dh1101

Copyright (c) 2011 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing [email protected].

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Bit Error Probability of Spatial Modulation (SM)MIMO over Generalized Fading Channels

Marco Di Renzo, Member, IEEE and Harald Haas, Member, IEEE

AbstractIn this paper, we study the performance of SpatialModulation (SM) MultipleInputMultipleOutput (MIMO)wireless systems over generic fading channels. More precisely,a comprehensive analytical framework to compute the AverageBit Error Probability (ABEP) is introduced, which can be usedfor any MIMO setups, for arbitrary correlated fading channels,and for generic modulation schemes. It is shown that, whencompared to stateoftheart literature, our framework: i) hasmore general applicability over generalized fading channels; ii) is,in general, more accurate as it exploits an improved unionboundmethod; and, iii) more importantly, clearly highlights interestingfundamental trends about the performance of SM, which aredifficult to capture with available frameworks. For example, byfocusing on the canonical reference scenario with independentand identically distributed (i.i.d.) Rayleigh fading, we introducevery simple formulas which yield insightful design informationon the optimal modulation scheme to be used for the signalconstellation diagram, as well as highlight the different roleplayed by the bit mapping on the signal and spatialconstellationdiagrams. Numerical results show that, for many MIMO setups,SM with Phase Shift Keying (PSK) modulation outperformsSM with Quadrature Amplitude Modulation (QAM), which isa result never reported in the literature. Also, by exploitingasymptotic analysis, closedform formulas of the performancegain of SM over other singleantenna transmission technologiesare provided. Numerical results show that SM can outperformmany singleantenna systems, and that for any transmission ratethere is an optimal allocation of the information bits onto spatialand signalconstellation diagrams. Furthermore, by focusing onthe Nakagamim fading scenario with generically correlatedfading, we show that the fading severity plays a very importantrole in determining the diversity gain of SM. In particular,the performance gain over singleantenna systems increases forfading channels less severe than Rayleigh fading, while it getssmaller for more severe fading channels. Also, it is shown thatthe impact of fading correlation at the transmitter is reduced forless severe fading. Finally, analytical frameworks and claims aresubstantiated through extensive Monte Carlo simulations.

Index TermsLargescale antenna systems, massivemultipleinputmultipleoutput (MIMO) systems, performanceanalysis, singleRF MIMO design, spatial modulation (SM).

Manuscript received August 25, 2011; revised December 15, 2011; acceptedJanuary 17, 2012. This paper was presented in part at the IEEE/ICST Int. Conf.Communications and Networking in China (CHINACOM), Beijing, China,August 2010. The review of this paper was coordinated by Dr. E. Au.

Copyright (c) 2012 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to [email protected].

M. Di Renzo is with the Laboratoire des Signaux et Syste`mes, UniteMixte de Recherche 8506, Centre National de la Recherche ScientifiqueEcole Superieure dElectriciteUniversite ParisSud XI, 91192 GifsurYvette Cedex, France, (email: [email protected]).

H. Haas is with The University of Edinburgh, College of Science andEngineering, School of Engineering, Institute for Digital Communications(IDCOM), Kings Buildings, Mayfield Road, Edinburgh, EH9 3JL, UnitedKingdom (UK), (email: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier XXX.XXX/TVT.XXX.XXX

I. INTRODUCTION

SPATIAL modulation (SM) is a digital modulation con-cept for MultipleInputMultipleOutput (MIMO) wire-less systems, which has recently been introduced to increasethe data rate of singleantenna systems by keeping a lowcomplexity transceiver design and by requiring no band-width expansion [1][4]. Unlike conventional spatial mul-tiplexing schemes [2], [5], in SM the multiplexing gainis realized through mapping a block of information bitsinto two informationcarrying units: the conventional signalconstellation diagram (e.g., Phase Shift Keying (PSK) orQuadrature Amplitude Modulation (QAM)) and the socalledspatialconstellation diagram, which is the antennaarray atthe transmitter. Like in conventional modulation schemes, thefirst subblock of information bits determines the point ofthe signalconstellation diagram that is actually transmitted.Specific to SM is that the second subblock identifies thesingle active transmitantenna. As a result, the point of thesignalconstellation diagram is transmitted through a singleactive antenna belonging to the spatialconstellation diagram.This simple modulation concept brings two main advantages:i) for each channel use, the data rate increases by a factor equalto the logarithm of the number of antennas at the transmitter[2], [5]; and ii) the receiver can detect the whole block of bitsthrough singlestream demodulation, as the second subblockof bits is only implicitly transmitted through the activationof the transmitantenna [6]. With respect to singleantennasystems, the net gain is a multiplexing gain for the samedecoding complexity, while the price to pay is the need of moreantennas at the transmitter. Recent results have showcased theperformance gain of SM with respect to other stateoftheart transmission technologies for single and multiantennasystems [5][19]. Finally, it is worth mentioning that SMseems to be an appealing transmission technology for highrate and lowcomplexity MIMO implementations that exploitthe recently proposed massive MIMO or largescale an-tenna systems paradigm [20], [21]. In fact, in these systems itis envisaged that improved performance and energy efficiencycan be achieved by using tens or hundreds antenna elements atthe base station, instead of exploiting base station cooperation.In this perspective, SM can be regarded as a lowcomplexitymodulation scheme that exploits the massive MIMO idea butwith a single active RF chain. The design of MIMO schemesthat retain the benefits of multipleantenna transmission whilehaving a single active RF element is another recent and majortrend in current and future MIMO research [22].Since its introduction, many researchers have been study-

ing the performance of SMMIMO over fading channels,



either through timeconsuming Monte Carlo simulations orthrough analytical modeling. Despite being more challenging,analytical modeling is, in general, preferred because: i) itallows a deeper understanding of the system performance;ii) it enables a simpler comparison with other competingtransmission technologies; and iii) it provides opportunitiesfor system optimization. A careful look at current stateoftheart reveals the following contributions. The vast majorityof analytical frameworks are useful for a special case ofSM, which is called SpaceShiftKeying (SSK) modulation[8]. SSK modulation is a lowcomplexity and lowdatarateversion of SM where only the spatialconstellation diagram isexploited for data modulation. This transmission technologyis extensively studied in [23][29] for various MIMO setupsand channel models. The analytical study conducted in [23][29] has highlighted fundamental properties of the spatialconstellation diagram with respect to fading severity, channelcorrelation, power imbalance, transmitdiversity, as well asrobustness to multipleaccess interference and channel estima-tion errors. However, these frameworks are of limited use tounderstand the performance of SM, as the signalconstellationdiagram is neglected. On the other hand, analytical modelingof SM is limited to a very few papers, which have variouslimitations. In [5] and [30], the authors study a suboptimalreceiver design and the Symbol Error Probability (SEP) iscomputed by resorting to numerical integrations, which are noteasy to compute and, in some cases, are numerically unstable.In [6], the authors study the Average Bit Error Probability(ABEP) of the MaximumLikelihood (ML) optimum receiverover independent and identically distributed (i.i.d.) Rayleighfading. The framework is based on the unionbound method.Due to the absence of a scaling factor in the final formula [31],this bound is rather weak. Furthermore, and, more importantly,the framework is valid for realvalued signalconstellationpoints, and, thus, it cannot be used, e.g., for PSK and QAM.In [9], the authors provide a first closedform frameworkto compute the ABEP of SMMIMO over generically cor-related Rician fading and for arbitrary modulation schemes.Also, the framework highlights some fundamental behaviorsof SM, such as its incapability to achieve transmitdiversity[17]. However, [9] has the following important limitations:i) the analysis is applicable to Rician fading only; ii) theframework is based on conventional unionbound methods,whose accuracy degrades for high modulation orders and smallnumbers of receiveantennas; and iii) signal and spatialconstellation diagrams are treated as a single entity, whichdoes not highlight the role played by each of them for variousfading channels and modulation schemes.In this depicted context, this paper is aimed at proposing

a comprehensive analytical framework to study the ABEP ofSMMIMO over generalized fading channels. More specifi-cally, we are interested in studying: i) the interplay of signaland spatialconstellation diagrams, and whether an optimalallocation of the information bits between them exists; ii)the effect of adding the spatialconstellation diagram on topof the signalconstellation diagram, and whether conventionalsignal modulation schemes (e.g., PSK and QAM) are the bestchoice for SM, or whether new optimal modulation schemes

should be designed to fully exploit the benefits of this hybridmodulation scheme; and iii) advantages and disadvantages ofSM with respect to conventional singleantenna PSK/QAMand SSK modulations, as a function of the MIMO setup andfading scenario. To this end, we propose a new analyticalframework that foresees to write the ABEP as the summationof three contributions: 1) a term that mainly depends on thesignalconstellation diagram; 2) a term that mainly dependson the spatialconstellation diagram; and 3) a joint term thatdepends on both constellation diagrams and highlights theirinteractions. This new analytical formulation allows us tointroduce an improved unionbound method, which is moreaccurate than conventional unionbound methods, and enablesa deeper understanding of the role played by both informationcarrying units for various channel models and MIMO setups.Some of the most important and general results of this paperare as follows: i) we show that SM outperforms singleantennaPSK/QAM schemes only for data rates greater than 2bpcu(bits per channel use), and that SM with QAMmodulatedpoints in the signalconstellation diagram is never superiorto singleantenna QAM for singleantenna receivers. On theother hand, for multiantenna receivers and higher data ratesSM can significantly outperform singleantenna PSK/QAM.Closedform expressions of this performance gain over i.i.d.Rayleigh fading are given; ii) unlike singleantenna systems,where QAM always outperforms PSK, we show that SM withPSKmodulated points in the signalconstellation diagramcan outperform SM with QAMmodulated points. This isdue to the interactions of signal and spatialconstellationdiagrams, and for i.i.d. Rayleigh fading we show analyticallythat the ABEP of SM does not depend only on the Euclideandistance of the points in the signalconstellation diagram.This provides important information on how to conceive newmodulation schemes that are specifically optimized for SM;iii) by considering, as a case study, Nakagamim fading, weshow that the fading severity, mNak, plays an important roleon the performance of SM. More specifically, like conven-tional modulation schemes, the ABEP gets worse for wirelesschannel with fading more severe (0:5 mNak < 1) thanRayleigh. However, with respect to singleantenna PSK/QAMmodulation, the performance gain of SM increases thanks tothe higher diversity gain experienced by the information bitsmapped onto the spatialconstellation diagram. On the con-trary, the performance gain decreases for less severe (mNak >1) fading because the diversitygain of the bits mapped ontothe spatialconstellation diagram is independent of the fadingparameter mNak. Also, it is shown that channel correlation atthe transmitter has a less impact when 0:5 mNak < 1.The remainder of this paper is organized as follows. In

Section II, the system model is described. In Section III, theimproved unionbound is introduced, and the specific contri-bution of spatial and signalconstellation diagrams is shown.In Section IV, closedform expressions of the ABEP forvarious fading channels and modulation schemes are provided.In Section V, the canonical i.i.d. Rayleigh fading channel isstudied in detail, and closedform expressions of the gain withrespect to singleantenna PSK/QAM and SSK modulationsare given. In Section VI, numerical results are shown to



substantiate claims and analytical derivations. Finally, SectionVII concludes this paper.

II. SYSTEM MODEL

We consider a generic Nt Nr MIMO system, where Ntand Nr denote the antennas at the transmitter and at thereceiver, respectively. We assume that the transmitter can senddigital information via M complex symbols. In SM literature[4], the set of Nt antennas (nt = 1; 2; : : : ; Nt) is calledspatialconstellation diagram, while the set of M complexpoints (l for l = 1; 2; : : : ;M ) is called signalconstellationdiagram. The basic idea of SM is to map blocks of informationbits into two information carrying units [5]: 1) a symbol, whichis chosen from the complex signalconstellation diagram; and2) a single active transmitantenna, which is chosen from thespatialconstellation diagram.More specifically, SM works as follows. At the transmitter,

the bitstream is divided into blocks containing log2 (Nt) +log2 (M) bits each, with log2 (Nt) and log2 (M) being thenumber of bits needed to identify a transmitantenna in thespatialconstellation diagram and a symbol in the signalconstellation diagram, respectively. Each block is split intotwo subblocks of log2 (Nt) and log2 (M) bits each. Thebits in the first subblock are used to select the transmitantenna that is switched on for transmission, while all theother antennas are kept silent. The bits in the second subblock are used to choose a symbol in the signalconstellationdiagram. At the receiver, the detector can recover the wholeblock of log2 (Nt) + log2 (M) information bits by solving anNtMhypothesis detection problem, which jointly estimatesthe transmitantenna that is not idle and the signal waveformthat is transmitted from it.In this paper, the generic block of log2 (Nt)+log2 (M) bits

is called message, and it is denoted by (nt; l), wherent = 1; 2; : : : ; Nt and l = 1; 2; : : : ;M univocally identifythe active transmitantenna, nt, and the complex symbol, l,transmitted from it, respectively. The Nt M messages areequiprobable.

A. Notation

Throughout this paper, we use the notation as follows.i) We adopt a complexenvelope signal representation. ii)j =

p1 is the imaginary unit. iii) () is the complexconjugate operator. iv) (x y) (t) = R +11 x () y (t ) dis the convolution of x () and y (). v) jj is the absolutevalue. vi) E fg is the expectation operator. vii) Re fg andIm fg are real and imaginary part operators. viii) (x) =R +10

x1 exp () d is the Gamma function. ix) Q (x) =1p

2 R +1

xexp

t22 dt is the Qfunction. x) Em isthe average energy per transmission. xi) Tm is the transmissiontimeslot of each message. xii) w () is the unitenergy, i.e.,R +11 jw (t)j2 dt = 1, elementary transmitted pulse waveformthat is nonzero only in [0; Tm]. xiii) The signal relatedto (nt; l) and transmitted from antenna nt is denotedby s ( tj (nt; l)) =

pEmlw (t). xiv) The generic point

of the signalconstellation diagram, l, is defined as l =Rl + j

Il = l exp (jl), where

Rl = Re flg, Il =

Im flg, l =q

Rl2

+Il2, and l = arctan

IlRl.

xv) Pr fg denotes probability. xvi) The noise nr at theinput of the nrth (nr = 1; 2; : : : ; Nr) receiveantenna is acomplex Additive White Gaussian Noise (AWGN) process,with power spectral density N0 per dimension. Across thereceiveantennas, the noises nr are statistically independent.xvii) We introduce =Em/(4N0). xviii) () is the Dirac deltafunction. xix) bxe is the function that rounds x to the closestinteger. xx) bc is the floor function. xxi) Gm;np;q

:j (ap)

(bq)

is the MeijerG function defined in [32, Ch. 8, pp. 519].xxii) MX (s) = E fexp (sX)g is the Moment GeneratingFunction (MGF) of Random Variable (RV) X . xxiii) X

d=Y

denotes that the RVs X and Y are equal in distribution or law,i.e., they have the same Probability Density Function (PDF).xxiv) (x!) is the factorial of x. xxv) ()1 is the inverse of asquare matrix. xxvi) Iv () is the modified Bessel function offirst kind and order v [33, Ch. 9]. xxvii)

is the binomial

coefficient. xxviii) NH~nt; ~l

! (nt; l) is the Hammingdistance of messages

~nt; ~l

and (nt; l), i.e., the number

of positions where the information bits are different, with0 NH

~nt; ~l

! (nt; l) log2 (NtM).B. Channel ModelWe consider the frequencyflat slowlyvarying fading chan-

nel model as follows: hnt;nr () = nt;nr ( nt;nr ) is the channel impulseresponse of the wireless link from the ntth transmitantenna to the nrth receiveantenna. nt;nr =

Rnt;nr +

jInt;nr = nt;nr exp (j'nt;nr ) is the complex chan-nel gain, and nt;nr is the propagation timedelay. Nospecific distribution for the channel envelopes, nt;nr =q

Rnt;nr2

+Int;nr

2, the channel phases, 'nt;nr =

arctanInt;nr

Rnt;nr

, and Rnt;nr = Re fnt;nrg,

Int;nr = Im fnt;nrg is assumed a priori. The delays nt;nr are assumed to be known at thereceiver, i.e., perfect timesynchronization is considered.Also, we assume 1;1 = 1;2 = : : : = Nt;Nr , whichis a realistic assumption when the distance betweentransmitter and receiver is much larger than the spacingof the antenna elements [24]. Due to these assumptions,the delays nt;nr are neglected in the next sections.

C. MLOptimum DetectorLet

~nt; ~l

be the transmitted message1. The signal

received by the nrth receiveantenna, if ~nt; ~l

is trans-

mitted, is:

znr (t) = sch;nrtj ~nt; ~l+ nr (t) (1)

where sch;nrtj ~nt; ~l =

s j ~nt; ~l h~nt;nr (t) = pEm~nt;nr~lw (t) for

~nt = 1; 2; : : : ; Nt, nr = 1; 2; : : : ; Nr, and ~l = 1; 2; : : : ;M .

1We emphasize that symbols with ~ identify the actual message that istransmitted, while symbols without ~ denote the trial message that is testedby the detector to solve the Nt Mhypothesis detection problem. Also,symbols with ^ denote the message estimated by the detector. This notationdoes not apply to the antennaindex, nr , at the receiver since there is nohypothesistesting in this case.



n^t; l^

= argmax

for nt=1;2;:::;Ntand l=1;2;:::;MfD (nt; l)g

= argmaxfor nt=1;2;:::;Ntand l=1;2;:::;M

(NrXnr=1

ZTm

znr (t) sch;nr ( tj (nt; l)) dt

1

2

ZTm

jsch;nr ( tj (nt; l))j2 dt) (2)

ABEP = E

8



8>>>>>>>>>>>>>:

ABEPsignal =1Nt

log2(M)log2(NtM)

NtPnt=1

ABEPMOD (nt)

ABEPspatial =1M

log2(Nt)log2(NtM)

MPl=1

ABEPSSK (l)

ABEPjoint =1

NtM1

log2(NtM)

NtPnt=1

MPl=1

NtP~nt 6=nt=1

MP~l 6=l=1

NH (~nt ! nt) +NH

~l ! l

nt; l; ~nt; ~l

(7)

8>>>>>:ABEPMOD (nt) =

1M

1log2(M)

MPl=1

MP~l=1

NH

~l ! l

E(nt)

Prl^ = l

~lABEPSSK (l) =

1Nt

1log2(Nt)

NtPnt=1

NtP~nt=1

[NH (~nt ! nt)l (nt; ~nt)](8)

Euclidean distance of the points in the signalconstellationdiagram, and, thus, ABEPsignal can be regarded as the termthat shows how the signalconstellation diagram affects theperformance of SM. 2) ABEPspatial is the summation of Maddends ABEPSSK (). From, e.g., [24, Eq. (35)], we observethat ABEPSSK () is the ABEP of an equivalent SSKMIMOscheme, where is replaced by 2l . Except for this scalingfactor, ABEPSSK () only depends on the Euclidean distanceof the points in the spatialconstellation diagram, and, thus,ABEPspatial can be regarded as the term that shows howthe spatialconstellation diagram affects the performance ofSM. 3) ABEPjoint has a more complicated structure, andit depends on the Euclidean distance of points belonging tosignal and spatialconstellation diagrams. Thus, it is calledjoint because it shows how the interaction of these two nonorthogonal diagrams affects the ABEP of SM.Finally, let us emphasize that: i) even though Proposition 1

might seem a simple and less compact rearrangement of (3),in Section IV and in Section V we show that (6)(8) allowus to get very simple, and, often, closedform expressionsfor specific modulation schemes and fading channels; andii) unlike ABEPSSK () and ABEPjoint, which are obtainedthrough conventional unionbound methods, ABEPMOD () isthe exact error probability related to the signalconstellationdiagram. In other words, no unionbound is used to com-pute this term. The exact computation of ABEPMOD ()avoids the inaccuracies of using the unionbound method forperformance analysis of conventional modulation schemes,especially for largeM and smallNr [34], [37]. For this reason,we call the framework in (6)(8) improved unionbound. Thebetter accuracy of this new bound is substantiated in SectionVI through Monte Carlo simulations. For the convenienceof the reader, in Table I we report the exact expressionof ABEPMOD () in (8) for PSK and QAM modulations.Formulas in Table I are useful for arbitrary fading channels,and when Gray coding is used to map the information bitsonto the signalconstellation diagram.

IV. SIMPLIFIED EXPRESSIONS OF THE ABEP

Proposition 1 provides a very general framework to com-pute the ABEP for arbitrary fading channels and modulationschemes. By direct inspection, we notice that (6)(8) can

be computed in closedform if the MGFs of the SignaltoNoiseRatios (SNRs) (nt), (nt;~nt), and (nt;l;~nt;~l) areavailable in closedform. If so, the ABEP can be obtainedthrough the computation of simple singleintegrals and sum-mations. More specifically, M(nt) () is available in [34] formany correlated fading channels, which allows us to computeABEPMOD (), and, eventually, ABEPsignal. On the otherhand, the computation of M(nt;~nt) () and M(nt;l;~nt;~l) ()deserves further attention, as they are not available in theliterature for arbitrary fading channels. Thus, the objective ofthis section is threefold: i) to compute closedform expressionsof M(nt;~nt) () and M(nt;l;~nt;~l) () for the most commonfading channel models; ii) to provide simplified formulas ofthe ABEP in (7) and (8) for specific modulation schemes andfading channels; and iii) to analyze the obtained formulas tobetter understand SM. To our best knowledge, and accordingto Section I, such a comprehensive study is not available inthe literature.

A. Identically Distributed Fading at the Transmitter

Let us consider the scenario with identically dis-tributed fading at the transmitter. We study uncorrelatedand correlated fading, where in the latter case the termidentically distributed means that all pairs of wire-less links are equicorrelated. In formulas, this implies:M(nt) (s) = MMOD (s), M(nt;~nt) (s) = MSSK (s), andM

(nt;l;~nt;~l)(s) =M

(l;~l)(s) for nt = 1; 2; : : : ; Nt and

~nt = 1; 2; : : : ; Nt, which means that the MGFs are the samefor each nt or for each pair (nt; ~nt). Accordingly, the ABEPin Proposition 1 can be simplified as shown in Corollary 1.Corollary 1: For identically distributed fading, (7) in

Proposition 1 simplifies as shown in (9) on top of the nextpage, where ABEPMOD is the error probability in TableI with M(nt) (s) = MMOD (s). If a constantmodulusmodulation is considered, i.e., l = 0 for l = 1; 2; : : : ;M ,then ABEPspatial in (9) reduces to (10) on top of the nexttwo pages. Likewise, if a constantmodulus modulation, i.e.,l = 0 for l = 1; 2; : : : ;M , and independent and uniformlydistributed channel phases are considered, then ABEPjoint in(9) simplifies to (11) on top of the next two pages.Proof : ABEPsignal in (9) follows immediately from



8>>>>>>>>>:

ABEPsignal =log2(M)

log2(NtM)ABEPMOD

ABEPspatial =1M

log2(Nt)log2(NtM)

Nt2

MPl=1

h1

R =20

MSSK

2l2 sin2()

di

ABEPjoint =1M

1log2(NtM)

MPl=1

MP~l 6=l=1

hNt log2(Nt)

2 +NH~l ! l

(Nt 1)

i 1

R =20

M(l;~l)

2 sin2()

d

(9)

TABLE IABEPMOD () OF PSK AND QAM MODULATIONS WITH MAXIMAL RATIO COMBINING (MRC) AT THE RECEIVER AND GRAY CODING. FOR QAM

MODULATION, WE CONSIDER A GENERIC RECTANGULAR MODULATION SCHEME WITH M = IM JM . SQUAREQAM MODULATION IS OBTAINED BYSETTING IM = JM =

pM . THE MGF OF (nt) =

PNrnr=1

jhnt;nr j2 ,M(nt) (), IS AVAILABLE IN CLOSEDFORM IN [34] FOR MANY CORRELATEDFADING CHANNELS. NOTE THAT FADING CORRELATION AT THE TRANSMITTER DOES NOT AFFECT ABEPMOD (). BUT FADING CORRELATION AT THE

RECEIVER DOES.

Generic Fading Channels

PSK

[34;Eq:(8:29)]

[38;Eq:(2);Eq:(7)]

8>>>>>>>>>:ABEPMOD (nt) =

1log2(M)

M1Pl=1

" 2 lM j lM m+ 2 log2(M)P

k=2

l2kjl2k

m!Pl (nt)#

Pl (nt) =12

R [1(2l1)=M ]0 T

l (;nt) d 12

R [1(2l+1)=M ]0 T

+l (;nt) d

Tl (;nt) =M(nt)2

sin2[(2l1)=M ]sin2()

; T+l (;nt) =M(nt)

2

sin2[(2l+1)=M ]

sin2()

QAM

[34;Eq:(4:2)]

[39;Eq:(22)]

8>>>>>>>>>>>>>>>:

ABEPMOD (nt) =1

log2(M)

"log2(IM )P

l=1Pl (IM ;nt) +

log2(JM )Pl=1

Pl (JM ;nt)

#

Pl (K;nt) =2K

12l

K1P

k=0

((1)

2l1kK

2l1

j2l1kK

+ 12

kTk (nt)

)Tk (nt) =

1

R =20 M(nt)

6(2k+1)2

(I2M+J2M2) sin2()

d

i.i.d. Rayleigh Fading (ABEPRayleighMOD = ABEPMOD (nt) = ABEPMOD)

R () =h12

1

q

2+

iNr Nr1Pnr=0

nNr+1rr

h12

1 +

q

2+

inro

PSK

8>>>>>:Pl (nt) = Pl = (1=2) INr

c; #

(1=2) INr c+; #+INr

c; #

is available in [34;Eq:(5A:24)]

c = 420 sin2 [ (2l 1) =M ] ; # = [(2l 1) =M ]

PSKapprox:

[34;Eq:(8:119))]ABEPMOD = 2maxflog2(M);2g

maxfM=4;1gPk=1

R420 sin

2h(2k1)

M

i

QAM

[34;Eq:(5A:4b)]Tk (nt) = Tk = R

2420(2k+1)

2

I2M+J2

M2

i.i.d. Rayleigh Fading HighSNR (ABEPsignal = [log2 (M) = log2 (NtM)] ABEPRayleighMOD )

PSK

[34;Eq:(8:119)]

[40;Eq:(14:4:18)]

8>: GPSKMOD (M) =

2maxflog2(M);2g

maxfM=4;1gPk=1

nsin

h(2k1)

M

io2NrABEPRayleighMOD

1= 22Nr

2Nr1Nr

GPSKMOD (M)

420

Nr

QAM

[40;Eq:(14:4:18)]

8>>>>>>>>>>>>>>>:

GQAMMOD (K; k) =2K

12l

K1P

k=0

((1)

2l1kK

2l1

j2l1kK

+ 12

k(2k + 1)2Nr

)

GQAMMOD (K) =

"1

log2(M)

6

I2M+J2

M2

Nr# log2(K)Pl=1

GQAMMOD (K; k)

ABEPRayleighMOD1= 2Nr

2Nr1Nr

hGQAMMOD (IM ) +G

QAMMOD (JM )

i 420

Nr



ABEPspatial =Nt2

log2 (Nt)

log2 (NtM)

"1

Z =20

MSSK

202 sin2 ()

d

#(10)

ABEPjoint =

M (Nt 1)

2

log2 (M)

log2 (NtM)+Nt (M 1)

2

log2 (Nt)

log2 (NtM)

"1

Z =20

MSSK

202 sin2 ()

d

#(11)

(nt;nr;~nt;~nr)Nak =

E f[nt;nr E fnt;nrg] [~nt;~nr E f~nt;~nrg]grEn[nt;nr E fnt;nrg]2

orEn[~nt;~nr E f~nt;~nrg]2

o (12)

(7) by taking into account that for identically distributedfading the M addends of the summation are all the same.ABEPspatial in (9) can be obtained by noticing that: i)for identically distributed fading, l (nt; ~nt) in (8) is thesame for nt = 1; 2; : : : ; Nt and ~nt = 1; 2; : : : ; Nt, and,thus, it can be moved out of the twofold summation; andii)PNt

nt=1

PNt~nt=1

NH (~nt ! nt) =N2t2log2 (Nt) for

any bit mapping. Finally, some algebraic manipulations leadto (9). Equation (10) follows from (9) with l = 0 forl = 1; 2; : : : ;M . ABEPjoint in (9) can be obtained as follows:i) for identically distributed fading,

nt; l; ~nt; ~l

in (7)

can be moved out of the twofold summation with indexes ntand ~nt because it is the same for each pair (nt; ~nt); and ii)PNt

nt=1

PNt~nt 6=nt=1

NH (~nt ! nt) +NH

~l ! l

=

N2t2log2 (Nt) + Nt (Nt 1)NH

~l ! l

for

any bit mapping. Finally, some simplifications leadto (9). Equation (11) can be obtained from (9)and the following considerations: i) if the channelphases are uniformly distributed, then (nt;l;~nt;~l) =PNr

nr=1

~nt;nr~l nt;nrl2 d=PNrnr=1 ~nt;nr~l nt;nrl2.In fact, since adding a constant phase term to auniformly distributed phase still yields a uniformlydistributed phase, i.e., ('nt;nr + l)

d='nt;nr , then

nt;nrl = [nt;nr exp (j'nt;nr )] [l exp (jl)] =

nt;nrl exp (j ('nt;nr + l))d=nt;nrl exp (j'nt;nr ) =

nt;nrl; ii) if l = 0 for l = 1; 2; : : : ;M , then

(nt;l;~nt;~l)

= 20PNr

nr=1j~nt;nr nt;nr j2 = 20(nt;~nt),

which for identically distributed fading impliesM

(l;~l)(s) = MSSK

20s. Thus, the integral in (9) can

be replaced by the integral in (11); and iii) for a constantmodulus modulation, the integral in (9) can be moved out ofthe twofold summation, which can be simplified using theidentity

PMl=1

PM~l=1NH

~l ! l

=M2

2log2 (M) for

any bit mapping. Finally, some algebraic manipulations leadto (11). This concludes the proof.

Corollary 1 leads to two important considerations about theperformance of SM: i) ABEPsignal and (9) shows that, foridentically distributed fading, the ABEP of SM is independentof the bit mapping of the spatialconstellation diagram. Thisresult is reasonable and agrees with intuition: if the channelsare statistically identical, on average the Euclidean distanceof pairs of channel impulse responses is the same. In this

case, the bit mapping has no role in determining the ABEP.On the other hand, the complexvalued points of the signalconstellation diagram have different Euclidean distances, andthis bit mapping plays an important role; and ii) under somerealistic assumptions (i.e., constantmodulus modulation anduniform channel phases), ABEPjoint in (9), which in the mostgeneral case depends on both spatial and signalconstellationdiagrams, depends only on the signalconstellation diagram.Thus, since there are no terms in Table I, (10), and (11)that depend on both constellation diagrams, we conclude thatthey can be optimized individually. In particular, the best bitmapping for the signalconstellation diagram turns out to bethe conventional one based only on the Euclidean distance.Finally, we notice that, e.g., (10) and (11) are very simple to

be computed, and avoid the computation of foldsummationson Nt and M . This is an important difference with respectto other frameworks available in the literature, where fourfold summations are always required, regardless of modulationscheme and channel model [6], [9]. Also, Corollary 1 sim-plifies the frameworks in [24] and [26] for SSK modulation,as the twofold summation can be avoided for some fadingchannels and modulation schemes. Furthermore, we mentionthat Corollary 1 provides closedform results if the MGFs,which depend on the specific fading channel model, areavailable in closedform, as well as if the related finite integralcan be computed explicitly. In Section IV-B and in SectionIV-C, we show some fading scenarios (Nakagamim and Ricefading with arbitrary fading parameters and correlation) wherethe MGFs can be obtained in closedform. Also, in SectionV we study the i.i.d. Rayleigh fading scenario where foldsummations can be avoided and integrals can be computedin closedform, thus leading to a very simple analyticalframework for system analysis and optimization.

B. Nakagamim Envelopes with Uniform Phases

In this section, the fading envelopes nt;nr areNakagamim RVs with fading severity m(nt;nr)Nak =E2nt;nr

2.En2nt;nr E

2nt;nr

2o and meansquare value (nt;nr)Nak = E

2nt;nr

. We adopt the notation

nt;nr Nm

(nt;nr)Nak ;

(nt;nr)Nak

. The amplitude correlation

coefficient, (nt;nr;~nt;~nr)Nak , is defined in (12) on top of thispage. Also, the channel phases, 'nt;nr , are independent and



M(nt;~nt) (s) =

1tridmNak2(4mNak4) (mNak)

+1Xk1=0

+1Xk2=0

+1Xk3=0

2414

k1+k2+k3 jp12j2k1 jp23j2k2 jp34j2k3F(p11;p33)k (s)F(p22;p44)k (s)(k1!) (k2!) (k3!) (k1 +mNak) (k2 +mNak) (k3 +mNak)

35 (13)

8>>>:F (p11;p33)k (s) = (1/4) s(mNak+k1)p11 s(mNak+k2+k3)p33 G1;22;2

s2sp11sp33

1mNak k2 k3 1mNak k10 0

F (p22;p44)k (s) = (1/4) s(mNak+k1+k2)p22 s(mNak+k3)p44 G1;22;2

s2sp22sp44

1mNak k3 1mNak k1 k20 0

(14)

uniformly distributed RVs in [0; 2). We adopt the notation'nt;nr U (0; 2). Finally, channel phases and fadingenvelopes are assumed to be independent.Given this fading model, let us analyze and explicitly

compute each term in (6).

1) ABEPsignal: M(nt) () has been widely studied in theliterature, and closedform expressions for nonidenticallydistributed and arbitrary correlated Nakagamim fading canbe found in [34, Sec. 9.6.4].

2) ABEPspatial: For Nakagamim fading, M(nt;~nt) ()is not wellknown in the literature, and, only recently, ithas been analyzed in [24] for singleantenna receivers, i.e.,Nr = 1. Thus, we need to generalize [24] for our systemmodel. Two case studies are considered: i) correlated fadingat the transmitter and independent fading at the receiver;and ii) correlated fading at both ends of the MIMO chan-nel. In the first case study, by exploiting the independenceof the fading at the receiver, we have M(nt;~nt) (s) =QNr

nr=1M(nt;~nt;nr) (s), whereM(nt;~nt;nr) () is the MGF of

(nt;~nt;nr) = j~nt;nr nt;nr j2. This latter MGF is availablein closedform in [24, Sec. III] for generic correlated fadingat the transmitter. The second case study is analytically morecomplicated, as M(nt;~nt) () requires the computation of theexpectation of 2Nr correlated RVs. Proposition 2 provides thefinal expression of M(nt;~nt) () for Nr = 2.Proposition 2: Given 2Nr arbitrary distributed and corre-

lated Nakagamim RVs with fading envelopes (nt;nr and~nt;nr ) distributed according to the multivariate NakagamimPDF in [41, Eq. (2)] and channel phases uniformly and i.i.d. in[0; 2), then M(nt;~nt) () for Nr = 2 is given in (13) on topof this page, where: i) is the 2Nr 2Nr correlation matrixof the Gaussian RVs associated to nt;nr and ~nt;nr , whichcan be computed from the amplitude correlation coefficients(nt;nr;~nt;~nr)Nak by using the method in [42]; ii) trid is the tri

diagonal approximation of , which can be obtained as de-scribed in [41, Sec. IV] and Appendix II; iii) pab = 1trid(a; b)are the entries of 1trid; iv) mNak = m

(nt;nr)Nak = m

(~nt;nr)Nak is

the fading parameter common to all links; and v) F (p11;p33)k (),F (p22;p44)k () are defined in (14) on top of this page, wheresp = s+ (p/2).

Proof : See Appendix II. Formulas for Nr > 2 can beobtained as described in Appendix II. For arbitrary Nr, thefinal formula is given by the (2Nr 1)fold series of theproduct of Nr terms F (;)k ().

It is worth mentioning that (14) gives an exact result when is tridiagonal, i.e., = trid. On the contrary, for arbitrarycorrelation, and by using the Green method [41, Sec. IV], itprovides a very tight approximation (see Section VI). Finally,we mention that the series in (14) converge very quickly thanksto the factorial and the Gamma functions in the denominator.3) ABEPjoint: To compute M

(nt;l;~nt;~l)() we need

Proposition 3.Proposition 3: For Nakagamim fading envelopes and

uniform phases, (nt;l;~nt;~l) reduces to (nt;l;~nt;~l) =

(nt;l;~nt;~l)=

PNrnr=1

(~l)~nt;nr (l)nt;nr 2, where(l)nt;nr

d=

(l)nt;nr exp (j'nt;nr ),

(l)nt;nr = lnt;nr , and

(l)nt;nr N

m

(nt;nr)Nak ;

(nt;nr;l)Nak

with (nt;nr;l)Nak =

2l

(nt;nr)Nak .

Proof : The equality in law (l)nt;nrd=

(l)nt;nr exp (j'nt;nr )

can be obtained by using the same analytical development usedfor (11) in Corollary 1, and, more specifically, the identity inlaw ('nt;nr + l)

d='nt;nr . This concludes the proof.

Proposition 3 points out that (nt;~nt) and (nt;l;~nt;~l)are related by a scaling factor in the mean power of eachchannel envelope, i.e., (nt;nr;l)Nak

.

(nt;nr)Nak =

2l . Accord-

ingly, M(nt;l;~nt;~l)

() can be computed by using the sameframeworks used to computeM(nt;~nt) (). In other words, forarbitrary correlation, M

(nt;l;~nt;~l)() is still given by (13)

but with a different correlation matrix .4) Diversity Analysis: The accurate analysis of

ABEPsignal, ABEPspatial, and ABEPjoint through closedform expressions of the MGFs allows us to provideimportant considerations about the diversity gain [43] ofSM in Nakagamim fading, as well as to understand theconstellation diagram that dominates the performance of SMfor highSNR. The main result is given in Proposition 4.Proposition 4: Let us assume, for the sake of simplic-

ity, mNak = m(nt;nr)Nak = m

(~nt;nr)Nak for each wireless link.

The diversity gain, DivSM, of SM is equal to DivSM =min fNr;mNakNrg.Proof : From (6), we have DivSM =

min fDivsignal;Divspatial;Divjointg, where Divsignal,Divspatial, and Divjoint are the diversity gains of ABEPsignal,ABEPspatial, and ABEPjoint, respectively. In fact, forhighSNR the worst term dominates the slope of the ABEP,and, thus, the diversity gain [43]. From Section IV-B.1 and



8>>>>>:ABEPspatial =

1Nt log2(NtM)

NtPnt=1

NtP~nt=1

h1

R =20

MSSK

202 sin2()

di

ABEPjoint =1

Nt log2(NtM)

NtPnt=1

NtP~nt 6=nt=1

nhM log2(M)

2 +NH (~nt ! nt) (M 1)i h

1

R =20

MSSK

202 sin2()

dio (15)

[34, Sec. 9.6.4], it follows that Divsignal = mNakNr. FromSection IV-B.2, Section IV-B.3, [24], and [44], it follows thatDivspatial = Divjoint = Nr. In fact, as analytically shownin [44], M(nt;~nt;nr) () and F

(;)k () have unit diversity

gain regardless of the fading severity mNak, and, thus, thediversity is determined only by the number Nr of antennas atthe receiver. This concludes the proof.

Proposition 4 unveils important properties of SM and pro-vides information about the best scenarios where SM shouldbe used. More specifically: i) in scenarios with less severefading than Rayleigh, i.e., mNak > 1, we have DivSM =Divspatial = Divjoint = Nr. We conclude that the ABEP ismainly determined by the spatialconstellation diagram (i.e.,ABEPspatial ABEPsignal and ABEPjoint ABEPsignal),and that the diversity gain is independent of fading sever-ity; ii) in scenarios with more severe fading than Rayleigh,i.e., 0:5 mNak < 1, we have DivSM = Divsignal =mNakNr. We conclude that the ABEP is mainly determinedby the signalconstellation diagram (i.e., ABEPsignal ABEPspatial and ABEPsignal ABEPjoint), and that thediversity gain strongly depends on fading severity; iii) due tothe increasing diversity gain of ABEPsignal with mNak [34], itis expected that ABEPsignal provides a negligible contributionfor increasing mNak, and that the ABEP gets better withmNak. This behavior is similar to conventional modulations[34], but different from SSK [24], [26]; iv) from [34], itis known that conventional singleantenna systems have thesame diversity gain as ABEPsignal, i.e., Divconventional =Divsignal = mNakNr. Thus, with respect to conventionalmodulations, the performance gain of SM is expected toincrease for 0:5 mNak < 1, while it is expected to decreasefor mNak > 1. This conclusion agrees with intuition, sinceSM encodes part of the information bits onto the spatialconstellation diagram, whose points are more closelyspacedif mNak > 1; and v) from [24], [44], it is known that thediversity gain of SSK modulation is the same as ABEPspatial,i.e., DivSSK = Divspatial = Nr, which is independent ofmNak. Thus, unlike conventional modulation schemes and SM,SSK modulation does not experience any diversity reductionwhen 0:5 mNak < 1, and it can be concluded that, thanksto the higher diversity gain, it turns out to be, among SMand conventional modulations, the best transmission schemein scenarios with fading less severe than Rayleigh. The priceto be paid is the need of many antennas at the transmitterto achieve the same rate. On the contrary, in more severefading channels, SSK modulation turns out to be worse thanconventional modulation. In conclusion, the performance ofSM in Nakagamim fading strongly depends on mNak, andthere is no clear transmission technology better than othersfor any mNak. This important result suggests the adoption of a

multimode adaptive transmission scheme, which can switchamong the best modulation according to the fading severityand the desired rate.Finally, we close this section with the following corollary.Corollary 2: For Nakagamim fading envelopes, uniform

channel phases, and a constantmodulus modulation, i.e., l =0 for l = 1; 2; : : : ;M , ABEPspatial and ABEPjoint in (7) canbe simplified as shown in (15) on top of this page.

Proof : Equation (15) can be obtained through analyticalsteps similar to (10) and (11), but without the assumption ofidentically distributed fading. Corollary 2 shows that, for a constantmodulus modu-

lation, ABEPspatial and ABEPjoint can be computed onlythrough M(nt;~nt) () = MSSK (). This makes even moreevident the connection established between M(nt;~nt) () andM

(nt;l;~nt;~l)() in Proposition 3. We note that in (15) nei-

ther ABEPspatial nor ABEPjoint depend on the bit mappingused for the signalconstellation diagram. Thus, the optimalityof usual bit mappings adopted for ABEPsignal seems to bepreserved.

C. Rician Fading

Let us consider a generic Rician fading [26], [45]. Inthis case, nt;nr are generically correlated complex GaussianRVs, and Rnt;nr and

Int;nr are independent by definition.

We adopt the notation Rnt;nr = ERnt;nr

, Int;nr =

EInt;nr

, and 2nt;nr = E

nRnt;nr Rnt;nr

2o=

EnInt;nr Int;nr

2o. Also, we use the shorthands

Rnt;nr GRnt;nr ;

2nt;nr

and Int;nr G

Int;nr ;

2nt;nr

.

The analysis of Rician fading is simpler than Nakagamimfading, and a unified framework can be used to compute (6).The main enabling result is summarized in Proposition 5.Proposition 5: Given a complex Gaussian RV nt;nr , then

(l)nt;nr = lnt;nr is still a complex Gaussian RV such

that Ren(l)nt;nr

o G

Rnt;nr;l;

2l

2nt;nr

, Im

n(l)nt;nr

o

GInt;nr;l;

2l

2nt;nr

, with Rnt;nr;l =

Rnt;nrl cos (l)

Int;nrl sin (l) and Int;nr;l

= Int;nrl cos (l) +

Rnt;nrl sin (l). Also, Ren(l)nt;nr

oand Im

n(l)nt;nr

oare

independent RVs.

Proof : By definition, (l)nt;nr =l exp (jl)

Rnt;nr + j

Int;nr

= Re

n(l)nt;nr

o+

jImn(l)nt;nr

owith Re

n(l)nt;nr

o= l

Rnt;nr cos (l)

lInt;nr sin (l) and Im

n(l)nt;nr

o= l

Int;nr cos (l) +

lRnt;nr sin (l). Then, by taking into account that

Rnt;nr

and Int;nr are Gaussian distributed, independent, and have



8>>>>>>>>>:

ABEPsignal =log2(M)

log2(NtM)ABEPRayleighMOD

ABEPspatial =1M

log2(Nt)log2(NtM)

Nt2

MPl=1

R 4202l ABEPjoint =

1M

1log2(NtM)

MPl=1

MP~l 6=l=1

hNt log2(Nt)

2 + (Nt 1)NH~l ! l

R220 2l + 2~l i(16)

8



8



TABLE IIISNR ( IN dB) DIFFERENCE (SEE TABLE II FOR DEFINITION) BETWEEN SMPSK/QAM AND SINGLEANTENNA PSK/QAM MODULATION, AS WELL AS

SMQAM AND SSK MODULATION. SM OUTPERFORMS (i.e., IT REQUIRES LESS TRANSMITENERGY PER SINGLE TRANSMISSION) THE COMPETINGTRANSMISSION TECHNOLOGY IF

(X=Y )SNR > 0. FOR A GIVEN RATE R IN bpcu, THE CONSTELLATION SIZE IS: I)M

(PSK;QAM) = 2R FOR

SINGLEANTENNA PSK/QAM MODULATION; II) NSSKt = 2R FOR SSK MODULATION; AND III)MNt = 2R FOR SM, WHERE Nt = 2; 4; 8 IN THE

FIRST/SECOND/THIRD LINE OF EACH ROW, RESPECTIVELY. N.A. MEANS NOT AVAILABLE.

Nr = 1

Rate (R) / (X=Y )SNR

2 bpcu 3 bpcu 4 bpcu 5 bpcu 6 bpcu

(PSK, SMPSK)2:4304N:A:

N:A:

0:96911:5761N:A:

0:5799

0:2803

0:0684

2:0412

2:2640

2:1512

3:2906

4:2597

4:3511

(QAM, SMQAM)2:4304N:A:

N:A:

1:09391:7009N:A:

3:31992:75422:9661

2:20553:34062:3416

4:14224:30644:9156

(SSK, SMQAM)0:5799

N:A:

N:A:

0:7918

0:1848

N:A:

0:28540:2803

0:0684

0:2460

0:88900:1100

0:34810:51231:1215

Nr = 2

Rate / (X=Y )SNR



N:A:

1:9011

1:6453

N:A:

4:5154

5:3471

5:2585

5:6931

8:8845

9:2429

5:9642

11:1650

13:1632


N:A:

1:7709

1:5152

N:A:

0:1040

2:0064

1:9177

2:3751

2:2836

4:2581

0:9242

2:6976

2:5484

(SSK, SMQAM)0:4509

N:A:

N:A:

0:3959

0:1401

N:A:

1:76220:1401

0:0515

1:82801:91960:0550

3:62421:85082:0000

Nr = 3

Rate / (X=Y )SNR



N:A:

3:0103

2:8560

N:A:

5:5248

7:2677

7:2144

5:9627

10:9352

11:8624

6:0094

11:9378

16:1295


N:A:

2:7651

2:6108

N:A:

0:9978

3:6577

3:6044

3:3520

3:8842

6:5402

1:6807

4:4339

4:7666

(SSK, SMQAM)0:3574

N:A:

N:A:

0:2639

0:1096

N:A:

2:56640:0934

0:0401

3:15162:61940:0367

5:74572:99262:6598

some MIMO setups and data rates, SM with PSKmodulatedpoints (SMPSK) outperforms SM with QAMmodulatedpoints (SMQAM) for the same average energy constraint.On the other hand, it is wellknown that ABEPsignal withQAM modulation is never worse than ABEPsignal with PSKmodulation. This result can be well understood with the helpof (18): unlike PSK, QAM has points with moduli that canbe either smaller or larger than one, which has an impacton (M;Nr)spatial ,

(M;Nr)joint , and

(M;Nr;H)joint . Since the ABEP of

SM is a weighted summation of all these terms, it turnsout that SMPSK might outperform SMQAM. This leadsto two important conclusions: 1) the best modulation scheme(between PSK and QAM) to use depends on M and Ntfor a given data rate; and 2) neither PSK nor QAM seemto be optimal signalmodulation schemes for SM. However,(18) provides the criterion to compute the optimal modulationscheme that minimizes the ABEP.



A. Comparison with SingleAntenna PSK/QAM and SSKModulations

By exploiting Corollary 5, in this section we aim atcomputing in closedform the SNR difference between SMand other transmission technologies with similar complex-ity, such as singleantenna PSK/QAM and SSK modula-tions. The highSNR framework for PSK/QAM can befound in Table I, while for SSK we get ABEPSSK =(Nt/2)R

420

1= 2Nr (Nt/2)

2Nr1Nr

420

Nr fromABEPspatial in (18).Due to space constraints, we cannot report all the details

of the analytical derivation, but we can only summarizethe main procedure used to compute the formulas in Ta-ble II. From (18) and Table I, for any transmission tech-nology, X , the ABEP is ABEPX = KX

20X

Nr=

KX (SNRX)Nr . Then, for any pair ABEPX and ABEPY ,

we have ABEPX = ABEPY ) KX (SNRX)Nr =KY (SNRY )

Nr . If we define the SNR difference (in dB) as(X/Y )

SNR= 10 log10 (SNRX/SNRY ), then we get

(X/Y )SNR

= (10/Nr) log10 (KY /KX) = (10/Nr) log10

(X/Y )

SNR

. If

(X/Y )SNR

> 0, then, for the same ABEP, Y needs (X/Y )SNR

dBless transmitenergy than X , i.e., (X/Y )

SNRis the energy gain

of Y with respect to X .Using Table II, in Table III we show some examples

about the SNR advantage/disadvantage of SM with respectto SSK and singleantenna PSK/QAM. Further comments arepostponed to Section VI.

VI. NUMERICAL AND SIMULATION RESULTS

The aim of this Section is to substantiate frameworks andclaims through Monte Carlo simulations. Two case studiesare considered: 1) i.i.d. Rayleigh fading (Section V); and2) identically distributed Nakagamim fading (Section IV-B).In the first case study, we focus our attention on the betteraccuracy provided by our upperbound, on the comparisonof SM with other modulations, and on understanding therole played by the signal and spatialconstellation diagrams.In the second case study, we turn our attention to analyzethe effect of fading correlation and fading severity on theachievable diversity. Without loss of generality, we considerthe identically distributed setup to keep the chosen parametersand variables reasonably low in order to maintain a sensibleset of simulation results. This allows us to focus our attentionon fundamental behaviors and to show the main trends. Inparticular, in the presence of channel correlation, we considerthe constant correlation model [41]. The reason of this choiceis twofold: i) to reduce the number of parameters neededto identify the correlation profile; and ii) to study a worstcase scenario, which arises when assuming that the constantcorrelation coefficient corresponds to the pair of antennas thatare most closelyspaced.

A. Better Accuracy of the Improved UpperBound

In Fig. 1 and Fig. 2, we study the accuracy of the improvedupperbound in Section III-A against Monte Carlo simula-tions and the conventional unionbound. The frameworks for

Fig. 1. ABEP of PSK (MPSK = 64) and SMPSK (M = 32, Nt =2) against Em=N0. Accuracy of proposed analytical framework (denotedby improved unionbound in the legend) and conventional unionbound(denoted by unionbound in the legend) for unitpower (20 = 1) i.i.d.Rayleigh fading (the rate is R = 6bpcu).

Fig. 2. ABEP of QAM (MQAM = 64) and SMQAM (M = 32,Nt = 2) against Em=N0. Accuracy of proposed analytical framework(denoted by improved unionbound in the legend) and conventional unionbound (denoted by unionbound in the legend) for unitpower (20 = 1)i.i.d. Rayleigh fading (the rate is R = 6bpcu).

singleantenna PSK/QAM are obtained from Table I. It canbe noticed that our framework is, in general, more accuratethan the conventional unionbound, and that it well overlapswith Monte Carlo simulations. In particular, our bound ismore accurate than the conventional unionbound for largeM and small Nr. Also, the figures compare the ABEP of SMand singleantenna PSK/QAM. In particular, the worstcasescenario with only Nt = 2 is considered. We observe twodifferent trends: i) in Fig. 1, SMPSK always outperformsPSK, regardless of Nr, and the gain increases with Nr; onthe other hand, ii) in Fig. 2, SMQAM is worse than QAMif Nr = 1 and it outperforms QAM if Nr = 3. This resultis substantiated by the highSNR framework in Table II. Thegeneral outcome of our study for i.i.d. Rayleigh fading is the



Fig. 3. ABEP of SMPSK against Em=N0. Performance comparisonfor various sizes of signal and spatialconstellation diagrams. Accuracy ofproposed analytical frameworks for unitpower (20 = 1) i.i.d. Rayleighfading (the rate is R = 4bpcu). The setup (M = 2, Nt = 8) is not shown,as it overlaps with the setup (M = 4, Nt = 4).

Fig. 4. ABEP of SMQAM against Em=N0. Performance comparisonfor various sizes of signal and spatialconstellation diagrams. Accuracy ofproposed analytical frameworks for unitpower (20 = 1) i.i.d. Rayleighfading (the rate is R = 4bpcu). The setup (M = 2, Nt = 8) is not shown,as it overlaps with the setup (M = 4, Nt = 4).

following: i) SMQAM never outperforms QAM for Nr = 1;and ii) SMQAM never outperforms QAM for data rates (R)less than R = 2bpcu. Further comments about this outcomeare given in Section VI-B.

B. Comparison with PSK, QAM, and SSK Modulations

Motivated by Fig. 2, we exploit the framework in Table IIto deeper understand the possible performance advantage ofSM with respect to SSK and singleantenna PSK/QAM. Theaccuracy of the frameworks in Table II has been validatedthrough Monte Carlo simulations, and a perfect match hasbeen found. In particular, the interested reader might verifythe accuracy of Table II by looking at the SNR differenceestimated through Monte Carlo simulations in Figs. 38. Table

Fig. 5. ABEP of SMPSK against Em=N0. Performance comparisonfor various sizes of signal and spatialconstellation diagrams. Accuracy ofproposed analytical frameworks for unitpower (20 = 1) i.i.d. Rayleighfading (the rate is R = 5bpcu). The setup (M = 2, Nt = 16) is notshown, as it overlaps with the setup (M = 4, Nt = 8).

Fig. 6. ABEP of SMQAM against Em=N0. Performance comparisonfor various sizes of signal and spatialconstellation diagrams. Accuracy ofproposed analytical frameworks for unitpower (20 = 1) i.i.d. Rayleighfading (the rate is R = 5bpcu). The setup (M = 2, Nt = 16) is notshown, as it overlaps with the setup (M = 4, Nt = 8).

III provides the following outcomes: i) if Nr = 1, SMQAMnever outperforms QAM, and the gap increases with the datarate; ii) whatever Nr is and if R < 3bpcu, SMPSK andSMQAM never outperform PSK and QAM, respectively; iii)except the former setups, SM always outperforms PSK andQAM, and the gain increases with R and if more antennasare available at the transmitter, i.e., more information bitscan be sent through the spatialconstellation diagram; andiv) the SNR gain increases with Nr, which means that SMis inherently able to exploit receiver diversity much betterthan PSK/QAM. It is important to emphasize here that inSection V-A we have pointed out that QAM might not be thebest modulation scheme for SM. This means that the optimalsignalconstellation diagram for SM is still unknown and,



Fig. 7. ABEP of SMPSK against Em=N0. Performance comparisonfor various sizes of signal and spatialconstellation diagrams. Accuracy ofproposed analytical frameworks for unitpower (20 = 1) i.i.d. Rayleighfading (the rate is R = 6bpcu). The setup (M = 2, Nt = 32) is notshown, as it overlaps with the setup (M = 4, Nt = 16).

thus, the ABEP of SM might be reduced further by lookingfor the signalconstellation diagram that optimizes the coef-ficients (M;Nr)spatial ,

(M;Nr)joint , and

(M;Nr;H)joint . In other words,

the noticeable gain offered by SM might be increased further,and possibilities of improvement for those setups where SMis worse than stateoftheart might be found as well. Furthercomments about the impact of the signal modulation schemeon the performance of SM is available in Section VI-C. Thisstudy corroborates our analytical findings, and confirms that anadaptive multimode modulation scheme might be a very goodchoice. Finally, in Table III we compare SMQAM with SSKas well. It can be noticed that, especially for high data rates,SSK outperforms SMQAM. This result shows that, when Rincreases, it is convenient to transmit the information bits onlythrough the spatialconstellation diagram, as this minimizesthe ABEP over i.i.d. fading channels. However, the price topay for this additional improvement is the need of largerantenna arrays at the transmitter. So, there is a clear tradeoff between the achievable performance and the number ofantennas that can be put on a transmitter, and still being ableto keep the i.i.d. assumption. In any case, these numericalexamples corroborate the potential performance and energygain benefits of exploiting SSK for lowcomplexity massiveMIMO implementations [20].

C. Interplay of Signal and SpatialConstellation Diagrams

In this section, we wish to give a deeper look at theperformance of SM for various configurations of signal andspatialconstellation diagrams, as well as at the effect of theadopted modulation scheme. More specifically, we seek toanswer two fundamental questions: 1) is there, for a given datarate R, an optimal pair (Nt;M) that minimizes the ABEP?and ii) is the optimal modulation scheme for singleantennasystems still optimal for SM? The results shown in Figs. 38 provide a sound answer to both questions. In particular, if

Fig. 8. ABEP of SMQAM against Em=N0. Performance comparisonfor various sizes of signal and spatialconstellation diagrams. Accuracy ofproposed analytical frameworks for unitpower (20 = 1) i.i.d. Rayleighfading (the rate is R = 6bpcu). The setup (M = 2, Nt = 32) is notshown, as it overlaps with the setup (M = 4, Nt = 16).

R = 4bpcu: i) the ABEP decreases by increasing Nt, but theimprovement is negligible for Nt > 4. Thus, Nt = 4 can beseen as the optimal choice in this scenario; ii) the SNR gainwith Nt is higher in SMQAM than in SMPSK; and iii) forNt = 2, SMPSK outperforms SMQAM, which substantiatesthe claims in Section V, while there is no difference betweenthem for Nt 4. In fact, in this latter case PSK and QAM leadto the same signalconstellation diagram. Thus, since PSKmodulation is, in general, simpler to be implemented as thepower amplifiers at the transmitter have less stringent linearityrequirements [46], then SMPSK seems to be preferred toSMQAM in all cases. If R = 5bpcu: i) Nt = 8 is thebest choice to minimize both the ABEP and the size of theantennaarray at the transmitter; ii) for SMPSK, the setupNt = 4 is a very appealing configuration as the ABEP is closeto the optimal value but the complexity of the transmitter isvery low; iii) for Nt = 2, SMQAM is definitely superiorto SMPSK, as the spatialconstellation diagram has a lowimpact on the overall performance; iv) for Nt = 4, SMPSKis much better than SMQAM, and, in particular, for SMQAM the net improvement when moving from Nt = 2 toNt = 4 is negligible; and v) for Nt 8, there is no differencebetween SMPSK and SMQAM since they have the samesignalconstellation diagram, and, thus, SMPSK is the bestchoice because simpler to implement. Also, if R = 6bpcu,we have a behavior similar to R = 4bpcu and R = 5bpcu.Thus, we focus only on two main aspects: i) the best ABEPis obtained when Nt = 16. By comparing the best MIMOsetup for different rates, we conclude that the best Nt increaseswith the rate, and the rule of thumb seems to be: double thenumber of transmitantennas for each 1bpcu increase of thedata rate. Even though this increase of the rate might appearto be small for every doubling of the number of antennasat the transmitter, this multiplexing gain is obtained with asingle active RF chain and with low (singlestream) decoding



0 5 10 15 20 25 30 35 40 45 50105

104

103

102

101

100AB

EP

Em

/N0 [dB]

SMQAM [M=2, Nt=32, Monte Carlo]SMQAM [M=2, Nt=32, Model]SMQAM [M=32, Nt=2, Monte Carlo]SMQAM [M=32, Nt=2, Model]QAM [M=64, Monte Carlo]QAM [M=64, Model]SSK [Nt=64, Monte Carlo]SSK [Nt=64, Model]

Fig. 9. ABEP against Em=N0 over i.i.d. Nakagamim fading (mNak = 1:0,i.e., Rayleigh, Nr = 2, and rate R = 6bpcu). Performance comparison andaccuracy of the analytical framework for SMQAM, QAM, and SSK.

complexity. These two features agree with current trends inMIMO research [20], [22], as mentioned in Section I; and ii)if Nt = 8, SMPSK is a very appealing choice to achieve verygood performance with lowcomplexity. Also, we emphasizethe good accuracy of our framework in all analyzed scenarios.Finally, we close this section by mentioning that the good

performance offered by SMPSK against SMQAM for someMIMO setups and rates brings to our attention that SMPSKmight be a good candidate for energy efficient applications. Asa matter of fact, in [46] it is mentioned that a nonnegligiblepercentage of the energy consumption at the base stations ofcurrent cellular networks is due to the linearity requirementsof the power amplifiers, which are needed to use highordermodulation schemes (such as QAM), and which result inthe low power efficiency of the amplifiers. Furthermore, in[47, Pg. 12] it is clearly stated that this power inefficiencysignificantly contributes to the socalled quiescent energy,which is independent of the amount of transmitted data, and,thus, should be reduced as much as possible.

D. Impact of Fading Severity

In Fig. 9 and Fig. 10, we study the impact of fading severityon the performance of QAM, SM, and SSK modulations.Figure 9 shows the basic scenario with i.i.d. Rayleigh fading(mNak = 1:0), where from Section IV-B.4 we know that allmodulations have the same diversity. Figure 10 highlights theeffect of more (mNak = 0:5) and less (mNak = 1:5) severefading. The figures provide three important outcomes, whichare well captured by the framework in Section IV-B.4: i)overall, the ABEP gets better for increasing values of mNak;ii) the SNR gain of SM with respect to QAM increases ifmNak = 0:5, as a consequence of the steeper slope of somecomponents of the ABEP of SM. Furthermore, we notice thatSSK is the only modulation scheme with no reduction ofthe diversity gain. If Nt = 32, SM has performance veryclose to SSK, but the different slope is noticeable even formoderate SNRs; and iii) if mNak = 1:5, QAM provides the

0 5 10 15 20 25 30 35 40 45 50105

104

103

102

101

100

ABEP

Em

/N0 [dB]

SMQAM [M=2, Nt=32, Monte Carlo]SMQAM [M=2, Nt=32, Model]SMQAM [M=32, Nt=2, Monte Carlo]SMQAM [M=32, Nt=2, Model]QAM [M=64, Monte Carlo]QAM [M=64, Model]SSK [Nt=64, Monte Carlo]SSK [Nt=64, Model]

mNak=0.5

mNak=1.5

Fig. 10. ABEP against Em=N0 over i.i.d. Nakagamim fading (mNak = 0:5and mNak = 1:5, Nr = 2, and rate R = 6bpcu). Performance comparisonand accuracy of the analytical framework for SMQAM, QAM, and SSK.

best diversity gain, but at lowSNR the high coding gainintroduced by SM and SSK is still advantageous. However,a crossing point can be observed for highSNR, which showsthat QAM should be preferred in this case. In conclusion, theseresults substantiate the diversity analysis conducted in SectionIV-B.4, and show, once again, that the characteristics of thefading are of paramount importance to assess the superiority ofa modulation scheme with respect to another one. An adaptivemultimode modulation scheme might be an appealing choicein order to use always the best modulation scheme for anyfading scenario.

E. Impact of Fading Correlation

Finally, in Figs. 1114 we study the impact of fading corre-lation at the transmitter and at the receiver over Nakagamimfading. The analytical framework is available in Section IV-B.4, and, in particular, in the analyzed scenario M(nt) (s) =M (s) can be found in [34, Eq. (9.173)]. We use a constantcorrelation model, and Nak denotes the correlation coefficientof pairs of Nakagamim envelopes. We consider two casestudies: i) channel correlation only at the transmitter (Fig. 11,Fig. 12); and ii) channel correlation only at the receiver (Fig.13, Fig. 14). The rationale of this choice is to investigate thedifferent effect that correlation might have at either ends ofthe communication link. In fact, according to (5), correlationmight have a different impact at the transmitter and at thereceiver: correlation at the transmitter affects the distance ofpoints in the spatialconstellation diagram, while correlationat the receiver reduces the diversity gain of Maximal RatioCombining (MRC) at the destination.In Fig. 11 and Fig. 12, we study the impact of correlation at

the transmitter. It can be noticed, as expected, that performancedegrades with channel correlation. Also, the impact of corre-lation increases with Nt, which is a reasonable outcome inour scenario. However, the SNR degradation with increasingvalues of Nak is tolerable if Nak < 0:6, while for highervalues a few dB loss can be observed. Very interestingly,



0 5 10 15 20 25 30 35 40 45 50105

104

103

102

101

100AB

EP

Em

/N0 [dB]

Model [i.i.d.]Monte Carlo [Nak=0.3]Model [Nak=0.3]Monte Carlo [Nak=0.6]Model [Nak=0.6]Monte Carlo [Nak=0.9]Model [Nak=0.9]

mNak=0.5

mNak=1.5

Fig. 11. ABEP of SMQAM against Em=N0 over correlated (at thetransmitter) and identically distributed Nakagamim fading (mNak = 0:5and mNak = 1:5, Nr = 2, and rate R = 6bpcu). Performance comparisonand accuracy of the analytical framework for M = 2 and Nt = 32.

0 5 10 15 20 25 30 35 40 45 50105

104

103

102

101

100

ABEP

Em

/N0 [dB]

Model [i.i.d.]Monte Carlo [Nak=0.6]Model [Nak=0.6]Monte Carlo [Nak=0.9]Model [Nak=0.9]

mNak=0.5

mNak=1.5

Fig. 12. ABEP of SMQAM against Em=N0 over correlated (at thetransmitter) and identically distributed Nakagamim fading (mNak = 0:5and mNak = 1:5, Nr = 2, and rate R = 6bpcu). Performance comparisonand accuracy of the analytical framework for M = 32 and Nt = 2.

Fig. 12 shows that channel correlation has a negligible effectif mNak = 0:5. This result is very interesting, especially ifcompared to the same curves in Fig. 11 and with the ABEPof QAM in Fig. 10 (QAM uses just one transmitantenna and,thus, it is not affected by fading correlation at the transmitter).In particular, we note that: i) if Nt = 2, SM is alwayssuperior to QAM, regardless of fading correlation; and ii) ifNt = 32, SM is much better that QAM, even for a high fadingcorrelation (Nak = 0:9). The net outcome is the following:for severe fading channels, correlation degrades the ABEPbut it does not offset the SNR gain that, for independentfading, SM has with respect to QAM. On the other hand,if mNak = 1:5 the superiority of QAM becomes even morepronounced if compared to the independent fading scenario.In conclusion, fading correlation at the transmitter poses no

0 5 10 15 20 25 30 35 40 45 50105

104

103

102

101

100

ABEP

Em

/N0 [dB]

Model [i.i.d.]Monte Carlo [Nak=0.6]Model [Nak=0.6]Monte Carlo [Nak=0.9]Model [Nak=0.9]

mNak=1.5

mNak=0.5

Fig. 13. ABEP of SMQAM against Em=N0 over correlated (at the receiver)and identically distributed Nakagamim fading (mNak = 0:5 and mNak =1:5, Nr = 2, and rate R = 6bpcu). Performance comparison and accuracyof the analytical framework for M = 2 and Nt = 32.

0 5 10 15 20 25 30 35 40 45 50105

104

103

102

101

100AB

EP

Em

/N0 [dB]

Model [i.i.d.]Monte Carlo [Nak=0.3]Model [Nak=0.3]Monte Carlo [Nak=0.6]Model [Nak=0.6]Monte Carlo [Nak=0.9]Model [Nak=0.9]

mNak=1.5

mNak=0.5

Fig. 14. ABEP of SMQAM against Em=N0 over correlated (at the receiver)and identically distributed Nakagamim fading (mNak = 0:5 and mNak =1:5, Nr = 2, and rate R = 6bpcu). Performance comparison and accuracyof the analytical framework for M = 32 and Nt = 2.

problems to SM in severe fading channels, while it should becarefully managed in other fading scenarios, especially if wewant to keep the performance advantage over singleantennaQAM (whose ABEP is not affected by this correlation). ForSM, solutions to counteract fading correlation have recentlybeen proposed in [9] and [14]. Once again, we emphasize that,because of the constant correlation model, Fig. 11 and Fig. 12show the worst case effect of fading correlation, especially forlarge Nt.In Fig. 13 and Fig. 14, we study the impact of correlation

at the receiver. Overall, the ABEP degrades for increasingNak. A higher robustness to fading correlation can be noticedfor mNak = 1:5. If mNak = 0:5, the diversity advantage ofSM with respect to QAM if kept in the presence of channelcorrelation too. For large antennaarrays at the transmitter



8>>>>>:ABEPboundsignal =

1Nt

log2(M)log2(NtM)

NtPnt=1

ABEPboundMOD (nt)

ABEPboundMOD (nt) =1M

1log2(M)

MPl=1

MP~l=1

"NH

~l ! l

E(nt)

(Q

s~l l2 NrP

nr=1jnt;nr j2

!)# (19)

(e.g., Nt = 32), the diversity loss in ABEPsignal has anegligible impact even for high correlated channels. IfmNak =1:5, we observe that the SNR degradation gets smaller forlarger antennaarrays at the transmitter. In other words, trans-mitting more information bits through the spatialconstellationdiagram (e.g., increasing Nt) can mitigate the effect of channelcorrelation at the receiver. However, Fig. 11 and Fig. 12 pointout a clear tradeoff: increasing Nt degrades the ABEP if wehave channel correlation at the transmitter. We believe that theexploitation of the proposed frameworks for an endtoendsystem optimization by taking into account all these tradeoffs might be a very important research issue: how to find theoptimal SM setup providing the best performance/complexitytradeoff, as a function of fading correlation, fading severity,etc.Finally, we wish to emphasize the good accuracy of our

framework for the very complicated fading scenario underanalysis. Our framework agrees with Monte Carlo simulationsin all scenarios. Only in some figures there are negligibleerrors, which are mainly due to the Green approximationdescribed in Section IV-B. Thus, our frameworks can beexploited for accurate system optimization.

VII. CONCLUSIONIn this paper, we have proposed a comprehensive frame-

work for the analysis of SMMIMO over generalized fadingchannels. The framework is applicable to a large variety ofcorrelated fading models and MIMO setups. Furthermore,and, more importantly, by carefully analyzing the obtainedformulas, we have derived important information about theperformance of SM over fading channels, including the effectof fading severity, the achievable diversity gain, along with theimpact of the signalconstellation diagram. It has been shownthat the modulation scheme used in the signalconstellationdiagram significantly affects the performance, and, for i.i.d.Rayleigh fading, closedform expressions for its optimizationhave been proposed. Finally, we have conducted an extensivesimulation campaign to validate the analytical derivation, andhave showcased important trends about the performance of SMfor a large variety of fading scenarios and MIMO setups. Webelieve that our frameworks can be very useful to understandfundamental behaviors and tradeoffs of SM, as well as canbe efficiently used for system optimization.

ACKNOWLEDGMENTWe gratefully acknowledge support from the European

Union (PITNGA2010264759, GREENET project) for thiswork. M. Di Renzo acknowledges support of the Laboratoryof Signals and Systems under the research project JeunesChercheurs. H. Haas acknowledges the EPSRC under grantEP/G011788/1 for partially funding this work.

APPENDIX IPROOF OF Proposition 1

Before going into the details of the proof, let usanalyze the Hamming distance, NH

~nt; ~l

! (nt; l),of messages

~nt; ~l

and (nt; l). In particular,

NH~nt; ~l

! (nt; l) is equal to the numberof different bits between the messages. Since a biterror might occur when: i) only the antennaindex iswrongly detected; ii) only the signalmodulated pointis wrongly detected; or iii) both antennaindex andsignalmodulated point are wrongly detected, then weconclude that total number of bits in error is given byNH

~nt; ~l

! (nt; l) = NH (~nt ! nt)+NH ~l ! l,where NH (~nt ! nt) and NH

~l ! l

are defined in

Proposition 1. This remark is used to compute (6)(8), and itis important to highlight the role played by the bitmappingin each constellation diagram. Proposition 1 can be obtainedas follows:

ABEPsignal is obtained from (4) by grouping togetherall the terms for which ~nt = nt and ~l 6= l, andby noticing that: i) NH (~nt ! nt) = 0 if ~nt =nt; ii) (5) reduces to APEP

~nt; ~l

! (nt; l) =E(nt)

Q

q~l l2PNrnr=1 jnt;nr j2. Then,

ABEPsignal = ABEPboundsignal in (4) reduces to (19)

on top of this page. It can readily be noticed thatABEPboundMOD (nt) is the unionbound of a conventionalmodulation scheme [34], where: i) only the ntthtransmitantenna is active; and ii) we have the sameconstellation diagram as the signalconstellation diagramof SM. More specifically, ABEPboundMOD (nt) is the ABEPof a singleinputmultipleoutput system with maximalratio combining. This ABEP is known in closedform formany modulation schemes and bit mappings, without theneed to using unionbound methods. Thus, to get moreaccurate estimates of the ABEP, ABEPboundMOD () can bereplaced by ABEPMOD (), as shown in (8), which isthe exact ABEP of a singleinputmultipleoutput systemwith maximal ratio combining.

Likewise, ABEPspatial is obtained from (4) by group-ing together all the terms for which ~nt 6= nt and~l = l, and by noticing that: i) NH

~l ! l

= 0 if

~l = l; ii) (5) reduces to APEP~nt; ~l

! (nt; l) =E(nt;~nt)

Q

q2l

PNrnr=1

j~nt;nr nt;nr j2

. Fi-

nally, from [34, Eq. (4.2)] we have l (nt; ~nt) =APEP

~nt; ~l

! (nt; l), where l (; ) is defined inSection III-A.

ABEPjoint in (7) collects all the terms that areneither in ABEPsignal nor in ABEPspatial. More



M(nt;~nt) (s) = E(exp

"s

NrXnr=1

j~nt;nr nt;nr j2#)

= E

(NrYnr=1

exps j~nt;nr nt;nr j2

)

= E

(NrYnr=1

exps2~nt;nr NrY

nr=1

exps2nt;nr E'

(NrYnr=1

exp [2s~nt;nrnt;nr cos ('~nt;nr 'nt;nr )]))(20)

J (s;~nt;nr ; nt;nr ) =

NrYnr=1

E' fexp [2s~nt;nrnt;nr cos ('~nt;nr 'nt;nr )]g =NrYnr=1

I0 (2s~nt;nrnt;nr ) (21)

M(nt;~nt) (s) = E(

NrYnr=1

exp

s2~nt;nr exp s2nt;nr I0 (2s~nt;nrnt;nr ))

=

Z

(NrYnr=1

exp

s2~nt;nr exp s2nt;nr I0 (2s~nt;nrnt;nr ))f () d

(22)

8>>>>>>>:M(nt;~nt) (s) =

R

nhexp

s21;1

exp

s22;1

I0 (2s1;12;1)

i hexp

s21;2

exp

s22;2

I0 (2s1;22;2)

iof () d

f () =

" 1tridmNak2(mNak1)(mNak)

21;122;2 exp

p44

222;2

#hjp12j(mNak1) 1;1 exp

p11

221;1

ImNak1 (jp12j1;11;2)

ihjp23j(mNak1) 1;2 exp

p22

221;2


ihjp34j(mNak1) 2;1 exp

p33

222;1


i(23)

8>>>:F(p11;p33)k (s) =

Z +10

Z +10

2mNak+2k111;1

2mNak+2k2+2k312;1 exp

hs+

p11

2

21;1

iexp

hs+

p33

2

22;1

iI0 (2s1;12;1) d1;1d2;1

F(p22;p44)k (s) =Z +10

Z +10

2mNak+2k1+2k211;2

2mNak+2k312;2 exp

hs+

p22

2

21;2

iexp

hs+

p44

2

22;2

iI0 (2s1;22;2) d1;2d2;2

(24)

specifically, (7) can be obtained from [34, Eq. (4.2)]:nt; l; ~nt; ~l

= APEP

~nt; ~l

! (nt; l) =E(nt;~nt)

Q

qPNr

nr=1

~nt;nr~l nt;nrl2,where (; ; ; ) is defined in Section III-A.

APPENDIX IIPROOF OF Proposition 2

By definition, M(nt;~nt) () is given by (20) on top of thispage, where the last equality explicitly shows the conditioningover fading envelopes and channel phases, and , ' are shorthands to denote the set of all fading envelopes and channelphases, respectively. Let us compute J (s;~nt;nr ; nt;nr ) =E'

nQNrnr=1

exp [2s~nt;nrnt;nr cos ('~nt;nr 'nt;nr )]o

in(20). It can be obtained as shown in (21) on top of thispage, where the first equality is due to the independenceof the channel phases, and the second equality is obtainedfrom [35, pp. 339, Eq. (366), Eq. (367)] and [24, Eq. (14)].Accordingly, M(nt;~nt) () simplifies as shown in (22) on topof this page, where f () is the multivariate NakagamimPDF in [41, Eq. (2)].

As an example, and without loss of generality, let usconsider Nr = 2. For ease of notation, we set nt = 1 and~nt = 2. Accordingly, (22) reduces to (23) shown on top ofthis page. Finally, by using the infinite series representationof Iv () in [33, Eq. (9.6.10)], and after lengthy algebraicmanipulations, M(nt;~nt) () can be rewritten as shown in(13) where the integrals shown in (24) on top of this pagehave been introduced. These latter integrals can be computedin closedform from [24, Sec. IIIB], thus obtaining the finalresult in (14). More specifically, the analytical procedure wehave used to compute (14) is as follows: i) first, the integral onvariable 2;1 is solved in closedform by using the identitiesin [32, Eq. (8.4.3.1)] and [32, Eq. (8.4.22)], as well as byapplying the MellinBarnes theorem in [32, Eq. (2.24.1.1)] onthe obtained integral; ii) second, the obtained singleintegralon variable 1;1 is solved in closedform by using again theidentity in [32, Eq. (8.4.3.1)] and by applying the MellinBarnes theorem in [32, Eq. (2.24.1.1)].

The analytical development can be generalized to arbitraryNr by simply inserting in (22) the general PDF in [41, Eq.(2)] and solving the integrals as in (21)(24).

Finally, a few comments about the Green approximation



= trid in (23). i) The PDF in (23) requires the correlationmatrix of the Gaussian RVs associated to the fadingenvelopes. This matrix can be computed from the amplitudecorrelation coefficient (nt;nr;~nt;~nr)Nak by using the procedure in[42, Sec. III]. ii) For arbitrary and unequal values of (nt;nr)Nak ,the Green method in [41], which is given under the assumptionthat (nt;nr)Nak = 1 for nt = 1; 2; : : : ; Nt and nr = 1; 2; : : : ; Nr,must be generalized. More specifically, the coefficients ui in[41, Eq. (9)], which are needed to computetrid, take the formui = (i; i)/vi, where (i; i) is the entry of located inthe ith row and in the ith column, and vi are the coefficientsto be computed by solving the nonlinear system of equationsin [41, Eq. (10)].

REFERENCES[1] Y. Chau and S.H. Yu, Space modulation on wireless fading channels,

IEEE Veh. Technol. Conf., vol. 3, pp. 16681671, Oct. 2001.[2] S. Song, Y. Yang, Q. Xiong, K. Xie, B.J. Jeong, and B. Jiao, A

channel hopping technique I: Theoretical studies on band efficiency andcapacity, IEEE Int. Conf. Commun., Circuits and Systems, vol. 1, pp.229233, June 2004.

[3] R. Y. Mesleh, H. Haas, C. W. Ahn, and S. Yun, Spatial modulation A new low complexity spectral efficiency enhancing technique, IEEEInt. Conf. Commun. Netw. in China, pp. 15, Oct. 2006.

[4] M. Di Renzo, H. Haas, and P. M. Grant, Spatial modulation formultipleantenna wireless systems A survey, IEEE Commun. Mag.,vol. 49, no. 12, pp. 182191, Dec. 2011.

[5] R. Y. Mesleh, H. Haas, S. Sinanovic, C. W. Ahn, and S. Yun, Spatialmodulation, IEEE Trans. Veh. Technol., vol. 57, no. 4, pp. 22282241,July 2008.

[6] J. Jeganathan, A. Ghrayeb, and L. Szczecinski, Spatial modulation:Optimal detection and performance analysis, IEEE Commun. Lett., vol.12, no. 8, pp. 545547, Aug. 2008.

[7] Y. Yang and B. Jiao, Informationguided channelhopping for highdata rate wireless communication, IEEE Commun. Lett., vol. 12, no. 4,pp. 225227, Apr. 2008.

[8] J. Jeganathan, A. Ghrayeb, L. Szczecinski, and A. Ceron, Spaceshift keying modulation for MIMO channels, IEEE Trans. WirelessCommun., vol. 8, no. 7, pp. 36923703, July 2009.

[9] T. Handte, A. Muller, and J. Speidel, BER analysis and optimizationof generalized spatial modulation in correlated fading channels, IEEEVeh. Technol. Conf. Fall, pp. 15, Sep. 2009.

[10] S. U. Hwang, S. Jeon, S. Lee, and J. Seo, Softoutput ML detector forspatial modulation OFDM systems, IEICE Electronics Express, vol. 6,no. 19, pp. 14261431, Sep. 2009.

[11] M. M. Ulla Faiz, S. AlGhadhban, and A. Zerguine, Recursive leastsquares adaptive channel estimation for spatial modulation systems,IEEE Malaysia Int. Conf. Commun., pp. 14, Dec. 2009.

[12] M. Di Renzo and H. Haas, Performance comparison of different spatialmodulation schemes in correlated fading channels, IEEE Int. Conf.Commun., pp. 16, May 2010.

[13] N. Serafimovski, M. Di Renzo, S. Sinanovic, R. Y. Mesleh, and H. Haas,Fractional bit encoded spatial modulation (FBESM), IEEE Commun.Lett., vol. 14, no. 5, pp. 429431, May 2010.

[14] R. Y. Mesleh, M. Di Renzo, H. Haas, and P. M. Grant, Trellis codedspatial modulation, IEEE Trans. Wire

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