit of the multi-cell coverage (probability) and how to maximize it. We then propose the volume spectral efficiency of the mmWave UAV network and analyze its limit in Section IV. Finally, Section V concludes our analytical findings and observations.

Ii System Model and Preliminaries

In this paper, we consider a cellular network consisting of two tiers of BSs: a tier of the BSs on the ground and a tier of the UAVs hovering in the sky and serving as aerial BSs. The ground BSs are of the same type and performance, and they are assumed to use the UHF band and form an independent homogeneous Poisson point process (HPPP) of intensity . In particular, they can be expressed as set given by

(1)

where denotes ground BS and its location. Also, we assume all the UAVs are also of the same type and performance and they only use the mmWave band. The ground projection points of the positions of all the UAVs in the sky form an independent HPPP of intensity . Specifically, the set of all the UAVs is expressed as

(2)

where is UAV and its location, is the ground projection point of , and denotes the (random) vertical height of . This two-tier mmWave UAV network model can be employed in the scenario that the UHF ground BSs have a much higher capability of signal penetration than the mmWave UAVs so that they can be viewed as macro BSs which have much larger transmit power than UAVs and are in charge of the control signaling plane of the network to send control signal information to users and UAVs in the network, whereas the UAVs can be viewed as small cell BSs taking charge of the data plane of the network to deliver payload data to users since they have a much wider bandwidth than the UHF ground BSs and are able to flexibly adapt their positions to the environment so as to enhance their channel quality. Such a (control-data) plane-split network architecture has the advantage of alleviating the frequent handover problem between small cell BSs [17]. Another feature of this two-tier mmWave UAV network is that it is easily extended to multi-tier planar heterogeneous networks in the literature as long as all the heights of the UAVs are set to zero, which means the modeling and analysis based on the proposed network model in this paper are more general than those in the prior works of multi-tier planar cellular networks. Moreover, all (mobile) users in the network also form an independent HPPP. Without loss of generality, we assume there is a typical user located at the origin and many following equations and analyses will be expressed and proceeded based on the location of this typical user111According to the Slivnyak theorem [18, 19], the statistical properties evaluated at the origin are the same as those evaluated at any particular point in an HPPP.. An illustrative example of the mmWave UAV network with a plane-split architecture is depicted in Fig. 1.

Fig. 1: An illustrative example of a mmWave UAV network with a data-control plane-split architecture. Each user in the network has to associate with a UHF ground BS and a mmWave UAV. Without loss of generality, a typical user is assumed to be located at the origin.

Ii-a Elevation Angle, Path-loss Model and Multi-cell Association

A channel in the network is generally considered as either a line-of-sight (LoS) or a non-line-of-sight (NLoS) one. If the channel is visually blocked from a user to a BS, it is NLoS and LoS otherwise. Since whether a channel is LoS or not highly depends on the network environment, we adopt the following low-altitude-platform (LAP) expression in [20] that generally characterizes the LoS probability of a channel between the typical user and UAV at height in different network environments:

(3)

where denotes the Euclidean distance between the typical user and point , and are environment-related constants (for rural, urban, etc.) [20]. All ’s for

are assumed to be independent random variables (RVs). Note that

is not a function of and any more, i.e., whether the channel between the typical user and a UAV on the ground is LoS does not depend on the distance between them. To generally characterize the position of each UAV in the sky, we propose the following height distribution model of UAV :

(4)

where and are constants. Namely, is a power-law function of the distance between the typical user and with parameters and . Such a height model is motivated by the idea of suppressing the interference, that is, the horizontally farther interfering UAVs can hover either at a lower height to make their channels have a higher NLoS probability or at a higher height to make their channel undergo more path loss. Note that the proposed height distribution is so general that it can characterize many position control scenarios of the UAVs. For the case of , for instance, all the UAVs are controlled to hover at the same height of . For the case of , all the UAVs are controlled to maintain the same elevation angle of from the typical user to them. Also note that the heights of the ground BSs are ignored in this paper since they are assumed to be fairly small compared with the heights of the UAVs.

The path-loss model between any BS and the typical user is given by

(5)

where is the path-loss exponent equal to if and equal to if , is the indicator function that is unity if event is true and zero otherwise, where () is equal to () if the channel between and the typical user is LoS and () otherwise. The physical meaning of () can be interpreted as the intercept of the LoS mmWave (UHF) channels, whereas the physical meaning of () can be interpreted as the integrated intercept and penetration loss of the NLoS mmWave (UHF) channels [2].

In the light of the plane-split architecture of the cellular network, each user should associate with a ground BS and a UAV at the same time by adopting the following multi-cell association scheme:

(6)

where and denote the ground BS and the UAV associated with the typical user, respectively. Namely, each user associates with the ground BS and the UAV that provide them with the minimum path loss. The path-loss distributions related to and are given in the following lemma.

Lemma 1.

Suppose all the channels between the typical user and the UAVs are spatially independent. If the penetration loss of all NLoS mmWave channels is infinitely large (i.e., considering

), then the complementary cumulative distribution function (CCDF) of

in (6) can be found as

(7)

where . The CCDF of can be found as

(8)

where and .

Proof:

See Appendix -A. ∎

Since the link between and the typical user must be LoS, we know , and thereby (7) leads to the following result:

(9)

where . This result reveals that the distribution of the square of the distance from the projection point of to the typical user is characterized by which can be interpreted as the intensity of the LoS UAVs. Such an LoS UAV intensity is not a constant (except the case of ) and this means the LoS UAVs in general are a non-homogeneous PPP. We can use two special cases to further explain this interesting and important observation. For example, if , then all the UAVs are at the height of in this case, and we thus have

(10)

where . Thus, all the LoS UAVs with the same height are not an HPPP any more in that their intensity can be equivalently viewed as which is location-dependent 222More precisely, all the UAVs hovering at the same height in the sky form a three-dimensional (3D) non-homogeneous PPP with a location-dependent intensity . As such, their ground projection points are a non-homogeneous PPP with the same intensity as well.. Another example is the scenario of in which all the elevation angles from the typical user to the LoS UAVs are the same and equal to so that we get

(11)

Hence, all the LoS UAVs form an HPPP of intensity because is an exponential RV with parameter [18, 19]. A similar observation can also be drawn for the ground BSs. For example, the result in (8) can alternatively be expressed as

(12)

which means that set can be equivalently viewed as an HPPP of intensity , as shown in Theorem 1 of our previous work in [21, 22]. The results in Lemma 1 will be employed to exploit the statistical properties of the signal-to-interference plus noise ratio (SINR) of the users in the downlink.

Ii-B Small-Scale Fading Channel Model with MISO Beamforming

In the previous subsection, the path-loss model between a BS and a user is specified, yet we would like to specify how to model the small-scale multiple-input-single-output (MISO) fading channel gain from a BS to a user in this subsection. For the tractability of analysis, we assume that all the users are equipped with a single antenna, while all the ground BSs and UAVs are equipped with and antennas, respectively. Hence, each downlink channel is a MISO channel and the fading channel gain of a channel from BS to the typical user can be modeled by

(13)

where denotes the beamforming fading channel gain333Please note that the penetration loss of the NLoS channels of the ground BSs is already modeled in their path-loss model so that their fading gain does not need to consider the LoS effect. from and it is a Gamma RV with shape and rate parameters , is the beamforming fading channel gain444According to [23]

, we know that the fading gain vector of a mmWave multiple-input-multiple-output (MISO) channel can be properly represented by a clustered channel model consisting of small-scale fading and angle-of-departure (AoD)-based transmit array gain vectors. Here we thus assume that all UAVs have a uniform linear array and are able to perfectly align their beam with the angle-of-departure (AoD) of their array in order to maximize their antenna array gain from them to their serving users.

from , is the fading gain from which is an exponential RV with unit mean, denotes the transmit antenna array gain of in which and are the azimuth and the inclination of the main lobe, respectively; and are the boresight azimuth and the inclination of , respectively (see Fig. 1); , and are the main lobe gain and side lobe gain of the antenna array of UAVs, respectively. Note that and since we assume

is uniformly distributed over

and is uniformly distributed over . In the following subsection, we will apply the previously proposed path-loss model and the small-scale fading channel model to model two incomplete shot signal processes generated by the UAVs and ground BSs and investigate their statistical properties.

Ii-C The Incomplete Shot Signal Process

Without loss of generality, we assume that is the UAV that provides the typical user with the th smallest path-loss among all the UAVs in . For the typical user, its 3D th-incomplete shot signal process generated by the UAVs in the network is defined as

(14)

where denotes the transmit power of a UAV. Note that does not contain the signal powers generated by the UAVs in that generate the first smallest path-loss signals received by the typical user so that it is called the th-incomplete shot signal process. Similarly, the path loss from to the typical user is assumed to be the th smallest one among all the path losses from all the ground BSs in . We then define the 2D th-incomplete shot signal process of the typical user as follows:

(15)

where is the transmit power of the ground BSs and stands for the equivalence in distribution. Likewise, does not include the first smallest path-loss signals among all path-loss signals generated by all the ground BSs. Note that we do not consider the void cell phenomenon in (14) and (15) because the user intensity is assumed to be so large that all the UAVs in the network are associated with at least one user with high probability [24, 21].

Studying the statistical properties of and is crucial because it helps us understand how the aggregated signal powers from the UAVs and the ground BSs are affected by the network parameters (such as LoS/NLoS channel modeling parameters, BS intensities, etc.). In particular, we are interested in the Laplace transforms of and in that they will facilitate our following analyses. The Laplace transform of a non-negative RV is defined as for , and thereby the Laplace transforms of and are found as shown in the following proposition.

Proposition 1.

Suppose all the signal powers from mmWave UAVs with an NLoS channel are so small that they can be completely ignored at the users. According to defined in (14) and defined in (15), the Laplace transform of conditioned on RV can be found as

(16)

where the probability density function (PDF) of

is given by

(17)

in which and for is defined as

(18)

The Laplace transform of for a given can be derived as

(19)

where is already defined in Lemma 1 and function for is given by

(20)

where . Hence, and .

Proof:

See Appendix -B. ∎

The objective of finding the above Laplace transforms of and is that they can be generally employed in many analytical situations. For example, if users associate with their nearest LoS UAV and they can cancel the signals from the first nearest interfering LoS UAVs, we can use to evaluate the statistical properties of the interference from other interfering UAVs so as to clarify the fundamental interplays between the height and intensity of the UAVs. Likewise, the formula of also helps us characterize the statistical properties of the interference from the ground BSs when the signals from the first nearest ground BSs are removed by cell association and/or interference cancellation. Although the results in Proposition 1 are somewhat complex, they are quiet general and able to reduce to a much simpler form for some special cases. For example, consider the case of a fixed elevation angle between the typical user and the UAVs (i.e., ) and in (16), (i.e., the mmWave side-lobe interference is too small to be considered), and (16) then reduces to

(21)

which has a closed-form result equal to if . This result is not a function of , i.e., the Laplace transform of scaled by does not depend on any more. Similarly, if , then is largely simplified and can be found as a nearly closed-form result given by

(22)

and it does not depend on the intensity of the ground BSs as well. In other words, the Laplace transform of does not change with if interference is scaled by . These above observations regarding and will be quite useful for the following analyses.

Iii The Multi-Cell Coverage: Limit Analysis and Optimization

In the network, users are able to associate with one ground BS that transmits control signals and one UAV that delivers downlink payload data. Therefore, they have to be simultaneously covered by the two BSs associated with them in order to successfully receive data from their serving UAV. Accordingly, we first propose the SINR models of a user in the mmWave and UHF spectra and use them to define the multi-cell coverage (probability) of a user. Afterwards, we will analyze the multi-cell coverage and study how to maximize it by optimally deploying and positioning the UAVs. Finally, some numerical results are provided to validate our analytical findings and observations.

Iii-a Analysis of the Multi-cell Coverage

According to the path-loss model in (5), the multi-cell association scheme in (6), the fading channel gain model in (II-B), and the th-incomplete shot signal process in (14), the SINR of the typical user associating with UAV is defined as

(23)

where is the noise power in the mmWave band and that is the 3D 1st-incomplete shot signal process based on (14) denotes the interference from all the interfering UAVs. Similarly, the SINR of the typical user associating with ground BS can be defined as

(24)

where is the 2D 1st-incomplete shot signal process accounting for the interference from all the interfering ground BSs. Note that there is no noise power term in (24) because the network in the UHF band is usually interference-limited. Using the SINRs defined in (23) and (24), we can define the multi-cell coverage (probability) of a user as follows:

(25)

in which is the SINR threshold for successful decoding. The probability is called the UAV coverage, whereas the probability is called the ground coverage. The equality in (25) holds due to the independence between and . This multi-cell coverage reflects how likely a user is able to successfully receive signals from its associated UAV and ground BS at the same time. Such a multi-cell coverage definition stems from the fact that the UAVs can successfully deliver their data to the users in the network only when the users can be simultaneously “covered” by their associated UAV as well as ground BS.

The multi-cell coverage of a user is derived as shown in the following proposition.

Proposition 2.

If all the signals from the NLoS UAVs are too small to affect the SINR model in (23) and the multi-cell association scheme in (6) is adopted, the multi-cell coverage defined in (25) can be expressed as in which the UAV coverage can be explicitly found as

(26)

in which is defined in (17) for and is defined in (18), and the ground coverage is explicitly derived as

(27)

where is defined in (20) and it does not depend on .

Proof:

See Appendix -C. ∎

From Proposition 2, we are able to learn a few important implications which are summarized as follows. First of all, since we devise a new technique to derive the coverage for multiple-input-single-output (MISO) channels as shown in Appendix -C, the two coverage expressions in (2) and (27) are much neater and more general than the existing results of MISO coverage probabilities in the literature (for example, see the coverage results of a mmWave network in [25, 22]). As such, they easily reduce to the results in some special cases. For the single transmit antenna and interference-limited network case, for example, in (2) significantly reduces to

(28)

while in (27) neatly simplifies to

(29)

Moreover, in (2) can readily reduce to the coverage for the noise-limited case by setting or as zero, and it can also be applied to evaluate the coverage for the case of ground mmWave BSs by setting equal to zero. Second of all, if all the UAVs and ground BSs are equipped with a massive antenna array and the network is interference-limited, we can further show that and in this case can be characterized by the following two expressions:

(30)

and

(31)

where represents the inverse Laplace transform of function . To the best of our knowledge, in (30) and in (31) are the most explicit coverage expressions firstly found in this paper for mmWave UAV networks with massive MISO beamforming. They can be effectively evaluated by the numerical techniques of the inverse Laplace transform even though they are not in closed-form. Hence, we conclude that the upper limit of the multi-cell coverage with massive MISO beamforming is given by

(32)

which can be analytically evaluated by using (30) and (31). Third of all, since does not depend on the intensity of the ground BSs, the multi-cell coverage is only influenced by the intensity and position of the UAVs that largely affect . In the following subsection, we will further look into how the intensity and position of the UAVs impact and discuss how to optimize them in order to maximize .

Iii-B Optimal Deployment and Position Control of UAVs for Maximizing Multi-cell Coverage

In this subsection, our focus is on how to maximize the multi-cell coverage by optimally deploying the ground BSs and UAVs as well as controlling the hovering positions of the UAVs. According to Proposition 2, we are able to see how the height distribution models of the UAVs with different values of and in (4) impact the coverage performance of the UAVs. Here we are particularly interested in two height control models of the UAVs, i.e., (constant) height control model and (constant) elevation angle control model. They are elaborated as follows.

Iii-B1 Height Control Model

For this model, all the UAVs are controlled to hover at the same height of , i.e., letting in (2). Such a height control model has an advantage, that is, it intrinsically suppresses the interference received by a user because the channels from the UAVs farther to the user not only undergo more path loss but also become NLoS with a higher probability due to the smaller elevation angles from the user to the farther UAVs. Another advantage of this model is to make the analysis of the UAV coverage much more tractable so that we are able to analytically understand how the height of the UAVs influences the UAV coverage performance. To further demonstrate this point, consider in (2) for and the interference-limited network case, and we thus get

(33)

which essentially indicates that dominates and it is dominated by . Since the ground coverage is not dependable upon , and , we need to maximize the UAV coverage by optimizing and so as to maximize the multi-cell coverage . To maximize the UAV coverage, we formulate the following optimization problem:

(34)

where is given in (2). This optimization problem enjoys the following property.

Proposition 3.

If all the UAVs are controlled to hover at the same height, there exists a unique optimal intensity of that maximizes the UAV coverage in (2). Likewise, there exists an optimal height of that maximizes the UAV coverage for a given UAV intensity.

Proof:

See Appendix -D. ∎

Proposition 3 reveals an important fact, that is, the multi-cell coverage can be maximized by either optimizing for a given or optimizing for a given . In addition, Proposition 3 is valid for any number of transmit antennas equipped at the UAVs so that the upper limit of the multi-cell coverage with massive MISO beamforming in (32) also can be maximized by optimizing and . Likewise, we can formulate the following optimization problem

(35)

Solving the above optimization problem and then evaluating at the optimal solution pair of and give rise to the fundamental maximal limit of the multi-cell coverage for the height control model.

Iii-B2 Elevation Angle Control Model

For this model, all the UAVs are controlled to maintain the same elevation angle from the typical user to them, i.e., setting in (2) makes all the elevation angles from the typical user to all the UAVs equal to . Although such an elevation angle control model may not increase the NLoS probability of the channels from all the interfering UAVs to the typical user, it may make the channels of all the interfering UAVs undergo more path loss if compared with the constant height model, especially when the network is in an environment with a large path-loss exponent. By considering the interference-limited case and using (2) with , the UAV coverage for this elevation angle control model is readily obtained as

(36)

which does not depend on the UAV intensity. This result manifests that deploying many UAVs does not ameliorate the UAV coverage once all elevation angles from a user to all the UAVs remain the same. Consequently, the UAVs can only rely on their massive antenna array to further ameliorate , that is, as goes to infinity, in (36) will converge up to

(37)

which is obtained by using (30) with . Furthermore, since we can show that does not depend on any more, we know that in (36) and in (37) cannot be maximized by optimizing and . In light of this, it is concluded that for the elevation angle control model cannot be maximized by optimizing the UAV intensity and the evaluation angle and it only can be enhanced through massive MISO beamforming so that its fundamental maximal limit is in (37). Moreover, it is worth pointing out that the height control model may not always outperform the elevation angle control model from the coverage perspective since in (33) may be smaller than in (36) for some values of and . In the following subsection, we will use some numerical results to verify these aforementioned analytical findings.

Iii-C Numerical Results

Parameter BS Type UHF Ground BS mmWave UAV (28 GHz)
Transmit Power (W) 20 1
Intensity (BSs/) (or see figures)
Number of Antennas 16 4
Intercept Coefficients (dB)
Azimuth of the Main Lobe (not applicable)
Inclination of the Main Lobe (not applicable)
Gain of the Main Lobe (not applicable)
Gain of the Side Lobe (not applicable)
Maximum Height (m) in (4) (not applicable) 200
Path-loss Exponent 4 3
Parameters in (3)
SINR Threshold 1
TABLE I: Network Parameters for Simulation[23, 2]
Fig. 2: Simulation results of coverage probabilities , and for the height control model: (a) Coverage versus Height of the UAVs for , (b) UAV Coverage versus and Height of the UAVs.
Fig. 3: Simulation results of coverage probabilities , and for the elevation angle control model: (a) Coverage versus Elevation Angle for , (b) UAV Coverage versus and Elevation Angle .

In this subsection, some numerical results are provided to validate our previous analytical results of the multi-cell coverage , UAV coverage and ground coverage . The network parameters for simulation are listed in Table I. To clearly and simply validate our previous analytical results and observations, we consider the mmWave UAV network is so densely deployed that it is interference-limited. We first present the simulation results of the coverage for the height control model in Fig. 2. As shown in Fig. 2(a), we can see how the three different coverage probabilities vary with the height of the UAVs. All the simulated coverage results perfectly coincide with their corresponding analytical results in Fig. 2(a), which validates the correctness of our previous analyses. The ground coverage, as expected, remains a constant about 0.651, whereas the UAV coverage significantly changes with height and finally reaches a maximum around 0.95 at m. Hence, there indeed exists an optimal height that maximizes the UAV coverage, as claimed in Proposition 3, and this phenomenon can also be further observed in Fig. 2(b). From Fig. 2(b), we see how the UAV coverage varies with and the height and how it maximizes at different optimal pairs of and . Since in general is not a convex function of and as shown in Fig. 2(b), there does not exist a unique global pair of and that minimizes , which supports the statement in Proposition 3.

Fig. 4: Simulation results of the coverage probabilities , and versus the number of transmit antennas and for and the height control model with m.

The simulation results of the coverage probabilities for the elevation angle control model are demonstrated in Fig. 3. The results of the coverage probabilities , and versus elevation angle are shown in Fig. 3(a) and all the analytical results perfectly coincide with their corresponding simulated results. Fig. 3(a) also shows that all the coverage probabilities do not depend the elevation angle , which supports our previous discussion. As a matter of fact, all the coverage probabilities are not dependable upon intensity either. This is demonstrated in 3(b), which plots the UAV coverage as a 2D horizontal plane located at the height of 0.865. As the figure shows, changing the elevation angle and deploying the UAVs in the sky do not benefit the coverage performance of the users in the network using the elevation angle control model. Hence, if the height of the UAVs is optimally controlled, the height control model certainly outperforms the elevation control model in terms of the UAV coverage, as shown in Fig. 2 and Fig. 3. Fig. 4 shows how the number of the transmit antennas of the UAVs and ground BSs impact the coverage probabilities. As shown in the figure, as the numbers of the transmit antennas and increase, the UAV coverage and ground coverage increase and eventually converge up to 0.975 and 0.66, respectively. These two maximum coverage values validate the correctness of the upper limits of the UAV coverage in (30) and ground coverage in (31) since they are exactly the same as the coverage values evaluated by (30) and (31).

Iv The Volume Spectral Efficiency: Limit Analysis and Optimization

In the previous section, we have learned that jointly optimizing the UAV intensity and position is able to maximize the multi-cell coverage. This inspires us to further investigate how the throughput of the UAV mmWave network is affected by the intensity and position of the UAVs. Specifically, we propose the volume spectral efficiency to characterize the network throughput of the UAV mmWave network per unit volume and bandwidth. The volume spectral efficiency will be analyzed and its explicit expression will be derived. We then will study how to maximize the volume spectral efficiency by optimizing the intensity and position of the UAVs.

Iv-a Analysis of Volume Spectral Efficiency

Recall that the ground BSs and UAVs are in charge of the control and data planes of the network, respectively. Thus, whether the UAVs are able to effectively deliver data to the users highly depends on whether the users are able to successfully receive the control signals sent by the ground BSs. As a result of this plane-split architecture, the throughput of this mmWave UAV network needs to consider the ground coverage impact. The link rate from UAV to the typical user can be characterized by the SINR in (23) and it is defined as

(38)

which indicates that is non-zero if and only if the users are covered by the ground BSs. It can be used to define the volume spectral efficiency of the mmWave UAV network as follows: