Bit Error Probability of Space Shift Keying MIMO over Multiple-Access Independent Fading Channels

Transcription

1 Bit Error Probability of Space Shift Keying MIMO over Mltiple-Access Independent Fading Channels Marco Di Renzo, Harald Haas To cite this version: Marco Di Renzo, Harald Haas. Bit Error Probability of Space Shift Keying MIMO over Mltiple- Access Independent Fading Channels. IEEE Transactions on Vehiclar Technology, Institte of Electrical and Electronics Engineers, 011, 60 (8), pp < /TVT >. <hal > HAL Id: hal Sbmitted on 0 Jan 01 HAL is a mlti-disciplinary open access archive for the deposit and dissemination of scientific research docments, whether they are pblished or not. The docments may come from teaching and research instittions in France or abroad, or from pblic or private research centers. L archive overte plridisciplinaire HAL, est destinée a dépôt et à la diffsion de docments scientifiqes de nivea recherche, pbliés o non, émanant des établissements d enseignement et de recherche français o étrangers, des laboratoires pblics o privés.

2 TRANSACTIONS ON VEHICULAR TECHNOLOGY 1 Bit Error Probability of Space Shift Keying MIMO over Mltiple Access Independent Fading Channels Marco Di Renzo, Member, IEEE and Harald Haas, Member, IEEE Abstract In this paper, we stdy the performance of Space Shift Keying (SSK) modlation for Mltiple-Inpt Mltiple Otpt (MIMO) wireless systems in the presence of mltiple access interference. More specifically, a synchronos mlti ser scenario is considered. The main technical contribtions of this paper are as follows. Two receiver strctres based on the Maximm Likelihood (ML) criterion of optimality are developed and analytically stdied, i.e., the single and mlti ser detectors. Accrate frameworks to compte the Average Bit Error Probability () over independent and identically distribted (i.i.d.) Rayleigh fading channels are proposed. Frthermore, simple and easy to se lower and pper bonds for performance analysis and system design are introdced. The frameworks accont for the near far effect, which significantly affects the achievable performance in mltiple access environments. Also, we extend the analysis to Generalized SSK (GSSK) modlation, which foresees mltiple active antennas at the transmitter. With respect to SSK modlation, GSSK modlation achieves higher data rates at the cost of an increased complexity at the transmitter. The performance of SSK and GSSK modlations is compared to conventional Phase Shift Keying (PSK) and Qadratre Amplitde Modlation (QAM) schemes, and it is shown that SSK and GSSK modlations can otperform conventional schemes for varios system setps and channel conditions. In particlar, the performance gain of SSK and GSSK modlations increases for increasing vales of the target bit rate and of the nmber of antennas at the receiver. Finally, we pt forth the concept of Coordinated Mlti Point (or network MIMO) SSK (CoMP SSK) modlation, as a way of exploiting network cooperation and the spatial constellation diagram to achieve high data rates. Analytical derivations and theoretical findings are sbstantiated throgh extensive Monte Carlo simlations for many setps. Index Terms Mltiple-inpt mltiple otpt (MIMO) wireless systems, space shift keying (SSK) modlation, spatial modlation (SM), mltiple access fading channels, performance analysis, bonds. I. INTRODUCTION SPACE SHIFT KEYING (SSK) is a recently proposed modlation scheme for Mltiple-Inpt Mltiple Otpt (MIMO) wireless systems [1], []. It encodes the information Manscript received May 19, 011; revised Agst 3, 011; accepted Agst 4, 011. This paper was presented in part at the IEEE Global Commnications Conference (GLOBECOM), Miami, FL, USA, December 010. The review of this paper was coordinated by Prof. D. Matolak. M. Di Renzo is with the Laboratoire des Signax et Systèmes, Unité Mixte de Recherche 8506, Centre National de la Recherche Scientifiqe École Spériere d Électricité Université Paris Sd XI, 9119 Gif sr Yvette Cedex, France, (e mail: marco.direnzo@lss.spelec.fr). H. Haas is with The University of Edinbrgh, College of Science and Engineering, School of Engineering, Institte for Digital Commnications (IDCOM), King s Bildings, Mayfield Road, Edinbrgh, EH9 3JL, United Kingdom (UK), (e mail: h.haas@ed.ac.k). Color versions of one or more of the figres in this paper are available online at Digital Object Identifier XXX.XXX/TVT.XXX.XXX bits onto the spatial position (i.e., the index) of the antennas at the transmitter, and enables data decoding by exploiting the differences in the Channel Implse Responses (CIRs) of the transmit to receive wireless links [3]. It is receiving an increasing attention de to its simple transmitter and receiver design, and, more important, becase it represents the fndamental bilding block of Spatial Modlation (SM) [4] [7]. SM is a low complexity hybrid modlation scheme for MIMO systems, which maps the information bits onto two information carrying nits: the signal constellation diagram, which is determined by conventional modlation schemes (e.g., Phase Shift Keying (PSK) and Qadratre Amplitde Modlation (QAM)), and the spatial constellation diagram, which is determined by SSK modlation [6]. By exploiting signal and spatial constellation diagrams, SM introdces a mltiplexing gain with respect to single antenna systems that increases logarithmically with the nmber of antennas at the transmitter. Frthermore, with respect to spatial mltiplexing MIMO systems, the mltiplexing gain is obtained with no inter channel interference. This enables very simple single stream and Maximm Likelihood (ML ) optimm decoding at the receiver [], [6], [7]. Recent analytical and simlation reslts have highlighted that SM and SSK modlation can provide better performance with redced decoding complexity than state of the art single and mlti antenna wireless systems [], [5] [15]. The reader can find a comprehensive and detailed overview of the contribtion of recent papers on SM and SSK modlation in [3], [13], and [16]. By careflly looking at recent research works on SM and SSK modlation, it is possible to notice that all the stdies available so far consider the point to point reference scenario. For example, in [] and [7], the Average Bit Error Probability () of SSK modlation and SM, respectively, is stdied over independent and identically distribted (i.i.d.) Rayleigh fading channels; in [8], the framework in [] is generalized to the so called Generalized SSK (GSSK) modlation, which is an improved version of SSK modlation where more than one antenna can be active at any time instance; in [10], the stdy in [] is extended by taking into accont channel (Trellis) coding to redce the error probability of detecting the active transmit antenna; in [11], the performance of SM over Nakagami m fading channels is investigated; in [13], the of SSK modlation with and withot transmit diversity is analyzed over generically correlated and distribted Rician fading channels; and, finally, in [3], [16], [17], the performance of SSK modlation is analyzed over correlated and non identically distribted Nakagami m fading with and withot perfect Channel State Information (CSI). However, to the best of the athors knowledge, none of these papers address

3 TRANSACTIONS ON VEHICULAR TECHNOLOGY receiver design and performance analysis of SSK modlation in the presence of mltiple access interference. The present research work is motivated by the fact that, de to the simltaneos transmission of varios sers over the same physical wireless channel, the vast majority of wireless commnication networks are interference limited [18]. Therefore, the potential merits of SSK modlation for its application to the next generation wireless commnication systems highly depends on its robstness to mltiple access interference. The main aim of this paper is to nderstand advantages and disadvantages of SSK modlation in mltiple access environments (i.e., mlti ser SSK modlation), and figre ot if the claimed benefits of SSK modlation in a single ser environment are retained. Similar to [], [3], and [16], we focs or attention only on SSK modlation as it enables a simple analytical derivation and insightfl nderstanding of the role played by the spatial constellation diagram, which is the main innovative enabler for performance improvement of SM/GSSK/SSK modlation [1]. In this context, we feel important to notice that the adoption of SM/GSSK/SSK modlation schemes inherently reqires that each ser is eqipped with mltiple antennas at the transmitter and at the receiver. Also, since in some cases, e.g., in SSK modlation, only a single antenna is active and the information is conveyed only by the spatial constellation diagram, the nmber of antennas in each device might be qite large to achieve good data rates. However, this large nmber of antennas seems not to be a critical bottleneck for the development of the next generation mltiple access celllar systems, as crrent research is moving towards the tilization of the millimeter wave freqency spectrm [19]. In fact, in this band compact horn antenna arrays with 48 elements and compact patch antenna arrays with more than 4 elements at the Base Station (BS) and at the mobile terminal, respectively, are crrently being developed to spport mlti gigabit transmission rates [0]. In detail, the main contribtions and technical novelties of this paper can be smmarized as follows: i) for the first time, the performance of modlation schemes exploiting the space modlation concept in a mlti ser interference environment is investigated and compared to traditional modlation schemes; ii) two ML optimm receiver strctres are investigated, i.e., the single ser detector that is interference naware and the joint mlti ser detector that is interference aware; iii) for analytical tractability, we limit or performance stdy to the synchronos case, which is often considered as the initial reference scenario for performance analysis in the presence of interference [1]; iv) accrate frameworks for SSK and GSSK modlations are developed to compte the over i.i.d. freqency flat Rayleigh fading channels. The frameworks can take into accont the so called near far effect that significantly affects the performance of wireless systems in mltiple access environments [1]; v) simple bonds are derived to compare varios modlation schemes (e.g., SSK, GSSK, PSK, QAM) and to nderstand advantages and disadvantages of each of them for varios MIMO configrations; vi) the proposed frameworks and bonds are sefl for an arbitrary nmber of antennas at the transmitter and at the receiver, as well as for any modlation order and bit to antenna index mapping; and vii) we pt forth the concept of Coordinated Mlti Point (or network MIMO) SSK (CoMP SSK) modlation, as a way of exploiting network cooperation and the spatial constellation diagram to achieve high bit rates. Or comprehensive analytical stdy highlights the following general reslts: i) it is shown that SSK and GSSK modlations can otperform conventional modlation schemes in the presence of mltiple access interference; ii) if a single ser detector is sed, SSK modlation provides better performance than PSK modlation for bit rates greater than bits/s/hz per ser, while it otperforms QAM for bit rates greater than bits/s/hz per ser if the receiver is eqipped with at least two antennas; iii) GSSK modlation is always worse than SSK modlation, bt it achieves higher bit rates for the same nmber of antennas at the transmitter; iv) the performance gain of SSK and GSSK modlations increases for increasing vales of the reqested bit rate per ser; v) the robstness of SSK and GSSK modlations to mlti ser interference increases by adding more antennas at the receiver; vi) when a mlti ser detector is sed, SSK and GSSK modlations seem to be more robst to mlti ser interference than conventional modlation schemes. For example, for bit rates greater than bits/s/hz per ser, SSK modlation always otperforms QAM regardless of the nmber of antennas at the receiver; and vii) it is shown that CoMP SSK modlation can provide very high bit rates at the cost of network cooperation, which can be realized throgh a backhal link. Also, it is shown that CSI at the transmitter is not needed to implement this scheme. The reminder of this paper is organized as follows. In Section II, the system model is introdced. In Section III, the of SSK modlation with ML optimm single ser detection is investigated. In Section IV, the performance of SSK modlation is compared to QAM and PSK modlation when a ML optimm single ser detector is sed. In Section V, the framework in Section III is extended to GSSK modlation, and SSK and GSSK modlations are compared. In Section VI, all the frameworks proposed in the previos sections are generalized to ML optimm mlti ser detection, and asymptotic analysis is sed to reveal advantages and disadvantages of SSK and GSSK modlations. In Section VII, the concept of CoMP SSK modlation is introdced. In Section VIII, nmerical reslts are shown to sbstantiate or analytical derivations and findings. Finally, Section IX concldes this paper. II. SYSTEM MODEL We consider a very general mlti ser MIMO setp with N active sers (i.e., transmitters) and N d possible receivers. Each transmitter/receiver is eqipped with an antenna array of /N r antennas. This setp can accommodate varios mltiple access sitations, sch as: i) the scenario in which a single receiver mst decode the information of more than one transmitter, and ii) the so called interference channel (e.g., the X channel) [18], where each receiver mst decode the information from a single transmitter and can disregard the information transmitted by the other sers. Withot loss of generality, among the possible transmitter/receiver pairs,

4 TRANSACTIONS ON VEHICULAR TECHNOLOGY 3 z (r) (t) = E h (x,r) w (t) + }{{} probe link N =1 [ E h (x,r) w (t) } {{ } interference ] + n (r) (t) }{{} AWGN (1) we focs or attention on a particlar link that is sally known as probe link or intended link (see, e.g., [, Fig. 1]). More specifically, we are interested in stdying the error probability of the data sent from a generic ser, with = 1,,...,N, to a generic receiver when the other N 1 sers are simltaneosly transmitting on the same physical channel. So, there is no loss of generality in considering N d =1. In or system model, the N sers exploit the differences (i.e., known as spatial signatres or channel fingerprints [3]) in the wireless channel of any transmit to receive link with a twofold objective: data modlation and mltiple access. Accordingly, the mlti ser SSK modlation scheme analyzed in this paper can be seen as a generalization of the so called Space Division Mltiple Access (SDMA) [3] [5] and Channel Division Mltiple Access (ChDMA) [6] schemes, which exploit ser specific CIRs only for differentiating simltaneosly transmitting sers, i.e., for mltiple access only. The fndamental difference between mlti ser SSK modlation and mlti ser SDMA/ChDMA is that the former scheme ses the differences in the CIRs for data modlation in addition to mltiple access, while the latter schemes rely on conventional (e.g., PSK and QAM) modlation for transmitting the data of the sers. In mlti ser SSK modlation, the stochastic differences in the CIRs are exploited in a twofold way: i) at the microscopic level, i.e., differences among co located transmit antennas of the same ser, for data modlation, and ii) at the macroscopic level, i.e., differences among spatially distribted antenna arrays associated to different sers, for mltiple access. To the best of the athors knowledge, the performance analysis of this mltiple access scheme has never been considered in literatre. A. Assmptions and Notation Throghot this paper, the following assmptions and notation are sed. i) A synchronos mlti access channel with perfect time synchronization at the receiver is considered [1]. Accordingly, for ease of notation, time delays can be neglected dring the analytical derivation. ii) The receiver is assmed to have perfect CSI. More specifically, if a single ser detector is sed, the receiver needs only the CSI of the probe link. While, if a joint mlti ser detector is sed, the receiver needs the CSI of all the active sers. iii) In all wireless links, freqency flat independent Rayleigh fading is assmed. In particlar, identically distribted fading is considered for wireless links related to co located antennas, while non identically distribted fading among the sers is considered. This allows s to inclde the near far effect in the analytical derivation. In formlas, we denote by h (t,r) the complex CIR from the t th (t =1,,..., ) transmit antenna of the th ( =1,,...,N ) ser to the r th (r =1,,...,N r ) receive antenna of { the destination. } { Moreover, } we se the notation h (t,r) =Re h (t,r) +jim h (t,r), where Re { } and Im { } are real and imaginary operators, and j = { } { } 1 is the imaginary nit. Re h (t,r) and Im h (t,r) are independent real valed Random Variables (RVs). iv) X N ( μ, σ ) denotes a Gassian RV with mean μ and standard deviation σ. v) { Owing } to the assmption of Rayleigh fading, we have Re h (t,r) N ( ) } 0,σ and Im {h (t,r) N ( 0,σ). vi) ( ), i.e., overbar, denotes complex conjgate. vii) E { } denotes the expectation operator. viii) Pr { } denotes probability. ix) Q (x) = ( 1 / π ) + exp ( t / ) dt is the Q fnction x and Q 1 ( ) is sed to denoted the inverse Q fnction. x) T s denotes the dration of the time slot where each information symbol is transmitted. xi) The noise n (r) at the inpt of the r th receive antenna (r =1,,...,N r ) is assmed to be an Additive White Gassian Noise (AWGN) process, with both real and imaginary parts having a power spectral density eqal to N 0. Across the receive antennas, the noise is statistically independent. xii) M denotes the modlation order of QAM and PSK modlation. The M symbols of the signal constellation diagram of ser are denoted by the complex nmbers s (m) for m =1,,...,M and =1,,...,N. For PSK modla- tion, we have s (m) =1. xiii) If GSSK modlation is sed, a denotes the nmber of simltaneosly active antennas at the transmitter, with 1 a. The effective size of the spatial constellation diagram is denoted by N = log ( a ) [8], where ( ) is the binomial coefficient and is the floor fnction. xiv) In SSK and GSSK modlations, E and E /a denote the average energy transmitted by each antenna of ser ( =1,,...,N ) that emits a non zero signal, respectively. In particlar, in GSSK modlation a niform energy allocation scheme is considered among the active transmit antennas. In QAM and PSK modlation, E is the average transmitted energy per information symbol of ser. xv) w ( ) is the real valed nit energy transmitted plse shape in each time slot T s. xvi) Γ(x) = + 0 t z 1 exp ( t) dt is the Gamma fnction. xvii) δ x,y is the Kronecker delta fnction, which is defined as δ x,y =1if x = y and δ x,y =0elsewhere. xviii) (a) =, (a), and (a) denote conventional eqality (=), ineqality ( ), and approximation ( ) operators, respectively, which have been labeled with (a) as a short hand to better identify them in the text, and provide comments on the analytical procedre that is sed for their comptation. III. SSK MODULATION WITH SINGLE USER DETECTION In SSK modlation, each ser encodes blocks of log ( ) data bits into the index of a single transmit antenna, which is

5 TRANSACTIONS ON VEHICULAR TECHNOLOGY 4 ˆx = { arg min y =1,,..., D (y ) } = arg min y =1,,..., { Nr T s z (r) (t) } E h (y,r) w (t) dt () (a) 1 = E h 1 N H (x,y )Pr{ ˆx = y x } log x =1 ( ) y =1 (b) 1 1 N H (x,y )E h {Pr {x y }} log x =1 ( ) }{{} y =1 APEP(x y ) (3) { { PEP (x y ) (a) { } = Pr D (x ) >D (y ) (b) Nr [ E ( h(y = Pr Re,r) Ω N N r E Re ( h(y,r) h ) N (x,r) =1 ( E h (x,r) h ) (x,r) η (r)]} N r >E ), 4N N r 0E h (y,r) h (y,r) } h (x,r) (4) h (x,r) (5) switched on for data transmission while all the other antennas are kept silent. Let, for =1,,...,N, be the ser of interest, i.e., the probe link, while let all the other N 1 sers be interfering sers. Also, let x, for x =1,,...,, be the antenna index of the generic ser that is actally switched on for transmission. Accordingly, the signal received after propagation throgh the wireless fading channel and impinging pon the r th (r =1,,...,N r ) receive antenna is given in (1) on top of the previos page. As mentioned in Section II, (1) confirms that in mlti ser SSK modlation only the differences in the CIRs are exploited for modlation and mltiple access. All the sers share the same time slot, freqency band, or spreading code [6]. By assming single ser detection, the interference is not exploited for optimal detection, and the ML optimm estimate, ˆx, of x is shown in () on top of this page [7], where y denotes the trial instance of x sed in the hypothesis testing problem, and D (y ) = Nr z T (r) s (t) E h (y,r) w (t) dt. The of the detector in () can be compted in closed form as shown in (3) on top of this page, where (a) = comes from [8, Eq. (4) and Eq. (5)], and (b) = is the well known asymptotically tight nion bond [9, Eq. (1.44)], which has recently been sed in [3, Eq. (35)] for point to point SSK modlation. Frthermore, N H (x,y ) is the Hamming distance between the bit to antenna index mappings of x and y ; E h { } is the expectation compted over all the fading channels (probe link and interference); and APEP (x y ) = E h {PEP (x y )} is the Average Pairwise Error Probability (APEP), i.e., the probability of estimating y when, instead, x is transmitted, nder the assmption that x and y are the only two antenna indexes possibly being transmitted. Accordingly, PEP (x y ) = Pr {x y } is conditioned on both antenna indexes x and y. Unlike [3], the main contribtion of this paper consists in taking into accont mltiple access interference when compting the APEPs. The PEPs in (3) can explicitly be written as shown in (4) on top of this page, where (a) = comes directly from (), and (b) = can be obtained after lengthly analytical maniplations. Frthermore, we have defined η (r) = ( N E ) =1 h (x,r) + n (r) w and n (r) w = T s n (r) (t) w (t) dt. By direct inspection, it is easy to show that n { w (r) is} a complex valed Gassian { } RV with distribtion Re n (r) w N(0,N 0 ) and Im n (r) w N (0,N 0 ). By conditioning pon all the fading channels (probe link and interference), it can be) proved ]} that Ω = η (r) { Nr [ E ( h(y,r) Re h (x,r) in (4) is a real valed Gassian RV with distribtion shown in (5) on top of this page. Ths, from the definition of Q fnction in Section II-A, the PEP in (4) has closed form expression given in (6) on top of the next page. It is worth mentioning that the PEP in (6) is very general and can be sed for channel models different from Rayleigh fading stdied in this paper. Frthermore, (6) is obtained withot making any assmptions abot the statistical distribtion of the interference. This enables s to se, with minor changes, this formla for modlation schemes different from SSK, sch as PSK/QAM and GSSK, which are stdied in Section IV and Section V, respectively. Ths, the assmption of Rayleigh fading is here made only for analytical tractability, and to get simple and insightfl closed form expressions and bonds, which: i) enable a fairly simple comparison among different state of the art modlation schemes; ii) provide gidelines for system design and optimization; and iii) shed

6 TRANSACTIONS ON VEHICULAR TECHNOLOGY 5 Ω P Ω I {}}{{ }}{ N r E h (y,r) h (x,r) N r E Re ( h(y,r) h ) N ( ) (x,r) E h (x,r) =1 PEP (x y )=Q N r 4N0 E h (y,r) h (x,r) }{{} Ω N (6) ( )} { ( )} Ω P (E APEP (x y )=E h {Q ) γ =E Ω N + σi h Q 4+4 N [ =1 (E σ) / ] N 0 [ ( )] Nr Nr { (a) 1 SINR (Nr ) [ ( )] r } (8) 1+r 1 SINR = SINR r + SINR lights on the robstness of SSK modlation to mltiple access interference. To compte the APEP in Rayleigh fading, i.e., to remove the conditioning over channel statistics, we se a two step procedre: i) first, we condition the PEP in (6) pon the channel gains of the probe link and remove the conditioning over the channel gains of the interference; and ii) then, we remove the conditioning over the channel gains of the probe link. By conditioning pon the probe link, Ω I in (6) is a conditional Gassian RV with distribtion Ω I h N 0,σI =4E PN i =1 `Eσ " P Nr,r) h(y h (x,r) #!, while Ω P and Ω N are conditional constant terms. Accordingly, from (6) we have: ( )} E h\ {PEP (x y )} (a) ΩP Ω = I E h\ {Q Ω ( N (b) ΩP = E h\ {Q σ )} I ΩI Ω N Ω (7) ( ) N (c) Ω = P Q Ω N + σi where E h\ { } denotes the expectation over all the fading gains except those of the probe link; (a) = comes from (6); (b) = is obtained by introdcing the RV Ω I, which is defined as Ω I =Ω I /σ I N(0, 1); and (c) = is a notable integral that involves the Q fnction and Gassian RVs, and is tablated in [1, Eq. (3.66)]. The last step consists in removing the conditioning over the channel gains of the probe link. From (7), we obtain (8) on top of this page, where we have defined: i) γ = Nr h (y,r) h (x,r) /( ) ; ii) SINR = SNR 1 + INR\ is the Signal to (Interference+Noise) Ratio ( )/ (SINR); iii) SNR = E σ N 0 is the Signal to Noise Ratio (SNR) of the probe link; and iv) INR \ = N [( ] =1 E σ)/ N0 is the aggregate Interference to Noise Ratio (INR) of all the interferers. The identity in (a) = is obtained as follows: i) γ is the smmation of the sqare absolte vale of Gassian RVs and, ths, is a Chi Sqare RV [30, Eq. ( 1 136), Eq. ( 1 137)]; and ii) the Q fnction is averaged over the reslting Chi Sqare RV [30, Eq. ( ), Eq. ( )]. Finally, the can be compted by sbstitting (8) in (3). By careflly looking at (8), we notice that the APEP is independent of x and y, i.e., the actal and trial antenna indexes, bt it only depends on SNR and INR of probe link and interference, respectively. Ths, by defining in (8) APEP (x y ) = APEP for all x =1,,..., and y =1,,...,, the in (3) simplifies as follows: where Nt x =1 (a) APEP log ( ) (a) = APEP x =1 y =1 N H (x,y ) (9) = comes from the identity Nt y =1 N H (x,y ) = ( / ) Nt log ( ), which can be derived via direct inspection for all possible bit to antenna index mappings. In conclsion, (8) and (9) provide a very simple analytical framework of the of mlti ser SSK modlation over Rayleigh fading channels. Very interestingly, from (9) we observe that the is independent of the bit to antenna index mapping. This stems from the assmption of i.i.d. Rayleigh fading for the wireless links of co located transmit antennas. Also, we note that (8) and (9) redce to known reslts if there is no mltiple access interference, i.e., INR \ =0(see, e.g., [] and [3]). A. Asymptotic Analysis In this section, we stdy some asymptotic case stdies to highlight some general behaviors when sing the spatial constellation diagram for modlation. 1) SNR 1 and INR \ 1 (noise limited scenario): This corresponds to a scenario in which mlti access interference can be neglected and large SNR analysis for the probe

7 TRANSACTIONS ON VEHICULAR TECHNOLOGY 6 z (r) (t) = E h (r) s(x ) w (t)+ N ŝ (x ) = arg min ( y s ) for y =1,,...,M s D =1! ( y ) [ E h (r) = s ( y ) s (x) ] w (t) + n (r) (t) arg min for y =1,,...,M { Nr T s z (r) (t) E h (r) s(y ) } w (t) (10) dt link can be considered. In this case, from [30, Sec ] the trns ot to be asymptotically eqal to (Nr+1)( N r 1) N Nt r SNR Nr. This formla shows that, if the mltiple access interference is negligible, the goes to zero for increasing SNR and that the diversity order is eqal to N r. These findings agree with [] and [3]. This formla highlights a trend that was not shown either in [] or [3]: the linearly increases with the nmber of transmit antennas. As the bit rate increases with log ( ), there is a trade off to consider. ) INR \ 1 and SIR = SNR /INR \ 1 (interference limited scenario): This corresponds to a scenario in which the AWGN is negligible, and the Signal to Interference Ratio (SIR) is high. In this case, the is asymptotically eqal to (Nr+1)( N r 1) / N Nt r SIR Nr with SIR = E σ N ( =1 E σ). This simple formla shows a nmber of interesting trends: i) in an interference limited scenario, we observe an error floor in the, which means that the does not go to zero as the AWGN goes to zero. This is becase the single ser detector is interference naware; ii) the error floor is lower (better performance) when either N r or the SIR increase; and iii) the error floor is higher (worse performance) when increases. To overcome the error floor, we need to increase either the transmit energy (E ) of the probe link or the nmber of antennas, N r, at the destination. 3) N r 1: We have jst remarked that mltiple access interference can be, in part, mitigated by adding more antennas at the receiver. We are interested in analyzing the asymptotic when N r is very large. To derive this reslt, we start from the last eqality in the first line of (8). From [30, Eq. ( 1 136) and Eq. ( 1 139)], it follows that the RV γ has mean and variance eqal to E { } { γ = ( σ [Γ (N r +0.5)/Γ (N r )] and E γ E { }) } γ = { 4σ N r [Γ (N r +0.5)/Γ (N r )] }, respectively. Since Γ(N r +0.5)/Γ (N r ) { = Nr if N r 1, then ( E γ E { }) } γ 0 and E { } γ σ Nr. Ths, if N r 1, the RV γ tends to a constant, i.e, γ σ Nr, and, from (8), the redces to ( /) Q ( N r SINR ). We notice that the has a typical waterfall behavior and the effect of fading is drastically redced. However, even thogh the AWGN is negligible with respect to mltiple access interference, we still have an error floor. Bt it is mch / redced. In particlar, if SNR 1 and SIR = SNR INR\ 1, then the nmber of receive antenna Nr that provide the target is eqal to N r = (1/SIR) [ Q 1 ( / Nt )]. This is a very simple formla that can be sed for a simple system design when the assmption N r 1 is reasonable. IV. SINGLE USER DETECTION: COMPARISON OF PSK, QAM, AND SSK MODULATION In this section, we aim at stdying the performance of conventional QAM and PSK modlation, and at comparing them with SSK modlation. We consider a detector similar to (), bt the search space is given by the signal constellation diagram rather than by the spatial constellation diagram. Also, the methodology we se for performance analysis is similar to Section III. For this reason, and de to space constraints, we omit the details of the analytical derivation and report only the final reslts. Instead, we focs or attention on trying to nderstand advantages and disadvantages of sing the spatial constellation diagram as a sorce of information. Finally, we note that for QAM and PSK modlation the transmitter is eqipped with a single antenna, i.e., =1. Received signal and ML optimm detector are shown! in s (y ) (10) on top of this page, where we have defined D Nr z T (r) s (t) E h (r) s(y ) w (t) dt, and have sed a notation similar to (1) and (). We emphasize that in (10) only the differences in the wireless channels are exploited to distingish the sers. Unlike, e.g., [1], no spreading seqences are sed. This is in agreement with the SDMA/ChDMA mltiple access schemes described in Section II. The final expressions of the can be fond in Table I, along with the asymptotic formlas that are valid for noise and interference limited scenarios. By comparing the reslts in Section III and Table I, the following general comments can be = made: i) for QAM and PSK modlation the eqality (a) = in (9) does not hold as the APEPs depend on the actal pair of points in the signal constellation diagram. Ths, when this distance becomes small we can expect worse performance than SSK modlation; ii) for QAM we notice that the depends on the actal symbols transmitted by the interfering sers. Ths, the signal constellation diagram affects both the power of probe link and the aggregate interference; and iii) even thogh the asymptotic APEPs of QAM and PSK modlation in the noise limited scenario look similar, we expect different reslts becase the actal signal constellation diagram is different. Let s now stdy, in detail, whether/when we can expect that SSK modlation otperforms either QAM or PSK modlation. As a case stdy, we provide some formlas for PSK modlation, as its is simpler to manage. However,

8 TRANSACTIONS ON VEHICULAR TECHNOLOGY 7 TABLE I OF QAM AND PSK MODULATION. THE LAST FOUR ROWS SHOW FORMULAS «RELATED TO THE ASYMPTOTIC ANALYSIS, SIMILAR TO SECTION III-A. A NOTATION SIMILAR TO SECTION III IS ADOPTED. N H s (x ),s (y ) IS THE HAMMING DISTANCE BETWEEN THE BIT TO SYMBOL MAPPING OF s (x ) AND s (y ). Modlation/Scenario PSK (nion bond) QAM (nion bond) PSK/QAM (noise limited) PSK (interference limited) QAM (interference limited) 8 >< 8 >< >: 1 M log (M) MP MP x =1 y =1 «N H s (x ),s (y ) APEP s (x ) " SINR = (1/) SNR s (y ) s (x ) # / `1 + INR \ 1 M N log (M) MP x 1 =1... MP x =1... MP MP x N =1 y =1 " SINR = (1/) SNR s (y ) s (x ) # / 1+ INR g \ «s (y ) {z } Eq. (8) «N H s (x ),s (y ) APEP s (x ) «s (y ) {z } Eq. (8) >: INR g \ = P» N =1 E σ s (x) ««APEP s (x ) s (y ) `N r 1 s (y ) N r s (x ) N r SNR Nr «APEP s (x ) s (y ) `N r 1 s (y ) N r s (x ) N r `SNR /INR \ Nr «APEP s (x ) s (y ) `N r 1 s (y ) N r s (x ) N r SNR / INR g Nr \ PSK SSK N t log ( ) x =1 y =1 { ( ) [ / s N H s (x ),s (y ) (y ) s (x ) ] } N r (11) similar conclsions can be drawn for QAM as well. For a fair comparison, i.e., to garantee the same bit rate, we assme = M and se the symbol in what follows. By sing the formlas valid in the asymptotic regime, the ratio in (11) on top of this page can be compted, which holds for both noise and interference limited scenarios. By looking into (11), the following conclsions can be made: i) SSK modlation will never be sperior to PSK modlation if s (y ) s (x ). This happens, e.g., when = M = or = M = 4. On the other hand, when = M > 4 we can expect that a crossing point exists and that SSK modlation otperforms PSK modlation. In other words, the higher the target bit rate is, the more advantageos SSK modlation is. This conclsion holds for QAM as well. This reslt was arged by simlation in [] for a noise limited scenario, bt no proof was given. Also, we have shown that the trend holds in the presence of mltiple access interference as well; ii) for those setps where SSK otperforms QAM/PSK modlation we have s (y ) s (x ) < for some signal constellation points. In this case, the ratio in (11) increases exponentially with N r : the larger N r is, the higher the performance gain provided by SSK modlation is. Also this reslt was arged by simlation in [] for a noise limited scenario, bt no proof was given. We have shown that the trend holds in the presence of mltiple access interference as well; and iii) by direct inspection of (11), we can compte the asymptotic SNR and SIR gains of a modlation scheme with ( respect to / the other) as Δ SNR = Δ SIR = (10/N r ) log 10 PSK SSK. V. GSSK MODULATION WITH SINGLE USER DETECTION The working principle of GSSK modlation is as follows ( [8]: i) each ser encodes blocks of log Nt ) a bits into a point of a spatial constellation diagram of size N = log ( a ), which enables Nta antennas to be switched on for data transmission while all the other antennas are kept silent, and ii) similar to SSK modlation, the receiver solves a N hypothesis testing problem to estimate the a antennas that are not idle, which reslts in the estimation of the message emitted by the encoder of the probe link. Similar to Section III, received signal and ML optimm detector are given in (1) on y (a) top of the next page, where D = Nr T s z(r) (t) [ N (E y (a) ta,r a=1 /a )h w (t)] dt;

9 TRANSACTIONS ON VEHICULAR TECHNOLOGY 8 z (r) (t) = Nta a=1 ˆx = arg min y Θ GSSK [ E { D x (a) a h,r y (a) ] w (t) + N } N r = arg min y Θ GSSK N ta =1 a=1 T s [ E a h,r) (x(a) [ z(r) (t) Nta E a=1 ] w (t) + n (r) (t) y (a),r a h w (t)] dt (1) 1 N log (N) x Θ GSSK SINR = 1 SNR N ta (x, y ) 1 + INR \ a N H (x, y ) APEP (x y ) }{{} y Θ GSSK Eq. (8) (13) [ ] x = x (1),x(),...,x(Nta) denotes the a dimensional vector of active antenna indexes of the probe link; ˆx is its ML optimm estimate; y is the trial instance of x sed in the hypothesis testing problem; and Θ GSSK is the spatial constellation diagram of GSSK modlation, i.e., the set of N = log ( a ) vectors of antenna indexes that can possibly be activated for transmission. Θ GSSK is a sbset of the set of all possible combinations, i.e., ( ) a, of antenna indexes. Frthermore, we notice that in (1) the available energy per transmission of each ser, E, is scaled by a to ensre a fair comparison with SSK modlation. In particlar, a niform power allocation strategy is assmed among the antennas. Improvement is possible by sing opportnistic power allocation [31], bt it is not here considered. The can be compted by sing an analytical derivation similar to SSK modlation. The main difference consists in taking into accont that, for each ser, the eqivalent channel seen by the intended receiver is the smmation of the channels originated from a antennas. Accordingly, by comparing (1) and (1) we notice that the signal models become the same if we replace h (x,r) in (1) with ( 1 / ) N h(x(a),r) Nta ta a=1 in (1) for = 1,,...,N. Ths, with the help of the formal sbstittion h (x,r) ( 1 / ) N h(x(a),r) Nta ta a=1, all the analytical steps in (4) (8) can formally be repeated. By doing so, the only modification trns ot to be the final expression of γ in (8), which can be generalized as γ = (1/a ) N r Nta a=1 h y (a),r a a=1 h x (a),r. De to space constraints, the details of the analytical derivation are here omitted. On the other hand, we focs or attention on analyzing advantages and disadvantages of SSK and GSSK modlations. The final expression of the is smmarized in (13) on top of this page, where N H (x, y ) is the Hamming distance between the bit to antenna index tple mapping of x and y ; N ta (x, y ) is the nmber of non shared antenna indexes of x and y. For example, if x = [1,, 3] and y = [1, 3, 4], then N ta (x, y )=, i.e., indexes and 4. Let s here emphasize that N ta (x, y ) comes from the generalized expression of γ given above; and the other symbols have the same definition as in Section III. Frthermore, if a =1, GSSK redces to SSK modlation. In this case, N = and N ta (x, y )=for each pair (x, y ) Θ GSSK Θ GSSK, and, as expected, (13) redces to (9). Finally, by direct inspection and whatever the effective spatial constellation diagram is, the ineqalities N ta (x, y ) a hold. The lower bond corresponds to SSK modlation, while the pper bond corresponds to a spatial constellation diagram where all the indexes in x and y are different. This reslt is exploited in Section V-A to compare SSK and GSSK modlations. Finally, we note that, nlike SSK modlation, in GSSK modlation the eqality (a) = in (9) does not hold becase the PEPs actally depend on (x, y ), even for i.i.d. fading channels. A. Asymptotic Analysis and Comparison with SSK Modlation Similar to SSK modlation, we can analyze the performance in noise and interference limited scenarios. In particlar, the APEP in (13) redces to APEP (x y ) / ] [a N Nr ( ta (x, y ) Nr 1) N r Υ N r, where Υ = SNR in a noise limited scenario and Υ = SIR in an interference limited scenario, respectively. Accordingly, from Section III-A we can compte the following ratio: APEP GSSK (x y ) APEP SSK [ a N ta (x, y ) ] Nr (14) From (14), the following conclsions can be drawn: i) since N ta (x, y ) a, it follows that APEP GSSK (x y ) APEP SSK, i.e., GSSK modlation is always worse than SSK modlation, regardless of the actal spatial constellation diagram; and ii) the extra SNR or SIR that we need for GSSK to get the same APEP as for SSK modlation is ( Δ Υ = (10/N r ) log 10 APEP GSSK (x y ) / APEP SSK which lies in the interval 0 Δ Υ 10 log 10 (a ). This reslt is very important becase it shows that the larger the nmber, a, of active antennas is, the worse GSSK modlation with respect to SSK modlation is. The intitive reason for this trend is as follows. If N ta (x, y ) < a,it means that x and y have some antenna indexes in common, which cancel ot in the hypothesis testing problem. Since the transmit energy is distribted among the active antennas, this reslts in a destrctive interference cancelation effect: ),

10 TRANSACTIONS ON VEHICULAR TECHNOLOGY 9 { ˆx = arg min D (y)} N r = arg min y=[y 1,y,...y N ] y=[y 1,y,...y N ] for y =1,,..., for y =1,,..., (a) 1 N N t x 1=1 x N =1 y 1=1... y N =1 T s N [ ] z(r) (t) E h (y,r) w (t) dt =1 [ ] NH (x,y ) log (N APEP t) x y (x y) APEP x y (x y) (b) = ( 1 δ x,y ) Eh {Pr {x y}} (c) = ( 1 δ x,y ) Eh {PEP (x y)} (15) (16) E h {PEP (x y)} = [ ( )]N r 1 AggrSNR 1 + AggrSNR N r ( Nr 1+r r ) [ 1 ( 1+ AggrSNR + AggrSNR )]r (18) we transmit power on the common indexes, bt it does not contribte to the ML optimm decision process. Ths, for better performance we shold keep the nmber of active antennas as small as possible. Finally, we emphasize that the conclsions in i) and ii) hold for the too, as it can be derived from (9) and (13). VI. MULTI USER DETECTION: ANALYSIS AND COMPARISON In the sections above, we have stdied the when the receiver is interference naware and exploits, for data detection, only the CSI of the probe link. The main advantage of this receiver is the low comptational complexity, while its main disadvantage is the error floor when mltiple access interference is the dominant effect. In this section, we stdy the performance of the ML optimm joint mlti ser detector [1], which exploits CSI of all the active sers, and, ths, is interference aware. We compte accrate nion bond estimates of the for all the modlation schemes (SSK, GSSK, PSK, and QAM) of interest, and, throgh asymptotic analysis, we provide some weaker bonds to better nderstand the behavior of the detector. De to space constraints, we provide a detailed derivation of the of SSK modlation, and give only the final reslt for the other modlation schemes. Finally, we emphasize that with respect to, e.g., [1] or stdy does not exploit any signatre code for mltiple access capabilities, bt exploits only the differences/randomness of the CIRs among the active sers. This agrees with the definition of SDMA/ChDMA mltiple access schemes described in Section II. Let s consider SSK modlation. The received signal is always given by (1), bt the detector is different. In particlar, the ML optimm joint mlti ser detector is given in (15) on top of this page [1], [7], where the following notation is sed: i) x = [x 1,x,...x N ] is the vector of antenna indexes that is actally active in the considered time slot; ii) ˆx = [ˆx 1, ˆx,...ˆx N ] is its ML optimm estimate; iii) y = [y 1,y,...y N ] is the trial instance of ˆx sed in the hypothesis testing problem; and iv) D (y) = Nr T s z (r) (t) N =1 [ E h (y,r) w (t)] dt. Using argments similar to (3), the of a generic ser, e.g.,, can be pper bonded as shown in (16) on top of this page, where: i) in (a) /, the scaling factor 1 takes into N N t accont that the N N t possible vectors x are eqiprobable; ii) in (a), N H (x,y )/log ( ) acconts for the percentage of wrong bits between x and y, which is related to bit to antenna index mapping. It is worth mentioning that, as far as ser is concerned, an error occrs if and only if x y. In other words, even thogh x y, this does not imply that we have an error for all the N sers. In the best case, and error occrs for one ser only; iii) in (a), the is conditioned pon the event x y to take into accont that we are interested in compting the of ser ; iv) in (b) =, the factor ( ) 1 δ x,y takes into accont that, as mentioned above, there is no contribtion to the if x = y,even thogh x y; and v) (c) = tells s that the is niqely determined by the PEPs of the pair (x, y), i.e., PEP (x y). The PEP, PEP (x y), conditioned pon all the fading channel gains, can be compted by sing analytical steps similar to Section III. The final reslt is as follows: 0v PEP (x y) =Q B t 1 N r X XN 4N E h (y,r) 0 =1 h (x,r) 1 (17) Finally, by exploiting, similar to (8), the properties of Chi Sqare RVs, we can remove the conditioning over all fading channel statistics. After some algebra, and sing [30, Eq. ( 1 136), Eq. ( 1 137)] and [30, Eq. ( ), Eq. ( )], we can obtain (18) on top of this page, where AggrSNR = N {[ =1 E σ (1 δ x,y ) ]/ } N 0 is the Aggregate SNR. We note that the delta fnction, δ x,y, in AggrSNR takes into accont that if x = y, then h (y,r) h (x,r) =0in (17) and, ths, it does not contribte to the SNR seen by the detector. In other words, the antenna indexes shared by x and y cancel ot in the hypothesis testing problem. By comparing (8) and (18), we notice, as expected, that the main difference between single and mlti ser detector is the absence of error floor for high SNRs in (18), i.e., E h {PEP (x y)} 0 if N 0 0. The price to be paid C A

11 TRANSACTIONS ON VEHICULAR TECHNOLOGY 10 TABLE II OF PSK, QAM, j AND GSSK MODULATIONS WITH MULTI USER DETECTION. FOR GSSK MODULATION, WE HAVE EXPLICITLY USED THE log Nt IDENTITY N = a k, AND x 1:N =[x 1, x,...,x» N ] IS USED AS A SHORT HAND FOR VECTORS OF VECTORS. FURTHERMORE, s (x) = s (x 1) 1,s (x ),...,s (x N) N. Modlation PSK/QAM GSSK 8 >< >: 8 >< >: 1 M N AggrSNR = N P =1 MP x 1 =1! 3 MP MP MP N H s (x ),s (y )... 6 APEP x N =1 y 1 =1 y N =1 4 log (M) x y s (x) s (y) 7 {z } 5 Eq. (18), Eq. (0) s (x ) " E σ N 0 s (y ) 1 P P P P N log ( a ) APEP x y (x 1:N y 1:N ) x 1 Θ GSSK x N Θ GSSK y 1 Θ GSSK y N Θ GSSK APEP x y (x 1:N y 1:N )= N H(x,y j ) k log Nt APEP x y (x 1:N y 1:N ) a {z } AggrSNR = N P =1» E σ N 0 N ta (x,y) (1 δ N x,y ta ) Eq. (18), Eq. (0) [ ] ( ) 1 N N Nr 1, t log ( ) Nr N r x y [( 1 δx,y ) NH (x,y ) AggrSNR Nr] (19) ( ) Δ SNR 10 log N 10 r SULB = 10 log N 10 r 1 N (N+1) t log ( ) ( ) 1 δx,y NH (x,y ) E σ x y N (0) [E σ (1 δ x,y )] =1 for this performance improvement is a higher comptational complexity, as the detector in (15) has a complexity that increases exponentially with N. By sing a similar analytical derivation, Table II smmarizes the of PSK, QAM, and GSSK modlations. The performance comparison is postponed to Section VIII for varios system setps. As far as QAM and PSK modlation are concerned, it is worth mentioning that: i) formally, the of both modlation schemes is the same, bt the signal constellation diagram is different; and ii) thanks to the assmptions of synchronos mltiple access interference and SDMA/ChDMA mltiple access scheme, the framework in Table II is sefl to model spatial mltiplexing MIMO systems with ML optimm detection as well. More specifically, in this case N streams are simltaneosly transmitted by a single ser eqipped with = N antennas [3]. The only difference is that in spatial mltiplexing MIMO we are interested in the average among all the streams, i.e., = (1/N ) N =1, which can be derived from Table II. Some simlation reslts abot spatial mltiplexing MIMO schemes are given in Section VIII. A. SSK Modlation: Asymptotic Analysis and Bonds In this section, we stdy the asymptotic and compte some bonds to shed lights on the performance of mlti ser detectors for SSK modlation. For ease of notation, in what follows we se the short hand N Nt t Nt x 1=1 x N =1 y... 1=1 y N =1 ( ) x y ( ). 1) AggrSNR 1: Similar to Section III-A, for high SNR the in (16) is asymptotically eqal to (19) on top of this page. If N =1, it can be shown that the (henceforth called Single User Lower Bond (SULB), as it provides the performance withot mltiple access interference) redces to SULB (Nr+1)( N r 1) N Nt r SNR Nr. As expected, SULB is eqivalent to the of the single ser detector compted in Section III-A for the noise limited scenario. It is interesting to nderstand the relation between the of the mlti ser detector and the SULB. By direct inspection, the extra SNR needed in a mlti ser scenario to achieve the same as in a noise limited scenario is given in (0) on top of this page. The formla in (0) provides a qite accrate estimate of the extra SNR to get the same as in a noise limited environment. However, it is not mch insightfl becase it explicitly depends on the bit to antenna index mapping. To deeper nderstand, we analyze some special cases and provide some weaker bonds, which better reveal the behavior of mlti ser detection for SSK modlation.

12 TRANSACTIONS ON VEHICULAR TECHNOLOGY 11 L U ( )( (Nr+1) Nr 1 E σ ) Nr L (Nr+1) log ( a ) N N r ( Nr 1) ( ) E σ Nr ta N r N N r N 0 0 }{{} LL (Nr+1) N log ( a ) ( N r 1) ( ) ( )( E σ Nr N r N 0 U (Nr+1) N log ( Nt a ) N N r Nr 1 E σ ) Nr ta N r N 0 }{{} UU (3) ) E w σw E σ =1,,...N (strong interference scenario): Let s assme that among the N sers there is a ser, henceforth called worst (w) ser, with the worst propagation channel. The of this ser can be readily estimated with argments similar to [1], as the ser experiences very strong aggregate interference from the remaining N 1 sers. In this case, it is known that the mlti ser detector can perfectly estimate and redce to zero the interference generated by the other sers. In this case, ( its tends to the SULB, i.e., w (Nr+1) N Nr 1)[( ] Nr. t N Ew r σw)/ N0 3) E b σb E σ = 1,,...N (weak interference scenario): Let s assme that among the N sers there is a ser, henceforth called best (b) ser, with the best propagation channel. This scenario is more complicated to stdy than the strong interference case. However, we provide a tight bond to estimate the. If E b σb ( E σ )/ =1,,...N, then from (18) we have AggrSNR Eb σb N0. Accordingly, for high SNR we get: ( )[ 1 Nr 1 Eb σ ] Nr b b N N t log ( ) N r N 0 (1) N H (x,y ) x y By direct inspection, it is easy to show that x y N ( / ) H (x,y ) = N (N 1) t N (N 1) t N t log ( ). Accordingly, ( (1) simplifies to b (Nr+1) N N Nr 1)[( )/ ] Nr, t N Eb r σb N0 which is a very simple and easy to compte formla. Ths, the SNR gap with respect to the( SULB / can be compted ) as Δ SNRb = (10/N r ) log 10 b SULB b = 10 [(N 1)/N r ] log 10 ( ). This formla is very insightfl, as it provides a simple relation among all the parameters of interests, and, so, can be sed for a qick system design. For example, for a given Δ SNRb, and N r, we can compte the maximm nmber of sers that can share the wireless medim to garantee the desired. Also, we notice that the larger N r is, the smaller Δ SNRb is, and the trns ot to be very close to the SULB. 4) Generic ser (arbitrary interference scenario): The analysis of the for a generic ser can accrately be performed by sing (16) and (18), or by sing the asymptotic reslt for AggrSNR 1. However, its performance can be easily lower and pper bonded as L U : ( )( L = (Nr+1) Nr 1 E σ N r N 0 ( )( U = (Nr+1) N N Nr 1 E σ t N r N 0 ) Nr ) Nr () The lower bond, L, comes from the fact that the cannot the better than the SULB. On the other hand, the pper bond, U, comes from the fact that the cannot be worse than the scenario with weak interference, which is the worst case sitation for any ser. The SNR gap between the two bonds is Δ SNR = (10/N r ) log 10 ( U / L ) = 10 [(N 1)/N r ] log 10 ( ). We notice that, for any ser, the larger N r is, the closer to the SULB the is. While, the larger N or (i.e., the rate) are, the frther from the SULB the is. These trends are reasonable, will be validated by simlation in Section VIII, and can be exploited for a simple design of very general and complicated MIMO systems. B. GSSK Modlation: Asymptotic Analysis and Bonds As far as GSSK modlation is concerned, we can perform a similar asymptotic analysis. In particlar, insightfl bonds can be obtained by combining the stdy in Section V-A and in Section VI-A. More specifically, for a generic ser, the is lower and pper bonded as shown in (3) on top of this page. In particlar, L corresponds to the SULB of GSSK modlation. It actally depends on the nmber, N ta (x, y ), of non shared antenna indexes of x and y. However, in Section V-A we have proved that N ta (x, y ) can be lower bonded by the SULB of SSK modlation, i.e., LL in (3), as well as pper bonded by considering the worst case scenario with N ta (x, y ) = for every x and y, as given in the right hand side of (3). On the other hand, U corresponds, similar to Section VI-A, to the worst case scenario with weak interference. Its lower and pper bond / shown in (3) can be / obtained by setting N ta (x, y ) a =and N ta (x, y ) a =/a for every x and y, respectively. In fact, in Section V we have shown that N ta (x, y ) a for every x and y. Overall, from (3) we conclde that the of GSSK modlation lies in the interval LL UU. More ( specifically, / the SNR ) gap is Δ SNR = (10/N r ) log 10 UU LL = 10 log 10 ( ) +

13 TRANSACTIONS ON VEHICULAR TECHNOLOGY 1 ( / (10/N r ) log 10 N log (Nt,Nta) ). Also in this case, Δ SNR provides reasonable and insightfl otcomes abot the behavior of GSSK modlation. Finally, we emphasize that LL is the SULB of SSK modlation, and, so, we can readily estimate the relation among the two modlation schemes for generic MIMO systems. VII. COORDINATED MULTI POINT (COMP) SPACE MODULATION In the previos sections, we have shown that exploiting the spatial constellation diagram can be beneficial to improve the performance. However, to better exploit the spatial constellation diagram and have a sbstantial performance improvement withot sacrificing the bit rate, large antenna arrays are needed. While for some emerging transmission freqency bands this might not be a problem, as many antennas can efficiently be packed in a device [19], [0], in general there might be physical limitations on the nmber of antennas of each array. Two soltions to overcome this problem are, e.g., GSSK modlation [8] and SM [6]. GSSK modlation enables a better exploitation of the available antennas at the transmitter by activating more antennas and increasing the bit rate. However, in Section V we have seen that, in general, the of GSSK modlation is worse than SSK modlation. SM is a hybrid modlation scheme where spatial and signal constellation diagrams are jointly exploited to find a good trade off between bit rate and performance. However, for small antenna arrays the benefit of the spatial constellation diagram can be exploited only in part, as many information bits need to be encoded into the signal constellation diagram. Also, Generalized SM (GSM) might be another soltion that combines GSSK modlation and SM for a better trade off. In this section, we wish to bring to the attention of the reader that another way of providing large antenna arrays in SSK modlation is to exploit the concept of virtal MIMO [33], also known as Distribted Antenna System (DAS), BS cooperation, or CoMP transmission [34] [38]. The main idea is to share the antenna arrays of mltiple transmitters, ths having a larger eqivalent (virtal) antenna array that can be sed to encode a larger nmber of information bits. The basic idea is the following. Let Nt BS be the nmber of BSs connected to, e.g., a Base Station Controller (BSC) via a reliable wired backhal link, sch that all of them can receive the message that the core network is intended to transmit to a remote handset. Also, let Nt AR be the nmber of transmit antennas available in each BS. Accordingly, the virtal MIMO system of Nt BS BSs has a total nmber of = Nt BS Nt AR antennas that form a virtal (distribted and very large) spatial constellation diagram. With this approach, ( CoMP SSK ) ( modlation ) can transmit log ( ) = log N BS t + log N AR t bits per time slot. Hybrid soltions sch as CoMP GSSK, CoMP SM, and CoMP GSM are possible with their own advantages and disadvantages. A comprehensive stdy of all these soltions is ot of the scope of this paper. In this paper, we are only interested in ptting forth the concept of CoMP SSK modlation as a practical way of achieving very large bit rates by exploiting jst the spatial constellation diagram. Also, we wish to nderstand the asymptotic performance gain of SSK modlation as increases withot bond. In Section VIII, we will show varios reslts for, e.g., > 64, which might not be achievable, in practice, with a single BS, bt can be obtained by resorting to a CoMP approach (e.g., with (Nt BS,Nt AR )=(4, 16) or (Nt BS,Nt AR )=(8, 8)). With respect to conventional BS cooperation methods [36], in CoMP SSK modlation the backhal has less stringent reqirements as the coordinated BSs do not have to exchange data for cooperative beamforming, bt the backhal is sed only for disseminating the information from the core network to the BSs. Frthermore, we emphasize that since the cooperative BSs do not perform distribted beamforming, no transmit CSI is reqired, even thogh it might be beneficial [31]. VIII. NUMERICAL AND SIMULATION RESULTS In this section, we provide some nmerical reslts to compare the performance of varios modlation schemes in the presence of mltiple access interference, and to sbstantiate or analytical findings. In particlar, the system model introdced and described in Section II is accrately reprodced in or simlation environment, and varios MIMO setps and interference scenarios are analyzed. The specific simlation parameters can be fond in the caption of each figre. The reslts are obtained by assming E = = 1,,...,N. So, the near far effect is modeled throgh the fading parameters σ. In Fig. 1, we observe the near far effect for two MIMO setps with a different nmber of receive antennas. As compted analytically in Section III, the gets worse as the interference increases. Also, the system is more robst to mltiple access interference as N r increases. We notice that or analytical model (nion bond) is very accrate. Only when, it starts being less accrate. This is a reasonable otcome as the nion bond is tight only for low vales of, while the interference introdces an error floor. However, the model can, in general, well track the error floor, as predicted in Section III. In Fig., we stdy the robstness of single ser detection to the nmber of active sers N. As predicted by the nion bond in Section III, the gets worse when increasing N, bt the receiver can work qite well when N r 3. We notice that the error floor increases with N. In Fig. 3 and Fig. 4, we compare the performance of SSK, GSSK, QAM, and PSK modlations for varios target bit rates. We remind the reader that for large vales of, SSK modlation is implemented throgh the CoMP approach. As predicted in Section IV and Section V, QAM and PSK modlation otperform SSK modlation only if the bit rate if less than bits/s/hz, and SSK modlation always otperforms GSSK modlation. Also, the higher the target bit rate is, the larger the gap is. This confirms the findings in Section IV, and highlights that sing the spatial constellation diagram is beneficial with and withot mltiple access interference. Overall the bonds can very well captre the behavior of all the modlation schemes if the error floor is not too high. This means that they are sefl for all scenarios of practical interest. In Fig. 5 and in Fig. 6, we

14 TRANSACTIONS ON VEHICULAR TECHNOLOGY 13 N r = 1 N r = 3 SSK GSSK σ = 1E 4 σ = 1E 3 N = t N = 4 t σ = 1E σ = 1E 1 σ = 1 SULB = 8 N = 16 t = 3 N = 64 t ( = 5, a = ) (N = 6, N = 3) t ta ( = 8, a = 4) Fig. 1. of SSK modlation with single ser detection. Setp: =8; N =; σ1 =1; Nr =1(left) and Nr =3(right). Markers show Monte Carlo simlations and solid lines the analytical model (i.e., (8) and (9)). The of ser 1 (probe/intended link) is shown. SULB stands for Single User Lower Bond, i.e., it represents the scenario with no mltiple access interference. Fig. 3. of SSK (left) and GSSK (right) modlations with single ser detection. Setp: N =3; σ1 =1; σ i =10 for i =, 3,...,N ; N r =. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., (8) and (9) for SSK modlation and (13) for GSSK modlation). The of ser 1 (probe/intended link) is shown. PSK QAM N r = 1 N r = 3 N = 1 N = N = 3 N = 4 N = Fig.. of SSK modlation with single ser detection. Setp: =8; σ 1 =1; σ i =10 for i =, 3,...,N ; N r =1and N r =3. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., (8) and (9)). The of ser 1 (probe/intended link) is shown. M = M = 4 M = 8 M = 16 M = 3 M = Fig. 4. of PSK (left) and QAM (right) modlations with single ser detection. Setp: N =3; σ1 =1; σ i =10 for i =, 3,...,N ; N r =. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., the nion bond in the first and second row of Table I). The of ser 1 (probe/intended link) is shown. analyze the performance of SSK modlation with respect to the nmber of receive antennas. If N r = 3 (Fig. 5), we observe a non negligible performance gain, if the bit rate if greater than bits/s/hz, provided by SSK modlation with respect to QAM. The price to be paid is, of corse, the need to exploit the CoMP principle to achieve very high bit rates, e.g., when = 64. However, the SNR gain is so significant to motivate the CoMP approach. On the other hand, if N r = 1 (Fig. 6), we notice that QAM is always sperior to SSK modlation, while SSK modlation is better than PSK and GSSK modlations. This reslt is not available in the literatre even for N = 1, as the vast majority of papers typically consider MIMO setps with N r > 1. Asa conseqence, if a single ser scenario is considered and the receiver can be eqipped with only one receive antenna, then SSK modlation is not the best choice and we shold se QAM. In all the other cases, SSK modlation is sperior to QAM. Frthermore, in a mlti ser scenario we can very nlikely design and se a MIMO system with N r = 1, since, as shown in Fig. 6 for N =, the rapidly gets worse with the target bit rate. In scenarios with mlti ser interference and single ser detection we are forced to increase N r to get adeqate performance. In these sitations, SSK modlation is always better than QAM. Finally, in Fig. 7 we stdy the for very high bit rates (CoMP SSK has a large nmber of cooperative BSs). We can observe a significant performance gain of SSK modlation with respect to all the other modlation schemes. Also, GSSK modlation

15 TRANSACTIONS ON VEHICULAR TECHNOLOGY 14 N = 1 /M = /M = 8 N = SSK and QAM QAM PSK and GSSK /M = 64 SSK GSSK SSK /M/N = 8 /M/N = 64 /M/N = /M/N = Fig. 5. of SSK (ble crves) and QAM (green crves) modlations with single ser detection. Setp: N =1(left) and N =(right); σ1 =1 and σ = 5 10 ; N r = 3. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., (8) and (9) for SSK modlation and the nion bond in the second row of Table I). The of ser 1 (probe/intended link) is shown. N = 1 N = Fig. 7. of, on the left, SSK (ble crves) and QAM (green crves), and, on the right, PSK (red crves) and GSSK (magenta crves) modlations with single ser detection. Setp: N =; σ 1 =1and σ =10 ; N r =3. For GSSK modlation we have: (,a) =(5, ) if N =8; (,a) = (8, 4) if N =64; (,a) = (11, 4) if N = 56; and (,a) = (1, 5) if N = 104. For SSK and GSSK modlations, markers show Monte Carlo simlations and solid lines the analytical model (i.e., (8) and (9) for SSK modlation and (13) for GSSK modlation). For QAM and PSK modlations, only Monte Carlo simlations (markers pls solid lines) are shown for ease of readability. The of ser 1 (probe/intended link) is shown. N = SSK PSK QAM GSSK /M/N = /M/N = 8 /M/N = Fig. 6. of SSK (ble crves), GSSK (magenta crves), PSK (red crves), and QAM (green crves) modlations with single ser detection. Setp: N =1(left) and N =(right); σ1 =1and σ =5 10 ; N r =1. Only Monte Carlo simlations (markers pls solid lines) are shown for ease of readability. The of ser 1 (probe/intended link) is shown. significantly otperforms QAM. Overall, or analysis and simlations confirm the potential benefits of sing the spatial constellation diagram in both single and mlti ser scenarios. In Figs. 3, 4, 7, we have compared the of SSK/GSSK modlations and single antenna PSK/QAM with the main goal of nderstanding the performance gap among these transmission technologies when ML optimm performance can be achieved with low complexity single stream decoding at the receiver. In other words, the receiver has almost the same complexity for all the modlation schemes. However, when considering the complexity of the transmitter, the comparison in Figs. 3, 4, 7 might appear a bit nfair for single antenna PSK/QAM SSK ( = 64) MIMO QAM (M = 64, = 1) MIMO QAM (M = 8, = ) MIMO QAM (M = 4, = 3) MIMO QAM (M =, N = 6) t E /N m 0 Fig. 8. of SSK modlation and spatial mltiplexing MIMO with QAM (MIMO QAM) for N =1. A single stream and a mlti stream decoders are sed at the receiver for SSK modlation and spatial mltiplexing MIMO, respectively. Setp: σ =1for all the wireless links; N r =3. Only Monte Carlo simlations are shown. The crves of SSK modlation with =64, and MIMO QAM with =1and M =64are the same as those shown in Fig. 5. systems, as SSK/GSSK modlations need large antenna arrays to achieve the same transmission rate. Motivated by this consideration, in Fig. 8 we stdy a complementary sitation in which PSK/QAM systems sing spatial mltiplexing MIMO [39, Sec. I A] are compared to SSK/GSSK modlations. In this case, PSK/QAM systems need a mlti stream decoder to garantee ML optimm performance. More specifically, in a point to point link the detector is the same as in Section VI, with the only exception that all the streams are siml-

16 TRANSACTIONS ON VEHICULAR TECHNOLOGY 15 taneosly transmitted from the same device nder the total transmit power constraint. Unlike Figs. 3, 4, 7, in this case the comparison is certainly nfair for SSK/GSSK modlations, which need mch less decoding complexity for ML optimm performance. However, we feel important to show this setp and to analyze all the possibilities. For simplicity, we limit the stdy to the setp with N = 1, since ML optimm decoding for spatial mltiplexing MIMO in the presence of mltiple access interference wold reqire a two fold mlti stream decoder to cope with inner (i.e., de to mltiplexing many streams at the same transmitter) and oter (i.e., de to the mlti ser scenario) interferences. This scenario wold reqire a more practical and sb optimal decoder, e.g., based on sphere decoding [40], to keep the complexity at a reasonable level. Accordingly, this stdy is postponed to ftre research. From Fig. 8, important considerations can be made. We notice that by increasing the nmber of antennas at the transmitter, spatial mltiplexing MIMO with QAM achieves, as expected, better performance than single antenna QAM. However, the price to pay for this performance improvement is, among the others, mlti stream decoding at the receiver. Very interestingly, we see that SSK modlation is never worse than spatial mltiplexing MIMO, even thogh SSK modlation needs jst low complexity single ser decoding. In this case the price to pay is the need to have mltiple radiating elements at the transmitter, even thogh jst one of them is active and, ths, no mltiple transmit chains are needed. Similar to [6] and [7], these reslts clearly show the potential advantages of sing the spatial constellation diagram when comparing SSK modlation to more complicated MIMO schemes. However, it shold be emphasized that, becase of the very different hardware and comptational complexity reqirements, no conclsive statements can be made abot the speriority of one transmission technology against the other. The only pragmatic conclsion that can be drawn is the clear potential gain that might come from sing the spatial constellation diagram, along with the inherent performance/complexity trade off among different modlation schemes. In Figs. 9 17, we show that with mlti ser detection. The setp is the same as in Figs So, the interested reader can readily compare single and mlti ser detection for the same operating conditions. In Fig. 9 and Fig. 10, we stdy the accracy of the lower and pper bond derived in Section VI-A. We observe that, for varios strong/weak interference scenarios, the bonds well track the behavior of the system. In particlar, Fig. 10 shows that the of a generic ser that is sbject to neither strong nor weak interference is well bonded by or analytical frameworks. For those sers where the assmption of strong/weak interference can be made, the bonds are asymptotically tight. Figre 11 clearly shows that no error floor is present with mlti ser detection, and the goes to zero if the noise is very small. This is an important reslt and the confirmation that both modlation and mltiple access can be garanteed by exploiting only the randomness of the wireless channels. Similar to Fig. 3 and Fig. 4, in Figs we compare the performance of varios modlation schemes. Also in this case, we notice the non negligible performance gain of σ = 1E 3 σ = 1E σ = 1E 1 σ = 1 σ = 1E+1 σ = 1E 4 σ = 1E Fig. 9. of SSK modlation with mlti ser detection. Setp: =8; N =; σ1 =1; Nr =3. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., (16) and (18)). Frthermore, dashed lines show the estimated lower bond (i.e., L in ()), which corresponds to the Single User Lower Bond (SULB) when no mltiple access interference is present; and dotted lines show the estimated pper bond (i.e., U in ()). The of ser (probe/intended link) is shown. User 1 User User Fig. 10. of SSK modlation with mlti ser detection. Setp: =8; N =3; σ1 =0.1, σ =1, σ 3 =10; Nr =3. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., (16) and (18)). Frthermore, dashed lines show the estimated lower bond (i.e., L in ()), which corresponds to the Single User Lower Bond (SULB) when no mltiple access interference is present; and dotted lines show the estimated pper bond (i.e., U in ()). The of all the sers is shown. SSK modlation. Overall, the predictions in Section VI are confirmed, and the bonds developed in Section VI-A and Section VI-B agree with Monte Carlo simlations. A very interesting reslt is shown in Fig. 16. Unlike Fig. 6, we observe that, for a bit rate grater than bits/s/hz, with mlti ser detection SSK modlation is not worse than QAM even if N r =1. In particlar, we observe a crossing point for high SNRs, where the of SSK modlation is at least the same as QAM. This reslt clearly highlights that SSK modlation with mlti ser detection is inherently more robst than QAM to mltiple access interference. Finally, Fig. 17 shows a reslt

17 TRANSACTIONS ON VEHICULAR TECHNOLOGY 16 N = 1 N = N = 3 N = 4 N = 5 N r = ( = 5, a = ) ( = 6, a = 3) ( = 8, a = 4) Fig. 11. of SSK modlation with mlti ser detection. Setp: =8; σ1 =1and σ i = for i =, 3,...,N ; N r =1and N r =3. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., (16) and (18)). Frthermore, dotted lines show the estimated pper bond (i.e., U in ()). The of ser 1 (probe/intended link) is shown. It is worth mentioning that some simlation reslts (markers) are not shown de to the long simlation time for medim/high vales of. Fig. 13. of GSSK modlation with mlti ser detection. Setp: N = 3; σ1 =1and σ i =10 for i =, 3,...,N ; N r =. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., the formla in the second row of Table II). The of ser 1 (probe/intended link) is shown. Frthermore, dashed lines show the estimated lower bond (i.e., LL in (3)) and dotted lines show the estimated pper bond (i.e., UU in (3)), which correspond to the bonds when the probe link is the best link. = = 4 = 8 = 16 = 3 = Fig. 1. of SSK modlation with mlti ser detection. Setp: N = 3; σ1 = 1 and σ i = for i =, 3,...,N ; N r =. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., (16) and (18)). Frthermore, dotted lines show the estimated pper bond (i.e., U in ()). The of ser 1 (probe/intended link) is shown. It is worth mentioning that some simlation reslts (markers) are not shown de to the long simlation time for medim/high vales of. M = M = 4 M = 8 M = 16 M = Fig. 14. of PSK modlation with mlti ser detection. Setp: N = 3; σ 1 =1and σ i =10 for i =, 3,...,N ; N r =. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., the formla in the first row of Table II). The of ser 1 (probe/intended link) is shown. It is worth mentioning that some simlation reslts (markers) are not shown de to the long simlation time for medim/high vales of. similar to Fig. 7 for high bit rates. The performance gain of SSK modlation is well confirmed in this case too. IX. CONCLUSION In this paper, we have proposed a comprehensive framework to stdy the of SSK/GSSK modlations over Rayleigh fading channels with mltiple access interference. The frameworks are sefl for single and mlti ser detectors. Frthermore, simple pper and lower bonds have been developed, and have been sed to get insightfl information abot advantages and disadvantages of sing the spatial constellation diagram as a sorce of information. Clear indications abot the system behavior for varios channel conditions, interference levels, and MIMO setps have been provided. Comprehensive performance comparisons with conventional modlation schemes for single and mlti ser detection have been given. Overall, or theoretical findings have been well sbstantiated by Monte Carlo simlations. Ongoing research is concerned with: i) the extension of the framework to the asynchronos mltiple access scenario and to SM/GSM; ii) the extension of the analysis to freqency

18 TRANSACTIONS ON VEHICULAR TECHNOLOGY 17 SSK and QAM PSK and GSSK /M/N = 8 /M/N = 64 N /M/N = 56 t M = M = 4 M = 8 M = 16 M = Fig. 15. of QAM with mlti ser detection. Setp: N =3; σ 1 =1 and σ i =10 for i =, 3,...,N ; N r =. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., the formla in the first row of Table II). The of ser 1 (probe/intended link) is shown. It is worth mentioning that some simlation reslts (markers) are not shown de to the long simlation time for medim/high vales of. SSK /M = /M = 8 /M = 64 N r = 3 SSK Fig. 16. of SSK (ble and green lines for N r =1and N r =3, respectively) and QAM (red and magenta lines for N r =1and N r =3, respectively) modlations with mlti ser detection. Setp: N =; σ 1 =1 and σ =5 10. Markers show Monte Carlo simlations and solid lines the analytical model (i.e., (16) and (18) for SSK modlation and the formla in the first row of Table II for QAM). The of ser 1 (probe/intended link) is shown. selective fading channels, and the possible application of space modlation to Ultra Wide Band (UWB) wireless systems [41], [4], as recently sggested in []; iii) the development of simple decoding algorithms, e.g., based on the Sphere Decoding principle [40], for the mlti ser detector; and iv) by exploiting the theory of Stochastic Geometry [], the development and analysis of detectors robst to network interference generated by many randomly distribted interferers. ACKNOWLEDGMENT We grateflly acknowledge spport from the Eropean Union (PITN GA , GREENET project) for this Fig. 17. of, on the left, SSK (ble crves) and QAM (green crves), and, on the right, PSK (red crves) and GSSK (magenta crves) modlations with mlti ser detection. Setp: N = ; σ1 = 1 and σ = 10 ; N r =3. For GSSK modlation we have: (,a) =(5, ) if N =8; (,a) =(8, 4) if N =64; and (,a) = (11, 4) if N = 56. For SSK and GSSK modlations, markers show Monte Carlo simlations and solid lines the analytical model (i.e., (16) and (18) for SSK modlation and the formla in the second row of Table II for GSSK modlation). For QAM and PSK modlations, only Monte Carlo simlations (markers pls solid lines) are shown for ease of readability. The of ser 1 (probe/intended link) is shown. Also, dashed lines and dashed lines pls markers show the performance estimated throgh the bonds (i.e., U in () for SSK modlation and UU in (3) for GSSK modlation). More specifically: for SSK modlation, dashed lines show the estimated pper bond if N =8 and N =64, and dashed lines with markers the pper bond if N =8; and, for GSSK modlation, dashed lines show the estimated pper bond if N = 8 and dashed lines with markers the pper bond if N = 64 and N = 56. This is done to improve the readability of the figre. Note that, for GSSK modlation the estimated lower bond corresponds exactly to the pper bond compted for SSK modlation and shown on the left hand side of the figre. It is worth mentioning that some simlation reslts (markers) are not shown de to the long simlation time for medim/high. work. M. Di Renzo acknowledges spport of the Laboratory of Signals and Systems (LS) nder the research project Jenes Cherchers. H. Haas acknowledges the EPSRC (EP/G011788/1) and the Scottish Fnding Concil spport of his position within the Edinbrgh Research Partnership in Engineering and Mathematics. REFERENCES [1] Y. Cha and S. H. Y, Space modlation on wireless fading channels, IEEE Veh. Technol. Conf., vol. 3, pp , Oct [] J. Jeganathan, A. Ghrayeb, L. Szczecinski, and A. Ceron, Space shift keying modlation for MIMO channels, IEEE Trans. Wireless Commn., vol. 8, no. 7, pp , Jly 009. [3] M. Di Renzo and H. Haas, A general framework for performance analysis of space shift keying (SSK) modlation for MISO correlated Nakagami-m fading channels, IEEE Trans. Commn., vol. 58, no. 9, pp , Sep [4] H. Haas, E. Costa, and E. Schltz, Increasing spectral efficiency by data mltiplexing sing antennas arrays, IEEE Int. Symp. Personal, Indoor, Mobile Radio Commn., vol., pp , Sep. 00. [5] Y. Yang and B. Jiao, Information gided channel hopping for high data rate wireless commnication, IEEE Commn. Lett., vol. 1, no. 4, pp. 5 7, Apr [6] R. Y. Mesleh, H. Haas, S. Sinanovic, C. W. Ahn, and S. Yn, Spatial modlation, IEEE Trans. Veh. Technol., vol. 57, no. 4, pp. 8 41, Jly 008.

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Marco Di Renzo (SM 05 AM 07 M 09) was born in L Aqila, Italy, in He received the Larea (cm lade) and the Ph.D. degrees in Electrical and Information Engineering from the Department of Electrical and Information Engineering, University of L Aqila, Italy, in April 003 and Janary 007, respectively. From Agst 00 to Janary 008, he was with the Center of Excellence for Research DEWS, University of L Aqila, Italy. From Febrary 008 to April 009, he was a Research Associate with the Telecommnications Technological Center of Catalonia (CTTC), Barcelona, Spain. From May 009 to December 009, he was a Research Fellow with the Institte for Digital Commnications (IDCOM), The University of Edinbrgh, Edinbrgh, United Kingdom (UK). Since Janary 010, he has been a Tenred Researcher ( Chargé de Recherche Titlaire ) with the French National Center for Scientific Research (CNRS), as well as a research staff member of the Laboratory of Signals and Systems (LS), a joint research laboratory of the CNRS, the École Spériere d Électricité (SUPÉLEC), and the University of Paris Sd XI, Paris, France. His main research interests are in the area of wireless commnications theory, signal processing, and information theory. Harald Haas (SM 98 AM 00 M 03) received the Ph.D. degree from the University of Edinbrgh in 001. His main research interests are in the areas of wireless system design/analysis and digital signal processing, with a particlar focs on interference coordination in wireless networks, spatial modlation and visible light commnications. He joined the International University Bremen (Germany), now Jacobs University Bremen, in September 00 where he is now Honorary Professor of electrical engineering. In Jne 007, he joined the University of Edinbrgh (Scotland/UK) where he is Professor of Mobile Commnications in the Institte for Digital Commnications (IDCOM). Professor Haas has been invited to present his work on visible light commnications at the TED conference in 011. Since 007, he has been a Reglar High Level Visiting Scientist spported by the Chinese 111 program at Beijing University of Posts and Telecommnications.