IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 70, …

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 70, NO. 7, JULY 2021 6881

Performance Analysis of Uplink NOMA SystemsWith Hardware Impairments and Delay Constraints

Over Composite Fading ChannelsNgoc Phuc Le , Le Chung Tran , Xiaojing Huang , Jinho Choi , Eryk Dutkiewicz , Son Lam Phung ,

and Abdesselam Bouzerdoum

Abstract—In this paper, we propose a mixture gamma distribu-tion based analytical framework for NOMA wireless systems overcomposite fading channels. We analyze the outage probability (OP),delay-limited throughput (TP) and effective capacity (EC) in uplinkNOMA with imperfect successive interference cancellation (SIC)due to the presence of residual hardware impairments and delayconstraints. A mixture gamma distribution is used to approximatethe probability density functions of fading channels. Based on this,we obtain closed-form expressions in terms of Meijer-G functionsfor the OP, the TP and the EC. We also perform asymptoticanalysis of these metrics to characterize system behaviors at thehigh signal-to-noise ratio regime. Moreover, upper-bounds for theEC is derived. Efficacy of NOMA over orthogonal multiple accessis analytically examined. Unlike the existing works, our analyticalexpressions hold for NOMA systems with an arbitrary number ofusers per cluster over a wide range of channel models, includinglognormal-Nakagami-m,KG,η − μ, Nakagami-q (Hoyt),κ − μ,Nakagami-n (Rician), Nakagami-m, and Rayleigh fading chan-nels. This unified analysis facilitates evaluations of impacts of theresidual interference, the power allocation among users, the delayquality-of-service exponent as well as the shadowing and small-scale fading parameters on the performance metrics. Simulationresults are provided to validate theoretical analysis.

Index Terms—Non-orthogonal multiple access (NOMA),hardware impairments, successive interference cancellation (SIC),outage probability, throughput, effective capacity, compositefading channels.

Manuscript received October 23, 2020; revised April 3, 2021; accepted May30, 2021. Date of publication June 2, 2021; date of current version July 20, 2021.This work was supported by the New South Wales (NSW) Defence InnovationNetwork (DIN) and the NSW State Government, Australia, through the DINPilot Project Grant (Project no: 888-006-985). The review of this article wascoordinated by Prof. Hongzi Zhu. (Corresponding author: Ngoc Phuc Le.)

Ngoc Phuc Le is with the Institute of Research and Development, Duy TanUniversity, Danang 550000, Vietnam, and also with the School of Electrical,Computer and Telecommunications Engineering, University of Wollongong,Wollongong, NSW 2522, Australia (e-mail: [email protected]).

Le Chung Tran and Son Lam Phung are with the School of Electrical, Com-puter and Telecommunications Engineering, University of Wollongong, Wollon-gong, NSW 2522, Australia (e-mail: [email protected]; [email protected]).

Xiaojing Huang and Eryk Dutkiewicz are with the School of Electrical andData Engineering, University of Technology Sydney (UTS), Ultimo, NSW 2007,Australia (e-mail: [email protected]; [email protected]).

Jinho Choi is with the School of Information Technology, Deakin University,Burwood, Victoria 3125, Australia (e-mail: [email protected]).

Abdesselam Bouzerdoum is with the School of Electrical, Computer andTelecommunications Engineering, University of Wollongong, Wollongong,NSW 2522, Australia, and also with the Division of Information and ComputingTechnology, College of Science and Engineering, Hamad Bin Khalifa University,Doha 34110, Qatar (e-mail: [email protected]).

Digital Object Identifier 10.1109/TVT.2021.3086045

I. INTRODUCTION

N EXT generations of wireless communication networksare envisioned to provide very high data-rates due to the

explosive growth of mobile data and Internet-of-Things (IoT)applications. One of the most promising techniques that hasattracted a great attention in recent years is non-orthogonalmultiple access (NOMA) [1]. In general, NOMA techniquescan be categorized into code-domain NOMA and power-domainNOMA systems. In power-domain NOMA systems, multipleusers are multiplexed in a power-domain by exploiting the chan-nel gain difference while performing successive interferencecancellation (SIC) at the receiver [2]. Since individual infor-mation of different users in NOMA systems can be transmittedsimultaneously on the same frequency spectrum, the spectrumefficiency can be enhanced in comparison with orthogonal mul-tiple access (OMA) methods [3]. NOMA was first consideredfor downlink in [1] and for uplink in [4]. In essence, there aredifferences in working principles between downlink and uplinkNOMA in terms of user clustering and power allocation [5].Performance analysis of downlink NOMA and uplink NOMAis also not identical due to the difference of the receive signalstructure. Until now, NOMA has been considered for severaltypes of wireless systems, including relay networks [6], cooper-ative relaying [7], multiple-input multiple-output (MIMO) andmassive MIMO systems [8]–[9], mmWave systems [10], andunmanned aerial vehicle (UAV) based communications [11].

A. Review of Related Papers

Many researchers have analyzed the performance in terms ofthe outage probability (OP) and ergodic capacity of NOMA sys-tems [12]–[19]. For downlink NOMA, Ding et al. [12] analyzedthe outage performance and the ergodic sum rate with randomlydeployed users. They shown that NOMA can achieve betteroutage performance than OMA subject to a careful selectionof the users’ data rates and power coefficients. Furthermore,they claimed that NOMA can attain superior ergodic capacityin comparison with OMA. Impacts of user pairing on individualrates and the sum rate in downlink NOMA with a fixed power al-location and cognitive-radio inspired NOMA were investigatedin [13]. For uplink NOMA, the authors in [14] analyzed theoutage probability and the achievable sum rate in a Rayleighfading channel. A simple uplink power back-up control scheme

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6882 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 70, NO. 7, JULY 2021

was also proposed for power allocation in this work. The outage-constrained min-max power allocation algorithm was developedfor both NOMA and OMA systems in [15]. In [17], the outageprobability of NOMA systems with an advanced SIC receiverwas analyzed. Additionally, a dynamic power-allocation schemefor downlink and uplink NOMA systems was considered in [16],where the outage probability and ergodic capacity were used asthe criteria to analyze the performance of the proposed scheme.The performance of NOMA system under more complex butpractical channel models was investigated in [18]–[19]. In par-ticular, the authors of [18] derived a closed-form expression ofthe outage probability of downlink NOMA for the case of aNakagami-m fading channel. Expressions for the OP, the er-godic capacity and the average bit-error rate (BER) of downlinkNOMA over κ− μ shadowed fading channels were derivedin [19]. These results unveiled the significance impacts of thefading characteristics on the performance of NOMA systems.We note that these studies considered only downlink NOMAsystems with ideal hardware transceivers and perfect SIC. More-over, the derived expressions are based on a specific channelmodel, which cannot be applied to other channel scenarios.

In many 5G/beyond 5G use cases (i.e., ultra-reliable lowlatency communication (URLLC) applications), limited latencyis one of the most crucial design parameters. Although theergodic capacity has been widely used in communication sys-tem analysis and designs, it is unable to explicitly characterizethe delay-constrained performance. Motivated by this, manyresearchers adopted the link-layer metric of effective capacity(EC) as a natural choice. This metric quantifies the maximumachievable rate that can be supported by a link given a delay-violation probability [20]. In fact, the EC has been consideredfor several types of wireless systems (see [21] and referencestherein). The performance analysis of downlink NOMA in termsof EC over Rayleigh fading channels was carried out in [22].In this work, the authors shown that NOMA can achieve ahigher sum EC over OMA at high signal-to-noise ratio (SNR)values. They also compared individual EC attained by each userin NOMA versus OMA. The EC analysis in downlink NOMAover α− μ fading channel environments was reported in [23].The results shown that while NOMA consistently outperformsOMA in the practical SNR range, the relative gain reduces undersevere fading conditions or a stringent delay quality-of-service(QoS) constraint. Optimal power allocation to maximize the ECin downlink NOMA with delay QoS constraints was investigatedin [24]. The author shown that this problem is non-convex and theconcept of partial EC was introduced to realize a sub-optimalallocation policy. Additionally, the EC performance in uplinkNOMA systems over Rayleigh fading channels was consideredin [25]. It is worth noting that these studies considered onlyRayleigh fading channels. Given that the EC performance de-pends on the fading characteristics, it is essential to examine thesystem performance under other fading channels. Furthermore,these works assumed ideal radio frequency (RF) front-ends.

In realistic communication systems, hardware transceiverssuffer from RF impairments due to in-phase/quadrature (I/Q)imbalance in modulators, resolution of analog-to-digital con-verter, phase noise in local oscillators, and non-linearity power

amplifiers, etc. [26]. These impairments have a non-negligibleand fundamental impact on performance of wireless systems.Although some algorithms have been introduced to mitigatethis impact, there are still residual hardware impairments. Inthe context of NOMA systems, the authors in [27] considereddownlink NOMA subject to RF impairments, where detrimentaleffects of the RF impairments were demonstrated through biterror-rate (BER) evaluations. The impacts of residual hardwareimpairments on the outage performance and the ergodic sum-rate of NOMA-based relay networks over Nakagami-m fadingchannels was analyzed in [28]. In [29], cooperative NOMA withhardware impairments and imperfect channel estimation overα− μ fading channels was examined. In particular, closed-formexpressions of the outage probability, the ergodic capacity, andthe energy efficiency were derived. Also, outage performanceof cooperative cognitive radio NOMA with hardware and in-terference limitations was studied in [30]. In addition, energyefficiency in downlink multicell multicarrier NOMA systemswith hardware impairments was maximized in [31]. We notethat these studies considered only downlink NOMA. Thus, howuplink NOMA affected by hardware impairments has not beenaddressed yet. Also, the considered channel models on thoseworks did not take into consideration impacts of shadowing.

B. Paper Contributions and Structure

In this paper, we consider an uplink NOMA system overcomposite fading channels. Practical implementation limitationsrelated to transceiver RF and buffer constraint are incorporatedin the system model. Also, the composite fading channel modeltakes into account both the large-scale fading and the multipathfading. By analyzing this system model, we aim to address theopen problems pointed out above. The main contributions of thiswork are as follows.

1) First, we propose an analytical framework for NOMAsystems by utilizing a mixture gamma (MG) distributionapproximation for the probability density functions (PDF)of fading channels. With the MG approach, closed-formexpressions of key performance metrics can be attainedin the NOMA systems with an arbitrary number of usersper cluster. This overcomes challenges when using con-ventional analytical approaches due to the complexity ofthe channel model.

2) Second, the proposed framework facilitates analysis ofthe NOMA system in the presence of residual impair-ments with imperfect SIC at the receiver, which is essen-tial from a practical perspective. Moreover, the obtainedclosed-form expressions hold for a wide range of fadingchannel models, such as lognormal-Nakagami-m, KG,η − μ, Nakagami-q (Hoyt), κ− μ, Nakagami-n (Rician),Nakagami-m, and Rayleigh fading channels.

3) Third, we focus on analyzing performance metrics ofdelay-limited throughput (TP) and effective capacity (i.e.,link-layer capacity) over general composite fading chan-nels to quantify delay-constrained performance, whichis missing from the open literature. These performance


LE et al.: PERFORMANCE ANALYSIS OF UPLINK NOMA SYSTEMS 6883

TABLE IRELATED CONTRIBUTIONS VERSUS THIS WORK ON NOMA SYSTEMS

*Covers several models, such as lognormal-Nakagami-m, KG, η − μ, Nakagami-q (Hoyt), κ− μ, Nakagami-n (Rician), Nakagami-m, and Rayleigh fading channels.

metrics are among the most crucial design criteria fordelay-sensitive applications in 5G/B5G networks.

A summary of key system characteristics considered in thiswork compared to other existing contributions is provided inTable I . The specific results of this paper is summarized below.� Derivation of the closed-form expressions for the OP, the

TP and the EC in NOMA systems with imperfect SIC undercomposite fading channels.

� Asymptotic analysis of the OP, the TP and the EC in thehigh SNR regime, which provides insights into impacts ofkey system parameters on the system behaviors.

� Derivation of the upper-bound for the EC in both NOMAand OMA systems.

� Analytical comparisons between the NOMA system andthe OMA system in terms of the TP and the EC undercomposite fading channel conditions.

� Evaluation of impacts of hardware impairments, powerallocation, delay QoS constraints and channel parameterson the system performance.

The rest of the paper is organized as follows. In Section II,we describe the uplink NOMA system model. Section III de-rives the OP and the TP in the NOMA system with two usersper cluster without hardware impairments, whereas Section IVanalyzes the EC. Extended analysis of the OP and the EC in theNOMA system suffering hardware impairments with imperfectSIC and/or more than two users per cluster are performed inSection V. Simulation results are provided in Section VI. Finally,Section VII concludes the paper.

II. SYSTEM MODEL

We consider an uplink power-domain NOMA wireless systemthat consists of a base station (BS) and U users with non-idealhardware over composite fading channels as shown in Fig. 1,1.All nodes are equipped with a single antenna. Users are groupinto K clusters, each of which consists of I users. The userswithin a cluster transmit their data to the BS via NOMA, whereas

1In this study, we focus on a power-domain NOMA system. An investigationon code-domain NOMA, e.g., [32] goes beyond the scope of this work. Also,the composite fading channel conditions undergo both shadowing fading (i.e.,large-scale fading) and multipath fading (i.e., small-scale fading) [33].

Fig. 1. An illustration of uplink NOMA sytem model (I = 3, U = 9).

transmissions among clusters are orthogonal to one another.The position of the uth user is denoted as (xu, yu, 0), u =1, 2, . . ., U , whereas the position of the BS is (xb, yb, h). Thus,a distance-based path gain between the uth user and the BSisGu = g0/[(h

2 + [(xu − xb)2 + (yu − yb)

2])β/2], where g0 isthe reference gain of the users at the reference distance andβ is the path-loss exponent. Also, the multipath channel co-efficient and the large-scale fading are denoted as hu and Su,respectively. The overall instantaneous channel gain, denoted asMu, is thus a superposition of these three types of gains, i.e.,Mu = GuSuHu, where Hu = |hu|2. We assume that randomvariables representing fading in all links are independent buthaving the same mean and variance.

In uplink NOMA, all the users within each cluster transmittheir signals to the BS over the same radio spectrum simultane-ously. We assume that perfect channel estimation is available atthe BS, which can be obtained via training using pilot sequences.Thus, the received signal for each cluster in the presence ofresidual hardware impairments can be expressed as [28]–[31]

y =

I∑i=1

√GiSihi(

√Pisi + ψi) + n, (1)



where si is the transmitted symbol of the ith user withE{|si|2} = 1 where E{.} denotes an expectation operator, Pi isthe transmit power of the ith user, and n denotes additive whiteGaussian noise (AWGN) which has a zero-mean and varianceN0, i.e., n ∈ CN (0, N0). Also, ψi represents the aggregateddistortion noise due to non-ideal transceiver associated withthe ith user and the BS. This component can be modeled asψi ∈ CN (0, κ2

iPi) [28]–[31], where κi denotes the level ofhardware impairments that can be measured in practice by theerror vector magnitude (EVM).

To minimize the impacts of propagation errors when SICis employed, the decoding often follows the order of decreas-ing channel gains [5]. Specifically, the user with the highestchannel gain will be decoded first. After decoding the signalsent by the first user, the BS reconstructs the codeword s1

and subtracts√P1M1s1 from the received signal y. Then, the

signal corresponding to the second user can be decoded, andso on. The user with the lowest channel gain will be decodedlast, and thus can enjoy interference-free in cases of perfectSIC. Without loss of generality, we assume that the users areindexed based on the order of G1 > G2 > . . . > GI . Then, theBS detects signals in the order of 1, 2, . . ., I2. Thus, the receivedsignal-to-interference-plus-noise ratio (SINR) associated withthe signal transmitted by the ith user is

γi =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

P1M1I∑

i=1κ2iPiMi+

I∑

j=2PjMj+N0

, i = 1

PiMiI∑

i=1κ2iPiMi+

I∑

j=1+i

PjMj+N0

, i = 2, . . ., I − 1

PIMII∑

i=1κ2iPiMi+N0

, i = I.

(2)

For notational convenience, let us denote γi � PiGi/N0 andMG,i � SiHi. Then, the SINR value in (2) is rewritten as

γi =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

γ1MG,1

κ21γ1MG,1+

I∑

j=2(1+κ2

j)γjMG,j+1, i = 1

γiMG,i

i∑

k=1κ2kγkMG,k+

I∑

j=1+i

(1+κ2j)γjMG,j+1

, i = 2, . . ., I − 1

γIMG,I

I∑

k=1κ2kγkMG,k+1

, i = I.

(3)

In this work, we assume that the instantaneous CSI is notavailable at users for reduced complexity of transmitters. Also,overhead signalling from the BS to users is kept as low as possi-ble. To this end, fixed power was considered. Specifically, the BSonly need to inform to an active user which clusters it is allocated(i.e., frequency bands) and the detection order of this user withinits cluster based on the distance from the user to the BS. Basedon the information, this user selects a frequency band and atransmit power level (via power back-off as in [14]) for uplink

2We considered fixed-order decoding based on distance in this work foranalytical tractability. A dynamic decoding-order based on instantaneous CSImay result better performance, which is left for future investigations. Interestedreaders are referred to [34] for an analysis of the accuracy of the distance-basedranking versus the instantaneous CSI ranking.

data transmission. The diverse received power is obtained byexploiting the differences in the path-loss and transmit powers.In particular, a power back-off step of PΔ(dB) is adopted, i.e.,the transmit power of the ith user is PΔ(dB) stronger than thatof the (i+ 1)th user. Thus, the transmit power of the ith useris Pi = αiP , where αi = 10−(i−1)PΔ/10 and P is the powerallocated to User 1. In cases that transmit powers of users areallocated based on instantaneous CSI, better performance canbe achieved at the cost of additional complexity. The optimalpower allocation in this case goes beyond the scope of thiswork. Interested readers are referred to [5], [15]-[16] and [23]for possible power allocation approaches.

III. ANALYSIS OF OUTAGE PROBABILITY AND THROUGHPUT:TWO USERS PER NOMA CLUSTER

In this section, we first describe a mixture gamma distributionapproximation for composite fading channels. After that, weanalyze the OP and TP in uplink NOMA with two users perNOMA cluster and without hardware impairments. Analysis forscenarios of more than two users per cluster as well as residualhardware impairments are performed in Section V.

A. Approximation of Fading Channel Distribution

In composite fading channels, the PDF of fading channel ingeneral has a complicated integral form. Thus, it is challengingto obtain closed-form expressions for several system calcula-tions. For example, the PDF of lognormal-Nakagami-m fadingchannels is described as [35]

fNL(x) =

∫ ∞

0

xm−1e−mx/�y

Γ(m)

(m

�y

)me−

(ln(y)−μ)2

2σ2

√2πσy

dy, (4)

where m is the severe fading parameter in Nakagami-m fading,� is the unfaded SNR, μ and σ represent the mean and thestandard deviation of lognormal shadowing, and Γ(.) denotesthe gamma function [36, Eq. (8.310.1)]. To overcome this issue,we consider a mixture gamma (MG) distribution approach forthe PDF approximation in this work. It was shown in [35]that a MG approximation can offer high accuracy for differ-ent kinds of fading channels, such as lognormal-Nakagami-m,KG, η − μ, Nakagami-q (Hoyt), κ− μ, Nakagami-n (Rician),Nakagami-m, and Rayleigh fading channels. By using a MGdistribution approximation, the PDF of the composite fadingchannel between the ith user and the BS is expressed as [35]

fMG,i(x) =

N∑n=1

ai,nxbi,n−1e−ci,nx, x ≥ 0, (5)

whereN is the number of terms for the MG distribution approxi-mation, andai,n, bi,n, and ci,n are the parameters associated withthe nth Gamma component. These parameters are determineddepending on particular types of composite fading channel mod-els. Detail calculations of these parameters for different channelmodels are provided in Table II [35].



TABLE IIPARAMETERS OF A MIXTURE GAMMA DISTRIBUTION FOR DIFFERENT CHANNEL MODELS

From (4), the cumulative distribution function (CDF) of theMG distribution is formulated as

FMG,i(x) =

N∑n=1

ai,nc−bi,ni,n γ(bi,n, ci,nx), (6)

where γ(., .) is the lower incomplete gamma function [36].In cases that bi,n is an integer, we have the following property.Lemma 1: The parameters of ai,n, bi,n and ci,n in the MG

approximation satisfies

N∑n=1

ai,nc−bi,ni,n (bi,n − 1)! = 1. (7)

Proof: From the PDF expression in (5), we have∫∞0 fMG,i

(x)dx =∑N

n=1 ai,n∫ +∞

0 xbi,n−1e−ci,nxdx =∑Nn=1 ai,nc

−bi,ni,n (bi,n − 1)!, where the integral is calculated

with the help of [36, Eq. (3.351.3)]. On the other hand, bydefinition, we have

∫ +∞0 fMG,i

(x)dx = 1. Consequently, theresult in (7) follows.

B. Outage Probability

1) Expressions of Outage Probability: We now analyze theoutage probability in the uplink NOMA wireless system withtwo NOMA users per cluster, i.e., a 2-user NOMA system model.This is in line with most of the existing works, e.g., [1], [6]–[7], [9]–[11], [13]–[14], [16], [19], [22]–[25]. Note that groupinga large number of users in NOMA is not preferable in practicebecause of performance degradation due to residual interferenceand increased complexity and power consumption. In the 2-userNOMA system with perfect SIC, the SINR values are given as(cf. (3))

γ1 =γ1MG,1

γ2MG,2 + 1, (8)

and

γ2 = γ2MG,2. (9)

The OP of User 1, which is the probability that the receivedSINR at the BS is below a given threshold, is calculated as

Pout,1(γth,1) = Pr(γ1 < γth,1), (10)

where γth,1 is the threshold for correct data detection, whichis related to the target rate Rc,1 as Rc,1 = log2(1 + γth,1). ForUser 2, its OP can be evaluated as [14]–[15]

Pout,2(γth,2) = 1 − [1 − Pr(γ1 < γth,1)][1 − Pr(γ2 < γth,2)],(11)

where γth,2 is the detection threshold, i.e., Rc,2 = log2(1 +γth,2) and Rc,2 is the target rate of User 2. By using theMG approximation in (5) and (6), we obtain the closed-formexpressions of the OP in the following theorem.

Theorem 1: The OPs of User 1 and User 2 in uplink NOMAunder composite fading channels are, respectively, given by

Pout,1(γth,1) = 1 −N∑

n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m)e− c1,mγth,1

γ1

×b1,m−1∑p=0

1p!

(c1,mγth,1

γ1

)p p∑k=0

(p

k

)γk2Γ(b2,n + k)(

c2,n +c1,mγth,1γ2

γ1

)b2,n+k

(12)

and

Pout,2(γth,2) = 1 − [1 − Pout,1(γth,1)]

×[

1 −N∑

n=1

a2,nc−b2,n

2,n γ (b2,n, c2,nγth,2/γ2)

].

(13)

Proof: See Appendix A.



In an OMA system, individual user transmits data over orthog-onal channels. The SNR of the User i is γi = γiMG,i, i = 1, 2.Thus, the OP expression is obtained as

POMAout,i (γth,i) =

N∑n=1

ai,nc−bi,ni,n γ

(bi,n, ci,nγ

OMAth,i /γi

). (14)

2) Asymptotic Analysis: In this subsection, we discuss theasymptotic behaviors of the OP. Let us denote α � γ2/γ1. In ahigh SNR region (i.e., γ1 → +∞, γ2 → +∞), the asymptoticOP expressions are given in the following corollary.

Corollary 1: The asymptotic OP expressions in uplink NOMAat the high SNR regime are given by

P∞out,1(γth,1) = 1 −

N∑n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m)

×(

1 − c1,mγth,1γ1

)×

b1,m−1∑p=0

1p!(c1,mγth,1α)

p

× Γ(b2,n + p)

(c2,n + c1,mγth,1α)

b2,n+p

, (15)

and

P∞out,2(γth,2) = 1

− [1 − P∞out,1(γth,1)]

[1 −

N∑n=1

a2,n

b2,n

(γth,2γ2

)b2,n].

(16)

Proof: The OP of User 1 are obtained by discarding the high-order terms of 1/γi in (12) and using the approximation of e−x ≈1 − x when x→ 0. For User 2, the OP is attained by using theresult of γ(a, x) ≈ xa/a when x→ 0.

For the case of the OMA system, we can approximate theresults in (14) as

POMA,∞out,i (γth,i) =

N∑n=1

ai,nbi,n

(γOMAth,i

γi

)bi,n

, i = 1, 2. (17)

C. Evaluation of Sum Throughput

In delay-limited (DL) transmission, the users transmit dataat a constant rate of Rc,i (bps/Hz). The achievable throughputof the ith user in the uplink NOMA system can be evalu-ated as τNOMA

i = (1 − Pout,i)Rc,i, where Pout,i is the OPfor the ith user obtained in the previous section. Thus, thesum throughput is τNOMA

sum = τNOMA1 + τNOMA

2 . Similarly,the sum throughput in the OMA system is obtained as τOMA

sum =τOMA

1 + τOMA2 , where τOMA

i = (1 − POMAout,i )Rc,i andRc,i =

12 log2(1 + γOMA

th,i ). Note that a factor 1/2 here accounts forthe fact that each user in the OMA system only has a half ofthe available channel resource for a fair comparison with theNOMA system. Additionally, asymptotic throughput is obtainedby replacing Pout,i by P∞

out,i in the above expressions. Numer-ical comparison between τNOMA

sum and τOMAsum is provided in

Section VI.

IV. ANALYSIS OF EFFECTIVE CAPACITY

A. Effective Capacity

1) Preliminaries on Effective Capacity: We consider thetransmission of the ith user, where upper-layer packets areencapsulated into frames stored at the transmit buffer of the linklayer. Denote Ai(t), Qi(t) and Ri(t) as the arrival process, thequeue state and the departure process at the time t, respectively.The state of the queue is determined by [24]

Qi(t+ 1) = [Qi(t) +Ai(t)−Ri(t)]+, (18)

where [x]+ = max(0, x). We assume that Ai(t) and Ri(t) arestationary and ergodic that satisfy E{Ai(t)} < E{Ri(t)}. Then,the queue state converges to a steady state Q∞

i [37]. Under adelay QoS constraint, we can consider the buffer overflow prob-ability, i.e., the probability that the steady-state queue exceedsa certain threshold. Based on the large deviation theory, we canexpress [22]

− limτ→∞

ln[Pr(Q∞i > τ)]

τ= θi, (19)

where τ is the overflow threshold and θi is called the de-lay quality-of-service (QoS) exponent. With the constraintspecified in (19), it is required that the arrival processand the departure process satisfy ΛAi

(θi) + ΛRi(−θi) = 0,

where ΛAi(θi) and ΛRi

(−θi) are the Gartner-Ellis lim-its of the arrival process and the service process, i.e.,ΛAi

(θi) = limT→∞ 1T ln(E{eθi

∑Tt=1 Ai(t)}) and ΛRi

(θi) =

limT→∞ 1T ln(E{eθi

∑Tt=1 Ri(t)}). When the arrive rate is a con-

stant, i.e.,Ai(t) = Ai, we haveAi = −ΛRi(−θi)

θi� CE,i, where

CE,i is named the effective capacity [20]. We assume that thewireless channels from the users to the BS follow a block fadingdistribution with a block-length duration of T (seconds). TheEC can be expressed as [22]

CE,i = − 1θiTB

ln(E{e−θiTBRi

})(bps/Hz), (20)

where B is the bandwidth. It is noted from (20) that the EC is afunction of the delay-related parameter θi. The condition of θi →∞ represents a system with a strict delay constraint, whereasθi → 0 corresponds to the system with no delay constraint [20].

2) Exact Expressions of Effective Capacity: In the NOMAsystem with imperfect SIC, by substituting Ri = log2(1 + γi)into (20), we can express the EC of the ith user as

CE,i = − 1Ωi ln 2

ln(E{(1 + γi)

−Ωi})

= − 1Ωi ln 2

ln

(∫ ∞

0(1 + x)−Ωifγi

(x)dx

), (21)

where Ωi � θiTBln 2 and fγi

(x) is the PDF of γi defined in (2).The expression of fγi

(x) is given in Appendix B. By evaluatingthe integral in (21), we obtain the following result.

Theorem 2: The expressions of the EC of User 1 and User 2 inuplink NOMA over composite fading channels are, respectively,



given by

CE,1 =−1

Ω1 ln 2ln

[N∑

n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m)

×b1,m−1∑p=0

1p!

(c1,m

γ1

)p

×p∑

k=0

(p

k

)γk2Γ(b2,n + k)

×((b2,n+k)

c1,mγ2

γ1QE,1(p, b2,n + k + 1)

− pQE,1(p− 1, b2,n + k)

+c1,m

γ1QE,1(p, b2,n + k)

)], (22)

and

CE,2 =−1

Ω2 ln 2ln

[N∑

n=1

a2,n

Γ(Ω2)cb2,n

2,n

× G1,22,1

(γ2

c2,n

1 − b2,n, 1 − Ω2

0

)],

(23)

where

QE,1(u, v) =1

cv2,nΓ(v)Γ(Ω1)

×G1,0:1,1:1,11,0:1,1:1,1

(u+ 1 1 − Ω1 1 − v− 0 0

γ1

c1,m,γ2

c2,n

),

(24)

and G(: | : | : |., .) denotes the extended generalized bivariateMeijer-G function (EGBMGF) [38], whereasG(.| :) is the Mei-jer G-function [36, Eq. (9.301)].

Proof: See Appendix B.Note that efficient implementations of the EGBMGF function

are readily available in Mathematical [39] and Matlab [40].Similar to (21), for the OMA system, we have

COMAE,i = − 1

Ωi ln 2ln(

E{(1 + γi)

−Ωi/2})

. (25)

Note that in (25) a factor Ωi/2 accounts for the fact that eachuser only accesses a half of the channel resource. By performingsimilar calculations as in the case of User 2 with perfect SIC inthe NOMA system, we have

COMAE,i =

−1Ωi ln 2

ln

[N∑

n=1

ai,n

Γ(Ωi/2)cbi,ni,n

× G1,22,1

(γici,n

1 − bi,n, 1 − Ωi/20

)]. (26)

3) Asymptotic Analysis: At the high SNR regime, i.e., γ1 →+∞ and γ2 → +∞, the SINR values defined in (8) becomes

γ∞1 =MG,1

αMG,2. From (21), the EC can be formulated as

C∞E,i = − 1

Ωi ln 2ln(E{(1 + γ∞i )−Ωi

})

= − 1Ωi ln 2

ln

(∫ ∞

0(1 + x)−Ωifγ∞

i(x)dx

), (27)

where fγ∞i(x) is the PDF of γ∞i , i = 1, 2. By evaluating this

integral, we obtain the following result.Corollary 2: The asymptotic expressions for the EC of User

1 and User 2 at a high SNR region are given by

C∞E,1 ≈

−1Ω1 ln 2

ln

{1

Γ(Ω1)

N∑n=1

N∑m=1

a2,na1,mc−b1,m1,m c

−b2,n2,n Γ(b1,m)

×b1,m−1∑p=0

1p!

(αc1,m

c2,n

)p [αc1,m

c2,n

×G2,22,2

(αc1,m

c2,n

−p,−b2,n − p

Ω1 − p− 1, ε

)

−pG2,22,2

(αc1,m

c2,n

−p+ 1,−b2,n − p+ 1Ω1 − p, ε

)]},

(28)

and

C∞E,2 = − 1

Ω2 ln 2ln

(N∑

n=1

a2,nΓ(b2,n − Ω2)

γΩ22 c

b2,n−Ω2

2,n

). (29)

where ε denotes an arbitrarily small number.Proof: See Appendix C.Similarly, the approximated EC of OMA users is

COMA,∞E,i = − 1

Ωi ln 2ln

(N∑

n=1

ai,nΓ(bi,n − Ωi/2)

γΩi/2i c

bi,n−Ωi/2i,n

). (30)

Remarks: The result of (28) unveil that the ECs of User 1 inthe NOMA system saturates at a finite value in the high SNRregion. It is also obvious from (29) that the EC of User 2 inthe NOMA system with perfect SIC increases with respect tolog2(γ2). Meanwhile, for the OMA system, the EC of each userCOMA,∞

E,i increases with log2(γi)/2.

B. Upper-Bound on the Effective Capacity

It can be shown from (20) that, when θ → 0, the effectivecapacity is upper-bounded by the conventional ergodic capacity,defined as CUB = E{log2(1 + γi)}. The closed-form expres-sion of CUB is given in the following theorem.

Theorem 3: The upper-bound for the EC of User 1 and User2 are, respectively, given by

CUB,1 =

N∑n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m)

×b1,m−1∑p=0

1p!

(c1,m

γ1

)p

×p∑

k=0

(p

k

)γk2Γ(b2,n + k)



×[(b2,n + k)

c1,mγ2

γ1QUB,1(p, b2,n + k + 1)

− pQUB,1(p− 1, b2,n + k) +c1,m

γ1QUB,1(p, b2,n + k)

],

(31)

and

CUB,2 =1

ln 2

N∑n=1

a2,nΓ(b2,n)

γb2,n

2

ec2,n/γ2

×b2,n∑v=1

Γ(−b2,n + v, c2,n/γ2)

(c2,n/γ2)v

, (32)

where

QUB,1(u, v) =1

cv2,nΓ(v) ln 2G1,0:1,2:1,1

1,0:2,2:1,1

×(u+ 1 1, 1 1 − v− 1, 0 0

γ1

c1,m,γ2

c2,n

), (33)

Proof: As a sketch of the proof, we start with the definition ofthe ergodic capacityCUB,i = E{log2(1 + γi)} =

∫∞0 log2(1 +

x)fγi(x)dx. By performing similar calculation steps as in Ap-

pendix B and using the result of ln(1 + z) = G1,22,2(z|

1, 11, 0

) [41],

we obtain (31). Also, the result of (32) is obtained with the helpof [42, Eq. (32), Eq. (78)].

For the case of the OMA system, the upper-bounds for theeffective capacities are obtained as

COMAUB,i =

12 ln 2

N∑n=1

ai,nΓ(bi,n)

γbi,ni

eci,nγi

×bi,n∑v=1

Γ(−bi,n + v, ci,n/γi)

(ci,n/γi)v

. (34)

C. Comparison of Sum Effective Capacities

In this subsection, we compare the sum EC in the NOMA sys-tem and the OMA system. Recall that the ECs in the NOMA andOMA systems areCNOMA

E,i = − 1Ωi ln 2 ln(E{(1 + γi)

−Ωi}) (cf.

(21)), and COMAE,i = − 1

Ωi ln 2 ln(E{(1 + γi)−Ωi/2}) (cf. (25)),

respectively. Thus, for the NOMA system, the sum EC isobtained as CNOMA

E,sum = CNOMAE,1 + CNOMA

E,2 . Meanwhile, thesum EC in the OMA system is COMA

E,sum = COMAE,1 + COMA

E,2 .The analytical results for CNOMA

E,sum and COMAE,sum are provided in

the following theorem.Theorem 4: The sum ECs in the NOMA system with imperfect

SIC and the OMA system satisfy1) CNOMA

E,sum is monotonically increasing with respect to(w.r.t.) γ, ranging from zero to a finite value of (C∞

E,1 +C∞

E,2), whereC∞E,1 andC∞

E,2 are given in (28) and (29), re-spectively. Meanwhile, COMA

E,sum is monotonically increas-ing w.r.t. γ from zero to infinite.

2) At a low SNR region, CNOMAE,sum increases faster

than COMAE,sum, i.e., limγ→0(

∂(CNOMAE,sum −COMA

E,sum)

∂γ ) ≥ 0.

Meanwhile, COMAE,sum increases faster than CNOMA

E,sum , i.e.,

limγ→∞(∂(CNOMA

E,sum −COMAE,sum)

∂γ ) ≤ 0 at high SNR.Proof: See Appendix D.From the Theorem 4, we note that at a low SNR region both

CNOMAE,sum andCOMA

E,sum increase at a rate dependent on the averagechannel gains, i.e., VL = 1

2 ln 2 E{MG,1} (cf. (76)). However,at the high SNR regime, the increasing speeds of CNOMA

E,sum

and COMAE,sum tend to diminish since limγ→∞

∂CNOMAE,i

∂γ = 0 and

limγ→∞∂COMA

E,i

∂γ = 0 (cf. (79)–(81)). This behavior is similar toa downlink NOMA/OMA system over Rayleigh fading channelsin [22]. Simulation results matching the analysis are provided inSection VI.

V. UPLINK NOMA WITH MORE THAN TWO USERS PER

CLUSTER AND RESIDUAL HARDWARE IMPAIRMENTS

The proposed analytical framework using the MG approxi-mation is useful for analyzing NOMA systems with more thantwo users per cluster and/or NOMA with imperfect SIC due topractical implementation limitations such as residual hardwareimpairments. In this section, we first introduce a mathematicalresult on weighted sum of mixture gamma variables, which is,to the best of our knowledge, not available in the literature.Based on this, we analyze the OP and the EC achieved inuplink NOMA systems suffering hardware impairments withan arbitrary number of users per cluster.

Suppose thatMG,i, i = 1, 2, . . ., I, follows a MG distributionwith parameters of ai,n, bi,n and ci,n with its PDF is given in(5). Let us define MG =

∑Ii=1 τiMG,i, where τi are positive

real numbers. We have the following property about a weightedsum of mixture gamma variables.

Lemma 2: The PDF and CDF of a weighted sum of mixturegamma variables MG are given, respectively, by

fMG(x) =

N∑n1=1

N∑n2=1

. . .

N∑nI=1

a{ni}xb{ni}−1e−c{ni}x, (35)

and

FMG(x) =

N∑n1=1

N∑n2=1

. . .

N∑nI=1

a{ni}c−b{ni}{ni} γ(b{ni}, c{ni}x),

(36)

where a{ni} =Ψ({ni})[c{ni}]

b{ni}

Γ(b{ni}), b{ni} =

Υ2({ni})Φ({ni}) ,

c{ni} =Υ({ni})Φ({ni}) , Υ({ni}) =

∑Ii=1 bi,ni

τici,ni

, Φ({ni}) =∑Ii=1 bi,ni

( τici,ni

)2, and Ψ({ni}) =∏I

i=1ai,ni

Γ(bi,ni)

cbi,nii,ni

.

Proof: See Appendix E.Note that (35) as well as (36) is a summation ofN I elements.

The accuracy of the expressions (35) and (36) are confirmed bysimulations as shown in Fig. 2.

Based on the result in Lemma 2, we can express the SINR ofthe ith user as (cf.(3))

γi =γiMG,i

κ2iγiMG,i + MG,i + 1

, (37)



Fig. 2. PDF and CDF of weighted sum of mixture gamma variables.

where

MG,i =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

I∑j=2

(1 + κ2j)γjMG,j , for i = 1

i−1∑k=1

κ2kγkMG,k+

I∑j=1+i

(1 + κ2j)γjMG,j , for i = 2, . . ., I − 1

I−1∑k=1

κ2kγkMG,k, for i = I.

(38)

We note that MG,i is a weighted sum of (I − 1) MG vari-ables, which is a MG variable. From (38), we can expressMG,i =

∑I−1u=1 τuMG,u for notational convenience. The OP

expression for the case of more than two NOMA users per clusterin the presence of hardware impairments is given below.

Theorem 5: The closed-form expression of the OP of User ith

in uplink NOMA under composite fading channels is given by

Pout,i(γth,i) = 1 −i∏

j=1

[1 − Po,j(γth,j)], (39)

where

Po,j(γth,j) = 1 −N∑

n1=1

N∑n2=1

. . .

N∑nI−1=1

N∑m=1

a{nu}aj,mc−bj,mj,m

× Γ(bj,m)e− cj,mγth,j

(1−κ2jγth,j)γj

bj,m−1∑p=0

1p!

(cj,mγth,j

(1 − κ2jγth,j)γj

)p

×p∑

k=0

(p

k

)Γ(b{nu} + k)(

c{nu} +cj,mγth,j

(1−κ2jγth,j)

γj

)b{nu}+k, (40)

and the parameters a{nu}, b{nu}, and c{nu}, are calculated basedon Lemma 2.

Proof: The result is obtained by performing similar calcula-tions as in Appendix A with the help of Lemma 2.

Based on the result in (39), the expression of the sum-throughput per cluster is obtained as

τNOMAsum =

I∑i=1

[1 − Pout,i(γth,i)]Rc,i. (41)

For the case of OMA system with hardware impairments overcomposite fading channels, the OP is given by

POMAout,i (γth,i) =

N∑n=1

ai,nc−bi,ni,n γ

(bi,n,

ci,nγOMAth,i

(1 − κ2iγ

OMAth,i )γi

),

(42)

With respect to the EC, it is challenging to derive a closed-form expression in the presence of residual hardware impair-ments. However, in case of ideal-hardware transceiver (i.e.,κi = 0), the closed-form expression in terms of the Meijer-Gfunction for the EC in uplink NOMA with an arbitrary numberof users per cluster is given in the following result.

Theorem 6: The EC of User i, i = 1, 2, .., I, in uplink NOMAwithout hardware impairments over composite fading channelsis given by

CE,i(i≤I−1) =−1

Ωi ln 2ln

⎡⎣ N∑n1=1

N∑n2=1

. . .

N∑nI−i=1

N∑m=1

a{nu}ai,m

× c−bi,mi,m Γ(bi,m)

bi,m−1∑p=0

1p!

(ci,mγi

)p p∑k=0

(p

k

)Γ(b{nu} + k)

×((b{nu} + k)

ci,mγi

QE,i(p, b{nu} + k + 1)−

pQE,i(p− 1, b{nu} + k) +ci,mγi

QE,i(p, b{nu} + k)

)],

(43)

where

QE,i(u, v)

=1

cv{nu}Γ(v)Γ(Ωi)G1,0:1,1:1,1

1,0:1,1:1,1

(u+ 1 1 − Ωi 1 − v

− 0 0γi

ci,m,

1c{nu}

),

(44)

and

CE,I

=−1

Ωi ln 2ln

[N∑

n=1

ai,n

Γ(Ωi)cbi,ni,n

G1,22,1

(γici,n

1 − bi,n, 1 − Ωi

0

)].

(45)

Proof: The result is obtained by performing similar calcula-tions as in Appendix B with the help of Lemma 2.

Also, similar to Theorem 3, the corresponding upper-boundof the EC is given in the following.

Corollary 3: The upper-bound for the EC of User i in uplinkNOMA over composite fading channels is given by



CUB,i(i≤I−1) =

N∑n1=1

N∑n2=1

. . .

N∑nI−i=1

N∑m=1

a{nu}ai,mc−bi,mi,m

× Γ(bi,m)

bi,m−1∑p=0

1p!

(ci,mγi

)p p∑k=0

(p

k

)Γ(b{nu} + k)

×((b{nu} + k)

ci,mγi

QUB,i(p, b{nu} + k + 1)−

pQUB,i(p− 1, b{nu} + k) +ci,mγi

QUB,i(p, b{nu} + k)

),

(46)

where

QUB,i(u, v)

=1

cv{nu}Γ(v) ln 2G1,0:1,2:1,1

1,0:2,2:1,1

(u+ 1 1, 1 1 − v− 1, 0 0

γici,m

,1

c{nu}

),

(47)

and

CUB,I

=1

ln 2

N∑n=1

ai,nΓ(bi,n)

γbi,ni

eci,nγi

bi,n∑v=1

Γ(−bi,n + v, ci,n/γi)

(ci,n/γi)v

.

(48)

For the OMA system, the EC expression and its upper-boundcan be easily obtained based on (26) and (34), respectively.Simulation results matching these expressions are provided innext section.

VI. SIMULATION RESULTS

In simulations, we assume that 12 active users are uniformlydistributed within a circle with radius 200 meters, centered atthe Origin (0,0,0). 2-user, 3-user and 4-user NOMA clusters areformed by using the method proposed in [5]. The BS is mountedon an unmanned aerial vehicle (UAV) at a position (0,0100m).The lognormal-Nakagami-m fading channel is adopted fordemonstrations. Here, the actual lognormal-Nakagami-m fadingchannel coefficients are obtained in which shadowing is gen-erated as lognormal variables, whereas the small-scale fadingis generated by Nakagami-m variables. Furthermore, TDMA(time-division multiple access) is used as an OMA benchmark.To have a fair comparison with NOMA, the transmit powersof OMA users are scaled up proportional to the number ofusers per cluster. For example, transmit power of OMA usersis double that of NOMA users for a 2-user NOMA cluster.Unless otherwise specified, the list of simulation parameters isdescribed in Table III. All results are averaged over all clustersand channel realizations.

Fig. 3 shows the OP in the 2-user NOMA system withoutresidual hardware impairments versus the transmit power PT .It can be seen that the OPs of the users in the NOMA systemare smaller than those in the OMA system in the low-to-midSNR region. However, in the high SNR region, the OMA system

TABLE IIISIMULATION PARAMETERS

Fig. 3. Outage probabilities versus the transmit power PT .

Fig. 4. Throughput versus the transmit power PT .

achieves better OP performance. Specifically, there exhibits anerror-floor at the high SNR region due to the existence of inter-ference in the NOMA system. Furthermore, the analytical curves(cf. (12)–(13)) match exactly with the simulation curves, whichconfirms the accuracy of the derived expressions in Section III.As expected, the asymptotic curves (cf. (15)–(16)) close to theexact curves in the high SNR region.

The achieved throughput in the 2-user NOMA system withouthardware impairments under different levels of power back-offis shown in Fig. 4. It can be seen that when the power back-off is large, a smaller amount of power is allocated to User 2,which leads to a less balance in terms of the achievable ratebetween two users. In Fig. 5, we plot the total throughput of thewhole network when 2-user, 3-user, 4-user cluster schemes inthe presence of residual hardware impairments are employed.Here, a power back-off step of 5 dB is used as in [14]. The total



Fig. 5. Throughput in 2-user, 3-user, and 4-user NOMA clusters with hardwareimpairments.

Fig. 6. Effective capacities versus the transmit power PT .

power PT of all transmitters in the network is kept the same.3

Also, analytical results are obtained based on (41) derived inSection V. The results show that the total TP is decreased whenthe number of users per NOMA cluster increases. This is becausethe detrimental effect of the inter-user interference introduced byadditional users outweighs the benefit in terms of the throughput.As a result, lower total throughput is attained with an increasingof the cluster size.

In Fig. 6, we plot the ECs as a function of the transmit powerPT for a 2-user NOMA cluster without hardware impairments.It can be seen that the ECs are higher when the transmit powerincreases. For the NOMA system, the EC of User 1 becomessaturated in the high SNR region due to the presence of theresidual interference. Meanwhile, the EC of User 2 and theECs in the OMA system without residual hardware impairmentsincrease linearly with log(P ) and log(P )/2, respectively. Thisagrees with our remarks based on analytical results presentedin Section IV. The results also unveil that the sum EC achievedin the 2-user NOMA cluster are higher than that in the OMAsystem in the low-to-mid SNR region, but lower in a very highSNR regime. Furthermore, the results obviously confirm ourconclusions in Theorem 4 regarding the characteristics of thesum ECs in the NOMA and OMA systems. It is also worthnoting that the simulation results match the analytical resultsderived in Section IV (cf. (22)–(34)). The total ECs achievedwhen 2-user, 3-user, and 4-user NOMA cluster schemes are

3We ignore impacts of bandwidth in this comparison for simplicity, i.e., 2-user,3-user, and 4-user clusters have an equal bandwidth.

Fig. 7. Total EC in 2-user, 3-user, and 4-user NOMA clusters.

Fig. 8. Effective capacities versus delay-QoS exponent θ.

implemented are shown in Fig. 7. It can be seen that the ECis reduced when the number of users per cluster increases.This behavior can be explained similar to the case of the totalthroughput discussed above. Additionally, the matching betweenthe analysis and simulation results confirms the accuracy of thederived EC expressions in Theorem 6.

The impact of the delay-QoS exponent θ on the ECs of a2-user NOMA scheme is shown in Fig. 8. It can be seen that theECs reach their upper-bounds when θ approaches zero, which isconsistent to our discussions in Section IV. On the other hand,when θ increases the EC values is decreased. This means thatmore stringent delay constraints lead to a reduction of the EC.The results also demonstrate that the EC gap between the theNOMA system and the OMA system is reduced with an increaseof θ, which indicates that the NOMA system is less robust to thedelay constraints than its counterpart.

In Fig. 9, we show the impact of the residual hardwareimpairments on the TP and the EC of the 2-user NOMA system.It is obvious that TP and EC are reduced when the value κiincreases. This is because an increase of the impairments resultsin a lower SINR value, which in turns increases the OP andreduces the TP and the ECs (cf. (2)). The results also demonstratethat User 2 in the NOMA system is more sensitive to this kind ofinterference than its counterparts. This can be explained by thefact that NOMA signals are superposed in time and frequencydomains, which makes the residual impairments of a particularuser affect not only itself but other users also. Moreover, largerpower is allocated to User 1 due to the power back-off policy.

The impact of the power back-off level on the TP and theEC of the 2-user NOMA scheme is shown in Fig. 10. It can be



Fig. 9. Impact of residual hardware impairments on TP and EC.

Fig. 10. Impact of a power back-off on TP and EC.

seen that the back-off value plays a significant impact on thesystem performance. In particular, the TP and EC of User 1 inthe NOMA system are improved when a larger power back-offstepPΔ is applied. This makes sense since higherPΔ leads to anenhanced SINR of User 1. Meanwhile, a larger power back-offimplies that lower power is allocated to User 2, which in turnsreduces the TP and EC of User 2. It is important to observe fromthe figure that there exists an optimal back-off value such thatthe throughput is maximal. Also, there exists a certain range PΔ

where the NOMA system is better than the OMA system.4 Thesesystem characteristics indicate that a careful choice of the powerallocation coefficient is important to realize an efficient NOMAsystem.

Finally, to show the impact of the lognormal-Nakagami-mchannel parameters on the system performance, we plot inFig. 11 and Fig. 12 the throughput and the effective-capacityof the 2-user NOMA system versus the standard deviation ofthe shadowing and the fading parameter, respectively. It can beseen from Fig. 11 that higher shadowing causes a reduction ofboth the TP and the EC in all systems. However, this effect isnot significant. Meanwhile, the results in Fig. 12 show that theachievable TP and EC are improved when the parameter m isincreased that indicates a better fading channel condition.

4In practice, a power back-off policy is not implemented in OMA systems.Instead, the arrived power from different OMA users at the BS is required equal.In simulations, we consider power back-off for all systems, where the power ofOMA users are multiplied by I for fair comparisons with NOMA. This providesinsights into behaviors of power-domain NOMA system associated with SICversus orthogonal transmission.

Fig. 11. Impact of shadowing standard deviation σ (dB).

Fig. 12. Impact of small-scale fading parameter m.

VII. CONCLUSION

We have proposed a mixture gamma distribution based analyt-ical framework for NOMA wireless systems with delay-sensitivetransmissions and imperfect SIC due to hardware impairmentsunder several fading channels. The analytical closed-forms ofthe OP, the TP and the EC, together with their asymptoticexpressions at the high SNR regime, have been derived. Theseanalytical results match well with the simulations results. Per-formance comparisons of the hardware impairments-sufferedNOMA system with the ideal NOMA system as well as theOMA system have also been addressed. Our approach can beapplied for various fading distributions in NOMA systems withan arbitrary number of users, so that the effects of the channelparameters, including shadowing and small-scale fading, on thesystem performance can be easily evaluated. The results haveshown that the presence of the residual hardware impairmentsreduces the TP and the EC of the NOMA system. Also, theTP and the EC became lower when the delay-QoS constraintbecomes stringent. The results also demonstrated that the powerallocation is important to achieve performance gain of theNOMA system over its counterpart. Our future works will studythe optimal power allocation and user-pairing to maximize thetotal sum-rate. A hybrid NOMA/OMA scheme would also bean interesting approach for future investigations.



APPENDIX ADERIVATIONS OF OUTAGE PROBABILITY

The OP of User 1 is calculated as (cf. (8))

Pout,1(γth,1) = Pr(γ1 < γth,1) = Pr

(γ1MG,1

γ2MG,2 + 1< γth,1

)

= Pr

(MG,1 <

γth,1(γ2MG,2 + 1)γ1

)

=

∫ +∞

0

∫ γth,1(γ2x2+1)

γ1

0fMG,1(x1)fMG,2(x2)dx1dx2

=

∫ +∞

0fMG,2(x2)FMG,1

(γth,1(γ2x2 + 1)

γ1

)dx2,

(49)

where fMG,2(x) and FMG,1(x) denote the PDF and CDF ofMG,2 and MG,1, respectively. Recall that MG,i � SiHi, i =1, 2, are composite fading variables, whose PDF and CDF can beapproximated by the MG distribution approach. By substitutingthe results in (5) and (6) into (49), we have

Pout,1(γth,1) =

N∑n=1

N∑m=1

a2,na1,mc−b1,m

1,m ×∫ +∞

0xb2,n−1e−c2,nxγ

(b1,m,

c1,mγth,1(γ2x+ 1)γ1

)dx. (50)

By expressing γ(n, x) = (n− 1)![1 − e−x∑n−1

m=0 xm/m!]

when n is an integer [36, Eq. (8.352.1)] and using the resultof∫ +∞

0 xne−ax = n!a−n−1 [36], we can rewrite (50) as

Pout,1(γth,1) =

N∑n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m)

[Γ(b2,n)

cb2,n

2,n

− I],

(51)

where

I �∫ +∞

0xb2,n−1e−c2,nxe

− c1,mγth,1(γ2x+1)

γ1 ×

b1,m−1∑p=0

1p!

(c1,mγth,1(γ2x+ 1)

γ1

)p

dx. (52)

With the help of Lemma 1, we can simplify (51) to

Pout,1(γth,1) = 1 −N∑

n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m)I. (53)

To calculate I, we first apply the binomial expansion of (a+x)n =

∑nk=1

(nk

)xkan−k, and then using the result in [36, Eq.

(3.351.3)]. After some manipulations, we arrive at

I = e− c1,mγth,1

γ1 ×b1,m−1∑p=0

1p!

(c1,mγthγ1

)p p∑k=0

(p

k

)γk2Γ(b2,n + k)(

c2,n +c1,mγth,1γ2

γ1

)b2,n+k.

(54)

By substituting (54) into (53) we obtain (12). With respectto Pout,2(γth), it is readily to show that Pr(γ2 < γth,2) =∑N

n=1 a2,nc−b2,n

2,n γ(b2,n, c2,nγth,2/γ2). Hence, (13) is obtainedby substituting this result and (12) into (11). The proof is thencompleted.

APPENDIX BDERIVATION OF EFFECTIVE CAPACITIES

We first calculate the PDF of the instantaneous SNR γ1. Bydefinition, we have fγ1(x) =

∂∂xPr(γ1 < x). Using the result

in (12), we obtain

fγ1(x) = −N∑

n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m)

b1,m−1∑p=0

1p!

(c1,m

γ1

)p

×p∑

k=0

(p

k

)γk2Γ(b2,n + k)

∂

∂x

⎛⎝ xpe

− c1,mx

γ1(c2,n +

c1,mxγ2γ1

)b2,n+k⎞⎠

=

N∑n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m)

b1,m−1∑p=0

1p!

(c1,m

γ1

)p p∑k=0

(p

k

)

× γk2Γ(b2,n + k)

⎡⎢⎣xpe

− c1,mx

γ1 (b2,n + k)c1,mγ2

γ1(c2,n +

c1,mxγ2γ1

)b2,n+k+1

−(pxp−1 − c1,mxp

γ1

)e− c1,mx

γ1(c2,n +

c1,mxγ2γ1

)b2,n+k

⎤⎥⎦ . (55)

Now, substituting the result in (55) into (21), we obtain

CE,1 =−1

Ω1 ln 2ln

[N∑

n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m − 1)

×b1,m−1∑p=0

1p!

(c1,m

γ1

)p p∑k=0

(p

k

)γk2Γ(b2,n + k)J

⎤⎦ ,

(56)

where

J � (b2,n + k)c1,mγ2

γ1QE,1(p, b2,n + k + 1)−

pQE,1(p− 1, b2,n + k) +c1,m

γ1QE,1(p, b2,n + k), (57)

and

QE,1(u, v) �∫ +∞

0

(1 + x)−Ω1xue− c1,mx

γ1(c2,n +

c1,mxγ2γ1

)v dx. (58)

To evaluate the integral in (58), we express the functions of e−ax,(1 + x)b, and (c+ dx)−v in terms of the Meijer-G function by

using the results in [41]. Specifically, we have e− c1,mx

γ1 =

G1,00,1(

c1,m

γ1x−0), (1 + x)−Ω1 = 1

Γ(Ω1)G1,1

1,1(x1 − Ω1

0) and



(c2,n +c1,mxγ2

γ1)−v = 1

cv2,nΓ(v)G1,1

1,1(c1,mγ2c2,nγ1

x1 − v

0). Then,

we obtain

QE,1(u, v) =1

cv2,nΓ(v)Γ(Ω1)

∫ +∞

0xuG1,0

0,1

(c1,m

γ1x−0

)

×G1,11,1

(x

1 − Ω1

0

)G1,1

1,1

(c1,mγ2

c2,nγ1x

1 − v0

)dx

=1

cv2,nΓ(v)Γ(Ω1)

× G1,0:1,1:1,11,0:1,1:1,1

(u+ 1 1 − Ω1 1 − v− 0 0

γ1

c1,m,γ2

c2,n

),

(59)

where the last step is obtained by using the integral result in [43]and G(: | : | : |., .) denotes the extended generalized bivariateMeijer-G function [38]. The calculation for CE,2 is done in asimilar way. In particular, the PDF of γ2 under perfect SIC as

fγ2(x) =

N∑n=1

a2,n

γb2,n

2

xb2,n−1e−c2,nx/γ2 . (60)

Substituting this result to (21), and then evaluating the result-ing integral by using the result in [36, Eq. (7.813.1)]. The result(23) is thus obtained.

APPENDIX CASYMPTOTIC DERIVATION OF EFFECTIVE CAPACITIES

We perform similar calculation steps as in Appendix A toobtain the CDF of γ∞1 as

Fγ∞1(x) = 1 −

N∑n=1

N∑m=1

a2,na1,mc−b1,m

1,m Γ(b1,m)×

b1,m−1∑p=0

(αc1,mx)p

p!

Γ(b2,n + p)

(c2,n + αc1,mx)b2,n+p. (61)

From (61), we can express the PDF of γ∞1 as

fγ∞1(x) =

N∑n=1

N∑m=1

a2,na1,mc−b1,m1,m Γ(b1,m)

b1,m−1∑p=0

(αc1,m)p

p!×

Γ(b2,n + p)

[(b2,n + p)αc1,mxp

(c2,n + αc1,mx)b2,n+p+1 − pxp−1

(c2,n + αc1,mx)b2,n+p

].

(62)

On the other hand, using the same approach as in Appendix Band the result in [36, Eq. (7.811.5)], we obtain∫ ∞

0

(1 + x)−Ωxu

(b + cx)vdx =

1bvΓ(v)Γ(Ω)

G2,22,2

(c

b

−u,−v + 1Ω− u− 1, 0

).

(63)

By substituting (62) into (25), and then evaluating the result-ing integral by using (63), we arrive at

C∞E,1≈

−1Ω1 ln 2

ln

{1

Γ(Ω1)

N∑n=1

N∑m=1

a2,na1,mc−b1,m

1,m c−b2,n

2,n Γ(b1,m)

×b1,m−1∑p=0

1p!

(αc1,m

c2,n

)p [αc1,m

c2,nG2,2

2,2

(αc1,m

c2,n

−p,−b2,n − pΩ1 − p− 1, ε

)

− pG2,22,2

(αc1,m

c2,n

−p+ 1,−b2,n − p+ 1Ω1 − p, ε

)]}. (64)

In the above expression, an arbitrarily small value of ε isused to approximate zero. This guarantees the conditions for theexistence of the Mejer-G function are always satisfied. Similarcalculation steps are performed to obtain the result in (29).

With respect to C∞E,2, we can use the approximation of (1 +

γ2) ≈ γ2 when γ2 → ∞ to simplify the EC expression of User2. In particular, we have

C∞E,2 = − 1

Ω2 ln 2ln

(∫ ∞

0x−Ω2fγ2(x)dx

), (65)

where fγ2(x) is given in (60). By evaluating (65) with the helpof [36, Eq. (3.381.4)], we obtain the result in (29).

APPENDIX DCOMPARISON OF EFFECTIVE CAPACITIES

A. Monotonic Increase

To prove the first part of Theorem 4, we showthat limγ→0 C

NOMAE,sum = 0, limγ→0 C

OMAE,sum = 0,

limγ→∞ CNOMAE,sum = (C∞

E,1 + C∞E,2), limγ→∞ COMA

E,sum = ∞,∂CNOMA

E,sum

∂γ > 0 and∂COMA

E,sum

∂γ > 0. The EC of User 1 and User2 in the NOMA system respectively are

CNOMAE,1 = − 1

Ω1 ln 2ln

[E

{(1 +

γ1MG,1

γ2MG,2 + 1

)−Ω1}]

= − 1Ω1 ln 2

ln

[E

{(1 +

γαNOMA1 MG,1

γαNOMA2 MG,2 + 1

)−Ω1}]

,

(66)

and

CNOMAE,2 = − 1

Ω2 ln 2ln[E{(

1 + γαNOMA2 MG,2

)−Ω2}]

.

(67)

It can be shown that limγ→0 CNOMAE,1 = 0 and

limγ→0 CNOMAE,2 = 0. Thus, limγ→0 C

NOMAE,sum =

limγ→0(CNOMAE,1 + CNOMA

E,2 ) = 0. Also, limγ→∞ CNOMAE,1

= C∞E,1 and limγ→∞ CNOMA

E,2 = C∞E,2 (cf. (28)–(29)).

Hence, limγ→∞ CNOMAE,sum = limγ→∞(CNOMA

E,1 + CNOMAE,2 ) =

(C∞E,1 + C∞

E,2).The first-order derivative of CNOMA

E,1 is calculated as

∂CNOMAE,1

∂γ= − 1

Ω1 ln 2

(E

{(1 +

γαNOMA1 MG,1

γαNOMA2 MG,2+1

)−Ω1})′

E

{(1 +

γαNOMA1 MG,1

γαNOMA2 MG,2+1

)−Ω1}



=

E

{(1 +

γαNOMA1 MG,1

γαNOMA2 MG,2+1

)−Ω1−1αNOMA

1 MG,1

(γαNOMA2 MG,2+1)2

}

ln 2E

{(1 +

γαNOMA1 MG,1

γαNOMA2 MG,2+1

)−Ω1} > 0.

(68)

Also, we have

∂CNOMAE,2

∂γ

=αNOMA

2 E{MG,2

(1 + γαNOMA

2 MG,2)−Ω2−1

}ln 2E

{(1 + γαNOMA

2 MG,2)−Ω2

} > 0.

(69)

Consequently, we obtain

∂CNOMAE,sum

∂γ=∂(CNOMA

E,1 + CNOMAE,2

)∂γ

> 0. (70)

For the OMA system, the EC of the ith user is given by

COMAE,i = − 1

Ωi ln 2ln[E{(

1 + γαOMAi MG,i

)−Ωi/2}]

.

(71)

It is readily that limγ→0 COMAE,i = 0 and limγ→∞ COMA

E,i = ∞.Thus, we have limγ→0 C

OMAE,sum = 0 and limγ→∞ COMA

E,sum = ∞.Additionally, we have

∂COMAE,i

∂γ=αOMAi E

{MG,i

(1 + γαOMA

i MG,i

)−Ωi/2−1}

2 ln 2E{(

1 + γαOMAi MG,i

)−Ωi/2}

> 0. (72)

Therefore, we obtain

∂COMAE,sum

∂γ=∂(COMA

E,1 + COMAE,2

)∂γ

> 0. (73)

This completes the first part of Theorem 4.

B. At the Low and High SNR Regimes

For the second part of Theorem 4, we will show

that VL � limγ→0(∂(CNOMA

E,sum −COMAE,sum)

∂γ ) ≥ 0 and VH �limγ→∞(

∂(CNOMAE,sum −COMA

E,sum)

∂γ ) ≤ 0. The value of VL at thelow SNR regime is rewritten as

VL = limγ→0

(∂CNOMA

E,1

∂γ+∂CNOMA

E,2

∂γ− ∂COMA

E,1

∂γ− ∂COMA

E,2

∂γ

).

(74)

From (68), (69), and (72), we have limγ→0∂CNOMA

E,i

∂γ

= 1ln 2 E{αNOMA

i MG,i} and limγ→0∂COMA

E,i

∂γ =1

2 ln 2 E{αOMAi MG,i}. Thus, (74) becomes

VL =1

ln 2

[E{αNOMA

1 MG,1}+ E

{αNOMA

2 MG,2}

−12

E{αOMA

1 MG,1}− 1

2E{αOMA

2 MG,2}]. (75)

Since E{MG,1} = E{MG,2}, (75) is reduced to

VL =

(αNOMA

1 + αNOMA2 − 1

2αOMA

1 − 12αOMA

2

)E {MG,1}

ln 2

=αNOMA

2

2 ln 2E {MG,1} . (76)

The average value of MG,1 can be easily evaluated by usingthe PDF function of a mixture gamma function in (5) and theintegral result in [36, Eq. (3.351.3)]. Therefore, we obtain

VL =αNOMA

2

2 ln 2

N∑n=1

a1,nc−b1,n−11,n b1,n! > 0. (77)

At the high SNR regime, we have

VH = limγ→∞

(∂CNOMA

E,1

∂γ+∂CNOMA

E,2

∂γ− ∂COMA

E,1

∂γ− ∂COMA

E,2

∂γ

).

(78)

From (68), (69), and (72), we obtain the results in (79)–(81),shown at the top of the next page. It can be concluded from (78)that VH < 0 when γ → ∞. This completes the proof.

APPENDIX EWEIGHTED SUM OF MIXTURE GAMMA FUNCTIONS

Let the random variableMG,i have the mixture gamma distri-bution with the PDFfMG,i

(x) =∑N

n=1 ai,nxbi,n−1e−ci,nx, x ≥

0 (cf.(5)). DefineYG,i = g(MG,i) = τiMG,i, where τi is a pos-itive real number. This transformation has an inverse MG,i =g−1(YG,i) = YG,i/τi and the Jacobian dMG,i/dYG,i = 1/τi.By using the transformation method, we can express the PDF ofYG,i as

fYG,i(y) = fMG,i

(g−1(y)

) ∣∣∣∣dxdy∣∣∣∣

=

N∑n=1

ai,n

(y

τi

)bi,n−1

e−ci,ny/τi ×∣∣∣∣ 1τi

∣∣∣∣=

N∑n=1

ai,nΓ(bi,n)

cbi,ni,n

Γ

(bi,n,

τici,n

), y > 0, (82)

where Γ(β, α) = xβ−1e−x/α

αβΓ(β), (x > 0), denotes the gamma func-

tion with shape β and scale α.Define the weighted sum of independent MG variables as

MG =∑I

i=1 YG,i. Then the PDF of MG is formulated as

fMG(y) = fYG,1(y) ∗ fYG,2(y) ∗ . . . ∗ fYG,I

(y)

=

[N∑

n=1

a1,nΓ(b1,n)

cb1,n

1,n

Γ

(b1,n,

τ1

c1,n

)]∗ . . .∗



limγ→∞

∂CNOMAE,1

∂γ=

1ln 2

limγ→∞

⎡⎢⎢⎣ 1γ2 ×

E

{(1 +

αNOMA1 MG,1

αNOMA2 MG,2+1/γ

)−Ω1−1αNOMA

1 MG,1

(αNOMA2 MG,2+1/γ)2

}

E

{(1 +

αNOMA1 MG,1

αNOMA2 MG,2+1/γ

)−Ω1}

⎤⎥⎥⎦ ∼ lim

γ→∞1γ2 (79)

limγ→∞

∂CNOMAE,2

∂γ=

1ln 2

limγ→∞

⎡⎣ 1γ×

E{αNOMA

2 MG,2(1/γ + αNOMA

2 MG,2)−Ω2−1

}E{(

1/γ + αNOMA2 MG,2

)−Ω2}

⎤⎦ ∼ lim

γ→∞1γ

(80)

limγ→∞

∂COMAE,i

∂γ=

12 ln 2

limγ→∞

⎡⎣ 1γ×

E{αOMAi MG,i

(1/γ + αOMA

i MG,i

)−Ωi/2−1}

E{(

1/γ + αOMAi MG,i

)−Ωi/2}

⎤⎦ ∼ lim

γ→∞1γ

(81)

[N∑

n=1

aI,nΓ(bI,n)

cbI,nI,n

Γ

(bI,n,

τIcI,n

)]

=

N∑n1=1

N∑n2=1

. . .

N∑nI=1

{I∏

i=1

ai,niΓ(bi,ni

)

cbi,nii,ni

×[Γ

(b1,n1 ,

τn1

c1,n1

)∗

Γ

(b2,n2 ,

τn2

c2,n2

)∗ . . . ∗ Γ

(bI,nI

,τnI

cI,nI

)]}, (83)

where * denotes the convolution. It is challenging to calculate(83) since it involves high-order convolution of gamma vari-ables. Thus, we apply the Welch-Satterthwaite approximation,which originally developed for independent chi-square randomvariables [44]. Specifically, we can express

Γ

(b1,n1 ,

τn1

c1,n1

)∗ Γ(b2,n2 ,

τn2

c2,n2

)∗ . . . ∗ Γ

(bI,nI

,τnI

cI,nI

)

≈ Γ

(Υ2 ({ni})Φ ({ni}) ,

Φ({ni})Υ ({ni})

), (84)

where Υ({ni}) �∑I

i=1 bi,ni

τici,ni

and Φ({ni}) �∑Ii=1 bi,ni

( τici,ni

)2. Thus, (83) is rewritten as

fMG(y) =

N∑n1=1

N∑n2=1

. . .

N∑nI=1

a{ni}yb{ni}−1e−c{ni}y, (85)

where a{ni}, b{ni}, and c{ni} are defined in Lemma 2. Thiscompletes the proof.

ACKNOWLEDGMENT

We thank the New South Wales (NSW) Defence InnovationNetwork (DIN) and the NSW State Government, Australia, forthe financial support of this project through the DIN Pilot ProjectGrant (ID: 888-006-985, Funding Years: 2019-2020).

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