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Spatial Modulation AidedSparse Code-Division Multiple Access

Yusha Liu,Student Member, IEEE, Lie-Liang Yang,Fellow, IEEE, and Lajos Hanzo,Fellow, IEEE

Abstract—In order to support high-user-load multiple-access(MA), we propose a non-orthogonal MA scheme based on abeneficial amalgam of spatial modulation (SM) and sparse code-division multiple-access (SCDMA), which is termed as the SM-SCDMA. Hence, SM-SCDMA inherits both the merits of SMwith single radio-frequency MIMO transceiver implementationand the advantages of SCDMA relying on low-complexity signaldetection. In this paper, we evaluate the potential of SM-SCDMAas well as its low-complexity near-optimum signal detection.Given these objectives, we consider both the maximum likelihooddetection (MLD) and the message passing algorithm aideddetection (MPAD) that is derived based on the maximuma-posteriori principles. In order to evaluate the performance of largeSM-SCDMA without relying on time-consuming simulations,we propose new approaches for analyzing the performance ofSM-SCDMA systems. A range of formulas that are valid inthe signal-to-noise ratio (SNR) region of practical interest arederived. Finally, the performance of SM-SCDMA systems isinvestigated by addressing diverse design concerns. Our studiesand performance results show that SM-SCDMA constitutes apromising MA scheme for the future ultra dense systems. Assistedby the MPAD, it is capable of supporting high-user-load MAtransmission associated with a normalized user-load factor oftwo.

Index Terms—CDMA, NOMA, spatial modulation, low densitysignature, maximum a posteriori detection, maximum-likelihooddetection, message passing algorithm, performance analysis.

I. I NTRODUCTION

During the 2010s, research interests in multiuser communi-cations have been focused on non-orthogonal multiple-access(NOMA) [1–15], with the motivation of supporting ultra-densedeployment, which faces the problem of supporting more users(or devices) than the number of resource units available. Foruplink communications, sparse code spread CDMA [16, 17], orsimply termed as sparse CDMA (SCDMA), has been proposedfor meeting these objectives. In SCDMA, each spreadingsequence contains only a small number of non-zero entries.Hence, each user’s transmitted symbol is represented by asmall number of non-zero chips in the spreading codes. Withinthis family, the proposed schemes include both low densitysignature (LDS) assisted CDMA [2] and sparse code multipleaccess (SCMA) [18–21]. The studies in [2, 18] show that byexploiting the sparsity of the signatures, SCDMA becomes ca-pable of supporting more users than conventional CDMA [22],while relying on relatively low-complexity detectors. Thereason behind this is that the employment of low density sig-nature codes allows us to invoke a message passing algorithm

Y. Liu, L.-L. Yang and L. Hanzo are with School of ElectronicsandComputer Science, University of Southampton, SO17 1BJ, UK. (E-mail:yl6g15, lly, lh@ecs.soton.ac.uk). The author would like toacknowledge withthanks the financial assistance from EPSRC of UK and ERC of EU.

(MPA) [23–26] assisted detector for achieving near-optimumperformance. By contrast, in the conventional CDMA systems,each user’s transmitted symbol is spread by fully populatedspreading codes, hence it is difficult to design practical low-complexity near-optimum detection schemes, especially, whensupporting a high normalized user-load.

As a parallel development in multiple-input multiple-output(MIMO) systems, spatial modulation (SM) has drawn a lotof research attention in recent years, as evidenced by [27–29] and the references therein. These schemes includespace-shift keying (SSK) [30, 31] conveying information solely by theindices of the transmit antennas (TAs). The SM of [27, 30, 32]activates one out of several TAs to convey a classic amplitude-phase modulation (APM) symbol, such as, quadrature am-plitude modulated (QAM) symbol. By contrast,generalizedSM (GSM) [33] simultaneously activates multiple TAs toconvey multiple APM symbols, whilespace-time shift-keying(STSK) [34] exploits all degrees of freedom in the joint space-time domain for information delivery. Additionally, owingtothe recent research interest in NOMA, the authors in [15]have proposed and investigated a NOMA-based SM scheme.Similarly, by activating one or a subset of receive antennaswith the aid of appropriate transmitter preprocessing, a rangeof preprocessing assisted SM (PSM) schemes have beenproposed. These include the preprocessing-aided SSK [35]solely using receive antenna indices for implicit informationdelivery, and the PSM of [35] that conveys information byactivating a single receive antenna out of several for conveyingan APM symbol. By contrast, the generalized PSM (GPSM)of [36] simultaneously activates multiple receive antennas andconveys multiple APM symbols, etc. Various extensions ofthe above-mentioned SM or PSM schemes have also beensurveyed in [28, 29].

Multiple-access (MA) communications has also been stud-ied in association with SM [37–43]. However, in these SM-assisted MA schemes, users are supported solely by thespatial-domain resources, which may require an excessivenumber of receive antennas at a base-station (BS) [38, 40–43]. This is because the number of BS antennas shouldlinearly increase with the number of users, in order to guar-antee the required quality-of-services (QoS) by employingpractically implementable detectors at the BS. Explictly,theimplementation of these schemes faces challenges in terms ofantenna deployment, channel estimation, pilot-overhead,etc.This situation becomes even severer, when a future MA systemhas to support a huge number of communication devices. Inthis case, SM amalgamated with other MA techniques [44–46] is more desirable for practical implementations. Therefore,

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we intrinsically amalgamate SM and NOMA, resulting in ourSpatial Modulated Sparse CDMA (SM-SCDMA) scheme con-ceived for supporting MA communications in heavily loadedcommunications environments. In summary, the contributionsof the paper are as follows:

• A SM-SCDMA scheme is proposed for supporting ahigh normalized user-load in uplink communications. Weassume that each user employs a few TAs for supportingthe SM, while the BS employs several receive antennasfor enhancing the detection reliability, where MA isachieved with the aid of SCDMA. Since our SM-SCDMArelies on the principle of NOMA, it has the potential ofsupporting the MA transmission, where the number of(active) users is higher than the total number of chips inthe spreading codes.

• It has been recognized that employing SM is beneficialfor the implementation of MIMO using a low numberof radio-frequency transceivers, and provides lower com-plexity receiver implementation options [27, 28]. How-ever, when there is a high number of users sharing theresources based on CDMA, the design of relatively low-complexity detectors that are capable of achieving near-optimum performance is challenging. In this paper, wefirst consider the classic maximum-likelihood detection(MLD) in order to study the best possible potential ofthe SM-SCDMA scheme. Then, we derive a maximuma-posteriori detector (MAPD), based on which we developa reduced complexity MPA-aided detector (MPAD) thatis suitable for SM-SCDMA systems, which employs SMand exploits receiver diversity.

• In most existing references on LDS-CDMA or SCMA,the performance study depends mainly on simulations,which can typically deal with tens of users. However,it is hard to simulate the performance of a MA systemsupporting hundreds or even thousands of users. However,future NOMA systems are expected to support a hugenumber of users, hence it is interesting to predict theachievable performance of such large-scale MA systems.With this motivation, in this paper, we mathematicallyanalyze the error performance of SM-SCDMA systemsby first deriving the single-user bit error rate (BER) boundand then analyzing the approximate BER of multi-userSM-SCDMA systems. By exploiting the specific structureof SM-SCDMA, a range of formulas are derived, whichallow us to estimate the BER performance of the SM-SCDMA systems of arbitrary size. Note that in [47],the authors have analyzed the pairwise error probabil-ity (PEP) and upper-bound average symbol error rate(ASER) of the general multidimensional constellationswith MLD, and considered the LDS-CDMA and SCMAas special cases. By contrast, the authors of [48] havederived a lower bound for the ASER of the SCMA.

• The performance of SM-SCDMA systems is investigatedboth by Monte-Carlo simulations and by the evaluationof the formulas derived. Based on the SM-SCDMAsystems of relatively small size, we verify the formulasderived by simulation and find their valid ranges. Then,

the performance of relatively large-scale SM-SCDMAsystems is investigated based on the numerical evaluationof our formulas derived. Furthermore, we address theimpact of the related parameters on the BER performanceattained, and demonstrate the efficiency of the MPAD.

• Finally, we propose a novel 8QAM scheme. When com-municating over Gaussian channels, it outperforms allthe existing 8QAM schemes in the relatively low signal-to-noise (SNR) (≤ 10 dB) region, while achieving asimilar BER performance to the best existing 8QAMschemes in the relatively high-SNR region. By contrast,when communicating over Rayleigh fading channels, itoutperforms all the existing 8QAM schemes within theSNR region considered.

The rest of this paper is structured as follows. Section IIdescribes the proposed SM-SCDMA system model. Then, thedetection algorithms are addressed in Section III. SectionIVanalyzes the BER of the SM-SCDMA system, while the BERperformance of SM-SCDMA systems is investigated in Sec-tion V. Finally, Section VI summarizes the main conclusionsof our research.

II. SM-SCDMA SYSTEM MODEL

In this section, we describe the transmitter and receivermodels in Sections II-A and II-B, respectively. Furthermore,our main assumptions and notations are detailed.

A. Transmitter Model

Let us consider a single-cell wireless uplink, whereKusers simultaneously transmit their information to a singleBS. In order to avoid dealing with the trivial cases, but tofocus our attention on the important principles, we assumethat all users employ the sameM1 number of TAs satisfyingM1 = 2b1 , where b1 is an integer. We assume that the BSemploysU receive antennas for attaining receiver diversity.Based on the principles of SM [27], we assume that withina symbol duration ofTs seconds, one of theM1 TAs of auser is activated for transmittingb1 bits, forming anM1SSKscheme [31]. Hence, the SSK symbol set can be expressed asS1 = {0, 1, ...,M1 − 1} and the correspondingb1-bit symbolscan be their natural binary representations. In addition tothe M1SSK, the SM also uses the activated TA to transmitb2 = log2 M2 bits, relying on aM2-ary APM (M2APM),such asM2-ary QAM (M2QAM). For convenience, the APMsymbol set is expressed asS2 = {s2,0, s2,1, ..., s2,M2−1},whose elements satisfy

∑M2−1i=0 |s2,i|2/M2 = 1. Therefore,

when considering both theM1SSK andM2APM, b = (b1+b2)bits per symbol can be transmitted by a user to the BS.Note that the SM model considered can be readily extendedto the generalized SM model of [28], where multiple TAsare activated for simultaneously transmitting multiple APMsymbols.

Fig. 1 depicts the transmitter schematic of thekth user ofthe SM-SCDMA system. During a symbol period, theb-bitsymbolbbbk of userk is first partitioned into two sub-symbols,the SSK symbolbbbk1 consisting of the firstb1 bits and the APMsymbolbbbk2 consisting of the remainingb2 bits. Let sk1 ∈ S1

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bbbk1 sk1

sk2

bbbk2

bbbk = bbbk1|bbbk2

V (bbbk1)

V (bbbk2)Sparse

Spreading

ccck

...

...

M1 − 1

sk1

0

sk2ccck

Fig. 1. Transmitter schematic diagram of thekth user in the SM-SCDMAsystem.

be an integer value obtained from the mappingV1(bbbk1),which determines the TA activated by userk. Furthermore,let sk2 ∈ S2 be obtained from the mappingV2(bbbk2), whichis an APM symbol. Following the principles of SCDMA [2],sk2 is spread to a maximum ofdx chips (dx << N ) usingthe spreading sequenceccck = [ck0, ck1, ..., ck(N−1)]

T assignedto user k, where ccck is of N -length and hasdx non-zerochips, satisfying‖ccck‖2 = 1. Furthermore, we assume that themaximum number of users sharing any of theN chips isdf .In the following analysis, for simplicity, we assume thatdxanddf are constants. The set of sparse codes achieving theseare referred to as the regular sparse codes. Finally, as shown inFig. 1, thekth user’s SM-SCDMA signal is transmitted fromthe sk1th TA, following some further transmitter processingthat is not shown in the figure.

Therefore, given the SM symbolsk1|sk2 of userk, the unity-energy baseband equivalent signal transmitted by userk canbe expressed as

tttk(sk1) = sk2ccck, k = 1, 2, · · · ,K (1)

wheretttk(sk1) indicates that the signal is transmitted by thesk1th antenna. For convenience of description, below we de-fineX = S1⊗S2, which is a set containing theM(= M1M2)combinations of the elements inS1 with those inS2.

B. Receiver Model

Assume that the channel between any user and the BSexperiences flat Rayleigh fading1. Let the channel impulseresponse (CIR) between thesk1th TA of user k and theUreceive antennas at the BS be expressed as

hhhsk1= [h(0)

sk1, h(1)

sk1, · · · , h(U−1)

sk1]T , sk1 = 0, . . . ,M1 − 1 (2)

whereh(u)sk1 are independent identically distributed (iid) com-

plex Gaussian random variables with zero mean and a varianceof 0.5 per dimension. Therefore, when there areK uplinkusers, theN -length observation vector received by theuthreceive antenna is given by

yyyu =

K∑

k=1

h(u)sk1

tttk(sk1) +nnnu =

K∑

k=1

h(u)sk1

sk2ccck +nnnu,

u = 0, 1, · · · , U − 1 (3)

1Frequency-selective fading may be eliminated by multicarrierSM-SCDMA systems.

wherennnu is the Gaussian noise vector of zero mean and havinga covariance matrix of2σ2IIIN , expressed asCN (0, 2σ2IIIN ),with σ2 = 1/(2γ0), andγ0 = bγb being the SNR per symbol,while γb the SNR per bit.

Let yyy = [yyyT0 , yyyT1 , · · · , yyyTU−1]

T and nnn =[nnnT

0 ,nnnT1 , · · · ,nnnT

U−1]T . Then, we can expressyyy as

yyy =K∑

k=1

(IIIU ⊗ ccck)hhhsk1sk2 +nnn, (4)

where⊗ denotes the Kronecker product [49]. Furthermore, wecan rewrite (4) in a compact form as

yyy =HHH(sss1)sss2 +nnn (5)

whereHHH(sss1) = [(IIIU ⊗ ccc1)hhhs11 , · · · , (IIIU ⊗ cccK)hhhsK1], sss1 =

[s11, s21, ..., sK1]T collects the SSK symbols andsss2 =

[s12, s22, ..., sK2]T collects the APM symbols transmitted by

theK users.

III. S IGNAL DETECTION

In this section, we address the detection of SM-SCDMAsignals. We first consider the optimal MLD in Section III-A,whereas Section III-B derives the MAPD. Then, based on thisthe MPAD is developed in Section III-C.

A. Maximum-Likelihood Detection (MLD)

Let us express the symbols transmitted by theK usersas a vectorxxx = [x1, x2, · · · , xK ]T , where we expressxk = sk1|sk2, xk ∈ X . We assume that the BS employsthe knowledge of both the channels and of the spreadingsequences. Then, the MLD finds the estimate ofxxx by solvingthe optimization problem of2

xxx =arg minxxx∈XK

{

∥

∥

∥

∥

yyy −K∑

k=1

(IIIU ⊗ ccck)hhhsk1sk2

∥

∥

∥

∥

2}

(6)

=arg maxxxx∈XK

{

K∑

k=1

ℜ{s∗k2hhhHsk1

(IIIU ⊗ ccck)Hyyy}

−1

2

K∑

k=1

K∑

i=1

s∗k2hhhHsk1

(IIIU ⊗ ccck)H(IIIU ⊗ ccci)hhhsi1 si2

}

(7)

where xk = sk1|sk2 determines the termhhhsk1sk2, andℜ{x}

returns the real part ofx. In the above equation, we can showthat

hhhHsk1

(IIIU ⊗ ccck)Hyyy =

U−1∑

u=0

(h(u)sk1

)∗cccHk yyyu (8)

Furthermore, letCk be a set containing thedx indices ofccckhaving non-zero entries. Then (8) can be further simplified to

hhhHsk1

(IIIU ⊗ ccck)Hyyy =

U−1∑

u=0

∑

m∈Ck

(h(u)sk1

)∗c∗kmyum (9)

2Throughout the paper,x, x and x represent the symbol transmitted, thespecific symbol being tested by the search algorithms and the final detectedsymbol, respectively.

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The second term of (7) can be simplified as

hhhHsk1

(IIIU ⊗ ccck)H(IIIU ⊗ ccci)hhhsi1 = hhhH

sk1(IIIU ⊗ cccHk ccci)hhhsi1

=hhhHsk1

(IIIU ⊗∑

m∈Ck∩Ci

c∗kmcim)hhhsi1

=

(

∑

m∈Ck∩Ci

c∗kmcim

)

hhhHsk1

hhhsi1

(10)

whereCk∩Ci gives a set containing the indices where bothccckandccci have non-zero entries. Upon substituting (9) and (10)into (7), we obtain

xxx =arg maxxxx∈XK

{

K∑

k=1

U−1∑

u=0

∑

m∈Ck

ℜ{s∗k2(h(u)sk1

)∗c∗kmyum}

−1

2

K∑

k=1

K∑

i=1

(

∑

m∈Ck∩Ci

c∗kmcim

)

s∗k2hhhHsk1

hhhsi1 si2

}

(11)

From (11) we know that althoughCk has onlydx non-zeroelements, which can be exploited for significantly reducingthecomputation, the MLD’s complexity is stillO(MK). There-fore, the MLD described by (11) is impractical, especially,when the SM-SCDMA systems are proposed for supportingnumerous users (devices). Next, we consider the symbol-basedMAPD, which facilitates the employment of the MPA, asshown in Section III-C.

B. Maximum A-PosterioriDetection (MAPD)

Given the observation ofyyy in (4), the symbol-by-symbolbased MAPD detects thelth user’s symbol via maximizingthe posteriori probability as

xl =arg maxxl∈X

{P (xl|yyy)} = arg maxxl∈X

{P (xl)p(yyy|xl)}

=arg maxxl∈X

{

∑

xxxl∈XK−1

P (xxxl, xl)p(yyy|xxxl, xl)

}

(12)

where p(x|a) is the conditional probability density function(PDF) of x given a, P (a) is the a-priori probability of a,while xxxl is a (K − 1)-element vector obtained fromxxx byremoving thelth user’s symbol. Since the observations{yun}are independent for given(xxxl, xl), (12) can be written in detailas

xl = arg maxxl∈X

∑

xxxl∈XK−1

P (xxxl, xl)

U−1∏

u=0

N−1∏

n=0

p(yun|xxxl, xl)

(13)

Let Kl be a set containing all the interfering users sharing atleast one (non-zero) chip with userl. Let furthermorexxx[n] bea df -length vector containing the symbols sent by the userssharing thenth chip of the spreading sequences. Then (13)can be further simplified to

xl = arg maxxl∈X

∑

xxx′l∈X |Kl|

P (xxx′l, xl)

U−1∏

u=0

∏

n∈Cl

p(yun|xxx[n])

(14)

where |Kl| is the cardinality ofKl, andxxx′l is of |Kl|-length.

In (14), whenxxx[n] is given, the PDF ofp(yun|xxx[n]) can beexpressed as

p(yun|xxx[n]) =1

2πσ2exp

(

−‖yun −∑k∈Dn

h(u)sk1sk2ckn‖2

2σ2

)

(15)

where we defineDn the set containing the indices of the usersconveying information using thenth chip of the spreadingcodes.

Equation (14) shows that the complexity of detecting userl is O(M |Kl|). When regular sparse codes are employed,we have |Kl| = dx(df − 1), which is a constant usuallymuch smaller thanK. Hence, employing sparse codes forspreading is capable of reducing the detection complexity ofMAPD. Furthermore, when the MAPD is implemented byan approximated algorithm, the detection complexity can befurther reduced [2].

C. Message Passing Algorithm Aided Detection (MPAD)

Similar to the classic low-density parity-check codes [50],the input-output relationship in the proposed SM-SCDMAsystem can be described by a factor graph [2, 24], as shownin Fig. 2 for the parameters specified in the caption. In thisfactor graph, theK symbols of the formxk = sk1|sk2 andsent byK users are represented byK variable nodes, whiletheUN observations inyyy act asN function (or check) nodes,with each havingU channel inputs provided by theU receiveantennas. As shown in the figure, users1, 6 and7 share the0thfunction node, which has two observations obtained from thefirst and second receive antennas respectively. Hence, whilethese three users do interfere with each other on the0th chip,information carried by the different chips activated by thethreeusers can also be transferred from one chip to another viathe 0th function node. By doing this, the overall detectionperformance of SM-SCDMA may be significantly improved.Below we detail the MPAD.

Let us define two sets as

Kj = {i : 0 ≤ i ≤ N − 1, cji 6= 0} , j = 1, 2, . . . ,K

Ci = {j : 1 ≤ j ≤ K, cji 6= 0} , i = 0, 1, . . . , N − 1 (16)

where we have|Kj | = dx, representing thedx connectionswith the variable nodej, and |Ci| = df , giving the dfconnections with the function nodei.

Let us explicitly express the symbol set asX ={a0, a1, . . . , aM−1}, whereM = M1M2. Let the probabilityηam,(n)j,i be a message sent from the variable nodexj to

the function nodefi (corresponding to theith chip of thespreading sequences) during thenth iteration. Note that hereηam,(n)j,i is the probability that we havexj = am, given all the

messages received byxj from all of its neighboring functionnodes, excludingfi. Similarly, let the probabilityδam,(n)

i,j

represent a message sent from the function nodei to thevariable nodej during thenth iteration. Explicitly, this is theprobability thatxj = am, given the specific messages receivedby fi from all its neighboring variable nodes, excludingxj .

5

y02, y12 y03, y13 y04, y14 y05, y15y00, y10 y01, y11

x4x1 x2 x3 x5 x6 x7 x8 x9

Fig. 2. Factor graph representation of the SM-SCDMA system with the parameters ofU = 2, N = 6, K = 9, anddx = 2, df = 3.

Then, following [51, 52] and with the aid of Fig. 2, the MPADcan be described as follows.

Initially, we setηam,(0)j,i = 1/M for all am ∈ X and for any

specific(j, i) pairs, wherej ∈ Kj andi ∈ Ci. Then, during thenth, n ≥ 0, iteration, the probabilityδam,(n)

i,j can be computedas

δam,(n)i,j =

∑

xxx[i]∈Xdf−1,xj=am

∏

xv∈xxx[i]\xj

ηxv,(n)j,i

×U−1∏

u=0

p(yui|xxx[i], xj = am),

m = 0, . . . ,M − 1 (17)

for i ∈ Ci and j ∈ Kj , where∏

xv∈xxx[i]\xjηxv,(n)j,i is the

a-priori probability of a givenxxx[i] with xj = am, whilep(yui|xxx[i], xj = am) is given by (15).

Then, during the(n+ 1)st iteration, the valuesδam,(n)i,j for

i ∈ Ci andj ∈ Kj are used to computeηam,(n+1)j,i for j ∈ Kj

and i ∈ Ci, using the formula of

ηam,(n+1)j,i = ξj,i

∏

v∈Kj\i

δam,(n)v,j , m = 0, 1, . . . ,M − 1 (18)

whereξj,i is applied to make∑M−1

m=0 ηam,(n+1)j,i = 1.

Finally, when the affordable number of iterations is ex-hausted, the data symbols sent from theK users are detectedas

xj = argmaxam

∏

v∈Kj

δam,(n)v,j , k = 1, 2, . . . ,K (19)

The complexity of the MPAD is dependent on the numberof users sharing a chip. When regular sparse codes are used,the complexity per user can be shown to beO(Mdf ) [52].Hence, provided thatdx, df ≥ 2, the MPAD has a lowercomplexity than the MAPD considered in Section III-B, whichhas the complexity ofO(Mdx(df−1)) per user for regularsparse codes.

IV. PERFORMANCEANALYSIS

This section contributes to the error performance analysisof SM-SCDMA systems. We first analyze the single-userBER performance, which is the lower-bound BER of SM-SCDMA systems. Then, based on the single-user BER, wederive formulas for approximating the BER of SM-SCDMAsystems supportingK users.

A. Analysis of Single-User Average BER

When the SM-SCDMA system supports only a single user,the MLD of (7) becomes

x =argmaxx∈X

{

ℜ{s∗2hhhHs1(IIIU ⊗ ccck)

Hyyy} − 1

2‖hhhs1 s2‖2

}

(20)

where the indexk (except ccck) is dropped for notationalconvenience. A detection error occurs, whenx 6= x, whichresulted from the event that there is at least onex 6= x, yielding

ℜ{s∗2hhhHs1(IIIU ⊗ ccck)

Hyyy} − 1

2‖hhhs1 s2‖2

> ℜ{s∗2hhhHs1(IIIU ⊗ ccck)

Hyyy} − 1

2‖hhhs1s2‖2 (21)

Let us arrange (21) as

ℜ{(hhhs1 s2 − hhhs1s2)H(IIIU ⊗ ccck)

Hyyy}

>1

2

(

‖hhhs1 s2‖2 − ‖hhhs1s2‖2)

(22)

Then, upon substitutingyyy from (4) for K = 1 into (22), weobtain

ℜ{

(hhhs1 s2 − hhhs1s2‖)H nnn′}

>1

2

(

‖hhhs1 s2 − hhhs1s2‖2)

(23)

where nnn′ = (IIIU ⊗ ccc)Hnnn, which obeys the PDF ofCN (0, 2σ2IIIU ). Hence, the probability that the transmittedsymbol x is detected asx, which is usually referred to asthe PEP, is given by

PEP (x → x) =EHHH

[

Q

(

√

1

4σ2‖hhhs1 s2 − hhhs1s2‖2

)]

=EHHH

[

Q

(√

γ02‖hhhs1 s2 − hhhs1s2‖2

)]

(24)

whereQ(x) is the GaussianQ-function defined asQ(x) =(2π)−1/2

∫∞

xe−t2/2dt, and EHHH [·] denotes the expectation

with respect to the MIMO channelHHH between the user andthe BS.

As shown in [36, 53], the average bit error rate (ABER) ofSM schemes may be approximately evaluated from the union-bound expressed as

PbS ≤ 1

M1M2b

M1−1∑

m1=0

M2−1∑

m2=0

M1−1∑

m1=0

M2−1∑

m2=0

×D(

bbb(m1)1 |bbb(m2)

2 , bbb(m1)

1 |bbb(m2)

2

)

× PEP

(

s(m1)1 |s(m2)

2 → s(m1)1 |s(m2)

2

)

(25)

6

whereD(·, ·) is the Hamming distance between the two binaryentries, andbbb(m)

i , s(m)i represent themth realizations in bits

and symbols. Furthermore, in [53], the closed-form formulasfor the PEP of (24) have been derived in the context ofdifferent fading channels. Specifically, in this paper, we areonly interested in the iid Rayleigh fading channels. In thiscase, following the analysis in [53],PEP (x → x) can beanalyzed by dividing it into the following three cases:

1) (M1SSK-Error-Only):PEP (x → x) = PEP (s1|s2 →s1|s2);

2) (M2APM-Error-Only): PEP (x → x) = PEP (s1|s2 →s1|s2);

3) (Both M1SSK andM2APM in Error): PEP (x → x) =PEP (s1|s2 → s1|s2).

Upon invoking the alternative representation for theQ-function [54]3, these probabilities can be explicitly expressedas

PEP (s1|s2 → s1|s2) =1

π

∫ π/2

0

(

2 sin2 θ

2 sin2 θ + γ0|s2|2)U

dθ

(26a)

PEP (s1|s2 → s1|s2) =1

π

×∫ π/2

0

(

4 sin2 θ

4 sin2 θ + γ0|s2 − s2|2)U

dθ (26b)

PEP (s1|s2 → s1|s2) =1

π

×∫ π/2

0

(

4 sin2 θ

4 sin2 θ + γ0(|s2|2 + |s2|2)

)U

dθ (26c)

Equations (26a)-(26c) show that the fading channel effectshave been averaged out in the PEP. This means that thecomputation of (25) does not need to consider the specificM1SSK symbols, only theM2APM symbols need to beconsidered. Therefore, (25) can be simplified to [53]

PbS ≤M1b12M2b

M2−1∑

m2=0

PEP (s1|s(m2)2 → s1|s(m2)

2 ) (27a)

+1

M2b

M2−1∑

m2=0

M2−1∑

m2 6=m2

D(

bbb(m2)2 , bbb

(m2)

2

)

× PEP (s1|s(m2)2 → s1|s(m2)

2 ) (27b)

+1

M2b

M2−1∑

m2=0

M2−1∑

m2 6=m2

[

b1M1

2+ (M1 − 1)

× D(

bbb(m2)2 , bbb

(m2)

2

)]

PEP

(

s(m1)1 |s(m2)

2 → s(m1)1 |s(m2)

2

)

(27c)

where the first and third terms at the righthand side areobtained by exploiting the fact that the average number oferroneous bits between a pair of(bbb1, bbb1) is b1M1/[2(M1−1)].Furthermore, inPEP (·), si → si or s(m)

i → s(m)i means that

the error event corrupts the transmittedsi (or s(m)i ) into si (or

s(m)i ).

3Q(x) = π−1∫ π/20

exp(

−

x2

2 sin2 θ

)

dθ

Note first that if we assume that theM2APM uses Graycoding [54], the ABER of theM2APM is given by

Pb(M2APM) =1

M2b1

M2−1∑

m2=0

M2−1∑

m2 6=m2

D(

bbb(m2)2 , bbb

(m2)

2

)

× PEP (s1|s(m2)2 → s1|s(m2)

2 )

=Ps(M2APM)/b1 (28)

wherePs(M2APM) represents the average symbol error prob-ability of M2APM. For different types of APMs, there areclosed-form formulas to evaluate their ABER and averagesymbol error probability, which can be found in references,such as, [54–56]. Hence, we can use the existing formulas forPb(M2APM) as well asPs(M2APM), and evaluate (27b) asb1Pb(M2APM)/b or Ps(M2APM)/b.

Note secondly that, sinceU in (26) is an integer, accordingto (64) of [57] or (5A.4b) of [54], the PEPs in (26a) - (26c)can be expressed in closed form as

PEP (x → x) =

(

1− µ

2

)U U−1∑

u=0

(

U

u

)(

1 + µ

2

)u

(29)

whereµ corresponding to (26a), (26b) and (26c) is respectivelydefined as

µ =

√

γ0|s2|2γ0|s2|2 + 2

, if (26a) for Case (a)√

γ0|s2 − s2|2γ0|s2 − s2|2 + 4

, if (26b) for Case (b)√

γ0(|s2|2 + |s2|2)γ0(|s2|2 + |s2|2) + 4

, if (26c) for Case (c)

(30)

The PEPs can be readily computed with the aid of (29).

Equation (27) gives a bound for the ABER or the approxi-mate ABER (if the SNR is sufficiently high) of the single-userSM-SCDMA system, without informing us whether the erroris dominated by theM1SSK or theM2APM. This informationis sometimes important in design. For example, if a designerknows that one is more reliable than the other one, the twomodulation schemes can be used for unequal protection ofmultimedia information. Another consideration in the designis that the appropriateM1SSK andM2APM may be chosen sothat both the modulation schemes give a similar ABER. Thisis because the overall ABER is always dominated by the lessreliable one. In this case, the reliability of the more reliableone may be slightly reduced by transmitting at a higher rate,without unduly increasing the overall ABER.

Similar to the derivation of (25) [54], the ABER of theM1SSK and that of theM2APM have the following union-

7

bounds:

PbS(M1SSK) ≤ 1

M1b1

M1−1∑

m1=0

M2−1∑

m2=0

M1−1∑

m1=0

M2−1∑

m2=0

×D(

bbb(m1)1 , bbb

(m1)

1

)

PEP

(

s(m1)1 |s(m2)

2 → s(m1)1 |s(m2)

2

)

(31)

PbS(M2APM) ≤ 1

M2b2

M1−1∑

m1=0

M2−1∑

m2=0

M1−1∑

m1=0

M2−1∑

m2=0

×D(

bbb(m2)2 , bbb

(m2)

2

)

PEP

(

s(m1)1 |s(m2)

2 → s(m1)1 |s(m2)

2

)

(32)

Again, considering that the PEPs shown in (26a) - (26c) forthe three cases are not dependent on the fading channels, (31)and (32) can be simplified to the formulas of

PbS(M1SSK) ≤ M1

2

M2−1∑

m2=0

PEP

(

s1|s(m2)2 → s1|s(m2)

2

)

+M1

2

M2−1∑

m2=0

M2−1∑

m2 6=m2

PEP

(

s1|s(m2)2 → s1|s(m2)

2

)

(33)

PbS(M2APM) ≤ M1

M2b2

M2−1∑

m2=0

M2−1∑

m2 6=m2

D(

bbb(m2)2 , bbb

(m2)

2

)

× PEP

(

s1|s(m2)2 → s1|s(m2)

2

)

+M1(M1 − 1)

M2b2

M2−1∑

m2=0

M2−1∑

m2 6=m2

D(

bbb(m1)2 , bbb

(m1)

2

)

× PEP

(

s1|s(m2)2 → s1|s(m2)

2

)

(34)

In (33) and (34), the PEPs are respectively given in (26a) -(26c), which can be computed using (29) associated with (30).

B. Analysis of Approximate BER of SM-SCDMA Systems

In this section, we analyze the ABER of SM-SCDMAsystems with MLD. In practice we are usually interested inthe performance of an uncoded system at the specific ABERof about10−3, which can be readily corrected by FEC codes.For this ABER or a lower ABER, we can realize that theprobability of two or more users being simultaneously in errorat the same transmission instant should be negligible, providedthat the number of (active) users is significantly smaller than1/ABER. Hence, in order to make the analysis tractable, weassume that there is at most one user among the(K − 1)interfering users that may be in error.

Before we start analyzing the ABER of SM-SCDMA sys-tems, we can readily obtain from (25) and (26a) - (26c) thatthe ASER of the single-user SM-SCDMA system is bounded

as

PsS ≤ 1

M1M2

M1−1∑

m1=0

M2−1∑

m2=0

M1−1∑

m1=0

M2−1∑

m2=0

× PEP

(

s(m1)1 |s(m2)

2 → s(m1)1 |s(m2)

2

)

=M1 − 1

M2

M2−1∑

m2=0

PEP

(

s1|s(m2)2 → s1|s(m2)

2

)

+M1 − 1

M2

M2−1∑

m2=0

M2−1∑

m2 6=m2

PEP

(

s1|s(m2)2 → s1|s(m2)

2

)

+M1 − 1

M2

M2−1∑

m2=0

M2−1∑

m2 6=m2

PEP

(

s1|s(m2)2 → s1|s(m2)

2

)

(35)

This probability will be used later for computing the ABERof the SM-SCDMA systems.

Let us now assume that there is a desired user indexed inthe same way as in Section IV-A. Then, provided that all the(K − 1) interfering users are correctly detected, the ABER ofthe reference user is the same as (27).

By contrast, when an interfering user indexed as ‘k’, whichcan be any of the(K − 1) interfering users, is in error, wehave the pairwise erroneous event of

‖yyy − (IIIU ⊗ ccc)hhhs1 s2 − (IIIU ⊗ ccck)hhhsk1sk2‖2

< ‖yyy − (IIIU ⊗ ccc)hhhs1s2 − (IIIU ⊗ ccck)hhhsk1sk2‖2 (36)

from which we can derive the PEP, given as

P(i)EP (x, xk → x, xk) =EHHH

Q

√

√

√

√2

U−1∑

u=0

γ(i)u

(37)

where x, x, xk and xk represent respectivelys1|s2, s1|s2,sk1|sk2 and sk1|sk2, while

γ(i)u =γ0‖ccch(u)

s1s2 + ccckh

(u)sk1

sk2 − ccch(u)s1 s2 − ccckh

(u)sk1

sk2‖2/4= γ0α/4 (38)

where a superscripti is added to distinguish the different casesto be considered later.

When expressing the realizations ofx, x, xk, xk ass(m1)1 |s(m2)

2 , s(m1)1 |s(m2)

2 , s(mk1)k1 |s(mk2)

k2 , s(mk1)k1 |s(mk2)

k2 , sim-ilar to (25), we can represent the ABER of the SM-SCDMAsystem under the condition of a given interfering userk beingincorrectly detected as

PbM (k) ≈ 1

M1M2b

M1−1∑

m1=0

M2−1∑

m2=0

M1−1∑

m1=0

M2−1∑

m2=0

M1−1∑

mk1=0

M2−1∑

mk2=0

M1−1∑

mk1=0

×M2−1∑

mk2=0

D(

bbb(m1)1 |bbb(m2)

2 , bbb(m1)

1 |bbb(m2)

2

)

× PEP

(

s(m1)1 |s(m2)

2 , s(mk1)k1 |s(mk2)

k2

→ s(m1)1 |s(m2)

2 , s(mk1)k1 |s(mk2)

k2

)

(39)

where we simply use ‘≈’ to replace ‘<’, since certain assump-tions resulting in approximation are invoked, as noted at the

8

beginning of this section. In order to evaluate (39), we haveto consider the following nine cases:

1) P(1)EP (s1|s2, sk1|sk2 → s1|s2, sk1|sk2): γ

(1)u =

γ0‖ccc(h(u)s1

− h(u)s1 )s2 + ccck(h

(u)sk1

− h(u)sk1)sk2‖2/4

2) P(2)EP (s1|s2, sk1|sk2 → s1|s2, sk1|sk2): γ

(2)u =

γ0‖ccch(u)s1 (s2 − s2) + ccck(h

(u)sk1

− h(u)sk1)sk2‖2/4

3) P(3)EP (s1|s2, sk1|sk2 → s1|s2, sk1|sk2): γ

(3)u =

γ0‖ccc(h(u)s1

s2 − h(u)s1 s2) + ccck(h

(u)sk1

− h(u)sk1)sk2‖2/4

4) P(4)EP (s1|s2, sk1|sk2 → s1|s2, sk1|sk2): γ

(4)u =

γ0‖ccc(h(u)s1

− h(u)s1 )s2 + ccckh

(u)sk1(sk2 − sk2)‖2/4

5) P(5)EP (s1|s2, sk1|sk2 → s1|s2, sk1|sk2): γ

(5)u =

γ0‖ccch(u)s1 (s2 − s2) + ccckh

(u)sk1(sk2 − sk2)‖2/4

6) P(6)EP (s1|s2, sk1|sk2 → s1|s2, sk1|sk2): γ

(6)u =

γ0‖ccc(h(u)s1

s2 − h(u)s1 s2) + ccckh

(u)sk1(sk2 − sk2)‖2/4

7) P(7)EP (s1|s2, sk1|sk2 → s1|s2, sk1|sk2): γ

(7)u =

γ0‖ccc(h(u)s1

− h(u)s1 )s2 + ccck(h

(u)sk1

sk2 − h(u)sk1sk2)‖2/4

8) P(8)EP (s1|s2, sk1|sk2 → s1|s2, sk1|sk2): γ

(8)u =

γ0‖ccch(u)s1 (s2 − s2) + ccck(h

(u)sk1

sk2 − h(u)sk1sk2)‖2/4

9) P(9)EP (s1|s2, sk1|sk2 → s1|s2, sk1|sk2): γ

(9)u =

γ0‖ccc(h(u)s1

s2 − h(u)s1 s2) + ccck(h

(u)sk1

sk2 − h(u)sk1sk2)‖2/4

In the above expression, the indices ofmi or mi are removedfor notational simplicity. The same notational simplificationsare used in our forthcoming discourse, whenever there is noconfusion. Furthermore, in the case of, such as PEP, which isindependent of the specific index, such as a specific TA, thereis also no index formi or mi.

As seen in (38) or the above formulas corresponding to thenine cases, the instantaneous SNRγ(i)

u is a function of thelinear combination of 2-to-4 independent Gaussian randomvariablesh(u)

s1, h

(u)s1 , h

(u)sk1

and/or h(u)sk1 . Hence, for givenccc,

ccck, s2, s2, sk2 and sk2, the SNR γ(i)u obeys the Gamma

distribution with the PDF expressed in the form of [54, 56]

fγ(i)u(x) =

1

Γ(βi)

(

βi

γi

)βi

xβi−1 exp

(

−βix

γi

)

,

0 ≤ x < ∞ (40)

where the average SNRγi and the shaping parameterβi aredetermined by the first and second order moments ofγ

(i)u ,

given by

γi = E[γ(i)u ], βi =

γ2i

E[

(γ(i)u )2

]

− γ2i

(41)

In order to computeγi and βi with respect to the dif-ferent cases, we need the second and fourth order momentsof complex Gaussian random variables. Let us assume thatX = a+ jb, where botha andb are Gaussian distributed withzero mean and a variance ofσ2. Then, we haveE[X2] = 2σ2

andE[X4] = 8σ2. With these results in mind, we can readilyderive γi and βi for the nine cases, which are expressed

respectively as

γ1 =γ0(|s2|2 + |sk2|2)/2,

β1 =(|s2|2 + |sk2|2)2

|s2|4 + |sk2|4 + 2|ρk|2|s2|2|sk2|2; (42a)

γ2 =γ0(|s2 − s2|2 + 2|sk2|2)/4,

β2 =(|s2 − s2|2 + 2|sk2|2)2

|s2 − s2|4 + 4|sk2|4 + 4|ρk|2|s2 − s2|2|sk2|2; (42b)

γ3 =γ0(|s2|2 + |s2|2 + 2|sk2|2)/4,

β3 =(|s2|2 + |s2|2 + 2|sk2|2)2

(|s2|2 + s22)2 + 4|sk2|4 + 4|ρk|2(|s2|2 + |s2|2)|sk2|2

;

(42c)

γ4 =γ0(2|s2|2 + |sk2 − sk2|2)/4,

β4 =(2|s2|2 + |sk2 − sk2|2)2

4|s2|4 + |sk2 − sk2|4 + 4|ρk|2|s2|2|sk2 − sk2|2(42d)

γ5 =γ0(|s2 − s2|2 + |sk2 − sk2|2)/4,

β5 =(|s2 − s2|2 + |sk2 − sk2|2)2

|s2 − s2|4 + |sk2 − sk2|4 + 2|ρk|2|s2 − s2|2|sk2 − sk2|2;

(42e)

γ6 =γ0(|s2|2 + |s2|2 + |sk2 − sk2|2)/4,

β6 =(|s2|2 + |s2|2 + |sk2 − sk2|2)2

(|s2|2 + |s2|2)2 + |sk2 − sk2|4+ 2|ρk|2(|s2|2 + |s2|2)|sk2 − sk2|2

; (42f)

γ7 =γ0(2|s2|2 + |sk2|2 + |sk2|2)/4,

β7 =(2|s2|2 + |sk2|2 + |sk2|2)2

4|s2|4 + (|sk2|2 + |sk2|2)2+ 4|ρk|2|s2|2(|sk2|2 + |sk2|2)

; (42g)

γ8 =γ0(|s2 − s2|2 + |sk2|2 + |sk2|2)/4,

β8 =(|s2 − s2|2 + |sk2|2 + |sk2|2)2

|s2 − s2|4 + (|sk2|2 + |sk2|2)2+ 2|ρk|2|s2 − s2|2(|sk2|2 + |sk2|2)

; (42h)

γ9 =γ0(|s2|2 + |s2|2 + |sk2|2 + |sk2|2)/4,

β9 =(|s2|2 + |s2|2 + |sk2|2 + |sk2|2)2

(|s2|2 + |s2|2)2 + (|sk2|2 + |sk2|2)2+ 2|ρk|2(|s2|2 + |s2|2)(|sk2|2 + |sk2|2)

(42i)

In the above equations,ρk = cccHccck is the cross-correlationbetweenccc andccck. From γi andβi shown in (42a) - (42i), itis not hard for us to infer the following observations:

• In the formulas (42a) - (42i), when we set|sk2|2 = 0 and|sk2|2 = 0, we always obtainβi = 1 for i = 1, . . . , 9.Correspondingly, the nine cases are reduced to the threecases, which we considered earlier in Section IV-A forthe single-user scenario or that all the(K−1) interferingusers are correctly detected.

• γi is always increased due to the error event of a seconduser, meaning that having two users simultaneously inerror is rare compared to having a single user in error ina SM-SCDMA system.

• As 0 ≤ |ρk| < 1, we can find 1 < βi ≤ 2 forall the nine cases. Since a higherβi value results in alower error probability, as seen in [54, 56] and below,again, this explains that the probability of two users beingsimultaneously in error is lower than that of only one user

9

being in error.• When the erroneous userk does not share any chips with

the reference user, we then haveρk = 0. Consequently,the value ofβi becomes higher than that of the case,where the erroneous userk and the reference user sharesome chips for transmission. Therefore, an interferinguser not sharing chips with the reference user imposesa lower impact on the error probability of the referenceuser than an interfering user sharing some chips with thereference user.

Having obtainedγi and βi for the different cases, thecorresponding PEPs can be obtained via averaging (37) using(40), yielding [54, 56]

P(i)EP (x, xk → x, xk) =

1

π

∫ π/2

0

(

βi sin2 θ

βi sin2 θ + γi

)βiU

dθ,

i = 1, . . . , 9 (43)

which, according to [54, 56–58], can be represented in aclosed-form as

P(i)EP (x, xk → x, xk)

=Γ(βiU + 1/2)

2√πΓ(βiU + 1)

√

γiβi + γi

(

βi

βi + γi

)βiU

× 2F1

(

1, βiU +1

2;βiU + 1;

βi

βi + γi

)

,

i = 1, . . . , 9 (44)

where 2F1 (a, b; c; z) is the hypergeometric function definedas [59] 2F1 (a, b; c; z) =

∑∞k=0

(a)k(b)kzk

(c)kk!and (a)k = a(a +

1) · · · (a+ k − 1), (a)0 = 1.

Below we provide the expressions for computing the ap-proximate error probabilities in some special cases of practicalinterest.

First, for the ABER of the SM-SCDMA systems supportingK users (one reference user plus(K−1) interfering users), weassume that there is at most one interfering user in error. Thisassumption is reasonable due to the fact that when the SNR issufficiently high, the probability of having two or more userssimultaneously detected in error in addition to the referenceuser is negligible. For example, let us assume that the errorrate of a user is about10−3, which is the error rate of interestin practice. Then, when an optimum detector is employed, theaverage number of users erroneously detected during a channeluse is aboutK × 10−3, provided thatK is comparable tothe total number of chips used by the SM-SCDMA system.However, we should note that when there are two or moreusers simultaneously detected in error, the erroneous eventsof these users generate some correlation, as implied by (42a)- (42i). Nevertheless, this correlation should not significantlyaffect the average number of erroneous users, because theprobability of having two or more users simultaneously inerror is insignificant, in comparison to that of having onlyone user in error. Based on the above assumption and (35),

the ABER of the SM-SCDMA systems can be expressed as

PbM ≈(1− PsS)K−1PbS + [1− (1− PsS)

K−1]

× 1

K − 1

K−1∑

k=1

PbM (k) (45)

where PbM (k) is the ABER of the reference user when thekth interfering user is in error, which is given by (39). Whentaking into account our nine cases, we have

PbM (k) ≈ M21 (M1 − 1)b1

2M2b

M2−1∑

m2=0

M2−1∑

mk2=0

× PEP

(

s1|s(m2)2 , sk1|s(mk2)

k2 → s1|s(m2)2 , sk1|s(mk2)

k2

)

+M1(M1 − 1)

M2b

M2−1∑

m2=0

M2−1∑

m2 6=m2

M2−1∑

mk2=0

D(

bbb(m2)2 , bbb

(m2)

2

)

× PEP

(

s1|s(m2)2 , sk1|s(mk2)

k2 → s1|s(m2)2 , sk1|s(mk2)

k2

)

+M1(M1 − 1)

M2b

M2−1∑

m2=0

M2−1∑

m2 6=m2

M2−1∑

mk2=0

[

b1M1

2+ (M1 − 1)

× D(

bbb(m2)2 , bbb

(m2)

2

)]

× PEP

(

s1|s(m2)2 , sk1|s(mk2)

k2 → s1|s(m2)2 , sk1|s(mk2)

k2

)

+M2

1 b12M2b

M2−1∑

m2=0

M2−1∑

mk2=0

M2−1∑

mk2 6=mk2

PEP

(

s1|s(m2)2 , sk1|s(mk2)

k2

→ s1|s(m2)2 , sk1|s(mk2)

k2

)

+M1

M2b

M2−1∑

m2=0

M2−1∑

m2= 6=m2

M2−1∑

mk2=0

M2−1∑

mk2 6=mk2

D(

bbb(m2)2 , bbb

(m2)

2

)

× PEP

(

s1|s(m2)2 , sk1|s(mk2)

k2 → s1|s(m2)2 , sk1|s(mk2)

k2

)

+M1

M2b

M2−1∑

m2=0

M2−1∑

m2 6=m2

M2−1∑

mk2=0

M2−1∑

mk2 6=mk2

[

b1M1

2+ (M1 − 1)

×D(

bbb(m2)2 , bbb

(m2)

2

)]

× PEP

(

s1|s(m2)2 , sk1|s(mk2)

k2 → s1|s(m2)2 , sk1|s(mk2)

k2

)

+M2

1 (M1 − 1)b12M2b

M2−1∑

m2=0

M2−1∑

mk2=0

M2−1∑

mk2 6=mk2

PEP

(

s1|s(m2)2 ,

× sk1|s(mk2)k2 → s1|s(m2)

2 , sk1|s(mk2)k2

)

+M1(M1 − 1)

M2b

M2−1∑

m2=0

M2−1∑

m2 6=m2

M2−1∑

mk2=0

M2−1∑

mk2 6=mk2

D(

bbb(m2)2 , bbb

(m2)

2

)

× PEP

(

s1|s(m2)2 , sk1|s(mk2)

k2 → s1|s(m2)2 , sk1|s(mk2)

k2

)

+M1(M1 − 1)

M2b

M2−1∑

m2=0

M2−1∑

m2 6=m2

M2−1∑

mk2=0

M2−1∑

mk2 6=mk2

[

b1M1

2

+(M1 − 1)D(

bbb(m2)2 , bbb

(m2)

2

)]

× PEP

(

s1|s(m2)2 , sk1|s(mk2)

k2 → s1|s(m2)2 , sk1|s(mk2)

k2

)

(46)

10

Note that, in (46), all the PEPs are in the form of (43) or(44), with the correspondingγi andβi are respectively givenin (42a) - (42i), which are determined by the specific symbolsof s

(m2)2 , s

(mk2)k2 , s

(m2)2 , s

(mk2)k2 as well asρk of the cross-

correlation between the sparse codes of the reference user andof the kth interfering user.

Secondly, in order to reduce the computations invoked,another ABER expression can be obtained by considering onlythe specific interfering users sharing at least one chip withthereference user. In this case, the ABER can be expressed as

P ′bM ≈(1− PsS)

K−1PbS + [1− (1− PsS)K−1]

× 1

K − 1

∑

k∈K

PbM (k) (47)

whereK = {k|Ck ∩ C 6= ∅ ∀k = 1, 2, . . . ,K − 1}, and∅ isan empty set.

Thirdly, we are interested in the ABER of the referenceuser, when there is a single erroneous interfering user and thisuser does not share any chips with the reference user. ThisABER can be formulated as

PIM ≈ PbM (k) (48)

wherek /∈ K can be any arbitrary user not inK. Note thatany k /∈ K has the same impact on the reference user.

Finally, the ABER of the reference user on condition thatthere is only a single erroneous interfering user and this usershares some chips with the reference user can be expressed as

P ′IM ≈ 1

∑

1k∈K

∑

k∈K

PbM (k) (49)

where∑

1k∈K denotes the total number of the interferingusers sharing some chips with the reference user. Note thatwhen regular sparse codes are employed, implying that thenumber of chips shared between any interfering user inK withthe reference user is the same, then the averaging operationin (49) is not required. Instead, we can computeP ′

IM byconsidering a single interfering user randomly chosen fromK.

V. PERFORMANCERESULTS

In this section, both the theoretical error probability boundsand the simulation results of SM-SCDMA systems are pre-sented. Firstly, we propose a new 8QAM constellation, whichis then considered in some of the other figures. Then, theperformance of SM-SCDMA systems associated with variousparameters are characterized, based on which we also findthe valid range of our formulas derived. Finally, we study theperformance of relatively large-scale SM-SCDMA systems,which are impervious to numerical simulations. Note that theparameters used for generating the results of a specific figureare directly detailed in conjunction with the figure.

Again, let us propose a new 8QAM constellation forour forthcoming investigation. In the literature, typically thethree 8QAM constellations of Fig. 3(b)-(d) are considered,all of which facilitate Gray coding [60]. Specifically, theconstellation of Fig. 3(b) is designed based on the classicsquare-16QAM. It has apeak-to-average ratio (PAR) of 1.8,

and the minimum distance of 0.63√E0, where E0 is the

average phasor energy [60]. Here, we propose a new 8QAMconstellation also designed based on square-16QAM, as shownin Fig. 3(a). It can be shown that this new constellation hasthe same PAR as that shown in Fig. 3(b), but aminimumdistance of 0.88

√E0. However, the Gray coding cannot be

fully applied, as each 8QAM symbol has only three bits, butthere are two points, namely, ‘000’ and ‘010’, each havingfour neighbors.

000

001

011 011

000

100

101

101 111

001 011

010110

110

010

100

000

111

111 100

011

110

010

101

(b) (c) (d)

101 001

011000

100010

110111

(a)

Fig. 3. Different constellations for 8QAM.

10-5

10-4

10-3

10-2

10-1

Bit

Err

orR

ate

0 3 6 9 12 15 18 21 24 27 30

Average SNR per Bit (dB)

Square 8QAM (b)8PSK (c)Star-8QAM (d)Proposed 8QAM (a)

AWGNRayleigh Fading

Fig. 4. BER performance of different 8QAM constellations over Gaussianchannels.

The BER performance of the above four 8QAM constella-tions when communicating over Gaussian or Rayleigh fadingchannels is depicted in Fig. 4. Explicitly, in Gaussian channels,the proposed 8QAM scheme marginally outperforms all theothers 8QAM schemes, when the SNR is lower than about10 dB, i.e. in the SNR range of interest. By contrast, whenconsidering Rayleigh fading channels, the proposed 8QAMscheme slightly outperforms the others 8QAM schemes acrossthe whole SNR region considered. Note that since the proposed8QAM constellation is embedded in the 16QAM constellation,it is beneficial for the implementation of adaptive modu-lation [61] in practice. Therefore, in our following results,we assume the constellation of Fig. 3(a), when 8QAM isemployed.

Fig. 5 shows the BER performance of SM-SCDMA sys-tems, when different number of modulation levels are assumedfor theM1SSK andM2QAM. In this figure, we also comparethe results obtained by simulations and those evaluated fromthe ABER bounds of (45) and (47). Furthermore, the single-user ABER bound of (27) is provided. Based on the results, wehave the following observations. Firstly, when the throughput

11

Rayleigh fading channel, N=12, K=16, dx = 3, U=1

10-5

10-4

10-3

10-2

10-1

1A

BE

R

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

Average SNR per bit (dB)

M1=2, M2 = 4M1=4, M2 = 4M1=2, M2 = 16

SimulationPbM

P’bM

Single-user ABER bound

Fig. 5. ABER performance of SM-SCDMA systems for different numberof modulation levels over Rayleigh fading channels.

Rayleigh fading channel, N=12, K=16, dx = 3, U=2

10-5

10-4

10-3

10-2

10-1

1

AB

ER

0 2 4 6 8 10 12 14 16 18 20

Average SNR per bit (dB)

M1=2, M2 = 4M1=4, M2 = 4M1=2, M2 = 16

Single-user ABER boundSimulationP’bM

Fig. 6. ABER performance of SM-SCDMA systems with different numberof modulation levels over Rayleigh fading channels.

is increased either by increasingM1 or M2, the BER per-formance degrades. Secondly, when the SNR increases, theBER performance of SM-SCDMA system usingN = 12chips to supportK = 16 users, which corresponds to anormalized user-load factor of4/3, converges to the single-user BER. Thirdly, the ABER bound evaluated by (45) andthat evaluated by (47) are indistinguishable, which requiresmuch less computation than (45). Furthermore, as expected,at the BER of about10−3, the ABER bounds evaluated from(45) or (47) are close to the corresponding results obtainedviasimulations, and they become closer, as the SNR increases.Therefore, when the SNR is sufficiently high, resulting in aBER of about10−3 or lower, the single user ABER boundof (27) and the ABER bound of (45) or (47) can be used topredict the achievable performance of SM-SCDMA systems.

In Fig. 5, we assumedU = 1. By contrast, we assumeU =2 in Fig. 6. The other parameters are the same for both figures.By comparing the results of these two figures, we can see thata significant performance improvement is available, when the

number of receive antennas is increased fromU = 1 to U = 2,which is an explicit benefit of the diversity gain and the powergain. Furthermore, similar to Fig. 5, the BER obtained viasimulations lies between the single-user ABER bound of (27)and the ABER bound of (47) (or (45)). Therefore, we may use(27) and (47) to predict the BER performance of SM-SCDMAsystems, or simply use (47), when the SNR is sufficiently high,resulting in a BER of10−3 or lower.

Rayleigh fading channel, N=12, K=16, dx=3, U=1

10-5

10-4

10-3

10-2

10-1

1

App

roxi

mat

edA

BE

R10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

Average SNR per bit (dB)

M1=2, M2 = 4M1=4, M2 = 4

PbM

PIM

P’IM

Fig. 7. Comparison of the conditional ABER ofPIM and P ′

IM , and theABER of PbM of SM-SCDMA systems.

In Section IV, we have considered the ABERPIM of areference user, when there is a single erroneous interferinguser without overlapping with the reference user, which is ex-pressed in (48). We have also considered the ABERP ′

IM of areference user, when there is a single erroneous interfering usersharing some chips with the reference user, which is expressedin (49). In Fig. 7, we compare these two conditional ABERsas well asPbM of (45). The results show that for a given SNR,P ′IM is much higher thanPIM , implying that the ABER is

dominated by those specific erroneous interfering users, whichshare some chips with the reference user. This also explain thatin Figs. 5,PbM andP ′

bM are indistinguishable.Fig. 8 depicts the performance of SM-SCDMA systems

for the user-load factors ofK = 16, 20 and 24, giving thenormalized user-load factors of4/3, 5/3 and 2. In additionto the observations from the previous figures, Fig. 8 showsthat for the other parameters considered, the performance ofthe SCDMA systems only slightly degrades, as the numberof users increases. This is the case, in particular when a SM-SCDMA system employsU = 2 receive antennas. As shownin Fig. 8, a relatively substantial performance drop is observed,when the normalized user-load factor is changed from4/3 to5/3. By contrast, there is only a marginal performance loss,when the normalized user-load factor is increased from5/3 to2.

Again, SM-SCDMA systems rely on two types of modula-tions schemes, i.e. the SSK and QAM. While SSK belongs tothe family of energy-efficient modulation schemes, QAM isa bandwidth-efficient modulation scheme [55]. Hence, whenfixing the bandwidth, there should be a trade-off between the

12

Rayleigh fading channel, N=12, dx = 3, M1=M2=4

10-5

10-4

10-3

10-2

10-1

1B

itE

rror

Rat

e

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Average SNR per bit (dB)

SimulationP’bM

K=16K=20K=24

U=2

U=1

Fig. 8. ABER performance of SM-SCDMA systems for different user-loads.

Rayleigh fading channel, N=12, K=16, dx=3, U=1

10-4

2

5

10-3

2

5

10-2

2

5

10-1

2

AB

ER

12 16 20 24 28 32 36 40 44

Average SNR per bit (dB)

16SSK, BPSK8SSK, QPSK4SSK, 8QAM (Proposed)4SSK, Star-8QAM2SSK, 16QAM

Fig. 9. ABER performance of the SM-SCDMA system for differentmodulation schemes transmittingb = 5 bits per symbol.

numbers of bits allocated to these two modulation schemes.Therefore, in Fig. 9, we demonstrate this trade-off, whenassuming that the total number of bits per symbol is fixed to5.Explicitly, the combination of8SSK and QPSK achieves thebest error performance among the four possible combinationsover the whole SNR region considered. Additionally, in Fig.9the proposed 8QAM shows slightly better ABER performancethan the star 8QAM, when communicating over Rayleighfading channels. This is also reflected in Fig. 9, where theproposed 8QAM scheme slightly outperforms the star 8QAMoperating in Rayleigh fading channels over the whole SNRrange considered.

So far, we have considered some relatively small SM-SCDMA systems associated withN = 12, so that we can usesimulation results to validate our mathematical analysis andgain insights into the characteristics of the various expressionsderived. Below we consider some relatively large-scale SM-SCDMA systems, whose performance is infeasible to studyby Monte-Carlo simulations.

In Figs. 10 and 11, we plot the approximated ABER evalu-

Rayleigh fading channel, N=128, dx = 2, M1 = 4, M2=8

10-5

10-4

10-3

10-2

10-1

1

App

roxi

mat

edA

BE

R

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

Average SNR per bit (dB)

K=128K=192K=256

U=2

U=1

Fig. 10. ABER performance of SM-SCDMA systems for different user-loads.

Rayleigh fading channel, N=256, dx = 2, M1 = 4, M2=8

10-5

10-4

10-3

10-2

10-1

1

App

roxi

mat

edA

BE

R

10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

Average SNR per bit (dB)

K=256K=384K=512

U=2

U=1

Fig. 11. ABER performance of SM-SCDMA systems for different user-loads.

ated fromP ′bM of (47) forN = 128 and256, respectively. For

both cases, we examine the performance, when the normalizeduser-load factors are respectively1, 1.5 and 2. We can seefrom the results that the error performance curves of all thethree loading factors are close to each other. From the resultsof Figs.8, 10 and 11 we infer that when a SM-SCDMA systembecomes larger in terms ofN , it is capable of supporting aheaver user-load at a given error rate.

VI. CONCLUSIONS

We have proposed a SM-SCDMA system for supportinghigh-user-load MA transmission. In order to show the potentialof SM-SCDMA and to allow it to achieve low-complexitynear-optimum detection, we have considered both the MLDand the MPAD developed based on the MAPD. We haveadded new ingredients to the error performance analysis ofSM systems and analyzed the ABER of SM-SCDMA systemsemploying MLD. A range of expressions have been obtainedfor estimating the ABER of SM-SCDMA systems operating indifferent situations. Finally, the performance of SM-SCDMA

13

systems has been investigated based on both simulations andnumerical evaluation of the expressions derived. Our studiesand performance results demonstrate that SM-SCDMA iscapable of supporting large-scale MA. It is also capable ofexploiting the space-time domain resources for improvingenergy efficiency. With the aid of the MAPD, it can supporteven a normalized user-load factor of2 without much perfor-mance degradation. Furthermore, when a SM-SCDMA systembecomes larger in terms of the spreading factor obtained byutilizing more bandwidth, it becomes more efficient, and theperformance only slightly degrades, when the normalized user-load factor is increased from one to two. Additionally, we notethat the error performance achieved by the MAPD is nearoptimum, which is close to that attained by the MLD.

Our future research will investigate SM-SCDMA on thebasis of multicarrier transmission over frequency-selectivefading channels.

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Yusha Liu received Her BEng degree in Communi-cation Engineering from Xiamen University, China,and her MSc degree in wireless communicationsfrom the University of Southampton, UK, whereshe is currently pursuing her PhD degree with theSouthampton Wireless Group. Her research interestsinclude spatial modulation, non-orthogonal mulitipleaccess (NOMA) and compressive sensing for wire-less communications.

Lie-Liang Yang (M’98, SM’02, F’16) received hisBEng degree in communications engineering fromShanghai TieDao University, Shanghai, China in1988, and his MEng and PhD degrees in communi-cations and electronics from Northern (Beijing) Jiao-tong University, Beijing, China in 1991 and 1997,respectively. From June 1997 to December 1997he was a visiting scientist to the Institute of RadioEngineering and Electronics, Academy of Sciencesof the Czech Republic. Since December 1997, he hasbeen with the University of Southampton, United

Kingdom, where he is the professor of wireless communications in theSchool of Electronics and Computer Science. He has research interest inwireless communications, wireless networks and signal processing for wirelesscommunications, as well as molecular communications and nano-networks.He has published over 350 research papers in journals and conferenceproceedings, authored/co-authored three books and also published severalbook chapters. The details about his research publicationscan be found athttp://www.mobile.ecs.soton.ac.uk/lly/. He is a fellow ofboth the IEEE and theIET, and a distinguished lecturer of the IEEE. He served as anassociate editorto the IEEE Trans. on Vehicular Technology and Journal of Communicationsand Networks (JCN), and is currently an associate editor to the IEEE Access.

Lajos Hanzo (http://www-mobile.ecs.soton.ac.uk)FREng, FIEEE, FIET, Fellow of EURASIP, DScreceived his degree in electronics in 1976 and hisdoctorate in 1983. In 2009 he was awarded anhonorary doctorate by the Technical University ofBudapest and in 2015 by the University of Edin-burgh. In 2016 he was admitted to the HungarianAcademy of Science. During his 40-year career intelecommunications he has held various researchand academic posts in Hungary, Germany and theUK. Since 1986 he has been with the School of

Electronics and Computer Science, University of Southampton, UK, wherehe holds the chair in telecommunications. He has successfullysupervised 111PhD students, co-authored 18 John Wiley/IEEE Press books onmobile radiocommunications totalling in excess of 10 000 pages, published1701 researchcontributions at IEEE Xplore, acted both as TPC and General Chair of IEEEconferences, presented keynote lectures and has been awarded a number ofdistinctions. Currently he is directing a 60-strong academic research team,working on a range of research projects in the field of wireless multimediacommunications sponsored by industry, the Engineering and Physical SciencesResearch Council (EPSRC) UK, the European Research Council’s AdvancedFellow Grant and the Royal Society’s Wolfson Research MeritAward. He isan enthusiastic supporter of industrial and academic liaison and he offers arange of industrial courses. He is also a Governor of the IEEEVTS. During2008 - 2012 he was the Editor-in-Chief of the IEEE Press and a ChairedProfessor also at Tsinghua University, Beijing. For further information onresearch in progress and associated publications please refer to http://www-mobile.ecs.soton.ac.uk Lajos has 30 000+ citations and an H-index of 70.