I Introduction
The telecommunications industry and academia have long agreed on the social benefits that can be brought by having cellularconnected unmanned aerial vehicles (UAVs) [1, 2, 3]. These include facilitating searchandrescue missions, acting as mobile small cells for providing coverage and capacity enhancements [4], and even automating logistics in indoor warehouses [5]. From a business standpoint, mobile network operators are chasing new revenue opportunities by offering cellular coverage to a heterogeneous population of terrestrial and aerial users [6, 7]. A certain consensus has been reached—both at 3GPP meetings and in the classroom—on the fact that presentday networks will be able to support cellularconnected UAVs up to a certain extent [8, 9, 10, 11, 12]. Besides, recent studies have shown that 5Gandbeyond hardware and software upgrades will be required by both mobile operators and UAV manufacturers to target large populations of UAVs flying at high altitudes [13, 14].
However, important usecases exist where direct communication between UAVs, bypassing ground network infrastructure, would be a key enabler. These include autonomous flight of UAV swarms, collision avoidance, and UAVtoUAV relaying, data transfer, and gathering. Similarly to ground devicetodevice (D2D) communications [15, 16], UAVtoUAV (U2U) communications may also bring benefits in terms of spectral and energy efficiencies, extended cellular coverage, and reduced backhaul demands [17, 18].
In this article, we investigate U2U communications underlaying a cellular network. In such a setup, UAVtoUAV transmitreceive pairs share the same spectrum with the uplink (UL) of cellular ground users (GUEs). Through stochastic geometry tools, we explicitly characterize the performance of both U2U and GUE UL, as well as their interplay. To the best of our knowledge, this work is the first one to do so by accounting for: (i) a realistic propagation channel model that depends on the UAV altitude, (ii) the impact of a practical base station (BS) antenna pattern, and (iii) a fractional power control policy implemented by all nodes. Our takeaways can be summarized as follows:

[leftmargin=*]

The presence of U2U links may degrade the GUE UL. However, such performance loss is not dramatic, since BSs perceive interfering UAVs through their antenna sidelobes, and UAVs can generally transmit at low power thanks to the favorable U2U channel conditions.

The performance of both U2U and GUE UL links degrades as UAVs fly higher. This is due to an increased probability of lineofsight (LoS)—and hence interference—on all UAVtoUAV, GUEtoUAV, and UAVtoBS interfering links. This negative effect outweighs the benefits brought by having larger GUEtoUAV and UAVtoBS distances.

The UAV power control policy has a significant impact on all links. A tradeoff exists between the performance of U2U and UL GUE communications, whereby increasing the UAV transmission power improves the former at the expense of the latter.

Smaller U2U distances can improve the performance of both U2U and GUE UL. Indeed, owed to a better U2U path loss, UAVs may employ a smaller transmission power and therefore reduce the interference they cause to other U2U links and to GUEs.
Ii System Model
In this section, we introduce the network topology, channel model, and power control mechanisms considered throughout the paper. The main parameters used in our study are given in Table I.
Iia Network Topology and Spectrum Sharing Mechanism
Ground cellular network
We consider the UL of a traditional ground cellular network as depicted in Fig. 1
, where BSs are uniformly distributed as a Poisson point process (PPP)
with density . All BSs are deployed at a height , and communicate with their respective sets of connected GUEs. Assuming that the number of GUEs is sufficiently large when compared to that of the BSs, the active GUEs on each timefrequency physical resource block (PRB) form an independent Poisson point process with density [16]. We further consider that GUEs associate to their closest BS, which generally also provides the largest reference signal received power (RSRP)^{1}^{1}1A GUE may connect to a BS other than the closest one if its link is in LoS with and not with . However, since the probability of LoS decreases with the distance, such event is unlikely to occur [19].. Therefore, the 2D distance between a GUE and its associated BS follows a Rayleigh distribution with a scale parameter given by . When focusing on a typical BS serving its associated GUE, the interfering GUEs form a nonhomogeneous PPP with density , where is the 2D distance between the interfering GUE and the typical BS [16, 20, 21].UAVtoUAV communications
As illustrated in Fig. 1, in this work we also consider that U2U transmitreceive pairs reuse the cellular GUE UL spectrum. We assume that U2U transmitters form a PPP with intensity , and that each U2U receiver is randomly and independently placed around its associated transmitter with distance distributed as . While our analysis holds for any transmit/receive UAV height, in the following we assume all UAVs to be located at the same height , to evaluate the impact of such parameter.
Spectrum sharing
We consider an underlay inband approach for resource sharing between GUE UL and U2U [15], where each PRB may be used by both link types. This results in four types of links: (i) GUEtoBS communication and/or interfering links, (ii) UAVtoBS interfering links, (iii) UAVtoUAV communication and/or interfering links, and (iv) GUEtoUAV interfering links.
IiB Propagation Channel and Power Control
We assume that any radio link between nodes and is affected by largescale fading (comprising path loss and antenna gain ) and smallscale fading .
Probability of LoS
We consider that links experience lineofsight (LoS) and nonLoS (NLoS) propagation conditions with probabilities and , respectively. In what follows, the superscripts and will denote system parameters under LoS and NLoS conditions, respectively. In our analysis we assume that is or can be approximated by a step function, i.e., is constant for an interval , where and .
Path loss
The distancedependent path loss between two nodes and is given by
(1) 
where denotes the reference path loss, is the path loss exponent, and , , and represent the 3D distance, 2D distance, and height difference between and , respectively. Table I lists the path loss parameters employed in our study, which depend on the nature of and . In the sequel, we employ the subscripts to denote UAV, GUE, and BS nodes, respectively.
Antenna gain
We assume that all GUEs and UAVs are equipped with a single omnidirectional antenna with unitary gain. On the other hand, we consider a realistic BS antenna radiation pattern to capture the effect of sidelobes, which is of particular importance in UAVtoBS links [12, 13]. We assume that each BS is equipped with a vertical, element uniform linear array (ULA), where each element has directivity
(2) 
as a function of the zenith angle . The total BS radiation pattern is obtained as the superposition of each element’s radiation pattern and by accounting for the array factor given by
(3) 
where denotes the electrical downtilt angle. The total antenna gain between a pair of nodes and is given by the product of their respective antenna gains.
Smallscale fading
On a given PRB, denotes the smallscale fading power between nodes and
. Given the different propagation features of groundtoground, airtoair, and airtoground links, we adopt the general Nakagamim smallscale fading model. As a result, the cumulative distribution function (CDF) of
is given by(4) 
where is the fading parameter, with LoS links typically exhibiting a larger value of than NLoS links.
Power Control
As per the 3GPP guidelines, we consider fractional power control for all nodes. Accordingly, the power transmitted by node while communicating to node is given by [22]
(5) 
where is the maximum transmit power at node , is a cellspecific parameter, is the fractional power control factor, and is the largescale fading between nodes and . The aim of (5) is to compensate for a fraction of the largescale fading, up to a limit imposed by [19].
Iii Analytical Results
Our U2U (resp. GUE UL) performance analysis is conducted for a typical BS (resp. UAV) receiver located at the origin. In what follows, uppercase and lowercase are employed to respectively denote random variables and their realizations, e.g.,
and . Throughout the derivations, we make use of the superscripts to denote LoS and NLoS conditions on a certain link.Iiia U2U Performance Analysis
We now derive the U2U link coverage, i.e., the complementary CDF (CCDF) of the signaltointerferenceplusnoise ratio (SINR) experienced by a UAV.
Theorem 1.
The U2U link coverage is given by
(6) 
where represents the ith derivative with respect to and is the typical U2U communication link distance. Also, by denoting the noise power with , we have
(7) 
In (6), the interference is characterized by its Laplacian, which is obtained as with
(8) 
where for
(9) 
Proof.
See Appendix A. ∎
IiiB GUE UL Performance Analysis
We now obtain the GUE UL coverage, i.e., the CCDF of the UL SINR experienced by a GUE in the presence of U2U communications sharing the same spectrum.
Theorem 2.
Proof.
See Appendix B. ∎
Iv Numerical Results and Discussion
Deployment  

BS distribution  PPP with / Km 
GUE distribution  One active GUE per cell, m 
UAV distribution  / Km, m, =100 m [19] 
Channel model  
Ref. path loss [dB]  ( in GHz) 
Path loss exponent  
Smallscale fading  Rayleigh^{2}^{2}2 After deriving analytical results under Nakagamim smallscale fading, we now consider Rayleigh as a special case. This has been shown not to change the qualitative performance trends [12, 10]., i.e., 
Prob. of LoS  ITU model as per (18) 
Thermal noise  174 dBm/Hz with 7 dB noise figure [19] 
PHY  
Spectrum  Carrier frequency: 2 GHz [19] 
System bandwidth: 10 MHz with 50 PRBs[19]  
BS antennas  See (2) for elements gain 
BS array configuration  Height: m, downtilt: , vertical array, 1 RF chain, element spacing: [19] 
Power control  Fractional power control based on GUEtoBS (resp. U2U) largescale fading for GUEs (resp. UAVs), with , dBm, and dBm [22] 
GUE/UAV antenna  Omnidirectional with 0 dBi gain [19] 
We now provide numerical results to evaluate the performance of U2U and GUE UL communications sharing the same spectrum. Specifically, we concentrate on characterizing the impact that the UAV altitude, the UAV power control, and the U2U distance have on the U2U and GUE UL links. Unless otherwise specified, the system parameters used in this section are provided in Table I.
We model the U2U link distance
via a truncated Rayleigh distribution with probability density function (PDF)
(17) 
where is the maximum U2U link distance, is the indicator function, and is the Rayleigh scale parameter, related to the mean distance through .
As for the probability of LoS between any pair of nodes x and y, we employ the well known ITU model [23, 8]:
(18) 
where are environmentdependent parameters set to to model an urban scenario [23].
Fig. 2 shows the CCDF of the SINR per PRB experienced by: (i) U2U links, (ii) the UL of GUEs in the presence of U2U links, and (iii) the UL of GUEs without any U2U links. For (i) and (ii), we consider two UAV heights, namely 50 m and 150 m. Fig. 2 also allows to make the following observations:

[leftmargin=*]

U2U communications degrade the UL performance of GUEs. However, such performance loss amounts to less than 3 dB in median, since (i) BSs perceive interfering UAVs through their antenna sidelobes, and (ii) UAVs generally transmit with low power due to the good U2U channel conditions.

The U2U performance degrades as UAVs fly higher, due to an increased UAVtoUAV and GUEtoUAV interference. The former is caused by a higher probability of LoS between a receiving UAV and interfering UAVs. The latter is caused by a higher probability of LoS between a receiving UAV and interfering GUEs, whose effect outweighs having larger GUEUAV distances.

The GUE UL performance also degrades as UAVs fly higher. However, this degradation is less significant than that experienced by the U2U links, since interference generated by GUEs in other cells is dominant.
Fig. 3 illustrates (i) the mean useful received power, (ii) the mean interference power received from GUEs, and (iii) the mean interference power received from UAVs, for both U2U and GUE UL links. These metrics are plotted as a function of the fractional power control factor employed by UAVs. We may observe the following:

[leftmargin=*]

The UAV power control policy has a significant impact on the performance of both U2U and UL GUE links.

In the scenario under consideration, the mean interference perceived by GUEs is dominated by the GUEgenerated transmissions from other cells for , where such interference also remains small compared to the mean useful received power. The interference generated by UAVs dominates for larger values of , and it saturates for , since almost all UAVs transmit with their maximum allowed power.

The mean interference perceived by UAVs is dominated by the GUEgenerated transmissions for , where such interference is also relatively large compared to the mean useful received power. For larger values of , the UAVtoUAV interference becomes dominant and keep growing alongside the useful signal up to , when almost all UAVs operate at maximum power.
Fig. 4 shows the probability of experiencing SINRs per PRB larger than 5 dB for both U2U and UL GUE links as a function of . We also consider three different values for the U2U distance parameter , namely 50 m, 100 m, and 150 m, corresponding to mean U2U distances of 63 m, 125 m, and 188 m, respectively. Fig. 4 allows us to draw the following conclusions:

[leftmargin=*]

There exists an inherent tradeoff between the performance of U2U and GUE UL, whereby increasing improves the former at the expense of the latter:

[leftmargin=*]

For , the U2U performance is deficient, since UAVs use a very low transmission power. In this range, the UL GUE performance is approximately constant, since the GUEgenerated interference is dominant.

For , the U2U performance increases at the expense of the UL GUE links.

For , the U2U performance saturates and that of the GUEs stabilizes, since almost all aerial devices reach their maximum transmit power.


Smaller U2U link distances—for fixed UAV density—correspond to a better U2U performance for all values of . This is because (i) UAVs perceive larger received signal powers for decreasing , since the path loss of the U2U links diminishes faster than the UAV transmit power when lessens, and (ii) the reduced UAVtoUAV interference due to the smaller transmission power employed by UAVs.

The UL GUE links also benefit from smaller U2U link distances when , since UAVs lower their transmit power and therefore reduce the UAVtoBS interference.
V Conclusion
In this article, we have provided an analytical framework to study U2U communications underlayed with the UL of a cellular network. When considering a realistic channel model, antenna pattern, and power control policy, we demonstrated that communications between pairs of closeby UAVs have a limited effect on the GUE UL, since the strong U2U channel gains allow UAVs to lower their transmit power. Our results also showed that both the U2U and GUE UL SINRs diminish as UAVs fly higher, since aerial equipment encounters LoS propagation conditions with a larger number of nodes, which leads to an overall interference growth. We also demonstrated how the UAV power control policy serves to find a performance tradeoff between U2U and UL GUE communications.
a Sketch of Proof of Theorem 1
To obtain the U2U link coverage we can write
(19) 
where
(20) 
obtained using the CDF of gammadistributed smallscale fading in (
4). As for the interference, one can write(21) 
where is the interference imposed by nodes x of condition on y. Each term in (21) can be characterized as follows:
(22) 
where accounts for the density of active users, and
(23) 
with . In the following we calculate the integral term in (23) by considering a change of variable as . Therefore we can rewrite
(24)  
where , , , and
(25) 
We also have
(26) 
where integration by parts is applied, and
(27) 
where we used the definition of incomplete gamma function. It follows that
(28)  
We note that for Nakagamim fading with parameter we have
(29) 
and
(30)  
Now through transformation properties of the hypergeometric function we can write
(31)  
Consequently, by using (28)–(31) we have
(32)  
Accordingly,
(33) 
By noting that
(34)  
the desired result is obtained.
B Sketch of Proof of Theorem 2
Similar to U2U coverage analysis, we can write
(35) 
where
(36) 
where the last equation is derived similarly to (20).
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