1 Impact of User Pairing on 5G Non-Orthogonal Multiple

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI10.1109/TVT.2015.2480766, IEEE Transactions on Vehicular Technology

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Impact of User Pairing on 5G Non-Orthogonal Multiple AccessDownlink Transmissions

Zhiguo Ding, Member, IEEE, Pingzhi Fan, Fellow, IEEE, and H. Vincent Poor, Fellow, IEEE

Abstract—Non-orthogonal multiple access (NOMA) representsa paradigm shift from conventional orthogonal multiple access(MA) concepts, and has been recognized as one of the keyenabling technologies for 5G systems. In this paper, the impactof user pairing on the performance of two NOMA systems,NOMA with fixed power allocation (F-NOMA) and cognitiveradio inspired NOMA (CR-NOMA), is characterized. For F-NOMA, both analytical and numerical results are provided todemonstrate that F-NOMA can offer a larger sum rate thanorthogonal MA, and the performance gain of F-NOMA overconventional MA can be further enlarged by selecting userswhose channel conditions are more distinctive. For CR-NOMA,the quality of service (QoS) for users with the poorer channelcondition can be guaranteed since the transmit power allocatedto other users is constrained following the concept of cognitiveradio networks. Because of this constraint, CR-NOMA exhibitsdifferent behavior compared to F-NOMA. For example, for theuser with the best channel condition, CR-NOMA prefers to pair itwith the user with the second best channel condition, whereas theuser with the worst channel condition is preferred by F-NOMA.

Keywords—Non-orthogonal multiple access (NOMA), cognitiveradio, power allocation, user pairing and outage probability.

I. INTRODUCTION

Multiple access (MA) in 5G mobile networks is an emergingresearch topic, since it is key for the next generation networkto keep pace with the explosive growth of mobile data andmultimedia traffic [1] and [2]. Non-orthogonal multiple access(NOMA) has recently received considerable attention as apromising candidate for 5G multiple access [3]–[6]. Partic-ularly, NOMA uses the power domain for multiple access,where different users are served at different power levels. InNOMA, the users with better channel conditions employ suc-cessive interference cancellation (SIC) to remove the messagesintended for other users before decoding their own [7]. Thebenefit of using NOMA can be illustrated by the followingexample. Suppose there is a user close to the edge of itscell, denoted by A, whose channel condition is very poor.For conventional MA, an orthogonal bandwidth channel, e.g.,a time slot, will be allocated to this user, and the other userscannot use this time slot. The key idea of NOMA is to squeeze

Z. Ding and H. V. Poor are with the Department of Electrical Engineering,Princeton University, Princeton, NJ 08544, USA. Z. Ding is also with theSchool of Computing and Communications, Lancaster University, LA1 4WA,UK. Pingzhi Fan is with the Institute of Mobile Communications, SouthwestJiaotong University, Chengdu, China.

The work of Z. Ding was supported by the UK EPSRC under grant numberEP/L025272/1. The work of P. Fan was supported by the National ScienceFoundation of China (NSFC, No. 61471302) and the 111 Project (No. 111-2-14). The work of H. V. Poor was supported by the U. S. National ScienceFoundation under Grants CNS-1456793 and ECCS-1343210.

another user with a better channel condition, denoted by B,into this time slot. Since A’s channel condition is very poor,the interference from B will not cause much performancedegradation to A, but the overall system throughput can besignificantly improved since additional information can bedelivered between the base station (BS) and B. The designof NOMA for uplink transmissions has been proposed in[4], and the performance of NOMA with randomly deployedmobile stations has been characterized in [5]. The user fair-ness in NOMA systems has been considered in [8], and thecombination of cooperative diversity with NOMA has beenconsidered in [9]. The application of multiple-input multiple-output (MIMO) technologies to NOMA has been proposed in[10] and [11].

Since multiple users are admitted at the same time, fre-quency and spreading code, co-channel interference will bestrong in NOMA systems, i.e., a NOMA system is interferencelimited. As a result, it may not be realistic to ask all the users inthe system to perform NOMA jointly. A promising alternativeis to construct a hybrid MA system, in which NOMA iscombined with conventional MA. In particular, the users inthe system can be divided into multiple groups, where NOMAis implemented within each group and different groups areallocated with orthogonal bandwidth resources. Obviously theperformance of this hybrid MA scheme is very dependent onwhich users are grouped together, and the aim of this paper isto investigate the effect of user pairing/grouping. Particularly,in this paper, we focus on a downlink communication scenariowith one BS and multiple users, where the users are orderedaccording to their connections to the BS, i.e., the m-th user hasthe m-th worst connection to the BS. We specifically considerthe situation in which two users, the m-th user and the n-thuser, are selected for performing NOMA jointly, where m < n.The impact of user pairing on the performance of NOMA willbe characterized in this paper, where two types of NOMA willbe considered. One is based on fixed power allocation, termedF-NOMA, and the other is cognitive radio inspired NOMA,termed CR-NOMA.

For the F-NOMA scheme, the probability that F-NOMAcan achieve a larger sum rate than conventional MA is firststudied, where an exact expression for this probability aswell as its high signal-to-noise ratio (SNR) approximation areobtained. These analytical results demonstrate that it is almostcertain for F-NOMA to outperform conventional MA, and thechannel quality of the n-th user is critical to this probability. Inaddition, the gap between the sum rates achieved by F-NOMAand conventional MA is also studied, and it is shown that thisgap is determined by how different the two users’ channelconditions are, as initially reported in [9]. For example, if



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n = M , it is preferable to choose m = 1, i.e., pairing theuser with the best channel condition with the user with theworst channel condition. This phenomenon can be explainedas follows. When m is small, the m-th user’s channel conditionis poor, and the data rate supported by this user’s channel isalso small. Therefore the spectral efficiency of conventionalMA is low, since the bandwidth allocated to this user cannotbe accessed by other users. The use of F-NOMA ensures thatthe n-th user will have access to the resource allocated tothe m-th user. If (n − m) is small, the n-th user’s channelquality is similar to the m-th user’s, and the benefit of usingNOMA is limited. But if n >> m, the n-th user can use thebandwidth resource much more efficiently than the m-th user,i.e., a larger (n −m) will result in a larger performance gapbetween F-NOMA and conventional MA.

The key idea of CR-NOMA is to opportunistically servethe n-th user on the condition that the m-th user’s qual-ity of service (QoS) is guaranteed. Particularly the transmitpower allocated to the n-th user is constrained by the m-thuser’s signal-to-interference-plus-noise ratio (SINR), whereasF-NOMA uses a fixed set of power allocation coefficients.Since the m-th user’s QoS can be guaranteed, we mainly focuson the performance of the n-th user offered by CR-NOMA.An exact expression for the outage probability achieved byCR-NOMA is obtained first, and then used for the study ofthe diversity order. In particular, we show that the diversityorder experienced by the n-th user is m, which means thatthe m-th user’s channel quality is critical to the performanceof CR-NOMA. This is mainly because of the imposed SINRconstraint, where the n-th user can be admitted into thebandwidth channel occupied by the m-th user, only if the m-th user’s SINR is guaranteed. As a result, with a fixed m,increasing n does not bring much improvement to the n-thuser’s outage probability, which is different from F-NOMA. Ifthe ergodic rate is used as the criterion, a similar differencebetween F-NOMA and CR-NOMA can be observed. Againtake the scenario described in the last paragraph as an example.If n = M , in order to yield a large gain over conventional MA,F-NOMA prefers the choice of m = 1, but CR-NOMA prefersthe choice of m = M − 1, i.e., pairing the user with the bestchannel condition with the user with the second best channelcondition.

II. NOMA WITH FIXED POWER ALLOCATION

Consider a classical cellular downlink communication sce-nario with one BS and M mobile users. All the users sharethe same bandwidth resources, such as time slots, spread-ing codes, and subcarrier channels. The received signals arecorrupted by additive white Gaussian noise. Without loss ofgenerality, assume that the users’ channels have been orderedas |h1|2 ≤ · · · ≤ |hM |2, where hm denotes the Rayleigh fadingchannel gain between the BS and the ordered m-th user. Wefurther assume that all users experience the same noise power.As discussed in the previous section, asking all the users in thenetwork to participate in NOMA is not preferable in practice,and in this paper we consider the situation in which the m-thuser and the n-th user, m < n, are paired to perform NOMA.

The insights obtained for the case with two selected userscan be used for the design of dynamic user pairing/groupingapproaches, which is beyond the scope of this paper.

In this section, we focus on F-NOMA, where the BSallocates a fixed amount of transmit power to each user. Inparticular, denote the power allocation coefficients for the twousers by am and an, where these coefficients are fixed anda2m+a2n = 1. According to the principle of NOMA, am ≥ ansince |hm|2 ≤ |hn|2. F-NOMA is applicable to the scenarioin which the base station knows only the order of users’channel conditions, but does not have perfect channel stateinformation. More sophisticated power allocation strategieswill be discussed in Section III. The rates achievable to theusers are given by

Rm = log

(1 +

|hm|2a2m|hm|2a2n + 1

ρ

), (1)

and

Rn = log(1 + ρa2n|hn|2

), (2)

respectively, where ρ denotes the transmit SNR. Note that then-th user can decode the message intended for the m-th usersuccessfully and Rn is always achievable at the n-th user, sinceRm ≤ log

(1 +

|hn|2a2m

|hn|2a2n+

1ρ

).

On the other hand, an orthogonal MA scheme, such as time-division multiple-access (TDMA), can support the followingdata rate:

R̄i =1

2log(1 + ρ|hi|2

), (3)

where i ∈ {m,n}. In the following subsections, the impactof user pairing on the sum rate and the individual user ratesachieved by F-NOMA is investigated.

A. Impact of user pairing on the sum rateIn this subsection, we focus on how user pairing affects

the probability that NOMA achieves a lower sum rate thanconventional MA schemes, which is given by

P(Rm +Rn < R̄m + R̄n). (4)

The following theorem provides an exact expression for theabove probability as well as its high SNR approximation.

Theorem 1. Suppose that the m-th and n-th ordered usersare paired to perform NOMA. The probability that F-NOMAachieves a lower sum rate than conventional MA is given by

P(Rm +Rn < R̄m + R̄n) = (5)

1−n−1−m∑

i=0

(n− 1−m

i

)(−1)iϖ1

m+ i

∫ ϖ2

ϖ4

f(y)(F (y))n−1−m−i

× (1− F (y))M−n

([F (y)]

m+i −[F

(ϖ2 − y

1 + y

)]m+i)dy

− ϖ3

ρ

n−1∑j=0

(n− 1

j

)(−1)j

ρ

M − n+ j + 1e−

(M−n+j+1)ϖ2ρ ,



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where f(x) = 1ρe

− xρ , F (x) = 1 − e−

xρ , ϖ1 =

M !(m−1)!(n−1−m)!(M−n)! , ϖ2 =

1−2a2n

a4n

, ϖ3 = M !(n−1)!(M−n)!

and ϖ4 =√1 +ϖ2 − 1. At high SNR, this probability can

be approximated as follows:

P(Rm +Rn < R̄m + R̄n) ≈1

ρn

(ϖ3ϖ

n2

n−ϖ1ϖ

), (6)

where ϖ =∑n−1−m

i=0

(n−1−m

i

) (−1)i

m+i

∫ϖ2

ϖ4yn−1−m−i

×(ym+i −

[ϖ2−y(1+y)

]m+i)dy, i.e., ϖ is a constant and

not a function of ρ.

Proof: See the appendix.It is important to point out that it is not true that F-NOMA

can always outperform orthogonal MA. Theorem 1 providesthe probability of the event that F-NOMA realizes a largersum rate than orthogonal MA. This result shows that it isvery likely that F-NOMA will outperform conventional MA,particularly at high SNR. Furthermore, the decay rate of theprobability P(Rm+Rn < R̄m+R̄n) is approximately 1

ρn , i.e.,the channel rank of the user with a strong channel conditiondetermines the decay rate of this probability. Therefore thechannel quality of the n-th user has an important impact todecrease the probability P(Rm +Rn < R̄m + R̄n).

B. Asymptotic studies of the sum rate achieved by NOMA

In addition to the probability P(Rm +Rn < R̄m + R̄n), itis also of interest to study how large of a performance gainF-NOMA offers over conventional MA, i.e.,

P(Rm +Rn − R̄m − R̄n < R),

where R is a targeted performance gain. The probabilitystudied in the previous subsection can be viewed as a specialcase by setting R = 0. An interesting observation for thecases with R > 0 is that there will be an error floor forP(Rm+Rn−R̄m−R̄n < R), regardless of how large the SNRis. This can be shown by studying the following asymptoticexpression of the sum rate gap:

Rm +Rn − R̄m − R̄n (7)

= log

(|hm|2 + 1

ρ

|hm|2a2n + 1ρ

)+ log

(1 + ρa2n|hn|2

)− 1

2log(1 + ρ|hn|2

) 12log(1 + ρ|hm|2

)→

ρ→∞log

(1

a2n

)+ log

(ρa2n|hn|2

)− log (ρ|hm||hn|)

= log |hn| − log |hm|,

which is not a function of SNR. Hence the probability can beexpressed asymptotically as follows:

P(Rm +Rn − R̄m − R̄n < R

)(8)

→ρ→∞

P (log |hn| − log |hm| < R) .

When R = 0, P(Rm +Rn − R̄m − R̄n < R

)→ 0, which is

consistent with Theorem 1, since

P(Rm +Rn < R̄m + R̄n

)∼ 1

ρn→

ρ→∞0.

When R ̸= 0, (8) implies that the probabilityP(Rm +Rn − R̄m − R̄n < R

)can be expressed asymptot-

ically as follows:


)→ P

(|hn|2

|hm|2< 22R

). (9)

Directly applying the joint probability density function (pdf)of the users’ channels in [12] yields a quite complicatedexpression. In [13], a simpler pdf for the ratio of two orderstatistics has been provided as follows:

f |hm|2|hn|2

(z) =M !

(m− 1)!(n−m− 1)!(M − n)!

m−1∑j1=0

n−m−1∑j2=0

(−1)j1+j2

(m− 1

j1

)(n−m− 1

j2

)(τ2 + τ1z)

−2,

where τ1 = j1 − j2 + n −m and τ2 = M − n + 1 + j2. Byusing this pdf, the addressed probability can be calculated asfollows:


)(10)

→ M !

(m− 1)!(n−m− 1)!(M − n)!

m−1∑j1=0

n−m−1∑j2=0

(−1)j1+j2

τ1

(m− 1

j1

)(n−m− 1

j2

)(1

τ2 + 2−2Rτ1− 1

τ2 + τ1

).

Recall that the probability P(Rm+Rn < R̄m+R̄n) is highlydependent on the channel rank of the user with a better channelcondition, as shown in the previous section. However, the prob-ability for the sum rate gap, P

(Rm +Rn − R̄m − R̄n < R

),

is affected by both users’ channel ranks, as demonstrated in(10). Simulation results will be provided in Section IV toconfirm the accuracy of (10), and also show that the sum rategap will be enlarged by scheduling two users whose channelconditions are very different.

C. Impact of user pairing on individual user ratesCareful user pairing not only improves the sum rate, but

also has the potential to improve the individual user rates, asshown in this section. We first focus on the probability thatF-NOMA can achieve a larger rate than orthogonal MA forthe m-th user which is given by

P(Rm > R̄m) (11)

=P

(1 + |hm|2a2m|hm|2a2n + 1

ρ

)2

> (1 + ρ|hm|2)

.

The following lemma provides a high SNR approximation ofthe above probability.



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Lemma 1. Suppose that the m-th and n-th ordered usersare paired to perform NOMA. The probability that F-NOMAachieves a larger data rate than conventional MA for the m-thuser can be approximated at high SNR as follows:

P(Rm > R̄m) ≈ ϖ5(1− 2a2n)

m

mρma4mn, (12)

where ϖ5 = M !(m−1)!(M−m)! .

Proof: See the appendix.Lemma 1 demonstrates that P(Rm > R̄m) decays at a rate

of 1ρm .Therefore if the m-th user’s data rate is used as the

performance criteria, it is important to have a smaller m, i.e.,scheduling a user with poorer channel conditions in order toincrease P(Rm > R̄m).

On the other hand, the probability that the n-th user canexperience better performance in a NOMA system than inorthogonal MA systems is given by

P(Rn > R̄n) = P

(log(1 + ρa2n|hn|2

)>

1

2log(1 + ρ|hn|2

).

Following similar steps as previously, we obtain the following:

P(Rn > R̄n) = P

(|hn|2 >

1− 2a2nρa4n

). (13)

Following steps similar to those in the proof of Lemma 1, anexact expression for the probability can be obtained as follows:

P(Rn > R̄n) = 1−n−1∑i=0

(n− 1

i

)(−1)iϖ3

M − n+ i+ 1(14)

×

(1− e

− (1−2a2n)(M−n+i+1)

ρa4n

),

and its high SNR approximation is given by

P(Rn > R̄n) ≈ 1−ϖ3(1− 2a2n)

n

nρna4nn. (15)

As can be seen from (12) and (15), the two users will havetotally different experiences in NOMA systems. Particularly, auser with a better channel condition is more willing to performNOMA since P(Rn > R̄n) → 1, which is not true for auser with a poorer channel condition. When individual userrates are used as criteria, it is preferable to pair two userswhose channel conditions are significantly distinctive, since(12) and (15) implies that m should be as small as possible andn should be as large as possible. Note that these conclusionsare consistent with the ones made in Section II.A and B. Forexample, a larger n leads to a decrease in P(Rm + Rn <R̄m + R̄n), i.e., it is more likely that F-NOMA realizes alarger sum rate than orthogonal MA, as shown in Section II.A.In order to enlarge the sum rate gap, it is ideal to schedule twousers whose channel conditions are distinctive, as discussed inSection II.B.

III. COGNITIVE RADIO INSPIRED NOMANOMA can be also viewed as a special case of cognitive

radio systems [14], [15], in which a user with a strong channelcondition, viewed as a secondary user, is squeezed into thespectrum occupied by a user with a poor channel condition,viewed as a primary user. Following the concept of cognitiveradio networks, a variation of NOMA, termed CR-NOMA, canbe designed as follows. Suppose that the BS needs to serve them-th user, i.e., a user with poor channel condition, due to eitherthe high priority of this user’s messages or user fairness, e.g.,this user has not been served for a long time. This user can beviewed as a primary user in a cognitive radio system. The n-thuser can be admitted into this channel on the condition that then-th user will not cause too much performance degradation tothe m-th user.

Consider that the targeted SINR at the m-th user is I , whichmeans that the choices of the power allocation coefficients, amand an, need to satisfy the following constraint:


ρ

≥ I. (16)

This implies that the maximal transmit power that can beallocated to the n-th user is given by

a2n = max

{0,

|hm|2 − Iρ

|hm|2(1 + I)

}, (17)

which means that an = 0 if |hm|2 < Iρ . Note that the choice of

an in (17) is a function of the channel coefficient hm, unlikethe constant choice of an used by F-NOMA in the previoussection.

Since the m-th user’s QoS can be guaranteed due to (16),we only need to study the performance experienced by the n-th user. Particularly the outage performance of the n-th useris defined as follows:

Pno , P

(log(1 + a2nρ|hn|2) < R

), (18)

and the following theorem provides an exact expression forthe above outage probability as well as its approximation.

Theorem 2. Suppose that the transmit power allocated to then-th user satisfies the predetermined SINR threshold, I , asshown in (17) at the top of the following page. The n-th user’soutage probability achieved by CR-NOMA is given by (19),where g(y) = e−y , G(y) = 1 − e−y , ϵ1 = 2R−1

ρ , b = Iρ ,

a = 1 + I and b ≤ aϵ1. The diversity order achieved by CR-NOMA is given by

limρ→∞

− logPno

log ρ= m.

Proof: See the appendix.Theorem 2 demonstrates an interesting phenomenon that, in

CR-NOMA, the diversity order experienced by the n-th useris determined by how good the m-th user’s channel quality is.This is because the n-th user can be admitted to the channeloccupied by the m-th user only if the m-th user’s QoS ismet. For example, if the m-th user’s channel is poor and its



5

Pon = ϖ5

M−n∑i=0

(M − n

i

)(−1)i

[G(b)]m+i

m+ i+

n−1−m∑i=0

(n− 1−m

i

)(−1)i

∫ aϵ1

b

g(y) (1−G(y))M−n G(y)n−1−m−iϖ1 (19)

×(G(y)m+i −G(b)m+i

)m+ i

dy +

n−1−m∑i=0

(n− 1−m

i

)(−1)i

∫ b+aϵ1

aϵ1

(1−G(y))M−n G(y)n−1−m−i

(G(y)m+i −G(b)m+i

)m+ i

×ϖ1g(y)dy +

n−1−m∑i=0

(n− 1−m

i

)(−1)i

∫ ∞

b+aϵ1

g(y) (1−G(y))M−n G(y)n−1−m−iϖ1

(G

(b

1− aϵ1|hn|2

)m+i

−G(b)m+i

)m+ i

dy.

targeted SINR is high, it is very likely that the BS allocatesall the power to the m-th user, and the n-th user might noteven get served.

Recall from the previous section that F-NOMA can achievea diversity gain of n for the n-th user, and therefore thediversity order achieved by CR-NOMA could be much smallerthan F-NOMA, particularly if n >> m. This performancedifference is again due to the imposed power constraint shownin (17).

It is important to point out that CR-NOMA can strictlyguarantee the m-th user’s QoS, and therefore achieve betterfairness compared to F-NOMA. In particular, the use of CR-NOMA can ensure that a diversity order of m is achievableto the n-th user, and admitting the n-th user into the samechannel as the m-th user will not cause too much performancedegradation to the m-th user.

The sum rate achieved by CR-NOMAWithout sharing the spectrum with the n-th user, i.e, all the

bandwidth resource is allocated to the m-th user, the followingrate is achievable:

R̃m = log(1 + ρ|hm|2

). (20)

It is easy to show that the use of CR-NOMA always achievesa larger sum rate since

Rm +Rn − R̃m (21)

= log

(1 +


ρ

)+ log

(1 + ρa2n|hn|2

)− log

(1 + ρ|hm|2

)= log

1 + ρa2n|hn|2

1 + ρa2n|hm|2≥ 0.

This superior performance gain is not surprising, since thekey idea of CR-NOMA is to serve a user with a strongchannel condition opportunistically, without causing too muchperformance degradation to the user with a poor channelcondition.

In addition, it is of interest to study how much the averagedrate gain CR-NOMA can yield, i.e., E {Rn}. This averagedrate gain can be calculated as follows:

E {Rn} =

∫ ∞

b

∫ ∞

x

log

(1 +

x− b

xaρy

)(22)

× f|hm|2,|hn|2(x, y)dydx.

In general, the evaluation of the above equation is difficult, andin the following we provide a case study when n − m = 1.Particularly, the joint pdf of the channels for this special casecan be simplified and the averaged rate gain can be calculatedas follows:

E {Rn} = ϖ1

∫ ∞

b

f(x)[F (x)]m−1

∫ ∞

x

log

(1 +

x− b

xaρy

)× f(y) (1− F (y))

M−ndydx (23)

=−ϖ1

M − n+ 1

∫ ∞

b

f(x)[F (x)]m−1

×∫ ∞

x

log

(1 +

x− b

xaρy

)d (1− F (y))

M−n+1dx.

After some algebraic manipulations, the above equation canbe rewritten as follows:

E {Rn} =ϖ1

M − n+ 1

∫ ∞

b

f(x)[F (x)]m−1

×(log

(1 +

x− b

aρ

)(1− F (x))

M−n+1

+1

ln 2

∫ ∞

x

(1− F (y))M−n+1

x−bxa ρ

1 + x−bxa ρy

dy

)dx.

Now applying Eq. (3.352.2) in [16], the average rate gain canbe expressed as follows:

E {Rn} =ϖ1

M − n+ 1

∫ ∞

b

f(x)[F (x)]m−1 (24)

×

log

(1 +

x− b

aρ

)(1− F (x))

M−n+1 − ex2a

ρ(x−b)

ln 2

×Ei(−(M − n+ 1)x− (M − n+ 1)xa

ρ(x− b)

))dx,

where Ei(x) denotes the exponential integral, i.e., Ei(x) ,∫ x

−∞et

t dt [16].

IV. NUMERICAL STUDIES

In this section, computer simulations are used to evaluatethe performance of the two NOMA schemes as well as theaccuracy of the developed analytical results.



6

5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR in dB

P(R

m+R

n<

R̄m+R̄

n)

Simulation resultsAnalytical results − Exact expressionAnalytical results − High SNR approximation

n=5, m=1

n=3,m=1

(a) m = 1

5 10 15 20 25 3010

−6

10−5

10−4

10−3

10−2

10−1

100

SNR in dB

P(R

m+R

n<

R̄m+R̄

n)

Simulation resultsAnalytical results − Exact expressionAnalytical results − High SNR approximation

n=5,m=2

n=3,m=2

(b) m = 2

Fig. 1. The probability that F-NOMA realizes a lower sum rate thanconventional MA. M = 5. The analytical results are based on Theorem 1.

A. NOMA with fixed power allocationIn Fig. 1, the probability that F-NOMA realizes a lower sum

rate than conventional MA, i.e., P(Rm + Rn < R̄m + R̄n),is shown as a function of SNR, with a2m = 4

5 and a2n = 15 .

As can be seen from both figures, F-NOMA can outperformconventional MA, particularly at high SNR. The simulationresults in Fig. 1 also demonstrate the accuracy of the analyt-ical results provided in Theorem 1. For example, the exactexpression of P(Rm + Rn < R̄m + R̄n) shown in Theorem1 matches perfectly with the simulation results, whereas thedeveloped approximation results become accurate at high SNR.

Another important observation from Fig. 1 is that increasingn, i.e., scheduling a user with a better channel condition, willmake the probability decrease at a faster rate. This observation

is consistent with the high SNR approximation results providedin Theorem 1 which show that the slope of the curve for theprobability P(Rm+Rn < R̄m+R̄n) is a function of n. In Fig.2, the probability P(Rm+Rn−R̄m−R̄n < R) is shown withdifferent choices of R. Comparing Fig. 1 to Fig. 2, one canobserve that P(Rm +Rn − R̄m − R̄n < R) never approacheszero, regardless of how large the SNR is. This observationconfirms the analytical results developed in (10) which showthat the probability P(Rm + Rn − R̄m − R̄n < R) is nolonger a function of SNR, when ρ → 0. It is interesting toobserve that the choice of a smaller m is preferable to reduceP(Rm + Rn − R̄m − R̄n < R), a phenomenon previouslyreported in [9].

5 10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

100

SNR in dB

P(R

m+R

n−R̄

m−R̄

n<

R)

Simulation results, m=1Asymptotic results, m=1Simulation results, m=2Asymptotic results, m=2

−Solid lines for the case with R=0.5 BPCU

−Dashed lines for the case with R=0.1 BPCU

Fig. 2. The probability that the sum rate gap between F-NOMA andconventional MA is larger than R. M = 5 and n = M . The analyticalresults are based on (10).

In Fig. 3, two different but related probabilities are showntogether. One is P(Rm > R̄m), i.e., the probability that itis beneficial for a user with a poor channel condition toperform F-NOMA, and the other is P(Rn < R̄n), i.e., theprobability that the user with a strong channel condition prefersconventional MA. Note that in Fig. 3, the dot-dash curvesare used to show the analytical results, and the solid curvesare used to show the simulation results. In addition, curvesfor analytical and numerical results are labeled with the samemarker if they are based on the same parameters (m and n). InSection II.C, analytical results have been developed to showthat both P(Rm > R̄m) and P(Rn < R̄n) are decreasingwith increasing SNR, which is confirmed by the simulationresults in Fig. 3. The reason that P(Rm > R̄m) is reducedat a higher SNR is that the m-th user’s rate in an F-NOMAsystem becomes a constant, i.e., log

(1 +

|hm|2a2m

|hm|2a2n+

1ρ

)→

ρ→∞

log(1 +

a2m

a2n

), which is much smaller than R̄m, at high SNR.

On the other hand, it is more likely for Rn to be larger thanR̄n since there is a factor of 1

2 outside of the logarithm of R̄n.For the numerical results shown in Figs. 1-3, the same set



7

of power allocation coefficients is used. In Fig. 4, the use ofdifferent choices of power allocation coefficients is considered.As shown in Fig. 4.a, when the power allocation coefficientam is decreasing, it becomes more likely that F-NOMA canachieve a larger sum rate than conventional MA. This can beexplained as follows. When am decreases, more power willbe given to the user with a better channel condition, whichleads to improvement in a system throughput. This is also thereason for the observation from Fig. 4.b that decreasing amdecreases the probability P(Rn < R̄n). However, a smalleram will bring performance degradation to the m-th user, sinceless power is allocated to this user. As a result, one can observein Fig. 4.b that P(Rm > R̄m) decreases with decreasing am,i.e., it is less likely that the m-th user experiences a largerdata rate in F-NOMA systems compared to orthogonal MAsystems, for a smaller am.

5 10 15 20 25 30

10−4

10−3

10−2

10−1

100

SNR in dB

Out

age

prob

abili

ty

Simulation results, m = 2, n = 3, P(Rm > R̄m)

Simulation results, m = 2, n = 3, P(Rn < R̄n)

Analytical results, m = 2, n = 3, P(Rm > R̄m)

Analytical results, m = 2, n = 3, P(Rn < R̄n)

Simulation results, m = 1, n = M , P(Rm > R̄m)

Simulation results, m = 1, n = M , P(Rn < R̄n)

Analytical results, m = 1, n = M , P(Rm > R̄m)

Analytical results, m = 1, n = M , P(Rn < R̄n)

Fig. 3. The behavior of individual data rates achieved by F-NOMA, P(Rn <R̄n) and P(Rm > R̄m). M = 5. The analytical results are based on (43)and (14).

B. Cognitive radio inspired NOMAIn Fig. 5 the n-th user’s outage probability achieved by

CR-NOMA is shown as a function of SNR. As can be seenfrom the figure, the exact expression for the outage probabilityPon , P(Rn < R) developed in Theorem 2 matches the

simulation results perfectly. Recall from Theorem 2 that thediversity order achievable for the n-th user is m. Or in otherwords, the slope of the outage probability is determined by thechannel rank of the user with a poorer channel condition, i.e.,the m-th user, which is also confirmed by Fig. 5. For example,when increasing m from 1 to 2, the outage probability issignificantly reduced, and its slope is also increased. To clearlydemonstrate the diversity order, we have provided an auxiliarycurve in the figure which is proportional to 1

ρm . As can beobserved in the figure, this auxiliary curve is parallel to theone for P(Rn < R), which confirms that the diversity orderachieved by CR-NOMA is m.

5 10 15 20 25 3010

−8

10−6

10−4

10−2

100

SNR in dB

P(R

m+R

n<

R̄m+R̄

n)

am = 1011

am = 911

am = 811

am = 711

(a) P(Rm +Rn < R̄m + R̄n)

5 10 15 20 25 3010

−8

10−6

10−4

10−2

100

SNR in dB

Out

age

prob

abili

ties

am = 711 , P(Rm > R̄m)

am = 711 ,P(Rn < R̄n)

am = 811 , P(Rm > R̄m)

am = 811 ,P(Rn < R̄n)

am = 911 ,P(Rm > R̄m)

am = 911 ,P(Rn < R̄n)

am = 1011 ,P(Rm > R̄m)

am = 1011 ,P(Rn < R̄n)

(b) P(Rm > R̄m) and P(Rn < R̄n)

Fig. 4. The impact of the power allocation coefficients on the sum rate andindividual data rates. M = 5. m = 1 and n = 3.

Since Theorem 2 states that the diversity order of P(Rn <R) is not a function of n, an interesting question is whethera different choice of n matters. Fig. 6 is provided to answerthis question. While the use of a larger n does bring somereduction in P(Rn < R), the performance gain of increasingn is negligible, particularly at high SNR. This is because thechannel quality of the m-th user becomes a bottleneck foradmitting the n-th user into the same channel.

In Fig. 7 the performance of CR-NOMA is evaluated byusing the ergodic data rate as the criterion. Due to the useof (17), the m-th user’s QoS can be satisfied, and thereforewe focus only on the n-th user’s data rate, which is theperformance gain of CR-NOMA over conventional MA. Fig.7 demonstrates that, by fixing (n − m), it is beneficial to



8

5 10 15 20 25 3010

−4

10−3

10−2

10−1

100

SNR in dB

P(R

n<

R)

Simulation resultsAnalytical results

An auxiliary line with 1ρm

m=1

m=2

Fig. 5. The outage probability for the n-th user achieved by CR-NOMA,when n = M . M = 5, R = 1 bit per channel use (BPCU) and I = 5. Theanalytical results are Theorem 2.

5 10 15 20 25 30

10−2

10−1

100

SNR in dB

P(R

n<

R)

n=Mn=M−1n=M−2n=M−3

I=1

I=5

Fig. 6. The outage probability for the n-th user achieved by CR-NOMA.m = 1, M = 5, and R = 1 BPCU.

select two users with better channel conditions. While Fig.6 shows that changing n with a fixed m does not affect theoutage probability, Fig. 7 demonstrates that user pairing hasa significant impact on the ergodic rate. Specifically, whenfixing the choice of m, pairing it with a user with a betterchannel condition can yield a gain of more than 1 bit perchannel use (BPCU) at 30dB. Another interesting observationfrom Fig. 7 is that with a fixed n, increasing m will improvethe performance of CR-NOMA, which is different from F-NOMA. For example, when n = M , Fig. 2 shows that theuser with the worst channel condition, m = 1, is the bestpartner, whereas Fig. 7 shows that the user with the second

5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

SNR in dB

E(R

n)

Simulation results, m=1, n=m+1Analytical results, m=1, n=m+1Simulation results, m=2, n=m+1Analytical results, m=2, n=m+1Simulation results, m=3, n=m+1Analytical results, m=3, n=m+1Simulation results, m=4, n=m+1Analytical results, m=4, n=m+1

(a) n = m+ 1

5 10 15 20 25 300

1

2

3

4

5

6

7

8

9

SNR in dB

E(R

n)

m=1, n=Mm=1, n=M−2m=2, n=Mm=2, n=M−2m=M−1, n=M

(b) General Cases

Fig. 7. The ergodic data rate for the n-th user achieved by CR-NOMA.M = 5 and I = 5. Analytical results are based on (24).

best channel condition, i.e., m = M − 1, is the best choice.

V. CONCLUSIONS

In this paper the impact of user pairing on the performanceof two NOMA systems, NOMA with fixed power allocation(F-NOMA) and cognitive radio inspired NOMA (CR-NOMA),has been studied. For F-NOMA, both analytical and numericalresults have been provided to demonstrate that F-NOMAcan offer a larger sum rate than orthogonal MA, and theperformance gain of F-NOMA over conventional MA can befurther enlarged by selecting users whose channel conditionsare more distinctive. For CR-NOMA, the channel quality of theuser with a poor channel condition is critical, since the transmit



9

power allocated to the other user is constrained following theconcept of cognitive radio networks. One promising futuredirection of this paper is that the analytical results developedin this paper can be used as criteria for design of distributedapproaches for dynamic user pairing/grouping.

APPENDIX

Proof for Theorem 1: Observe that the sum rate achieved byNOMA can be expressed as follows:

Rm +Rn = log

(1 + ρ|hm|2

ρ|hm|2a2n + 1

)(1 + ρa2n|hn|2

).

On the other hand the sum rate achieved by conventional MAis given by

R̄m + R̄n = log(1 + ρ|hm|2

) 12(1 + ρ|hn|2

) 12 . (25)

Now the addressed probability can be written as follows:

P(Rm +Rn > R̄m + R̄n) (26)

= P

((1 + ρ|hm|2

ρ|hm|2a2n + 1

)(1 + ρa2n|hn|2

)>(1 + ρ|hm|2

) 12

×(1 + ρ|hn|2

) 12

)= P

(1 + ρ|hm|2

(1 + ρa2n|hm|2)2>

1 + ρ|hn|2

(1 + ρa2n|hn|2)2

).

After some algebraic manipulations, this probability can berewritten as follows:

P(Rm +Rn > R̄m + R̄n) (27)

= P

(ρ(|hm|2 + |hn|2) + ρ2|hm|2|hn|2 >

1− 2a2na4n

).

The right-hand side of the above inequality is non-negativesince the n-th user will get less power than the m-th user, i.e.,a2n ≤ 1

2 . Note that the joint pdf of ρ|hm|2 and ρ|hn|2 is givenby [12]

f|hm|2,|hn|2(x, y) = ϖ1f(x)f(y)[F (x)]m−1 (1− F (y))M−n

× (F (y)− F (x))n−1−m

. (28)

In addition, the marginal pdf of |hn|2 is given by

f|hn|2(y) = ϖ3f(y) (F (y))n−1

(1− F (y))M−n

. (29)

By applying the above density functions, the addressedprobability can be expressed as follows:

P(Rm +Rn > R̄m + R̄n) =

∫ ∞

ϖ2

f|hn|2(y)dy︸︷︷︸Q2

(30)

+

∫ ∫(x+y)+xy>ϖ2,x<y<ϖ2

f|hm|2,|hn|2(x, y)dxdy

︸︷︷︸Q1

.

Note that the integral range for x in Q1 is ϖ2−y1+y < x <

y, and this range implies that ϖ2−y1+y < y, which causes an

additional constraint on y, i.e., y >√1 +ϖ2−1. By applying

the binomial expansion, the joint pdf can be further written asfollows:

f|hm|2,|hn|2(x, y) = ϖ1

n−1−m∑i=0

(n− 1−m

i

)(−1)if(x)

× f(y)[F (x)]m−1+i (1− F (y))M−n

(F (y))n−1−m−i.

Therefore the probability Q1 can now be evaluated as follows:

Q1 = ϖ1

n−1−m∑i=0

(n− 1−m

i

)(−1)i

m+ i

∫ ϖ2

ϖ4

f(y)(F (y))n−1−m−i

× (1− F (y))M−n

([F (y)]m+i −

[F

(ϖ2 − y

1 + y

)]m+i)dy.

(31)

On the other hand, Q2 can be calculated as follows:

Q2 =

∫ ∞

ϖ2

ϖ3f(y) (F (y))n−1

(1− F (y))M−n

dy (32)

=

∫ ∞

ϖ2

ϖ31

ρe−

(M−n+1)yρ

(1− e−

yρ

)n−1

dy.

By applying the binomial expansion, Q2 can be written asfollows:

Q2 =ϖ3

ρ

n−1∑j=0

(n− 1

j

)(−1)j

∫ ∞

ϖ2

e−(M−n+j+1)y

ρ dy (33)

=ϖ3

ρ

n−1∑j=0

(n− 1

j

)(−1)j

ρ

M − n+ j + 1e−

(M−n+j+1)ϖ2ρ .

Combining (31) with (33), the first part of the theorem isproved.

To find the high SNR approximations for Q1 and Q2, firstobserve that the integral in (31) is calculated for the rangeof 0 ≤ y < ϖ2. At high SNR, the two functions f(y) andF (y) can be approximated as follows: f(y) = 1

ρe− y

ρ ≈ 1ρ and

F (y) = 1− e−yρ ≈ y

ρ , since 0 ≤ y ≤ ϖ2 and ρ → ∞.

Define u(y) = ϖ2−y1+y . It is straightforward to show that

0 ≤ u(y) ≤ ϖ2,

for 0 ≤ y ≤ ϖ2 , since dg(y)dy < 0. Therefore at high SNR, we

have the following approximation:

F

(ϖ2 − y

1 + y

)= 1− e−

ϖ2−y

ρ(1+y) ≈ ϖ2 − y

ρ(1 + y).



10

Now the probability Q1 can be approximated as follows:

Q1 ≈ ϖ1

n−1−m∑i=0

(n− 1−m

i

)(−1)i

m+ i

∫ ϖ2

ϖ4

1

ρ

(y

ρ

)n−1−m−i

×

([y

ρ

]m+i

−[ϖ2 − y

ρ(1 + y)

]m+i)dy

≈ ϖ1

ρn

n−1−m∑i=0

(n− 1−m

i

)(−1)i

m+ i

∫ ϖ2

ϖ4

yn−1−m−i

×

(ym+i −

[ϖ2 − y

(1 + y)

]m+i)dy. (34)

The high SNR approximation for Q2 is more complicated.After applying the series expansion of the exponential func-tions in (33), we have

Q2 =

∞∑i=0

ϖ3

ρ

n−1∑j=0

(n− 1

j

)(−1)i+j (M−n+j+1)i−1ϖi

2

ρi−1

i!(35)

=

∞∑i=0

(−1)iϖ3ϖi2

i!ρi

n−1∑j=0

(n− 1

j

)(−1)j(M − n+ j + 1)i−1.

Consider Q2 as a function of ϖ2, and Q2 = 1 is true forϖ2 = 0, as can be seen from the definition of Q2 in (30).On the other hand by letting ϖ2 = 0 in (33), we obtain thefollowing equality:

ϖ3

n−1∑j=0

(n− 1

j

)(−1)j

1

M − n+ j + 1= 1. (36)

Consequently Q2 can be rewritten as follows:

Q2 = 1 +∞∑i=1

(−1)iϖ3ϖi2

i!ρi

n−1∑j=0

(n− 1

j

)(−1)j (37)

×i−1∑l=0

(i− 1

l

)(M − n+ 1)i−1−ljl.

Recall the following sums of the binomial coefficients (Eq.(0.154.3) in [16]):

n−1∑j=0

(n− 1

j

)(−1)jjl = 0, (38)

for n− 2 ≥ l ≥ 1 and

n−1∑j=0

(n− 1

j

)(−1)jjn−1 = (−1)n−1(n− 1)!. (39)

Therefore all the components in (37) containing jl, l < (n−1),can be removed, since they are equal to zero by using (38).Furthermore, all the components containing jl, l > (n−1) canalso be ignored, since the one with j = n− 1 is the dominant

factor. With these steps, the probability can be approximatedas follows:

Q2 ≈ 1 +(−1)nϖ3ϖ

n2

n!ρn(−1)n−1(n− 1)! (40)

= 1− ϖ3ϖn2

nρn.

Combining (34) and (40), the second part of the theorem isalso proved. �

Proof for Lemma 1: Recall that the probability that F-NOMA achieves a larger rate than orthogonal MA for the m-thuser is given by

P(Rm > R̄m) = P

(1 + |hm|2a2m|hm|2a2n + 1

ρ

)2

> (1 + ρ|hm|2)

.

(41)

After some algebraic manipulations, the above probability canbe further rewritten as follows:

P(Rm > R̄m) = P

(|hm|2 <

1− 2a2nρa4n

)(42)

=

∫ 1−2a2n

ρa4n

0

ϖ5

ρe−

(M−m+1)yρ

(1− e−

yρ

)m−1

dy

=

m−1∑i=0

(m− 1

i

)(−1)iϖ5

M −m+ i+ 1

(1− e

− (1−2a2n)(M−m+i+1)

ρa4n

).

By applying a series expansion, the above probability canbe rewritten as follows:

P(Rm > R̄m) =m−1∑i=0

(m− 1

i

)(−1)i+1ϖ5 (43)

×∞∑k=1

(−1)k(1− 2a2n)

k(M −m+ i+ 1)k−1

k!ρka4kn.

Again applying the results in (38) and (39), the above equationcan be approximated as follows:

P(Rm > R̄m) ≈ ϖ5(1− 2a2n)

m

mρma4mn. (44)

The lemma is proved. �

Proof for Theorem 2: Recall that the outage performance ofthe n-th user is given by

P(log(1 + a2nρ|hn|2) < R

)(45)

=P

(log

(1 +

|hm|2 − Iρ

|hm|2(1 + I)ρ|hn|2

)< R, |hm|2 >

I

ρ

)︸︷︷︸

Q3

+ P

(|hm|2 <

I

ρ

)︸︷︷︸

Q4

.



11

The first factor in the above equation can be calculated asfollows:

Q3 =P

(|hm|2 − I

ρ

|hm|2(1 + I)|hn|2 < ϵ1, |hm|2 >

I

ρ

). (46)

Recall that the users’ channels are ordered, i.e., |hm|2 < |hn|2,which brings additional constraints to the integral range in theabove equation. The constraints can be written as follows:

b < |hm|2 < min

{|hn|2,

b

1− aϵ1|hn|2

}. (47)

The outage events due to these constraints can be classified asfollows:

1) If |hn|2 < aϵ1, we have the following:

P

(|hm|2 − I

ρ

|hm|2(1 + I)|hn|2 < ϵ1

)(48)

= P(|hm|2(|hn|2 − ϵ1a) < b|hn|2

)= 1.

Therefore the probability Q3 can be expressed as fol-lows: 1

Q3 = P(b ≤ |hn|2 < aϵ1, |hn|2 > |hm|2 > b

).

2) If |hn|2 > aϵ1, there are two possible events:a) If |hn|2 > b + aϵ1, we have b

1− aϵ1|hn|2

< |hn|2,

and Q3 can be written as follows:

Q3 = P

(|hn|2 > b+ aϵ1, b < |hm|2 <

b

1− aϵ1|hn|2

).

b) If |hn|2 < b + aϵ1, we have b1− aϵ1

|hn|2> |hn|2,

and Q3 can be written as follows:

Q3 = P(aϵ1 < |hn|2 < b+ aϵ1, b < |hm|2 < |hn|2

),

which is again conditioned on b < aϵ1.Therefore, the probability Q3 can be written as follows:

Q3 = P(b ≤ |hn|2 < aϵ1, |hn|2 > |hm|2 > b

)(49)

+ P(|hn|2 < b+ aϵ1, b < |hm|2 < |hn|2

)+ P

(|hn|2 > b+ aϵ1, b < |hm|2 <

b

1− aϵ1|hn|2

).

1It is assumed that b ≤ aϵ1 here. For the case of b > aϵ1, the outageprobability can be calculated in a straightforward way, since there will befewer events to analyze. Note that the same diversity order will be obtainedregardless of the choice of b and aϵ1.

The first probability in (49) can be calculated by applying (29)as follows:

P(b ≤ |hn|2 < aϵ1, |hn|2 > |hm|2 > b

)=

n−1−m∑i=0

(n− 1−m

i

)(−1)i

∫ aϵ1

b

g(y) (1−G(y))M−n

×G(y)n−1−m−i

∫ y

b

ϖ1g(x)[G(x)]m−1+idxdy

=

n−1−m∑i=0

(n− 1−m

i

)(−1)i

∫ aϵ1

b

g(y) (1−G(y))M−n

×G(y)n−1−m−iϖ1

(G(y)m+i −G(b)m+i

)m+ i

dy.

Following similar steps, the second probability in (49) can beexpressed as

P(aϵ1 < |hn|2 < b+ aϵ1, b < |hm|2 < |hn|2

)(50)

=

n−1−m∑i=0

(n− 1−m

i

)(−1)i

∫ b+aϵ1

aϵ1

g(y) (1−G(y))M−n


(G(y)m+i −G(b)m+i

)m+ i

dy.

The third probability in (49) can be calculated as follows:

P

(|hn|2 > b+ aϵ1, b < |hm|2 <

b

1− aϵ1|hn|2

)(51)

=

n−1−m∑i=0

(n− 1−m

i

)(−1)i

∫ ∞

b+aϵ1

g(y) (1−G(y))M−n

×G(y)n−1−m−i

∫ b

1− aϵ1|hn|2

b

ϖ1g(x)[G(x)]m−1+idxdy

=n−1−m∑

i=0

(n− 1−m

i

)(−1)i

∫ ∞

b+aϵ1

g(y) (1−G(y))M−n


(G

(b

1− aϵ1|hn|2

)m+i

−G(b)m+i

)m+ i

dy.

Note that Q4 can be obtained easily by applying (29) and thefirst part of the theorem is proved.

Recall that the first probability in (49) can be expressed asfollows:

P(b ≤ |hn|2 < aϵ1, |hn|2 > |hm|2 > b

)= ϖ1

n−1−m∑i=0

(n− 1−m

i

)(−1)i

∫ aϵ1

b

g(y)

× (1−G(y))M−n

G(y)n−1−m−i

(G(y)m+i −G(b)m+i

)m+ i

dy,

where the integral range is 0 ≤ y ≤ (aϵ1). Note that whenρ → ∞, ϵ1 approaches zero, which means y → 0, g(y) ≈ 1



12

and G(y) ≈ 1 − y. Therefore the above probability can beapproximated as follows:

P(b ≤ |hn|2 < aϵ1, |hn|2 > |hm|2 > b

)(52)

≈ ϖ1

n−1−m∑i=0

(n− 1−m

i

)(−1)i

×∫ aϵ1

b

yn−1−m−i

(ym+i − bm+i

)m+ i

dy

≈ ϖ1

n−1−m∑i=0

(n− 1−m

i

)(−1)i

×

((aϵ1)

n−bn

m+i+1 − bm+i(

(aϵ1)n−m−i−bn−m−i

n−m−i

))m+ i

→ ρ−n.

Following similar steps, the second probability in (49) canbe approximated as follows:

P(aϵ1 < |hn|2 < b+ aϵ1, b < |hm|2 < |hn|2

)→ ρ−n. (53)

The exact diversity order of the third probability in (49)is difficult to obtain. Particularly the expression in (51) isdifficult to use for asymptotic studies, since the range of y isnot limited and those manipulations related to the high SNRapproximations cannot be applied here. We first rewrite (51)in an alternative form as follows:

P

(|hn|2 > b+ aϵ1, b < |hm|2 <

b

1− aϵ1|hn|2

)

= P

(b < |hm|2 < b+ aϵ1, b+ aϵ1 < |hn|2 <

|hm|2aϵ1|hm|2 − b

).

Note that b + aϵ1 < |hm|2aϵ1|hm|2−b always holds since |hm|2 <

b1− aϵ1

|hn|2.

Now applying the joint pdf of the two channel coefficients,we obtain the following expression:

P

(b < |hm|2 < b+ aϵ1, b+ aϵ1 < |hn|2 <


)=

n−1−m∑i=0

ϖ1

(n− 1−m

i

)(−1)i

∫ b+aϵ1

b

g(x)[G(x)]m−1+i

×∫ G( xaϵ1

x−b )

G(b+aϵ1)

[G(y)]n−1−m−i

(1−G(y))M−n

dG(y)dx.

Again applying the binomial expansion, the above probability

can be further expanded as follows:

P

(b < |hm|2 < b+ aϵ1, b+ aϵ1 < |hn|2 <


)=

n−1−m∑i=0

ϖ1

(n− 1−m

i

)(−1)i

∫ b+aϵ1

b

g(x)[G(x)]m−1+i

×M−n∑j=0

(M − n

j

)(−1)j

∫ G( xaϵ1x−b )

G(b+aϵ1)

[G(y)]n−1−m−i+j dG(y)dx

=

n−1−m∑i=0

ϖ1

(n− 1−m

i

)(−1)i

∫ b+aϵ1

b

g(x)[G(x)]m−1+i

×M−n∑j=0

(M − n

j

)(−1)j

n− 1−m− i+ j(54)

×

([G

(xaϵ1x− b

)]n−m−i+j

− [G(b+ aϵ1)]n−m−i+j

)dx.

Compared to (51), the above equation is more complicated;however, this expression is more suitable for asymptotic stud-ies, as explained in the following.

Recall that the integral range in (54) is b < x < b + aϵ1.When ρ → 0, we have b → 0 and b + aϵ1 → 0, whichimplies x → 0. Therefore the following approximation can beobtained:

P

(b < |hm|2 < b+ aϵ1, b+ aϵ1 < |hn|2 <


)(55)

≈n−1−m∑

i=0

ϖ1

(n− 1−m

i

)(−1)i

×M−n∑j=0

(M − n

j

)(−1)j

n− 1−m− i+ j

∫ b+aϵ1

b

xm−1+i

×

([G

(xaϵ1x− b

)]n−m−i+j

− [b+ aϵ1]n−m−i+j

)dx.

First focus on the following integral which is from the aboveequation:∫ b+aϵ1

b

xm−1+i

[G

(xaϵ1x− b

)]n−m−i+j

dx (56)

≈∫ aϵ1

0

(b+ z)m−1+i[1− e−

abϵ1z

]n−m−i+j

dz.

We can find the following bounds for the above integral:∫ b+aϵ1

b

xm−1+idx (57)

≥∫ b+aϵ1

b

xm−1+i

[G

(xaϵ1x− b

)]n−m−i+j

dx

≥∫ aϵ1

0

(b+ z)m−1+i

[1− 1

1 + abϵ1z

]n−m−i+j

dz,



13

where the lower bound is obtained due to the inequality

e−abϵ1

z ≤ 1

1 + abϵ1z

,

when 0 ≤ z ≤ aϵ1. The upper bound in (57) can beapproximated at high SNR as follows:∫ b+aϵ1

b

xm−1+idx (58)

=(b+ aϵ1)

m+i − bm+i

m+ i→ 1

ρm+i.

On the other hand, the lower bound in (57) can be approxi-mated as follows:∫ aϵ1

0

(b+ z)m−1+i

[1− 1

1 + abϵ1z

]n−m−i+j

dz (59)

= (abϵ1)n−m−i+j

∫ aϵ1+abϵ1

abϵ1

(w + b− abϵ1)m−1+i

wn−m−i+jdz

= (abϵ1)n−m−i+j

m−1+i∑k=0

(b− abϵ1)m−1+i−k

×∫ aϵ1+abϵ1

abϵ1

wk−(n−m−i+j)dz ,m−1+i∑k=0

ξk.

At high SNR, we can show that

ξk →{

ρ−(n+j) ln ρ, for k + 1 = n−m− i+ jρ−(n+j), otherwise

. (60)

Since log log ρlog ρ → 0 for ρ → ∞, the lower bound in (57) can

be approximated as follows:∫ aϵ1

0

(b+ z)m−1+i

[abϵ1

z + abϵ1

]n−m−i+j

dz → ρ−(n+j). (61)

Based on the upper and lower bounds in (58) and (61)and after some algebraic manipulation, we have the followinginequality:

ρ−n≤̇P

(|hn|2 > b+ aϵ1, b < |hm|2 <

b

1− aϵ1|hn|2

)≤̇ρ−m, (62)

where a≤̇b denotes(− log a

log ρ

)≤(− log b

log ρ

)when ρ → ∞ [17].

Combining (52), (53) and (62), we can obtain the followingasymptotic bounds:

ρ−n≤̇Q2≤̇ρ−m. (63)

Following similar steps as above, we can also find thatQ3

.= ρ−m, which is dominant in Pn

o , and the proof for thesecond part of the theorem is completed. �

REFERENCES

[1] Q. Li, H. Niu, A. Papathanassiou, and G. Wu, “5G network capacity:Key elements and technologies,” IEEE Veh. Technol. Mag., vol. 9, no. 1,pp. 71–78, Mar. 2014.

[2] “5G: A technology vision,” Huawei Technologies Co., Ltd., Shenzhen,China, Whitepaper Nov. 2013.

[3] Y. Saito, A. Benjebbour, Y. Kishiyama, and T. Nakamura, “System levelperformance evaluation of downlink non-orthogonal multiple access(NOMA),” in Proc. IEEE Annual Symposium on Personal, Indoor andMobile Radio Communications (PIMRC), London, UK, Sept. 2013.

[4] M. Al-Imari, P. Xiao, M. A. Imran, and R. Tafazolli, “Uplink non-orthogonal multiple access for 5G wireless networks,” in Proc. 11th In-ternational Symposium on Wireless Communications Systems (ISWCS),Barcelona, Spain, Aug 2014, pp. 781–785.

[5] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, “On the performance ofnon-orthogonal multiple access in 5G systems with randomly deployedusers,” IEEE Signal Process. Letters, vol. 21, no. 12, pp. 1501–1505,Dec 2014.

[6] J. Choi, “Non-orthogonal multiple access in downlink coordinated two-point systems,” IEEE Commun. Letters, vol. 18, no. 2, pp. 313–316,Feb. 2014.

[7] T. Cover and J. Thomas, Elements of Information Theory, 6th ed. Wileyand Sons, New York, 1991.

[8] S. Timotheou and I. Krikidis, “Fairness for non-orthogonal multipleaccess in 5G systems,” IEEE Signal Process. Letters, vol. 22, no. 10,pp. 1647–1651, Oct. 2015.

[9] Z. Ding and H. V. Poor, “Cooperative non-orthogonal multiple accessin 5G systems,” IEEE Commun. Letters, (to appear in 2015) Availableon-line at arXiv:1410.5846.

[10] J. Choi, “Minimum power multicast beamforming with superpositioncoding for multiresolution broadcast and application to NOMA sys-tems,” IEEE Trans. Commun., vol. 63, no. 3, pp. 791–800, Mar. 2015.

[11] Q. Sun, S. Han, C.-L. I, and Z. Pan, “On the ergodic capacity of MIMONOMA systems,” Wireless Communications Letters, IEEE, to appear in2015.

[12] H. A. David and H. N. Nagaraja, Order Statistics. John Wiley, NewYork, 3rd ed., 2003.

[13] K. Subranhmaniam, “On some applications of Mellin transforms tostatistics: Dependent random variables,” SIAM Journal on AppliedMathematics, vol. 19, no. 4, pp. 658–662, Dec. 1970.

[14] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, “Breakingspectrum gridlock with cognitive radios: An information theoreticperspective,” Proceedings of the IEEE, vol. 97, no. 5, pp. 894–914,May 2009.

[15] G. Zheng, S. Ma, K.-K. Wong, and T.-S. Ng, “Robust beamformingin cognitive radio,” IEEE Trans. Wireless Commun., vol. 9, no. 2, pp.570–576, Feb. 2010.

[16] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series andProducts, 6th ed. New York: Academic Press, 2000.

[17] L. Zheng and D. N. C. Tse, “Diversity and multiplexing : A fundamentaltradeoff in multiple antenna channels,” IEEE Trans. Inform. Theory,vol. 49, pp. 1073–1096, May 2003.



14

Zhiguo Ding (S’03-M’05) received his B.Eng inElectrical Engineering from the Beijing Universityof Posts and Telecommunications in 2000, and thePh.D degree in Electrical Engineering from ImperialCollege London in 2005. From Jul. 2005 to Aug.2014, he was working in Queen’s University Belfast,Imperial College and Newcastle University. SinceSept. 2014, he has been with Lancaster Universityas a Chair Professor. From Oct. 2012 to Sept. 2016,he has been also with Princeton University as anAcademic Visitor.

Dr Ding’ research interests are 5G networks, game theory, cooperativeand energy harvesting networks and statistical signal processing. He isserving as an Editor for IEEE Transactions on Communications, IEEETransactions on Vehicular Networks, IEEE Wireless Communication Letters,IEEE Communication Letters, and Journal of Wireless Communications andMobile Computing. He received the best paper award in IET Comm. Conf.on Wireless, Mobile and Computing, 2009, IEEE Communication LetterExemplary Reviewer 2012, and the EU Marie Curie Fellowship 2012-2014.

Pingzhi Fan (M93SM99-F15) received his PhDdegree in Electronic Engineering from the Hull Uni-versity, UK. He is currently a professor and directorof the institute of mobile communications, SouthwestJiaotong University, China. He is a recipient of theUK ORS Award, the Outstanding Young ScientistAward by NSFC, and the chief scientist of a na-tional 973 research project. He served as generalchair or TPC chair of a number of internationalconferences, and is the guest editor-in-chief, guesteditor or editorial member of several international

journals. He is the founding chair of IEEE VTS BJ Chapter and ComSocCD Chapter, the founding chair of IEEE Chengdu Section. He also servedas a board member of IEEE Region 10, IET(IEE) Council and IET Asia-Pacific Region. He has over 200 research papers published in various academicEnglish journals (IEEE/IEE/IEICE, etc), and 8 books (incl. edited), and is theinventor of 20 granted patents. His research interests include high mobilitywireless communications, 5G technologies, wireless networks for big data,signal design & coding, etc.

H. Vincent Poor (S’72, M’77, SM’82, F’87) re-ceived the Ph.D. degree in EECS from PrincetonUniversity in 1977. From 1977 until 1990, he wason the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the fac-ulty at Princeton, where he is the Michael HenryStrater University Professor and Dean of the Schoolof Engineering and Applied Science. He has alsoheld visiting appointments at several universities,including most recently at Stanford and ImperialCollege. Dr. Poor’s research interests are in the area

of wireless networks and related fields. Among his publications in these areasis the recent book Mechanisms and Games for Dynamic Spectrum Allocation(Cambridge University Press, 2014).

Dr. Poor is a member of the U. S. National Academy of Engineering and theU. S. National Academy of Sciences, and is a foreign member of AcademiaEuropaea and the Royal Society. He is also a fellow of the American Academyof Arts and Sciences, the Royal Academy of Engineering (U. K), and theRoyal Society of Edinburgh. He received the Marconi and Armstrong Awardsof the IEEE Communications Society in 2007 and 2009, respectively. Recentrecognition of his work includes the 2014 URSI Booker Gold Medal, the 2015EURASIP Athanasios Papoulis Award and honorary doctorates from AalborgUniversity, Aalto University, HKUST and the University of Edinburgh.

Documents

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