Ris + Uav

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 39, NO.
10, OCTOBER 2021 3051
Intelligent Reflecting Surface Enhanced

Multi-UAV NOMA Networks
Xidong Mu , Graduate Student Member, IEEE, Yuanwei Liu , Senior Member, IEEE,
Li Guo , Member, IEEE, Jiaru Lin , Member, IEEE,
and H. Vincent Poor , Life Fellow, IEEE
Abstract— Intelligent reflecting surface (IRS) enhanced I. I NTRODUCTION

multi-unmanned aerial vehicle (UAV) non-orthogonal multiple
access (NOMA) networks are investigated. A new transmission
framework is proposed, where multiple UAV-mounted base sta-
tions employ NOMA to serve multiple groups of ground users
with the aid of an IRS. The three-dimensional (3D) placement and
W ITH the rapid development of manufacturing technol-
ogy and the continuous reduction of costs, unmanned
aerial vehicles (UAVs), also known as drones, have received
transmit power of UAVs, the reflection matrix of the IRS, and the significant attention for potential use in civil applications,
NOMA decoding orders among users are jointly optimized for such as cargo delivery, search and rescue, and traffic
maximization of the sum rate of considered networks. To tackle monitoring [1], [2]. Among others, UAV-enabled communi-
the formulated mixed-integer non-convex optimization problem
with coupled variables, a block coordinate descent (BCD)-based cations is one of the most appealing applications. Equipped
iterative algorithm is developed. Specifically, the original problem with communication devices, UAVs can act as aerial base sta-
is decomposed into three subproblems, which are alternately tions (BSs) to provide wireless communication service in many
solved by exploiting the penalty-based method and the successive practical scenarios (e.g., communication recovery after natural
convex approximation technique. The proposed BCD-based algo- disasters or traffic offloading for temporary hotspots). Com-
rithm is demonstrated to be able to obtain a stationary point of
the original problem with polynomial time complexity. Numerical pared with conventional terrestrial communications, on the one
results show that: 1) the proposed NOMA-IRS scheme for hand, the air-to-ground (A2G) channel has a high probability
multi-UAV networks achieves a higher sum rate compared to the of being dominated by line-of-sight (LoS) links [3], which
benchmark schemes, i.e., orthogonal multiple access (OMA)-IRS helps to establish high data rates and reliable transmissions.
and NOMA without IRS; 2) the use of IRS is capable of providing On the other hand, the mobility of UAVs is controllable and
performance gain for multi-UAV networks by both enhancing
channel qualities of UAVs to their served users and mitigating can be exploited to improve the performance of communi-
the inter-UAV interference; and 3) optimizing the UAV placement cations. For instance, the UAV can fly closer to its intended
can make the sum rate gain brought by NOMA more distinct ground user to achieve better channel conditions.
due to the flexible decoding order design. Despite the above benefits, one prominent challenge of facil-
Index Terms— Intelligent reflecting surfaces, non-orthogonal itating UAV-enabled communications is how to mitigate the
multiple access, placement optimization, unmanned aerial severe interference caused by the LoS dominated A2G chan-
vehicles. nel, especially when there are multiple UAV-mounted BSs [4].
To address this issue, intelligent reflecting surfaces (IRSs)
Manuscript received October 22, 2020; revised February 5, 2021; accepted have been recently proposed as a promising solution [5]–[8].
April 12, 2021. Date of publication June 14, 2021; date of current version An IRS is a thin man-made surface, which is equipped with
September 16, 2021. This work was supported in part by the Beijing Natural
Science Foundation under Grant L192032, in part by the National Key a large number of low-cost and passive reflecting elements.
Research and Development Program of China under Grant 2019YFB1406500, Each of these elements can be reconfigured via amplitudes
in part by the Key Project Plan of Blockchain in Ministry of Education of and phase shifts, thus modifying the propagation of inci-
the People’s Republic of China under Grant 2020KJ010802, in part by the
Shandong Province Key Research and Development Program, China, under dent signals. By optimizing the reflection coefficients of the
Grant 2019JZZY020901, in part by the National Natural Science Foundation IRS, reflected signals can be combined coherently with the
of China under Grant 61771066, and in part by the U.S. National Science non-reflected signal to enhance the desired signal strength
Foundation under Grant CCF–1908308. (Corresponding author: Li Guo.)
Xidong Mu, Li Guo, and Jiaru Lin are with the Key Laboratory of Universal or destructively to suppress interference [5]. Furthermore,
Wireless Communications, Ministry of Education, Beijing University of Posts IRSs can be flexibly deployed on various structures, such as
and Telecommunications, Beijing 100876, China, and also with the School of building facades, roadside billboards, and indoor walls [6], [8],
Artificial Intelligence, Beijing University of Posts and Telecommunications,
Beijing 100876, China (e-mail: muxidong@bupt.edu.cn; guoli@bupt.edu.cn; which makes it trivial to integrate IRSs into existing wireless
jrlin@bupt.edu.cn). networks.
Yuanwei Liu is with the School of Electronic Engineering and Computer During the application of UAV-enable communications,
Science, Queen Mary University of London, London E1 4NS, U.K. (e-mail:
yuanwei.liu@qmul.ac.uk). UAV-mounted BSs usually need to simultaneously serve a
H. Vincent Poor is with the Department of Electrical Engineering, Princeton large number of ground users with stringent communication
University, Princeton, NJ 08544 USA (e-mail: poor@princeton.edu). requirements, especially for future beyond-5G (B5G) net-
Color versions of one or more figures in this article are available at
https://doi.org/10.1109/JSAC.2021.3088679. works. To tackle these challenges, advanced multiple access
Digital Object Identifier 10.1109/JSAC.2021.3088679 techniques are essential. Particularly, non-orthogonal multiple
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3052 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 39, NO. 10, OCTOBER 2021
access (NOMA) has been regarded as a promising candi- 2) Studies on IRS-Enhanced Communications: The IRS per-
date for integrating UAV into B5G networks due to the formance gain to wireless communication networks has been
advantages of enhancing spectral efficiency and supporting investigated in various aspects, such as energy consumption
massive connectivity [9]. By invoking superposition cod- reduction and sum rate enhancement. The authors of [20]
ing (SC) and successive interference cancellation (SIC) tech- minimized the transmit power for satisfying specific commu-
niques, NOMA1 allows multiple users to share the same nication requirements in IRS-aided communication systems,
time/frequency resources and distinguishes them using power where an alternating optimizing algorithm was proposed for
levels. The employment of NOMA in IRS-enhanced UAV optimizing the active beamforming at the BS and the passive
communications is highly attractive and conceived to be a beamforming at the IRS. In [21], the authors maximized the
win-win strategy due to the following reasons: energy efficiency in an IRS-assisted multi-user communication
• On the one hand, compared with conventional orthogonal scenario, where a power consumption model for IRS was
multiple access (OMA), NOMA can provide more flex- proposed. The authors of [22] formulated a transmit power
ible and efficient resource allocation for IRS-enhanced minimization problem in an IRS-assisted multi-user network,
UAV communications. Thus, diversified communication where the performances of OMA and NOMA were compared
requirements of users can be satisfied and the spectrum and a time-selective property of the IRS was employed for time
efficiency can be further enhanced. division multiple access (TDMA). In [23], the authors investi-
• On the other hand, in conventional NOMA transmission, gated an IRS-enhanced multiple-antenna NOMA network with
the SIC decoding orders among users are generally deter- the aim of maximizing the system sum rate. The authors
mined by the “dumb” channel conditions [10]. Note that of [24] proposed a novel double-IRS assisted communication
UAVs and IRSs are both “channel changing” technolo- system, where the cooperative passive beamforming can be
gies. The channel conditions of users can be enhanced employed. In [25], the authors jointly optimized the UAV
or degraded by exploiting UAVs’ mobility and/or adjust- trajectory and IRS phase shifts to maximize the average rate of
ing IRS reflection coefficients, thus enabling a “smart” the ground user. The authors of [26] proposed a UAV-assisted
NOMA operation to be carried out. multiple IRSs symbiotic radio system, where the weighted
sum-rate maximization problem and the max-min optimization
A. Prior Work problem were investigated.
1) Studies on UAV-Enabled Communications: Exten-
sive research contributions have studied UAV-enabled B. Motivation and Contributions
communications, which can be loosely classified into two While the aforementioned research contributions have laid
main categories, namely placement optimization [11]–[13] and a solid foundation on UAV-enabled and IRS-enhanced com-
trajectory design [14]–[19]. The authors of [11] investigated munications, the investigations on the adoption of IRS in
the placement optimization problem with the goal of using a UAV-enabled communications are still quite open, espe-
minimum number of UAV-mounted BSs to provide wireless cially for multi-UAV and multi-user scenarios. Although
coverage for given ground terminals. The authors of [12] some research contributions have investigated the joint UAV
studied the placement optimization in a downlink NOMA UAV trajectory and IRS reflection coefficient optimization prob-
network. In [13], the authors focused on a multi-UAV data lem [25], [26], the considered system models are limited to
collection network by employing uplink NOMA. Furthermore, single-UAV and/or single-user scenarios without considering
the authors of [14] studied the trajectory design of multiple multiple access schemes. To the best of our knowledge, there
UAVs, where the max-min average communication rate of is no existing work that investigates the potential performance
ground users was optimized. With the proposed rotary-wing gain of IRS-enhanced multi-UAV networks employing NOMA
UAV energy consumption model, the authors of [15] mini- transmission. The main challenges are identified as follows:
mized the energy required by the UAV for completing the 1) For a multi-UAV scenario, the communication rate of each
information transmission mission. In [16], the authors pro- user depends on not only the desired signal power strength
posed a novel UAV-enabled wireless power transfer system, but also the interference level. The optimization of UAV
where an asymptotically optimal solution for UAV trajectory placement needs to strike a balance between desired signal
design was derived. Considering multiple antennas at the UAV, strengths transmitted to served users and inter-UAV interfer-
the authors of [17] studied a robust trajectory and resource ence imposed to unintended users, which is a non-trivial task.
allocation design, where both optimal and low complexity 2) For a multi-user scenario, the optimal IRS configuration
suboptimal algorithms are proposed. The authors of [18] is not just to align the phases of reflected signals with the
further optimized the three-dimensional (3D) UAV trajectory non-reflected signals, as did in the single-user scenario [25],
to maximize the system throughput in a simultaneous uplink [26]. The IRS reflection coefficients need to be shared by
and downlink transmission scenario with two UAVs. In [19], multiple users at the same time, which makes the design of
the authors optimized the max-min average communication IRS reflection coefficients much complicated. 3) The employ-
rate through trajectory design in a UAV-enabled downlink ment of NOMA introduces additional channel condition-based
NOMA communication system. decoding order design [10], which causes UAV placement, IRS
1 In this article, we use “NOMA” to refer to “power-domain NOMA” for reflection coefficients, and NOMA decoding order design to
simplicity. be highly coupled. Therefore, efficient algorithms should be
MU et al.: IRS-ENHANCED MULTI-UAV NOMA NETWORKS 3053
carefully developed to fully reap the benefits of employing

IRS and NOMA for UAV-enabled communications.
Against the aforementioned background, the main contribu-
tions of this paper are summarized as follows:
• We propose a novel transmission framework for
multi-UAV communication networks, in which NOMA is
employed at each UAV-mounted BS for serving ground
users, and an IRS is deployed to enhance the transmission
from UAVs to their intended users while mitigating the
interference caused to other unintended users. Based on
this framework, we formulate the sum rate maximization
problem for joint optimization of the 3D placement and Fig. 1. Illustration of IRS-enhanced multi-UAV NOMA networks.
transmit power at UAVs, the reflection matrix at the IRS,
and the NOMA decoding orders at each user group. with the elements of vector a on the main diagonal. 1m×n
• We develop a block coordinate descent (BCD)-based denotes an all-one matrix of size m × n. HN denotes the set of
iterative algorithm, where the original problem is decom- all N -dimensional complex Hermitian matrices. rank(A) and
posed into three subproblems to be alternately solved. Tr(A) denote the rank and the trace of matrix A, respectively.
For the first two subproblems, namely, the joint UAV A 0 indicates that A is a positive semidefinite matrix.
placement and NOMA decoding order design, and the ⊗ denotes the Kronecker product.
IRS reflection matrix design, we efficiently solve them
by invoking the penalty-based method and the successive II. S YSTEM M ODEL
convex approximation (SCA) technique. Then, we opti-
Fig. 1 illustrates the considered narrow-band IRS-enhanced
mize the UAV transmit power with other variables fixed
multi-UAV NOMA networks operating over frequency-flat
by applying SCA. We further demonstrate that the pro-
channels, where K rotary-wing UAVs are deployed to serve
posed BCD-based iterative algorithm is guaranteed to
K user groups with the aid of an IRS of N reflecting
converge to a stationary point of the original problem
elements. One practical application of the considered setup is
with polynomial time complexity.
deploying UAV-mounted BSs to provide communication ser-
• Our numerical results show that the proposed
vice for temporary hotspots in rural or suburban environments,
IRS-enhanced UAV-NOMA scheme is capable of
where the users are assumed to be static or low-mobility2.
significantly improving the achieved sum rate, compared
In this paper, UAVs and ground users are equipped with
to several benchmark schemes. It also confirms that
single antenna. UAVs and corresponding served user groups
deploying the IRS cannot only improve the channel
are indexed by the set K = {1, · · · , K}. Users in each group
quality from UAVs to the served users but also
are indexed by the set Mk = {1, · · · , Mk }, ∀k ∈ K, where
mitigate the interference caused to other unserved users.
Mk denotes the number of users in the kth group. Without loss
Moreover, the performance gain of NOMA over OMA is
of generality, a 3D Cartesian coordinate system is considered.
greatly enhanced by optimizing the placement of UAVs
Let (k, i) denote the index of the ith user in the kth group.
since it enlarges users’ channel differences and enables
The locations of the (k, i)th user and the IRS are fixed at
a flexible NOMA decoding order design.
wik = [xki , yik , zik ]T , ∀i ∈ Mk , k ∈ K, and u = [xs , ys , zs ]T ,
C. Organization and Notation respectively. Let qk = [xk , yk , zk ]T denote the kth UAV’s
location. To ensure the safety of operation and avoid collisions,
The rest of this paper is organized as follows. Section II
the UAV’s flying height and the distance between any two
presents the system model for IRS-enhanced multi-UAV
UAVs should satisfy the following constraints:
NOMA networks. In Section III, the considered performance
metric is introduced and the sum rate maximization problem Zmin ≤ zk ≤ Zmax , ∀k ∈ K, (1)
is formulated. In Section IV, a BCD-based iterative algorithm qk − qj ≥ Δmin , ∀k = j ∈ K, (2)
is developed for solving the formulated joint optimization
problem. Numerical results are presented in Section V to verify where [Zmin , Zmax ] denotes the allowed range of the UAV
the effectiveness of the proposed algorithm compared with flying height, and Δmin denotes the minimum inter-UAV
other benchmark schemes. Finally, Section VI concludes the distance required for collision avoidance.
paper. In practice, the IRS is usually equipped with a smart
Notation: Scalars, vectors, and matrices are denoted by controller (e.g., a field-programmable gate array (FPGA)) for
lower-case, bold-face lower-case, and bold-face upper-case configuring reflection coefficients and exchanging information
letters, respectively. CN ×1 denotes the space of N × 1 between the IRS and UAVs [5]. Due to the fact that the
complex-valued vectors. The transpose and conjugate trans-
2 For high-mobile users, mobile UAVs and the dynamic IRS configuration
pose of vector a are denoted by aT and aH , respectively.
are required to guarantee performance gain. The resulting joint UAV trajectory
[a]n and a denote the nth element and the Euclidean norm and IRS dynamic configuration design problem is beyond the scope of the
of vector a, respectively. diag(a) denotes a diagonal matrix current work.
UAV-IRS-user link experiences substantial path loss, a large Moreover, for the UAV-IRS channel, gk is assumed to be
number of reflecting elements are required for the reflection LoS channel and can be expressed as
link to have a comparable path loss as the unobstructed direct
ρ0
UAV-user link [27]. This, however, causes a prohibitively gk = 2 gk
qk − u
high overhead/complexity for channel acquisition and reflec-
ρ0 −j 2πd 2πd
tion coefficient design/reconfiguration. To address this issue, = λ cos ϕk , . . . , e−j λ (N −1) cos ϕk ]T ,
2 [1, e
similar to [22], [28], adjacent IRS reflecting elements with qk −u
high channel correlation are grouped into a sub-surface. (6)
For instance, suppose that each sub-surface consists of N xu −xk
where cos ϕk = u−q k
is the cosine of the angle of
reflecting elements, the N reflecting elements are divided into arrival (AoA) from the kth UAV to the IRS.
M = N /N sub-surfaces3. Moreover, reflecting elements in Based on the aforementioned channel models, the effective
the same sub-surface are assumed to have the same reflection channel power gain between the jth UAV and the (k, i)th user
coefficients [22], [28]. Fig. 1 illustrates an example where with the aid of the IRS is given by
N = 6 reflecting elements are grouped into a sub-surface.
Since the narrow-band transmission is considered, the reflec- cjk,i = |hjk,i + rH
k,i Θgj | , ∀k, j ∈ K, i ∈ Mk .
2
(7)
tion coefficients of the IRS are assumed to be approximately
B. NOMA Transmission
constant across the entire signal bandwidth. The frequency-flat
IRS reflection matrix is denoted by Θ = diag(θ ⊗ 1N ×1 ) ∈ In this paper, UAVs are assumed to share the same fre-
CN ×N , where θ = [ejθ1 , ejθ2 , . . . , ejθM ]T , and θm ∈ quency band and each of them employs NOMA to pro-
[0, 2π), ∀m ∈ M = {1, . . . , M } denotes the corresponding vide communication service for ground users. To facilitate
phase shift4 of the mth sub-surface of the IRS. NOMA transmission, the transmitted signal of the kth UAV
to
Mits served group by invoking SC is given by sk =
A. Channel Model k √
pk,i sk,i , where pk,i and sk,i are the transmitted power
i=1 Mk
Let hjk,i ∈ C1×1 , rk,i ∈ CN ×1 , and gk ∈ CN ×1 denote the and signal for the (k, i)th user. We have i=1 pk,i ≤
channels between the jth UAV and the (k, i)th user, between Pmax,k , ∀k ∈ K, where Pmax,k denotes the maximum transmit
the IRS and the (k, i)th user, and between the kth UAV and power of the kth UAV. Then, the received signal at the (k, i)th
the IRS, respectively. As UAVs usually fly at a relatively user can be expressed as
high altitude and the IRS is also carefully deployed to avoid √
yk,i = (hkk,i + rH k,i Θgk ) pk,i sk,i
signal blockage (e.g., on a high roadside billboard in Fig. 1),
the channels hjk,i and rk,i are assumed to follow the Rician desired signal
channel model, which can be expressed as Mk √
+ (hkk,i + rH
k,i Θgk ) pk,t sk,t
ρ0 K1 1 j t=1,t=i
hjk,i = ( h
j
+ hk,i ),(3)
β1 k,i
qj − wi
k K 1 + 1 K 1+1 intra−group interference
K Mj √
ρ0 K2 1 + (hjk,i + rH
k,i Θgj ) pj,l sj,l +nk,i ,
rk,i = β2
( rk,i +
rk,i ), (4)
j=1,j=k

l=1
u − w k K2 + 1 K2 + 1 inter−group interference
i
where ρ0 is the path loss at the reference distance of 1 meter, (8)
β1 ≥ 2 and β2 ≥ 2 denote the path loss exponents of where nk,i denotes the additive white Gaussian noise (AWGN)
the UAV-user and IRS-user links, K1 and K2 denote the with zero mean and variance σ 2 .
j
Rician factors, hk,i = 1 and rk,i denote the deterministic LoS According to NOMA protocol, each user employs SIC to
components, and hj and rk,i denote the random Rayleigh
k,i
remove the intra-group interference. In particular, the user with
distributed non-LoS (NLoS) components. Specifically, similar the stronger channel power gain first decodes the signal of the
to [25], [26], a uniform linear array (ULA) is considered for user with the weaker channel power gain, before decoding
the IRS5 and rk,i is given by its own signal [10]. From (7), the channel power gains of
2π 2π(N −1) users can be manually modified in this work, which results
rk,i = [1, e−j λ d cos φk,i , . . . , e−j λ d cos φk,i
]T , (5) in Mk ! possible combinations of NOMA decoding orders in
where λ denotes the carrier wavelength, d denotes the element each group [22], [23]. We introduce a set of binary variables,
xk −xu
spacing, and cos φk,i = wi k −u is the cosine of the angle of αkt,i ∈ {0, 1}, ∀k ∈ K, ∀t, i ∈ Mk , to specify the decoding
i
orders among users in each group. For users served by the
departure (AoD) from the IRS to the (k, i)th user. kth UAV, if the effective channel power gain of the (k, t)th
3 For simplicity, we assume that M = N /N is an integer. user is larger than that of the (k, i)th user, we have αkt,i = 1;
4 It is worth noting that the assumption of continuous phase shifts provides a otherwise, αkt,i = 0. Therefore, for all k ∈ K, i = t ∈ Mk ,
theoretical performance upper bound for systems employing practical discrete {αkt,i } need to satisfy the following conditions:
phase shifts. The obtained results of continuous phase shifts can be quantized
into discrete ones and the resulting performance degradation is small for
1, if ckk,t ≥ ckk,i
sufficiently high phase shift resolutions [23].
5 It is worth noting that the results of this paper can be extended to the IRS
αkt,i = , (9)
0, otherwise
with uniform planar array (UPA) by considering the corresponding antenna
array response. αkt,i + αki,t = 1. (10)
In addition, for given decoding orders, the allocated power Based on Theorem 1 and Lemma 1, we approximate E{Rk,i }
should satisfy the following condition: as (15), where (15) is shown at the bottom of the next page.
Such approximation can be verified to achieve high accuracy
pk,i ≥ αkt,i pk,t , ∀i = t ∈ Mk , k ∈ K, (11)
in UAV-assisted communications [18]. From (15), it can be
which ensures that higher powers are allocated to the users observed that Rk,i depends on the deterministic LoS com-
with weaker channel power gains [10], i.e., pk,i ≥ pk,t , ponents, the large-scale path losses, and the reflection matrix
if αkt,i = 1. By doing so, a non-trivial communication rate of the IRS. In other words, Rk,i only requires the estima-
can be achieved at the weaker users and better user fairness tion of statistical channel state information (CSI) rather than
can be guaranteed. instantaneous CSI. This is more practical for IRS-enhanced
Therefore, the received signal-to-noise-plus-interference communications since the acquisition of instantaneous CSI is
ratio (SINR) of the (k, i)th user after carrying out SIC is given quite challenging due to the nearly passive working mode of
by IRSs [5]. In this paper, we assume that perfect statistical CSI
ckk,i pk,i can be obtained via recently proposed CSI channel estimation
γk,i = intra
Ik,i + Ik,i inter + σ 2 , ∀i ∈ Mk , k ∈ K, (12) methods for IRS-enhanced communication systems [29], [30].
The results in this work actually provide a theoretical per-
Mk
where Ik,i intra
= ckk,i t=1,t k inter
=i αt,i pk,t and Ik,i = formance upper bound for the considered network with CSI
K j Mj estimation error and overhead.
j=1,j=k ck,i l=1 pj,l . The achievable communication rate
Furthermore, from (9), we can observe that the decod-
of the (k, i)th user is given by Rk,i = log2 (1 + γk,i ),
ing orders among users in each group are also deter-
∀i ∈ Mk , k ∈ K.
mined by random variables, {cjk,i }. To facilitate our design,
III. P ROBLEM F ORMULATION we approximate (9) as follows:
In this section, we first introduce the considered perfor- 1, if qk − wtk ≤ qk − wik ;
mance metric and then formulate the joint optimization prob- αkt,i = . (16)
0, otherwise
lem for maximization of the sum rate of all users in considered
networks. Here, (16) means that the decoding orders among users in
each group are determined by the distances between users and
A. Performance Metrics their paired UAVs. The approximation is practically valid since
Note that {cjk,i } are random variables due to the involved 1) the effective channel power gains of users are dominated
random NLoS components. Therefore, the corresponding com- by the direct UAV-user link due to the substantial path loss
munication rate, Rk,i , is also a random variable. In this paper, experienced by the UAV-IRS-user link; and 2) for the direct
we are interested in the expected/average achievable commu- UAV-user link, the small scale fading is on the different order
nication rate, defined as E{Rk,i }. However, it is challenging of the magnitude compared to the distance-dependent large-
to derive a closed-form expression for E{Rk,i }, since its scale path loss [31]. As a result, the effective channel power
probability distribution is difficult to obtain. To tackle this gains of users are in general decided by the distances to the
issue, we approximate the expected achievable communication paired UAVs, i.e., a shorter distance leads to a higher channel
rate, E{Rk,i }, using the following theorem and lemma. power gain.
Theorem 1: If X and Y are two independent positive
random variables, for any a > 0 and b > 0, the following B. Joint Optimization Problem Formulation
approximation result holds We aim to maximize the sum rate of all users by jointly
⎧ ⎛ ⎞⎫ optimizing the UAV 3D placement and transmit power, the
a ⎨ a ⎬
E log 1 + ≈ E log ⎝1 + ⎠ (13) IRS reflection matrix, and the NOMA decoding orders among
b+ Y X ⎩ b+ E{X} ⎭ users of each group. Define Q = {qk , ∀k ∈ K}, P =
E{Y }
{pk,i , ∀k ∈ K, i ∈ Mk } and A = {αkt,i , ∀k ∈ K, i = t ∈ Mk }.
Proof: The proof is similar to that of [Theorem 1, 18] Then, the joint optimization problem can be formulated as
and hence it is omitted for brevity. follows:
Lemma 1: The expected effective channel power gain K Mk
between the jth UAV and the (k, i)th user is given by max Rk,i (17a)
Q,Θ,P,A k=1 i=1
E{cjk,i } ηk,i
j s.t. Zmin ≤ zk ≤ Zmax , ∀k ∈ K, (17b)
ρ0 − κ 1 qk − qj 2 ≥ Δ2min , ∀k = j ∈ K, (17c)
= |
hjk,i +
rH
k,i Θgj | +
2
qj − wik
β1
θm ∈ [0, 2π), ∀m ∈ M, (17d)
τk,i pk,i ≥ 0, ∀k ∈ K, i ∈ Mk , (17e)
+ 2, (14)
qj − u Mk
pk,i ≤ Pmax,k , ∀k ∈ K, (17f)
i=1
where
j
hjk,i = κ1
h , rH = κ2
rk,i , pk,i ≥ αkt,i pk,t , ∀i = t ∈ Mk , k ∈ K,
qj −wik β1 k,i k,i u−wik β2 (17g)
N λ0 (λ0 −κ2 ) K1 λ0 K2 λ0
τk,i = u−wk β2
, κ1 = K1 +1 , and κ2 = K2 +1 . 1, if qk − wtk ≤ qk − wik
i αkt,i = ,
Proof: See Appendix A. 0, otherwise
∀k ∈ K, i = t ∈ Mk , (17h) A. Optimizing {Q, A} for Given {Θ} and {P}

αkt,i + αki,t = 1, ∀k ∈ K, i = t ∈ Mk , (17i) For given {Θ} and {P}, the joint UAV placement and
NOMA decoding order optimization problem can be written
where (17b) denotes the feasible range of the UAV flying as
height, (17c) ensures a safe inter-distance between any two K Mk
UAVs, (17d) denotes the phase shift constraint of each IRS max Rk,i (18a)
Q,A k=1 i=1
sub-surface, (17e)-(17g) denote constraints on UAV transmit
s.t. (17b), (17c), (17g), (17h), (17i). (18b)
power, (17h) indicates that the user with a shorter distance to
the paired UAV is assigned as the stronger user, and (17i) However, problem (18) is still a mixed-integer non-convex
prevents both users being stronger users or weaker users optimization problem due to the complicated objec-
when qk − wtk = qk − wik . It is worth mentioning tive function, non-convex constraint (17c), and integer
that the joint optimization problem (17) is an offline design, constraints (17g)-(17i). To deal with the non-concave objective
i.e., determining the optimization variables, {Q, P, A, Θ}, function, we first introduce a series of auxiliary variables. Let
by assuming that the locations of ground users and the statistic j
{ukk,i , ∀k ∈ K, i ∈ Mk } and {lk,i > 0, ∀k = j ∈ K, i ∈ Mk }
CSI are perfectly known. denote the upper bound of the distance between the UAV
However, problem (17) is challenging to solve due to the and its served users, and the lower bound of the distance
following reasons. On the one hand, the involved optimiza- between the UAV and its unserved users, respectively. Let
tion variables are highly coupled, and the objective func- {uuk , ∀k ∈ K} and {llk > 0, ∀k ∈ K} denote the upper bound
tion is neither concave nor convex with respect to (w.r.t.) and lower bound of the distance between the UAV and the IRS.
the optimization variables. On the other hand, the NOMA Thus, we have
decoding order design introduces binary variables, which
make (17g)-(17i) involve integer constraints. As a result, (ukk,i )2 ≥ qk − wik 2 , ∀k ∈ K, i ∈ Mk , (19a)
problem (17) is a mixed-integer non-convex optimization j 2
qj − wik 2 ≥ (lk,i ) , ∀k = j ∈ K, i ∈ Mk , (19b)
problem, which is difficult to find the globally optimal solu-
tion. In the following, we propose an efficient iterative algo- (uuk ) ≥ qk − u , ∀k ∈ K,
2 2
(19c)
rithm to find a high-quality suboptimal solution by invoking qk − u ≥ (llk ) , ∀k ∈ K.
2 2
(19d)
BCD method [32].
Accordingly, the lower bound of the expected effective
channel power gain of the UAV to its served users, denoted
IV. BCD-BASED I TERATIVE A LGORITHM
by {η kk,i , ∀k ∈ K, i ∈ Mk }, and the upper bound of the
In this section, we develop a BCD-based iterative algorithm, expected effective channel power gain of the UAV to its
where the coupled optimization variables are divided into unserved users, denoted by {η jk,i , ∀k = j ∈ K, i ∈ Mk },
several blocks and the optimization variables in each block can be respectively expressed as (20) and (21),
are iteratively optimized with variables in the other blocks where (20) and (21) are shown at the bottom of the
fixed. To facilitate the application of the BCD method, the next page, respectively. Here, Bk,i k
= ρ0 |rHk,i Θgk | + τk,i ,
2
√ j
optimization variables in problem (17) are divided into three Ck,i = 2 Re{ κ1 ρ0
k
rk,i Θgk }, Dk,i = ρ0 |
H
rk,i Θgj |2 + τk,i ,
H
blocks: {Q, A}, {Θ}, and {P}. Specifically, for given IRS j √
Ek,i = 2 Re{ κ1 ρ0 rHk,i Θg j }, ∀k = j ∈ K, i ∈ Mk , and
reflection matrix and UAV transmit power, we first jointly opti- gk = [1, e λ −j 2πd cos ϕk 2πd
, . . . , e−j λ (N −1) cos ϕk ]T , ∀k ∈ K
mize the UAV placement, Q, and the NOMA decoding orders denotes the array response of the UAV-IRS channel in (6).
among users, A. Then, for given NOMA decoding orders, Moreover, we introduce auxiliary variables
UAV placement, and UAV transmit power, we optimize the {Uk,i ,∀k ∈ K,i ∈ Mk } and {Wk,i ,∀k ∈ K,i ∈ Mk } such
IRS reflection matrix, Θ. To handle these two subproblems, that
we employ the penalty-based method and SCA [33] to handle
K Mj
the involved integer constraints and the non-convex rank-one (Uk,i )2 = η jk,i pj,l + σ 2 , (22)
constraint. Next, we optimize the UAV transmit power, P, j=1,j=k l=1
2
for given UAV placement, IRS reflection matrix, and NOMA Mk (Uk,i )
Wk,i = αkt,i pk,t + . (23)
decoding orders by applying SCA. t=1,t=i η kk,i
⎧ ⎛ ⎞⎫
⎪
⎪ ⎪
⎪
⎨ ⎜ pk,i ⎟⎬
E{Rk,i } ≈ E log2 ⎜
⎝1 +
⎟
⎪ Mk 2 ⎠⎪
Mj
⎪
⎩ αkt,i pk,t +
E{ K
j=1,j=k cjk,i l=1 pj,l +σ } ⎪
⎭
t=1,t=i E{ckk,i }
⎛ ⎞
⎜ pk,i ⎟
= log2 ⎜
⎝1 + M
⎟ Rk,i . (15)
pj,l +σ2 ⎠
K j Mj
k k j=1,j=k ηk,i l=1
t=1,t=i αt,i pk,t + k
ηk,i
Therefore, the objective function of problem (18) is lower verified that problems (26) and (27) are equivalent when ξα →
bounded by ∞. To demonstrate this, suppose that at the optimal solution
K Mk K Mk pk,i to (27) with ξα → ∞, if any of the optimization variables
Rk,i ≥ log2 (1 + ), (24) {αkt,i } belong to (0, 1) (i.e., the inequality constraint (25a)
k=1 i=1 k=1 i=1 Wk,i
is not satisfied), the corresponding objective function’s value
where the equality holds when all equations in (19) are will be infinitely large. Then, we can always make {αkt,i }
satisfied with equality. become binary variables and the corresponding penalty term
To handle the binary variables, we first transform the integer is zero, which in turn achieves a finite objective function’s
constraint (17h) equivalently into the following constraints value and also ensures the inequality constraint (25a) to be
with continuous variables between 0 and 1: satisfied. However, if the initial value of ξα is sufficiently
large, the objective function of (27) is dominated by the
αkt,i − (αkt,i )2 ≤ 0, ∀k ∈ K, i = t ∈ Mk , (25a)
penalty term, and the effectiveness of optimizing the sum
0≤ αkt,i ≤ 1, ∀k ∈ K, i = t ∈ Mk , (25b) rate is negligible. To avoid this, we can first initialize ξα
qk − wtk 2 ≤ πk,t , ∀k ∈ K, t ∈ Mk , (25c) with a small value to find a good starting point, which may
αkt,i πk,t ≤ qk − wi , ∀k ∈ K, i = t ∈ Mk , (25d)
k 2 be infeasible for the original problem (26). Then, we can
gradually increase the value of ξα to a sufficiently larger value
where {πk,t , ∀k ∈ K, t ∈ Mk } are introduced auxiliary vari- to obtain a feasible binary solution. For any given penalty
ables, which represent the upper bound of qk − wtk 2 . In par- coefficient ξα , problem (27) is still a non-convex problem due
ticular, (25a) and (25b) jointly ensure that continuous variables to the non-convexity of the objective function and non-convex
{αkt,i } should be 0 or 1. (25c) and (25d) jointly ensure that constraints (17c), (19a)-(19d), (25d) and (26c)-(26e). In the
αkt,i = 0 when qk − wtk 2 > qk − wik 2 , which in turns following, we invoke SCA to obtain a suboptimal solution
makes αki,t = 1 due to the constraint (17i). of (27) iteratively.
Therefore, with the above introduced auxiliary variables, Let g({Wk,i , αkt,i }) denote the objective function of (27).
problem (18) can be equivalently written as problem (26) Note that g({Wk,i , αkt,i }) is concave w.r.t. {Wk,i , αkt,i }. In the
shown at the bottom of the next page, where X = (n)
nth iteration of the SCA, for given points {Wk,i , αt,i },
k(n)
j
{ukk,i , lk,i , uuk , llk , η kk,i , η jk,i , Uk,i , Wk,i , πk,i } denotes the set a global upper bound by applying the first-order Taylor expan-
of all introduced auxiliary variables for all k = j ∈ K, sion is given by
i ∈ Mk . The equivalence between problems (18) and (26) g({Wk,i , αkt,i })
can be demonstrated as follows: At the optimal solution K Mk K Mk Mk
W
to (26), if any of the constraints in (19a)-(19d) is satisfied ≤− Rk,i + ξα Ψkt,i ,
k=1 i=1 k=1 i=1 t=i
with strict inequality. Then, we can decrease the corresponding
(28)
values of {ukk,i , uuk } or increase the corresponding values
j W
of {lk,i , llk } to make all constraints in (19a)-(19d) satis- where Rk,i = log2 (1 +
pk,i
(n) ) −
pk,i log2 (e)
(n) (n)
Wk,i Wk,i (Wk,i +pk,i )
fied with equality. By doing so, the corresponding values (n) k(n) 2
of {η jk,i , Uk,i , Wk,i } or {η kk,i } can be further decreased or (Wk,i − Wk,i ) and Ψkt,i = αkt,i − [(αt,i ) +
k(n) k(n)
increased to make constraints (26b)-(26e) satisfied with equal- 2αt,i (αkt,i − αt,i )], ∀k ∈ K, i = t ∈ Mk .
ity, which also increases the value of the objective function. For non-convex constraints (17c), (19a)-(19d), and (26c),
As a result, at the optimal solution to (26), all constraints it is noted that the left-hand-side (LHS) of each constraint
of (19a)-(19d) and (26b)-(26e) must be satisfied with equality. is a convex function w.r.t. the corresponding optimization
Thus, problems (18) and (26) are equivalent. variables. Based on the first-order Taylor expansion, by replac-
To solve problem (26), we employ a penalty-based method ing the LHS of each constraint with its global lower bound,
and rewrite (26) into problem (27) shown at the bottom of the we have the following constraints for all k = j ∈ K, i ∈ Mk :
next page, where the inequality constraint (25a) is relaxed as
a penalty term in the objective function, and ξα > 0 is the (n) (n) (n) (n)
penalty coefficient which penalizes the objective function for −qk − qj 2 + 2(qk − qj )T (qk − qj )
any optimization variables αkt,i that belong to (0, 1). It can be ≥ Δ2min , (29a)
2

η kk,i = κ1 (ukk,i )−β1 hk,i + ρ0 (uuk ) rk,i Θgk + (ρ0 − κ1 )(ukk,i )−β1 + τk,i (uuk )−2
k −2 H
= ρ0 (ukk,i )−β1 + Bk,i

k
(uuk )−2 + Ck,i
k
(ukk,i )−β1 /2 (uuk )−1 , (20)
2

j −β1 j −2 H j −β1
η jk,i = κ1 (lk,i ) hk,i + ρ0 (llj ) rk,i Θgj + (ρ0 − κ1 )(lk,i ) + τk,i (llj )−2

j −β1
= ρ0 (lk,i ) j
+ Dk,i (llj )−2 + Ek,i
j j −β1 /2
(lk,i ) (llj )−1 , (21)
k(n) k(n) k(n)

(uk,i )2 + 2uk,i (ukk,i − uk,i ) where δmax denotes the maximum allowed displacement of
≥ qk − wik 2 , (29b) UAVs after each iteration of the SCA. The value of δmax
(n) (n) (n) needs to be relatively small such that we can assume that
qj − wik 2 + 2(qj − wik )T (qj − qj ) the AoAs are approximately unchanged in each iteration of
(n) (n)
j 2
≥ (lk,i ) , (29c) the SCA, i.e., gk ≈ gk , ∀k ∈ K, where gk denotes the
(n)
(n) 2 (n)
(uuk ) + 2uuk (uuk − uuk )
(n) antenna array response at the location qk . Therefore, the
j j
corresponding values of {Bk,i k k
, Ck,i , Dk,i , Ek,i } also remain
≥ qk − u2 , (29d) unchanged. The placement of UAVs in the (n + 1)th iteration
(n) (n) (n)
qk − u2 + 2(qk − u)T (qk − qk ) of the SCA is optimized based on the AoAs obtained in the nth
≥ (llk )2 , (29e) iteration, as assumed in [17]. To guarantee a certain accuracy
(n) (n) (n) of the approximation, according to [14], one method is to let
(Uk,i )2 + 2Uk,i (Uk,i − Uk,i )
the ratio of the maximum allowed displacement, δmax , to the
K Mj
≥ η jk,i pj,l + σ 2 , (29f) UAVs minimum height, Zmin , below a threshold εmax , i.e.,
zk ≤ εmax . As a result, the maximum value of δ under the
j=1,j=k l=1 δ
(n) k(n) (n) (n)
where {qk , uk,i , uuk , Uk,i , ∀k ∈ K, i ∈ Mk } are given accuracy threshold εmax is given by δ ≤ Zmin εmax . Note that
points in the nth iteration of the SCA. though a sufficiently small εmax increases the accuracy of the
Then, we rewrite the non-convex constraint (25d) as approximation, it degrades the effectiveness of optimizing the
follows: UAV placement and increases the computational complexity
2 2 due to the prohibitively large number of iterations needed for
(αkt,i + πk,t ) (αkt,i − πk,t ) (n)
− ≤ qk − wik 2 , (30) convergence. Therefore, an appropriate value of εmax needs
4 4 to be chosen to balance between accuracy and complexity.
for all k ∈ K, i = t ∈ Mk . It is observed that, for (30), Based on the above approximation and the intro-
the LHS is in form of a difference of convex functions, duced constraint (32), the RHSs of (26d) and (26e)
j
and the right-hand-side (RHS) is a convex function w.r.t. only depend on {ukk,i ,lk,i ,uuk ,llk}. Before dealing with
k(n) (n) (n)
qk . Therefore, for given points {αt,i , πk,t , qk } in the constraints (26d) and (26e), we first have the following
nth iteration of the SCA, (30) using the first-order Taylor lemma.
expansion can be replaced by Lemma 2: For any b1 > 0, b2 > 0, and b3 > 0, functions
(n) (n) (n) g1 (x, y) = b1 x−β1 + b2 y −2 and g2 (x, y) = b3 x−β1 /2 y −1 are
k,t,i ≤ qk −wi +2(qk −wi ) (qk −qj ),
dub k 2 k T
(31)
convex jointly w.r.t. x > 0 and y > 0.
(n) 2 (n)
k(n) k(n)
(αt,i −πk,t ) −2(αt,i −πk,t )(αk
t,i −πk,t )
Proof: Lemma 2 can be proved by showing the
where dub
k,t,i = 4 + Hessian matrices of functions g1 (x, y) and g2 (x, y) are posi-
2
(αk
t,i +πk,t )
4 , ∀k ∈ K, i = t ∈ Mk . tive semidefinite when x > 0 and y > 0. Therefore, g1 (x, y)
Furthermore, for non-convex constraints (26d) and (26e), and g2 (x, y) are convex functions.
note that the AoA in gk depends on the location of the kth For all k ∈ K, i ∈ Mk , let fk,i k
= ρ0 (ukk,i )−β1 +
UAV, qk , which makes the RHS of (26d) and (26e) intractable. Bk,ik
(uuk )−2 and k
gk,i = (ukk,i )−β1 /2 (uuk )−1 . The RHS
(n)
To tackle this obstacle, let {qk , ∀k ∈ K} denote the given k
of (26d) is given by fk,i + Ck,i k k
gk,i . Based on Lemma 2,
UAVs’ placement in the nth iteration of the SCA, we introduce
if Ck,i ≥ 0, fk,i + |Ck,i |
k k k k
gk,i is a convex function; otherwise,
the following constraint:
fk,i
k
− |Ck,i
k
|k
gk,i is in form of a difference of convex functions.
(n)
qk − qk 2 ≤ δmax
2
, ∀k ∈ K, (32) k(n) (n)
For given points {uk,i , uuk }, a global lower bound of
K Mk pk,i
max log2 (1 + ) (26a)
Q,A,X k=1 i=1 Wk,i
2
Mk (Uk,i )
s.t. Wk,i ≥ αkt,i pk,t + , ∀k ∈ K, i ∈ Mk , (26b)
t=1,t=i η kk,i
K Mj
(Uk,i )2 ≥ η jk,i pj,l + σ 2 , ∀k ∈ K, i ∈ Mk , (26c)
j=1,j=k l=1
η kk,i ≤ ρ0 (ukk,i )−β1 + k
Bk,i (uuk )−2 + Ck,i
k
(ukk,i )−β1 /2 (uuk )−1 , ∀k ∈ K, i ∈ Mk , (26d)
j −β1
η jk,i
≥ ρ0 (lk,i ) j
+ Dk,i (llj )−2 + Ek,i
j j −β1 /2
(lk,i ) (llj )−1 , ∀k = j ∈ K, i ∈ Mk , (26e)
(17b), (17c), (17g), (17i), (19a) − (19d), (25a) − (25d), (26f)
K Mk pk,i K Mk Mk 2
min − log2 (1 +
) + ξα (αkt,i − (αkt,i ) ) (27a)
Q,A,X k=1 i=1 Wk,i k=1 i=1 t = i
s.t. (17b), (17c), (17g), (17i), (19a) − (19d), (25b) − (25d), (26b) − (26e), (27b)
fk,i
k k k
+ Ck,i
gk,i using the first-order Taylor expansion is given where (bjk,i )H = rH k,i diag(gj ) ∈ C
1×N
denotes the cascaded
by LoS UAV-IRS-user channel before the reconfiguration of the
IRS, and [b j ∈ CM×1 ]m = N [bj ]
lb [fk ]lb + |Ck,i k
|[ k lb
gk,i ] , if Ck,i k
≥0 k,i n=1 k,i n+(m−1)N , ∀m ∈
fk,i
k k k
+ Ck,i
gk,i = k,i , M denotes the corresponding combined composite channel
[fk,i ] − |Ck,i |
k lb k k
gk,i , otherwise
associated with the mth sub-surface [28]. Based on the expres-
(33) j )H
sion of (37), let (hjk,i )H = [(b k,i hjk,i ], ∀k, j ∈ K, i ∈ Mk
k lb
where [fk,i ] and [ k lb
gk,i ] are shown at the bottom of the next j
and v = [θ T 1]T , the expected channel power gain ηk,i can
page, respectively. be rewritten as
Similarly, for all k = j ∈ K, i ∈ Mk , let fk,i j j −β1
= ρ0 (lk,i ) + H ρ0 − κ 1 τk,i
j
j −2 j
Dk,i (llj ) and gk,i = (lk,i ) j −β1 /2 −1
(llj ) . The RHS of (26e) is ηk,i = |(hjk,i ) v|2 + β1
+ 2
qj − wi k qj − u
given by fk,ij j j j(n) (n)
+Ek,i gk,i . For given points {lk,i , llj }, a global
= Tr(Hjk,i V) + k,i
j
, (38)
upper bound of fk,i j
+ Ek,i j j
gk,i using the first-order Taylor
λ0 −κ1
expansion is given by where Hjk,i = hjk,i (hjk,i )H , k,i
j
+
qj −wik β1
fj + |Ek,i j
| j j
≥0 τk,i
ub gk,i , if Ek,i , ∀k, j ∈ K, i ∈ Mk , and V = vv . In partic- H
fk,i
j j
+ Ek,i j
gk,i = k,i j j j lb , qj −u2
f − |E |[
k,i k,i g ] , otherwise
k,i ular, V needs to satisfy the following conditions: V 0,
(34) rank(V) = 1, and [V]mm = 1, m = 1, 2, . . . , M + 1.
where [j lb
gk,i ] is shown at the bottom of the next page. Then, the expected communication rate in (15) can be rewrit-
Therefore, for any given points {Qn , An , X n }, ten as (39) shown at the
bottom of the next page, where
Mk k K j Mj
problem (27) is approximated as the following problem: k,i = k,i t=1 αt,i pk,t + j=1,j=k k,i l=1
σ k
pj,l +σ 2 and
k
Mk k
K j Mj
K Mk W K Mk Mk σ k,i = k,i t=1,t=i αt,i pk,t + j=1,j=k k,i l=1 pj,l +
min − Rk,i +ξα Ψkt,i
Q,A,X k=1 i=1 k=1 i=1 t=i σ 2 , ∀k ∈ K, i ∈ Mk . Here, for ease of exposition, we define
(35a) αki,i = 1, ∀k ∈ K, i ∈ Mk .
s.t. η kk,i ≤ [fk,i
k

k k lb
+ Ck,i gk,i ] , (35b) Accordingly, problem (36) can be rewritten as
K Mk
η jk,i ≥ [fk,i
j j
+ Ek,i j ub
gk,i ] , (35c) max (fk,i −
gk,i ) (40a)
V k=1 i=1
(17b), (17g), (17i), (25b), (25c), (26b), (29a)−(29f), s.t. [V]mm = 1, m = 1, 2, . . . , M + 1, (40b)
(31), (32). V 0, V ∈ HM+1 , (40c)
(35d) rank(V) = 1. (40d)
Problem (35) is a convex optimization problem, the opti- For the non-convex rank-one constraint (40d), it can be
mal solution of which can be obtained using the standard equivalently transformed into the follow constraint:
convex program solvers such as CVX [34]. The proposed
penalty-based algorithm for solving problem (27) is summa- V∗ − V2 ≤ 0, (41)
rized in Algorithm 1, which contains double loops. In the
where V∗ = i σi (V) and V2 = σ1 (V) denote the
outer loop, we gradually increase the penalty coefficient as nuclear norm and spectral norm, respectively, and σi (V) is the
follows: ξα = ωξα , where ω > 1. In the inner loop, ith largest singular value of matrix V. For any V ∈ HM+1 ,
we optimize {Q, A, X } by iteratively solving problem (35) for
the given penalty coefficient. The objective function of (35)
is monotonically non-increasing after each iteration and a Algorithm 1 Proposed penalty-based algorithm for solving
stationary point of (27) can be obtained [33]. problem (27)
B. Optimizing {Θ} for given {Q, A} and {P} 1: Initialize {Q0 , A0 , X 0 }.
For given {Q, A} and {P}, the IRS reflection matrix 2: repeat
optimization problem can be written as 3: Set iteration index n = 0.
K Mk 4: repeat
max Rk,i (36a) 5: Solve problem (35) for given {Qn , An , X n }.
Θ k=1 i=1
s.t. (17d). (36b) 6: Update {Qn+1 , An+1 , X n+1 } with the obtained opti-
mal solutions, and n = n + 1.
Problem (36) is non-convex due to the non-concave objective
7: until convergence or reach the predefined number of
function and the non-convex unit-modulus constraint (17d).
iterations.
Before solving this problem, we first rewrite the first term of
j 8: Update {Q0 , A0 , X 0 } with the obtained optimal solu-
the expected channel power gain ηk,i in (14) as follows:
2 tions.
j
|
H
hjk,i +
rH j
k,i Θgj | = hk,i + (bk,i ) (θ ⊗ 1N ×1 )
2 9: Update ξα = ωξα .
2 10: until convergence or reach the predefined number of
j j )H θ ,
= hk,i + (b k,i (37) iterations.
we have V∗ − V2 ≥ 0 and the equality holds if and Algorithm 2 Proposed penalty-based algorithm for solving
only if V is a rank-one matrix. However, (41) is still a problem (42)
non-convex constraint. To solve problem (40), we add (41)
1: Initialize V(0) .
into the objective function as a penalty term, and the resulting
2: repeat
optimization problem yields 3: Set iteration index n = 0.
K Mk
min gk,i − fk,i ) + ξV (V∗ − V2 )
( 4: repeat
V k=1 i=1 5: Solve problem (45) for given V(n) .
(42a) 6: Update V(n+1) with the obtained optimal solution, and
s.t. (40b), (40c), (42b) n = n + 1.
7: until convergence or reach the predefined number of
where ξV ≥ 0 is the penalty coefficient. The equivalence iterations.
between problem (42) with ξV → ∞ and the original 8: Update V(0) with the obtained optimal solution.
problem (40) can be shown similarly as done in the previous 9: Update ξV = ωξV .
subsection. For any given ξV , although the objective function 10: until convergence or reach the predefined number of
of (42) is non-convex, it is in form of a difference of convex iterations.
functions. Next, we employ SCA to obtain a stationary point
of (42). As gk,i is a concave function w.r.t. V, a global upper
bound based on the first-order Taylor expansion is given by
Now, problem (45) is a convex optimization problem, which

gk,i (V) ≤ [
gk,i (V, V (n) ub
)] gk,i (V (n)
) can be efficiently solved by existing convex optimization
H solvers such as CVX [34]. The proposed algorithm for solving
+Tr((∇V gk,i (V (n)
)) (V − V(n) )), (43)
(42) is summarized in Algorithm 2. By iteratively solving
where ∇V gk,i (V(n) ) is shown at the bottom of the next problem (45), the objective function of (45) is monotonically
page and V(n) is the given point at the nth iteration of the non-increasing and a stationary point of (42) can be obtained
SCA. Similarly, a lower bound of the convex function, V2 , as the penalty coefficient increases to sufficiently large.
is given by
V2 ≥ V
(n)
V(n) 2 C. Optimizing {P} for given {Q, A} and {Θ}
H
+Tr[umax (V(n) )(umax (V(n) )) (V − V(n) )], (44) We first rewrite the expected communication rate in (15) as
(46), where (46) is shown at the bottom of the next page. For
where umax (V(n) ) denotes the eigenvector corresponding to given {Q, A} and {Θ}, the UAV transmit power optimization
the largest eigenvalue of V(n) . problem can be written as follows:
Therefore, for any given V(n) , the upper bound
n K Mk
gk,i (V, V(n) )]ub , and the lower bound V , problem (42) is
[ min (g k,i − f k,i ) (47a)
P k=1 i=1
approximated as the following problem:
s.t. (17e) − (17g). (47b)
K Mk
− fk,i ) + ξV (V∗ − V
ub (n)
min ([
gk,i ] )
V k=1 i=1 Note that g k,i is a concave function w.r.t. P, for given points
(45a) (n)
Pn = {pk,i , ∀k ∈ K, i ∈ Mk }, a global upper bound can be
s.t. (40b), (40c), (45b) expressed as (48), where (48) is shown at the bottom of the
[fk,i ] = ρ0 (uk,i )−β1 − β1 ρ0 (uk,i )−β1 −1 (ukk,i −uk,i ) + Bk,i (uuk )−2 − 2Bk,i (uuk )−3 (uuk − uuk ),
k lb k(n) k(n) k k k(n) (n) (n) (n)
β1 k(n) −β1 /2−1

] = (uk,i )−β1 /2 (uuk )−1 − (uuk )−1 (ukk,i −uk,i ) − (uk,i )−β1 /2 (uuk )−2 (uuk −uuk ).
k lb k(n) (n) (n) k(n) k(n) (n) (n)
[
gk,i (u )
2 k,i
β1 j(n)
] = (lk,i )−β1 /2 (llj )−1 − (lk,i )−β1 /2−1 (llj )−1 (lk,i −lk,i ) − (lk,i )−β1 /2 (llj )−2 (llj − llj ).
j lb j(n) (n) (n) j j(n) j(n) (n) (n)
[
gk,i
2
pk,i
Rk,i = log2 (1 + )
Mk k
K
j=1,j=k (Tr(Hjk,i V)+k,i
j
)
Mj
l=1 pj,l +σ2
t=1,t=i αt,i pk,t + Tr(Hkk,i V)+ k
k,i
Mk K Mj
= log2 (Tr(Hkk,i V) αkt,i pk,t + Tr(Hjk,i V) k,i )
pj,l + σ
t=1 j=1,j=k l=1

fk,i
Mk K Mj
− log2 (Tr(Hkk,i V) αkt,i pk,t + Tr(Hjk,i V) pj,l + σ k,i ), (39)
t=1,t=i j=1,j=k l=1

g k,i
Algorithm 3 Proposed SCA based algorithm for solving Algorithm 4 Proposed BCD-based algorithm for solving
problem (47) problem (17)
1: Initialize P0 , and set iteration index n = 0. = 1 to N
1: for n do
2: repeat 2: Initialize the n th set of {Q0 , A0 , Θ0 , P0 }, and set iter-
3: Solve problem (49) for given Pn . ation index n = 0.
4: Update Pn+1 with the obtained optimal solutions, and 3: repeat
n = n + 1. 4: Solve problem (26) for given Θn and Pn by applying
5: until convergence. Algorithm 1, and obtain Qn+1 and An+1 .
5: Solve problem (40) for given Pn , Qn+1 , and An+1 by
applying Algorithm 2, and obtain Θn+1 .
page. By replacing g k,i with its upper bound, problem (47) is 6: Solve problem (49) for given Qn+1 , An+1 , and Θn+1
approximated as the following problem: by applying Algorithm 3, and obtain Pn+1 .
K Mk
7: n = n + 1.
ub
min ([g k,i ] − f k,i ) (49a) 8: until convergence.
P k=1 i=1
s.t. (17e) − (17g). (49b)
9: Record
the

optimal
solutions
{Q∗(n) , A∗(n) , Θ∗(n) , P∗(n) } and the corresponding
as
It can be verified that problem (49) is a convex optimization objective function value Γ(n) .
problem, which can be solved by CVX [34]. The proposed 10: end
∗

11: Select the final solution as Γ(n ) = arg max Γ(n) .
SCA based algorithm for solving problem (47) is summarized
in Algorithm 3, which is guaranteed to converge to a locally =1,...,N
n
optimal solution of (47).

Based on the algorithms developed in the previous
subsections, the overall BCD-based algorithm for solving analyzed as follows: The computational complexity
problem (17) is summarized in Algorithm 4. In particular, N of applying Algorithm 1 is O(N1,out N1,in I13.5 ) [35],
K
sets of {Q , A , Θ , P } are randomly initialized as the input
0 0 0 0
where I1 = 5K + (2K + 3) k=1 Mk is the number
points of the developed algorithm, the converged solution of optimization variables, and N1,in and N1,out denote
with the highest sum rate is selected as the final solution. the number of iterations needed for convergence in the
The details of the initialization scheme will be presented in inner loop and outer loop of Algorithm 1, respectively.
Section V. Recall that the objective function of (27), (42), For Algorithm 2, since the computational complexity
and (49) is monotonically non-increasing in each iteration of of solving the semidefinite program (SDP) problem
the corresponding algorithm. Therefore, the objective function (45) is O((M + 1)4.5 ) [36], the complexity of applying
4.5
of (17) is non-decreasing after each iteration of Algorithm 4, Algorithm 2 is O(N2,out N2,in (M + 1) ), where N2,in
i.e., Γ{Qn , An , Θn , Pn } ≤ Γ{Qn+1 , An+1 , Θn+1 , Pn+1 }, and N2,out denote the number of iterations needed
K Mk
where Γ{Q, A, Θ, P} k=1 i=1 Rk,i is a function for convergence in the inner loop and outer loop of
w.r.t. Q, A, Θ, and P. As the objective function of Algorithm 2, respectively. Similarly, the computational
K of applying Algorithm 3 is O(N3 I3 ), where
3.5
(17) is upper bounded by a finite value due to the complexity
limited transmit power, the proposed Algorithm 4 is I3 = k=1 Mk is the number of optimization variables and
guaranteed to converge to a stationary point of (17). N3 denote the number of iterations needed for convergence.
The computational complexity of Algorithm 4 is Therefore, let NBCD denote the number of iterations
k k H
K H Mj
( M k
t=1,t=i αt,i pk,t (Hk,i ) +
j
j=1,j=k (Hk,i ) l=1 pj,l )log2 (e)
∇V gk,i (V (n)
)= Mk K j Mj
Tr(Hk,i V ) t=1,t=i αt,i pk,t + j=1,j=k Tr(Hk,i V ) l=1
k (n) k (n) pj,l + σ k,i
Mk K j Mj
k
Rk,i = log2 (ηk,i αkt,i pk,t + ηk,i pj,l + σ 2 )
t=1 j=1,j=k l=1

f k,i
Mk K j Mj
− log2 (ηk,i
k
αkt,i pk,t + ηk,i pj,l + σ 2 ) (46)
t=1,t=i j=1,j=k l=1

gk,i
Mk (n) K j Mj (n)

t=1,t=i αt,i (pk,t − pk,t ) + l=1 (pj,l − pj,l )
k k
ηk,i j=1,j=k ηk,i
g k,i (P) ≤ [g k,i (P, P )] n ub
gk,i (P ) +n
Mk K j Mj
. (48)
k
(ηk,i k 2
t=1,t=i αt,i pk,t + j=1,j=k ηk,i l=1 pj,l + σ )/log2 (e)
sub-surfaces, M , and the maximum transmit power of UAVs,

Pmax . Specifically, we consider the following three cases: 1)
M = 20 and Pmax = 20 dBm; 2) M = 60 and Pmax =
20 dBm; 3) M = 60 and Pmax = 30 dBm; As illustrated in
Fig. 2, for the three cases, the proposed BCD-based algorithm
converges as the number of iterations increases. It is also
observed that, for Case 2 and Case 3, it requires around 5 more
extra iterations for convergence since a larger M increases the
computational complexity of Algorithm 2.
B. Benchmark Schemes
In the following, we investigate the sum rate performance
Fig. 2. Convergence of the proposed BCD-based algorithm for different obtained by the proposed scheme. For comparison, we also
values of simulation parameters. consider two benchmark schemes as follows:
needed for convergence of the developed BCD-based • OMA: In this case, all UAVs are assumed to share
algorithm, the total computational complexity is given by the same frequency band and serve ground users in
O(NBCD (N1,out N1,in I13.5 + N2,out N2,in (M +1)4.5 +N3 I33.5 )), orthogonal time slots of equal size with transmit power
which is polynomial. It is also worth noting that an offline joint 0 ≤ pk ≤ Pmax , ∀k ∈ K. The achievable communication
optimization is considered, the potentially high computational rate of the (k, i)th user is given by
complexity of the BCD-based algorithm is acceptable given
1 ckk,i pk
the available computing power. OMA
Rk,i = log2 (1 + K ), (50)
Mk j 2
V. N UMERICAL R ESULTS j=1,j=k ck,i pj + σ
In this section, numerical results are provided to verify for all k ∈ K, i ∈ Mk . The expected achievable
the effectiveness of the proposed algorithm. We consider a communication rate for OMA is approximated by
network with two user groups served by K = 2 UAVs. Each OMA 1 k
ηk,i pk
group consists of 3 users that are randomly and uniformly Rk,i log2 (1 + K j
).
Mk j=1,j=k ηk,i pj + σ
2
distributed in two adjacent areas of 250 × 250m2 . The pre-
sented results in the following are obtained based on one • Interference Free (IF): In this case, all UAVs are
random realization of users’ distributions, as illustrated in assumed to be allocated with orthogonal frequency bands
Fig. 3. The simulated parameters are set as follows: The IRS of equal size and serve ground users in orthogonal time
is located at (0, 250, 20) meters, and the number reflecting slots of equal size. As the interference does not exist,
elements of each sub-surfaces is set to N = 20. The referenced all UAVs serve ground users with the maximum transmit
channel power gain is set to ρ0 = −30 dB, and the noise power. Therefore, the corresponding communication rate
power is σ 2 = −80 dBm. The path loss exponents for the of the (k, i)th user is given by
UAV-user link and IRS-user are set to β1 = β2 = 2.2, and
1 ckk,i Pmax
the corresponding Rician factors are K1 = K2 = 10 dB. The IF
Rk,i = log2 (1 + 1 2 ), (51)
minimum and maximum allowed flying height of UAVs are set KMk Kσ
to Zmin = 60 meter and Zmax = 100 meter, respectively. For for all k ∈ K, i ∈ Mk . The expected achievable
simplicity, we assume that all UAVs have the same maximum communication rate for IF is approximated by
transmit power, i.e., Pmax,k = Pmax , ∀k ∈ K. The accuracy k
threshold in Algorithm 1 for optimizing UAVs placements is IF 1 ηk,i Pmax
Rk,i log2 (1 + 1 2 ).
set to εmax = 0.1, and the corresponding δmax = 5 meter. KMk Kσ
The number of sets of initial points of Algorithm 4 is set It is worth noting that the proposed BCD-based algorithm can
to N = 10. For each initialization, the initial horizontal
be also applied to the two benchmark schemes. In particular,
locations of UAVs are randomly and uniformly generated the optimization problem for OMA can be solved with the
in each area of 250 × 250m2 with the initial flying height proposed algorithm without the intra-group interference terms
of zk = (Zmin + Zmax )/2, ∀k ∈ K, and then the NOMA and the NOMA decoding order design. For IF, the optimization
decoding orders among users in each group are initialized problem can be solved without considering the interference
based on their distances to the paired UAVs. The transmit terms, the NOMA decoding order design, and the UAV trans-
power of each UAV is initialized by the maximum transmit mit power design.
power, which is equally allocated to all served users. The
phase shift of each IRS sub-surface is randomly and uniformly
C. Optimal UAV Placement for Different Schemes
generated in [0, 2π).
In Fig. 3, we provide the optimal UAV placement obtained
A. Convergence of BCD-Based Algorithms by the proposed BCD-based algorithm for different schemes.
In Fig. 2, we provide the convergence of the devel- The maximum UAV transmit power is set to Pmax = 20 dBm
oped BCD-based algorithm for different numbers of IRS and the number of IRS sub-surfaces is set to M = 40. It is
and their served users to enhance the desired signal power

strengths, but also degrades the channel power gains between
UAVs and unserved users to mitigate the caused interference.
For user (1, 3), it shows that deploying the IRS simultaneously
improves the desired channel power gain with UAV 1 by more
than 10% and reduces the interference channel power gain
with UAV 2 by more than 40%. A similar phenomenon is
also observed at user (2, 1). This is because the two users
in the simulation setup are closest to the IRS, which allows
them to fully reap the benefits of the IRS. Moreover, a clear
degradation of the interference channel power gains can be
also observed at users (1, 1) and (2, 3). As the two users are
closest to the two UAVs for NOMA, see Fig. 3, the achieved
sum rate is dominated by the communication rates of the two
Fig. 3. Optimal placement of the two UAVs for different schemes.
users due to the good channel conditions. Therefore, mitigating
the inter-group interference caused to the two users is an
observed that the optimal horizontal placement of the two effective way to increase the sum rate. In Fig. 4(b), for OMA,
UAVs for NOMA and OMA is to hover near user (1, 1) and the “double effect” of the IRS can be also observed at each
user (2, 3), respectively. This is because, for the two schemes, user and is the most pronounced at user (1, 3), which is nearest
the locations of UAVs determine not only the received sig- to the IRS. However, in Fig. 4(c), we can observe that the IRS
nal strengths of their served users, but also the inter-group for IF only enhances the desired channel power gains of all
interference to unserved users. As a result, the two UAVs users. This is expected since the interference does not exist
prefer to be deployed near some of the served users while due to the completely orthogonal transmission. Similarly, the
keeping considerable far distances to other unserved users. improvement of channel power gain brought by the IRS is
The optimal flying heights of the two UAVs for the two noticeable for users (1, 3) and (2, 1), which are near to the
schemes are zk = Zmin, ∀k ∈ K. Though a lower flying IRS. This also allows UAVs for IF to be deployed closer to
height causes stronger interference to unserved users, it is more the other two users in each group, as illustrated in Fig. 3. The
beneficial for enhancing the channel qualities of served users above results underscore the benefits of deploying the IRS.
since the horizontal distances of UAVs to served users are
much smaller than those to unserved users. In contrast, for IF,
the optimal horizontal placement of the two UAVs tends to E. Sum Rate versus M and Pmax
be symmetric among all served users in each group with the In this subsection, we investigate the achieved sum rate of
optimal flying height of Zmin . As UAVs for IF do not need to the proposed algorithm. For comparison, we also consider the
control inter-group interference, they prefer to be deployed to following setups:
enhance the received signal strengths of all served users. • Without IRS: In this case, there is no IRS deployed in
the multi-UAV communication network. The optimiza-
D. Effect of Deploying the IRS tion problem is solved by only considering the direct
UAV-user links with the proposed BCD-based algorithm.
In this subsection, we investigate the effect on channel
• Fixed Location (FL): In this case, the horizontal location
qualities brought by deploying the IRS. We calculate the
of each UAV is fixed at the mean location of ground users
variety ratio between the expected channel power gain of the
2 in each group, as assumed in [13]. We only optimize
direct UAV-user link, E{|hjk,i | }, and the effective channel the height of each UAV with the proposed BCD-based
power gain between the UAV and the user reconfigured by algorithm.
j
the IRS, ηk,i , which is given by
For each setup, we also consider the proposed NOMA scheme
! 2
"
j
ηk,i − E |hjk,i | and two benchmark schemes.
j
ζk,i = ! 2
" , ∀k, j ∈ K, i ∈ Mk . (52) Fig. 5 shows the achieved sum rate versus the number of
E |hjk,i | IRS sub-surfaces, M , for Pmax = 20 dBm. It is first observed
that the achieved sum rates of all schemes with IRS increase
j
In particular, ζk,i > 0 means that the channel quality between with M , since higher reflecting array gains can be exploited
the jth UAV and the (k, i)th user is enhanced by the optimized for a larger size of IRS. However, for schemes without IRS,
IRS reflection matrix compared to the scheme without IRS. the achieved sum rates remain unchanged, which also demon-
j
ζk,i < 0 means that the corresponding channel quality is strates the benefits of the IRS. Among the three transmission
degraded by the IRS. schemes, the proposed NOMA achieves the best sum rate
Fig. 4 presents the variety ratio of channel qualities for performance. This is because, on the one hand, NOMA allows
different schemes with Pmax = 20 dBm and M = 40. In each all users to be simultaneously served in all resource blocks,
scheme, the two UAVs are deployed at the obtained optimal which facilitates flexible resource allocations to improve spec-
placements in Fig. 3. In Fig. 4(a), it is observed that the IRS tral efficiency. On the other hand, the optimization of UAV
not only enhances the channel power gains between UAVs placement and IRS reflection matrix provides a new degree-of-
Fig. 4. Effect of IRS for different schemes.
Fig. 5. Sum rate versus number of IRS sub-surfaces.

Fig. 6. Sum rate versus maximum transmit power.
with the optimized scheme. This is because the optimal UAV

freedom (DoF) for implementing NOMA, i.e., decoding order placement for IF is similar to the fixed scheme (see Fig. 3).
design via the placement of UAVs and inter-group interference Fig. 6 shows the achieved sum rate versus the maximum
mitigation via deploying the IRS. OMA achieves the worst UAV transmit power, Pmax , for M = 60. The achieved
performance since it provides limited resource blocks for sum rates of all schemes increase as Pmax increases. For
each user compared with NOMA and experiences inter-group NOMA and OMA, the achieved sum rate seems to be upper
interference compared with IF. We also observe that the IRS bounded by a finite value as Pmax increases. This is because
gain for NOMA is more pronounced than those for other the existence of inter-group interference makes the network
transmission schemes, which confirms the effectiveness of for the two schemes become interference-limited when the
the proposed IRS-enhanced multi-UAV NOMA transmission transmit power is large. In this case, optimizing the placement
framework. For NOMA, compared with the scheme with fixed of UAVs and the reflection matrix of the IRS has less effect
UAV placement, a significant sum rate gain can be achieved by on the interference mitigation. Owing to this reason, we can
optimizing the UAV placement even without IRS. For OMA, observe that IF outperforms NOMA and OMA when Pmax
the scheme with fixed UAV placement only outperforms the is larger than a certain value. The obtained results reveal that
scheme without IRS when the size of IRS is large. This the proposed NOMA scheme is preferable for limited transmit
is because the effective channel power gains of users are power and IF is superior for large transmit power.
dominated by the UAV placement. As a result, optimizing
the placement of UAVs is more effective than optimizing the VI. C ONCLUSION
IRS reflection matrix. This also underscores the importance of IRS-enhanced multi-UAV NOMA networks have been
optimizing the placement of UAVs. Moreover, the performance investigated. The 3D placement and transmit power of UAVs,
gain of NOMA over OMA is greatly enhanced by optimizing the reflection matrix of the IRS, and the NOMA decoding
the UAV placement. This is because adjusting the UAV orders of each user group were jointly optimized for maxi-
placement not only enlarges the channel disparity of users, mization of the sum rate of all users. To tackle the resulting
where NOMA yields higher performance gain than OMA, mixed-integer non-convex optimization problem, a BCD-based
but also enables a flexible NOMA decoding order design algorithm was developed to iteratively find a suboptimal
to further improve the sum rate. However, for IF, the fixed solution. Our numerical results showed that the achieved sum
UAV placement scheme achieves a comparable performance rate can be significantly improved by optimizing the UAV
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Xidong Mu (Graduate Student Member, IEEE)

received the B.S. degree in information engineering
from the Beijing University of Posts and Telecom-
munications (BUPT), Beijing, China, in 2015, where
he is currently pursuing the Ph.D. degree.
Since September 2020, he has been a Visiting Stu-
dent (an Online Enrolment) with the School of Elec-
tronic Engineering and Computer Science, Queen
Mary University of London, U.K. His research Jiaru Lin (Member, IEEE) received the B.S. and
interests include non-orthogonal multiple access, Ph.D. degrees from the School of Information Engi-
IRSs/RISs aided communications, UAV communi- neering, Beijing University of Posts and Telecom-
cations, and optimization theory. He received the Exemplary Reviewer Cer- munications (BUPT), China, in 1987 and 2001,
tificate of the IEEE T RANSACTIONS ON C OMMUNICATIONS in 2020. respectively. He studied at the Swiss Federal Institute
of Technology, Zurich, from 1991 to 1994.
Yuanwei Liu (Senior Member, IEEE) received the He is currently a Professor and the Ph.D. Supervi-
B.S. and M.S. degrees from the Beijing University sor with the School of Artificial Intelligence, BUPT.
of Posts and Telecommunications in 2011 and 2014, His research interests include wireless communica-
respectively, and the Ph.D. degree in electrical engi- tion, personal communication, and communication
neering from the Queen Mary University of London, networks.
U.K., in 2016.
He was with the Department of Informatics, Kings
College London, from 2016 to 2017, where he was
a Post-Doctoral Research Fellow. He has been a
Lecturer (an Assistant Professor) with the School
of Electronic Engineering and Computer Science,
Queen Mary University of London, since 2017. His research interests include
non-orthogonal multiple access, 5G/6G networks, machine learning, and
stochastic geometry. He has served as a TPC member for many IEEE
conferences, such as GLOBECOM and ICC. He received the IEEE ComSoc
Outstanding Young Researcher Award for EMEA in 2020. He also received
the Exemplary Reviewer Certificate of IEEE W IRELESS C OMMUNICA - H. Vincent Poor (Life Fellow, IEEE) received the
TIONS L ETTERS in 2015, IEEE T RANSACTIONS ON C OMMUNICATIONS Ph.D. degree in electrical engineering and computer
in 2016 and 2017, and IEEE T RANSACTIONS ON W IRELESS C OMMU - sciences (EECS) from Princeton University in 1977.
NICATIONS in 2017 and 2018. He has served as the Publicity Co-Chair From 1977 until 1990, he was on the faculty of the
for VTC 2019-Fall. He is the Leading Contributor for “Best Readings for University of Illinois at Urbana-Champaign. Since
Non-Orthogonal Multiple Access (NOMA)” and the Primary Contributor for 1990, he has been on the faculty at Princeton Univer-
“Best Readings for Reconfigurable Intelligent Surfaces (RIS).” He also serves sity, where he is currently the Michael Henry Strater
as the Chair for the Special Interest Group (SIG) in Signal Processing and University Professor. From 2006 to 2016, he has
Computing for Communications (SPCC) Technical Committee on the topic of served as the Dean for the Princeton’s School of
signal processing Techniques for next generation multiple access (NGMA), Engineering and Applied Science. He has also held
the Vice Chair for the SIG Wireless Communications Technical Commit- visiting appointments at several other universities,
tee (WTC) on the topic of Reconfigurable Intelligent Surfaces for Smart Radio including most recently at Berkeley and Cambridge. His research interests
Environments (RISE), and the Tutorials and Invited Presentations Officer for include information theory, machine learning and network science, and their
Reconfigurable Intelligent Surfaces Emerging Technology Initiative. He is applications in wireless networks, energy systems, and related fields. Among
currently an Editor on the Editorial Board of the IEEE T RANSACTIONS his publications in these areas is the forthcoming book Machine Learning
ON W IRELESS C OMMUNICATIONS, the IEEE T RANSACTIONS ON C OM - and Wireless Communications (Cambridge University Press, 2021). He is
MUNICATIONS , and IEEE C OMMUNICATIONS L ETTERS . He also serves also a member of the National Academy of Engineering and the National
as the leading Guest Editor for IEEE J OURNAL ON S ELECTED A REAS Academy of Sciences. He is also a Foreign Member of the Chinese Academy
IN C OMMUNICATIONS (JSAC) Special Issue on Next Generation Multiple of Sciences, the Royal Society, and other national and international academies.
Access, and a Guest Editor for IEEE J OURNAL OF S ELECTED T OPICS IN Recent recognition of his work includes the 2017 IEEE Alexander Graham
S IGNAL P ROCESSING (JSTSP) Special Issue on Signal Processing Advances Bell Medal and a D.Eng. (honoris causa) from the University of Waterloo,
for Non-Orthogonal Multiple Access in Next Generation Wireless Networks. awarded in 2019.

Ris + Uav

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 39, NO.

10, OCTOBER 2021 3051

Intelligent Reflecting Surface Enhanced

Abstract— Intelligent reflecting surface (IRS) enhanced I. I NTRODUCTION

carefully developed to fully reap the benefits of employing

∀k ∈ K, i = t ∈ Mk , (17h) A. Optimizing {Q, A} for Given {Θ} and {P}

= ρ0 (ukk,i )−β1 + Bk,i

k(n) k(n) k(n)

β1 k(n) −β1 /2−1

optimal solution of (47).

Mk (n) K j Mj (n)

sub-surfaces, M , and the maximum transmit power of UAVs,

and their served users to enhance the desired signal power

Fig. 4. Effect of IRS for different schemes.

Fig. 5. Sum rate versus number of IRS sub-surfaces.

with the optimized scheme. This is because the optimal UAV

Xidong Mu (Graduate Student Member, IEEE)

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Ris + Uav

Uploaded by

Copyright:

Available Formats

Ris + Uav

Uploaded by

Document Information

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Ris + Uav

Uploaded by

Copyright:

Available Formats

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 39, NO.

10, OCTOBER 2021 3051

Intelligent Reflecting Surface Enhanced

Abstract— Intelligent reflecting surface (IRS) enhanced I. I NTRODUCTION

carefully developed to fully reap the benefits of employing

∀k ∈ K, i = t ∈ Mk , (17h) A. Optimizing {Q, A} for Given {Θ} and {P}

= ρ0 (ukk,i )−β1 + Bk,i

k(n) k(n) k(n)

β1 k(n) −β1 /2−1

optimal solution of (47).

Mk (n) K j Mj (n)

sub-surfaces, M , and the maximum transmit power of UAVs,

and their served users to enhance the desired signal power

Fig. 4. Effect of IRS for different schemes.

Fig. 5. Sum rate versus number of IRS sub-surfaces.

with the optimized scheme. This is because the optimal UAV

Xidong Mu (Graduate Student Member, IEEE)

You might also like

Mk (n) K j Mj (n)