# Joint Location, Bandwidth and Power Optimization for THz-enabled UAV
Communications

Luyao Xu, Ming Chen, Mingzhe Chen, Zhaohui Yang, Christina Chaccour, Walid
Saad, and Choong Seon Hong L. Xu and M. Chen are with National Mobile
Communications Research Laboratory, Southeast University, Nanjing, 210096,
China, Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>Chen is with the
Electrical Engineering Department of Princeton University, NJ, 08544, USA,
Email<EMAIL_ADDRESS>Yang is with the Department of Engineering,
King’s College London, WC2R 2LS, UK, Email<EMAIL_ADDRESS>Chaccour
and W. Saad are with the Wireless@VT, Bradley Department of Electrical and
Computer Engineering, Virginia Tech, Blacksburg, VA, USA, Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>C. Hong is with the Department of Computer
Science and Engineering, Kyung Hee University, Yongin-si, Gyeonggi-do 17104,
Rep. of Korea, Email<EMAIL_ADDRESS>
###### Abstract

In this paper, the problem of unmanned aerial vehicle (UAV) deployment, power
allocation, and bandwidth allocation is investigated for a UAV-assisted
wireless system operating at terahertz (THz) frequencies. In the studied
model, one UAV can service ground users using the THz frequency band. However,
the highly uncertain THz channel will introduce new challenges to the UAV
location, user power, and bandwidth allocation optimization problems.
Therefore, it is necessary to design a novel framework to deploy UAVs in the
THz wireless systems. This problem is formally posed as an optimization
problem whose goal is to minimize the total delays of the uplink and downlink
transmissions between the UAV and the ground users by jointly optimizing the
deployment of the UAV, the transmit power and the bandwidth of each user. The
communication delay is crucial for emergency communications. To tackle this
nonconvex delay minimization problem, an alternating algorithm is proposed
while iteratively solving three subproblems: location optimization subproblem,
power control subproblem, and bandwidth allocation subproblem. Simulation
results show that the proposed algorithm can reduce the transmission delay by
up to $59.3\%$, $49.8\%$ and $75.5\%$ respectively compared to baseline
algorithms that optimize only UAV location, bandwidth allocation or transmit
power control.

###### Index Terms:

UAV communication, THz communication, resource allocation.

##  I Introduction

Unmanned aerial vehicles ( UAVs ) [1] are an effective solution to provide
temporary wireless connectivity [2, 3, 4] due to their flexible
maneuverability and convenient deployment. Compared to terrestrial base
stations, UAVs can provide higher energy efficiency, improved coverage, and
higher wireless capacity. However, deploying UAVs for wireless communications
also faces several challenges in terms of deployment, trajectory design, and
system optimization.

A number of existing works in [5, 6, 7, 8, 9, 10, 11] studied the use of UAVs
for wireless communications. The authors in [5] maximized the number of users
that are served by one UAV. In [6], the altitude and location of a UAV were
jointly optimized so as to minimize the UAV transmit power used to service
ground users. The problem of user-UAV association is studied in [7] with the
goal of minimizing transmit power. The authors in [8] studied the problem of
optimal trajectory design and transmit power of the UAV with the goal of
maximizing the minimum achievable rate among all users under their harvested
energy constraints. In [9], the optimal deployment of multiple UAVs was
analyzed to maximize the downlink coverage. Meanwhile, the work in [10]
studied the problem of three-dimensional (3D) coverage maximization for UAV
networks. Moreover, for an UAV enabled mobile edge computing network, the
authors in [11] jointly optimized the location and transmit power of UAVs to
minimize the energy consumption. However, none of these existing works such as
in [5, 6, 7, 8, 9, 10, 11] considered the use of the terahertz (THz) band for
UAVs. The THz band (0.1-10 THz) is considered to be one of the 6G driving
trends owing to its ultra-wide bandwidth [12]. Due to the ultra-wide bandwidth
of the THz band and the flexible location of the UAV, the combination of UAV
and THz can support unprecedentedly high data rate for future services.
Nevertheless, given the high uncertainty at higher frequency bands such as
THz, it is important to provide more degrees of freedom and control in network
management. It is generally known that UAVs can provide line-of-sight (LoS)
transmission links to the ground users given their ability to fly. To reap the
benefits of UAV deployment in THz systems, it is necessary to optimize the
position of the UAVs in a way to provide continual LoS links to the ground
users [13]. In [14], the authors proposed the use of a UAV and an intelligent
re?ective surface (IRS) to support THz communications while jointly optimizing
the UAV’s trajectory, the phase shift of the IRS, the allocation of THz sub-
bands, and the transmit power. Different from the work in [14], the authors in
[15] optimized the location of IRS rather than considering the location of the
UAV. However, the works in [14] and [15] only considered the problem of
resource allocation for downlink communication in UAV assisted THz
communication, without taking into account the joint downlink and uplink
design. Besides, the works in [14] and [15] only obtained a suboptimal
solution for the location of UAV or IRS.

The main contribution of this paper is a first fundamental analysis of the
optimal deployment of a UAV that operates at THz frequencies. We consider the
downlink and uplink of a wireless network in which THz-enabled UAVs can serve
the ground users. Our goal is to minimize the total delays of the uplink and
downlink transmissions between the UAV and the users, by jointly optimizing
the location of the operating UAV, and the bandwidth of the users, as well as
minimizing the transmit power of the users. To solve this optimization
problem, we propose an iterative algorithm that divides the original
optimization problem into three subproblems: A location optimization
subproblem, a power control subproblem, and a bandwith allocation subproblem
that are solved iteratively. All the subproblems are shown to be convex, and
hence, they can be effectively solved using standard optimization methods.
Numerical results show that the proposed algorithm can yield up to $59.3\%$,
$49.8\%$ and $75.5\%$ reduction in terms of the transmission delay compared
to, respectively, a baseline which only optimizes the UAV location, a baseline
which only optimizes the bandwidth of each user, and a baseline which only
optimizes the transmit power.

##  II System Model and Problem Formulation

Consider a UAV-assisted wireless communication system that consists of one UAV
and $N$ users. The coordinates of the UAV is denoted by $\bm{u}_{0}=(x,y,H)$,
while the coordinates of each user $n$ is given by
$\bm{u}_{n}=(x_{n},y_{n},0)$. Here, the altitude $H$ of the UAV is assumed to
be constant.

The THz channel gain between the UAV and user $n$ can be modeled as [16]:
$h_{n}=d_{n}^{-2}e^{-a\left(f\right)d_{n}}$,
where?$d_{n}=\|\bm{u}_{n}-\bm{u}_{0}\|$?is the distance between user $n$ and
the UAV. $e^{-a\left(f\right)d_{n}}$ is the path loss caused by the molecular
absorption, and $a\left(f\right)$ is a molecular absorption coefficient that
depends on the operating frequency $f$ and concentration of water vapor
molecules. For notational simplicity, hereinafter, we use $a$ to represent
$a\left(f\right)$. Based on $h_{n}$, the achievable rate for the uplink
transmission from user $n$ to the UAV is [17]

$\vspace{-.5em}r_{n}^{U}={w_{n}\log_{2}\left(1+\frac{p_{n}h_{0}}{w_{n}d_{n}^{2}e^{ad_{n}}\sigma^{2}}\right)},$
(1)

where $w_{n}$ is the bandwidth allocated to user $n$, $p_{n}$ is the transmit
power of user $n$, $\sigma^{2}$ is the Gaussian noise power, and $h_{0}$?is
the channel gain at a reference distance $d_{0}=1$ m. According to (1), the
uplink transmission delay of user $n$ is

$\vspace{-.5em}t_{n}^{U}\left(p_{n},w_{n},x,y\right)=\frac{D_{n}}{w_{n}\log_{2}\left(1+\frac{p_{n}h_{0}}{w_{n}d_{n}^{2}e^{ad_{n}}\sigma^{2}}\right)},$
(2)

where $D_{n}$ is the size of the data that user $n$ needs to transmit to the
UAV. Similarly, the achievable rate for the downlink transmission from the UAV
to user $n$ is

$\vspace{-.5em}r_{n}^{D}={w_{n}\log_{2}\left(1+\frac{qh_{0}}{w_{n}d_{n}^{2}e^{ad_{n}}\sigma^{2}}\right)},$
(3)

where $q$ is the transmit power of the UAV. Based on (3), the downlink
transmission delay from the UAV to user $n$ is

$\vspace{-.5em}t_{n}^{D}\left(q,w_{n},x,y\right)=\frac{E_{n}}{w_{n}\log_{2}\left(1+\frac{qh_{0}}{w_{n}d_{n}^{2}e^{ad_{n}}\sigma^{2}}\right)},$
(4)

where $E_{n}$ is the data that the UAV needs to transmit to user $n$.

Our goal is to minimize the total communication time between the UAV and its
served users while satisfying the energy constraints of all users.
Mathematically, this optimization problem can be posed as follows:

$\displaystyle\min_{\bm{P},\bm{W}x,y}$
$\displaystyle\sum_{n=1}^{N}\left(t_{n}^{U}\left(p_{n},w_{n},x,y\right)+t_{n}^{D}\left(q,w_{n},x,y\right)\right),$
(5) s.t. $\displaystyle\sum_{n=1}^{N}w_{n}=B_{W},$ (5a) $\displaystyle
p_{n}\leq P,\quad n=1,\cdots,N,$ (5b) $\displaystyle\quad 0$
$\displaystyle\leq t_{n}^{U}\left(p_{n},w_{n},x,y\right)p_{n}\leq Q_{n},\quad
n=1,\cdots,N,$ (5c)

where $Q_{n}$ is maximum total energy of user $n$, $P$ is maximum transmit
power of user $n$, $B_{W}$ is the total bandwidth of all users,
$\bm{P}=\left[p_{1},\cdots,p_{N}\right]$, and
$\bm{W}=\left[w_{1},\cdots,w_{N}\right]$. Constraint (5b) and (5c) indicates
that each user’s energy and transmit power used for data transmission are
limited, respectively. Problem (5) is novel due to the following two reasons.
First, the channel model in (1) takes the molecular absorption into account.
Second, the molecular absorption coefficient is a function of distance, which
further makes problem (5) different from previous UAV location optimization
problems in [5, 6, 7, 8, 9, 10, 11] without THz channel.

##  III Proposed Algorithm

Problem (5) is a non-convex problem because of the non-convex optimization
function and constraints. It is generally challenging to obtain the globally
optimal solution for problem (5). To obtain a suboptimal solution of problem
(5), we propose an algorithm which optimizes $p_{n}$, $w_{n}$, $x$, and $y$ in
an alternating manner.

###  III-A User Transmission Power Optimization

With fixed $w_{n}$, $x$ and $y$, problem (5) is formulated as

$\displaystyle\min_{\bm{P}}$
$\displaystyle\sum_{n=1}^{N}\left(\frac{D_{n}}{w_{n}\log_{2}\left(1+k_{n}(w_{n},x,y)p_{n}\right)}+A_{n}\right),$
(6) s.t. $\displaystyle\sum_{n=1}^{N}w_{n}=B_{W},$ (6a) $\displaystyle
p_{n}\leq P,\quad n=1,\cdots,N,$ (6b)
$\displaystyle\frac{\log_{2}\left(1+k_{n}(w_{n},x,y)p_{n}\right)}{p_{n}}\geq
l_{n},\quad n=1,\cdots,N,$ (6c)

where $k_{n}(w_{n},x,y)=\frac{h_{0}}{w_{n}d_{n}^{2}e^{ad_{n}}\sigma^{2}}$,
$l_{n}=\frac{D_{n}}{w_{n}Q_{n}}$, and
$A_{n}=\frac{E_{n}}{w_{n}\log_{2}\left(1+k_{n}(w_{n},x,y)q\right)}$. It is
obvious that constraints and the objective function are convex. Therefore,
problem (6) is convex.

Next, we find the optimal transmit power of each user given the location of
the UAV and the bandwidth of each user, as shown in the following proposition.

###### Proposition 1

Given the UAV location $(x,y,H)$ and the bandwidth of each user $w_{n}$, the
optimal transmit power of each user $n$ is given by:

$\vspace{-.8em}p_{n}^{*}=-\frac{W\left(-\frac{l_{n}\ln
2}{k_{n}(w_{n},x,y)}2^{-\frac{l_{n}}{k_{n}(w_{n},x,y)}}\right)}{l_{n}\ln
2}-\frac{1}{k_{n}(w_{n},x,y)},$ (7)

where function $W\left(x\right)$ is the inverse function of $xe^{x}$.

Proof: Since the objective function of problem (6) is a decreasing function
w.r.t. $p_{n}$, the optimal $p_{n}^{*}$ is the maximum value in the feasible
region. According to (6c), we find that
$\frac{\log_{2}\left(1+k_{n}(w_{n},x,y)p_{n}\right)}{p_{n}}$ is a decreasing
function w.r.t. $p_{n}$. Hence, the optimal transmit power $p_{n}^{*}$
satisfies the equation
$\frac{\log_{2}\left(1+k_{n}(w_{n},x,y)p_{n}^{*}\right)}{p_{n}^{*}}=l_{n}$.
This completes the proof. $\Box$

From Proposition 1, we can see that the optimal transmit power of each user
$n$ depends on the molecular absorption coefficient, the bandwidth of each
user, and the location of the UAV.

###  III-B UAV Location Optimization

Given $p_{n}$ and $w_{n}$, the optimization problem in (5) can be given by:

$\displaystyle\min_{x,y}\sum_{n=1}^{N}$
$\displaystyle\frac{D_{n}}{w_{n}\log_{2}\left(1+k_{n}(w_{n},x,y)p_{n}\right)}$
$\displaystyle\quad+\sum_{n=1}^{N}$
$\displaystyle\frac{E_{n}}{w_{n}\log_{2}\left(1+k_{n}(w_{n},x,y)q\right)},$
(8) s.t. $\displaystyle\sum_{n=1}^{N}w_{n}=B_{W},$ (8a) $\displaystyle
p_{n}\leq P,\quad n=1,\cdots,N,$ (8b)
$\displaystyle\frac{D_{n}p_{n}}{w_{n}\log_{2}\left(1+k_{n}(w_{n},x,y)p_{n}\right)}\leq
Q_{n}.$ (8c)

To solve problem (8), we need to prove that constraint (8c) and the
optimization function in (8) are convex, we first introduce an important
theorem as follows.

###### Theorem 1

Given $e_{n}$ = $1+k_{n}(w_{n},x,y)p_{n}$ $<$ 24, function
$\frac{D_{n}p_{n}}{w_{n}\log_{2}\left(1+k_{n}(w_{n},x,y)p_{n}\right)}$ is
convex w.r.t. $x$ and $y$.

###### Proof:

The proof is in the Appendix. ∎

Since both the objective function and constraint are convex according to
Theorem 1, problem (8) is convex.

Algorithm 1 Proposed Algorithm for Problem (5)

1: Initialize $(x^{(1)},y^{(1)},w_{n}^{(1)},p_{n}^{(1)})$, the iteration
number $k=1$.

2: repeat

3: With given $(x^{(k)},y^{(k)},w_{n}^{(k)})$, obtain $p_{n}^{(k+1)}$ by
solving problem (6).

4: With given $(w_{n}^{(k)},p_{n}^{(k+1)})$, obtain $(x^{(k+1)},y^{(k+1)})$ by
solving problem (8).

5: With given $(x^{(k+1)},y^{(k+1)},p_{n}^{(k+1)})$, obtain $(w_{n}^{(k+1)})$
by solving problem (9).

6: Set $k=k+1$.

7: until objective function converges

8: return $x,y,p_{n},w_{n},n=1,2,\cdots,N$

###  III-C User Bandwidth Optimization

Given $p_{n}$ and $(x,y,H)$, the optimization problem in (5) can be given by:

$\displaystyle\min_{\bm{W}}\sum_{n=1}^{N}$
$\displaystyle\frac{D_{n}}{w_{n}\log_{2}\left(1+k_{n}(w_{n},x,y)p_{n}\right)}$
$\displaystyle\quad+\sum_{n=1}^{N}$
$\displaystyle\frac{E_{n}}{w_{n}\log_{2}\left(1+k_{n}(w_{n},x,y)q\right)},$
(9) s.t. $\displaystyle\sum_{n=1}^{N}w_{n}=B_{W},$ (9a) $\displaystyle
p_{n}\leq P,\quad n=1,\cdots,N,$ (9b)
$\displaystyle\frac{D_{n}p_{n}}{w_{n}\log_{2}\left(1+k_{n}(w_{n},x,y)p_{n}\right)}\leq
Q_{n}.$ (9c)

It is obvious that constraint (9c) and the optimization function in (9) are
convex. Therefore, the problem (9) is convex.

###  III-D Algorithm Complexity

The overall algorithm used to solve problem (5) is summarized in Algorithm 1
and Algorithm 1 must converge due to the fact that the objective function with
finite lower bound is monotonically nonincreasing. From Algorithm 1, the main
complexity of solving problem (5) lies in solving the power optimization
problem in (6) and the bandwidth allocation problem in (9). The optimal
solution of problem (6) is obtained in closed-form in (7), whose complexity is
$\mathcal{O}(N)$. Similarly, the complexity of solving problem (9) also is
$\mathcal{O}(N)$. As a result, the total complexity of solving problem (5) is
$\mathcal{O}(2T_{0}{N})$, where $T_{0}$ is the total number of iterations
needed to run Algorithm 1.

##  IV Simulation Results

For our simulations, a 50 m $\times$ 50 m square area is considered with $N$ =
14 randomly distributed users (unless stated otherwise) and one UAV. Other
parameters are listed in Table I [18]. All statistical results are averaged
over a large number of simulation runs.

TABLE I: System Parameters Parameters | Values
---|---
Gaussian noise power $\sigma^{2}$ | -174 dBm
Channel gain at $d_{0}=1$ m $h_{0}$ | -40 dB
Molecular absorption coefficient $a$ | 0.005 m-1
Operating frequency $f$ | 1.2 THz
Data that user $n$ transmits to the UAV $D_{n}$ | [10,8,6,4] Tbits
Data that the UAV transmits to user $n$ $E_{n}$ | [8,6.4,4.8,3.2] Tbits
The transmit power of the UAV $q$ | 2 Watts
The total bandwidth of all users $B_{W}$ |  $100$ GHz
Maximum total energy of each user $Q_{n}$ | $Q_{1}=\cdots=Q_{N}=Q$
Maximum transmit power of each user $P$ | 0.1 Watts

Fig. 1 shows how the transmission delay changes as the molecular absorption
coefficient $a$ varies when the total bandwidth of users is $B_{W}$ = 100 GHz,
the maximum total energy of each user $Q$ is 8 J. From Fig. 1, we can see that
the transmission delay increases with the molecular absorption coefficient
$a$. This is due to the fact that as $a$ increases, the molecular absorption
loss increases. From Fig. 1, we can also see that the proposed algorithm can
reduce the transmission delay by up to $59.3\%$, $49.8\%$ and $75.5\%$
compared to a first baseline (labeled ‘OL’) that only optimizes the UAV
deployment, a second baseline (labeled ‘OP’) that only optimizes the transmit
power of the users, and a third baseline (labeled ‘OW’) that only optimizes
the bandwidth of each user. Besides, we use the exhaustive search method
(labelled ’EXH’) to obtain a near global optimal solution, which uses interior
point method to jointly optimize transmit power, location and bandwidth with
multiple initial solutions. From Fig. 1, the EXH algorithm achieves the best
performance at the cost of computation capacity. The gap between the proposed
algorithm and the EXH algorithm is small, which indicates that the proposed
algorithm can approach the optimal solution.

Fig. 2 shows how the transmission delay changes as the total bandwidth of
users vary when the maximum total energy of each user $Q$ is 2 J. From Fig. 2,
we can see that the transmission delay decreases as the total bandwidth of
users increases. This is because as $B_{W}$ increases, the achievable rate
between the user and UAV increases. We can also see that the transmission
delay increases as the UAV’s altitude increases under a fixed bandwidth. This
is due to the fact that, as the distance between the UAV and users increases,
the channel gain decreases.

Fig. 3 shows how the transmission delay changes as the number of users vary
when the total bandwidth of users $B_{W}$ is 100 GHz and the maximum total
energy of each user $Q$ is 2 J. From Fig. 3, we can see that the transmission
delay increases as the number of users increases and the larger the number of
users, the faster the growth rate. This is because as $N$ increases, the
distances between the UAV and individual users will be very far.

Fig. 1: Transmission delay (s) versus $a$ $(m^{-1})$ ($B_{W}$ = 100 GHz, $H$ =
20 m, $Q$ = 8 J).

?

Fig. 2: Transmission delay (s) versus the bandwidth (GHz) of users ($Q$ = 2
J).

?

Fig. 3: Transmission delay (s) versus the number of users ($B_{W}$ = 100 GHz,
$Q$ = 2 J).

?

##  V Conclusion

In this letter, we have proposed a novel approach to deploy a UAV for
providing wireless THz connectivity. We have explicitly considered the
molecular absorption effect in the THz-enabled UAV channel gain model. Then,
we have formulated an optimization problem that seeks to minimize the
transmission delay between the UAV and the ground users while meeting the
communication requirements of each user. To solve this problem, we have
proposed an iterative algorithm that divides the original optimization problem
into three subproblems to optimize three variables: the location of the UAV,
the users’ transmit power and the bandwidth of each user. Simulation results
show that the proposed algorithm can significantly reduce the transmission
delay compared to three baselines.

## Appendix

We rewrite the function
$\frac{D_{n}p_{n}}{w_{n}\log_{2}\left(1+k_{n}(x,y,w_{n})p_{n}\right)}$ as

$\vspace{-.5em}B\left(e_{n}\right)=\frac{D_{n}p_{n}}{w_{n}\log_{2}(e_{n})}.$
(10)

For notational simplicity, hereinafter, we use $B$ to represent
$B\left(e_{n}\right)$. The Hessian matrix $\bm{G}$ of (10) can be expressed as

$\vspace{-.5em}\bm{G}=\left[\begin{array}[]{cc}\frac{\partial^{2}B}{\partial
x^{2}}&\frac{\partial^{2}B}{\partial x\partial y}\\\
\frac{\partial^{2}B}{\partial x\partial y}&\frac{\partial^{2}B}{\partial
y^{2}}\end{array}\right].$ (11)

Then, to prove that the function $B$ is convex, we need to prove that the
Hessian matrix $\bm{G}$ is positive semi-definite. Hence, we first prove that
the first-order determinant of $\bm{G}$ is positive and then we prove that the
second-order determinant of $\bm{G}$ is positive.

Based on (11), the first order determinant of $\bm{G}$ is expressed as
follows:

$\displaystyle\frac{\partial^{2}B}{\partial x^{2}}=$
$\displaystyle\frac{\partial^{2}B}{\partial d_{n}^{2}}\left(\frac{\partial
d_{n}}{\partial x}\right)^{2}+\frac{\partial B}{\partial
d_{n}}\frac{\partial^{2}d_{n}}{\partial x^{2}},$ (12) $\displaystyle=$
$\displaystyle\frac{\partial^{2}B}{\partial e_{n}^{2}}\left(\frac{\partial
e_{n}}{\partial d_{n}}\right)^{2}\left(\frac{x-x_{n}}{d_{n}}\right)^{2}$
$\displaystyle+$ $\displaystyle\frac{\partial B}{\partial
e_{n}}\left(\frac{\partial^{2}e_{n}}{\partial
d_{n}^{2}}\left(\frac{x-x_{n}}{d_{n}}\right)^{2}+\frac{\partial
e_{n}}{\partial
d_{n}}\frac{d_{n}^{2}-\left(x-x_{n}\right)^{2}}{d_{n}^{3}}\right),$

where $\frac{\partial B}{\partial e_{n}}=-\frac{D_{n}p_{n}\ln
2}{w_{n}e_{n}\ln^{2}e_{n}}$, $\frac{\partial^{2}B}{\partial
e_{n}^{2}}=\frac{D_{n}p_{n}\ln 2\left(\ln
e_{n}+2\right)}{w_{n}e_{n}^{2}\ln^{3}e_{n}}$, $\frac{\partial e_{n}}{\partial
d_{n}}=-\frac{h_{0}p_{n}\left(ad_{n}+2\right)}{w_{n}\sigma^{2}d_{n}^{3}e^{ad_{n}}}$,
and $\frac{\partial^{2}e_{n}}{\partial
d_{n}^{2}}=\frac{h_{0}p_{n}\left(a^{2}d_{n}^{2}+4ad_{n}+6\right)}{w_{n}\sigma^{2}d_{n}^{4}e^{ad_{n}}}$.
Based on (12), we rewrite $\frac{\partial^{2}B}{\partial x^{2}}$ as

$\vspace{-.5em}\frac{\partial^{2}B}{\partial x^{2}}=I\frac{D_{n}p_{n}\ln
2\left(e_{n}-1\right)}{w_{n}e_{n}^{2}d_{n}^{4}\ln^{3}e_{n}},$ (13)

where $I=I_{1}z+\left(ad_{n}+2\right)d_{n}^{2}e_{n}\ln e_{n}$,
$z=\left(x-x_{n}\right)^{2}$ and $I_{1}=\left(2e_{n}-\ln
e_{n}-2\right)\left(ad_{n}+2\right)^{2}-e_{n}\ln
e_{n}\left(ad_{n}+2\right)-2e_{n}\ln e_{n}$. To prove that
$\frac{\partial^{2}B}{\partial X^{2}}$ is positive, we only need to prove that
$I>0$.

First, from the expression of $I$, we can treat it as a linear function with
slope $I_{1}$. To determine if $I$ is greater than 0, we consider the
following two cases:

_Case 1_ : If $I_{1}\geq 0$, we can directly obtain $I>0$.

_Case 2_ : If $I_{1}<0$, function $I$ is a decreasing function w.r.t $z$.
Since $z=(x-x_{n})^{2}<d_{n}^{2}$, we have $0\leq z<z_{\max}$, where
$z_{\max}=\frac{p_{n}h_{0}}{(e_{n}-1)w_{n}\sigma^{2}}$. Thus, to prove that
$I>0$ in its feasible region, we only need to prove that $z_{\max}<z^{*}$,
where $z^{*}$ is the zero point of $I$, and the expression of $z^{*}$ is given
by:

$\vspace{-.5em}z^{*}=\frac{\left(ad_{n}+2\right)d_{n}^{2}e_{n}\ln
e_{n}}{-I_{1}}.$ (14)

From (14), we find that the minimum value of $z^{*}$ is
$z_{\min}^{*}=\frac{\left(ad_{n}+2\right)d_{n}^{2}e_{n}\ln
e_{n}}{-I_{1\min}}$, where $I_{1\min}$ is the minimum value of $I_{1}$. If
$z_{\min}^{*}>z_{\max}$, then we have $z^{*}>z_{\min}^{*}>z_{\max}$ so that
$I>0$. Thus, we first find $I_{1\min}$, then compare $z_{\min}^{*}$ with
$z_{\max}$.

From the expression of $I_{1}$, we can see that $I_{1}$ is a function of
$\left(ad_{n}+2\right)$. When $e_{n}>1$, we find that $\left(2e_{n}-\ln
e_{n}-2\right)$ is positive. Hence, the function $I_{1}$ has two solutions,
i.e., $\frac{e_{n}\ln e_{n}\pm\sqrt{(e_{n}\ln e_{n})^{2}+8e_{n}\ln
e_{n}(2e_{n}-\ln e_{n}-2)}}{2\left(2e_{n}-\ln e_{n}-2\right)}$.
$\frac{e_{n}\ln e_{n}}{2\left(2e_{n}-\ln e_{n}-2\right)}$ is the minimum point
of function $I_{1}$. Fig. 4(a) shows how the value of function $I_{1}$ varies
as $\left(ad_{n}+2\right)$ changes. To determine the minimum point of function
$I_{1}$ whether exists in the feasible region, we must compare $\frac{e_{n}\ln
e_{n}}{2\left(2e_{n}-\ln e_{n}-2\right)}$ with $2$ by creating a new function
$\frac{e_{n}\ln e_{n}}{2\left(2e_{n}-\ln e_{n}-2\right)}-2$. Fig. 4(b) shows
how the value of function $\frac{e_{n}\ln e_{n}}{2\left(2e_{n}-\ln
e_{n}-2\right)}-2$ changes as $e_{n}$ varies. According to the increasing
trend and its feasible region, we observe that $\frac{e_{n}\ln
e_{n}}{2\left(2e_{n}-\ln e_{n}-2\right)}$ is less than $2$ when
$1<e_{n}<2940.74$. Since the distance between the user and UAV is usually
larger than 10 m, $e_{n}<24$111For the typical value, $d_{n}\geq$ 10 m, $a$ =
0.005 m-1 at $f$ = 1.2 THz, $h_{0}$ = -40 dB, $\sigma^{2}$ = -174 dBm, $w_{n}$
= 10 GHz and $p_{n}$ = 0.001 W, we always have $e_{n}\leq 23.781$.. As a
result, $\frac{e_{n}\ln e_{n}}{2\left(2e_{n}-\ln e_{n}-2\right)}<2$. Thus,
according to Fig. 4(a), we observe that $I_{1}$ increases as
$\left(ad_{n}+2\right)$ increases in its feasible region.

(a)

(b)

(c)

Fig. 4: Illustrations of three functions needed for the analysis of the first
order of the determinant.

Then, we have:

$\displaystyle z_{\min}^{*}$
$\displaystyle=\frac{\left(ad_{n}+2\right)d_{n}^{2}e_{n}\ln
e_{n}}{-\left.I_{1}\right|_{\min}}$ (15)
$\displaystyle>\frac{p_{n}h_{0}e_{n}\ln
e_{n}}{2w_{n}\sigma^{2}\left(e_{n}-1\right)\left(\ln e_{n}+2+e_{n}\ln
e_{n}-2e_{n}\right)}.$

To compare the value of $z_{\min}^{*}$ with $z_{\max}$, we have:

$\displaystyle z_{\min}^{*}-z_{\max}>$
$\displaystyle\frac{h_{0}p_{n}g\left(e_{n}\right)}{2w_{n}\sigma^{2}\left(e_{n}-1\right)\left(2+\ln
e_{n}+e_{n}\ln e_{n}-2e_{n}\right)},$ (16)

where $g\left(e_{n}\right)=4e_{n}-4-e_{n}\ln e_{n}-2\ln e_{n}$. From (16), we
can see that the values of $z_{\min}^{*}$ and $z_{\max}$ depend on the value
of $g\left(e_{n}\right)$. Therefore, we next analyze the function
$g\left(e_{n}\right)$.

By finding the first and second derivatives of $g\left(e_{n}\right)$, we
obtain the trend of $g\left(e_{n}\right)$ in Fig. 4(c). As shown in Fig. 4(c),
we define $e_{n}^{*}=41.4125$ as the non-zero point of function
$g\left(e_{n}\right)$ and $e_{n\max}$ as the maximum value of $e_{n}$. Since
$e_{n}<41$, we have $e_{n}<e_{n}^{*}$. As a result, we have
$g\left(e_{n}\right)>0$. Since $g\left(e_{n}\right)>0$, then
$z_{\min}^{*}>z_{\max}$ and hence we have $I>0$. Finally, based on (13), we
obtain that the first order determinant of $\bm{G}$ is positive.

Next, we prove that the second-order determinant of $\bm{G}$ is positive. The
second order determinant of $\bm{G}$ is expressed as

$\displaystyle G_{1}$
$\displaystyle=\left|\begin{array}[]{cc}\frac{\partial^{2}B}{\partial
x^{2}}&\frac{\partial^{2}B}{\partial x\partial y}\\\
\frac{\partial^{2}B}{\partial x\partial y}&\frac{\partial^{2}B}{\partial
y^{2}}\end{array}\right|,$ (17)
$\displaystyle=\frac{\left(D_{n}p_{n}\left(e_{n}-1\right)\ln
2\right)^{2}\left(2+ad_{n}\right)L}{w_{n}^{2}e_{n}^{3}\ln^{5}e_{n}d_{n}^{6}},$

where $L=(d_{n}^{2}-H^{2})S+H^{2}e_{n}\ln e_{n}(2+ad_{n})$ and
$S=\left(2e_{n}-2-\ln e_{n}\right)\left(2+ad_{n}\right)^{2}-2e_{n}\ln e_{n}$.
To prove $G_{1}>0$, we only need to prove that $L$ is positive. Since $S$ is
an increasing function w.r.t. $a$, we have
$L>\left(d_{n}^{2}-H^{2}\right)S_{1}+H^{2}e_{n}\ln e_{n}\left(2+ad_{n}\right)$
where $S_{1}=S|_{a=0}=2g\left(e_{n}\right)>0$. As a result, we have $G_{1}>0$.

Since both the first and second order determinants of $\bm{G}$ are positive,
the Hessian matrix $\bm{G}$ is positive semi-definite. As a result, function
$\frac{D_{n}p_{n}}{w_{n}\log_{2}\left(1+k_{n}(x,y,w_{n})p_{n}\right)}$ is
convex w.r.t. $x,y$. This completes the proof.

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