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arXiv:2005.00996v1 [eess.SP] 3 May 2020
1
Downlink and Uplink Intelligent Reflecting
Surface Aided Networks: NOMA and OMA
Yanyu Cheng, Student Member, IEEE, Kwok Hung Li, Senior Member, IEEE,
Yuanwei Liu, Senior Member, IEEE, Kah Chan Teh, Senior Member, IEEE, and
H. Vincent Poor, Life Fellow, IEEE
Abstract
Intelligent reflecting surfaces (IRSs) are envisioned to provide reconfigurable wireless environments
for future communication networks. In this paper, both downlink and uplink IRS-aided non-orthogonal
multiple access (NOMA) and orthogonal multiple access (OMA) networks are studied, in which an
IRS is deployed to enhance the coverage by assisting a cell-edge user device (UD) to communicate
with the base station (BS). To characterize system performance, new channel statistics for the BS-
IRS-UD link with Nakagami-mfading are investigated. For each scenario, closed-form expressions
for the outage probability and the ergodic rate are derived. To gain further insight, the diversity order
and the high signal-to-noise ratio (SNR) slope for each scenario are obtained according to asymptotic
approximations in the high-SNR regime. It is demonstrated that the diversity order is affected by the
number of IRS elements and Nakagami-mfading parameters, but the high-SNR slope is not related to
these parameters. Simulation results validate our analysis and reveal the superiority of the IRS over the
full-duplex decode-and-forward relay.
Index Terms
Intelligent reflecting surface (IRS), non-orthogonal multiple access (NOMA), orthogonal multiple
access (OMA).
Part of this work has been submitted to IEEE Global Communications Conference (GLOBECOM) 2020 [1].
Y. Cheng, K. H. Li, and K. C. Teh are with the School of Electrical and Electronic Engineering, Nanyang Technological
University, Singapore 639798 (e-mail: ycheng022@e.ntu.edu.sg; ekhli@ntu.edu.sg; ekcteh@ntu.edu.sg).
Y. Liu is with the School of Electronic Engineering and Computer Science, Queen Mary University of London, London E1
4NS, UK (e-mail: yuanwei.liu@qmul.ac.uk).
H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA (e-mail:
poor@princeton.edu).
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I. INT RODU CTI ON
Recently, intelligent reflecting surfaces (IRSs) have been proposed as a cost-effective solution
to enhance the spectral and energy efficiency of future wireless communication networks [2],
[3]. Specifically, an IRS consists of a large number of reconfigurable passive elements, and
each element can induce a change of amplitude and phase for the incident signal [4], [5]. By
appropriately adjusting amplitude-reflection coefficients and phase-shift variables, it can improve
the link quality and enhance the coverage significantly [6]–[8]. Compared with the conventional
communication assisting techniques such as relays, the IRS consumes less energy due to passive
reflection and is able to operate in full-duplex (FD) mode without self-interference [9].
IRSs have attracted intensive research interest in both academia and industry [10], [11]. In
[12], the superiority of an IRS was indicated as compared with a half-duplex (HD) decode-and-
forward (DF) relay. In [13], the performance of a system aided by multiple IRSs was evaluated,
and it was revealed that the IRS-aided system outperforms the FD DF relay (FDR)-aided system
when the number of IRSs exceeds a certain value. Besides wireless communication systems,
IRSs can also be applied to optical communication systems. In [14], it was revealed that the
outage probability (OP) for an optical communication system can be reduced by introducing an
IRS. Passive-beamforming design for IRSs is critical to system performance. In [15], multiple
IRSs were deployed in a single-input single-output (SISO) system, and the phase shifts of
IRSs were optimized by minimizing the OP. Active and passive beamforming were jointly
optimized for IRS-aided multiple-input single-output (MISO) systems in [16]–[18] with different
objectives. All aforementioned works assumed the ideal phase-shift model, which is difficult to
realize in practice. Hence, the phase shift was assumed to be discrete in [19], and a phase-shift
model in which the reflection coefficient is related to the phase shift was proposed in [20]. For
physical layer security, IRSs can also enhance system performance by blocking the signals of
eavesdroppers [21], [22].
On the other hand, non-orthogonal multiple access (NOMA) has been proposed as a candidate
technique for next-generation wireless communication networks as it can improve the spectral
efficiency by allocating multiple user devices (UDs) to a single resource block [23], [24]. In [25],
the authors presented a fair power allocation algorithm and demonstrated that NOMA always
has a larger capacity than orthogonal multiple-access (OMA) under different channel conditions.
Allocating many UDs to a single carrier is not practical since it leads to high computational
3
complexity [26]. Hence, UD pairing is a practical solution that can strike a balance between
the performance and the computational complexity [27]. In [28], the impact of UD pairing in
NOMA systems was investigated, and two UDs with distinct channel conditions were paired
to improve system performance. In [29], a two-tier NOMA system was proposed to ensure the
distinction of channel conditions for two UDs in a pair.
Since both IRS and NOMA are promising candidate techniques for future wireless commu-
nication networks, their combination has been investigated recently [30]–[36]. In [30], an IRS
was deployed to improve the coverage by assisting a cell-edge UD in data transmission, where
this cell-edge UD is paired with a cell-center UD under the NOMA scheme. The authors further
investigated the impact of random phase shifting and coherent phase shifting for an IRS-aided
NOMA system in [31]. In [32] and [33], beamforming vectors of the base station (BS) and the
IRS were optimized for an IRS-assisted NOMA system. The aforementioned works assumed that
the BS-IRS-UD channel is non-line-of-sight (NLoS). Since the IRS can be pre-deployed, the path
between the BS and the IRS can be line-of-sight (LoS) [34]–[36]. In [34], the authors assumed
the BS-IRS-UD link to be LoS and optimized the active and passive beamforming vectors for
a NOMA system aided by an IRS. In [35], the authors studied an IRS-assisted NOMA system
by considering ideal and non-ideal IRSs. In [36], the IRS’s parameters were designed for a
prioritized UD in an IRS-assisted NOMA network.
A. Motivation and Contributions
For IRS-aided NOMA networks, the channel statistics for the BS-IRS-UD link are crucial for
the performance evaluation. Most of the existing works have assumed that the BS-IRS-UD link
is NLoS, such as [30]–[33]. Some works have assumed that the BS-IRS-UD link is LoS, but
without deriving exact channel statistics, e.g., [34]–[36]. This motivates us to consider the BS-
IRS-UD link to be either LoS or NLoS, and derive the channel statistics for further analysis. In
addition, since the aim of deploying the IRS is to improve the coverage by enhancing the power
of cell-edge UDs, we consider a NOMA network in which a cell-edge UD cannot communicate
with the BS directly and needs the assistance from an IRS. The cell-edge UD is paired with a
cell-center UD to improve system performance [28], [29]. As compared with [1], this paper is
more comprehensive, and the contributions can be summarized as follows:
•We study both downlink and uplink transmissions for IRS-aided NOMA and OMA networks.
In particular, we investigate two aspects for each scenario, namely, the OP and the ergodic
4
rate (ER).
•We adopt the Nakagami-mfading model for the BS-IRS-UD link so that this link can be
either LoS or NLoS. Correspondingly, we derive new channel statistics for this link based
on the central limit theorem (CLT). Since the CLT-based channel statistics are inaccurate
near 0, we further derive exact channel statistics near 0by utilizing the Laplace transform.
•We derive closed-form expressions for the OP and the ER for each scenario. To gain further
insight, we further derive asymptotic approximations of the OP and the ER at a high signal-
to-noise ratio (SNR) to obtain the diversity order and the high-SNR slope, respectively. We
demonstrate that the number of IRS elements and Nakagami-mfading parameters affect
the diversity order but have no influence on the high-SNR slope.
•Finally, we compare the performance of IRS-aided NOMA networks with that of FDR-
aided NOMA networks. Simulation results reveal the superiority of the IRS over FDR in
the high-SNR regime.
B. Organization and Notation
The remainder of this paper is organized as follows: In Section II, the model of the IRS-aided
NOMA network is described. New channel statistics of the BS-IRS-UD link are presented in
Section III. The performance analysis for the downlink is conducted in Section IV, followed
by the analysis for the uplink in Section V. Furthermore, numerical and simulation results are
presented in Section VI. Finally, our conclusions are drawn in Section VII.
In this paper, scalars are denoted by italic letters. Vectors and matrices are denoted by bold-
face letters. For a vector v, diag(v)denotes a diagonal matrix in which each diagonal element
is the corresponding element in v, respectively. vTdenotes the transpose of v.arg(·)denotes
the argument of a complex number. Cx×ydenotes the space of x×ycomplex-valued matrices.
Pr(·)and E(·)denote the probability and the expectation, respectively.
II. MO DEL OF IRS-AIDED NOMA NET WO RKS
We consider an IRS-aided NOMA network with a single-antenna BS and two single-antenna
UDs, denoted by N and F, respectively, as shown in Fig. 1. More specifically, N is the cell-center
UD that can communicate with the BS directly, while F is the cell-edge UD that needs help
from an IRS for communications since there is no direct link between the BS and F due to long
distance and blocking objects. The IRS has Kreflection elements, and its reflection-coefficient
5
Fig. 1: Model of the IRS-aided NOMA network.
matrix is denoted by Θ=diag β1ejθ1, β2ejθ2,···, βKejθK, where βk∈[0,1] is the amplitude-
reflection coefficient and θk∈[0,2π)is the phase-shift variable of the kth element that can be
adjusted by the IRS (k= 1,2,···, K).
A. Channel Model
All channels experience quasi-static flat fading, and the channel state information (CSI) of all
channels is assumed to be perfectly known at the BS. The link between the BS and N is assumed
to be NLoS as it is a link between the BS and the ground-UD. Hence, the small-scale fading
between the BS and N follows the Rayleigh fading model and is denoted by h∼CN(0,1),
where CN(·,·)is the complex Gaussian distribution. The BS-IRS and IRS-F links can be either
LoS or NLoS for different scenarios. Due to the severe path loss, the signals that are reflected by
the IRS twice or more times are ignored. The small-scale fading vector between the BS and the
IRS is denoted by G∈C1×K. The small-scale fading vector between the IRS and F is denoted by
g∈CK×1. Particularly, they are G= [G1, G2,···, GK]and g= [g1, g2,···, gK]T, respectively.
All elements in Gand gfollow the Nakagami-mfading model with fading parameters, mGand
mg, respectively 1. In particular, it is NLoS for mG= 1 and is LoS for mG>1 (G ∈ {G, g}).
1The Nakagami-mfading model and the Rician fading model can be transformed into each other. It obeys the rule of
m=(K+1)2
2K+1 , where mand Kare fading parameters for the Nakagami-mand the Rician models, respectively [37, eq. (3.38)].
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B. Signal Model
1) Downlink: The BS transmits the signal x=√α1Pbsd
1+√α2Pbsd
2, where Pbdenotes the
transmit power of the BS, sd
1and sd
2denote the transmitted signals to N and F, respectively, and
α1and α2denote the power allocation coefficients for N and F, respectively (α1≪α2). The
received signals at N and F are given by
yN=hd−αh
2
Nx+n1,(1)
and
yF=GΘgd−αG
2
Rd−αg
2
Fx+n2,(2)
respectively, where dNand dRdenote the distances from the BS to N and the IRS, respectively,
dFdenotes the distance between the IRS and F, αh,αG, and αgdenote the path loss exponents for
BS-N, BS-IRS, and IRS-F links, respectively, and n1and n2denote the additive white Gaussian
noises (AWGNs) at N and F, respectively, with the same variance σ2
n.
At N, the signal of F is detected first, and the corresponding signal-to-interference-plus-noise
ratio (SINR) is given by
SINRd
N,F =|h|2d−αh
Nα2
|h|2d−αh
Nα1+ 1/ρ,(3)
where ρ=Pb/σ2
ndenotes the transmit SNR of the BS. After implementing the successive
interference cancellation (SIC), the signal of N is decoded, and the corresponding SNR is given
by
SNRd
N=|h|2d−αh
Nα1ρ. (4)
At F, its signal is decoded directly by regarding N’s signal as interference, and its SINR is given
by
SINRd
F=|GΘg|2d−αG
Rd−αg
Fα2
|GΘg|2d−αG
Rd−αg
Fα1+ 1/ρ.(5)
Remark 1. For the adaptive-rate transmission, the SINR at F is min SINRd
N,F ,SINRd
F.
However, in the considered system, since F has much more severe path loss than N (dN≪dF),
we have min SINRd
N,F ,SINRd
F≈SINRd
F.
7
2) Uplink: The received signal at the BS is given by
y=hd−αh
2
NpPusu
1+GΘgd−αG
2
Rd−αg
2
FpPusu
2+n, (6)
where Pudenotes the transmit power of each UD, su
1and su
2denote the transmitted signals from
N and F, respectively, and ndenotes the AWGN at the BS with variance σ2.
At the BS, the signal of N is decoded first by regarding the signal from F as interference, and
the corresponding SINR is given by
SINRu
N=|h|2d−αh
N
|GΘg|2d−αG
Rd−αg
F+ 1/ρ′,(7)
where ρ′=Pu/σ2denotes the transmit SNR of each UD. After carrying out the SIC, F’s signal
is detected with the SNR given by
SNRu
F=|GΘg|2d−αG
Rd−αg
Fρ′.(8)
III. NEW CHANNEL STATIS TIC S OF THE BS-IRS-F LIN K
In this section, we will present the optimized IRS’s parameters and the channel statistics for
the BS-IRS-F link.
A. Parameters of the IRS
For the BS-IRS-F link, we aim to provide the best channel quality to F by adjusting parameters
of the IRS. That is to maximize |GΘg|=PK
k=1 βkGkgkejθkj=√−1, where Gkand gk
are the kth element of Gand g, respectively. This can be achieved by intelligently adjusting
the phase-shift variable θkfor each element, i.e., the phases of all Gkgkejθkare set to be the
same. Therefore, there is not only one solution for {θk}(k= 1,2,···, K), and the generalized
solution is given by θk=˜
θ−arg(Gkgk), where ˜
θis an arbitrary constant ranging in [0,2π).
After adopting the optimal {θk}, we have
|GΘg|2=β2 K
X
k=1 |Gk||gk|!2
,(9)
where we assume that βk=β∀kwithout loss of generality.
8
0 50 100 150 200 250
X
0
0.01
0.02
fX(x)
Exact - Simulation
Approximation - CLT
0 50 100 150 200 250
X
0
0.5
1
FX(x)
Exact - Simulation
Approximation - CLT
Fig. 2: The PDF and the CDF of Xderived by the CLT when K= 30,mG= 3, and mg= 2.
B. New Channel Statistics
New channel statistics can be obtained by using the CLT as shown in the following lemma,
which has been verified by Monte Carlo simulations as shown in Fig. 2.
Lemma 1. Denote that X=(PK
k=1 |Gk||gk|)2
K(1−ξ), where ξ=1
mGmgΓ(mG+1
2)
Γ(mG)2Γ(mg+1
2)
Γ(mg)2
. When
the number of IRS elements Kis large, Xtends to follow a noncentral chi-square distribution
as X˙∼χ′2
1(λ), where λ=Kξ
1−ξ. Its probability density function (PDF) and cumulative distribution
function (CDF) are given by
fX(x) = λ1
4
2e−x+λ
2x−1
4I−1
2√λx=e−x+λ
2
∞
X
i=0
λixi−1
2
i!22i+1
2Γi+1
2,(10)
for x≥0and
FX(x) = 1 −Q1
2√λ, √x=e−λ
2
∞
X
i=0
λiγi+1
2,x
2
i!2iΓi+1
2,(11)
for x≥0, respectively, where Iv(·)is the modified Bessel function of the first kind, Qv(·,·)is
the Marcum Q-function, Γ(·)is the gamma function, and γ(·,·)is the lower incomplete gamma
function.
Proof. Please refer to Appendix A.
9
Remark 2. The use of CLT results in that fX(x)is not accurate for x→0+. However, we can
still carry out meaningful analysis based on the approximated PDF since it is accurate except
in the case of R0+
0g(x)fX(x)dx, where g(x)is any function of x.
To address the issue stated in Remark 2, we further derive exact channel statistics near 0
without using the CLT as shown in the following lemma.
Lemma 2. Denote that Z=PK
k=1 |Gk||gk|. When mG6=mg, the PDF and the CDF of Zfor
z→0+are given by
f0+
Z(z) = ˜mK
Γ(2msK)z2msK−1e−2√msmlz,(12)
for z≥0and
F0+
Z(z) = ˜mK(4msml)−msK
Γ(2msK)γ(2msK, 2√msmlz),(13)
for z≥0, respectively, where ˜m=√π4ms−ml+1(msml)msΓ(2ms)Γ(2ml−2ms)
Γ(ms)Γ(ml)Γ(ml−ms+1
2)with ms= min{mG, mg}
and ml= max{mG, mg}.
Proof. Please refer to Appendix B.
IV. PERFORM ANC E ANA LYSIS FOR DOWNLI NK TRANS MIS SIO N
In this section, we will investigate the performance of downlink NOMA and OMA networks.
In particular, for each scenario, the OP for the fixed-rate transmission and the ER for the adaptive-
rate transmission are derived.
A. NOMA
1) Outage Probability: For the fixed-rate transmission system, OP is a widely used metric to
measure system performance. The OPs of N and F are given by
Pd
N= 1 −PrSINRd
N,F ≥˜γF,SNRd
N≥˜γN,(14)
and
Pd
F= PrSINRd
F<˜γF,(15)
10
respectively, where ˜γN= 2 ˜
RN−1and ˜γF= 2 ˜
RF−1with ˜
RNand ˜
RFbeing the target rates
of N and F, respectively. Based on the new channel statistics in Lemma 1, the OPs of N and F
are derived as shown in the following theorem.
Theorem 1. In the considered IRS-aided NOMA network, the OPs of N and F for the downlink
are given by
Pd
N= 1 −e−˜ρm,(16)
and
Pd
F≈e−λ
2
∞
X
i=0
λiγi+1
2,˜ρ
2
i!2iΓi+1
2,(17)
respectively, where ˜ρm= max n˜γF
(α2−α1˜γF)aρ ,˜γN
aα1ρoand ˜ρ=˜γF
(α2−α1˜γF)bρ with a=d−αh
Nand
b=Kβ2(1 −ξ)d−αG
Rd−αg
F. Note that when we set power allocation coefficients, we need to
ensure that α2−α1˜γF>0.
Proof. First, we denote that Y=|h|2, and its PDF and CDF are given by fY(y) = e−yfor
y≥0and FY(y) = 1 −e−yfor y≥0, respectively. Then, Pd
Ncan be derived as
Pd
N= 1 −Praα2Y
aα1Y+ 1/ρ ≥˜γF, aα1ρY ≥˜γN=FY( ˜ρm).(18)
On the other hand, Pd
Fcan be derived as
Pd
F= Prbα2X
bα1X+ 1/ρ <˜γF≈FX( ˜ρ).(19)
This completes the proof.
Remark 3. According to Remark 2, the OP of F is accurate in the low-SNR regime but is
inaccurate in the high-SNR regime. Because the OP of F for ρ→ ∞ is derived by using
R0+
0fX(x)dx.
To solve the aforementioned problem, we can use Lemma 2 to derive the high-SNR approx-
imation of F’s OP as shown in the following proposition.
Proposition 1. In the high-SNR regime, Pd
Ncan be approximated as
Pd,∞
N= ˜ρm.(20)
11
When mG6=mg, the high-SNR approximation of Pd
Fis given by
Pd,∞
F=˜mK˜c2msK
1
Γ(2msK+ 1)ρ−msK,(21)
where ˜c1=q˜γF
c(α2−α1˜γF)with c=β2d−αG
Rd−αg
F.
Proof. By expanding the exponential function in (16) and extracting the leading-order term, we
can obtain (20). On the other hand, by expanding the lower incomplete gamma function in (13),
we have
F0+
Z(z) = ˜mK∞
X
l=0
(2√msml)lzl+2msK
Γ(l+ 2msK+ 1) e−2√msmlz.(22)
Then, by using the expansion of the exponential function, we have
F0+
Z(z) = ˜mK∞
X
l=0
(2√msml)lzl+2msK
Γ(l+ 2msK+ 1)
∞
X
i=0 −2√msmlizi
i!.(23)
Meanwhile, the OP of F for ρ→ ∞ can be derived as
Pd,∞
F= Prcα2Z2
cα1Z2+ 1/ρ <˜γF=F0+
Z˜c1ρ−1
2.(24)
Finally, by substituting z= ˜c1ρ−1
2into (23) and extracting the leading-order term, we can
approximate Pd,∞
Fas (21). This completes the proof.
Corollary 1. In the considered IRS-aided NOMA network, the diversity orders of N and F for
the downlink are given by dd
N= 1 and dd
F=msK, respectively.
Proof. Based on Proposition 1, we can obtain that the diversity order of F is msKwhen
mG6=mg. For the case of mG=mg, it is also msKaccording to the limit. The proof is
completed.
Remark 4. The diversity order of F for the downlink NOMA is affected by the number of IRS
elements and Nakagami-mfading parameters.
2) Ergodic Rate: For the adaptive-rate transmission system, ER is a commonly used metric
to measure system performance. The ERs of N and F are given by
Rd
N=Elog21 + SNRd
N,(25)
12
and
Rd
F=Elog21 + SINRd
F,(26)
respectively. After some mathematical manipulations, the ERs of N and F are obtained as
presented in the following theorem.
Theorem 2. In the considered IRS-aided NOMA network, the ERs of N and F for the downlink
are, respectively, given by
Rd
N=−e1
aα1ρ
ln(2)Ei −1
aα1ρ,(27)
and
Rd
F≈log2(1 + ˜α)−e−λ
2
ln(2)
∞
X
i=0
λi
i!2iΓi+1
2
u1
X
l=1
ω1,lJ1(tl),(28)
where Ei(·)is the exponential integral, ˜α=α2
α1,ω1,l =π
u1,tl=cos 2l−1
2u1π, and
J1(t) = γi+1
2,˜α(1 + t)
4bρα2−2bρα1˜α(1 + t)√1−t2
1 + 2
˜α+t.(29)
Proof. Please refer to Appendix C.
Remark 5. The ER of F is accurate as the calculation is not related to R0+
0g(x)fX(x)dx.
Because of the principle of downlink NOMA, the ER of N increases for the increase of the SNR,
while the ER of F approaches a ceiling when ρ→ ∞.
To provide insight into the performance, the high-SNR slope is considered that is defined as
S= lim
ρ→∞
R(ρ)
log2(ρ)[38]. To obtain it, the asymptotic expression for N’s ER and the ceiling for F’s
ER are derived in the following proposition.
Proposition 2. In the high-SNR regime, Rd
Ncan be approximated as
Rd,∞
N= log2(aα1ρ)−Ec
ln(2),(30)
where Ecdenotes the Euler constant. On the other hand, the ceiling for Rd
Fin the high-SNR
regime is given by
Rd,∞
F= log2(1 + ˜α).(31)
13
Proof. By using lim
x→0ex= 1 and Ei(−x)≈ln(x) + Ecfor x→0[39, eq. (8.214.2)], we can
approximate (27) as (30) when ρ→ ∞. On the other hand, since we have lim
ρ→∞
bα2X
bα1X+1/ρ = ˜α,
we can approximate (26) as (31) when ρ→ ∞. This completes the proof.
Corollary 2. In the considered IRS-aided NOMA network, the high-SNR slopes of N and F for
the downlink are given by Sd
N= 1 and Sd
F= 0, respectively.
Proof. We have Sd
N=dRd,∞
N
dlog2(ρ)= ln(2)ρdRd,∞
N
dρ = 1, and this completes the proof.
Remark 6. The high-SNR slope of F for the downlink NOMA is not related to the number of
IRS elements and Nakagami-mfading parameters.
B. OMA
OMA can be regarded as a special case of NOMA. The data rates of N and F under the OMA
scheme are given by 1
2log21 + |h|2d−αh
Nρand 1
2log21 + β2PK
k=1 |Gk||gk|2d−αG
Rd−αg
Fρ,
respectively. Here, we assume that each UD share half of the resource block.
1) Outage Probability: The OPs of N and F for the downlink are presented in the following
theorem.
Theorem 3. When N and F are under the downlink OMA scheme, their OPs are given by
Pdo
N= 1 −e−˜γo
N
aρ ,(32)
and
Pdo
F≈e−λ
2
∞
X
i=0
λiγi+1
2,˜γo
F
2bρ
i!2iΓi+1
2,(33)
respectively, where ˜γo
N= 22˜
RN−1and ˜γo
F= 22˜
RF−1.
Proof. Pdo
Nand Pdo
Fcan be derived as follows:
Pdo
N= Pr(aρY < ˜γo
N) = FY˜γo
N
aρ ,(34)
and
Pdo
F= Pr(bρX < ˜γo
F)≈FX˜γo
F
bρ .(35)
This completes the proof.
14
Remark 7. According to Remark 2,Pdo
Fis accurate in the low-SNR regime but is inaccurate in
the high-SNR regime. Because Pdo
Ffor ρ→ ∞ is derived by using R0+
0fX(x)dx.
To address the problem in Remark 7, we use Lemma 2 and obtain the high-SNR approxi-
mation of Pdo
Fthat is shown in the following proposition.
Proposition 3. In the high-SNR regime, Pdo
Ncan be approximated as
Pdo,∞
N=˜γo
N
aρ−1.(36)
When mG6=mg, the high-SNR approximation of Pdo
Fis given by
Pdo,∞
F=˜mK˜c2msK
2
Γ(2msK+ 1)ρ−msK,(37)
where ˜c2=q˜γo
F
c.
Proof. Similar to the proof of Proposition 1.
Corollary 3. When N and F are under the downlink OMA scheme, the diversity orders of N
and F are given by ddo
N= 1 and ddo
F=msK, respectively.
Remark 8. The diversity order of F for the downlink OMA is affected by the number of IRS
elements and Nakagami-mfading parameters. Moreover, the diversity orders of N and F for the
downlink OMA are the same as these for the downlink NOMA, respectively.
2) Ergodic Rate: The ERs of N and F for OMA are presented in the following theorem.
Theorem 4. When N and F are under the downlink OMA scheme, their ERs are, respectively,
given by
Rdo
N=−e1
aρ
2 ln(2)Ei −1
aρ,(38)
and
Rdo
F≈λ1
4
4e−λ
2
u2
X
l=1
ω2,lJ2(x2,l ),(39)
where x2,l is the lth root of Laguerre polynomial Lu2(x),ω2,l =x2,l
(u2+1)2(Lu2+1(x2,l ))2, and
J2(x) = x−1
4ex
2log2(1 + bρx)I−1
2√λx.(40)
15
Proof. For N, the analysis is similar to the proof of Theorem 2. For the ER of F, it can be
expressed as
Rdo
F≈Z∞
0
1
2log2(1 + bρx)fX(x)dx =λ1
4
4e−λ
2Z∞
0
x−1
4e−x
2log2(1 + bρx)I−1
2√λxdx
|{z }
J2
.
(41)
Next, J2can be approximated by using the Gauss-Laguerre quadrature. As such, we have
J2≃
u2
X
l=1
ω2,lJ2(x2,l ).(42)
This completes the proof.
Remark 9. The ER of F is accurate as the calculation is not related to R0+
0g(x)fX(x)dx. On
the other hand, both ERs of N and F increase as the SNR increases.
Then, the approximations in the high-SNR regime are given in the following proposition.
Proposition 4. In the high-SNR regime, the approximations of Rdo
Nand Rdo
Fare given by
Rdo,∞
N=1
2log2(aρ)−Ec
ln(2),(43)
and
Rdo,∞
F=1
2log2(1 + bρ(1 + λ)) ,(44)
respectively.
Proof. For N, the analysis is similar to the proof of Proposition 2. For F, we have E(X) = 1+ λ
according to the property of the noncentral chi-square distribution. Then, since 1
2log2(1 + bρX)
with respect to Xis concave, with the aid of the Jensen’s inequality [40], we have
Rdo
F=E1
2log2(1 + bρX)≤1
2log2(1 + bρE(X)) = 1
2log2(1 + bρ(1 + λ)) .(45)
When ρ→ ∞,Rdo
Fcan be approximated as (44), and the proof is completed.
Corollary 4. When N and F are under the downlink OMA scheme, their high-SNR slopes are
given by Sdo
N= 0.5and Sdo
F= 0.5, respectively.
Proof. Similar to the proof of Corollary 2.
16
Remark 10. The high-SNR slope of N for the downlink OMA is half of that for the downlink
NOMA. On the other hand, the high-SNR slope of F in downlink OMA networks, which is not
related to the number of IRS elements and Nakagami-mfading parameters, is larger than that
in downlink NOMA networks.
V. PERFO RMA NCE ANALYS IS FO R UPLI NK TRANS MISSIO N
In this section, we will investigate the performance of the uplink transmission. The analyses
of OP and ER will be presented in the following.
A. NOMA
1) Outage Probability: The OPs of N and F are, respectively, given by
Pu
N= Pr(SINRu
N<˜γN),(46)
and
Pu
F= 1 −Pr(SINRu
N≥˜γN,SNRu
F≥˜γF).(47)
Then, the closed-form expressions for OPs can be derived, and the results are shown in the
following theorem.
Theorem 5. In the considered IRS-aided NOMA network, the OPs of N and F for the uplink
are given by
Pu
N≈1−e−˜γN
aρ′−λ
2∞
X
i=0
λi
i!22i+1
2b˜γN
a+1
2i+1
2
,(48)
and
Pu
F≈1−e−˜γN
aρ′−λ
2∞
X
i=0
λiΓi+1
2,˜γN˜γF
aρ′+˜γF
2bρ′
i!22i+1
2b˜γN
a+1
2i+1
2Γi+1
2,(49)
respectively, where Γ(·,·)is the upper incomplete gamma function.
Proof. The OP of N can be transformed into
Pu
N= PraY
bX + 1/ρ′<˜γN≈Z∞
0Zb˜γNx
a+˜γN
aρ′
0
fY(y)dyfX(x)dx
= 1 −e−˜γN
aρ′−λ
2∞
X
i=0
λi
i!22i+1
2Γi+1
2Z∞
0
e−b˜γN
a+1
2xxi−1
2dx.
(50)
17
Furthermore, by referring to [39, eq. (3.381.4)], (48) can be derived. Similarly, the OP of F can
be first transformed into
Pu
F= 1 −PraY
bX + 1/ρ′≥˜γN, bρ′X≥˜γF≈1−Z∞
˜γF
bρ′Z∞
b˜γNx
a+˜γN
aρ′
fY(y)dyfX(x)dx
= 1 −e−˜γN
aρ′−λ
2∞
X
i=0
λi
i!22i+1
2Γi+1
2Z∞
˜γF
bρ′
e−b˜γN
a+1
2xxi−1
2dx.
(51)
Then, by referring to [39, eq. (3.381.3)], we can derive the closed-form expression as (49). This
completes the proof.
Remark 11. The OP of N is accurate since it does not involve the integral of R0+
0g(x)fX(x)dx.
Although the calculation of F’s OP is related to R0+
0g(x)fX(x)dx, it only has a small bias in
the medium-SNR regime. This is because F’s OP must be higher than N’s OP and converges to
a floor in the high-SNR regime with N’s OP due to the principle of uplink NOMA.
The floor for OPs of both UDs is presented in the following proposition.
Proposition 5. In the high-SNR regime, Pu
Nand Pu
Fapproach the same floor as
Pu,∞
both = 1 −e−λ
2
∞
X
i=0
(λ)i
i!22i+1
2b˜γN
a+1
2i+1
2
.(52)
Proof. We have that lim
ρ′→∞
e−˜γN
aρ′= 1 and lim
ρ′→∞
Γi+1
2,˜γN˜γF
aρ′+˜γF
2bρ′= Γ i+1
2. Thus, the
limit of both Pu
Nand Pu
Ffor ρ′→ ∞ can be obtained, and the proof is completed.
Corollary 5. In the considered IRS-aided NOMA network, both diversity orders of N and F for
the uplink are 0, i.e., du
N= 0 and du
F= 0.
Remark 12. The diversity order of F for the uplink NOMA is not related to the number of IRS
elements and Nakagami-mfading parameters.
2) Ergodic Rate: The ERs of the N and F are given by
Ru
N=E(log2(1 + SINRu
N)) ,(53)
and
Ru
F=E(log2(1 + SNRu
F)) ,(54)
18
respectively. Then, the ERs of N and F can be obtained, and the results are provided in the
following theorem.
Theorem 6. In the considered IRS-aided NOMA network, the ER of N for the uplink is given
by
Ru
N≈ − λ1
4
2 ln(2)e
1
aρ′−λ
2
u3
X
l=1
ω3,lJ3(x3,l ),(55)
where x3,l is the lth root of Laguerre polynomial Lu3(x),ω3,l =x3,l
(u3+1)2(Lu3+1(x3,l ))2is the weight,
and
J3(x) = x−1
4e(b
a+1
2)xEi −b
ax−1
aρ′I−1
2√λx.(56)
On the other hand, the ER of F for the uplink is given by
Ru
F≈λ1
4
2e−λ
2
u4
X
l=1
ω4,lJ4(x4,l ),(57)
where x4,l is the lth root of Laguerre polynomial Lu4(x),ω4,l =x4,l
(u4+1)2(Lu4+1(x4,l ))2, and
J4(x) = x−1
4ex
2log21 + bρ′xI−1
2√λx.(58)
Proof. Please refer to Appendix D.
Remark 13. Both ERs are accurate as they are not related to the integral of R0+
0g(x)fX(x)dx.
Due to the principle of uplink NOMA, the ER of N approaches a ceiling when ρ′→ ∞, while
the ER of F increases as the SNR increases.
The ceiling for N and the high-SNR approximation for F are presented in the following
proposition.
Proposition 6. In the high-SNR regime, Ru
Napproaches a ceiling that is given by
Ru,∞
N≈ − λ1
4
2 ln(2)e−λ
2
u3
X
l=1
ω3,lJ∞
3(x3,l),(59)
where
J∞
3(x) = x−1
4e(b
a+1
2)xEi −b
axI−1
2√λx.(60)
On the other hand, the high-SNR approximation of Ru
Fis given by
Ru,∞
F= log21 + bρ′(1 + λ).(61)
19
Proof. For N, when ρ′→ ∞, we have 1
aρ′→0. Hence, based on (55) and (56), we can obtain
(59). For F, the analysis is similar to the proof of Proposition 4.
Corollary 6. In the considered IRS-aided NOMA network, the high-SNR slopes of N and F for
the uplink are given by Su
N= 0 and Su
F= 1, respectively.
Proof. Similar to the proof of Corollary 2.
Remark 14. The high-SNR slope of F for the uplink NOMA is not affected by the number of
IRS elements and Nakagami-mfading parameters.
B. OMA
For OMA, the performance analyses for the uplink and the downlink are similar, and we can
obtain all results for the uplink by referring to Section IV-B in a simple method, i.e., replacing
the transmit SNR of the BS (ρ)by the transmit SNR of the UD (ρ′).
After completing all analyses for downlink and uplink networks, all results related to diversity
order and high-SNR slope are summarized in Table I for ease of reference.
TABLE I: Diversity order and high-SNR slope for each scenario
Multiple-access scheme UD Downlink Uplink
d S d S
NOMA N1 1 0 0
FmsK0 0 1
OMA N1 0.5 1 0.5
FmsK0.5msK0.5
VI. NU MERIC AL RESULTS A ND DISCU SS I ON
In this section, numerical results are presented for the performance evaluation of the consid-
ered network. Meanwhile, Monte Carlo simulations are conducted to verify the accuracy. The
parameters are shown in Table II.
For comparisons, we regard an FDR-aided NOMA network as the benchmark. Specifically,
an FDR under the classic protocol is deployed at the place of the IRS to help F to communicate
with the BS. The FDR works under a realistic assumption that is the same as [41]. Specifically,
the self-interference channel is denoted by hrand follows CN(0, λr). Since the reflection at the
20
TABLE II: Parameters setting
Bandwidth B= 1 MHz
Amplitude-reflection coefficient of the IRS β= 0.9
Distances dN= 20 m, dR= 80 m, and dF= 20 m
Path-loss exponents αh= 3.5,αG= 2.5, and αg= 2.5
Nakagami-mfading parameters mG= 3 and mg= 1.5
Target data-rates for the fixed-rate transmission ˜
RN=˜
RF= 1 Kbps
Power-allocation coefficients for the downlink α1= 0.1and α2= 0.9
Number of points for Chebyshev-Gauss and Gauss-Laguerre quadratures u1=u2=u3=u4= 100
20 25 30 35 40 45 50
Transmit SNR (dB)
10-4
10-3
10-2
10-1
100
Outage probability
Simulation
N, Anal., CLT
F, Anal., CLT
N, FDR-NOMA
F, FDR-NOMA
NOMA
OMA
Fig. 3: OPs versus the transmit SNR in downlink networks when K= 10.
IRS is passive without consuming the energy, for fairness, we assume that the transmit power
at the BS and the FDR is Pr
b= 0.5Pbfor the downlink, and the transmit power at F and the
FDR is Pr
u= 0.5Pufor the uplink.
A. Downlink Networks
In Fig. 3, the OPs versus the transmit SNR in IRS-aided NOMA, IRS-aided OMA, and FDR-
aided NOMA networks of downlink are plotted. First, the analyses of N for both NOMA and
OMA are accurate as all simulation results coincide with the corresponding analytical results that
are derived from (16) and (32). For F, the analyses under these two schemes that are derived
from (17) and (33) are accurate in the low-SNR regime but are inaccurate in the high-SNR
21
30 35 40 45 50 55 60 65 70
Transmit SNR (dB)
10-3
10-2
10-1
100
Outage probability
Simulation
NOMA, Asymptotic
OMA, Asymptotic
F, K=1
N
F, K=3
Fig. 4: High-SNR approximations of OPs in downlink networks when K= 1 and K= 3.
regime, which results from the use of the CLT-based channel statistics. As a benchmark, the
OP curves for the FDR-aided system are plotted for comparisons. We observe that N in the
IRS-aided NOMA system always has better performance than that in the FDR-aided NOMA
system, since the transmit power of the BS in the former system is twice that in the latter. For
F, the FDR-aided system has better performance than the IRS-aided system in the low-SNR
regime, since the IRS transmission experiences severe path loss, which has been pointed out in
[31]. Nevertheless, in the high-SNR regime, the IRS-aided system has much better performance.
One reason is that the OP of F in the FDR-aided system converges to a floor in the high-SNR
regime due to the self-interference. On the other hand, F in the IRS-aided system has a large
diversity order, which is the advantage of the IRS over the FDR.
In Fig. 4, we plot the high-SNR approximate curves when K= 1 and K= 3. We observe
that the OPs of N and F for NOMA and OMA gradually approach their respective asymptotic
curves derived from (20), (21), (36), and (37), which validates our analysis. Furthermore, by
observing slopes, both diversity orders of N under NOMA and OMA schemes are 1when K= 1
and K= 3 since the value of Khas no influence on the OP of N. For F, the diversity orders
under these two schemes are the same. They are equal to 1.5and 4.5for K= 1 and K= 3,
respectively, which is consistent with Corollary 1 and Corollary 3.
In Fig. 5, the ER curves for different downlink networks are depicted. First, it is observed
that the simulation points for IRS-aided NOMA and OMA networks match well with the
22
30 35 40 45 50 55 60 65 70 75 80
Transmit SNR (dB)
0
1
2
3
4
5
6
Ergodic rate (Mbps)
Simulation
N, Anal., CLT
F, Anal., CLT
N, FDR-NOMA
F, FDR-NOMA
Asymptotic OMA
NOMA
Fig. 5: ERs versus the transmit SNR in downlink networks when K= 30.
corresponding analytical results that are derived from (27), (28), (38), and (39). It is also observed
that the high-SNR approximations that are derived from (30), (31), (43), and (44) are accurate,
and hence, it can be used to derive high-SNR slopes. We observe that the high-SNR slope of
N for NOMA is 12, while that for OMA is 0.5. Due to the principle of downlink NOMA, the
ER of F approaches a ceiling in the high-SNR regime, while that under the OMA scheme has
a high-SNR slope of 0.5, which coincides with Corollary 2 and Corollary 4. As a benchmark,
the ER curves of the FDR-aided NOMA are plotted for comparisons. It is observed that N in the
IRS-aided system always has a higher ER than that in the FDR-aided system, since the transmit
power of the BS in the former system is twice that in the latter. On the other hand, for F, the
ER in the IRS-aided system is lower than that in the FDR-aided system in the low-SNR regime.
This is because the IRS transmission experience severe path loss, which has been pointed out
in [31]. As the SNR increases, the ER of F in the IRS-aided system approaches a higher ceiling
than that in the FDR-aided system due to the existence of the self-interference channel when
using the FDR, which demonstrates the superiority of IRS again.
23
20 25 30 35 40 45 50 55 60 65 70
Transmit SNR (dB)
10-5
10-4
10-3
10-2
10-1
100
Outage probability
Simulation
N, Anal., CLT
F, Anal., CLT
N, FDR-NOMA
F, FDR-NOMA
Asymptotic
NOMA
OMA
Fig. 6: OPs versus the transmit SNR in uplink networks when K= 10.
B. Uplink Networks
In Fig. 6, the OPs versus the transmit SNR in different uplink networks are plotted. For
the IRS-aided NOMA network, it is observed that the simulation points of N match well with
the analytical results that are derived from (48). The simulation results of F have a small bias
with the analytical results from (49) in the medium-SNR regime, which is consistent with our
explanation in Remark 11. Then, we observe that both OPs of N and F decrease as the increase
of the transmit SNR and approach a floor gradually that is derived from (52). This is because of
the principle of uplink NOMA that F’s signal is regarded as interference to decode N’s signal.
Thus, the diversity orders of both UDs are 0, which is consistent with Corollary 5. For OMA,
since there is no interference, the OPs of both UDs decrease as the transmit SNR increases. The
diversity orders of N and F are 1and 15, respectively, by observing the slopes of asymptotic
curves. Finally, it is observed that the OPs for the FDR-aided NOMA network also approach a
floor. In Fig. 6, the floor for the IRS-aided network is lower than that for the FDR-aided network.
However, the former will be higher than the latter if we increase K. Thus, for uplink IRS-aided
networks, the contribution of the IRS to OP is not obvious.
In Fig. 7, the ER curves for different uplink networks are depicted. For the IRS-aided NOMA
network, we observe that the simulated results coincide with the corresponding analytical results
2The high-SNR slope of 1corresponds to the slope of 0.1 log210 ≈0.33 in Fig. 5.
24
45 50 55 60 65
Transmit SNR (dB)
0
0.5
1
1.5
2
2.5
3
3.5
4
Ergodic rate (Mbps)
Simulation
N, Anal., CLT
F, Anal., CLT
N, FDR-NOMA
F, FDR-NOMA
Asymptotic NOMA
OMA
Fig. 7: ERs versus the transmit SNR in uplink networks when K= 30.
that are derived from (55) and (57). In addition, the approximations in the high-SNR regime that
are derived from (59) and (61) are also asymptotic exact. Following that, it is observed that the
high-SNR slopes of N and F are 0and 1, respectively, which is consistent with Corollary 6. It
is also observed that both UDs have the same high-SNR slope of 0.5under the OMA scheme.
Lastly, we observe that N in the FDR-aided NOMA network also converges to a ceiling. For F,
it remains increasing with the increase of the transmit SNR in the IRS-aided network, while it
has a ceiling in the FDR-aided network due to the self-interference, revealing the advantage of
the IRS.
VII. CON CLU SI ONS
In this paper, we have studied IRS-aided NOMA and OMA networks for downlink and uplink
transmissions. With the derivation of new channel statistics, closed-form expressions for the OP
and the ER have been derived. Furthermore, the diversity order and the high-SNR slope for each
scenario have been obtained to provide insight into the performance. It has been demonstrated that
the diversity order is related to the number of IRS elements and Nakagami-mfading parameters,
while the high-SNR slope is not affected by these parameters. Since this work focuses on SISO
networks, IRS-aided MISO networks are worthy of investigation for future work.
25
APPEN DIX A
PROO F O F LEMMA 1
Based on the property of the Nakagami-mfading model, the expectation and variance of |Gk|
are µG=1
mG1
2Γ(mG+1
2)
Γ(mG)and VarG= 1 −1
mGΓ(mG+1
2)
Γ(mG)2
, respectively, where G ∈ {G, g}.
Then, the expectation and variance of |Gk||gk|are µp=µgµG=√ξand Varp=Varg+
µ2
gVarG+µ2
G−µ2
gµ2
G= 1 −ξ, respectively. Next, since all |Gk||gk|(k= 1,2,···, K)are
independent and identically distributed (i.i.d.), based on the CLT, PK
k=1 |Gk||gk|tends towards
a Gaussian distribution as
K
X
k=1 |Gk||gk|˙∼N(Kµp, KVarp).(A.1)
Furthermore, we unify the variance and have
PK
k=1 |Gk||gk|
pKVarp
˙∼N √K µp
pVarp
,1!.(A.2)
Hence, X=(PK
k=1 |Gk||gk|)2
KVarpfollows a noncentral chi-square distribution χ′2
1(λ)with a PDF given
by
fX(x) = λ1
4
2e−x+λ
2x−1
4I−1
2√λx,(A.3)
where λ=Kµ2
p
Varpand
I−1
2√λx= √λx
2!−1
2∞
X
i=0 λx
4i
i!Γ(i+1
2).(A.4)
The CDF of Xis given by
FX(x) = 1 −Q1
2√λ,√x,(A.5)
where
Q1
2√λ, √x= 1 −e−λ
2
∞
X
i=0 λ
2iγi+1
2,x
2
i!Γ i+1
2.(A.6)
This completes the proof.
26
APPEN DIX B
PROO F O F LEMMA 2
Denote that Qk=|Gk||gk|. According to [42], the PDF of Qk(the product of two Nakagami-m
random variables) is given by
fQk(q) = 4(msml)ms+ml
2
Γ(ms)Γ(ml)qms+ml−1Kms−ml(2√msmlq),(B.1)
for q≥0, where Kv(·)is the modified Bessel function of the second kind. The Laplace transform
of fQkis derived as
LfQk(s) = 4(msml)ms+ml
2
Γ(ms)Γ(ml)Z∞
0
qms+ml−1e−sqKms−ml(2√msmlq)dq. (B.2)
Furthermore, by referring to [39, eq. (6.621.3)], we have
LfQk(s) = φ(s+ 2√msml)−2msF2ms, ms−ml+1
2;ms+ml+1
2;s−2√msml
s+ 2√msml,(B.3)
where φ=√π4ms−ml+1(msml)msΓ(2ms)Γ(2ml)
Γ(ms)Γ(ml)Γ(ms+ml+1
2)and F(·,·;·;·)is the hypergeometric series. When
s→ ∞, since ms< mlsatisfies the condition of [39, eq. (9.122.1)], we have
L∞
fQk(s) = ˜m(s+ 2√msml)−2ms.(B.4)
Since all Qk(k= 1,2,···, K)are i.i.d., the Laplace transform of the PDF of Z=PK
k=1 Qk
for s→ ∞ is given by
L∞
fZ(s) =
K
Y
k=1 L∞
fQk(s) = ˜mK(s+ 2√msml)−2msK.(B.5)
Thus, by conducting the inverse Laplace transform for (B.5), the PDF of Zfor z→0+can be
derived as (12) based on [39, eq. (17.13.3)]. Following that, according to [39, eq. (3.351.1)], the
CDF of Zfor z→0+can be derived as (13). This completes the proof.
APPEN DIX C
PROO F O F THEOREM 2
First, we have
Rd
N=−Z∞
0
log2(1 + aα1ρy)d(1 −FY(y)) = 1
ln(2) Z∞
0
e−y
y+1
aα1ρ
dy. (C.1)
27
Then, by referring to [39, eq. (3.352.4)], we can derive (27). To obtain Rd
F, we denote that
˜
X=bα2X
bα1X+1/ρ , and the corresponding CDF is given by
F˜
X(˜x) = FX˜x
bρ(α2−˜xα1)=e−λ
2
∞
X
i=0
λiγi+1
2,˜x
2bρ(α2−˜xα1)
i!2iΓi+1
2.(C.2)
Hence, we have
Rd
F≈ −Z˜α
0
log2(1 + ˜x)d(1 −F˜
X(˜x)) = 1
ln(2) Z˜α
0
1−F˜
X(˜x)
1 + ˜xd˜x
= log2(1 + ˜α)−1
ln(2)e−λ
2
∞
X
i=0
λi
i!2iΓ(i+1
2)Z˜α
0
γi+1
2,˜x
2bρ(α2−˜xα1)
1 + ˜xd˜x
|{z }
J1
,
(C.3)
where ˜α=α2
α1. To further transform J1, we denote that t=2˜x
˜α−1. Then, we have
J1=Z1
−1
γi+1
2,˜α(1+t)
2bρ(2α2−α1˜α(1+t))
1 + 2
˜α+tdt. (C.4)
Next, by using the Chebyshev-Gauss quadrature, we can approximate J1as
J1≃
u1
X
l=1
ω1,lJ1(tl).(C.5)
This completes the proof.
APPEN DIX D
PROO F O F THEOREM 6
First, the ER of N can be transformed into
Ru
N≈Z∞
0−Z∞
0
log21 + ay
bx + 1/ρd(1 −FY(y))fX(x)dx
=1
ln(2) Z∞
0 Z∞
0
e−y
y+b
ax+1
aρ
dy!fX(x)dx.
(D.1)
Then, by referring to [39, eq. (3.352.4)], (D.1) can be rewritten as
Ru
N≈ − 1
ln(2) Z∞
0
eb
ax+1
aρ Ei −b
ax+1
aρfX(x)dx
=−λ1
4
2 ln(2)e1
aρ −λ
2Z∞
0
x−1
4e(b
a−1
2)xEi −b
ax−1
aρI−1
2√λxdx
|{z }
J3
.(D.2)
Next, by using the Gauss-Laguerre quadrature, we have
J3≃
u3
X
l=1
ω3,lJ3(x3,l ).(D.3)
28
For the ER of F, it can be expressed as
Ru
F≈Z∞
0
log2(1 + bρx)fX(x)dx =λ1
4
2e−λ
2Z∞
0
x−1
4e−x
2log2(1 + bρx)I−1
2√λxdx
|{z }
J4
.(D.4)
Next, J4also can be approximated by adopting the Gauss-Laguerre quadrature. As such, we
have
J4≃
u4
X
l=1
ω4,lJ4(x4,l ).(D.5)
This completes the proof.
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