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Secure RIS Deployment Strategies for
Wireless-Powered Multi-UAV Communication
Danyu Diao, Buhong Wang, Member, IEEE, Kunrui Cao, Member, IEEE, Beixiong Zheng, Senior Member, IEEE,
Jiang Weng, and Jingyu Chen
Abstract—Reconfigurable intelligent surface (RIS) is viewed
as a promising technique that can be utilized to improve the
performance of systems by reconfiguring signal propagation en-
vironments. This paper investigates green and secure unmanned
aerial vehicle (UAV) Internet-of-Things (IoT) communications
with the aid of RIS, where multiple UAVs harvest energy from a
power beacon (PB) and send information uplink to access point
(AP) with non-orthogonal multiple access (NOMA). In particular,
communication can be divided into two phases during each time
frame: energy transfer and information transmission. Three RIS
deployment strategies are proposed. In mode I, RISs are deployed
between UAVs and AP to enhance the information transmission.
In mode II, RISs are deployed between PB and UAVs to enhance
the energy transfer. In mode III, RISs are deployed between
UAVs and a hybrid access point (HAP) to enhance energy transfer
and information transmission simultaneously. Considering phase
compensation error caused by imperfect conditions, we define
and evaluate ergodic capacity (EC), ergodic capacity probability
(ECP) and ergodic secrecy capacity (ESC) of three modes to
measure the reliability and security of the system. The asymptotic
expressions are also derived for further insights. Numerical
results are presented to validate the correctness of theoretical
derivations. Results demonstrate that the passive beamforming
gain promised by RIS can significantly enhance the performance
of systems. Mode III outperforms other modes in terms of
reliability and security. When the transmission power and the
number of UAVs increase, the ESCs of modes I and III converge
to the same performance floor.
Index Terms—Reconfigurable intelligent surface (RIS), un-
manned aerial vehicle (UAV), non-orthogonal multiple access
(NOMA), performance analysis.
This work was supported in part by the National Natural Science Foun-
dation of China under Grant 62301600, 62331022, 62201214, 62101560 and
61902426, in part by the Natural Science Basic Research Program of Shaanxi
under Grant 2022JQ-619, in part by the Natural Science Foundation of Guang-
dong Province under Grant 2023A1515011753, in part by the Research Funds
of Guangzhou Science and Technology under Grant SL2023A04J00790, in
part by the open research fund of the State Key Laboratory of ISN under
Grant ISN23-04, in part by the Fundamental Research Funds for the Central
Universities under Grant 2023ZYGXZR106, in part by China Postdoctoral
Science Foundation under Grant 2021M692502, and in part by the National
University of Defense Technology Research Fund under Grant ZK21-44.
(Corresponding authors: Buhong Wang; Kunrui Cao.)
D. Diao, B. Wang, and J. Weng are with the School of Information and
Navigation, Air Force Engineering University, Xi’an 710077, China, and D.
Diao is also with the State Key Laboratory of Integrated Services Networks,
Xidian University, Xi’an 710071, China. (email:diaodanyu2021@sina.com;
hongwks@aliyun.com; wengjiang858@163.com).
K. Cao and J. Chen are with the School of Information and Communica-
tions, National University of Defense Technology, Wuhan 430035, China, and
K. Cao is also with the State Key Laboratory of Integrated Services Networks,
Xidian University, Xi’an 710071, China. (e-mail: krcao@nudt.edu.cn and
chenjingyu@nudt.edu.cn).
B. Zheng is with the School of Microelectronics, South China University
of Technology, Guangzhou 511442, China (e-mail: bxzheng@scut.edu.cn).
I. INTRODUCTION
Unmanned aerial vehicle (UAV) communication has become
a key technology in the sixth generation (6G) era due to
its high channel quality, wide area coverage and flexible
deployment [1]. As a typical energy-constrained Internet-of-
Things (IoT) terminal, the depletion of UAV’s battery may
result in network node failure [2]. Therefore, it is essential to
investigate the charging strategies for UAVs in order to reduce
maintenance costs and improve energy efficiency. Wireless-
powered communications use a “harvest-then-transmit” proto-
col to collect energy downlink and then transmit information
uplink, which enables energy harvesting in a controllable
manner [3], [4]. It makes sense in UAV communications
owing to its capability to enhance endurance and sustain
networks. In accordance with this attractive solution, some
research has been devoted to addressing energy issues of
UAV communications. The authors of [5] proposed a joint
energy transfer and artificial noise (ETAN) scheme to utilize a
power beacon (PB) to supply energy to multiple UAVs, thereby
increasing UAVs’ endurance time. Aiming at maximizing the
system energy efficiency, the wireless power transfer time,
power allocation and position of the UAV were optimized in
a wireless-powered UAV system [6]. Considering the energy
constraints of UAV and terrestrial users, the sum energy
harvested by all terrestrial users was maximized in [7], where
a UAV is deployed to charge the terrestrial users after being
charged by a base station. Aiming at minimizing the energy
consumed by the UAV, time durations, the UAV’s transmitting
powers and trajectory were jointly optimized in [8].
On the other hand, the security of UAV communications is
also a critical issue that needs to be addressed collaboratively
with energy efficiency due to the broadcast nature of air-to-
ground channels [9]. Physical layer security (PLS), which
makes use of wireless channel characteristics to safeguard
transmission, is a good complementary to conventional cryp-
tography encryption technology [10], [11]. Based on the
broadcast nature and energy-constraint characteristic of UAVs,
it becomes necessary and urgent to improve the secrecy
performance and endurance of UAV IoT communications
simultaneously. Reconfigurable intelligent surface (RIS), as a
promising technology, can intelligently control the wireless
propagation medium by manipulating the phase shift of in-
coming electromagnetic [12], [13]. Specifically, a RIS consists
of multiple passive reflecting elements, each of which can
intelligently regulate the amplitude and phase changes of the
incident signal. RISs can therefore be flexibly deployed to
2
improve the PLS performance and energy transfer without
dedicated radio frequency (RF) components. This ground-
breaking development is anticipated to unlock the full potential
of UAV communications in terms of security [14]–[19], and
energy efficiency [20]–[23]. In [14], a UAV-mounted RIS
framework was proposed to safeguard transmission in the
presence of multiple eavesdroppers. Furthermore, a multi-
functional UAV-mounted RIS was exploited to effectively
combat eavesdropping, where the RIS element can switch
between the amplification mode and jamming mode [15]. In
[16], the authors evaluated the secrecy outage probability of
UAV-RIS systems to unveil the influence of key parameters.
Considering the legitimate user data rate and total power
constraints, the authors of [17] formulated a transmition power
minimization problem for secure UAV active RIS systems
and demonstrated the superiority of active RIS. In [18],
the trajectory and reflection coefficients of UAV-mounted
RIS were jointly optimized to maximize the secrecy rate
by deep reinforcement learning methods. Unlike [18], the
reliability constraint of the system was further considered
in [19]. The transmit beamforming, artificial noise, UAV-
mounted RIS placement, and RIS’s passive beamforming were
jointly optimized while satisfying the quality of service (QoS)
requirement of legitimate users. These findings collectively
highlight the fact that RIS plays a vital role in elevating
the secrecy performance of UAVs compared with the no-
RIS method. Moreover, RISs can also facilitate the harvested
energy in wireless-powered UAV communications [20]–[23].
In [20], the authors exploited a UAV as the energy source
to transfer energy to terrestrial nodes with the aid of RIS.
The energy consumption was minimized by jointly optimizing
the UAV’s trajectory, hovering time, and the RIS’s reflection
coefficients. Considering both hovering and mobile UAVs, the
authors of [21] proposed two deep reinforcement learning
methods to maximize the throughput of wireless-powered UAV
systems with the aid of RIS. In [22], a RIS was deployed
to enhance both energy harvesting and information transmis-
sion, where a power station and a UAV are responsible for
providing energy and collecting information from terrestrial
devices respectively. In [23], the authors formulated a sum-
rate maximization problem in UAV-RIS system by jointly
optimizing UAV trajectory and transmitting power, successive
interference cancellation (SIC) decoding order, power splitting
ratio and RIS reflection coefficient. However, it is worth noting
that the above work is devoted to utilizing UAVs to power
energy-constrained devices on the ground, ignoring the energy
constraints of UAVs themselves.
On a parallel avenue, non-orthogonal multiple access
(NOMA) has emerged as a potential candidate for 6G com-
munications due to its high spectral efficiency and user fair-
ness [24], [25]. Unlike traditional orthogonal multiple access
(OMA), in the uplink NOMA system, users with different
channel conditions can transmit information to the base station
in the same resource block (e.g., time, code, and frequency
domain) [26]. For power-domain NOMA, SIC is implemented
at the receiver to achieve multiplexing in the power domain,
thus efficiently utilizing the available spectrum. In this context,
NOMA schemes are superior to OMA schemes in the multi-
UAV system, which has been verified in [27]–[29]. In [27],
the authors minimized the total energy consumption of ground
users while satisfying data rate constraints. Considering large-
scale cellular networks, a 3-D UAV framework for providing
service to terrestrial NOMA users was proposed in [28].
Analytical expressions of outage probability and the ergodic
rate were derived by utilizing the stochastic geometry method.
NOMA has proven to fit in collaborative multi-UAV communi-
cations and can be used to enhance the reliability performance
[29].
A. Motivation
As mentioned above, RIS can facilitate PLS by adjusting
channel propagation conditions [14]–[19], and can also alle-
viate energy consumption by enhancing sustained and steady
radio-frequency energy harvested [20]–[23]. There are many
separate studies regarding these two aspects, but research
gaps still exist in utilizing RIS to simultaneously address
the aforementioned two issues. Moreover, existing literature
tends to focus on the use of UAV-RIS to power terrestrial IoT
devices, ignoring the energy constraints of UAVs themselves
[20], [21]. The authors of [17] proposed a mathematical
framework to reduce the energy consumption of UAVs, but
do not fundamentally address the energy constraints of UAVs.
Leveraging RIS to combine both secure transmission and
energy transfer can further address the risks of UAV security
and energy constraints in a more efficient manner, which is
still a major research gap. Corresponding RIS deployment
strategies are also far from being understood. Moreover, the
above studies tend to adopt the OMA schemes. Due to scarce
aerial spectrum resources, multi-UAV systems tend to be more
suited to the NOMA schemes.
On the other hand, the existing literature considers perfect
phase shift and does not take into account phase shift errors of
RISs [14]–[16], [18]–[23]. However, the performance of RIS
systems depends highly on the reliability of phase estimation
and alignment. In fact, due to factors such as UAV jittering, air
turbulence effect, RIS phase alignment is usually imperfect in
UAV systems. It is worth analyzing the performance of UAV-
RIS systems with imperfect phase compensation to obtain
important insights into network design and implementation.
In view of this, this paper focuses on investigating the RIS
deployment strategies to achieve green and secure UAV sys-
tems, taking account into phase compensation errors.
B. Contribution
This paper presents a theoretical framework for green and
secure multi-UAV systems. In the presence of eavesdropping
attacks, exploiting RIS to enable secure communication and
maintain sufficient energy simultaneously for UAV systems
has not been reported in the literature yet. Corresponding
RIS deployment strategies are also far from being understood,
which motivates this work. In this paper, the potential beam-
forming gain of RIS is exploited to enhance energy transfer
or adjust secure propagation. By conducting a comprehensive
analysis of system performance, the influence of key param-
eters is revealed, providing guidance for multi-UAV system
3
design. The main contributions of this paper are summarized
as follows.
•To address both energy shortage and security problems
of UAV communications, we introduce RIS into wireless-
powered UAV NOMA communication. Depending on the
different deployment strategies, we propose three RIS-
assisted wireless-powered UAV transmission modes. 1)
In mode I, RISs are deployed between UAVs and access
point (AP) to enhance the information transmission be-
tween UAVs and AP; 2) In mode II, RISs are deployed
between PB and UAVs to enhance the energy transfer; 3)
In mode III, RISs are deployed between UAVs and hybrid
access point (HAP) in order to enhance both energy
transfer and information transmission.
•To be practical, we take into account the air-to-ground
channels with Nakagami-mfading. In addition, the im-
pact of phase compensation error caused by imperfect
conditions such as UAV jitter is evaluated. The reliable
and secrecy performance under three modes is analyzed
in terms of ergodic capacity (EC), ergodic capacity
probability (ECP) and ergodic secrecy capacity (ESC).
Specifically, to characterize the reliable performance of
the system, the closed-form expression of EC under three
modes is derived. Furthermore, we define and evaluate
asymptotic ECP to further investigate the asymptotic re-
liable performance of the system. To evaluate the secrecy
performance of the system, we obtain the closed-form
expressions of ESC under three modes. The asymptotic
expressions are also derived to gain further insights.
The derived expressions characterize the impact of key
parameters such as time-splitting factor, the number of
UAVs, and phase compensation errors. The proposed
theoretical framework can provide a theoretical reference
for practical system design and deployments.
•Numerical results show that 1) All three modes outper-
form the benchmark mode without the aid of RIS. 2)
From the perspective of reliability, mode III achieves the
best performance compared with other modes. 3) From
the perspective of security, mode III is superior to other
modes when the transmission power is low. When the
transmission power and the number of UAVs increase,
mode I and mode III converge to the same performance
floor. 4) It is advisable to choose a higher time-splitting
factor with the increase in the number of UAVs in the
system. 5) The diversity orders under three modes are
determined by the number of UAVs and the quality of
the channel. 6) Phase compensation errors deteriorate the
secrecy performance of modes I and III, but the effect on
mode II is negligible.
The rest of the paper is organized as follows. In Section
II, we describe the system model and propose three RIS
deployment modes. In Section III, the reliability and security
performance under these three proposed modes is investigated
in terms of EC, ECP and ESC. The asymptotic expression of
ESC is also derived for intuitive insights. Numerical results
are presented in Section IV. Finally, we conclude the paper in
Section V.
0RGH,0RGH,,0RGH,,,
Energy transfer Information transmission
T
t
(
)
1
T
t
-
Energy transfer Information transmission Eavesdropping link
AP
PB
E
HAP
E
U1
PB AP
U1
U2
UK
E
U1
U2
UK
UK
U2
Fig. 1. System Model.
II. SY ST EM MO DE L AN D PROP OS ED MO DE S
A. System Model
We consider an energy-constrained multi-UAV system con-
sisting of a low-latitude UAV swarm, Kterrestrial RISs, a
passive eavesdropper (E), a PB, and an AP, where PB and
AP can be co-located or integrated as a HAP. 1All RISs,
denoted by Rk(k∈1,2, ..., K), have Nreflecting elements
and are pre-deployed at a certain distance from each other.
The interference signals reflected from neighboring RISs can
be ignored due to severe path loss [31]. Kselected UAVs,
denoted by Uk, form a NOMA group and each UAV is assisted
by the corresponding RIS to transfer information or energy. 2
To address the energy-constraint problem, UAVs are not only
powered by onboard batteries but also harvest energy from
PB. The communication process Tcan be divided into the
1The base station can detect the eavesdropper from the unintentional local
oscillator power leaked by the RF front end [30].
2In the proposed UAV-RIS system, RISs are preconfigured on the walls
of high buildings. The distance between RISs is sufficiently large and the
interference signals reflected from the neighboring RISs are negligible owing
to obstacles. Within the coverage service range of each RIS, only one UAV
with high priority is selected for service, i.e., the phase shift parameters of
RISs are optimized based on the channel information of the prioritized UAV.
In this paper, we focus on analyzing the performance of priority UAVs in
each service range [31].
4
energy harvesting (EH) phase and information transmission
(IT) phase with a time-splitting factor τ. We assume that
the onboard batteries equipped at UAVs are used to support
maneuvering, and the energy harvested from PB is used for
information processing as widely adopted in [32]–[34]. In the
EH phase, HAP (or PB) transfers energy downlink to UAVs.
In the IT phase, UAVs exploit the harvested energy to transmit
information uplink to HAP (or AP) with NOMA protocol.
As shown in Fig. 1, three RIS deployment modes are
proposed: 1) In mode I, RISs are deployed between UAVs and
AP to enhance the information transmission between UAVs
and AP; 2) In mode II, RISs are deployed between PB and
UAVs to enhance the energy transfer; 3) In mode III, RISs are
deployed between UAVs and HAP in order to enhance both
energy transfer and information transmission.
In mode I, the channels between AP/Ukand Rkare de-
noted by har,k =hh1
ar,k, h2
ar,k,...hN
ar,kiT∈CN×1and
hur,k =hh1
ur,k, h2
ur,k,...hN
ur,ki∈C1×N, respectively. In
mode II, the channels between PB/Ukand Rkare denoted
by hpr,k =hh1
pr,k, h2
pr,k,...hN
pr,kiT∈CN×1and hur,k . In
mode III, the channels between HAP/Ukand Rkare denoted
as the same as in mode I. In the above three modes, the
direct channel between PB and Ukis denoted by hpu,k. The
direct channel between HAP (or AP) and Ukis denoted
by hua,k. The channels between Rk/Ukand E are denoted
as her,k =hh1
er,k, h2
er,k,...hN
er,kiT∈CN×1and hue,k ,
respectively. As widely adopted in literature [35]–[37], all
UAVs have hovering ability, therefore all channels can be mod-
eled as quasi-static fading links. Each entry of hpr,k,hur,k,
har,k,her,k , as well as hpu,k,hua,k are assumed to undergo
independent and identically distributed (i.i.d.) Nakagami-m
small-scale fading with fading severity parameter mwand
average channel gain Ωw(w∈ {pr, ur, ar, er, pu, ua, ue}),
i.e., hw∼Nakagami (mw,Ωw)[5], [37].
B. Proposed Schemes
During the EH phase τ T , PB transfers energy directly to
Ukin mode I, while PB (or HAP) transfers energy to Ukvia
Rkin mode II (or mode III). The harvested energy by Ukin
each mode can be respectively expressed as
EI
k=ητ T Ps|hpu,k|2,(1)
EII
k=ητ T Ps|hur,kΘkhpr,k +hpu,k |2,(2)
EII I
k=ητ T Ps|hur,kΘkhar,k +hua,k |2,(3)
where ηand Psdenote the energy conversion efficiency and
the transmitting power of HAP (or PB), respectively. Θk
∆
=
diag ejφk,1, ej φk,2, ..., ejφk,N is the reflection-coefficient ma-
trix of Rk.{φk,n}N
n=1 denotes the phase shift of the nth
reflecting element of Rk.
During the IT phase (1 −τ)T, Ukexploits the harvested
energy to transmit confidential messages to AP (or HAP) via
Rkin mode I (or mode III), while Uktransmits confidential
messages directly to AP in mode II. The transmitting power
of Ukcan be respectively given by
PI
k=EI
k
(1 −τ)T=Pt|hpu,k|2,(4)
PII
k=EII
k
(1 −τ)T=Pt|hur,kΘkhpr,k +hpu,k |2,(5)
PII I
k=EIII
k
(1 −τ)T=Pt|hur,kΘkhar,k +hua,k |2,(6)
where Pt=ητ
1−τPs. According to the uplink NOMA
protocol, HAP (or AP) first decodes the signal of the
UAV with better channel conditions and performs SIC
to subtract it. Without loss of generality, assume that
|har,1Θ1hur,1+hua,1|2≥ |har,2Θ2hur,2+hua,2|2≥. . . ≥
|har,K ΘKhur,K +hua,K|2. The SINRs of UAV Ukunder
three schemes are obtained as
γI
k=|har,kΘkhur,k +hua,k |2PI
k
PK
i=k+1 |har,iΘihur,i +hua,i|2PI
i+σ2,(7)
γII
k=|hua,k|2PI I
k
PK
i=k+1 |hua,i|2PI I
i+σ2,(8)
γIII
k=|har,kΘkhur,k +hua,k |2PIII
k
PK
i=k+1 |har,kΘkhur,k +hua,k |2PIII
i+σ2,(9)
where σ2denotes the variance of additive white Gaussian
noise (AWGN) at receivers. As a special case, when k=K,
γI
K=|har,K ΘKhur,K +hua,K|2PI
K
σ2,(10)
γII
K=|hua,K |2PII
K
σ2,(11)
γIII
K=|har,K ΘKhur,K +hua,K|2PI II
K
σ2.(12)
The sum achievable data rate of UAVs can be formulated as
Rsum = (1 −τ)
K
P
k=1
log2(1 + γk). After some mathematical
manipulations, the sum achievable data rate under three modes
can be formulated as
RI
sum = (1 −τ) log2 1 + ρt
K
X
k=1
|hpu,k|2|har,k Θkhur,k +hua,k|2!,
(13)
RII
sum = (1 −τ) log2 1 + ρt
K
X
k=1
|hur,kΘkhpr,k +hpu,k |2|hua,k|2!,
(14)
RII I
sum = (1 −τ)×
log2 1 + ρt
K
X
k=1
|hur,kΘkhar,k +hua,k |2|har,kΘkhur,k +hua,k|2!,
(15)
where ρt=Pt
σ2. We consider the worst-case scenario where E
knows the decoding order so that it can perform SIC to decode
the target signal similar to legitimate users [38]. As such, the
sum eavesdropping rate under three modes can be written as
5
RI
e= (1 −τ) log2 1 + ρt
K
X
k=1
|hpu,k|2|hur,k Θkher,k +hue,k|2!,
(16)
RII
e= (1 −τ)×
log2 1 + ρt
K
X
k=1
|hur,kΘkhpr,k +hpu,k |2|hur,kΘkher,k +hue,k|2!,
(17)
RII I
e= (1 −τ)×
log2 1 + ρt
K
X
k=1
|hur,kΘkhar,k +hua,k |2|hur,kΘkher,k +hue,k|2!.
(18)
C. Phase Shift Design
The phase shift of RIS should be matched with the phases
of legitimate channels to maximize the harvested energy and
information transmission in three modes. Taking mode I as
an example, the sum achievable data rate can be improved by
maximizing |har,kΘkhur,k +hua,k |2. Ideally, optimal phase
φopt
k,n should be set as φopt
k,n = arg hn
ua,k−arg hn
ur,khn
ar,k.
However, it is worth noting that RIS phase compensation is
typically imperfect in UAV-RIS communications due to UAV
jitter, air turbulence effects, and so on. The phase compensa-
tion error φerr
ncan be modeled as von Mises distribution with
zero mean and concentration parameter κ[35]. κis inversely
proportional to the phase compensation error. When the UAV
experiences significant jitter caused by airflow effects, the
phase compensation error increases and κdecreases. The PDF
and eigenfunction of φerr
nare respectively expressed as
fφerr
n(x) = eκcos(x)
2πI0(κ),(19)
ϕp=Eejpφerr
n=Ip(κ)
I0(κ),(20)
where I0(·)and Ip(·)denote the modified Bessel function of
the first kind with order zero and p.
III. PERFORMANCE ANALYSIS FOR THREE MODES
In this section, we analyze the reliable and secure perfor-
mance of the proposed three modes under phase compensation
errors. 1) To holistically characterize the reliable performance
of the system, the closed-form expressions of ergodic capacity
(EC) under three modes are derived. Furthermore, we define
and evaluate asymptotic ergodic capacity probability (ECP) to
further investigate the asymptotic performance of the system.
2) To evaluate the secrecy performance of the system, we
derive the closed-form expressions of ergodic secrecy capacity
(ESC) under three modes. The asymptotic expressions are also
obtained to gain further insights.
Before starting the investigation of the system performance,
new channel statistics for the legitimate and eavesdropping
channels under mode I are derived as examples to provide a
theoretical basis for the subsequent performance evaluation.
A. New Channel Statistics
Under phase compensation errors, HI=
K
P
k=1 |hpu,k|2|har,k Θkhur,k +hua,k|2and He
I=
K
P
k=1 |hpu,k|2|hur,k Θkher,k +hue,k|2can be rewritten as
HI=
K
X
k=1 |hpu,k|2
N
X
n=1 hn
ur,khn
ar,kejφerr
n+|hua,k|
2
,
(21)
He
I=
K
X
k=1 |hpu,k|2
N
X
n=1 hn
ur,khn
er,kejφerr
n+|hue,k|
2
.(22)
In order to evaluate the performance of three modes, the
statistical distribution of HIand He
Ineed to be obtained first,
which are given in the following lemmas.
Lemma 1. The probability density function (PDF) and cumu-
lative density function (CDF) of HIare expressed as
fHI(x) = xKmI−1
Γ (KmI) (βI)K mIexp −x
βI,(23)
FHI(x) = 1
Γ (KmI)γK mI,x
βI,(24)
where γ(·,·)is lower incomplete gamma function,
mI=(EG1)2
E|G1|2−(EG1)2,βI=E|G1|2−(EG1)2
EG1,
EG1= ΩpuEg1,k ,E|G1|2=Γ(mpu+2)
Γ(mpu)(mpu /Ωpu)2E|g1,k |2,
Eg1,k =f1µw1, σ2
w1, σ2
v1,Ωua, mua and
E|g1,k|2=f2µw1, σ 2
w1, σ2
v1,Ωua, mua ,µw1=Nξ2
1ϕ1,
σ2
w1=N
2(1 + ϕ2−2ξ4
1ϕ12),µv1= 0,σ2
v1=N
2(1 −ϕ2),
ξ1=pE(|hur,k|)E(|har,k |),E(|hur,k|) = Γ(mur+0.5)
Γ(mur)√mur /Ωur
,
E(|har,k|) = Γ(mar+0.5)
Γ(mar)√mar /Ωar
,f1(·)and f2(·)are expressed
as (53) and (54) in Appendix A.
Proof. See Appendix A.
Lemma 2. The PDF and CDF of He
Iare given by
fHe
I(x) = xKme
I−1
Γ (Kme
I) (βe
I)Kme
I
exp −x
βe
I,(25)
FHe
I(x) = 1
Γ (Kme
I)γKme
I,x
βe
I,(26)
where me
I=(EGe1)2
E|Ge1|2−(EGe1)2,βe
I=E|Ge1|2−(EGe1)2
EGe1,
EGe1= Ωpu (NΩurΩer + Ωue )and E|Ge1|2=
2Γ(mpu+2)
Γ(mpu)(mpu /Ωpu)2(NΩur Ωer + Ωue)2.
Proof. See Appendix B.
B. Ergodic Capacity
In this subsection, we analyze the EC of the system under
three modes, which indicates the average achievable rate of
the system.
According to the definition, the EC under mode i(i∈
{I, II , II I}) can be denoted as
Ri
EC =ERi
sum =E[(1 −τ) log2(1 + ρtHi)] ,(27)
6
where i∈ {I, II , II I}. The closed-form expressions of EC
under three modes are given as follows.
Theorem 1. The EC under mode I is expressed as
RI
EC =(1 −τ)
ln 2 Λ (βI) +
KmI−1
X
n=1
(−1)nΛ (βI) + Ξ (βI)
n!(ρtβI)n!,
(28)
where Λ (x) = exp 1
ρtx−Ei −1
ρtx,Ξ (x) =
n
P
l=1
Cl
n(−1)n−lexp 1
ρtx 1
ρtx−lΓl, 1
ρtx,xrounds the
closet integer of x,Γ (·,·)is upper incomplete gamma func-
tion, mIand βIare given by Lemma 1.
Proof. See Appendix C.
Theorem 2. The EC under mode II is expressed as
RII
EC =(1 −τ)
ln 2 Λ (βII ) +
KmI I −1
X
n=1
(−1)nΛ (βII ) + Ξ (βI I )
n!(ρtβII )n!,
(29)
where mII =(EG2)2
E|G2|2−(EG2)2,βII =E|G2|2−(EG2)2
EG2,
EG2= ΩuaEg2,k ,E|G2|2=Γ(mua+2)
Γ(mua)(mua /Ωua)2E|g2,k |2,
Eg2,k =f1µw2, σ2
w2, σ2
v2,Ωpu, mpu ,E|g2,k|2=
f2µw2, σ2
w2, σ2
v2,Ωpu, mpu ,µw2=Nξ2
2ϕ1,
σ2
w2=N
2(1 + ϕ2−2ξ4
2ϕ12),µv2= 0,σ2
v2=N
2(1 −ϕ2),
ξ2=pE(|hur,k|)E(|hpr,k |),E(|hpr,k|) = Γ(mpr+0.5)
Γ(mpr)√mpr /Ωpr
.
Proof. Under phase compensation errors, we
denote g2,k =|hur,kΘkhpr,k +hpu,k |2=
N
P
n=1 hn
ur,khn
pr,kejφer r
n+|hpu,k|
2
. The first and
second moments of g2,k can be formulated as
Eg2,k =f1µw2, σ2
w2, σ2
v2,Ωpu, mpu ,E|g2,k|2=
f2µw2, σ2
w2, σ2
v2,Ωpu, mpu . The distribution of
HII =
K
P
k=1 |hua,k|2g2,k obeys HI I ∼Γ (KmII , βI I ).
Following similar proof steps in Appendix C, the EC under
mode II can be readily obtained. We skip the proof for
brevity.
Theorem 3. The EC under mode III is given by
RII I
EC =(1 −τ)
ln 2 Λ (βII I ) +
KmIII −1
X
n=1
(−1)nΛ (βII I ) + Ξ (βII I )
n!(ρtβII I )n!,
(30)
where mII I =(EG3)2
E|G3|2−(EG3)2,βII I =E|G3|2−(EG3)2
EG3,EG3=
(Eg1,k)2,E|G3|2=E|g1,k |22
.
Proof. Under phase compensation errors,
|hur,kΘkhar,k +hua,k |2and |har,kΘkhur,k +hua,k|2
are independent and identically dis-
tributed. The distribution of HIII =
K
P
k=1 |hur,kΘkhar,k +hua,k |2|har,kΘkhur,k +hua,k|2obeys
HIII ∼Γ (KmIII , βII I ). Following similar proof steps in
Appendix C, the EC under mode III can be readily obtained.
We skip the proof for brevity.
C. Ergodic Capacity Probability
To characterize the asymptotic performance of ergodic ca-
pacity, we define a new evaluation metric ECP in this sub-
section. It is denoted as the probability that the transmission
capacity is below a set threshold.
According to the definition, ECP can be expressed as
Pi
EC P = Pr {(1 −τ) log2(1 + ρtHi)< Rth},(31)
where Rth denotes a set threshold. We characterize the asymp-
totic ECP in a high signal-to-noise ratio (SNR) region as
shown in the following proposition.
Proposition 1. The asymptotic expression of ECP under three
modes can be expressed as
PI
EC P asy = Γ (|mpu −mg1|) Γ (m∗
1)vm∗
1
1
Γ (mpu) Γ (mg1)!Kξm∗
1K
Γ (m∗
1K+ 1),
(32)
PII
EC P asy = Γ (|mua −mg2|) Γ (m∗
2)vm∗
2
2
Γ (mua) Γ (mg2)!Kξm∗
2K
Γ (m∗
2K+ 1),
(33)
PII I
EC P asy =vmg1
1
Γ (mg1)Kξmg1K
Γ (mg1K+ 1),(34)
where ξ=2
Rth
(1−τ)−1
ρt,v1=mpu
Ωpuβg1
,v2=mua
Ωuaβg2
,
mg1=(Eg1,k)2
E|g1,k|2−(Eg1,k )2,mg2=(Eg2,k )2
E|g2,k|2−(Eg2,k )2,
βg1=E|g1,k|2−(Eg1,k )2
Eg1,k ,βg2=E|g2,k|2−(Eg2,k )2
Eg2,k ,m∗
1=
min {mpu, mg1}and m∗
2= min {mua, mg2}.
Proof. See Appendix D.
Remark 1. According to the above proposition, we can
formulate the diversity orders under three modes as GI=
Kmin {mpu, mg1},GI I =Kmin {mua, mg2},GI II =
Kmg1, which are determined by the number of UAVs and
quality of the channels.
D. Ergodic Secrecy Capacity
The ESC indicates the statistical average achievable secrecy
rate of the system, which is defined as
Ri
sec =EhRi
sum −Ri
e+i,(35)
where [x]+= max {x, 0}and i∈ {I , II, II I}.
1) Exact Expression: We first analyze the closed-form
expressions of ESC under three modes. The ESC in (35) can
be rewritten as
Ri
sec =EhRi
sum −Ri
e+i
(a)
≥ERi
sum −Ri
e+
=ERi
sum −ERi
e+,(36)
where inequality (a)is obtained by exploiting Jensen’s in-
equality, i.e., E[max{x, y}]≥max {E[x],E[y]}.
7
Theorem 4. The ESC under mode I can be expressed as
RI
sec
=(1 −τ)
ln 2 (Λ (βI)−Λ (βe
I))
+(1 −τ)
ln 2
KmI−1
X
n=1
1
n!(ρtβI)n((−1)nΛ (βI) + Ξ (βI))
−(1 −τ)
ln 2
Kme
I−1
X
n=1
1
n!(ρtβe
I)n((−1)nΛ (βe
I) + Ξ (βe
I))
+
,
(37)
Proof. Based on Lemma 1 and Lemma 2, the distribution
of HIand He
Iare formulated as HI∼Γ (KmI, βI)and
He
I∼Γ (Kme
I, βe
I). The closed-form expression of ERI
sum
is obtained as shown in Theorem 1. By following (59) in
Appendix C, the closed-form expression of ERI
eis readily
obtained as
ERI
e=(1 −τ)
ln 2
Λ (βe
I) +
Kme
I−1
X
n=1
(−1)nΛ (βe
I) + Ξ (βe
I)
n!(ρtβe
I)n
.
(38)
By substituting (28) and (38) into (36), Theorem 4 is proved
straightforwardly.
Theorem 5. The ESC under mode II can be expressed as
RII
sec
=(1 −τ)
ln 2 (Λ (βI I )−Λ (βe
II ))
+(1 −τ)
ln 2
KmI I −1
X
n=1
1
n!(ρtβII )n((−1)nΛ (βI I ) + Ξ (βII ))
−(1 −τ)
ln 2
Kme
II −1
X
n=1
1
n!(ρtβe
II )n((−1)nΛ (βe
II ) + Ξ (βe
II ))
+
,
(39)
where me
II =(EGe2)2
E|Ge2|2−(EGe2)2,βe
II =E|Ge2|2−(EGe2)2
EGe2,
EGe2=Eg2,k (NΩurΩer + Ωue ),E|Ge2|2=
2E|g2,k|2(NΩur Ωer + Ωue )2.
Proof. By denoting He
II =
K
P
k=1
g2,k|hur,k Θkher,k +hue,k|2,
ERII
ecan be computed as ERII
e= (1 −τ)×
log2(1 + ρtHe
II ). The distribution of He
II obeys
He
II ∼Γ (K me
II , β e
II ). Following similar steps as in
the proof of Theorem 4, RIII
sec in (39) can be obtained. We
skip the proof for brevity.
Theorem 6. The ESC under mode III can be expressed as
RII I
sec
=(1 −τ)
ln 2 (Λ (βI II )−Λ (βe
II I ))
+(1 −τ)
ln 2
KmIII −1
X
n=1
1
n!(ρtβIII)n((−1)nΛ (βI II ) + Ξ (βII I ))
−(1 −τ)
ln 2
Kme
III−1
X
n=1
1
n!(ρtβe
II I )n((−1)nΛ (βe
II I ) + Ξ (βe
III))
+
,
(40)
where me
II I =(EGe3)2
E|Ge3|2−(EGe3)2,βe
III =E|Ge3|2−(EGe3)2
EGe3,
EGe3=Eg1,k (NΩurΩer + Ωue ),EGe3=
2Eg1,k(NΩur Ωer + Ωue )2.
Proof. Following similar steps as in the proof of Theorem 4,
the proof of Theorem 6 can be achieved. We skip the proof
for brevity.
2) Asymptotic Expression: To obtain deep insights, we
further evaluate the asymptotic ESC of the system in a high
SNR region, i.e., Ps→ ∞.
Proposition 2. The asymptotic ESC under mode i(i∈
{I, II , II I}) can be expressed as
Ri
asy =1−τ
ln 2 {ψ(Kmi)−ψ(Kme
i) + ln (βi)−ln (βe
i)}+,
(41)
where ψ(·)denotes Euler psi function [39].
Proof. See Appendix E.
Remark 2. It can be observed from Proposition 2 that
the asymptotic ESC of three modes is independent of the
transmitting power Ps, which indicates that the ESC of each
mode converges to a performance floor with the increase of Ps.
Pscan be moderately designed to achieve secure and energy-
efficient communications.
Proposition 3. When Kis large, the asymptotic ESC can be
further expressed as
RI
asy =RIII
asy =1−τ
ln 2 ln Eg1,k
NΩurΩer + Ωue ,(42)
RII
asy =1−τ
ln 2 ln Ωua
NΩurΩer + Ωue .(43)
Proof. We have ψ(v)→ln vwhen v→ ∞ [39]. By
substituting miβi,me
i, and βe
iinto (41), and after some
mathematical manipulations, asymptotic ESC can be further
formulated as (42) and (43). The proof is completed.
Remark 3. Utilizing Proposition 3, one can observe that the
asymptotic ESC is independent of the number of UAV K
with the increase of K. Asymptotic ESCs of modes I and
III converge under large Kand Ps. According to (53), since
Eg1,k is an increasing function of N, increasing Nleads to
secrecy improvement for mode I and mode III. However, RII
asy
is deteriorated conversely with the increase of Nbased on
(43). The reason is that the increase of Nwill inevitably result
8
-50 -40 -30 -20 -10 0
Ps(dBm)
0
1
2
3
4
5
6
7
8
EC
Mode I
Mode II
Mode III
No RIS
Simulation
-21.8 -21.4
3.48
3.5
3.52
3.54
3.56
3.58
Fig. 2. The EC versus Psunder different modes.
in a larger wiretap channel capacity during the IT phase, while
the legitimate channel capacity will not be affected for mode
II.
IV. SIMULATION RESULTS AN D DISCUSSION
In this section, numerical results are presented to validate
the analysis and offer some useful insights for three modes.
Without loss of generality, the simulation parameters are set as
Ps=−30 dBm, N= 20,K= 3,η= 1,Rth = 5,σ2=−60
dBm, κ= 60,τ= 0.5,dpr =dur =dar = 5,dpu =dua =
10,der =due = 20,mpr =mur =mar =mer =mpu =
mua = 3,Ωw=d−2
w(w∈ {pr, ur, ar, er, pu, ua, ue}).
The Monto Carlo simulations are conducted over 105channel
realizations.
In Fig. 2, we investigate the EC versus Psunder the
proposed three modes. The derived expressions of EC match
well with the simulation results and the correctness of the
theoretical derivation is verified. The traditional UAV wireless-
powered mode without the aid of RIS (‘No RIS’) is considered
a benchmark for comparison. All the proposed three modes
outperform the traditional ‘No-RIS’ scheme, demonstrating
the benefits of potential passive beamforming gain of RIS in
improving the reliability of wireless-powered communications.
Mode III outperforms modes I and II due to the simultaneous
enhancement of information reception and energy transfer.
Moreover, as can noticed, phase compensation errors nega-
tively affect the EC. When κ≥60, the influence of phase
compensation errors becomes negligible and can be regarded
as a perfect phase shift. As a result, it is recommended to
design the phase shift such that κ≥60 is satified to achieve
the desired performance.
Fig. 3 characterizes the asymptotic reliable performance of
the system under the three modes. As shown in the figure,
the derived asymptotic expressions of ECP match well with
the corresponding simulation results, verifying the correctness
of theoretical derivation. Mode I and mode II achieve similar
reliability performance. When Pstends to infinity, the ECP
of mode III is significantly lower than that of mode I and
-10 -5 0 5 10
Ps(dBm)
10-10
10-8
10-6
10-4
10-2
100
ECP
Mode I
Mode II
Mode III
No RIS
Simulation
Asymptotic
Fig. 3. The ECP versus Psunder three modes, where N= 5.
-50 -40 -30 -20 -10 0 10
Ps(dBm)
0
0.5
1
1.5
2
2.5
3
3.5
4
ESC
Mode I
Mode II
Mode III
No RIS
Simulation
Asymptotic
Fig. 4. The ESC versus Psunder different modes.
mode II, which validates the advantages of exploiting RIS to
facilitate both energy transfer and information transfer.
Fig. 4 highlights the significance of RIS deployment and
the effectiveness of the proposed modes. It is observed that
the derived accurate and asymptotic expressions of ESC match
well with the simulation results, which verifies the correctness
of theoretical derivation. We observe that the ESC increases
with Ps, but the improvement tends to saturate in the high SNR
region. It can be interpreted that increasing Pspromotes the
quality of the legitimate channel while expanding the capacity
of the eavesdropping channel, which leads to a performance
floor. The performance of mode III outperforms mode I in the
low SNR region while they achieve the same ESC floor in
the high SNR region. This phenomenon is confirmed by the
insights given in Remark 3. Moreover, it is found that mode
II can achieve a higher ESC than ‘No RIS’ mode at low Ps,
while the opposite is true at high Ps. This observation suggests
that it would be advisable to deploy RISs in mode III for better
secrecy performance.
Fig. 5 plots the ESC versus Nunder different modes.
9
10 20 30 40 50 60
N
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
ESC
Mode I
Mode II
Mode III
No RIS
Solid lines: K=3
Dashed lines: K=10
Fig. 5. The ESC versus Nunder different modes.
-50 -40 -30 -20 -10 0 10
Ps (dBm)
0.5
1
1.5
2
2.5
3
3.5
4
ESC
Mode I, dpu=10
Mode I, dpu=5
Mode II, dpu=10
Mode II, dpu=5
Mode III, dua=10
Mode III, dua=5
Simulation
Fig. 6. The ESC versus Psfor different distances.
As expected, the ESCs of modes I and III improve with
increasing N. However, the secrecy performance of mode II
degrades with an increase in N. This is attributed to the fact
that an increase in reflecting elements under mode II results
in increased information leakage to eavesdroppers, while the
legitimate channels remain unimproved during the IT phase.
In addition, it is found that increasing the number of UAVs,
K, has a positive impact on secrecy performance under mode
I. For modes II and III, an increase in Kpromotes secrecy
performance when Nis small, while the opposite is true when
Nis large.
Fig. 6 shows the ESC versus Psfor different distances.
Note that the ESCs under the three modes increase with Ps
and converge to a stable value. This is consistent with Fig. 4.
Moreover, it is observed that the asymptotic value of modes
I and II is independent of dpu, which can be verified by (42)
and (43), respectively. However, the asymptotic value of mode
III is dependent on dua. The reason for this is that, as shown
in (42), the asymptotic value of mode III depends on Eg1,k,
which is a variable about dua. A reduction in dua can markedly
2345678
K
0
0.5
1
1.5
2
2.5
3
3.5
4
ESC
Mode I
Mode II
Mode III
Solid lines:Ps=-10dBm
Dashed lines: Ps=-50dBm
Fig. 7. The ESC versus Kfor different Ps.
-50 -40 -30 -20 -10 0 10
Ps(dBm)
0
0.5
1
1.5
2
2.5
3
3.5
4
ESC
=60
=5
=0
Mode III
Mode I
Mode II
Fig. 8. The ESC versus Psfor different concentration parameter κ.
improve the ESC for mode III. This implies that when Psis
sufficiently large, it is preferable to deploy UAVs as close as
possible to HAP (or AP), regardless of the distance between
the UAVs and the PB.
Fig. 7 illustrates the impact of Kon the ESC. We observe
that an increase of Kenhances the secrecy performance for
small Ps. Nevertheless, the secrecy performance deteriorates
under the condition of large Ps. It is also observed that the
ESCs of modes I and III gradually converge as Kincreases,
verifying the correctness of Proposition 3.
In Fig. 8, we evaluate the impact of the concentration
parameter κon the ESC. As κdecreases, indicating an increase
in the phase compensation error, we observe a deterioration
in the secrecy performance of the three modes. Regardless of
the value of κ, we observe that the asymptotic ESC values
of modes I and III converge. The asymptotic ESC achieved
by perfect phase shifts (i.e., κ→ ∞) is about 2.3 times the
ESC achieved by random phase shifts (i.e., κ= 0) for modes
I and III. However, it is worth noting that the effect of phase
compensation error is negligible for mode II.
10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.5
1
1.5
2
2.5
3
3.5
4
ESC
Mode I
Mode II
Mode III
Solid lines: K=3
Dashed lines: K=4
Fig. 9. The ESC versus τfor different K, where N= 10.
Fig. 9 shows the relationship between ESC and the time-
splitting factor τ. As can be observed from the figure, the ESCs
under three modes are significantly enhanced with an increase
in τ, but are compromised as τcontinues to increase. This
is due to the fact that an increase in τyields greater energy
collected by UAVs at the expense of a shorter information
transmission time. The simulation results demonstrate that τ
can be optimized to enhance the secrecy performance of UAV
wireless-powered communications. It is noteworthy that the
optimal τincreases with an increment of the number of UAVs
K. Considering the above, it is of salient significance to select
appropriate τto maximize ESC for different modes.
V. CONCLUSION
In this paper, three UAV wireless-powered NOMA transmis-
sion modes are proposed based on different RIS deployment
strategies. Taking into account practical Nakagmi-mchannels
and phase compensation errors caused by imperfect conditions
such as UAV jitter, we conducted a comprehensive investi-
gation of the system’s reliability and security. Specifically,
we define and evaluate the system’s EC, ECP and ESC.
The simulation results validated the correctness of theoretical
derivation and demonstrated the superiority of the proposed
three modes over traditional modes without the aid of RIS.
From a reliability perspective, mode III outperforms other
modes. From a security perspective, mode III is superior
to other modes when Psis low. When Psis large, modes
I and III achieve similar secrecy performance. The secrecy
performance of mode II deteriorates with increasing Ps. While
phase compensation errors significantly deteriorate the secrecy
performance of modes I and III, their effect on mode II is
negligible.
APPENDIX A
PROO F OF LEMMA 1
By denoting g1,k =
N
P
n=1 hn
ur,khn
ar,kejφer r
n+|hua,k|
2
,
we have HI=
K
P
k=1
G1=
K
P
k=1 |hpu,k|2g1,k .
First, we derive the distribution of g1,k. Let us
define
N
P
n=1 hn
ur,khn
ar,kejφer r
n=|Wn+jVn|,
we have Wn=
N
P
n=1 hn
ur,khn
ar,kcos φerr
nand
Vn=
N
P
n=1 hn
ur,khn
ar,ksin φerr
n. For large N,Wnand
Vncan be approximated as independent RVs, where
Wn∼ N µw1, σ2
w1and Vn∼ N µv1, σ2
v1[40]. The
expectations and variances of Wnand Vncan be respectively
expressed as
µw1=E(Wn) = NEhn
ur,khn
ar,kcos φerr
n=Nξ2
1ϕ1,
(44)
σ2
w1=V(Wn) =NEhn
ur,khn
ar,kcos φerr
n2
−NE2hn
ur,khn
ar,kcos φerr
n
=N
2(1 + ϕ2−2ξ4
1ϕ12),(45)
µv1=E(Vn) = NEhn
ur,khn
ar,ksin φerr
n= 0,(46)
σ2
v1=V(Vn) =NEhn
ur,khn
ar,ksin φerr
n2
−NE2hn
ur,khn
ar,ksin φerr
n
=N
2(1 −ϕ2),(47)
where ξ1=pE(|hur,k|)E(|har,k |),E(|hur,k|) =
Γ(mur+0.5)
Γ(mur)√mur /Ωur
and E(|har,k|) = Γ(mar+0.5)
Γ(mar)√mar /Ωar
.
According to [41], the pth moment of Vn∼ N 0, σ2
v1is
given by
EVnp=(p!
(p/2)! σ2
v1
√2p,for an even p,
0,for an odd p. (48)
Wncan be regarded as Wn=W0
n+µ1, where W0
n∼
N0, σ2
w1. The pth moment of Wnis given by
EWnp=EW0
n+µ1p=
p
X
i=0
Ci
pEW0
n
iµp−i
1,(49)
where EW0
n
ican be obtained by (48). The pth moment of
Nakagami RV |hua,k|is given by [42]
E|hua,k|p=Γmua +p
2
Γ (mua) (mua /Ωua)p
2
.(50)
Owing to g1,k =
N
P
n=1 hn
ur,khn
ar,kejφer r
n+|hua,k|
2
=
(Wn+|hua,k|)2+Vn2, the first and second moments of g1,k
can be expressed as
Eg1,k =EWn2+E|hua,k|2+EVn2+ 2EWnE|hua,k |,(51)
E|g1,k|2=EWn4+ 4EWn3E|hua,k |+ 6EWn2E|hua,k |2
+4EWnE|hua,k|3+E|hua,k |4+EVn4+ 2EVn2EWn2
+4EVn2E|hua,k|EWn+ 2EVn2E|hua,k |2.(52)
11
By substituting Eq.(48), Eq.(49), and Eq.(50) into Eq. (51) and
Eq. (52), the first and second moments of g1,k can be further
calculated as
Eg1,k =f1µw1, σ2
w1, σ2
v1,Ωua, mua
=µ2
w1+σ2
w1+ Ωua +σ2
v1+ 2µw1
Γ(mua + 0.5)
Γ(mua)pmua /Ωua
,
(53)
E|g1,k|2=f2µw1, σ 2
w1, σ2
v1,Ωua, mua
= 4µw1
Γ(mua + 0.5)
Γ(mua)pmua /Ωua σ2
v1+ 3σ2
w1+µ2
w1+µ4
w1
+ 6σ2
w1µ2
w1+ 3 σ4
w1+σ4
v1+6Ωua + 2σ2
v1σ2
w1+µ2
w1
+ 4µw1
Γ(mua + 1.5)
Γ(mua)(mua /Ωua)1.5+ 2σ2
v1Ωua.(54)
Next, we derive the distribution of G1=|hpu,k|2g1,k.
Owing to |hpu,k| ∼ Nakagami (mpu,Ωpu ),|hpu,k|2∼
Γ (mpu,Ωpu /mpu)holds. The first and second moments of
G1are given by
EG1=E|hpu,k|2Eg1,k = Ωpu Eg1,k,(55)
E|G1|2=E|hpu,k|4E|g1,k |2=Γ (mpu + 2)
Γ (mpu) (mpu /Ωpu)2E|g1,k |2.
(56)
By exploiting moment-matching approach [42], we have
G1∼Γ (mI, βI), where mI=(EG1)2
E|G1|2−(EG1)2and βI=
E|G1|2−(EG1)2
EG1. Based on the additivity of Gamma function,
the distribution of HI=
K
P
k=1
G1can be denoted as HI∼
Γ (KmI, βI). The proof of Lemma 1 is completed.
APPENDIX B
PROO F OF LEMMA 2
Due to the fact that phase shifts are designed according to
the legitimate channel,
N
P
n=1 hn
ur,khn
er,kejφer r
nis the sum of
complex-valued random variables, and its real and imaginary
parts follow independent Gaussian distributions. Moreover,
the direct channel hue,k between Ukand E is modeled
by Rayleigh fading owing to terrestrial obstacles, i.e.,
mue = 1. Hence,
N
P
n=1 hn
ur,khn
er,kejφer r
n+|hue,k|
2
follows an exponential distribution with parameter
NΩurΩer + Ωue . The first and second moments of
N
P
n=1 hn
ur,khn
er,kejφer r
n+|hue,k|
2
are NΩurΩer + Ωue
and 2(NΩurΩer + Ωue )2. Therefore, the distribution of
He
I=
K
P
k=1 |hpu,k|2
N
P
n=1 hn
ur,khn
er,kejφer r
n+|hue,k|
2
can be calculated as He
I∼Γ (Kme
I, βe
I). Based on the
aforementioned results, we prove Lemma 2.
APPENDIX C
PROO F OF TH EO RE M 1
The closed-form expression of RI
EC under mode I can be
expressed as follows
RI
EC =E[(1 −τ) log2(1 + ρtHI)]
=(1 −τ)
ln 2 Z∞
0
1−FHIx
ρt
1 + xdx. (57)
Furthermore, the lower incomplete Gamma function can be
developed as Eq.(8.352.4) in [39].
FHI(x)=1−exp −x
βIKmI−1
X
n=0
xn
n!(βI)n,(58)
Then, by substituting (58) into (57), we have
RI
EC
=(1 −τ)
ln 2
KmI−1
X
n=0
1
n!(ρtβI)nZ∞
0
exp −x
ρtβIxn
1 + xdx
(b)
=(1 −τ)
ln 2 Z∞
0
exp −x
ρtβI
1 + xdx
| {z }
I1
+(1 −τ)
ln 2
KmI−1
X
n=1
1
n!(ρtβI)n
× n
X
l=1
Cl
n(−1)n−lZ∞
1
exp −t−1
ρtβItl−1dt
| {z }
I2
+ (−1)nZ∞
1
exp −t−1
ρtβI
tdt
| {z }
I3
,(59)
where step (b)is obtained by making variable substitution
t=x+ 1. With the aid of Eq.(3.352.4), Eq.(3.381.3) and
Eq.(3.352.2) in [39], I1,I2and I3can be rewritten as
I1=−exp 1
ρtβIEi −1
ρtβI,(60)
I2=
n
X
l=1
Cl
n(−1)n−lexp 1
ρtβI 1
ρtβI−l
Γl, 1
ρtβI,
(61)
I3= (−1)nexp 1
ρtβI−Ei −1
ρtβI.(62)
By substituting I1,I2and I3into (59), the closed-form
expression of RI
EC is obtained. The proof is completed.
12
APPENDIX D
PROO F OF PROPOSITION 1
The asymptotic expression of ECP under mode ican be
calculated as
Pi
EC P = Pr {(1 −τ) log2(1 + ρtHi)< Rth}.
= Pr (Hi<2
Rth
(1−τ)−1
ρt)
=F0+
Hi(ξ),(63)
where ξ=2
Rth
(1−τ)−1
ρt. First, we derive the asymptotic ECP
under mode I. According to Appendix A, we have |hpu,k|2∼
Γ (mpu,Ωpu /mpu)and g1,k =|hur,k Θkhar,k +hua,k|2∼
Γ (mg1, βg1). Based on [43], the PDF of G1=|hpu,k|2g1,k
can be given by
fG1(x) = 2
Γ (mpu) Γ (mg1)x1
2(mpu+mg1)−1v
1
2(mpu+mg1)
1
×Kmpu−mg1(2√xv1),(64)
where v1=mpu
Ωpuβg1
and Kv(·)is the modified Bessel function
of the second kind. According to Eq. (8.432.6) in [39] and
Kv(x) = K−v(x), we have
lim
x→0+Kmpu−mg1(2√xv1)≈1
2Γ (|mpu −mg1|) (xv1)−|mpu−mg1|
2.
(65)
When x→0+,fG1(x)can be rewritten as
fG1(x) = Γ (|mpu −mg1|)
Γ (mpu) Γ (mg1)vm∗
1
1xm∗
1−1,(66)
where m∗
1= min {mpu, mg1}. The Laplace transform of
fG1(x)is derived as
L∞
fG1(s) = Γ (|mpu −mg1|)
Γ (mpu) Γ (mg1)vm∗
1
1
∞
Z
0
xm∗
1−1e−sxdx
(c)
=Γ (|mpu −mg1|)
Γ (mpu) Γ (mg1)vm∗
1
1s−m∗
1Γ (m∗
1).(67)
With the help of Eq. (3.381.4) in [39] ∞
R
0
xv−1e−µxdx =
µ−vΓ (v), step (c)is obtained. Owing to HI=
K
P
k=1
G1, the
Laplace transform of fHI(x)can be further derived as
L∞
fHI(s) =
K
Y
k=1
L∞
fG1(s)
= Γ (|mpu −mg1|) Γ (m∗
1)vm∗
1
1
Γ (mpu) Γ (mg1)!K
vm∗
1K
1s−m∗
1K.
(68)
Based on Eq. (17.13.3) in [39], the PDF of HIfor x→0+
can be denoted as
f0+
HI(x) = Γ (|mpu −mg1|) Γ (m∗
1)vm∗
1
1
Γ (mpu) Γ (mg1)!Kxm∗
1K−1
Γ (m∗
1K).
(69)
By integrating (69), the CDF of HIfor x→0+can be
obtained as
F0+
HI(x) = Γ (|mpu −mg1|) Γ (m∗
1)vm∗
1
1
Γ (mpu) Γ (mg1)!Kxm∗
1K
Γ (m∗
1K+ 1).
(70)
Following similar derivation steps, F0+
HII can be derived as
F0+
HII (x) = Γ (|mua −mg2|) Γ (m∗
2)vm∗
2
2
Γ (mua) Γ (mg2)!Kxm∗
2K
Γ (m∗
2K+ 1),
(71)
where m∗
2= min {mua, mg2}. Under mode III, owing to
m∗
3= min {mg1, mg1}=mg1,F0+
HII I can be calculated as
F0+
HIII (x) = vmg1
1
Γ (mg1)!Kxmg1K
Γ (mg1K+ 1).(72)
By substituting (70), (71), and (72) into (63), asymptotic ECP
under mode iis obtained. The proof is completed.
APPENDIX E
PROO F OF PROPOSITION 2
When Ps→ ∞, by substituting (23) and (25) into (35), we
have
Ri
asy =E(1 −τ) log2Hi
He
i+
=1−τ
ln 2 Z∞
0
ln zfHI(z)dz −Z∞
0
ln yfHe
I(y)dy
=1−τ
ln 2 1
Γ (KmI) (βI)K mIZ∞
0
zKmI−1e−z
βIln zdz
−1
Γ (Kme
I) (βe
I)Kme
IZ∞
0
yKme
I−1e
−y
βe
Iln ydy!,(73)
With the help of Eq. (4.352.1) in [39], we have
R∞
0xv−1e−µx ln xdx =1
µvΓ (v) [ψ(v)−ln µ]. After some
manipulations, the asymptotic closed-form expression of ESC
in Proposition 2 can be proved. The proof is completed.
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