outage probability analysis for the multi-carrier noma downlink … · 2020. 4. 23. · carrier...

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Outage Probability Analysis for the Multi-CarrierNOMA Downlink Relying on Statistical CSI

Shenhong Li, Student Member, IEEE, Mahsa Derakhshani Member, IEEE, Sangarapillai Lambotharan, SeniorMember, IEEE, Lajos Hanzo, Fellow, IEEE

Abstract—In this treatise, we derive tractable closed-formexpressions for the outage probability of the single cell multi-carrier non-orthogonal multiple access (MC-NOMA) downlink,where the transmitter side only has statistical CSI knowledge.In particular, we analyze the outage probability with respectto the total data rates (summed over all subcarriers), given aminimum target rate for the individual users. The calculationof outage probability for the distant user is challenging, sincethe total rate expression is given by the sum of logarithmicfunctions of the ratio between two shifted exponential randomvariables, which are dependent. In order to derive the closed-formoutage probability expressions both for two subcarriers and for ageneral case of multiple subcarriers, efficient approximations areproposed. The probability density function (PDF) of the productof shifted exponential distributions can be determined for thenear user by the Mellin transform and the generalized upperincomplete Fox’s H function. Based on this PDF, the correspond-ing outage probability is presented. Finally, the accuracy of ouroutage analysis is verified by simulation results.

Index Terms—Outage probability, Non-orthogonal multipleaccess, MC-NOMA, Channel state information (CSI), Shiftedexponential distribution, Mellin transform, Generalized upperincomplete Fox’s H function

I. INTRODUCTION

Non-orthogonal multiple access (NOMA) has been recentlyintroduced as a potential technique of improving the spectralefficiency for next generation systems beyond that of orthog-onal multiple access (OMA) [1], [2]. As opposed to OMAschemes, NOMA is capable of supporting a large number ofusers via non-orthogonal resource allocation by the simulta-neous exploitation of the time, frequency and code domainsas well as multiplexing them at different power levels [3], [4],[5]. The superposition of signals is performed by ensuringthat the inter-user interference can be successfully removedby applying successive interference cancellation (SIC) at thereceiver. The authors of [4], [6], [7] introduced the concepts,challenges, applications and research trends of NOMA sys-tems. The authors of [4] introduced the basics of NOMA,

S. Li, M. Derakhshani, and S. Lambotharan are with the Signal Processingand Networks Research Group, Wolfson School, Loughborough Univer-sity, Leicestershire, LE11 3TU, United Kingdom (email: [email protected];[email protected]; [email protected]).

L. Hanzo is with the School of Electronics and Computer Science, Uni-versity of Southampton, Southampton, SO17 1BJ, United Kingdom (email:[email protected]).

This work was supported in part by the Engineering and Physical SciencesResearch Council under Grant EP/R006385/1, EP/Noo4558/1, EP/PO34284/1,COALESCE, of the Royal Society’s Global Challenges Research Fund Grantas well as of the European Research Council’s Advanced Fellow GrantQuantCom.

including their power allocation policies, the application ofSIC, and compared NOMA and conventional OMA. This studyalso discussed the recent developments in NOMA, such asthe combination of NOMA and MIMO schemes, cooperativeNOMA, and NOMA in cognitive radio networks. Finally,research challenges and promising future directions were pro-vided. The authors of [6] provided a comprehensive reviewof power-domain NOMA and discussed its performance withthe aid of numerical results. In [7], the authors highlightedthe basic principles, key features, receiver complexity andengineering feasibility of NOMA systems.

The achievable performance of NOMA systems signifi-cantly relies on the accuracy of the channel state information(CSI). Therefore, the outage probability of failing to supporta minimum target rate is a critical performance metric. Theoutage analysis of NOMA has received considerable researchattention both under the assumptions of only having statisticalCSI1 (e.g., [8]–[12]), imperfect CSI (e.g., [13], [14]), andalso imperfect SIC (e.g., [15], [16]). These studies have beenfocused mainly on single-carrier NOMA systems. Hence, tosolve this open problem, we focus on the outage analysis ofthe multi-carrier NOMA downlink in the face of only havingstatistical knowledge of the CSI available at the transmitter.Furthermore, having the exact knowledge of the instantaneousCSI at the transmitter constitutes a strong idealized simplifyingassumption. Hence, the main motivation of this treatise is toanalyze the outage probability w.r.t the data rates, (summedover all subcarriers) in the multi-carrier NOMA downlinkunder a more realistic model. The outage probability of multi-carrier NOMA systems has been studied by ensuring that theoutage probability associated with a certain minimum data rateof each user on each subcarrier does not exceed a certaintolerable maximum value. By contrast, in this work, we defineand analyze the outage probability in terms of the total datarate of each user under the assumption that channel coding isemployed and thus the decoding is performed after combiningthe signals gleaned from all subcarriers. Assuming that theforward error correction (FEC) codewords extend over a singleblock, the outage probability for a given rate is the probabilitythat the mutual information of these parallel channels falls(sum rate here) below that rate [17]. In such a case, outage(or failure in decoding) would occur when the detection errorsover different subcarriers cannot be identified and corrected,which is the case if the sum rate is lower than the target.

1Statistical CSI represents the long-term channel state information, whichmeans that only the distribution of the channel gain (including its type andits statistical properties) is known, but not the actual realization.

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Moreover, we focus our attention on the case when only theCSI statistics are known at the transmitter and thus an outageevent may happen when user k is unable to detect either itsown message or the messages of the weaker users for SICpurposes.

A. Literature review

Previously, to explore the potential gains of applyingNOMA, many contributions on NOMA systems assumedthat perfect CSI knowledge is available at both the receiverand transmitter sides [3], [18]–[21]. The authors of [18]conceived resource allocation algorithm for full-duplex mul-ticarrier NOMA (MC-NOMA) systems. As a further devel-opment, the authors of [19]–[21] investigated dynamic powerallocation, optimal resource allocation (power allocation anduser pairing) as well as distributed power control in a multi-cell NOMA system, respectively. However, having the exactknowledge of instantaneous CSI at the transmitter is anidealized simplifying assumption, since finite-delay, finite-accuracy quantized feedback links must be used. Hence, amore realistic research trend has emerged, focusing on NOMAwhen only imperfect knowledge of the CSI is available. Forexample, the authors of [22] studied the power allocationproblem of the NOMA downlink in the face of imperfect CSI,while the authors of [13] improved the energy efficiency of adownlink NOMA single-cell network by considering imperfectCSI. They assumed a channel estimation error model wherethe BS only knows the estimated channel gain and has apriori knowledge of the variance of the estimation error. Inthese scenarios, the outage probability is an important metric,which is widely used for performance analyses, especiallywhen the communication channel is characterized by randomparameters. The existing contributions exploring the outageanalysis of NOMA mainly focus on single-carrier scenarios.

As for the single-carrier NOMA downlink, Ding et al. [23]first derived the outage probability for the downlink of aNOMA system in conjunction with the idealized simplifyingassumption of perfect channel state information (CSI), assum-ing that all mobile users were uniformly distributed withina cell. This work provided a closed-form expression of theoutage probability, under the limitation of a single-carrier case.Later Hou et al. [9] characterized the outage performance ofthe downlink in a NOMA system with perfect CSI and fixedpower allocation for transmission over Nakagami-m fadingchannels. Again, this work is also limited to the perfect CSIscenario and single-carrier systems. Timotheous and Krikidis[24] studied the power allocation problem from the viewpointof fairness under full CSI and average CSI knowledge byapplying outage probability (OP) constraints. The effect ofimperfect CSI and of second order statistics (SOS)-based CSIon the performance of the NOMA downlink was studied byYang et al. [25]. By contrast, Shi et al. [26] investigatedthe outage performance of the downlink in a NOMA systemwith only statistical CSI knowledge at the transmitter. Theyderived the optimal power allocation factor in closed formand determined the optimal detection order to avoid excessiveoutage probabilities. Then Wang et al. [27] also analyzed the

outage performance of the downlink in a NOMA system withonly statistical CSI knowledge at the transmitter. Yang et al.[28] derived the outage probability for the paired users of adynamic power allocation based hybrid NOMA systems. Yueet al. [29] also investigated the outage performance of a unifiedNOMA framework by invoking stochastic geometry. Finally,Cui et al. [30] applied multiple input and multiple output(MIMO) techniques to a NOMA system and analysed the out-age probability by considering both the channel’s distributioninformation (CDI) and the channel estimation uncertainty.

As for the single-carrier NOMA uplink, its outage analysiswas disseminated in [31]–[33] under statistical CSI, but theanalysis was limited to at most three users. In [10], Liu etal. extended the outage analysis of the single-carrier NOMAuplink to an arbitrary number of users. Furthermore, Wanget al. [11] analyzed the outage probability of NOMA forindependent but not identically distributed (i.ni.d) Rayleighfading environments. A closed-form outage expression wasderived for both the dynamic and the fixed detection order.Xia et al. [8] proposed an advanced multiuser SIC detectorfor the NOMA uplink and analyzed the outage performanceof this scenario.

The NOMA techniques have evolved from single-carrierNOMA (SC-NOMA) to multi-carrier NOMA (MC-NOMA)[34]. In order to fully realize the potential of NOMA, effi-cient scheduling techniques are also needed. In conventionalmulticarrier systems, the total bandwidth is partitioned intomultiple subcarriers and each subcarrier is allocated to a singleuser in order to avoid multiuser interference. Sun et al. [35]have demonstrated that the MC-NOMA provides an excellentsystem performance compared to the MC-OMA.

However, as for the multi-carrier NOMA downlink, thereis paucity of research contributions focusing on its outageanalysis. The current approach is to ensure that the outageprobability associated with the given data rate of each useron each subcarrier does not exceed a certain value. Forinstance, Wei et al. [36] proposed a power-efficient resourceallocation scheme for MC-NOMA relying on imperfect CSI,while Tweed et al. [16] studied outage constrained resourceallocation in conjunction with imperfect SIC, but both used theoutage per subcarrier. However, it is not practically attractiveto have a specific outage probability for the data rate on eachsubcarrier, since the ultimate aim of MC-NOMA is to ensureachieving a specific total rate for each user. Hence, in thispaper, we consider a more realistic MC-NOMA downlinksystem where the outage probability is defined in terms ofthe total data rate of each user averaged over all subcarriersbeing below a certain threshold. We also examine the outageperformance under statistical CSI provision.

B. Contributions

The key novelty in our work is that of deriving closed-formexpressions for the outage probability of multi-carrier NOMAdownlink for an arbitrary number of users and subcarriers.Indeed, the outage probability relying on statistical propertieshas been studied before, but not by unveiling the intricacies ofmulti-carrier NOMA. To the best of our knowledge, this work

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is the first one to analyze outage in the MC-NOMA downlink,where the outage probability of decoding the transmission ofuser k at any receiver (at user k or other users closer to theBS) is defined based on the sum rate over all subcarriersas the probability that the transmission of user k cannot bedecoded with negligible error probability. There are only afew studies on the OP analysis of MC-NOMA [16], [34],where the OP was defined as the probability of failing tosatisfy the minimum required data rate of each user on eachsubcarrier. By contrast, the OP of each user is defined hereas the probability of failing to detect its own message orthe messages of the weaker users based on the total rate ofan individual user over all subcarriers. As NOMA is moreattractive for the case of two users per subcarrier [34], we firstconsider the case for two users at each subcarrier and analysethe outage probability of both the users who share multiplesubcarriers using NOMA. Then, we extend the results to amore general case of an arbitrary number of users. Our maincontributions are summarized as follows:

1) We analyze the OP of the far user of the multi-carrierNOMA downlink. The challenge is then to obtain thestatistics required for the OP analysis, where the total rateexpression is given by the sum of logarithmic functionsof the ratio between two shifted exponential random vari-ables, which are dependent. In order to derive a closed-form solution, first we transform the sum of logarithmicfunctions into the product of the ratio between two shiftedexponential random variables and then approximate theproduct by ignoring the high-order terms. The simulationresults show that the approximation is quite tight, hence itwas used for our subsequent derivations. Then, in the caseof two subcarriers, we analytically calculate the PDF andCDF of the inverse shifted exponential random variables,and derive the corresponding OP expression. For thegeneral case of an arbitrary number of subcarriers, thesum of inverse shifted exponential random variables maybe accurately modelled by a Gamma distribution. Hencethe corresponding closed-form expression is provided forthe OP.

2) We also investigate the OP of the near user. We considerthat an outage event may happen if the near user cannotdetect either its own message or the message of the faruser. Thus, first the probability of successfully detectingthis user and the probability of successfully detecting thefar user are studied. The former is characterized by thecomplementary CDF (CCDF) of the product of shiftedexponential random variables in the case of successfulSIC detection. To obtain this CCDF, by applying theMellin transform [35], we determine the PDF of theproduct of shifted exponential random variables, whichis represented in the form of the generalized upperincomplete Fox’s H function. Subsequently, to derive theclosed-form OP expression of the near user, we analyzethe integral of the generalized upper incomplete Fox’sH function. The Mathematica programming code is alsoprovided for implementing the generalized upper incom-plete Fox’s H function. Our simulation results confirm

the accuracy of the OP expression derived.3) We extend the OP analysis to the case of an arbitrary

number of users in each subcarrier and derive the OPexpression of each user.

4) We compare the OP of each user in our MC-NOMAsystem to that of an OMA system, which confirmed thatour MC-NOMA indeed outperforms the correspondingOMA system.

C. Organisation

The remainder of this paper is organized as follows. SectionIII presents the system model. In Section IV, we study theOP for the minimum rate of two individual users. We extendthe OP for the general case of multiplexing an arbitrarynumber of users per subcarrier in Section V. Simulation resultsare presented in Section VI, followed by our conclusions inSection VII.

II. SYSTEM MODEL

Consider a downlink NOMA system supporting K single-antenna users by a single-antenna base station (BS). Thechannel’s power gain between the BS and user k at subcarriern is formulated by hk,n = χk,nd

−λk , where χk,n denotes the

squared magnitude of channel coefficient that is assumed toundergo small scale fading, dk is the distance between thekth user and the BS, k = 1, 2, . . . ,K, and λ is the pathloss exponent. We assume Rayleigh fading channels and χk,nare i.i.d. exponential random variables with a mean of one,for every user and subcarrier. Without loss of generality, thedistances are sorted as d1 > d2 > · · · > dk > · · · > dK , sothat user 1 is the farthest user from the BS.

Successive interference cancellation (SIC) is utilized at thereceiver side (users) to separate the signals. Explicitly, the SICreceiver first decodes the strongest signal by treating all othersignals as interference. After this stage, the strongest signal isremoved from the composite signal, and the second strongestsignal is decoded and canceled and so on, until the weakestsignal is decoded. Generally, the SIC detection order is basedon the ordered channel gain information. Relying on statisticalCSI knowledge, according to [37] and [38], the optimaldetection order is determined based on the average channelgains of the users. If various users have different outageprobability requirements, detection order should be based onthe weighted channel gains as described in [37]. In our work,we consider identical requirement for outage probability, henceSIC detection order can be based on statistical knowledge ofthe channels at the transmitter, which in our case is based onthe distances of users.

In this system, user 1 is the farther user from the BS, henceuser 1 attempts coherent detection of its own signal withoutperforming SIC, because the strongest signal has the lowestinterference. For user k, where k > 1, at the first step of theSIC process, the signal of user 1 which is the strongest signalis detected and removed. After this, the signal of user 2 isdetected and canceled. This iterative process is repeated untilthe signal of user k−1 is detected and canceled. At this point

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user k detects its own signal without the interference of users1, 2, . . . , k − 1.

Therefore, we can directly formulate the signal-to-interference-plus-noise ratio (SINR) for user k at subcarriern as

SINRk,n =

βk,nhk,n

hk,nK∑

j=k+1

βj,n+σ2

k ∈ 1, 2, . . .K

βK,nhK,nσ2 k = K.

(1)

where βk,n is the transmit power allocation of user k atsubcarrier n and σ2 is the additive white Gaussian noise(AWGN) power. In addition, we assume that the AWGN noisepower is the same for all users assigned to each subcarrier.

III. OUTAGE PROBABILITY ANALYSIS FOR TWO USERS

We derive tractable OP expressions of the multi-carrierNOMA downlink when only the statistical knowledge of theCSI is available at the transmitter side. We assume user 1 isthe far user while user 2 is closer to the BS.

Assuming perfect interference cancellation at the receiverside, based on (1), the data rate of each user in each subcarrieris expressed as

R1,n = log(1 +β1,nh1,n

σ2 + β2,nh1,n), R2,n = log(1 +

β2,nh2,n

σ2).

(2)

Therefore, the total data rate of each user over all the sharedsubcarriers is expressed as

R1 =

N∑n=1

log(1 +β1,nh1,n

σ2 + β2,nh1,n), (3)

R2 =

N∑n=1

log(1 +β2,nh2,n

σ2).

A. Outage Probability of User 1

Again we define the OP of user 1 as the probability that thetotal rate of this user is less than a threshold. Consequently,the OP of user 1 over N subcarriers is

Prout1,N = Pr(R1 ≤ r1), (4)

where r1 denotes the target data rate of user 1.Let us now reformulate the data rate of user 1 by introducing

signal power, Rayleigh fading, and path loss coefficients,yielding:

R1 =

N∑n=1

log(1 +β1,nd

−λ1 χ1,n

σ2 + β2,nd−λ1 χ1,n

) (5)

=

N∑n=1

log(σ2 + (β1,n + β2,n)d−λ1 χ1,n

σ2 + β2,nd−λ1 χ1,n

).

Upon defining An = (β1,n + β2,n)d−λ1 and Bn = β2,nd−λ1 ,

the data rate of user 1 is expressed as

R1 =

N∑n=1

log(σ2 +Anχ1,n

σ2 +Bnχ1,n), (6)

where χ1,n follows an exponential distribution with a meanof one. Again, we aim for calculating the OP of user 1.

Based on (4) and (6), we rewrite the OP of user 1 as

Prout1,N = Pr

(N∑n=1

log

(σ2 +Anχ1,n

σ2 +Bnχ1,n

)≤ r1

). (7)

Upon defining Zn =Anχ1,n+σ2

Bnχ1,n+σ2 , the OP can be expressed as

Prout1,N = Pr

(N∏n=1

Zn ≤ er1). (8)

To calculate (8), we have to find the distribution of∏Nn=1 Zn,

i.e. the product of multiple ratios between two shifted expo-nential random variables, which are dependent. To derive atractable expression for the PDF of this product, we proposea novel approximation method for accurately representing∏Nn=1 Zn. First, let us transform Zn as follows:

Zn = un −σ2mn

Bnχ1,n + σ2= mn

[unmn− σ2

Bnχ1,n + σ2

],

(9)

where un = AnBn

and mn = AnBn− 1. Hence,

∏Nn=1 Zn can be

written asN∏n=1

Zn =

N∏n=1

mn ×N∏n=1

[unmn− σ2

Bnχ1,n + σ2

]. (10)

In (10),∏Nn=1 Zn is represented as a product of binomials.

As N increases, the product will result in higher-order termsrelated to the product of the σ2

Bnχ1,n+σ2 terms. This willincrease the complexity of computations. Therefore, since

σ2

Bnχ1,n+σ2 is smaller than one, we approximate∏Nn=1 Zn by

removing the higher-order terms, yielding

N∏n=1

Zn ≈N∏n=1

un −N∑n=1

∏n′ 6=n

un′

mnσ2

Bnχ1,n + σ2. (11)

To confirm the accuracy of this approximation, we havecompared the PDF and CDF of the approximated repre-sentation of

∏Nn=1 Zn in (11) to the original one in (10)

under different settings2. Figure 1 represents the PDF andCDF results for the cases of both two subcarriers and foursubcarriers. It is apparent that the shape of histogram PDFand empirical CDF for the approximation coincides with thatof the original ones. Hence, the new approximation methodof the product of the ratios between two shifted exponentialrandom variables is considered to be satisfactory for furtheranalyses. Furthermore, we have investigated the sensitivity ofthe approximation to different parameters, such as t =

β1,n

β2,n,

ρ = d1d2

, SNR2,n =β2,nd

−λ2

σ2 for the case of four subcarriers.Figure 2 demonstrates the CDF results considering differentvalues of t, ρ, SNR2,n aiming to evaluate the accuracy of OPapproximation under different cases. It is apparent that theapproximation is accurate enough in all cases.

2Note by means of the properties of downlink NOMA system, the valueof Bn ≤ An −Bn, thus, 2Bn ≤ An and Bn ≤ An

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a. Two subcarriers b. Four subcarriers

Fig. 1. The PDF and CDF (ρ = 1.5, t = 2, SNR2,n = 18dB)

-40 -20 0 20 40 60 80

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(x

)

Empirical CDF

Original

Approximation

(a) ρ = 1.5, t = 2

-1 -0.5 0 0.5 1 1.5

x ×104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(x

)

Empirical CDF

Original

Approximation

(b) ρ = 1.5, t = 10

-2 -1.5 -1 -0.5 0 0.5 1 1.5

x ×104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(x

)

Empirical CDF

Original

Approximation

(c) ρ = 2, t = 10

Fig. 2. The Empirical CDF (SNR2,n = 18dB)

Having established accuracy of this approximation, weanalyze the CDF and PDF of the approximated parameters.Let us define Yn =

(∏n′ 6=n un′

)mnσ

2

Bnχ1,n+σ2 . Based on (8)and (11), the OP of user 1 becomes

Prout1,N = Pr

(N∏n=1

un −N∑n=1

Yn ≤ er1)

(12)

= Pr

(N∏n=1

un − er1 ≤N∑n=1

Yn

).

To calculate the OP based on (12), first we derive the proba-bility distribution of Yn. Considering that

∏Nn=1 un has a fixed

value and χ1,n obeys the exponential distribution, the CDF andPDF of Yn can be easily calculated. Introducing a new variablesn =

(∏n′ 6=n un′

)mn, we can write Yn = σ2sn

Bnχ1,n+σ2 .Therefore, the CDF of Yn may be expressed as

FYn(yn) =Pr(Yn ≤ yn) = Pr(

σ2snBnχ1,n + σ2

≤ yn)

(13)

= 1− Pr(χ1,n ≤

σ2 (sn − yn)

Bnyn

)

=

eσ2(yn−sn)Bnyn , 0 ≤ yn ≤ sn,

0, yn < 0,

1, yn > sn.

Consequently, we can calculate the PDF of Yn as

fYn(yn) =

{eσ2(yn−sn)Bnyn × σ2sn

Bny2n, 0 ≤ yn ≤ sn,

0, otherwise.(14)

Next, based on (13) and (14), we will discuss the OP ofdifferent number of subcarriers.

1) The Case of Two Subcarriers: According to (12), forN = 2, we have

Prout1,2 = 1− Pr(Y1 + Y2 ≤ C), (15)

where C = u1u2 − er1 . To calculate the OP of user 1 for thecase of N = 2 according to (15), we proceed by deriving theprobability of Pr(Y1 + Y2 ≤ C) (based on (13) and (14)) andproceed by defining:

Pr(Y2 ≤ C − y1) =

eσ2(C−y1−s2)

(C−y1)B2 , C − s2 ≤ y1 ≤ C,0, y1 > C,

1, y1 < C − s2.

(16)

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Subsequently, we can represent Pr(Y1 + Y2 ≤ C) based on(16) as

Pr(Y1 + Y2 ≤ C) =

∫ C−s2

−∞fY1

(y1)dy1 (17)

+

∫ C

C−s2eσ2(C−y1−s2)

(C−y1)B2 fY1(y1)dy1.

Thus, the probability of Pr(Y1 + Y2 ≤ C) consists of twointegrations. The upper limit of

∫ C−s2−∞ fY1(y1)dy1 is C − s2,

and∫ CC−s2 e

σ2(C−y1−s2)

(C−y1)B2 fY1(y1)dy1 is defined over a closedinterval [C − s2, C]. In the next step, we determine the prob-ability of Pr(Y1 + Y2 ≤ C) by discussing different ranges ofC compared to s1 and s2.(a) For C < 0, we have Pr(Y1 + Y2 ≤ C) = 0, because

the probability of a positive value being smaller than anegative value is equal to zero.

(b) For 0 ≤ C ≤ s2, we have Pr(Y1 + Y2 ≤ C) =∫ min(C,s1)

0eσ2(C−y1−s2)

(C−y1)B2 fY1(y1)dy1

(c) For s2 < C ≤ s1 + s2, we have Pr(Y1 + Y2 ≤ C) =∫ C−s20

fY1(y1)dy1 +∫ min(C,s1)

C−s2 eσ2(C−y1−s2)

(C−y1)B2 fY1(y1)dy1

(d) For C > s1 + s2, we have Pr(Y1 + Y2 ≤ C) = 1.Let us now summarize all the conditions as

Pr(Y1 + Y2 ≤ C) = (18)

0, C < 0,∫ min(C,s1)

0eσ2(C−y1−s2)

(C−y1)B2 fY1(y1)dy1, 0 ≤ C ≤ s2,∫ C−s2

0fY1

(y1)dy1+∫ min(C,s1)

C−s2 eσ2(C−y1−s2)

(C−y1)B2 fY1(y1)dy1, s2 < C ≤ s1 + s2,

1, C > s1 + s2.

(19)

According to (18), to derive an expression for Pr(Y1 +Y2 ≤ C) and hence for the OP, we have to calculate∫eσ2(C−y1−s2)

(C−y1)B2 fY1(y1)dy1. Hence, we discuss the derivationof this integral. First, substituting fY1

(y1) based on (14) into∫eσ2(C−y1−s2)

(C−y1)B2 fY1(y1)dy1 , we have∫

eσ2(C−y1−s2)

(C−y1)B2 fY1(y1)dy1 = (20)

eσ2

B1 eσ2

B2σ2s1

B1

∫1

y21

e−σ2s2B2

1C−y1 e

−σ2s1B1

1y1 dy1.

According to (20), we only have to calculate the integral of∫y−2

1 e−a2C−y1 e

−a1y1 dy1 where a1 = σ2s1

B1and a2 = σ2s2

B2. Since

the integral cannot be solved analytically, we evaluate it bynumerical integration in Matlab. Therefore, we can expressthe OP as

Prout1,2 = (21)

1, C < 0

1− eσ2

B1 eσ2

B2σ2s1B1

I (min(C, s1)) , 0 ≤ C ≤ s2

1− eσ2(C−s2−s1)

B1(C−s2) − eσ2

B1 eσ2

B2σ2s1B1

I (min(C, s1))

−eσ2

B1 eσ2

B2σ2s1B1

I (C − s2) , s2 ≤ C ≤ s1 + s2

0, C > s1 + s2,

where we have I(t) =∫ t

0y−2

1 e−a2C−y1 e

−a1y1 dy1.

2) General Case of N Subcarriers: Calculating the OP fora larger number of subcarriers using the same technique as fortwo subcarriers to derive Pr

(∏Nn=1 un − er1 ≤

∑Nn=1 Yn

)would be challenging, due to several numerical integrationsinvolved. Therefore, for the general case, by using mean-squared error curve fitting for different numbers of subcarriers,the Gamma distribution was found to have a good agreementwith our simulation results in the model

∑Nn=1 Yn.

Figure 3 confirms that although a distribution fitting testwould not provide a high confidence level,

∑Nn=1 Yn can

be reasonably approximated by a Gamma distributed randomvariable, Υn(k, θ), whose shape and scale parameters may becalculated based on the mean value and variance of

∑Nn=1 Yn

as follows:

θ =Var

[∑Nn=1 Yn

]E[∑N

n=1 Yn

] , κ =E[∑N

n=1 Yn

]θ

. (22)

In order to obtain the required statistics (i.e., κ and θ), wewill calculate the mean and variance of

∑Nn=1 Yn, respectively.

Specifically, the mean can be written as

E[ N∑n=1

Yn

]=

N∑n=1

E[Yn] =

N∑n=1

(∫ sn

0

ynfYn(yn)dyn

).

(23)

Subsequently, based on (14), we have

E[ N∑n=1

Yn

]=

N∑n=1

(σ2snBn

eσ2

Bn

∫ sn

0

1

yne−σ2snBnyn dyn

)(24)

=

N∑n=1

(σ2snBn

eσ2

BnE1

(σ2

Bn

)),

where E1 represents the exponential integral calculation. Sub-sequently, we can write the variance as

Var[ N∑n=1

Yn

]=

N∑n=1

Var[Yn] (25)

=

N∑n=1

(∫y2nf(yn)dyn − E[Yn]2

),

where we have E[Yn] = σ2snBn

eσ2

BnE1

(σ2

Bn

). Consequently,

based on (14), we have

Var[ N∑n=1

Yn

]= (26)

N∑n=1

((σ2snBn

)2(Bnσ− e

σ2

BnE1

(σ2

Bn

)− e

2σ2

Bn E21

(σ2

Bn

))).

By substituting (24) and (26) into (22), the required statisticsof the Gamma distributed random variable are obtained andthe OP can be written as

Prout1,N = Pr

(N∏n=1

un − er1 ≤ Υn(κ, θ)

)(27)

= 1− Pr (Υn(κ, θ) ≤ CN ) ,

7

0 100 200 300 400 500 600 700 800

0

200

400

600

800

1000

1200

1400

n=3

Sum of Yn

Gamma distribution

a. Three subcarriers

0 1 2 3 4 5 6 7 8

×104

0

100

200

300

400

500

600

700

800

900

1000

n=5

Sum of Yn

Gamma distribution

b. Five subcarriers

Fig. 3. The approximation for the sum of Yn

where we have CN =∏Nn=1 un − er1 . Finally, the OP is

calculated as

Prout1,N = 1− FΥn(CN ) (28)

= 1− 1

Γ(κ)γ

(κ,CNθ

),

where γ(κ, CNθ

)=∫ Cn

θ

0tκ−1e−tdt is the lower incomplete

Gamma function.

B. Outage Probability of User 2

In this subsection, we analyze the OP of user 2. We definethe outage expression of user 2 as

Prout2,N = 1− Prsuc

2,N × Prsuc1→2,N . (29)

where Prsuc2,N = Pr(R2 > r2) represents the successful

probability of user 2 conditioned on successful SIC of user1’s signal and Prsuc

1→2,N = Pr(R1→2,N > r1→2,N ) denotes theprobability of the success event of user 2 detecting the messageof user 1. Therefore, in order to calculate the outage expressionof user 2, we first need to derive Prsuc

2,N and Prsuc1→2,N .

First, to derive Prsuc2,N , we represent it as

Prsuc2,N = Pr

(N∑n=1

log(1 +β2,nh2,n

σ2) > r2

)(30)

= Pr

(N∏n=1

(1 +β2,nd

−λ2 χ2,n

σ2) > er2

).

where R2 =∑Nn=1 log(1 +

β2,nh2,n

σ2 ) is the achievable rate ofuser 2, conditioned on the successful SIC of user 1’s signal.Let us now define ln =

β2,nd−λ2

σ2 and Wn = 1 + lnχ2,n, thenwe have:

Prsuc2,N = Pr

(N∏n=1

Wn > er2

). (31)

In order to calculate Prsuc2,N , we first determine the probability

distribution of W =∏Nn=1Wn. Since χ2,n has an exponential

distribution and ln has a fixed value, Wn is a shifted exponen-tial random variable. According to [39], the PDF of a shiftedexponential random variables is defined as f(x) = γe−γ(x−α)

for α > 0, where γ is a rate parameter and α is a shiftparameter.

Since Wn is a shifted exponential random variable withγn = 1

lnand αn = 1, the PDF of Wn given by fWn

(wn) =1lne

1ln

(wn−1). The Mellin transform [40] of Wn is defined as

MWn{fWn

(wn)|s} =

∫ ∞0

ws−1n fWn

(wn)dxn (32)

= e1/ln ls−1n Γ (s, 1/ln) ,

where Γ (s, 1/ln) =∫∞

1/lnts−1e−tdt is the upper incomplete

Gamma function. Based on the inherent property of the Mellintransform, namely that the Mellin transform of the product ofindependent random variables can be expressed as the productof the individual Mellin transforms, we arrive at:

MW {fW (w)|s} =

N∏n=1

MWn{fWn

(wn)|s}. (33)

Consequently, based on (32), the Mellin transform of W =∏Nn=1Wn will be

MW {fW (w)|s} =

N∏n=1

(e1/ln ls−1

n Γ(s, 1/ln))

(34)

= KNCs−1N

N∏n=1

Γ(s, 1/ln),

where we have KN =∏Nn=1 e

1/ln and CN =∏Nn=1 ln.

Therefore, having established the Mellin transform of W ,the corresponding PDF of W is calculated by integratingMW {fW (w)|s} as

fW (w) =1

2πi

∫ c+i∞

c−i∞MW {fW (w)|s}w−sds (35)

=KN

CNHN,00,N

[w

CN

∣∣∣−−−−−(0,1,1/l1),...,(0,1,1/lN )

],

8

where∏Nn=1 αn < w < ∞, and Hm,n

p,q [r] represents thegeneralized upper incomplete Fox’s H function [41] . Thisfunction is defined as

Hm,np,q [r] , Hm,n

p,q

[r∣∣∣(ai,αi,Ai)1,p(bj ,βj ,Bj)1,q

](36)

, Hm,np,q

[r∣∣∣(a1,α1,A1),...,(ap,αp,Ap)

(b1β1,B1),...,(bqβq,Bq)

],

where m,n, p and q are integers such that 0 ≤ m ≤ q, 0 ≤n ≤ p, ai, bj ∈ C where C is the field of complex number, andαi, Ai, βj and Bj ∈ R+ (1 ≤ i ≤ p, 1 ≤ j ≤ q) where R+

is the set of positive real numbers. Fox defined the H-functionin his celebrated studies of symmetrical Fourier kernel as theMellin-Barnes type integral [42] as follows

Hm,np,q [r] =1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]r−sds, (37)

where we have:

Mm,np,q [s] = (38)∏mj=1 Γ(bj + βjs,Bj)

∏ni=1 Γ(1− ai − αis,Ai)∏p

n+1 Γ(ai + αis,Ai)∏qj=m+1 Γ(1− bj − βjs,Bj)

,

while C is a suitable contour of Barnes type [42] such thatthe poles of Γ(bj + βjs,Bj), j = 1, . . . ,m lie on the righthand side of the contour and those of Γ(1−ai−αis,Ai), i =1, . . . , n lie on the left hand side of the contour. Furthermore,the contour C runs from c− i∞ to c+ i∞, so that we have:(a) sb = (−bj − k)/βj (j = 1, . . . ,m; k = 0, 1, 2, . . . )(b) sa = (1− ai + k)/αi (i = 1, . . . , n; k = 0, 1, 2, . . . ).

Therefore, the value of C is chosen between sb and sa. Ourobjective is to find the probability of success event for user 2.According to (31), we have

Prsuc2,N = Pr (W > er2) =

∫ ∞er2

fW (w)dw. (39)

Subsequently, based on fW (w) in (35), we have

Prsuc2,N = 1− KN

CN

∫ er2

0

HN,00,N

[w

CN|−−−−−(0,1,1/l1),...,(0,1,1/lN )

]dw.

(40)

The difficulty in calculating this probability is in the inte-gration of Fox’s H function, which is relegated to AppendixA. Finally, based on (64), the corresponding probability ofsuccess for user 2 may be expressed as:

Prsuc2,N = (41)

1−KN

(H0,N+1N+1,1

[er2

CN

∣∣∣(1,1,0)(0,1,0),(1,1,1/l1),...,(1,1,1/lN )

]).

In addition, to derive the OP of user 2, it is necessaryto calculate the probability of the success event of user 2detecting the message of user 1, the corresponding probabilityis defined as Prsuc

1→2,N = Pr(R1→2 > r1→2), where R1→2 =∑Nn=1 log

(1 +

β1,nh2,n

σ2+β2,nh2,n

)represents the achievable rate of

detecting user 1’s message at user 2 and r1→2 is the targetrate. By applying the same analysis method for OP of user 1,the success probability for the case of two subcarriers canbe expressed as (42), where B1 = β2,nd

−λ2 and we have

I(t) =∫ t

0y−2

1 e−a2C−y1 e

−a1y1 dy1. For the case of N subcarriers,

this success probability will be Prsuc1→2,N = 1

Γ(k)γ(k, Cn

θ

).

IV. OUTAGE PROBABILITY ANALYSIS FOR AN ARBITRARYNUMBER OF USERS

In this section, we extend the OP analysis for the generalcase of multiplexing an arbitrary number of users per subcar-rier. Let us assume that we have K users in this system and thedetection order is based on the distance from the base stationto the users, d1 > d2 > · · · > dk > · · · > dK . Thus, the totaldata rate of each user over all the shared subcarriers can beexpressed as

Rk = (43)N∑n=1

log

(1 +

βk,nhk,nσ2+

∑Kj=k+1 βj,nhk,n

), k ∈ {1, 2, ..,K − 1}

N∑n=1

log(

1 +βK,nhK,n

σ2

), k = K.

Each subcarrier can be allocated to K users. When relying on afixed power allocation method, the transmit power allocationsof different users will follow β1,n > β2,n > · · · > βK,n.Hence, the outage probability of user 1 (the farthest user) overN subcarriers is defined as

Prout1,N = Pr(R1 ≤ r1) (44)

= Pr

(N∑n=1

log

(1 +

β1,nh1,n

σ2 +∑Kj=2 βj,nh1,n

)≤ r1

).

Let us now reformulate the data rate of user 1 by introducingsignal power, Rayleigh fading, and path loss coefficients,yielding:

R1 =

N∑n=1

log

(1 +

β1,nh1,n

σ2 +∑Kj=2 βj,nh1,n

)(45)

=

N∑n=1

log

σ2 +(β1,n +

∑Kj=2 βj,n

)d−λ1 χ1,n

σ2 +∑Kj=2 βj,nd

−λ1 χ1,n

.

Upon defining A1n = (β1,n +

∑Kj=2 βj,n)d−λ1 and B1

n =∑Kj=2 βj,nd

−λ1 , the data rate of user 1 is expressed as

R1 =

N∑n=1

log(σ2 +A1

nχ1,n

σ2 +B1nχ1,n

),

where χ1,n follows an exponential distribution with a mean ofone. Comparing the expression of the achievable data rate ofuser 1 in the case of K users to the case of two users (i.e., (6)),it is clear that it has a similar form, except that the parametersof A1

n and B1n hold different values. Therefore, the derivation

of the OP expressions of user 1 over two or N subcarriers forthe case of K users will follow the same steps (i.e., (7)-(20)& (21)-(27)) as that of the case of two users.

For user k, when 1 < k < K, to analyse the outageprobability, it is necessary to calculate the success probabilityof the event of user k detecting both its own message as wellas the message of users who are farther from user k. In otherwords, the corresponding outage probability of user k can beobtained as

Proutk,N = 1− Prsuc

k,N ×k−1∏j=1

Prsucj→k,N , (46)

9

Prsuc1→2,2 =

0, C < 0

eσ2

B1 eσ2

B2σ2s1B1

I(

min(C, s1)), 0 ≤ C ≤ s2

eσ2(C−s2−s1)

B1(C−s2) − eσ2

B1 eσ2

B2σ2s1B1

(I(

min(C, s1))− I

(C − s2

)), s2 ≤ C ≤ s1 + s2

1, C > s1 + s2,

(42)

where Prsuck,N is given as

Prsuck,N = Pr(Rk > rk).

The rate of user k can be written as Rk =N∑n=1

log(σ2+Aknχ1,n

σ2+Bknχ1,n)

where Akn =

(βk,n +

K∑j=k+1

βj,n

)d−λk and Bkn =

K∑j=k+1

βj,nd−λk . Moreover, Prsuc

j→k,N represents the success

probability of the event of user k detecting the message ofuser j < k who is farther away from user k as

Prsucj→k,N = Pr(Rj→k > rj→k),

where the achievable rate of user k detecting the message ofuser j is

Rj→k =

N∑n=1

log

(1 +

βj,nhk,n

σ2 +∑Ki=j+1 βi,nhk,n

).

This rate can be rewritten as Rj→k =N∑n=1

log(σ2+Aj→kn χ1,n

σ2+Bj→kn χ1,n)

where Aj→kn =

(βj,n +

K∑i=j+1

βi,n

)d−λk and Bj→kn =

K∑i=j+1

βi,nd−λk . Therefore, the same approach as in (7)-(20)

& (21)-(27) can be applied for deriving the closed-formexpressions of Prsuc

j→k,N since it follows the same form as in(6).

The outage probability of user K is obtained as

ProutK,N = 1− Prsuc

K,N ×K−1∏j=1

Prsucj→K,N , (47)

The success probability of user K detecting the messages ofother users can be derived by adopting the same method asfor the case of user k when 1 < k < K. What is differenthowever, is the analysis of Prsuc

K,N , which is expressed as

PrsucK,N = Pr(RK > rK) (48)

= Pr

(N∑n=1

log

(1 +

βK,nd−λK χK,nσ2

)> rK

)

= Pr

(N∏n=1

(1 +

βK,nd−λk χK,nσ2

)> erK

),

The achievable rate of user K is given by the sum of thelogarithm of shifted exponential random variables. Hence, wecan apply the generalized upper incomplete Fox’s H function

to obtain the success event of user K, as seen in steps (29)-(40). Here, we have ln =

βK,nd−λK

σ2 , KN =∏Nn=1 e

1/ln andCN =

∏Nn=1 ln. Therefore,

PrsucK,N = Pr(RK > rK) (49)

= 1− KN

(H0,N+1N+1,1

[erk

CN

∣∣∣(1,1,0)

(0,1,0),(1,1,1/l1),...,(1,1,1/lN )

]).

V. SIMULATION RESULTS

We present several simulation results for the OP for thecase of two users and three users. In the simulation results,generally, the SNR of User K (the user closest to the BS)is treated as a unified benchmark SNRK,n =

βK,nd−λK

σ2 . Weassume that both the distance and power changes follow thesame scale, i.e., the distance ratio variable is defined as ρ =d1d2

= · · · = dK−1

dK, and the power ratio coefficient of t =

β1,n

β2,n= · · · = βK−1,n

βK,n. Therefore, the average received SNR of

user k can be formulated as

SNRk,n =βk,nd

−λk

σ2(50)

= tK−k ×(ρK−k

)−λ × SNRK,n, k = 1, . . .K − 1

For the case of two users, we have provided numerical re-sults for both users to validate the accuracy of the expressionsderived. The corresponding data rate of user 2 can be writtenas R2 =

∑Nn=1 log(1+χ2,nSNR2,n). In order to guarantee that

SIC is successful, more transmission power is needed to thefar user that experiences the worst-case channel condition [43].Hence, we only explore the power ratio coefficients which arelarger than one, i.e., t =

β1,n

β2,n> 1.

Hence, the data rate for user 1 can be written as

R1 =

N∑n=1

log(σ2 + (β1,n + β2,n)d−λ1 χ1,n

σ2 + β2,nd−λ1 χ1,n

) (51)

=

N∑n=1

log(σ2 + (t+ 1)SNR2,nσ

2ρ−λχ1,n

σ2 + SNR2,nσ2ρ−λχ1,n),

where the corresponding average received SNR of user 1 atsubcarrier n is defined as SNR1,n =

β1,nd−λ1

σ2 = t × ρ−λ ×SNR2,n. As expected, when ρ increases, the data rate of user1 decreases and the corresponding OP increases. Without lossof generality, in our simulations, we assume that each userhas the same power level over the different subcarriers. In thefollowing, we analyze the OP of user 1 and user 2, respectively.

10

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Target data rate (nats/s/Hz)

10-3

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, t=1.2

Approximation, t=1.2

Original, t=1.4


Original, t=1.6


(a) Two subcarriers

1.8 2 2.2 2.4 2.6 2.8 3

Target rate (nats/s/Hz)

10-3

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, t=1.2


Original, t=1.4


Original, t=1.6


(b) Three subcarriers

3.5 4 4.5 5


10-3

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, t=1.2


Original, t=1.4


Original, t=1.6


(c) Five subcarriers

Fig. 4. The outage probability of user 1 vs target rate, with different power ratio coefficients t (SNR2,n = 18dB , ρ = 1.5)

0.6 0.8 1 1.2 1.4 1.6


10-3

10-2

10-1

100

Outa

ge p

robabili

ty

Original, ρ=2

Approximation, ρ=2

Original, ρ=3

Approximation, ρ=3

Original, ρ=4

Approximation, ρ=4

(a) SNR2,n = 18dB

0.6 0.8 1 1.2 1.4 1.6


10-5

10-4

10-3

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, ρ=2

Approximation, ρ=2

Original, ρ=3

Approximation, ρ=3

Original, ρ=4

Approximation, ρ=4

(b) SNR2,n = 25dB

Fig. 5. The outage probability of user 1 vs target rate, with different distance ratios ρ (t = 1.2)

A. User 1

Figure 4 depicts the relationship between the target datarate and the OP for three cases having different numbersof subcarriers. The value of SNR2,n is fixed to 18dB. Itis observed that the OP of user 1 increases with the targetrate and decreases with the power ratio coefficient t. Thisobservation implies that NOMA systems achieve a goodperformance when the power levels of the two users are ratherdifferent. This is helpful, when determining the optimal powerallocation policy considering a specific user distribution. Theapproximation matches closely to the true ones, however, thegap between the approximated and original OP increases witht for all three cases, particularly in the lower target rate region.Likewise, as the number of subcarriers increases, the accuracyof the approximation decreases. This confirms accuracy of theOP analysis for the case of two subcarriers. In Figure 5, wefocus our attention on the effects of the distance ratio ρ onthe OP of two subcarriers. As the target rate and ρ increase,the OP of user 1 increases. Obviously, the accuracy of theapproximated OPs is similar for different values of ρ. As thevalue of SNR2,n increases, the gap between the approximatedand original OPs shrinks. The OP approximation error versus

SNR2,n is demonstrated in Figure 6. The OP approximationerror is defined as the difference between the original OP basedon numerical results and the approximated OP based on theanalytical results.

12 14 16 18 20 22 24 26 28 30

SNR2,n

(dB)

10-5

10-4

10-3

10-2

10-1

100

OP

Ap

pro

xim

atio

n E

rro

r

ρ=2. t=1.2

ρ=3, t=1.2

ρ=4, t=1.2

Fig. 6. OP approximation error versus SNR2,n (r = 0.8)

11

5 10 15 20

SNR2,n

(dB)

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, t=1.2


Original, t=1.4

Apptoximation, t=1.4

Original, t=1.6


(a) Two subcarriers (r = 1.5)

5 10 15 20

SNR2,n

(dB)

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, t=1.2


Original, t=1.4


Original, t=1.6


(b) Three subcarriers (r = 2.25)

5 10 15 20

SNR2,n

(dB)

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, t=1.2


Original, t=1.4


Original, t=1.6


(c) Five subcarriers(r = 3.75)

Fig. 7. The outage probability of user 1 vs SNR2,n, with different power ratio coefficients t (ρ = 1.5)

Figure 7 and Figure 8 illustrate the effects of SNR2,n on theOP of user 1 for different number of subcarriers. As the valueof SNR2,n increases, the OP decreases. These two figuresanalyze the impact of various power ratios t and distance ratiosρ on the OP considering different numbers of subcarriers.The OP decreases with the power ratio and increases withthe distance ratio. These figures highlight the necessity ofcarefully choosing t taking into account the SNR (implyingthe total available power budget) and ρ (implying the userdistributions). For instance, at SNR2,n = 20dB and ρ = 1.5,opting for t = 1.6 reduces the OP by a factor of 25 over thatof t = 1.2. Moreover, the difference between the original andthe approximated OP becomes larger for high values of powerratio and distance ratio. The accuracy of the approximation isreduced upon increasing the number of subcarriers.

B. User 2

In this subsection, we verify the accuracy of the probabilityof success expression derived in (41) for the second user. Thisis defined as Prsuc

2 = Pr(∑N

n=1 log(1 +β2,nh2,n

σ2 ) ≥ r2

), i.e.,

the probability of satisfying the total rate of an individualuser over all subcarriers, which is calculated by using thegeneralized upper incomplete Fox’s H function. It is shownthat the simulation results closely match the analytical resultsfor all three cases.

Figure 9 shows the relationship between the probabilityof success event and the target rate. The value of averagereceived SNR for user 2 on subcarrier n is set to 18 dB. Asthe target data rate increases, the probability of success eventdecreases. By contrast, the higher the number of subcarriers,the higher the success rate is. In Figure 10, we observe thatas SNR2,n increases, the probability of success also increases.Additionally, the success rate of a larger number of subcarriersis higher than that of a lower number of subcarriers N . Thisfigure demonstrates that the derived probability of successfor user 2 in terms of detecting its own message (i.e. Prsuc

2,N )matches perfectively to the simulation results.

Let us now analyze the OP of user 2 and compare this to thatof user 1. For the sake of simplicity, we assume an identicalnumber of subcarriers for both users. Observe from Figure

11, that as expected, the OP of both users increases with thetarget rate. The OP of user 1 is always higher than that ofuser 2 since user 1 suffers from more interference than user2. The diversity gain of having higher degrees of freedom inthe case of five subcarriers is clear from the lower OP that isachievable for a fixed target rate. The diversity order will befurther discussed in Figure 13.

C. Comparison with OMA

For an OMA system, each subcarrier is used exclusivelyby a single user only. If Nk denotes the set of subcarriersallocated to user k, its rate is given by

ROMAk =

∑n∈Nk

log

(1 +

βk,nhk,nσ2

), (52)

Hence, the corresponding outage probability is

Pr(ROMAk ≤ rk) (53)

=KNkCNk

∫ erk

0

HNk,00,Nk

[w

CNk|−−−−−(0,1,1/l1),...,(0,1,1/lNk )

]dw,

In the following simulations, we compare the OMA andNOMA systems for the case of two users and two subcarriersas well as for the case of three users and three subcarriers.It is observed that the outage performance for each user ofNOMA system is superior to that of the OMA system for bothcases. This is because the diversity order of the NOMA systemis higher than that of the OMA system. It should be notedhowever that the outage performance of NOMA is sensitiveto the values of power ratio t and distance ratio ρ.

Moreover, we investigate the diversity gain (which is definedas limSNR→∞

log(Prout)log(SNR) , where SNR is the ratio of the total

transmit power and the receiver noise power) of MC-NOMAfor various cases and compare it with that of the OMA scheme.Fig 13 shows that MC-NOMA having two subcarriers achievesa diversity order of approximately two while the diversityorder of OMA is one. Furthermore, it can be observed that thediversity order of MC-NOMA associated with N = 3 is closeto three, even though two users share the three subcarriers.

12

5 10 15 20 25 30

SNR2,n

(dB)

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, ρ=2

Approximation, ρ=2

Original, ρ=3

Approximation, ρ=3

Original, ρ=4

Approximation, ρ=4

(a) Two subcarriers (r = 1.5)

10 12 14 16 18 20 22 24 26 28 30

SNR2,n

(dB)

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, ρ=2

Approximation, ρ=2

Original, ρ=3

Approximation, ρ=3

Original, ρ=4

Approximation, ρ=4

(b) Three subcarriers (r = 2.25)

10 12 14 16 18 20 22 24 26 28 30

SNR2,n

(dB)

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original, ρ=2

Approximation, ρ=2

Original, ρ=3

Approximation, ρ=3

Original, ρ=4

Approximation, ρ=4

(c) Five subcarriers (r = 3.75)

Fig. 8. The outage probability of user 1 vs SNR2,n, with different distance ratios ρ (t = 1.2)

0 2 4 6 8 10 12 14 16 18 20

Target rate(nats/s/Hz)

10-4

10-3

10-2

10-1

100

The p

robabili

ty o

f success e

vent

Original, N=2

Approximation, N=2

Original, N=3

Approximation, N=3

Original, N=5

Approximation, N=5

Fig. 9. Success probability of user 2 vs target rate

2 4 6 8 10 12 14 16 18 20

SNR2,n

(dB)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

The p

robabili

ty o

f success e

vent

Original, N=2

Approximation, N=2

Original, N=3

Approximation, N=3

Original, N=5

Approximation, N=5

Fig. 10. Success probability of user 2 vs SNR2,n

0.6 0.8 1 1.2 1.4 1.6


10-4

10-3

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original outage for user 1

Approximated outage for user 1



(a) Two subcarriers

1.2 1.4 1.6 1.8 2 2.2 2.4


10-5

10-4

10-3

10-2

10-1

100

Ou

tag

e p

rob

ab

ility





(b) Three subcarriers

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4


10-3

10-2

10-1

100

Ou

tag

e p

rob

ab

ility





(c) Five subcarriers

Fig. 11. Outage probability of both users vs target rate (SNR2,n = 18dB, t = 1.2, ρ = 1.5)

D. OP analysis for K = 3

Fig. 14 compares the derived OP against data rate withsimulation results for the case of K = 3 users in eachsubcarrier. Clearly, the analytical results closely match thesimulation results. Compared to Fig 11 (a), this figure showsthat the OP for the case of three users sharing each subcarrieris slightly higher than that of two users allocated to each

subcarrier.

E. Joint Outage Probability

Based on the separate outage probabilities, the joint outageprobability can be calculated as

Prjointout = Pr (R1 +R2 ≤ r) = Pr (R1 ≤ r −R2) (54)

13

5 10 15 20 25

SNR2,n

(dB)

10-3

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original OMA user 1

Approximated OMA user 1

Original NOMA user 1

Approximated NOMA user 1

Original OMA user 2

Approximated OMA user 2

Original NOMA user 2

Approximated NOMA user 2

(a) Two users and two subcarriers

5 10 15 20 25

SNR3,n

(dB)

10-6

10-5

10-4

10-3

10-2

10-1

100

Ou

tag

e P

rob

ab

ility

Original OMA user 1Approximated OMA user 1Original NOMA user 1Approximated NOMA user 1Original OMA user 2Approximated OMA user 2Original NOMA user 2Approximated NOMA user 2Original OMA user 3Approximated OMA user 3Original NOMA user 3Approximated NOMA user 3

(b) Three users and three subcarriers

Fig. 12. Outage probability comparison between OMA and NOMA systems (r = 1, t = 1.2, ρ = 1.5)

-1 0 1 2 3 4 5 6 7

log(SNR)

-12

-10

-8

-6

-4

-2

0

log

(PO

1)

OMA (N=1)

NOMA (N=2)

NOMA (N=3)

X: 5.413

Y: -9.991

X: 6.272

Y: -6.041

X: 3.512

Y: -9.927

Fig. 13. log(Prout) versus log(SNR) (r = 1, t = 1.2, ρ = 1.5)

0.4 0.5 0.6 0.7 0.8 0.9 1


10-4

10-3

10-2

10-1

100

Ou

tag

e p

rob

ab

ility

Original outage for user 1Approximated outage for user 1Original outage for user 2Approximated outage for user 2Original outage for user 3Approximated outage for user 3

Fig. 14. Outage probability of each user versus target rate (SNR2,n =18dB, t = 1.2, ρ = 1.5)

=

∫ r

0

Prout1 (r −R2)fR2(R2)dR2

where fR2(R2) is the corresponding PDF of R2 =∑Nn=1 log

(1 +

β2,nd−λ2 χ2,n

σ2

). There is no simple or direct way

of deriving the PDF of the sum of the logarithm of shiftedexponential random variables, although the pdf of the productof shifted exponential random variables is known. Thus, wepresent R2 as R2 = log

[∏Nn=1

(1 +

β2,nd−λ2 χ2,n

σ2

)]. Upon

assuming W =∏Nn=1Wn and Wn = 1 +

β2,nd−λ2 χ2,n

σ2 , wehave R2 = logW . By changing the variables, we transformthe integral (55) as

Prjointout =

∫ er

−∞

1

WProut

1 (r − logW )fW (W )dW, (55)

where fW (W ) represents the PDF of W .

The outage probability of user 1 is obtained as Prout1,N (r −

logW ) = 1 − 1Γ(κ)γ

(κ, CNθ

)for the case of N subcarriers

according to the steps ranging from (4) to (27), assum-ing CN =

∏Nn=1 un − er−logW . Moreover, the PDF of

W is obtained following the steps (29)-(34) as fW (W ) =KNCNHN,00,N

[wCN

∣∣∣−−−−−(0,1,1/l1),...,(0,1,1/lN )

].

Therefore, the joint outage probability will be

Prjointout =

∫ er

−∞

1

W

(1− 1

Γ(κ)γ

(κ,CNθ

))× (56)

KN

CNHN,00,N

[w

CN

∣∣∣−−−−−(0,1,1/l1),...,(0,1,1/lN )

]dW

where we have KN =∏Nn=1 e

1/ln , CN =∏Nn=1 ln and

ln =β2,nd

−λ2

σ2 . This integral can be numerically solved. Here,

14

we provide a set of simulation results to examine the joint OPperformance of MC-NOMA system. Fig 15 depicts the jointOP for different numbers of users and subcarriers. Apparently,increasing the number of subcarriers significantly decreasesthe outage probability as a benefit of increasing the diversitygain achieved by a higher number of subcarriers. The diversityorder has already been discussed in Figure 13.

8 10 12 14 16 18 20


10-3

10-2

10-1

100

Join

t outa

ge p

robabili

ty

Two users (N=2)

Three users (N=2)

Five users (N=2)

Two users (N=3)

Three users (N=3)

Five users (N=3)

Fig. 15. Joint outage probability versus target rate (SNR2,n = 18dB, t = 1.2,ρ = 1.5)

VI. CONCLUSIONS

In this paper, tractable OP expressions have been derivedfor the single cell multi-carrier NOMA downlink only relyingon the statistical CSI based on approximation techniques.The PDF and CDF of the inverse shifted exponential randomvariables were formulated for the far user and the OP wasderived for the case of two subcarriers. To obtain the OPof far user for the general case of an arbitrary number ofsubcarriers, the sum of the inverse shifted exponential randomvariables was approximated by a Gamma distribution. On theother hand, the PDF of the product of shifted exponentialrandom variables was derived for the near user using theMellin transform and the generalized upper incomplete Fox’sH function. Our analysis and simulation results demonstrate agood accuracy for the approximations.

APPENDIX ATHE INTEGRAL OF FOX’S H FUNCTION

Here, a solution for the integral of Fox’s H function in (40)is formulated as:∫ r

0

Hm,np,q

[u∣∣∣(ai,αi,Ai)1,p(bj ,βj ,Bi)1,q

]du (57)

=

∫ r

0

1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]u−sdsdu

=1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]

∫ r

0

u−sdu ds

=1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]u1−s

1− s

∣∣∣∣r0

ds

=1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]r1−s

1− sds− 1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]01−s

1− sds;

The integral of the H function depends on the value of s.Hence we analyze this for three different cases of s.

(a) s > 1;∫ r

0

Hm,np,q


]du (58)

=1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]r1−s

1− sds+∞;

(b) s = 1; ∫ r

0

Hm,np,q


]du = 0; (59)

(c) s < 1; ∫ r

0

Hm,np,q


]du (60)

=1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]r1−s

1− sds;

According to the definition and properties of the incompleteGamma function, we obtain the recurrence relations throughintegration by parts

Γ(2− s, 0) =

∫ ∞0

t1−se−tdt (61)

= 0 +

∫ ∞0

(1− s)t−se−tdt = (1− s)Γ(1− s, 0);

Hence, we have

1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]r1−s

1− sds (62)

= r1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]Γ(1− s, 0)

Γ(2− s, 0)r−sds

= r1

2πi

∫ c+i∞

c−i∞Mm,n+1

p+1,q+1[s]r−sds

= rHm,n+1p+1,q+1

[r∣∣∣(ai,αi,Ai)1,p,(0,1,0)

(bj ,βj ,Bi)1,q,(−1,1,0)

];

By exploiting basic properties of H functions asrkHm,np,q

[r∣∣∣(ai,αi,Ai)1,p(bj ,βj ,Bi)1,q

]= Hm,np,q

[r∣∣∣(ai+kαi,αi,Ai)1,p(bj+kλj ,λj ,Bi)1,q

],

the last integral can be written as

1

2πi

∫ c+i∞

c−i∞Mm,n

p,q [s]r1−s

1− sds (63)

= Hm,n+1p+1,q+1

[r∣∣∣(ai+αi,αi,Ai)1,p,(1,1,0)

(bj+βj ,βj ,Bi)1,q,(0,1,0)

];

In our case, the value of s is always less than the one basedon the value of bj = 0 and λj = 1. Hence, the integral of thegeneralized upper incomplete Fox’s H function becomes:∫ er2

0

HN,00,N

[w

CN|−−−−−(0,1,1/l1),...,(0,1,1/lN )

]dw (64)

= er2H0,N+1N+1,1

[er2

CN

∣∣∣(0,1,0)(−1,1,0),(0,1,1/l1),...,(0,1,1/lN )

]= CNH0,N+1

N+1,1

[er2

CN

∣∣∣(1,1,0)(0,1,0),(1,1,1/l1),...,(1,1,1/lN )

];

15

APPENDIX BMATHEMATICA CODE FOR GENERALIZED UPPER

INCOMPLETE FOX’S H FUNCTION

Algorithm 1 Pseudo code of generalized upper incompleteFox’s H (GUIFoxH)

Input A,B, z,R,W. z = er2, and R,W are determined according to the con-vergence region, so that the contour integration path in thecomplex s-plane running from R − IW to R + IW and Wtends infinity.

Output V = GUIFoxH(A,B, z,R,W )

Function GUIFoxH (A,B, z,R,W )⇒ Auxiliary function

a. Gamma functionG = Γ(s, α) =

∫∞αts−1e−tdt (The upper incomplete

Gamma function)b. Mellin moment generating function

M =Mm,np,q [s] (65)

=

∏mj=1 Γ(bj + βjs,Bj)

∏ni=1 Γ(1− ai − αis,Ai)∏p

n+1 Γ(ai + αis,Ai)∏qj=m+1 Γ(1− bj − βjs,Bj)

return M

V = H0,N+1N+1,1

[er2

CN

∣∣∣(1,1,0)(0,1,0),(1,1,1/l1),...,(1,1,1/lN )

](66)

=1

2πi

∫ R+Wi

R−Wi

M0,N+1N+1,1[s](

er2

CN)−sds

return V

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Shenhong Li (S’18) received M.Sc. degree in mo-bile communications from the Loughborough Uni-versity, UK in 2014. She was an Intern at Nokia BellLabs, France. She is currently working toward thePh.D degree with the Wolfson School of Mechanical,Electrical and Manufacturing Engineering, Lough-borough University, UK. Her research interests in-clude non-orthogonal multiple access, resource allo-cation in wireless networks and optimization prob-lems.

Mahsa Derakhshani (S’10, M’13) is a Lecturer(Assistant Professor) in Digital Communicationswith the Wolfson School of Mechanical, Electri-cal and Manufacturing Engineering, LoughboroughUniversity, UK. Prior to joining Lboro, she was anHonorary NSERC Postdoctoral Fellow with Depart-ment of Electrical and Electronic Engineering, Impe-rial College London (2015 to 2016), a PostdoctoralResearch Fellow with the Department of Electricaland Computer Engineering, University of Toronto,Toronto, Canada and a Research Assistant with

the Department of Electrical and Computer Engineering, McGill University(2013 to 2015). She received her Ph.D. in electrical engineering degreefrom McGill University, Montral, Canada, in 2013. Her research interestsinclude radio resource management for wireless networks, software-definedwireless networking, applications of machine learning and optimization forcommunication systems. Dr. Derakhshani has received several awards andfellowships including the John Bonsall Porter Prize, the McGill EngineeringDoctoral Award, the Fonds de Recherche du QuebecNature et Technologies(FRQNT) and Natural Sciences and Engineering Research Council of Canada(NSERC) Postdoctoral Fellowships. Currently, she serves as an Editor of IEEEWireless Communications Letters and IET Signal Processing Journal.

Sangarapillai Lambotharan (SM’06) received thePh.D. degree in signal processing from ImperialCollege London, U.K., in 1997. He was a VisitingScientist with the Engineering and Theory Centre,Cornell University, USA, in 1996. Until 1999, hewas a Post-Doctoral Research Associate with Impe-rial College London. From 1999 to 2002, he waswith the Motorola Applied Research Group, U.K.,where he investigated various projects, includingphysical link layer modeling and performance char-acterization of GPRS, EGPRS, and UTRAN. He was

with Kings College London and Cardiff University as a Lecturer and a SeniorLecturer, respectively, from 2002 to 2007. He is currently a Professor ofdigital communications and the Head of the Signal Processing and NetworksResearch Group, Wolfson School of Mechanical, Electrical and ManufacturingEngineering, Loughborough University, U.K. His current research interestsinclude 5G networks, MIMO, blockchain, machine learning, and networksecurity. He has authored more than 200 journal articles and conference papersin these areas. He serves as an Associate Editor for IEEE TRANSACTIONSON SIGNAL PROCESSING.

Lajos Hanzo (http://www-mobile.ecs.soton.ac.uk,https://en.wikipedia.org/wiki/Lajos Hanzo) FREng,FIEEE, FIET, Fellow of EURASIP, DSc holds anhonorary doctorate by the Technical University ofBudapest (2009) and by the University of Edinburgh(2015). He is a Foreign Member of the HungarianAcademy of Sciences and a former Editor-in-Chiefof the IEEE Press. He has served as Governor ofboth IEEE ComSoc and of VTS. He has published1900+ contributions at IEEE Xplore, 18 Wiley-IEEEPress books and has helped the fast-track career of

119 PhD students. Over 40 of them are Professors at various stages of theircareers in academia and many of them are leading scientists in the wirelessindustry.

outage probability analysis for the multi-carrier noma downlink … · 2020. 4. 23. · carrier...

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