spatial modulation-aided cooperative noma: performance ... · performance analysis and comparative...

IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 13, NO. 3, JUNE 2019 715

Spatial Modulation-Aided Cooperative NOMA:Performance Analysis and Comparative Study

Qiang Li , Miaowen Wen , Senior Member, IEEE, Ertugrul Basar , Senior Member, IEEE,H. Vincent Poor , Fellow, IEEE, and Fangjiong Chen , Member, IEEE

Abstract—The combination of non-orthogonal multiple access(NOMA) and multiple-input multiple-output (MIMO) signaling iscapable of providing significantly improved spectral efficiency andeffective multiple access. However, conventional MIMO-NOMAentails significantly increased power consumption and implemen-tation complexity. Inspired by the essence of spatial modulation(SM), in this paper, a novel three-node cooperative relaying system(CRS) using SM-aided NOMA, termed CRS-SM-NOMA, is pro-posed, in which the messages for two users are conveyed by two dif-ferent information-bearing units of SM. As the reference scheme,conventional CRS-NOMA is further re-investigated with multiplereceive antennas. Besides the achievable rate, which is adoptedby most of the existing NOMA works as the performance metric,the bit error rate (BER) of CRS-SM-NOMA and CRS-NOMA isanalyzed assuming maximum-likelihood detection. Approximatebit error probability expressions are derived in closed form forboth users. Simulation results verify the analysis and show thatthe proposed CRS-SM-NOMA outperforms conventional CRS-NOMA and SM-OMA in terms of the BER as well as ergodicsum rate.

Index Terms—Bit error rate, cooperative relaying, non-orthogonal multiple access, spatial modulation, sum rate.

I. INTRODUCTION

W ITH the dramatic growth of mobile data traffic andrequired connectivity among very large numbers of

Manuscript received September 7, 2018; revised December 24, 2018; ac-cepted February 1, 2019. Date of publication February 7, 2019; date of currentversion May 22, 2019. This work was supported in part by the National Nat-ural Science Foundation of China under Grant 61871190, Grant 61671211,and Grant U1701265, in part by the Natural Science Foundation of Guang-dong Province under Grant 2018B030306005, Grant 2016A030311024, andGrant 2016A030308006, in part by the U.S. National Science Foundation underGrants CCF-093970 and CCF-1513915, in part by the Turkish Academy ofSciences (TUBA) GEBIP Programme, in part by the State Key Laboratory ofIntegrated Service Networks of Xidian University under Grant ISN20-16, andin part by the China Scholarship Council under Grant 201806150074. The guesteditor coordinating the review of this paper and approving it for publication wasDr. Yuanwei Liu. (Corresponding author: Miaowen Wen.)

Q. Li and M. Wen are with the School of Electronic and Information En-gineering, South China University of Technology, Guangzhou 510640, China,and also with the State Key Laboratory of Integrated Service Networks, Xi-dian University, Xi’an 710071, China (e-mail:, [email protected];[email protected]).

F. Chen is with the School of Electronic and Information Engineering,South China University of Technology, Guangzhou 510640, China (e-mail:,[email protected]).

E. Basar is with the Communications Research and Innovation Laboratory,Department of Electrical and Electronics Engineering, Koc University, Sariyer34450, Turkey (e-mail:,[email protected]).

H. V. Poor is with the Department of Electrical Engineering, Princeton Uni-versity, Princeton, NJ 08544 USA (e-mail:,[email protected]).

Digital Object Identifier 10.1109/JSTSP.2019.2898099

devices, wireless resources are becoming increasingly scarce.Moreover, in conventional orthogonal multiple access (OMA)systems, each user should occupy a distinct time/frequencychannel, which limits the spectral efficiency (SE) of the sys-tem. To improve the SE, the concept of non-orthogonal multipleaccess (NOMA) has been proposed, in which multiple usershaving different channel conditions are served by one base sta-tion (BS) through the same resource block [1]. The users inNOMA are typically multiplexed in the power domain by usingsuperposition coding at the transmitter and successive interfer-ence cancellation (SIC) at the receiver [2]. It has been shownthat compared with OMA techniques, NOMA achieves largersystem throughput, lower latency, and greater fairness [3]–[5].Due to these advantages, NOMA has been widely recognizedas one of the key enabling technologies to meet the stringentrequirements in fifth generation (5G) wireless networks [6], [7],and various aspects of NOMA have been extensively studied inthe past few years.

Since near users with better channel conditions need to decodenot only their own signals but also the signals intended for farusers with poorer channel conditions in NOMA systems, theycan act as relays to help far users, thus forming the paradigm ofcooperative NOMA [8]–[15]. Specifically, in [8], a cooperativeNOMA system with one BS and an arbitrary number of users isproposed, in which all users can achieve the maximum diversitygain. The study of [9] applies the technique of simultaneouswireless information and power transfer to NOMA networks, inwhich a near user serves as an energy harvesting node as well asan information relay. In [10] and [11], NOMA is introduced intoa three-node cooperative relaying system (CRS) that comprises asource, a relay, and a destination to enhance the SE. The outageprobability and ergodic sum rate of a NOMA-based relayingnetwork over Nakagami-m fading channels are investigated in[12] and [13], where the BS communicates with multiple cell-edge mobile users simultaneously with the help of a cell-centerrelay. Unlike the studies in [8]–[13] where relays operate inthe half-duplex mode, [14] and [15] analyze downlink NOMAsystems with cooperative full-duplex relaying. Besides one-wayrelaying, NOMA-based two-way relay networks are discussedin [16] and [17].

On the other hand, multiple-input multiple-output (MIMO)in the form of spatial multiplexing is another important SEenhancing technology. It has been shown that at high signal-to-noise ratios (SNRs), the capacity of MIMO systems increaseslinearly with the minimum number of transmit and receive

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716 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 13, NO. 3, JUNE 2019

antennas. Hence, NOMA has been studied in multi-antennascenarios. In [18], multi-antenna NOMA is categorized intobeamformer-based and cluster-based structures, depending onwhether one beamformer serves one user or multiple users.For the beamformer-based structure, [19] proposes two novelcoordinated beamforming techniques for multi-cell MIMO-NOMA, in which beamforming vectors of two base stationsare jointly optimized to enhance the performance of cell-edgeusers. For the cluster-based structure, [20] presents a framework,in which users are randomly grouped into several clusters, andprecoding and detection matrices are carefully designed to can-cel the inter-cluster interference. However, due to zero-forcingdetection, the number of antennas at each user is required to begreater than that at the BS, which limits the application of thisframework to massive MIMO-NOMA [21]. To remove this con-straint, a novel cluster-based MIMO-NOMA design is proposedin [22] by allowing the existence of inter-cluster interference.In the above-mentioned MIMO-NOMA systems, however, thepower and spatial domains should be jointly exploited to elimi-nate the user interference via beamforming, which complicatesthe beamforming design and SIC [23]. Besides, the require-ment of multiple radio-frequency (RF) chains at the transmitterfurther lowers the energy efficiency and increases the implemen-tation complexity. Recently, spatial modulation (SM), which ac-tivates only a single transmit antenna for each transmission, hasbeen regarded as a competitive multi-antenna technique for 5Gand beyond networks [24]–[26]. SM completely avoids inter-antenna interference and the requirement of multiple RF chains.In SM systems, the information is conveyed by the index of anactive antenna as well as an M -ary modulated symbol. As aspecial realization of SM, space shift keying (SSK) solely uti-lizes the index of an active antenna to carry information [27].Applying SM to NOMA systems is capable of improving theSE without increasing the power consumption and implemen-tation complexity. Moreover, in current multi-user SM systems,the frequently used space division multiple access (SDMA) suf-fers from severe inter-user interference. Recalling that NOMAcan effectively mitigate inter-user interference, it appears to bea promising multiple access method for SM-based multiusercommunications. Hence, the combination of SM and NOMAhas been promoted by the attempt to exploit the potential ben-efits of both SM and NOMA [28]–[31]. In [28] and [29], thebit stream for each user is split into two parts: the first partis used to activate a specific antenna and the second part ismapped to an M -ary modulated symbol. This, however, ne-cessitates the use of multiple RF chains, which contradicts theinherent principle of SM. In order to keep a single transmit an-tenna active, in [30], the active antenna is selected according tothe data for one specific user, and then the superposition codedsymbol of all users is emitted by that antenna. It is worth not-ing that, in [28]–[30], superposition coding and SIC are stillneeded, thus imposing challenges on practical implementationsof SM-NOMA. The study of [31] investigates a different pro-totype of downlink SM-NOMA, which avoids the utilization ofSIC at the receiver. In this setup, the antennas at the BS arepartitioned into several groups, each of which serves a specificuser pair through mapping two independent bit streams to differ-ent information-bearing units. However, [31] provides relatively

limited computer simulation results and performance analysis,and it fails to investigate the potential of cooperative relaying inSM-NOMA systems.

Against the background, not only to overcome the disadvan-tages of the existing techniques but also to enhance the systemperformance further through joint SM and cooperative relaying,we propose a novel CRS for Rayleigh flat-fading channels us-ing SM-NOMA in this paper, termed CRS-SM-NOMA, whichconsists of one BS and two users. Two information-bearingunits of SM are utilized to convey the messages for two users,respectively.1 The user closer to the BS also acts as a relay to im-prove the performance of the other user. All nodes are equippedwith multiple transmit/receive antennas. The main contributionsof this paper are summarized as follows:

� The performance of CRS-SM-NOMA is theoretically an-alyzed in terms of bit error rate (BER) and achievablerate. Approximate closed-form bit error probability (BEP)expressions are derived for both users employing M -arypulse amplitude modulation (PAM)/quadrature amplitudemodulation (QAM). The asymptotic achievable rate ofCRS-SM-NOMA is analyzed and compared with that ofCRS-NOMA.

� Conventional CRS-NOMA [11] is re-investigated as thereference scheme. Different from the case in [11], multi-ple receive antennas are assumed for each user. Moreover,besides the sum rate, general BEP expressions are derivedin closed form for both users, where the effects of superpo-sition coding and SIC are properly taken into account. Tothe best of our knowledge, only [32] and [33] try to analyt-ically evaluate the BER performance of NOMA systemsin the literature. However, all derived results are restrictedto some specific constellation combinations and additivewhite Gaussian noise (AWGN) channels. From this per-spective, this work also provides a comprehensive back-ground on the theoretical BEP analysis of NOMA-basedsystems.

� Extensive computer simulations are performed to validatethe theoretical analysis and compare the performance ofCRS-SM-NOMA, CRS-NOMA, and SM-OMA. Simula-tion results accurately agree with the analysis and showthat CRS-SM-NOMA outperforms CRS-NOMA and SM-OMA in terms of BER and ergodic sum rate.

The remainder of this paper is organized in the following. Thesystem model of CRS-SM-NOMA is presented in Section II,followed by the performance analysis in Section III. Section IVdescribes the reference scheme, namely CRS-NOMA, and eval-uates its BER and achievable rate performance. Computer sim-ulation results are given in Section V, and finally Section VIconcludes the paper.

Notation: Scalar variables are in italic letters. Column vectorsand matrices are denoted by lowercase and uppercase boldfaceletters, respectively. Superscripts T and H stand for transposeand Hermitian transpose, respectively. hij and hi represent the

1CRS-SM-NOMA still belongs to NOMA family in the sense that two userssimultaneously occupy the same time-and-frequency resources. Further, due tothe advantages of SM, the messages for one user are implicitly embedded in thesignal for the other user, getting rid of the requirement of power allocation andSIC.

LI et al.: SPATIAL MODULATION-AIDED COOPERATIVE NOMA: PERFORMANCE ANALYSIS AND COMPARATIVE STUDY 717

Fig. 1. System model of CRS-SM-NOMA.

(i, j)-th element and the i-th column of H, respectively. Then × n identity matrix is symbolized by In . The probability ofan event and the probability density function (PDF) are denotedby Pr(·) and p(·), respectively. CN (μ, σ2) denotes the com-plex Gaussian distribution with mean μ and variance σ2 . Q(·),Γ(·), and E[·] denote the Gaussian Q-function, gamma func-tion, and expectation operation, respectively.

(··)

and ‖·‖ standfor the binomial coefficient and Frobenius norm, respectively.!! is the double factorial notation denoting the product of onlyodd integers. �·� denotes the floor function. mod(x, y) returnsthe remainder after division of x by y.

II. SYSTEM MODEL

The system model of CRS-SM-NOMA is given in Fig. 1,in which a BS (S) serves two users (A and B) simultaneously.All nodes operate in the half-duplex mode. Both the BS anduser A have Nt transmit antennas, while users A and B areequipped with NA

r and NBr receive antennas, respectively. There

is no constraint on Nt and NAr . The channel matrices of S-

to-A, S-to-B, and A-to-B links are denoted by H ∈ CN Ar ×Nt ,

G ∈ CN Br ×Nt , and F ∈ CN B

r ×Nt , respectively. The entries ofH, G, and F are modeled as independent and identically dis-tributed (i.i.d) complex Gaussian random variables followingthe distributions CN (0, λSA ), CN (0, λSB ), and CN (0, λAB ),respectively. Provided that A is the near user and B is the faruser, since the variance of channel coefficients is typically in-versely proportional to the propagation distance [4], we haveλSA > λSB .

In CRS-SM-NOMA, there are two phases involved in eachtransmission of b1 + b2 bits to users A and B, where b1 (b2) bitsare dedicated for user A (B). During the first time slot, the BSgenerates an SM symbol

x = [0, . . . , 0,︸︷︷︸

l−1

s, 0, . . . , 0︸︷︷︸

Nt −l

]T , (1)

and broadcasts it to users A and B, where s is an M -aryPAM/QAM symbol determined by the b1 bits with E[|s|2 ] = 1,and l ∈ {1, . . . , Nt} denotes the active antenna index deter-mined by the b2 bits. Obviously, we have b1 = log2(M) and

b2 = log2(Nt).2 By normalizing the transmit power to unity,the received signals at users A and B can be expressed as

ySA = Hx + nSA (2)

and

ySB = Gx + nSB , (3)

respectively, where nSA ∈ CN Ar ×1 and nSB ∈ CN B

r ×1 are theAWGN vectors at users A and B, which follow the distributionsCN (0, N0IN A

r) and CN (0, N0IN B

r), respectively. We define

the transmit SNR as ρ = 1/N0 . User B preserves the receivedsignal ySB for decoding purposes at the second phase, whileuser A adopts the maximum-likelihood (ML) detection to jointlydecode s and l, namely

(s, l) = arg mins,l

∥∥ySA − Hx

∥∥2

= arg mins,l

∥∥ySA − hls

∥∥2

, (4)

where s and l are the estimates of s and l at user A, respec-tively. User A then recovers its own information bits from s andpreserves l for cooperation.

At the second phase, user A forwards l to B via an SSK signalas

yAB = fl + nAB , (5)

where nAB ∈ CN Br ×1 is the AWGN vector at user B distributed

according to CN (0, N0IN Br

). By combining yAB in (5) andySB in (3), user B obtains its own message from

l = arg minl

{∥∥yAB − fl

∥∥2

+ mins

∥∥ySB − gls

∥∥2}

, (6)

where l is the estimate of l at user B.

III. PERFORMANCE ANALYSIS

In this section, we theoretically analyze the performance ofthe proposed CRS-SM-NOMA scheme in terms of BER andsum rate.

A. BEP Analysis

In this subsection, approximate BEP expressions are derivedin closed form for users A and B in CRS-SM-NOMA systems.

1) BEP of User A: User A extracts s from the received signalySA according to the ML detection in (4). Obviously, the errorevents for user A can be classified into two complementarytypes, depending on whether the index of the active antenna isdetected correctly or not. Thus, the overall BEP of user A canbe evaluated as

PA ≈ 12PIA + (1 − PIA ) PAb, (7)

where PIA is the error probability of the active antenna indexdetection at user A, PAb is the BEP of M -ary PAM/QAM de-modulation over NA

r i.i.d. Rayleigh fading channels, and thefactor 1/2 results from the fact that there is no symbol transmit-ted from inactive antennas.

2Nt in this paper is assumed to be an arbitrary integer power of two. Hence, lcan be easily obtained by converting the b2 bits into the decimal representation.


We first study the conditional pairwise error probability (PEP)from (4), which is the probability of detecting x as x conditionedon H, s, and s, namely

Pr (x → x |H, s, s ) = Pr(∥∥ySA − hls

∥∥2

>∥∥ySA − hl s

∥∥2)

= Q

(√κ

2N0

), (8)

where κ =∥∥hls − hl s

∥∥2 =

∑2N Ar

i=1 α2i with αi ∼ CN (0, λSA

(|s|2 + |s|2)/2). Hence, κ is chi-squared distributed with 2NAr

degrees of freedom, whose PDF is given by

pκ (x) =1

2N Ar Γ (NA

r ) σ2N Ar

xN Ar −1 exp

(− x

2σ2

), (9)

where σ2 = λSA (|s|2 + |s|2)/2. By averaging Pr(x →x|H, s, s) over κ, we arrive at [34, Eq. (64)]

Pr (x → x |s, s ) =∫ +∞

0Q

(√x

2N0

)pκ (x) dx

= [P (c)]NAr

N Ar −1∑

k=0

(NA

r − 1 + k

k

)[1 − P (c)]k , (10)

where

P (c) =12

(1 −√

c

1 + c

)(11)

with c = σ2/(2N0). According to the union bounding tech-nique, an upper bound on the error probability of active antennaindex detection at user A is given by

PIA ≤ 1NtM

∑

l,l

l �= l

∑

s,s

Pr (x → x |s, s )

=Nt − 1

M

∑

s,s

Pr (x → x |s, s ). (12)

For the calculation of PAb , let us first consider the M -PAMconstellation. The conditional error probability for the m-th bit

(m = 1, . . . , log2(M)) can be formulated as [35]

PAb

(m∣∣∣{γSA

k

}N Ar

k=1

)=

2M

(1−2−m )M −1∑

i=0

{

(−1)

⌊i ·2 m −1

M

⌋

×(

2m−1 −⌊

i · 2m−1

M+

12

⌋)

× Q

((2i + 1) d

√2γSA

tot

)}

,

(13)

where d =√

3/(M 2 − 1) is the half of the minimum distancebetween two signal points of the normalized M -PAM constel-

lation, and γSAtot =

∑N Ar

k=1 γSAk is the total instantaneous receive

SNR with γSAk = |hkl |2/N0 . To facilitate the calculation of the

average error probability, i.e., PAb (m), the Gaussian Q-functionin (13) is replaced with the alternative representation [34]

Q (x) =1π

∫ π/2

0exp(− x2

2sin2φ

)dφ, (14)

which leads to (15), shown at the bottom of this page. By statis-

tically averaging (15) over the joint PDF of {γSAk }N A

r

k=1 which isgiven by

p(γSA

1 , . . . , γSAN A

r

)=

N Ar∏

k=1

pγ S Ak

(γSA

k

)

=N A

r∏

k=1

1γSA

exp(−γSA

k

γSA

)(16)

with γSA = λSA/N0 , we have [36, Eq. (5A.4b)]

PAb (m) =2M

(1−2−m )M −1∑

i=0

{

(−1)

⌊i ·2 m −1

M

⌋

×(

2m−1 −⌊

i · 2m−1

M+

12

⌋)· T (i)

}, (17)

PAb

(m∣∣∣{γSA

k

}N Ar

k=1

)=

2M

(1−2−m )M −1∑

i=0

⎧⎨

⎩(−1)

⌊i ·2 m −1

M

⌋ (2m−1 −

⌊i · 2m−1

M+

12

⌋)

× 1π

∫ π/2

0

N Ar∏

k=1

exp

(

− ((2i + 1) d)2γSAk

sin2φ

)

dφ

⎫⎬

⎭. (15)

T (i) =1π

∫ π/2

0

[∫ ∞

0exp

(

− ((2i + 1) d)2γSAk

sin2φ

)

pγ S Ak

(γSA

k

)dγSA

k

]N Ar

dφ

=1π

∫ π/2

0

[sin2φ

((2i + 1) d)2 γSA + sin2φ

]N Ar

dφ

=(

1 − μc(i)2

)N Ar

N Ar −1∑

k=0

(NA

r − 1 + k

k

)(1 + μc(i)

2

)k

(18)


where T (i) is shown at the bottom of the previous page as (18)with

μc(i) =

√λSA ((2i + 1) d)2

N0 + λSA ((2i + 1) d)2 . (19)

Finally, a closed-form BEP expression for M -PAM over NAr

i.i.d. Rayleigh fading channels can be obtained by averaging(17) over all constellation points, yielding

PAb =1

log2(M)

log2 (M )∑

m=1

PAb (m). (20)

For the M -QAM constellation with M = MI MQ , since M -ary QAM consists of two independent PAM signals, namelyMI -ary PAM of the in-phase (I-) signal and MQ -ary PAM of thequadrature (Q-) signal, the derivation of PAb for M -ary QAM issimilar to that for M -ary PAM. Particularly, the probabilities thatthe mI -th bit (mI = 1, . . . , log2(MI )) in the I- component andthe mQ -th bit (mQ = 1, . . . , log2(MQ )) in the Q- componentare in error, denoted by PAb−I (mI ) and PAb−Q (mQ ), can beformulated as (17) except that M and m should be replaced withMI (MQ ) and mI (mQ ), respectively, for the I (Q)- dimension.Note that the parameter d in μc(i) for M -QAM should also berevised as d =

√3/(M 2

I +M 2Q −2). Finally, we are led to PAb for

M -QAM, namely

PAb =1

log2(M)

⎛

⎝log2 (MI )∑

mI =1

PAb−I (mI )

+log2 (MQ )∑

mQ =1

PAb−Q (mQ )

⎞

⎠ . (21)

After obtaining PIA and PAb , the overall BEP expressionfor user A can be derived by substituting (12) and (20) or (21)into (7). We observe from (10) and (18) that both PIA and PAb

achieve a diversity order of NAr , such that the overall diversity

order of user A is also given by NAr .

2) BEP of User B: In the active antenna index detection, anerror happens once the index of the active antenna is detectedincorrectly as one of the remaining Nt − 1 indices. A differentincorrect index may result in a different number of erroneousbits, and we observe that there are in total of

(b2t

)error events

that correspond to t erroneous index bits, where t ∈ {1, . . . , b2}.Thus, on average, the erroneous detection of the index of theactive antenna produces

ne =1

Nt − 1

b2∑

t=1

t ·(

b2

t

)(22)

erroneous bits out of b2 bits. Recalling that the bit errors of userB result from two complementary conditions: (a) user A detectsl correctly; (b) user A detects l incorrectly, we can obtain theapproximate BEP expression for user B as

PB ≈ ((1 − PIA ) PIB + PIA ) · ne

b2, (23)

where PIB is the error probability of active antenna index de-tection at user B on condition that user A correctly detects thatindex. Note that in (23), we assume that an error in active antennaindex detection at user A always causes an error at user B, whichis a common behavior for decode-and-forward relaying-basedsystems.

To calculate PB in (23), we turn to calculate PIB since PIA

has been given in (12). Let us consider (6) to derive the condi-tional PEP, which becomes

Pr (x → x |F,G, s, s ) = Pr(∥∥yAB − fl

∥∥2

+∥∥ySB − gls

∥∥2

>∥∥yAB − fl

∥∥2

+∥∥ySB − gl s

∥∥2)

= Q

(√η

2N0

), (24)

where η =∥∥fl − fl

∥∥2 +

∥∥gls − gl s

∥∥2

. We observe that η is thesum of two independent central chi-square random variablesboth of which are with 2NB

r degrees of freedom. Hence, themoment generating function (MGF) of η is given by [37]

Mη (ω) =[

1(1 − 2ωσ2

1 ) (1 − 2ωσ22 )

]N Br

, (25)

where σ21 = λAB and σ2

2 = λSB (|s|2 + |s|2)/2. Resortingto the approximation of Q-function, namely Q(x) ≈ 1/12 ·e−x2 /2 + 1/4 · e−2x2 /3 , and the MGF approach, we have

Pr (x → x |s, s ) ≈ 112

⎡

⎣ 1(1 + σ 2

12N0

)(1 + σ 2

22N0

)

⎤

⎦

N Br

+14

⎡

⎣ 1(1 + 2σ 2

13N0

)(1 + 2σ 2

23N0

)

⎤

⎦

N Br

.

(26)

Therefore, the erroneous active antenna index detection proba-bility at user B in case (a) can be expressed as

PIB ≈ 1NtM

∑

l,l

l �= l

∑

s,s

Pr (x → x |s, s )

=Nt − 1

M

∑

s,s

Pr (x → x |s, s ). (27)

Finally, we obtain the BEP of user B by substituting (12), (22),and (27) into (23). Since the diversity order of PIB in (26) is2NB

r , user B achieves a diversity order of min{2NBr ,NA

r } from(23).

B. Sum Rate Analysis Under Gaussian Input

In this subsection, we analyze the achievable sum rate ofCRS-SM-NOMA systems with Gaussian input.

1) Achievable Rate of User A: Since the information of userA is only conveyed by classical amplitude phase modulation(APM) symbols, the instantaneous achievable rate of user A


can be easily given by [38]

CA =12I(X;Y SA |Xch

)=

12Nt

Nt∑

l=1

log2

(1 + ρ‖hl‖2

),

(28)

where X and Xch denote APM and antenna symbol spaces,respectively, Y SA is the output symbol space, and 1/2 is due totwo-phase transmission.

2) Achievable Rate of User B: Since the index of the activeantenna should be detected at user A as well as user B, theachievable rate associated with user B is obtained as

CB =12

min{I(Xch ;Y SA

), I(Xch ;Y SB , Y AB

)}. (29)

Next, we deal with the calculations of I(Xch ;Y SA

)and

I(Xch ;Y SB , Y AB

). Specifically,

I(Xch ;Y SA

)= H (Xch) − H

(Xch

∣∣Y SA

)

= log2 (Nt) +1Nt

Nt∑

l=1

∫p(ySA |Xch= l

)

× log2(p(Xch= l

∣∣ySA

))dySA

= log2 (Nt) +1Nt

Nt∑

l=1

∫p(ySA |Xch= l

)

× log2

(p(ySA |Xch= l

)

Nt · p (ySA )

)

dySA , (30)

where

p(ySA |Xch= l

)=

1πN A

r det (Σl)exp{−(ySA

)HΣ−1

l ySA}

(31)

and

p(ySA

)=

1Nt

Nt∑

l=1

p(ySA |Xch= l

)(32)

with Σl = N0IN Ar

+ hlhHl . Similarly, we have

I(Xch ;Y SB , Y AB

)= H (Xch) − H

(Xch

∣∣Y SB , Y AB

)

= log2 (Nt) +1Nt

Nt∑

i= l

∫p(ySB ,yAB |Xch= l

)

× log2(p(Xch= l

∣∣ySB ,yAB

))dySB dyAB

= log2 (Nt) +1Nt

Nt∑

l=1

∫p(ySB ,yAB |Xch= l

)

× log2

(p(ySB ,yAB |Xch= l

)

Nt · p (ySB ,yAB )

)

dySB dyAB , (33)

where

p(ySB ,yAB |Xch= l

)=

1π2N B

r det (Ξl) det (Ωl)

× exp{−(ySB

)HΞ−1

l ySB − (yAB)H

Ω−1l yAB

}(34)

and

p(ySB ,yRB

)=

1Nt

Nt∑

l=1

p(ySB ,yAB |Xch= l

)(35)

with Ξl = N0IN Br

+ glgHl and Ωl = N0IN B

r+ flfH

l . Unfortu-nately, both (30) and (33) cannot be exactly calculated in closedform. For this reason, we employ numerical evaluation to es-timate them. Finally, the instantaneous achievable sum rate ofCRS-SM-NOMA systems can be readily obtained as

CCRS-SM-NOMA = CA + CB . (36)

At extremely high SNRs, since the index of the active antennais likely to be detected correctly, we have

CB ≈ 12log2 (Nt) , (37)

and

CCRS-SM-NOMA≈ 12Nt

Nt∑

l=1

log2

(ρ‖hl‖2

)+

12log2 (Nt) . (38)

IV. ON CONVENTIONAL CRS-NOMA

In this section, we take the conventional CRS-NOMA [11] asthe reference scheme. The considered system model is the sameas that of [11] except that users A and B are equipped with mul-tiple receive antennas. Besides the sum rate performance, weinvestigate the BER performance of CRS-NOMA and deriveclosed-form BEP expressions for both users. Without incurringconfusions, we use the same notations for received signals, chan-nel matrices and noise vectors as those in Section II. Note thatsince each node has one transmit antenna and multiple receiveantennas, channel matrices are degenerated into column vectors.For fair comparisons, channel coefficients and noise vectors fol-low the same distributions as those in CRS-SM-NOMA.

At the first phase in CRS-NOMA, the BS transmits the su-perposition coded symbol sC =

√asA +

√1 − asB to users A

and B instantaneously, where 0 < a ≤ 0.5 is the power alloca-tion factor for user A, and sA ∈ SA (sB ∈ SB ) is intended foruser A (B). Here, SA (SB ) is an MA -ary (MB -ary) normal-ized PAM/QAM constellation, and the signal constellation ofsC thus can be expressed as SC = {√asA +

√1 − asB |sA ∈

SA , sB ∈ SB }, which is an irregular MAMB -ary constellation.The total transmit power is normalized to unity and thus thereceived signals at users A and B are respectively expressed as

ySA = h(√

asA +√

1 − asB

)+ nSA (39)

and

ySB = g(√

asA +√

1 − asB

)+ nSB . (40)

User A first detects symbol sB by treating symbol sA as userinterference from (39) and then cancels sB using SIC to obtain


the desired symbol sA from

ySA = h(√

asA +√

1 − a (sB − sB ))

+ nSA , (41)

where sB is the estimate of sB at user A. User B conserves ySB

and does not detect any symbols during the first phase.At the second phase, user A relays sB to user B with full

transmit power such that the received signal at user B is givenby

yAB = f sB + nAB . (42)

According to the maximal ratio combining (MRC), userB combines ySB in (40) and yAB in (42) with weightsgH

√1 − a/(‖g‖2a + N0) and fH /N0 , respectively, resulting

in the combined signal

yB =gH

√1 − a

‖g‖2a + N0ySB +

fH

N0yAB

=‖g‖2 (1 − a)‖g‖2a + N0

sB +‖f‖2

N0sB +

‖g‖2√a (1 − a)‖g‖2a + N0

sA

+√

1 − agH nSB

‖g‖2a + N0+

fH nAB

N0. (43)

From (43), user B estimates its own symbol sB assuming correctdetection of sB at user A via

sB = arg minsB

∣∣∣∣∣yB −

(‖g‖2 (1 − a)‖g‖2a + N0

+‖f‖2

N0

)

sB

∣∣∣∣∣

2

. (44)

A. BEP Analysis for CRS-NOMA

In this subsection, the analytical BEP expressions are derivedfor users A and B in closed form. Due to SIC, the BEP of userA is highly affected by the error probability of detecting sB atuser A. According to the law of total probability, the BEP ofuser A can be approximated as

PA ≈ (1 − PAsB

)PAb +

PAsB

log2 (MA ), (45)

where PAsB

is the symbol error probability (SEP) of sB at user A,and PAb denotes the BEP of M -ary PAM/QAM demodulationover NA

r i.i.d. Rayleigh fading channels with transmit power a.Therefore, PAb is the same as (20) for PAM and (21) for QAMexcept that μc(i) in (19) should be revised as

μc(i) =

√aλSA ((2i + 1) d)2

N0 + aλSA ((2i + 1) d)2 . (46)

Similar to (45), the BEP of user B can be given by

PB ≈(1 − PA

sB

) · PsB

log2 (MB )+

PAsB

log2 (MB ), (47)

where PsBdenotes the SEP of detecting sB from (44) under the

condition that user A correctly detects sB . Since it is difficultto exactly evaluate the error propagation when sB is detectedincorrectly, we assume that an incorrectly detected sB yields oneerroneous bit for sA at user A in (45) and for sB at user B in (47).

Fig. 2. Signal constellation of superposition coded symbols, where sA andsB are 2-PAM and 4-PAM symbols, respectively.

Further, in (47), when sB is detected correctly at user A, the BEPof user B is approximated as PsB

divided by log2(MB ). Theseapproximations lead to the slight deviations between analyticalresults and their simulation counterparts, which will be shownin Section V.

1) PAM Constellation: We first consider the scenario inwhich both sA and sB are PAM symbols transmitted over theI- channel, i.e., MA = MAI and MB = MBI . They are inde-pendently selected from the constellations SA = {±dA ,±3dA ,±5dA , . . . ,±(MAI − 1)dA} and SB = {±dB ,±3dB ,±5dB ,. . . ,±(MBI − 1)dB }, respectively, where dA =

√3/(M 2

A I −1)

and dB =√

3/(M 2B I −1). We observe that SC is an irregular

MAI MBI -PAM constellation, and the power allocation factor,i.e., a, should satisfy (MAI − 1)dA

√a < dB

√1 − a to ensure

that all MAI superposition coded symbols associated with eachsB are located in the decision region for that sB . An example ofSC is given in Fig. 2, where MAI = 2, MBI = 4, and verticallines denote decision boundaries for sB .

Due to the superposition coding, the SEP of sB is equal to thatof sC with the decision boundaries for sB . Furthermore, becausethe constellation diagram of SC is symmetrical, we only needto consider the error probability of those constellation pointsin the left half of the plane. To begin with, let us define twovectors that respectively contain the decision boundaries andpositions of sC from left to right in the left half of the plane,namely

uBI =[uBI (0) , . . . , uBI (MBI /2 − 1)]T

=dB

√1 − a[− (MBI − 2) ,− (MBI − 4) , . . . , 0]T, (48)

vC I = [vC I (0) , . . . , vC I (MAI MBI /2 − 1)]T , (49)

where vC I (α) =√

a · vAI (mod (α,MAI )) +√

1 − a · vBI

(�α/MAI �), α = 0, . . . ,MAI MBI /2 − 1. Here, [vAI (0), . . . ,vAI (MAI − 1)]T Δ= vAI and [vBI (0), . . . , vBI (MBI − 1)]T Δ=vBI are two vectors containing the positions of signal pointsfrom left to right in MAI -PAM and MBI -PAM constellations,respectively.

Note that there is only a single decision boundary,i.e., uBI (0), for vC I (α), α ∈ {0, . . . ,MAI − 1}. By con-trast, there are two decision boundaries which lie atuBI (�α/MAI � − 1) and uBI (�α/MAI �), respectively, forvC I (α), α ∈ {MAI , . . . ,MAI MBI /2 − 1}. The distances be-tween each signal point and the corresponding decision bound-aries can be easily obtained from (48) and (49), and the errorprobability of each sC can also be derived by considering thatan error occurs if the noise exceeds either distance. Therefore, itcan be shown that the conditional SEP of sB at user A for PAM


is formulated as

PAsB −I |γ S A

t o t=

2MAI MBI

{MA I −1∑

α=0

Q

(√2γSA

tot dα0−I

)

+MA I MB I /2−1∑

α=MA I

{Q

(√2γSA

tot dα1−I

)

+ Q

(√2γSA

tot dα2−I

)}}

, (50)

where dα0−I = |vC I (α) − uBI (0)|, dα1−I = |vC I (α) − uBI

(� αMA I

� − 1)|, and dα2−I = |vC I (α) − uBI (� αMA I

�)|; γSAtot =

∑N Ar

k=1 γSAk is the total instantaneous receive SNR with γSA

k =|hk |2/N0 . The average SEP, i.e., PA

sB −I , then can be derived by

averaging (50) over the joint PDF of {γSAk }N A

r

k=1 . After severalmathematical manipulations similar to (14)–(19), we obtain

PAsB −I =

2MAI MBI

⎧⎨

⎩

MA I −1∑

α=0

T0−I (α)

+MA I MB I /2−1∑

α=MA I

[T1−I (α) + T2−I (α)]

⎫⎬

⎭, (51)

where

Ti−I (α) =(

1 − μi−I (α)2

)N Ar

N Ar −1∑

k=0

[(NA

r − 1 + k

k

)

×(

1 + μi−I (α)2

)k]

, i = 0, 1, 2 (52)

with

μi−I (α) =

√λSAd2

αi−I

N0 + λSAd2αi−I

. (53)

Finally, substituting (51) into (45) yields a closed-form expres-sion for PA .

For the calculation of PB , we first address the deriva-tion of PsB

as follows. Since the MRC at user B is theoptimal diversity combiner, it can be shown that the SEPconditioned on γSB

tot and γABtot is given at the bottom

of this page as (54), where zα0−I = |vBI (0) − uBI (0)|,zα1−I = |vBI (� α

MA I�) − uBI (� α

MA I� − 1)|, and zα2−I =

|vBI (� αMA I

�) − uBI (� αMA I

�)|; dα0−I , dα1−I , and dα2−I are

defined in (50); γSBtot =

∑N Br

k=1 γSBk =

∑N Br

k=1 |gk |2/N0 and

γABtot =

∑N Br

τ =1 γABτ =

∑N Br

τ =1 |fτ |2/N0 . Since random vari-

ables {γSBk }N B

r

k=1 and {γABτ }N B

rτ =1 are assumed to be independent,

the average SEP, i.e., PsB −I , becomes

PsB −I =∫ ∞

0· · ·∫ ∞

0︸︷︷︸2N B

r −f old

PBs−I |γ S Bt o t ,γ A B

t o t

×N B

r∏

k=1

pγ S Bk

(γSB

k

) ·N B

r∏

τ =1

pγ A Bτ

(γAB

τ

)

× dγSB1 · · · dγSB

N Br

dγAB1 · · · dγAB

N Br

, (55)

where pγ S Bk

(γSBk )=exp (−γSB

k /γSB )/γSB and pγ A Bτ

(γABτ )=

exp(−γABτ /γAB )/γAB are the PDFs of γSB

k and γABτ , respec-

tively, with γSB = λSB /N0 and γAB = λAB /N0 . By using thealternative representation of Gaussian Q-function in (14), (55)can be expressed as [36, Eq. (5A.58)]

PsB −I =2

MAI MBI

⎧⎨

⎩

MA I −1∑

α=0

W0−I (α)

+MA I MB I /2−1∑

α=MA I

[W1−I (α) + W2−I (α)]

⎫⎬

⎭, (56)

where Wi−I (α) is given at the bottom of the next page as (57)with

c1 = λSB d2αi−I /N0 , c2 = λAB z2

αi−I /N0 , (58)

Bk =Ak(2N Br −1k

) , Ck =N B

r −1∑

n=0

(kn

)

(2N Br −1n

)An, (59)

Ak = (−1)N Br −1+k

(N B

r −1k

)

(NBr − 1)!

N Br∏

n=1n �=k+1

(2NBr − n), (60)

Ik (c) = 1 −√

c

1 + c

[

1 +k∑

n=1

(2n − 1)!!n!2n (1 + c)n

]

. (61)

Afterwards, we can formulate PB based on (51) and (56).2) QAM Constellation: In this case, we consider that sA

and sB are drawn from MAI × MAQ (= MA ) and MBI ×

PsB −I |γ S Bt o t ,γ A B

t o t=

2MAI MBI

{MA I −1∑

α=0

Q

(√2(γSB

tot d2α0−I + γAB

tot z2α0−I

))

+MA I MB I /2−1∑

α=MA I

[Q

(√2(γSB


tot z2α1−I

))

+ Q

(√2(γSB


tot z2α2−I

))]}

(54)


TABLE IOBTAINED aop t VALUES FROM COMPUTER SEARCH FOR DIFFERENT CONSTELLATION COMBINATIONS OF sA AND sB

MBQ (= MB ) rectangular QAM constellations, respectively.Since an MA -QAM (MB -QAM) constellation is composed ofthe quadrature combination of MAI -PAM (MBI -PAM) in theI- channel and MAQ -PAM (MBQ -PAM) in the Q- channel, theBEP analysis for QAM constellations is similar to that for PAMconstellations addressed above. We redefine some parametersfor the Q- dimension including the decision boundaries, posi-tions of sA , sB and sC , in a similar fashion. Based on PA

sB −I in(51), the error probability for the Q- dimension, i.e., PA

sB −Q , isstraightforward. Note that for QAM constellations, dA and dB

become dA =√

3/(M 2A I +M 2

A Q −2) and dB =√

3/(M 2B I +M 2

B Q −2),respectively. Since a symbol error of QAM demodulation oc-curs when an incorrect symbol decision is made on the I- or Q-channel, the SEP of sB at user A for QAM is given by

PAsB

= 1 − (1 − PAsB −I

) (1 − PA

sB −Q

). (62)

Likewise, PsBfor QAM is obtained as

PsB= 1 − (1 − PsB −I ) (1 − PsB −Q ) . (63)

Finally, the BEP expressions for users A and B with QAMcan be derived by substituting (62) into (45) and (63) into (47),respectively. Similar to the observation in CRS-SM-NOMA, thediversity orders of users A and B for CRS-NOMA are given byNA

r and min{2NBr ,NA

r }, respectively.Since the value of power allocation factor, i.e., a, highly im-

pacts the BER performance of both users, it should be optimizedto minimize their BER values. Unfortunately, it seems impossi-ble to obtain a value of a that optimizes the BER performanceof both users simultaneously. As a compromise, we select

aopt = arg mina

max {PA, PB } (64)

as the optimal a in the sense of minimizing the system BER val-ues. Due to the complexity of PA and PB , we perform exhaus-tive computer search to derive aopt for different constellationcombinations, whose results are summarized in Table I.

B. Sum Rate Analysis for CRS-NOMA Under Gaussian Input

In this subsection, the achievable sum rate of CRS-NOMAsystems is studied under Gaussian input.

During the first time slot, symbol sB is first decoded with thereceive SNR

γAB =

(1 − a)ρ‖h‖2

aρ‖h‖2 + 1, (65)

and then symbol sA is detected after the SIC at user A with thereceive SNR

γA = aρ‖h‖2 . (66)

On the condition of correct detection of sB by user A at thefirst phase, it can be figured out from (43) that the receive SNRfor sB at user B during the second time slot is given by

γBB =

(1 − a) ρ‖g‖2

aρ‖g‖2 + 1+ ρ‖f‖2 , (67)

which is the sum of SNRs provided by ySB and yAB . There-fore, the instantaneous sum rate of CRS-NOMA is explicitlyexpressed as

CCRS-NOMA = CA + CB , (68)

where

CA =12log2 (1 + γA ) , (69)

and

CB =12log2

(1 + min

(γA

B , γBB

)). (70)

To see how the power allocation factor affects the ergodic sumrate in CRS-NOMA, we present the curves of the ergodic sumrate versus a with different system setups in Fig. 3, where NA

r =NB

r = 2 and λSB = 1. As seen from Fig. 3, a higher transmitSNR yields a larger ergodic sum rate as expected. Furthermore,the ergodic sum rate increases with a, especially when λSA ≥λAB . Therefore, we select aopt = 0.5 as the optimal power

Wi−I (α) =1π

∫ π/2

0

[∫ ∞

0exp(−γSB

k d2αi

sin2φ

)pγ S B

k

(γSB

k

)dγSB

k

]N Br[∫ ∞

0exp(−γAB

τ z2αi

sin2φ

)pγ A B

τ

(γAB

τ

)dγAB

τ

]N Br

dφ

=1π

∫ π/2

0

(sin2φ

γSB d2αi−I + sin2φ

)N Br(

sin2φ

γAB z2αi−I + sin2φ

)N Br

dφ

=

(c1c2

)N Br −1

2(1 − c1

c2

)2N Br −1

⎧⎨

⎩

N Br −1∑

k=0

(c2

c1− 1)k

BkIk (c2) − c1

c2

N Br −1∑

k=0

(1 − c1

c2

)k

CkIk (c1)

⎫⎬

⎭(57)


Fig. 3. Ergodic sum rate versus a for CRS-NOMA (NAr = N B

r = 2 andλS B = 1).

allocation coefficient in the sense of maximizing the ergodicsum rate.

When a = 0.5, at high SNR values, we have

CB ≈ 12, (71)

such that (68) reduces to

CCRS-NOMA ≈ 12log2

(0.5ρ ‖h‖2

)+

12

=12log2

(ρ‖h‖2

). (72)

By comparing (38) with (72), we conclude that our proposedCRS-SM-NOMA almost achieves an increase of log2(Nt)/2bps/Hz in sum rate over CRS-NOMA due to the use of spatialdomain in data transmission.

V. SIMULATION RESULTS AND DISCUSSION

In this section, we perform Monte Carlo simulations to ex-amine the uncoded BER and the achievable rate performanceof CRS-SM-NOMA, and compare them with those of CRS-NOMA and SM-OMA. In SM-OMA, the BS transmits twoSM signals to users A and B, respectively, in two consec-utive time slots, and there is no cooperation between twousers. From the received signals, users A and B extract theM -ary symbol and the index bits as the desired information,respectively. In all simulations, the channels are assumed tobe Rayleigh flat-faded and perfectly known to users A andB. The average powers of S-to-A, S-to-B, and A-to-B linksare set to be λSA = 10, λSB = 1, and λAB = 10, respec-tively. For brevity, we will refer to “CRS-SM-NOMA/SM-OMA

Fig. 4. BER performance comparison between “CRS-SM-NOMA (2-PAM,2)” and “CRS-NOMA (2-PAM, 2-PAM)”, where NA

r = 1, 2, N Br = 1, 2, and

a = 0.22.

(M -PAM/QAM, Nt)” as the CRS-SM-NOMA/SM-OMAscheme in which the BS employs M -PAM/QAM with Nt

transmit antennas, and “CRS-NOMA (MA -PAM/QAM, MB -PAM/QAM)” as the CRS-NOMA scheme in which the BS mapsthe information bits for users A and B into MA -PAM/QAM andMB -PAM/QAM symbols, respectively.

A. Uncoded BER Performance

In this subsection, the uncoded BER performance of CRS-SM-NOMA and CRS-NOMA is compared, assuming that usersA and B in all considered schemes employ ML or joint MLdetection. Since user A of SM-OMA performs the same processas that of CRS-SM-NOMA, the BER curves of user A for SM-OMA are not presented.

Fig. 4 shows the BER versus ρ curves of users A and B for“CRS-SM-NOMA (2-PAM, 2)” and “CRS-NOMA (2-PAM, 2-PAM)”, where NA

r = 1, 2, NBr = 1, 2, and a = 0.22. To see the

effects of the number of receive antennas clearly, the theoreticalBEP curves are not included. As seen from Fig. 4, since for bothCRS-SM-NOMA and CRS-NOMA, the diversity orders of usersA and B are given by NA

r and min{2NBr ,NA

r }, respectively,when NA

r ≤ 2NBr , increasing NA

r improves the diversity ordersof both users, while a larger NB

r merely enhances the BERperformance of user B in the low SNR region slightly. Althoughthe curve with NA

r = NBr = 2 is not provided in the figure, it is

expected to be the same as that with NAr = 2 and NB

r = 1 forboth CRS-SM-NOMA and CRS-NOMA, except that a minorimprovement can be observed in the low SNR region for userB. Because user A is influenced by the error propagation of SIC


Fig. 5. BER performance comparison among “CRS-SM-NOMA (2-PAM,2)”, “CRS-NOMA (2-PAM, 2-PAM)”, and “SM-OMA (2-PAM, 2)”, whereN A

r = N Br = 1 and a = 0.22.

in CRS-NOMA, we observe that user A of CRS-SM-NOMAoutperforms that of CRS-NOMA, achieving SNR gains of about3.5 dB and 4.5 dB when NA

r = 1 and NAr = 2, respectively, at

a BER value of 10−4 . On the contrary, user B of CRS-SM-NOMA performs worse than that of CRS-NOMA. Fortunately,the performance loss is less than 1.5 dB.

In Fig. 5, we compare the BER performance of “CRS-SM-NOMA (2-PAM, 2)”, “CRS-NOMA (2-PAM, 2-PAM)”, and“SM-OMA (2-PAM, 2)”, where NA

r = NBr = 1 and theoreti-

cal BEP results are also presented. It can be seen from Fig. 5that the theoretical BEP curves accurately match with their sim-ulation counterparts over the considered SNR region. In CRS-SM-NOMA, the BER performance of user A is superior to thatof user B, while the situation is reversed in CRS-NOMA. Al-though not both users of CRS-SM-NOMA perform better thanthose of CRS-NOMA, we observe that the performance gap be-tween two users in CRS-SM-NOMA, which is about 1 dB, ismuch smaller than that in CRS-NOMA, which is about 3 dB.This means that CRS-SM-NOMA provides a higher user fair-ness in error performance than CRS-NOMA. Moreover, user Bof CRS-SM-NOMA achieves about 10 dB SNR gain over thatof SM-OMA.

Fig. 6 depicts the performance comparisons among “CRS-SM-NOMA (4-QAM, 4)”, “CRS-NOMA (4-QAM, 4-QAM)”,and “SM-OMA (4-QAM, 4)”, where NA

r = NBr = 2 and a =

0.22. As shown in Fig. 6, all theoretical BEP results agree withtheir computer simulation counterparts quite well in the entireSNR region. User B of SM-OMA performs the worst among allschemes without cooperating with user A. Different from theobservations in Figs. 4 and 5, both users of CRS-SM-NOMA

Fig. 6. BER performance comparison among “CRS-SM-NOMA (4-QAM,4)”, “CRS-NOMA (4-QAM, 4-QAM)”, and “SM-OMA (4-QAM, 4)”, whereN A

r = N Br = 2 and a = 0.22.

outperform those of CRS-NOMA. More specifically, users Aand B in CRS-SM-NOMA obtain SNR gains of about 5 dB and2.5 dB over those in CRS-NOMA, respectively, at a BER valueof 10−5 . Moreover, a better BER fairness between two users isprovided by CRS-SM-NOMA.

Fig. 7 illustrates the comparison results among “CRS-SM-NOMA (4-QAM, 8)”, “CRS-NOMA (4-QAM, 8-QAM)”, and“SM-OMA (4-QAM, 8)”, where NA

r = NBr = 2 and a = 0.09.

Similar to Fig. 6, the approximate analytical BEP curves predictthe simulation results accurately for all SNR values, and the BERperformance of CRS-SM-NOMA is superior to that of CRS-NOMA for both users. User B of CRS-SM-NOMA preformssignificantly better than that of SM-OMA, achieving an SNRgain of about 10 dB. Particularly, by comparing with Fig. 6,we observe that the performance improvements achieved byCRS-SM-NOMA over CRS-NOMA become more prominentin Fig. 7 due to higher constellations. Approximately 6 dB SNRgain is obtained by user A and 4 dB SNR gain by user B, ata BER value of 10−5 . Meanwhile, CRS-SM-NOMA providesa higher fairness between two users, since their BER curvesalmost overlap.

B. Achievable Rate Performance

In Fig. 8, we make comparisons among CRS-SM-NOMA,CRS-NOMA, and SM-OMA in terms of achievable rate, whereNA

r = NBr = 2, Nt = 2, 4, and a = 0.5. As shown in Fig. 8,

for both CRS-SM-NOMA with Nt = 2 and CRS-NOMA, theachievable rate of user B is saturated to 0.5 bps/Hz, whichcorroborates the analysis in (37) and (71). For all considered


Fig. 7. BER performance comparison among “CRS-SM-NOMA (4-QAM,8)”, “CRS-NOMA (4-QAM, 8-QAM)”, and “SM-OMA (4-QAM, 8)”, whereN A

r = N Br = 2 and a = 0.09.

Fig. 8. Achievable rate comparison among CRS-SM-NOMA, CRS-NOMA,and SM-OMA, where N A

r = N Br = 2, Nt = 2, 4, and a = 0.5.

schemes, the sum rate as well as the achievable rate of user Agrows linearly with ρ. The ergodic sum rate of CRS-NOMAnearly equals the achievable rate of user A in CRS-SM-NOMAwith Nt = 2, and CRS-SM-NOMA achieves a sum rate im-provement of about log2(Nt)/2 bps/Hz over CRS-NOMA fora fixed ρ, which can be explained by (38) and (72). Further, as

Fig. 9. Ergodic sum rate comparison among CRS-SM-NOMA, CRS-NOMA,and SM-OMA, where Nt = 2, N A

r = N Br = 1, 2, 4, and a = 0.5.

indicated by (38), the sum rate of CRS-SM-NOMA improveswith Nt , and an increase of about 0.5 bps/Hz can be obtainedwhen Nt is doubled. The ergodic sum rate of CRS-SM-NOMAis larger than that of SM-OMA in the low SNR region, while athigh SNR, they are nearly equal.

Fig. 9 gives the comparison results among the ergodicsum rates of CRS-SM-NOMA, CRS-NOMA, and SM-OMA,where Nt = 2, NA

r = NBr = 1, 2, 4, and a = 0.5. It can be

observed from Fig. 9 that increasing the number of receiveantennas promotes significantly the ergodic sum rates forall schemes. However, different from the effect of doublingNt in Fig. 8, the increments in ergodic sum rate graduallyreduce as NA

r or NBr doubles. Moreover, with NA

r = NBr = 1,

CRS-SM-NOMA performs slightly better than CRS-NOMA.By contrast, CRS-SM-NOMA achieves an increase of aboutlog2(Nt)/2 = 0.5 bps/Hz in ergodic sum rate over CRS-NOMAwhen NA

r = NBr = 2, 4. These can be explained as follows.

With NAr = NB

r = 1, the index of the active antenna is verylikely to be detected erroneously at user B of CRS-SM-NOMA,which is very common for IM-based systems with single receiveantenna, and thus its achievable rate is far less than log2(Nt)/2.Increasing NA

r and NBr to 2 or 4 greatly improves the reliability

of the active antenna index detection such that the approximationin (37) makes sense. Similar to the observation in Fig. 8, CRS-SM-NOMA obtains a larger ergodic sum rate than SM-OMAat low SNR, and the gap narrows with increasing SNR.

VI. CONCLUSION

In this paper, we have proposed a new three-node CRS us-ing SM-NOMA, in which the information of two users per


transmission is carried by the index of the active antenna andthe ordinary APM symbol, respectively. The near user also func-tions as a relay to assist the far user. As the reference scheme,conventional CRS-NOMA has been re-investigated with mul-tiple receive antennas. For both CRS-SM-NOMA and CRS-NOMA, approximate BEP expressions have been derived inclosed form for both users employing PAM/QAM over Rayleighflat fading channels, and the system sum rates under Gaussianinput have also been developed. Computer simulations haveverified the performance analysis, and shown that the proposedCRS-SM-NOMA achieves better BER performance and a largersum rate than CRS-NOMA. We conclude that the proposedCRS-SM-NOMA can be considered as a candidate for a fair,reliable and high throughput multiple access technique towardsnext-generation wireless networks.

This work focuses only on the two-user scenario. Actu-ally, CRS-SM-NOMA can be extended to more than two usersthrough power multiplexing or SDMA, which involves resourceallocation, user pairing, beamforming design, etc. We leave thisextension for future study.

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Qiang Li received the B.S. degree from the InnerMongolia University of Science and Technology,Baotou, China, in 2013, and the M.S. degree fromthe Nanjing University of Aeronautics and Astronau-tics, Nanjing, China, in 2016. He is currently workingtoward the Ph.D. degree with the South China Uni-versity of Technology, Guangzhou, China.

He is currently a Visiting Student Research Collab-orator with Princeton University, Princeton, NJ, USA.His recent research interests include index modula-tion and non-orthogonal multiple access.

Miaowen Wen (M’14–SM’18) received the B.S.degree from Beijing Jiaotong University, Beijing,China, in 2009, and the Ph.D. degree from PekingUniversity, Beijing, China, in 2014. From 2012 to2013, he was a Visiting Student Research Collabo-rator with Princeton University, Princeton, NJ, USA.He is currently an Associate Professor with the SouthChina University of Technology, Guangzhou, China.He has authored or coauthored a book entitled In-dex Modulation for 5G Wireless Communications(Springer) and 100+ research papers, which include

60+ journal papers and 40+ conference papers. His research interests includeindex modulation, non-orthogonal multiple access, physical layer security, andmolecular communications.

He was an Exemplary Reviewer for the IEEE COMMUNICATIONS LETTERS in2017. He was the symposium Co-Chair for the IEEE ICNC’2019 and a work-shop Co-Chair for the IEEE/CIC International Conference on Communicationsin China (ICCC’2018), and is currently a Guest Editor for the IEEE JOURNAL ON

SELECTED AREAS IN COMMUNICATIONS (Special Issue on Spatial Modulationfor Emerging Wireless Systems), the IEEE JOURNAL OF SELECTED TOPICS IN

SIGNAL PROCESSING (Special Issue on Index Modulation for Future WirelessNetworks: A Signal Processing Perspective), and the IEEE ACCESS (SpecialSection on Advances in Signal Processing for Non-Orthogonal Multiple Ac-cess). He has served on the editorial boards of several international journals,including the IEEE ACCESS, the EURASIP Journal on Wireless Communicationsand Networking, the ETRI Journal, and the Physical Communication (Elsevier).He was the recipient of the Excellent Doctoral Dissertation Award from PekingUniversity and the Best Paper Awards from the IEEE International Conferenceon Intelligent Transportation Systems Telecommunications in 2012, the IEEEInternational Conference on Intelligent Transportation Systems in 2014, and theIEEE International Conference on Computing, Networking and Communica-tions in 2016.

Ertugrul Basar (S’09–M’13–SM’16) received theB.S. degree (Hons.) from Istanbul University, Istan-bul, Turkey, in 2007, and the M.S. and Ph.D. degreesfrom Istanbul Technical University, Istanbul, Turkey,in 2009 and 2013, respectively.

He is currently an Associate Professor with the De-partment of Electrical and Electronics Engineering,Koc University, Sariyer, Turkey, and the Director ofthe Communications Research and Innovation Labo-ratory. His primary research interests include MIMOsystems, index modulation, waveform design, visible

light communications, and signal processing for communications.Dr. Basar is currently an Editor for the IEEE TRANSACTIONS ON COMMUNI-

CATIONS and Physical Communication (Elsevier), and an Associate Editor forthe IEEE COMMUNICATIONS LETTERS. He was an Associate Editor for the IEEEACCESS from 2016 to 2018. Recent recognition of his research includes theScience Academy (Turkey) Young Scientists (BAGEP) Award in 2018, MustafaParlar Foundation Research Encouragement Award in 2018, Turkish Academyof Sciences Outstanding Young Scientist (TUBA-GEBIP) Award in 2017, andthe first-ever IEEE Turkey Research Encouragement Award in 2017.

H. Vincent Poor (S’72–M’77–SM’82–F’87) re-ceived the Ph.D. degree in electrical engineering andcomputer science from Princeton University, Prince-ton, NJ, USA, in 1977. From 1977 to 1990, he wason the faculty of the University of Illinois at Urbana-Champaign. Since 1990, he has been on the faculty atPrinceton, where he is currently the Michael HenryStrater University Professor of Electrical Engineer-ing. During 2006 to 2016, he was the Dean of Prince-ton’s School of Engineering and Applied Science. Hehas also held visiting appointments at several other

universities, including most recently at Berkeley and Cambridge. His researchinterests are in the areas of information theory and signal processing, and theirapplications in wireless networks, energy systems and related fields. Among hispublications in these areas is the recent book Information Theoretic Securityand Privacy of Information Systems (Cambridge Univ. Press, 2017).

Dr. Poor is a member of the National Academy of Engineering and the Na-tional Academy of Sciences, and is a Foreign Member of the Chinese Academyof Sciences, the Royal Society, and other national and international academies.He was the recipient of the Marconi and Armstrong Awards of the IEEE Com-munications Society in 2007 and 2009, respectively. Recent recognition of hiswork includes the 2017 IEEE Alexander Graham Bell Medal, Honorary Profes-sorships at Peking University and Tsinghua University, both conferred in 2017,and a D.Sc. honoris causa from Syracuse University also awarded in 2017.

Fangjiong Chen (M’06) received the B.S. de-gree in electronics and information technology fromZhejiang University, Hangzhou, China, in 1997, andthe Ph.D. degree in communication and informationengineering from the South China University of Tech-nology, Guangzhou, China, in 2002.

In 2002, he joined the School of Electronics andInformation Engineering, South China University ofTechnology, where he was a Lecturer from 2002 to2005, an Associate Professor from 2005 to 2011, andis currently a full-time Professor. He is also the Direc-

tor of the Department of Underwater Detection and Imaging, Mobile UltrasonicDetection National Research Center of Engineering Technology. His current re-search interests include signal detection and estimation, array signal processing,and wireless communication.

Dr. Chen was the recipient of the National Science Fund for OutstandingYoung Scientists in 2013, and was elected in the New Century Excellent TalentProgram of MOE, China, in 2012.

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