Outage performance of UAV-assisted AF relaying with hardware impairments

Abstract

The use of unmanned aerial vehicle (UAV) as aerial relay, i.e. UAV-assisted relaying, has the advantages of easy deployment, agility, and mobility compared to traditional ground relaying. In this paper, our goal is to acquire the universal outage probability (OP) analytical expression for multi-rotor UAV-assisted AF relaying with transceiver hardware impairments (HIs) while considering a practical large-scale fading. Contrary to the existing works on the UAV-assisted relaying, where the OP was derived by assuming a specific small-scale fading, we make a more general assumption of arbitrary small-scale fading and thereafter apply an integral region shape-based approximate method (IRS-AM) to deduce a closed-form OP expression with adjustable error in a more common way. The derivation process is free of the concrete model of the small-scale fading while the finial OP expression has a concise and intuitive form, which is determined by the cumulative distribution function (CDF) of the small-scale fading in both two links. Based on the derived results, we demonstrate the ceiling effect caused by HIs on the multi-rotor UAV-assisted AF relaying and obtain the end-to-end signal-to-noise-and-distortion ratio (SNDR) bound, which depends on the aggregate HIs levels and determines whether the considered relay network is always in outage or not. Computer simulations validate our studies and reveal the influence of HIs on the OP and the optimal UAV altitude.

Full text

Outage Performance of UAV-Assisted AF Relaying
With Hardware Impairments
Jinmei Zan, Guangyue Lu, Yinghui Ye∗
Shaanxi Key Laboratory of Information Communication Network and Security in Xi’an
University of Posts &Telecommunications
Abstract
The use of unmanned aerial vehicle (UAV) as aerial relay, i.e. UAV-assisted
relaying, has the advantages of easy deployment, agility, and mobility compared
to traditional ground relaying. In this paper, our goal is to acquire the univer-
sal outage probability (OP) analytical expression for multi-rotor UAV-assisted
AF relaying with transceiver hardware impairments (HIs) while considering a
practical large-scale fading. Contrary to the existing works on the UAV-assisted
relaying, where the OP was derived by assuming a specific small-scale fading, we
make a more general assumption of arbitrary small-scale fading and thereafter
apply an integral region shape-based approximate method (IRS-AM) to deduce
a closed-form OP expression with adjustable error in a more common way. The
derivation process is free of the concrete model of the small-scale fading while
the finial OP expression has a concise and intuitive form, which is determined
by the cumulative distribution function (CDF) of the small-scale fading in both
two links. Based on the derived results, we demonstrate the ceiling effect caused
by HIs on the multi-rotor UAV-assisted AF relaying and obtain the end-to-end
signal-to-noise-and-distortion ratio (SNDR) bound, which depends on the aggre-
gate HIs levels and determines whether the considered relay network is always
in outage or not. Computer simulations validate our studies and reveal the
influence of HIs on the OP and the optimal UAV altitude.
∗Corresponding author
Email addresses: zanjm0416@163.com (Jinmei Zan), tonylugy@163.com (Guangyue Lu),
connectyyh@126.com (Yinghui Ye)
Preprint submitted to Journal of L
A
T
E
X Templates May 30, 2023

Keywords: hardware impairments, multi-rotor UAV-assisted AF relaying,
outage probability, small-scale fading
1. Introduction1
Due to the small size, low price and high manoeuvrability, unmanned aeri-2
al vehicles (UAVs) have become a hot topic in recent years and with a wide3
range of applications [1–4]. With the breakthrough of key technologies related4
to UAVs, the load capacity, flight altitude and endurance of UAVs have been5
greatly improved, and the use of UAVs to enhance the quality of existing com-6
munications and provide wireless communication services to ground users has7
been realized [5, 6]. Compared with traditional ground communications, UAV-8
enhanced communications are characterized by easy deployment, no complex9
terrain restrictions, strong applicability of communication equipment, and high10
probability of line-of-sight (LoS) communication [7, 8]. Thanks to the above11
advantages, UAVs can be deployed as aerial hybrid access points to improve12
the coverage and capacity of existing networks [9, 10]. Of particular interest is13
UAV-assisted relaying, where UAV acts as the aerial relay to overcome the trans-14
mission obstacles between ground nodes raised by large buildings or mountains15
[11].16
There are two types of UAVs, namely fixed-wing and multi-rotor UAVs,17
in UAV-assisted relay networks [12–18]. The fixed-wing UAVs glide over the18
air but cannot hover over a fixed location, while the multi-rotor UAV permits19
vertical take-off and landing and hovers over a fixed location [19]. Thus, most20
of the existing works on fixed-wing UAV-assisted relay networks mainly focused21
on the trajectory optimization so as to boost the achievable performance, while22
for the multi-rotor UAV-enabled relay networks, authors mainly worked towards23
the derivation of the expressions for the outage probability (OP) or the ergodic24
capacity so that the optimal altitude can be determined by the derived results.25
Compared to the fixed-wing UAVs, multi-rotor UAVs not only are cheaper,26
but also serve as a static aerial relay, and thus enjoy a low complexity for27
2

practical designs and calls for in-depth study. Driven by this observation, this28
paper focuses on the multi-rotor UAV-assisted relaying instead of the fixed-wing29
UAV-based relaying.30
In [13], the authors derived the OP expression of the multi-rotor UAV-31
assisted relaying under Nakagami-m fading, and then used the derived results32
to determine the optimal altitude to minimize the OP. Authors of [14] intro-33
duced the energy harvesting into the multi-rotor UAV-assisted relay network34
and evaluated the outage performance under Rician fading. In the case of k−µ35
fading [15], the outage performance of multi-rotor UAV-assisted relaying un-36
der the condition of multiple antennas in both source and target nodes was37
investigated. In [16], the authors considered a UAV-assisted relay network with38
multiple source nodes and co-channel interferences, and studied the outage per-39
formance in the case of Nakagami-m fading. In [17], the authors extended the40
single UAV relay into the multi-UAV relays, and derived the closed-form OP41
expression under Nakagami fading. In [18], the authors not only evaluated the42
OP but also derived asymptotic OP expression at high signal-to-noise ratio to43
evaluate the achievable diversity gain. Considering the presentence of ground44
eavesdropper, authors in [20] modeled the air-to-ground (A2G) channel as Rice45
fading and studied the secrecy outage performance, where the source node sends46
confidential information to the destination with the help of a UAV relay. Simi-47
larly, in the presence of a ground eavesdropper, authors of [21] investigated the48
intercept probability and the ergodic secrecy rate for a multi-rotor UAV-assisted49
relay network.50
We note that most of the existing works on UAV-assisted relaying made51
the assumption of ideal hardware at all nodes [13–18, 20, 21], that is, they52
considered the perfect hardware condition at all transceivers. Nevertheless, the53
radio frequency (RF) transceivers underwent hardware impairments (HIs) in54
practical communications, among which the most common are phase noise [22],55
I/Q imbalance [23] and high power amplifier (HPA) nonlinearities [24], and56
most existing works on communications have considered HIs to better fit the57
practical model [25–27]. Although compensation algorithms have been explored58
3

to alleviate the adverse effects caused by HIs, residual HIs still occupy the59
dominance and this brings about the performance impairments to the multi-60
rotor UAV-assisted relay network [28]. Accordingly, it is quite important to re-61
examine the performance achieved by the UAV-assisted relaying in the presence62
of HIs and investigate the impacts introduced by HIs. However, to the best of63
our knowledge, only one paper [29] related to UAV-assisted relaying have taken64
the HIs into account so far, where the performance analysis was done under65
the general Nakagami-mfading. Except for the lack of HIs, existing works also66
exist limitations in the following ways.67
•The obtained OP expressions for various UAV-assisted relay scenarios were68
based on the specific small-scale fading, thus the application scope and69
flexibility of the OP expression remain limitations. There is an urgent70
need for a unified and accurate method, which is independent of specific71
fading distribution, to obtain a universal OP expression of multi-rotor72
UAV-assisted relaying.73
•Due to the existence of the probability of LoS and nonline-of-sight (NLoS)74
propagation group, the A2G channel is distinctive to the traditional chan-75
nel and is more complex and variable [30], but this point has been missed76
in most existing works. Therefore, a practical large-scale fading should be77
considered for deriving the OP.78
In this paper, we take the above observations into account and assume all79
transceivers in a multi-rotor UAV-assisted relay network are afflicted by HIs.80
On this basis, the OP is derived in a closed form.81
Our main contributions are as follows.82
•Exploiting the idea from Riemann integral, we propose an integral region83
shape-based approximate method (IRS-AM) to assist the derivation of the84
universal OP expression, which approximates the double integral as the85
product of two single integrals. Using the cumulative distribution function86
(CDF) of the small-scale fading, we derive a universal OP expression that87
has a very concise form.88
4

•Based on the obtained OP expression, we obtain the maximum endurable89
HIs level for maintaining an acceptable OP requirement at a given trans-90
mission data rate threshold. In particular, if the HIs levels reach the max-91
imum level, the network goes into outage; otherwise, the network works92
normally for information transmission, but the achievable outage perfor-93
mance can be degraded. In addition, we observed the ceiling effect caused94
by HIs on the UAV-Assisted AF relaying.95
•Simulation results are provided to validate the derived results and explore96
the effect of HIs on optimal UAV altitude. More specifically, the optimal97
UAV altitude that minimizes the OP is robust to HIs.98
The rest of this paper is organized as follows. Section II presents the system99
model, block diagram, and obtains the end-to-end signal-to-noise-and-distortion100
ratio (SNDR). In Section III, based on the derived SNDR, a closed-form univer-101
sal OP analytical expression is derived with the assistance of the IRS-AM for102
the multi-rotor UAV-assisted relay network. Section IV presents the comput-103
er simulation results to verify our theoretical derivation and summarize some104
useful conclusions while Section V concludes this paper.105
Notation:x∼ CN (ϑ, ψ ) represents a complex Gaussian random variable x106
with mean vaule ϑand variance ψ. Pr (·) denotes the probability of a random107
variable. FX(x) and fX(x) respectively denote the cumulative distribution func-108
tion (CDF) and the probability density function (PDF) of the random variable109
X.110
2. System model111
Consider a multi-rotor UAV-assisted AF relaying as depicted in Figure 1,112
which is composed of one source node (S), an amplify-and-forward (AF) UAV113
relay node (R) and one destination node (D). All nodes are assumed to be114
equipped with a single antenna and are capable of half-duplex communications.115
Since the direct transmission link between S and D is blocked by mountains and116
5

large buildings, the transmission from the source node to the destination node117
should rely on the UAV relay node. The multi-rotor UAV with vertical take-off118
and landing is considered in this model, which hovers at the height of Hin the119
center of the target area.120
Both ground nodes are located at the horizontal distance ri(i= 1,2) from
the projection of UAV on the ground, the distance between ground nodes and
UAV relay is di=pr2
i+H2,θi= arctan(H
ri) represents the elevation angle
between UAV and ground nodes. All wireless channels experience large-scale
fading Qiand small-scale fading gi. Let hidenote the channel coefficient and the
channel gain can be represented as |hi|2=|gi|2
Qi. Please note that the distribution
followed by small-scale fading is arbitrary in our model, and the CDF of small-
scale fading channel gain |gi|2is denoted by F|gi|2(·). We adopt the power loss
Qi= 10P Li
10 to describe the large-scale fading, where P Lirepresents the path
loss and it can be expressed as [13]
P Li=ηi
LoS −ηi
NLoS
1 + aiexp (−bi(θi−ai)) +Ai,(1)
where Ai= 20log10(di) + 20log10(4πf /c) + ηi
NLoS ,fis the frequency of the121
carrier, cis the speed of the light, ηi
LoS and ηi
NLoS respectively mean the ex-122
cessive path loss corresponding to the LoS propagation group and the NLoS123
propagation group in A2G link, and their values depend on the propagation124
environment. Similarly, aiand biare constant parameters related to the link125
propagation environments.126
Based on above and the block diagram in Figure 2, we analyze the entire
signal transmission process in two phases. In the first phase, S sends the trans-
mitted signal s1to R, and thereafter the received signal at R can be expressed
as
y1=h1(pP1s1+ηt,1) + ηr,1+n1,(2)
where s1is the unit power signal sent by S and it obeys the Gaussian distribution127
6

1
r
2
r
2
d
H
Source
UAV
Destination
1
d
SR→
RD→
1

2

Figure 1: System Model.
S
R
D
,1t

,1r

1
n
,2t

,2r

2
n
2
h
Source
UAV-assisted Relay
Destination
1
s
1
y
2
s
2
y
AF
1
g
( )
1
2
1
Q−
2
g
( )
1
2
2
Q−
1
h
Figure 2: Block diagram of UAV-assisted AF relaying with HIs.
of mean value 0 and variance 1, P1is the transmit power of S, n1∼ C N (0, N1)128
is the additive Gaussian white noise (AWGN) at R, ηt,1∼ CN (0, k2
t,1P1) and129
ηr,1∼ CN (0, k2
r,1P1|h1|2) respectively describe distortion noises from HIs at the130
transmitter S and receiver R. Here the parameters k2
t,1and k2
r,1characterize the131
HIs level at the transmitter and receiver in the S→Rlink [31].132
In the second phase, based on the amplify-then-forward protocol, R firstly
amplifies y1using an amplification gain G=qP2
|h1|2P1(1+k2
t,1+k2
r,1)+N1and for-
wards the amplified signal Gy1to D. The intended signal received by D from R
can be written as
y2=h2(Gy1+ηt,2) + ηr,2+n2,(3)
where P2is the transmit power of R, n2∼ C N (0, N2) models the received133
AWGN at D, ηt,2∼ CN (0, k 2
t,2P2) and ηr,2∼ C N (0, k2
r,2P2|h2|2) model the134
distortion noises from HIs at R and D in the second-hop link, respectively. Here135
7

the parameters k2
t,2and k2
r,2characterize the HIs level at the transmitter and136
receiver in the R→Dlink. Note that higher HIs level means larger values of the137
corresponding parameters, lower hardware costs and poorer hardware quality.138
According to eqs.(2) and (3), the instantaneous end-to-end signal-to-noise-
and-distortion ratio (SNDR) of the UAV-assisted relay network can be derived
as
γAF =|g1|2|g2|2
|g1|2|g2|2(k2
1+k2
2+k2
1k2
2) + 1
γ1|g2|2Q1(1 + k2
2) + 1
γ2|g1|2Q2(1 + k2
1) + Q1Q2
γ1γ2
=γSR γRD
γSR +γRD +1
(a)
≈γSR γRD
γSR +γRD
=|g1|2|g2|2
|g1|2|g2|2(k2
1+k2
2) + 1
γ2|g1|2Q2+1
γ1|g2|2Q1
,(4)
where γSR =γ1|g1|2
γ1|g1|2k2
1+Q1and γRD =γ2|g2|2
γ2|g2|2k2
2+Q2represent the SNDR in two139
links S→Rand R→D,γ1=P1
N1and γ2=P2
N2denote the transmit signal-to-140
noise ratio (SNR) at S and R, k1=qk2
t,1+k2
r,1and k2=qk2
t,2+k2
r,2represent141
the aggregate HIs levels in two links, respectively [31].142
In eq.(4), step (a) holds for the fact that transmit SNR is far greater than143
1 in practical communications. For ease of analysis, we use the approximate144
SNDR expression to deduce the OP in the next section.145
3. Outage Probability Analysis146
In this section, with the assistance of the unified IRS-AM, we present the147
derivation of the universal OP analytical expression and conclude with the anal-148
ysis of the OP expression and the impact of HIs on network outage performance.149
OP is defined as the probability that system SNDR lower than a certain150
SNDR threshold γth. Mathematically speaking, Pout = Pr(γAF < γth ), where151
γth = 22Rth −1 is the SNDR threshold, Rth is the transmission data rate thresh-152
old in bps/Hz.153
8

In the following, the derivation of Pout can be divided into two cases.154
(i): when γth ≥1
k2
1+k2
2,Pout can be written as
Pout = Pr (γAF < γth)
= Pr |g1|2|g2|2
|g1|2|g2|2(k2
1+k2
2) + 1
γ2|g1|2Q2+1
γ1|g2|2Q1
< γth!
= Pr 1
(k2
1+k2
2) + Q2
γ2|g2|2+Q1
γ1|g1|2
< γth!
(b)
=1,(5)
where step (b) is derived from 1
(k2
1+k2
2)+ Q2
γ2|g2|2+Q1
γ1|g1|2
<1
k2
1+k2
2. Therefore,155
1
k2
1+k2
2+Q2
γ2|g2|2+Q1
γ1|g1|2
< γth is satisfied and Pout equals to 1 no matter what156
the small-scale or large-scale fading is.157
(ii): when γth <1
k2
1+k2
2,Pout can be rewritten as
Pout = 1 −Pr(γAF > γth )
= 1 −Pr γAγB
γA+γB
> γth
= 1 −Pr 1
|g1|2<γ11−γth(k2
1+k2
2)
Q1γth
−1
|g2|2
γ1Q2
γ2Q1!
(c)
= 1 −Pr X1<γ11−γth (k2
1+k2
2)
Q1γth
−X2
γ1Q2
γ2Q1!
| {z }
Θ
,(6)
where step (c) is derived from the variable substitution Xi=1
|gi|2(i= 1,2).158
Based on the definition of CDF, the CDF of Xican be denoted as
FXi(x) = Pr (Xi< x)
= Pr 1
|gi|2< x
= 1 −Pr |gi|2<1
x
= 1 −F|gi|21
x.(7)
9

1
X
2
X
1th−
2th−
K th−
M th−
A
B
C
D
E
F
P
G
H
( )
( )
22
1
1 1 2 12
12
1 2 1
kk Q
th
xx
QQ
th
 

−+
=−
O
( )
( )
22
1 1 2
1
1
: 0 , th
th
kk
AQ



−+



( )
( )
22
2 1 2
2
1
: , 0
th
th
kk
BQ



−+



( )
( )
( )
22
2 1 2
2
11
: , 0
th
th
K k k
CMQ



− − +



( )
( )
22
2 1 2
2
1
: , 0
th
th
K k k
DMQ



−+



( )
( )
( )
( )
( )
( )
2 2 2 2
2 1 2 1 1 2
21
1 1 1 2 2 1
: ,
2
th th
th th
K k k M K k k
GMQ MQ
   


− − + + − − +



( )
( )
22
1 1 2
1
1
: 0 , th
th
kk
AQ



−+



( )
( )
22
2 1 2
2
1
: , 0
th
th
kk
BQ



−+



( )
( )
( )
22
2 1 2
2
11
: , 0
th
th
K k k
CMQ



− − +



( )
( )
22
2 1 2
2
1
: , 0
th
th
K k k
DMQ



−+



( )
( )
( )
( )
( )
( )
2 2 2 2
2 1 2 1 1 2
21
1 1 1 2 2 1
: ,
2
th th
th th
K k k M K k k
GMQ MQ
   


− − + + − − +



Figure 3: Original and approximate integral region of Θ.
159
Since Θ is the probability that X1less than a function γ1(1−γth (k2
1+k2
2))
Q1γth −
X2γ1Q2
γ2Q1associated with X2, the region of function x1=γ1(1−γth (k2
1+k2
2))
Q1γth −x2γ1Q2
γ2Q1
in the first quadrant is the original integral region (∆AOB) of fX1,X2(x1, x2)
according to Figure 3. Because of the statistical independence of both links,
random variables X1and X2are independent of each other, and the joint prob-
ability density function (pdf) of X1and X2always satisfies fX1,X2(x1, x2) =
fX1(x1)fX2(x2). Thus, we rewrite Θ as follows
Θ = Pr X1<γ11−γth (k2
1+k2
2)
Q1γth
−X2
γ1Q2
γ2Q1!
=ZZ

∆AOB
fX1,X2(x1, x2)dx1dx2
=
γ2(1−γth(k2
1+k2
2))
Q2γth
Z
0
γ1(1−γth(k2
1+k2
2))
Q1γth −x2γ1Q2
γ2Q1
Z
0
fX1,X2(x1, x2)dx1dx2
(d)
=
γ2(1−γth(k2
1+k2
2))
Q2γth
Z
0
γ1(1−γth(k2
1+k2
2))
Q1γth −x2γ1Q2
γ2Q1
Z
0
fX1(x1)fX2(x2)dx1dx2.(8)
For the eq.(8) corresponding to step (d), there exists a challenge if the160
10

derivation is carried out directly. In this way, step (d) can be transformed as161
FX2γ2(1−γth(k2
1+k2
2))
Q2γth −FX2(0)×
γ1(1−γth(k2
1+k2
2))
Q1γth −x2γ1Q2
γ2Q1
R
0
fX1(x1)dx1. We162
note that the double integral is denoted as the form of the CDF multiplied by163
a variable upper bound integral, which is not conducive to derive the universal164
OP expression. In order to overcome this challenge, we introduce the IRS-AM,165
which can transform the shape of the integral region and approximate the dou-166
ble integral as the product of two independent single integrals. As the integral167
region depicted in Figure 3, we describe the IRS-AM as follows.168
Step 1: Divide the original integral region ∆AOB into M(M≥1) small169
rectangular regions with equal width but unequal height, where the height of170
the first small rectangle is the length of the line AB.171
Step 2: Approximate again for all small rectangular regions. Take the K−th172
small rectangle as an example: make a line parallel to the horizontal axis through173
the midpoint P of segment EF, and the line intersects the small rectangle at G174
and H. Do the same for the rest of the small rectangular regions, and we take175
the total gray regions as the approximate integral region.176
Step 3: Calculate the integral value ΘKof function fX1,X2(x1, x2) on the
11

K−th gray region DK, which can be expressed as
ΘK=ZZ

DK
fX1(x1)fX2(x2)dx1dx2
=
Kγ2(1−γth (k2
1+k2
2))
MQ2γth
Z
(K−1)γ2(1−γth(k2
1+k2
2))
MQ2γth
(1+2M−2K)γ1(1−γth(k2
1+k2
2))
2MQ1γth
Z
0
fX1(x1)fX2(x2)dx1dx2
=
Kγ2(1−γth (k2
1+k2
2))
MQ2γth
Z
(K−1)γ2(1−γth(k2
1+k2
2))
MQ2γth
fX2(x2)dx2
(1+2M−2K)γ1(1−γth(k2
1+k2
2))
2MQ1γth
Z
0
fX1(x1)dx1
="FX2 Kγ21−γth (k2
1+k2
2)
MQ2γth !−FX2 (K−1)γ21−γth(k2
1+k2
2)
MQ2γth !#
×FX1 (1 + 2M−2K)γ11−γth(k2
1+k2
2)
2MQ1γth !
=F|g2|2MQ2γth
(K−1)γ2(1 −γth(k2
1+k2
2))−F|g2|2M Q2γth
Kγ2(1 −γth (k2
1+k2
2))
×1−F|g1|22MQ1γth
(1 + 2M−2K)γ1(1 −γth(k2
1+k2
2)).(9)
177
Step4: Integrate the function fX1,X2(x1, x2) on the whole approximate inte-
12

gral region (D1, D2,· · · , DK,· · · , DM), which can be expressed as
Θ = ZZ

D1+···+DK+···+DM
fX1(x1)fX2(x2)dx1dx2
=ZZ

D1
fX1(x1)fX2(x2)dx1dx2+· · · +ZZ

DK
fX1(x1)fX2(x2)dx1dx2
+· · · +ZZ

DM
fX1(x1)fX2(x2)dx1dx2
=
M
X
K=1 ZZ

DK
fX1(x1)fX2(x2)dx1dx2
=
M
X
K=1
ΘK.(10)
Combing (5), the OP can be denoted as Pout = 1 −
M
P
K=1
ΘKand the derivation178
is complete.179
In summary, the core of IRS-AM lies in finding the approximate rectangu-180
lar integral region, which can approximate the double integral as the product181
of two single integrals. Besides, there exists a compensation relationship be-182
tween triangular region HPF and EPG as depicted in Figure 3, which makes183
the approximate integral region closer to the original integration region.184
On the basis of the above derivation, we can represent the closed-form ana-185
lytical OP expression as summarized in Theorem 1.186
Theorem 1. Consider that small fading channel gains |g1|2and |g2|2
are independent and non-negative random variables with CDF of F|g1|2(·) and
F|g2|2(·), respectively. Then, the outage probability of multi-rotor UAV-assisted
AF relaying under imperfect hardware impairments can be given as
Pout(γth ) = 




1−
M
P
K=1
ΘKγth <1
k2
1+k2
2
1γth ≥1
k2
1+k2
2
.(11)
Remark 1. As exhibited in theorem 1, the aggregate HIs levels impose a187
13

constraint on the SNDR threshold, which produces a ceiling effect on the UAV-188
assisted relaying. Thus, parameters k1,k2and γth should be set reasonably189
to ensure normal information transmission. In other words, no matter which190
distribution the small-scale fading obeys, the communication will be always in191
outage if the SNDR threshold γth exceeds the bound 1
k2
1+k2
2, which decreases192
with HIs levels increase.193
Remark 2. Based on the above derivations, a universal and concise OP194
expression with adjustable error is obtained, which is independent of the specific195
channel distribution. On the one hand, according to the specific CDF of different196
small-scale fading distributions F|gi|2(·), the corresponding OP can be easily197
calculated and the accuracy of OP can be adjusted by changing the segmentation198
times M. On the other hand, the outage performance of the multi-rotor UAV-199
assisted relaying with HIs can be analyzed efficiently and flexibly by applying200
theorem1, for example, the lower bound of OP can be acquired in the case of201
M= 1 while the upper bound can be deduced when M→ ∞.202
Table 1
several traditional channel models
fading models CDF of the channel gain
F|gi|2(x)(i= 1,2)
channel fading coefficient
Rayleigh-fading 1 −exp −x
Ωi
Nakagami-fading 1 −Γmi,mi
Ωix
Γ(mi)mi
Rice-fading 1 −Q√2Ki,q2(1+Ki)
ΩixKi=ϑ2
i
2σ2
i
, Ωi=ϑ2
i+ 2σ2
i
4. Simulation Results203
In this section, Monte Carlo simulation experiments are used to validate204
our theoretical expression and evaluate the effect of key parameters on the205
considered network. We set parameters r1= 400 m, r2= 600 m, H= 300206
14

-10 -5 0 5 10 15 20 25 30
Transmit Power P [dBm]
0
0.2
0.4
0.6
0.8
1
Outage Probability (OP)
Rayleigh: 1= 2=1
Nakagami: m1=m2=2
Rice: K1=K2=2
Simulation
1 2 3 4 5 6 7 8 9 10 11 12
Segmentation Times M
0
0.05
0.1
0.15
0.2
0.25
0.3
Relative Approximation Error
Rayleigh: 1=2=1
Nakagami: m1=m2=2
Rice: K1=K2=2
Figure 4: Validate of the effectiveness, generality and accuracy of the OP expression.
m, k1=k2=k= 0.1, Rth = 1 bps/Hz, c= 3 ×108m/s, f= 2 ×109Hz,207
N1=N2=N=−100 dBm, P1=P2=P= 10 mW, η1
LoS =η2
LoS = 0.1 dB,208
η1
NLoS =η2
NLoS = 21 dB, a1=a2= 5.0188, b1=b2= 0.3511, M= 10, and209
assume them are fixed unless otherwise specified. To maximize the SNDR bound210
and the outage performance for the fixed hardware cost, the HIs levels at each211
transceiver link are selected to be identical [31]. We choose several traditional212
fading channels, namely, the Rayleigh fading |gi|2∼exp (1/λi), Nakagami-213
m fading |gi|2∼Gamma (mi,Ωi/mi) and Rice fading to present computer214
simulation and performance evaluation to demonstrate the utility of our results.215
The CDF of each fading are listed in Table I, where Ωi= E|gi|2{|gi|2}is average216
fading power, Γ (n)=(n−1)! is the complete Gamma function, Γ (n, z ) =217
R∞
zun−1e−udu is the incomplete Gamma function and Q(·,·) is the first-order218
Marcum Q function [32].219
The upper plot in Figure 4 shows the relative approximation error of OP
15

0 200 400 600 800 1000 1200 1400 1600 1800 2000
UAV Altitude H [m]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Outage Probability (OP)
Rayleigh: 1=2=1
Nakagami: m1=m2=2
Rice: K1=K2=2
Simulation
Figure 5: OP versus UAV altitude with different fading channels.
expression versus M, the relative approximation error is defined as
δ=|analytical value −simulation value|
|simulation value|,(12)
where the analytical and simulation values are respectively obtained from the-220
orem1 and Monte Carlo simulations. It can be observed that the relative ap-221
proximation error under all fading channels decreases with the increase of M,222
and the error is close to 0 around M= 10. For Rayleigh fading, it can be seen223
that even if M= 1, δis extremely small, equal to 0.08957. This trend manifests224
the effectiveness and accuracy of our OP expression derived with the assistance225
of IRS-AM, thus, it is valuable to apply the results in theorem1 to analyze and226
evaluate the system outage performance efficiently and flexibly. It can be seen227
from the bottom picture in Figure 4 that the outage performance is better with228
the increase of transmit power, and the analytical values obtained from theo-229
rem1 agrees well with the Monte Carlo simulation values for all selected fading230
channels, which reflects the correctness of our obtained expression in theorem1.231
16

0 0.05 0.1 0.15 0.2 0.25 0.3
HIs Levels k
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Outage Probability (OP)
Rayleigh: 1=2=1
Nakagami: m1=m2=2
Rice: K1=K2=1
Simulation
P=15 dBm
P =10 dBm
Figure 6: Pout(15) versus the HIs levels k1=k2under different fading channels with two
transmit power.
By selecting several representative small-scale fading model, the effectiveness,232
generality and accuracy of the OP expression are verified.233
Figure 5 plots the curves of OP as a function of UAV altitude. For different234
small-scale fading channels, we observe that the optimal UAV altitude is unique,235
meaning that the UAV should be positioned at an optimal altitude so as to236
maximize the system outage performance. This phenomenon can be explained237
in two ways: when the UAV altitude is too low (below the optimal altitude), OP238
increases due to the decreasing LoS component in A2G link and the shadowing239
effect; when the UAV altitude surpasses the optimal altitude, although the LoS240
component in A2G channel increases, it leads to increased path loss due to the241
increased distance between the UAV and the ground nodes, which also affects242
the outage performance. In this way, we validate the utility of our results in243
different small-scale fading where OP is the factor of interest.244
Figure 6 shows the OP versus the HIs levels k1=k2under two transmit245
power, where we set Rth = 2 bit/s/Hz, corresponding to SNDR threshold γth =246
17

0 10 20 30 40 50 60 70 80
SNDR threshold th
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Outage Probability (OP)
Rayleigh: 1=2=1
Nakagami: m1=m2=2
Rice: K1=K2=1
Simulation
k1=k2=0
k1=k2=0.1
Figure 7: Pout versus the SNDR threshold γth under different fading channels with P1=
P2= 25 dBm.
15 dB. Here we observe that Pout(15) always coverage to the ceiling 1 with the247
HIs levels approach a certain threshold, and this can be explained from theorem248
1. According to the maximum tolerable OP requirement in practical system249
design, this may provide a rough estimate of the range of HIs levels. Since250
the lower-cost hardware has higher levels of impairments, the hardware quality251
which can provide a compromise between outage performance and hardware252
cost can be selected properly in practical system design.253
In Figure 7, the relationship between OP and SNDR threshold is investigated254
under three fading channels, where two different HIs levels are considered :(i)255
k1=k2= 0.1; (ii) k1=k2= 0. As in the theoretical analysis, the maximum256
SNDR threshold γth = 50 dB (γth ≥1
k2
1+k2
2) exists for different channels at257
the given HIs levels k1=k2= 0.1. When the SNDR threshold exceeds the258
maximum threshold (γth ≥50 dB), the considered network goes in outage, i.e.259
Pout = 1. In particular, it can be observed that the difference between the OP260
in the ideal hardware assumption and the non-ideal hardware case is relatively261
18

0.05 0.1 0.15 0.2 0.25 0.3
HIs Levels k
286
287
288
289
290
291
292
Optimal UAV altitude H [m]
SNDR threshold th = 2
SNDR threshold th = 3
SNDR threshold th = 4
Figure 8: Optimal UAV altitude versus the level of symmetric impairments k1=k2with
different SNDR threshold.
small in the low SNDR region, but the difference between them becomes larger262
as the SNDR threshold increases. This phenomenon indicates that the negative263
impact of HIs on the considered network outage performance is more severe at264
high data rates.265
Figure 8 displays that HIs cause a slight drop in optimal UAV altitude266
and the effect of HIs on optimal UAV altitude is not significant, especially267
in the case of lower HIs levels. Specifically, for γth = 2, the optimal UAV268
altitude presents one descent point only when HIs levels increase to 0.25. For269
γth = 3, the optimal UAV altitude is not affected by hardware impairments.270
For γth = 4, with the levels of impairments increase to 0.2, the optimal UAV271
altitude presents one descent point while the descent altitude is 2 m. This272
phenomenon reveals that the change of optimal UAV altitude caused by HIs273
is more pronounced in higher SNDR threshold, especially for higher HIs levels.274
Although the effect of impairments on optimal UAV altitude is not obvious as275
that of outage performance, its negative effect on outage performance can not276
19

0 200 400 600 800 1000 1200 1400 1600 1800 2000
UAV Altitude H [m]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Outage Probability (OP)
Rayleigh: 1=2=1
Nakagami: m1=m2=2
Rice: K1=K2=1
AF SR RD (SR+RD)
AF=SR RD (SR+RD+1)
Figure 9: The OP gap before and after the SNDR approximation.
be overlooked.277
Figure 9 compares the OP gap before and after the SNDR approximation to278
verify the effect of the SNDR approximation on the accuracy of OP. It is clear279
that the pre-approximation and post-approximation simulations show the same280
trend for all three channels, with the pre-approximation outage performance in-281
ferior to the post-approximation. Although the OP difference increases slightly282
with increasing UAV altitude, this difference is within an acceptable range and283
has no impact on the optimal UAV altitude. Based on the above analysis, it284
can be concluded that the approximation to SNDR is reasonable and its impact285
on the accuracy of the OP is acceptable.286
Based on the above analysis, we summarize the main observations as follows.287
1) A general expression with the effectiveness, generality and accuracy that288
can be used to calculate the OP of the considered network in the case of any289
small-scale fading model is validated.290
2) There exists an optimal UAV altitude that minimizes the OP, i.e., the UAV291
20

should be placed at a suitable altitude to maximize the outage performance of292
the considered network.293
3) The consideration of HIs imposes a constraint on the SNDR threshold and294
hence cause the ceiling effect on the network, which can be explained by the295
fact that when the SNDR threshold surpasses the bound 1
k2
1+k2
2, UAV-assisted296
AF relaying will be in outage.297
4) Although the introduction of HIs has an obvious impact on the outage298
performance of the considered network, its presence produces a weak effect299
on the optimal UAV altitude. Corresponding to different HIs levels, although300
smaller HIs levels produce better outage performance compared with higher HIs301
levels, they correspond to almost identical optimal UAV altitude.302
5. Conclusions303
In this paper, we have proposed a unified way to deduce the universal OP304
expression in multi-rotor UAV-assisted relaying with HIs while considering a305
practical large-scale fading. Specifically, a universal OP expression under ar-306
bitrary channel distribution has been obtained with the help of IRS-AM. Our307
results were not only concise and intuitive but also easy to change the fading308
models and adjust the error immediately. Simulation results have verified the309
effectiveness, generality and accuracy of the OP expression. Based on the de-310
rived OP expression, we not only analyzed the ceiling effect induced by HIs but311
also investigated the existence of optimal UAV altitude and the fluctuation of312
it caused by HIs in simulation. Some factors have been revealed in simulation.313
Firstly, in order to achieve the optimal outage performance of the considered314
network, the UAV altitude should be carefully designed. Secondly, the value315
between data rate and HIs levels should be set reasonably to ensure that the316
network can work properly for information transmission. Thirdly, although HIs317
degrade the outage performance, it has few impacts on the optimal UAV alti-318
tude.319
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