On the Ergodic Secrecy Capacity of Intelligent Reflecting Surface Aided Wireless Powered Communication Systems

Abstract

This letter studies physical layer security (PLS) of an intelligent reflecting surface (IRS) aided wireless powered communication (WPC) system in the presence of a passive eavesdropper, and proposes three secure IRS-WPC modes. In mode-I, the IRS is deployed between hybrid access point (HAP) and wireless user (U) for co-located power station (PS) and access point (AP). In mode-II, the IRS is deployed between AP and U for separate PS and AP, while in mode-III, the IRS is deployed between PS and U for separate PS and AP. For each mode, the optimal phase shift is designed to maximize the reception of energy and information at legitimate receiver. The accurate and asymptotic ergodic secrecy capacity (ESC) of each mode is analyzed, and various new closed-form expressions are obtained. The results reveal that mode-I and mode-II significantly outperform benchmark without IRS, while mode-III outperforms benchmark without IRS under lower transmission power or small number of IRS elements.

Full text

1
On the Ergodic Secrecy Capacity of Intelligent Reflecting
Surface Aided Wireless Powered Communication Systems
Kunrui Cao, Haiyang Ding, Wei Li, Lu Lv, Mei Gao, Fengkui Gong, and Buhong Wang
Abstract—This letter studies physical layer security (PLS) of
an intelligent reflecting surface (IRS) aided wireless powered
communication (WPC) system in the presence of a passive
eavesdropper, and proposes three secure IRS-WPC modes. In
mode-I, the IRS is deployed between hybrid access point (HAP)
and wireless user (U) for co-located power station (PS) and
access point (AP). In mode-II, the IRS is deployed between
AP and U for separate PS and AP, while in mode-III, the IRS
is deployed between PS and U for separate PS and AP. For
each mode, the optimal phase shift is designed to maximize the
reception of energy and information at legitimate receiver. The
accurate and asymptotic ergodic secrecy capacity (ESC) of each
mode is analyzed, and various new closed-form expressions are
obtained. The results reveal that mode-I and mode-II significantly
outperform benchmark without IRS, while mode-III outperforms
benchmark without IRS under lower transmission power or small
number of IRS elements.
Index Terms—Reconfigurable intelligent surface, wireless pow-
ered communication, physical layer security.
I. INTRODUCTION
The recent advance of microwave wireless power transfer
technology enables wireless powered communication (WPC)
systems to be built, which eliminates the need for frequent
manual battery replacement/recharging, and achieves higher
throughput, longer device lifetime and lower system oper-
ating cost than conventional battery-powered communication
systems. Meanwhile, intelligent reflecting surface (IRS), also
termed reconfigurable intelligent surface, is a novel technique
to create smart radio environments to improve the performance
of wireless communications and mitigate the negative effects
introduced by natural wireless propagation. Thanks to the
advantages of IRS in terms of improving the reception quality
of signals, applying IRS into WPC (i.e., IRS-WPC) has
recently drawn increasing attention [1]–[4].
On the other hand, due to the openness of wireless commu-
nications, the WPC systems are vulnerable to eavesdropping
attacks, which makes information security an important con-
cern. Physical layer security (PLS), from the perspective of
information theory, provides a secure information transmission
against eavesdropping attacks without keys or complicated
encryption/decryption algorithms. Particularly, IRS has a great
potential in improving the physical layer security of the
This work was supported in part by the National Natural Science Foundation
of China under Grants 62101560, 61871387 and 61901313, in part by the
Natural Science Basic Research Program of Shaanxi under Grant 2022JQ-
619, in part by the National University of Defense Technology Research Fund
under Grant ZK21-44, in part by the Open Research Fund of the State Key
Laboratory of ISN under Grant ISN23-04, and in part by the China Post-
doctoral Science Foundation under Grants BX20190264 and 2019M650258.
(Corresponding authors: Haiyang Ding; Buhong Wang)
Kunrui Cao, Haiyang Ding, Wei Li, and Mei Gao are with the School of In-
formation and Communications, National University of Defense Technology,
Wuhan 430035, China, and Kunrui Cao is also with the State Key Laboratory
of Integrated Services Networks, Xidian University, Xi’an 710071, China.
Lu Lv and Fengkui Gong are with the State Key Laboratory of Integrated
Services Networks, Xidian University, Xi’an 710071, China. Buhong Wang
is with the School of Information and Navigation, Air Force Engineering
University, Xi’an 710077, China.
systems as it can reconfigure the wireless channels of legit-
imate receivers and malicious eavesdroppers to enhance the
reception of legitimate receivers and weaken the reception of
eavesdroppers [5], which motivates the research on PLS of
IRS-WPC.
However, the investigation on PLS of IRS-WPC is in its
infancy currently [6], and there is still a major research gap in
the investigation of the IRS’s effects on the PLS of WPC. In
particular, in the presence of eavesdropping attacks, a secrecy
performance study for IRS-WPC system is not reported in the
literature yet, and the impacts of key system parameters on
ergodic secrecy capacity (ESC) of the system are still far from
being understood, which motivates this work. In this paper, we
focus on the PLS of a typical IRS-WPC system in the presence
of an eavesdropper, and analyze the accurate and asymptotic
ESC for the system to obtain valuable insights. Moreover, the
effects of key system parameters are revealed.
II. SY ST EM MO DE L
We consider a typical WPC system consisting of a power
station (PS), an access point (AP), a wireless device (U) and a
passive eavesdropper (E), where PS and AP can be co-located
or integrated as a hybrid AP (HAP). The energy-constrained
U harvests the radio frequency energy from HAP (or PS), and
then exploits the harvested energy to transmit the information
to HAP (or AP). As shown in Fig. 1, we propose three IRS-
WPC modes: 1) In mode-I, an IRS is deployed between HAP
and U for co-located PS and AP to enhance the communication
link between HAP and U; 2) In mode-II, an IRS is deployed
between U and AP for separate PS and AP to enhance the
communication link between U and AP; 3) In mode-III, an
IRS is deployed between PS and U for separate PS and AP
to enhance the communication link between PS and U. In the
above three modes, it is assumed that the IRS has Nlow-cost
reconfigurable reflecting elements and each element can reflect
a phase shifted version of the incident signal independently to
enhance the signal reception power at receivers. There is no
direct link between nodes with the aid of the IRS, due to
the obstacles, which motivates the deployment of an IRS to
establish reliable communication links. Particularly, E attempts
to intercept the wireless information from U.1
In mode-I, the channel vectors between HAP/U and IRS
are denoted by har and hur, respectively. In mode-II, the
denotations of the channel vectors between AP/U and IRS
are the same as mode-I, and the channel coefficient between
PS and U is denoted by hpu. In mode-III, the channel
vectors between PS/U and IRS are denoted by hpr and hur,
respectively, and the channel coefficient between U and AP is
denoted by hua. In the above three modes, the channel vector
and coefficient between IRS/U and E are denoted by hre and
hue, respectively. Each entry of har,hur ,hpr and hre, as well
1In addition to PLS, covert communication technique can provide stealth or
low probability fo detection for secure communications. Several recent studies
have exploited the IRS to facilitate covert communication, e.g., [7], [8].

2
Information transmission
Energy transfer
Eavesdropping link
Energy transfer Information transmission HAP
U
E
IRS PS U
E
AP PS U
E
AP
Mode-I Mode-II Mode-III
IRS IRS
Fig. 1: System model.
as hpu,hua and hue are assumed to be independent complex
Gaussian distribution with zero mean and unit variance.
A. Transmission Schemes
During a time frame T, the communication is accomplished
over two phases, i.e., energy transfer (ET) and information
transmission (IT) phases. Specifically, during the time θT (0<
θ < 1), the HAP (or PS) transfers wireless energy to U via IRS
in mode-I (or mode-III), while PS directly transfers wireless
energy to U in mode-II. Mathematically, the energy harvested
by U in each mode is expressed, respectively, as
HI=ηθT PshT
urΘ1har 
2βaru,(1)
HII =η θT Psβpu|hpu|2,(2)
HII I =ηθT Psβpr u hT
urΘ3hpr 
2,(3)
where Θ1= diag[ejφI ,1, ..., ejφI,N ]and Θ3=
diag[ejφI II,1, ..., ej φII I,N ]are N×Ndiagonal reflecting
matrices of IRS in mode-I and mode-III, respectively, ηis
the energy conversion efficiency, and Psis the transmission
power of HAP (or PS) at the ET phase. Furthermore,
βaru =GaGuS2
dτ
ardτ
ur is the path loss from HAP to U via the nth
element of IRS, where Gaand Guare the antenna gains of
HAP and U, respectively, Sis the size of each element, dar
and dur are the distances from HAP to IRS and from IRS
to U, and τis path loss exponent. Similarly, βpu =GpGu
dτ
pu is
the path loss from PS to U, Gpis the antenna gain of PS,
dpu is the distance from PS to U, βpru =GpGuS2
dτ
prdτ
ru is the path
loss from PS to U via IRS, and dpr and dru are the distances
from PS to IRS and from IRS to U, respectively.
In the remaining time (1−θ)T, U uses the energy harvested
from ET phase to transmit its information to HAP (or AP) via
IRS in mode-I (or mode-II), while U uses the harvested energy
to directly transmit its information to AP in mode-III. In IT
phase, the transmission power of U in three modes can be
written as
PI
u=HI
(1 −θ)T=PthT
urΘ1har 
2βaru,(4)
PII
u=HII
(1 −θ)T=Pt|hpu|2βpu ,(5)
PII I
u=HII I
(1 −θ)T=PthT
urΘ3hpr 
2βpru,(6)
where Pt=Psηθ
1−θ. Then, the received signal at HAP (or AP)
in three modes can be given by
yI
a=qPI
uβaruhT
arΘ1hur s+na,(7)
yII
a=qPII
uβaruhT
arΘ2hur s+na,(8)
yII I
a=qPIII
uβuahua s+na,(9)
where Θ2= diag[ejφII,1, ..., ejφI I,N ],nais the additive white
Gaussian noise (AWGN) at HAP (or AP), sis the intended
signal, and βua =GuGa
dτ
ua . As a result, the signal-to-noise ratio
(SNR) of the received signal at HAP (or AP) in three modes
can be given by
SNRI
a=ρtβ2
aru hT
arΘ1hur 
2hT
urΘ1har 
2,(10)
SNRII
a=ρt|hpu|2hT
arΘ2hur 
2βpuβaru ,(11)
SNRII I
a=ρthT
urΘ3hpr 
2|hua|2βpru βua,(12)
where ρt=Pt
σ2and σ2is the variance of the AWGN.
On the other hand, the received signal at E in three modes
can be expressed as
yI
e=qPI
upβurehT
reΘ1hur +pβue hues+ne,(13)
yII
e=qPII
upβurehT
reΘ2hur +pβue hues+ne,(14)
yIII
e=qPIII
upβurehT
reΘ3hur +pβue hues+ne,(15)
where βure =GuGeS2
dτ
urdτ
re ,βue =GuGe
dτ
ue ,Geis the antenna gain
of E, dre and due are the distances from IRS to E and from
U to E, respectively, and neis the AWGN at E. As such,
the SNR of the received signal at E for three modes can be
respectively given by
SNRI
e=ρtβaru hT
urΘ1har 
2pβurehT
reΘ1hur+pβuehue 
2
,
(16)
SNRII
e=ρtβpu|hpu|2pβure hT
reΘ2hur +pβue hue
2
,(17)
SNRII I
e=ρtβpru hT
urΘ3hpr 
2pβurehT
reΘ3hur+pβuehue 
2
.
(18)
B. Optimal Phase Shift Designs
In this subsection, the inherent beamforming gain of IRS
is exploited to improve the communication quality of IRS-
WPC systems and simultaneously reduce the information
leakages from IRS to E. It is assumed that the channel state
information of all legitimate channels is known, which can
be obtained by using some channel estimation approaches in
IRS aided communication systems [1], [7]. But the channel
state information of E is unknown. Hence, the phase shifts at
the IRS are matched with the phases of legitimate channels
to maximize the harvested energy of U or (and) the reception
quality of AP (HAP) in three modes. It needs to be pointed
out that this letter focuses on continuous phase shifts, and the
design of discrete phase shifts [9] is left for our future work.
To be specific, the SNR to receive signal at HAP (or AP)
in three modes can be rewritten as
SNRI
a=ρtβ2
aru 
N
X
n=1
har,nhur,nej φI,n 
4
,(19)

3
SNRII
a=ρtβaruβpu |hpu|2
N
X
n=1
hur,nhar,nej φII,n 
2
,(20)
SNRIII
a=ρtβpruβua |hua|2
N
X
n=1
hur,nhpr,nej φIII ,n 
2
.(21)
To maximize the received SNR, the optimal phase shifts
φopt
I,n (or φopt
II ,n) should be perfectly matched with the phases
of har,n =|har,n|ejφhar,n and hur,n =|hur,n|ejφhur,n ,
i.e., φopt
I,n =φopt
II ,n =−φhar,n +φhur,n, where φhar,n
and φhur,n are the phases of har,n and hur,n, respectively.2
Meanwhile, the optimal phase shifts φopt
II I,n should be designed
as φopt
II I,n =−φhur,n +φhpr,n , where φhur,n and φhpr,n are
the phases of hur,n and hpr,n, respectively. With the optimal
phase shifts, the SNR at HAP (or AP) can be given by
SNRI
a=ρtβ2
aru N
X
n=1 |har,n||hur,n|!4
,(22)
SNRII
a=ρtβaruβpu |hpu|2 N
X
n=1 |hur,n||har,n|!2
,(23)
SNRIII
a=ρtβpruβua |hua|2 N
X
n=1 |hur,n||hpr,n|!2
.(24)
III. ERGODIC SEC RE CY CA PACITY ANA LYSIS
The ergodic secrecy capacity denotes the statistical aver-
age of the secrecy rate over fading channels, which can be
mathematically expressed as
Ei=Eh(1−θ) log21+SNRi
a−(1−θ) log21 +SNRi
e+i,
(25)
where E[x]is the expectation of x,i∈ {I, I I, I II }, and
{x}+= max{x, 0}.
A. Accurate Analysis
In order to derive the ESC of three modes, the statistics of
IRS cascaded channels with optimal phase shifts need to be
obtained first, which is given in the following lemma.
Lemma 1: The probability density function (PDF) of IRS
cascaded channel Z=PN
n=1 |har,n||hur,n|can be given by
fZ(z) = zv−1
Γ(v)ϕve−z
ϕ,(26)
where v=Nπ 2
16−π2,ϕ=16−π2
4π, and Γ(·)is the Gamma
function [10, Eq. (8.31)].
Proof: Generally, it is quite challenging to obtain an exact
PDF for Z. To overcome this difficulty, we use the Gamma
distribution to find a quite tight approximation for the PDF of
Z. In particular, by using [11, Lemma 1], it is concluded that
the product |har,n||hur,n|can be approximated as a Gamma
distribution with parameters π2
16−π2and 16−π2
4π. Then, a sum
of Nindependent and identically distributed Gamma random
variables still follows a Gamma distribution with parameters
Nπ 2
16−π2and 16−π2
4π. After some variable transformation, fZ(z)
in (26) can be obtained.
2In this paper, the channel reciprocity of the uplink (UL) and downlink
(DL) is assumed, which means that the UL and DL have the same optimal
phase shifts of IRS in mode-I. In fact, the same phase shifts lead to lower
feedback overhead and the reduction of implementation complexity [2].
With the help of Lemma 1, the ESC of the proposed three
modes is given as follows.
Theorem 1: The ESC of mode-I is given by
EI=(π2(1 −θ)
4KΓ(v) ln 2
K
X
k=1 q1−ω2
ksec2uk Γv, 1
ϕ4
qtan uk
ρtβ2
aru 
1 + tan uk
+(tan uk)v−1
ϕvexp 1
ρtβaru(N βure +βue) tan2uk
−tan uk
ϕEi−1
ρtβaru(N βure +βue) tan2uk!)+
,(27)
where uk=π(ωk+1)
4,ωk= cos 2k−1
2Kπ,Kis the accuracy
versus complexity parameter, Γ(·,·)is the upper incomplete
Gamma function [10, Eq. (8.35)], and Ei is the exponential
integral function [10, Eq. (8.211)].
Proof: See Appendix A.
Theorem 2: The ESC of mode-II is derived as
EII =(−π2(1 −θ)
4Kln 2
K
X
k=1 q1−ω2
ksec2uk (tan uk)v−1
Γ(v)ϕv
×exp 1
ρtβaruβpu tan2uk−tan uk
ϕEi −1
ρtβaruβpu tan2uk
+s4 tan uk
ρtβpu(N βur e +βue)
K1q4 tan uk
ρtβpu(N βure +βue)
1 + tan uk!)+
,
(28)
where K1(·)is the first order modified Bessel function of the
second kind [10, Eq. (8.432)], and the definitions of other
parameters are the same as (27).
Proof: See Appendix B.
Theorem 3: The ESC of mode-III is derived as
EII I =(π2(1 −θ)
4Kln 2
K
X
k=1 q1−ω2
ksec2uk
(tan uk)v−1
Γ(v)ϕve−
tan uk
ϕ
× e
1
ρtβpru(N βure
+βue) tan2ukEi −1
ρtβpru(N βure+βue) tan2uk
−e
1
ρtβpruβua tan2ukEi −1
ρtβpruβua tan2uk!)+
.(29)
Proof: With the optimal phase shift design of Θ3, one can
obtain hurΘ3hpr =PN
n=1 |hur,n||hpr,n|, where the PDF of
PN
n=1 |hur,n||hpr,n|is the same as Zin Lemma 1. Following
similar steps in the proof of Theorem 1, and with the helps of
Gaussian-Chebyshev quadrature and [10, Eq. (3.352.4)], the
theorem can be proved.
B. Asymptotic Analysis
In the following, the asymptotic ESC is analyzed for three
modes to obtain more intuitive insights into the impacts of key
system parameters. Specifically, when the transmission power
at HAP/PS is sufficiently large, i.e., Ps→ ∞, the asymptotic
ESC of three modes can be given as follows.
Proposition 1: The asymptotic ESC of mode-I can be
expressed as
EI
asy =1−θ
ln 2 ln βaruϕ2
Nβure +βue
+2ψ(v)+C+
,(30)

4
where ψ(·)is the psi function [10, Eq. (8.36)] and C=
0.5772... is Euler’s constant [10, Eq. (8.367)].
Proof: By substituting (10) and (16) into (25), when
Ps→ ∞, we have EI'En(1 −θ) log2βaruZ2
Ye,1o+
, where
Ye,1=hT
reΘ1hur √βure +hue √βue
2. Then, by using [10,
Eq. (4.352.1)] and [10, Eq. (4.331.1)], Eq. (30) is obtained.
Proposition 2: The asymptotic ESC of mode-II EII
asy is the
same as EI
asy.
Proof: Similar to the analysis of (37), Ye,2=
hT
reΘ2hur √βure +hue √βue
2subjects to an exponential
random variable with parameter Nβure +βue. Then, following
similar steps in the proof of (30), the proposition can be
proved.
Proposition 3: The asymptotic ESC of mode-III can be
given by
EII I
asy =(1 −θ) log2
βua
Nβure +βue +
.(31)
Proof: Following similar steps as in the proof of Propo-
sition 1, the proposition can be readily proved.
Corollary 1: When Ps→ ∞ and N→ ∞, it is ob-
tained that EI=EII ' {(1 −θ) log2N}+and EI II '
(1 −θ) log2(N−1)+.
Proof: We have ψ(v)→ln v[10, Eq. (8.362.2)] when
v=→ ∞. Hence, when N→ ∞,EI
asy in (30) can be further
expressed as
EI
asy '1−θ
ln 2 (ln N−1+ln N2)+
={(1−θ) log2N}+.(32)
Owing to EII
asy =EI
asy, one can obtain EI I
asy ' {(1 −
θ) log2N}+when N→ ∞. Following similar steps, EII I
asy '
{(1 −θ) log2(N−1)}+can be derived when N→ ∞. The
proof is completed.
Remark 1: It is observed from these propositions that
the asymptotic ESC of three modes is independent of the
transmission power Ps, indicating that the ESC of each mode
converges to a performance floor with the increase of Ps. This
is due to the fact that in addition to improving the information
reception at the legitimate receiver, a higher transmission pow-
er is helpful to the information reception at the eavesdropper.
Remark 2: It is concluded from Corollary 1 that the ESC
of mode-I and mode-II can be continuously improved by
increasing the number of IRS elements N. This means that
when the performance floor of ESC appears under a high Ps,
the ESC of mode-I and mode-II is still improved by increasing
N, showing the effectiveness of IRS with optimal phase shift.
In particular, the asymptotic ESC of mode-III is deteriorated
as Nincreases. This is because the increased Nenhances the
link from U to E but not enhances the link from U to AP in
information transmission phase of mode-III.
IV. SIMULATION RESULTS AN D DISCUSSION
This section presents numerical results to verify the theoret-
ical analysis. Without loss of generality, we assume dua = 20
m, dpu = 20 m, dur =dar = 10 m, dpr =dur = 10 m,
due =dre = 30 m, τ= 2.7,T= 1,θ= 0.4,η= 0.8,
Gu=Ga=Gp=Ge= 10 dBi, S= 0.1 m2,σ2=−58
dBm and K= 50. The traditional WPC without the aid of an
IRS is considered as a benchmark for comparison.
0 10 20 30 40 50 60
Ps (dBm)
0
0.5
1
1.5
2
2.5
3
3.5
4
Ergodic Secrecy Capacity (bps/Hz)
Mode-I
Mode-II
Mode-III
Benchmark without IRS
Asymptotic
Simulation
Fig. 2: The ESC versus Psfor different modes, where N= 32.
16 32 48 64 80 96 112 128 144 160 176
N
1
2
3
4
5
6
7
Ergodic Secrecy Capacity (bps/Hz)
Mode-I
Mode-II
Mode-III
Benchmark without IRS
Fig. 3: The ESC versus Nfor different modes, where Ps= 30 dBm.
Fig. 2 shows the ESC versus the transmission power of
HAP/PS Psfor different modes. It is seen that the derived
accurate expressions of ESC for three modes match well with
the simulation results, and the asymptotic ESC coincides with
the simulation results under high Ps, which demonstrates cor-
rectness of our ESC analysis. In particular, mode-I outperforms
mode-II with a low or moderate Ps, while mode-I and mode-II
achieve the same ESC floor with a high Ps. Moreover, both
mode-I and mode-II significantly outperform the benchmark
without the aid of IRS, indicating the superiority of the
IRS deployed over the information transmission link from
the perspective of enhancing ESC. And mode-III, deploying
the IRS to energy transfer link, outperforms the benchmark
without IRS under low and moderate Ps. But the increased
Psleads to more information leakage from the U-IRS-E link,
which makes the ESC in mode-III worse than that without IRS
under high Ps.
Fig. 3 shows the ESC versus the number of IRS elements N
based on different modes. It is observed that by increasing N,
the ESCs of mode-I and mode-II are continuously improved.
Note that the increased Nmakes the size and noise of IRS
larger in practice. The tradeoff between performance and cost
is an interesting investigation point, where Nis minimized
within an ESC constraint. In mode-III, the ESC is deteriorated
with the increase of N. This indicates that an IRS with optimal
phase shift should be deployed between U and HAP/AP, and
the beamforming gain of the IRS is helpful to the enhancement
of the ESC for IRS-WPC systems.
APPENDIX A
By using Jensen’s inequality (i.e., E[max{a, b}]≥
max{E[a],E[b]}), the ESC in (25) can be rewritten as
EI=En(1−θ) log2(1 +SNRI
a)−(1−θ) log2(1 +SNRI
e)o+
≥nE[(1−θ) log2(1 +SNRI
a)−(1−θ) log2(1 +SNRI
e)]o+
=nEI
a−EI
eo+
,(33)

5
where EI
a=E[(1−θ) log2(1+X1)],EI
e=E[(1−θ) log2(1+
Y1)],X1= SNRI
a, and Y1= SNRI
e.
We first derive EI
ain (33), which can be written as
EI
a=Z∞
0
(1 −θ) log2(1 + x1)fX1(x1)dx1
=1−θ
ln 2 Z∞
0
1−FX1(x1)
1 + x1
dx1.(34)
By substituting (22) into (34), and with the helps of Lemma
1 and [10, Eq. (3.381.3)], we have
1−FX1(x1) = Pr ρtβ2
aruZ4> x1
=Z∞
4
rx1
ρtβ2
aru
zv−1
Γ(v)ϕve−z
ϕdz =
Γv, 1
ϕ4
qx1
ρtβ2
aru 
Γ(v).(35)
After substituting (35) into (34), it is quite difficult to
obtain a closed-form expression for EI
a, and hence we apply
Gaussian-Chebyshev quadrature [7] to yield a close approxi-
mation. Specifically, by changing the variable of x1= tan uk,
EI
acan be further derived as
EI
a=1−θ
Γ(v) ln 2 Z∞
0
Γv, 1
ϕ4
qx1
ρtβ2
aru 
1 + x1
dx1
=π2(1 −θ)
4KΓ(v) ln 2
K
X
k=1 q1−ω2
ksec2uk
Γv, 1
ϕ4
qtan uk
ρtβ2
aru 
1 + tan uk
.(36)
Similarly, we have
EI
e=1−θ
ln 2 Z∞
0
1−FY1(y1)
1 + y1
dy1
=1−θ
ln 2 Z∞
0
Pr ρtβaruZ2Ye,1> y1
1 + y1
dy1,(37)
where Ye,1=hT
reΘ1hur √βure +hue √βue
2.
In order to derive EI
ein (37), we need to determine
the statistic of Ye,1. In particular, hT
reΘ1hur in Ye,1is the
sum of complex-valued random variables, and its real and
imaginary parts are correlated. Accordingly, we can take
hT
reΘ1hur as a complex Gaussian random distribution with
zero mean and variance of N. Based on this result, Ye,1=
hT
reΘ1hur √βure +hue √βue
2subjects to an exponential
random variable with parameter Nβure +βue. Hence, we have
Pr ρtβaruZ2Ye,1> y1=Z∞
0
FYe,1y1
ρtβaruz2fZ(z)dz
=Z∞
0
e−y1
ρtβaru(N βure +βue)z2e−z
ϕzv−1
Γ(v)ϕvdz
=π2
4K
K
X
k=1 q1−ω2
ksec2uk
(tan uk)v−1
Γ(v)ϕve−tan uk
ϕ
×e−y1
ρtβaru(N βure +βue) tan2uk,(38)
where the last result in (38) is with the help of Gaussian-
Chebyshev quadrature. Substituting (38) into (37), and using
[10, Eq. (3.352.4)], we have
EI
e=π2(1 −θ)
4Kln 2
K
X
k=1 q1−ω2
ksec2uk
(tan uk)v−1
Γ(v)ϕve−tan uk
ϕ
×Z∞
0
e−y1
ρtβaru(N βure +βue) tan2uk
1 + y1
dy1
=−π2(1 −θ)
4Kln 2
K
X
k=1 q1−ω2
ksec2uk
(tan uk)v−1
Γ(v)ϕve−tan uk
ϕ
×e
1
ρtβaru(N βure +βue) tan2ukEi−1
ρtβaru(N βur e+βue) tan2uk.
(39)
Substituting (36) and (39) into (33), the theorem is proved.
APPENDIX B
By using Jensen’s inequality, we have EII ≥ {EII
a−
EII
e}+, where EII
a=E[(1 −θ) log2(1 + SNRII
a)],EII
e=
E[(1 −θ) log2(1 + SNRII
e)]. Similar to the derivation in
Theorem 1, with the helps of Gaussian-Chebyshev quadrature
and [10, Eq. (3.352.4)], EII
acan be derived as
EII
a=Z∞
0
(1 −θ) log2(1 + x2)fX2(x2)dx2
=1−θ
ln 2 Z∞
0
1−FX2(x2)
1 + x2
dx2
=π2(1 −θ)
4Kln 2
K
X
k=1 q1−ω2
ksec2uk
(tan uk)v−1
Γ(v)ϕve−tan uk
ϕ
×Z∞
0
e
−x2
ρtβaruβpu tan2uk
1 + x2
dx2
=−π2(1 −θ)
4Kln 2
K
X
k=1 q1−ω2
ksec2uk
(tan uk)v−1
Γ(v)ϕve−tan uk
ϕ
×e
1
ρtβaruβpu tan2ukEi −1
ρtβaruβpu tan2uk,(40)
where X2= SNRII
ain (23).
Following similar steps in (37), and using [10, Eq. (3.324.1)]
and Gaussian-Chebyshev quadrature, we have
EII
e=1−θ
ln 2 Z∞
0
1−FY2(y2)
1 + y2
dy2
=1−θ
ln 2 Z∞
0s4y2
ρtβpu(N βure+βue)
K1
q4y2
ρtβpu(N βure
+βue)
1 + y2
dy2
=π2(1 −θ)
4Kln 2
K
X
k=1 q1−ω2
ksec2uks4 tan uk
ρtβpu(N βure +βue )
×
K1q4 tan uk
ρtβpu(N βure +βue)
1 + tan uk
,(41)
where Y2= SNRII
ein (17).
Combining the results of EII
aand EII
e, the ESC of mode-II
in closed-form can be obtained. Hence, the proof is completed.
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