MULTIPLE input multiple output (MIMO) systems offer

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1 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1 Generalised Sphere Decoding for Spaial Modulaion Abdelhamid Younis, Sinan Sinanović, Marco Di Renzo, Raed Mesleh and Harald Haas arxiv: v3 [cs.it] 3 May 13 Absrac In his paper, Sphere Decoding (SD) algorihms for Spaial Modulaion (SM) are developed o reduce he compuaional complexiy of Maximum Likelihood (ML) deecors. Two SDs specifically designed for SM are proposed and analysed in erms of Bi Error Raio (BER) and compuaional complexiy. Using Mone Carlo simulaions and mahemaical analysis, i is shown ha by carefully choosing he iniial radius he proposed sphere decoder algorihms offer he same BER as ML deecion, wih a significan reducion in he compuaional complexiy. A igh closed form expression for he BER performance of SM SD is derived in he paper, along wih an algorihm for choosing he iniial radius which provides near o opimum performance. Also, i is shown ha none of he proposed SDs are always superior o he ohers, bu he bes SD o use depends on he arge specral efficiency. The compuaional complexiy rade off offered by he proposed soluions is sudied via analysis and simulaion, and is shown o validae our findings. Finally, he performance of SM SDs are compared o Spaial Muliplexing (SMX) applying ML decoder and applying SD. I is shown ha for he same specral efficiency, SM SD offers up o 84% reducion in complexiy compared o SMX SD, wih up o 1 db beer BER performance han SMX ML decoder. Index Terms Muliple inpu muliple oupu (MIMO) sysems, spaial modulaion (SM), spaial muliplexing (SMX), sphere decoding (SD), large scale MIMO. I. INTRODUCTION MULTIPLE inpu muliple oupu (MIMO) sysems offer a significan increase in specral efficiency in comparison o single anenna sysems [1]. An example is Spaial Muliplexing (SMX) [], which ransmis simulaneously over all he ransmi anennas. This mehod achieves a specral efficiency ha increases linearly wih he number of ransmi anennas. However, hese sysems canno cope wih he exponenial increase of wireless daa raffic, and a larger number The associae edior coordinaing he review of his paper and approving i for publicaion was Prof. G. Bauch. Manuscrip received July 9, 1; revised December 4, 1. This work was presened in par a he IEEE GLOBECOM 1, Miami, USA, and IEEE ICC 11, Kyoo, Japan. A. Younis and H. Haas are wih The Universiy of Edinburgh, College of Science and Engineering, Insiue for Digial Communicaions, Join Research Insiue for Signal and Image Processing, King s Buildings, Mayfield Road, Edinburgh, EH9 3JL, UK (e mail: a.younis, h.haas}@ed.ac.uk). S. Sinanović is wih Glasgow Caledonian Universiy, School of Engineering and Buil Environmen, George Moore Building, Cowcaddens Road, Glasgow, G4 BA, UK (e mail: sinan.sinanovic@gcu.ac.uk). M. Di Renzo is wih he Laboraory of Signals and Sysems (LS), French Naional Cener for Scienific Research (CNRS), École Supérieure d Élecricié (SUPÉLEC), Universiy of Paris Sud XI (UPS), 3 rue Jolio Curie, 9119 Gif sur Yvee (Paris), France (e mail: marco.direnzo@lss.supelec.fr). R. Mesleh is wih he Elecrical Engineering Deparmen and SNCS research cener, Universiy of Tabuk, P.O.Box: Tabuk, Saudi Arabia (e mail: rmesleh.sncs@u.edu.sa). Digial Objec Idenifier 1.119/TCOMM /1$5. c 13 IEEE of ransmi anennas (large scale MIMO) should be used [3]. Large scale MIMO sudied in [4] [6], offers higher daa raes and beer bi error rae (BER) performance.however, his comes a he expense of an increase in: 1) Compuaional complexiy: A SMX maximum likelihood (ML) opimum receiver searches across all possible combinaions, and ries o resolve he iner channel inerference (ICI) caused by ransmiing from all anennas simulaneously, on he same frequency. Sphere decoder (SD) was proposed o reduce he complexiy of he SMX ML algorihm while reaining a near opimum performance [7], [8]. The SD reduces he complexiy of he ML decoder by limiing he number of possible combinaions. Only hose combinaions ha lie wihin a sphere cenred a he received signal are considered. However, even hough SMX SD offers a large reducion in complexiy compared o SMX ML, i sill has a high complexiy which increases wih he increase of he number of ransmi anennas. ) Hardware complexiy: In SMX he number of radio frequency (RF) chains is equal o he number of ransmi anennas. From [9], RF chains are circuis ha do no follow Moore s law in progressive improvemen. Therefore, increasing he number of ransmi anennas and consequenly he number of RF chains increases significanly he cos of real sysem implemenaion [1]. 3) Energy consumpion: RF chains conain Power Amplifiers (PAs) which are responsible for 5 8% of he oal power consumpion in he ransmier [11]. Therefore, increasing he number of RF chains resuls in a decrease in he energy efficiency [1]. Thus, SMX may no always be feasible and a more energy efficien and low complexiy soluion should be considered. Spaial Modulaion (SM) is a ransmission echnology proposed for MIMO wireless sysems. I aims o increase he specral efficiency, (m), of single anenna sysems while avoiding ICI [1]. This is aained hrough he adopion of a new modulaion and coding scheme, which foresees: i) he acivaion, a each ime insance, of a single anenna ha ransmis a given daa symbol (consellaion symbol), and ii) he exploiaion of he spaial posiion (index) of he acive anenna as an addiional dimension for daa ransmission (spaial symbol) [13]. Boh he consellaion symbol and he spaial symbol depend on he incoming daa bis. An overall increase by he base wo logarihm of he number of ransmi anennas of he specral efficiency is achieved. This limis he number of ransmi anennas o be a power of wo unless fracional bi encoding SM (FBE SM) [14], or generalised SM (GSM) [15]

2 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION are used. Acivaing only one anenna a a ime means ha only one RF chain is needed, which significanly reduces he hardware complexiy of he sysem [16]. Moreover, as only one RF chain is needed, SM offers a reducion in he energy consumpion which scales linearly wih he number of ransmi anennas [1], [17]. Furhermore, he compuaional complexiy of SM ML is equal o he complexiy of single inpu muliple oupu (SIMO) sysems [18], i.e. he complexiy of SM ML depends only on he specral efficiency and he number of receive anennas, and does no depend on he number of ransmi anennas. Recenly he poenial benefis of SM have been validaed no only by simulaions bu also via experimens [19], []. Moreover, in [1] for he firs ime he performance of SM is analysed using real world channel measuremens. Accordingly, SM appears o be a good candidae for large scale MIMO [] [5]. In spie of is low complexiy implemenaion, here is sill poenial for furher reducions, by limiing he number of possible combinaions using he SD principle. However, exising SD algorihms in lieraure do no consider he basic and fundamenal principle of SM, ha only one anenna is acive a any given ime insance. Therefore, wo modified SD algorihms based on he ree search srucure ha are ailored o SM are proposed. The firs SD will be called receiver cenric SD (SM Rx), which was firs presened in [6]. The algorihm in [6] combines he received signal from he muliple receive anennas, as long as he Euclidean disance from he received poin is less han a given radius. This SD based deecor is especially suiable when he number of receive anennas is very large. This echnique reduces he size of he search space relaed o he muliple anennas a he receiver (we denoe his search space as receive search space ). I will be shown laer ha here is no loss in eiher he diversiy order or he coding gain, i.e. he BER is very close o ha of he ML deecor. However, he main limiaion is ha i does no reduce he search space relaed o he number of possible ransmied poins (we denoe his as ransmi search space ). This limiaion prevens he deecor from achieving a significan reducion in compuaional complexiy when high daa raes are required. The second SD, which is called Transmi cenric (SM Tx) was firs presened in [7]. I aims a reducing he ransmi search space by limiing he number of possible spaial and consellaion poins. The SM Tx algorihm avoids an exhausive search by examining only hose poins ha lie inside a sphere wih a given radius. However, SM Tx is limied o he overdeermined MIMO seup (N N r ), where N and N r are he number of ransi and receiver anennas respecively. In [8], [9], i is shown ha SM Tx in [7] can sill be used for he case of (N r 1) N > N r, where SM Tx is referred o as enhanced Tx SD (E Tx SD). Moreover, in [8], [9] a deecor for he case of N > N r referred o as generalised Tx SD (G Tx SD) is proposed. By using he division algorihm he G Tx SD echnique: 1) Divides he se of possible anennas o a number of subses. ) Performs E Tx SD over each subse. 3) Takes he minimum soluion of all he ses. However, in his paper we propose a simple soluion, in which all ha is needed is o se a consan ϕ o for N N r and ϕ = σ n for N > N r, where σ n is he noise variance. In [8], [9], he normalised expeced number of nodes visied by he SM Tx algorihm is used o compare is complexiy wih he complexiy of he SM ML algorihm. This does no ake ino accoun pre compuaions needed by SM Tx. In his paper, when comparing he complexiy of SM Tx wih he complexiy of SM ML and SM Rx, we ake ino accoun he pre compuaions needed by he SM Tx. We show ha because of hose pre compuaions, he SM Tx echnique is no always he bes soluion, where in some cases i is even more complex han SM ML. The performance of boh SDs is closely ied o he choice of he iniial radius. The chosen radius should be large enough for he sphere o conain he soluion. On he one hand, he larger he radius is, he larger he search space, which increases he complexiy. On he oher hand, a small radius may cause he algorihm o fail in finding a poin inside he sphere. In his paper, a careful sudy of he performance of hese wo deecors is provided, along wih an accurae comparison of heir compuaional complexiy. Numerical resuls show ha wih no loss in he BER performance, he proposed soluions provide a subsanial reducion in compuaional complexiy wih respec o he SM ML decoder. We also derive a closed form expression for he BER performance of SM SD and show ha he iniial radius can be chosen such ha SM SD gives an opimum performance. Furhermore, i is shown ha SM Rx is less complex han SM Tx for lower specral efficiencies, while SM Tx is he bes soluion for higher specral efficiencies. Finally, using numerical resuls we show ha SM SD offers a significan reducion and nearly he same performance when compared o SMX wih ML decoder or SD. The remainder of his paper is organised as follows: In secion II, he sysem model along wih he ML opimum deecor is summarised. In secion III, he new SM Rx and SM Tx receivers are described. In secion IV, an accurae analysis of he compuaional complexiy of boh SM Rx and SM Tx is performed. In secion V, he analyical BER performance for SM SDs is derived along wih he iniial radius selecion mehod. Numerical resuls are presened in secion VI, and he paper is concluded in secion VII. A. SM Modulaor II. SYSTEM MODEL In SM, he bi sream emied by a binary source is divided ino blocks conaining m = log (N ) +log (M) bis each, where M is he consellaion size. Then he following mapping rule is used [1]: The firs log (N ) bis are used o selec he anenna which is swiched on for daa ransmission, while he oher ransmi anennas are kep silen. In his paper, he acual ransmi anenna which is acive for ransmission is denoed by l wih l 1,,...,N }. The second log (M) bis are used o choose a symbol in he signal consellaion diagram. Wihou loss of generaliy, Quadraure Ampliude Modulaion (QAM) is considered. In his paper, he acual complex symbol emied by he ransmi anenna l is denoed by s, wih s s 1,s,...,s M }.

3 YOUNIS e al.: GENERALISED SPHERE DECODING FOR SPATIAL MODULATION 3 Accordingly, he N 1 ransmied vecor is: x l,s = [ 1 (l 1),s, 1 (N l )] T, (1) where [ ] T denoes ranspose operaion, and p q is a p q marix wih all zero enries. Noe, a power consrain on he average energy per ransmission of uniy is assumed (i.e.e s = E[x H x] = 1), where E } is he expecaion operaor. From above, he maximum achievable specral efficiency by SM is, However, for SMX, m SM = log (N )+log (M) () m SMX = N log (M) (3) From () and (3), we can see ha he specral efficiency of SM does no increase linearly wih he number of ransmi anennas as SMX does. Therefore, SM needs a larger number of ransmi anennas/ larger consellaion size o arrive a he same specral efficiency as SMX. However, because in SM only one anenna is acive: The compuaional complexiy of SM does no depend on he number of ransmi anennas. Unlike SMX where he compuaional complexiy increases linearly wih he number of ransmi anennas, he compuaional complexiy of SM is he same as he compuaional complexiy of SIMO sysems. The number of RF chains needed by SM is significanly less han he number of RF chains needed by SMX. In fac, only one RF chain is required for SM. For hese reasons we believe ha SM is a srong candidae for large scale MIMO sysems, which srongly moivaes his work. B. Channel Model The modulaed vecor, x l,s, in (1) is ransmied hrough a frequency fla N r N MIMO fading channel wih ransfer funcion H, where N r is he number of receive anennas. In his paper, a Rayleigh fading channel model is assumed. Thus, he enries of H are modelled as complex independen and idenically disribued (i.i.d.) enries according o CN(, 1). Moreover, a perfec channel sae informaion (CSI) a he receiver is assumed, wih no CSI a he ransmier. Thus, he N r 1 received vecor can be wrien as follows: y = Hx l,s +n = h l s +n (4) where n is he N r dimensional Addiive Whie Gaussian Noise (AWGN) wih zero mean and variance σ per dimension a he receiver inpu, and h l is he l h column of H. Noe, he signal-o-noise-raio is SNR = E s /N o = 1/σ n. C. ML Opimum Deecor The ML opimum receiver for MIMO sysems can be wrien as, ˆx (ML) } = argmin y Hx x Q m F where Q m is a m space conaining all possible (N 1) ransmied vecors, F is he Frobenius norm, andˆ denoes he esimaed spaial and consellaion symbols. Noe, in SM only one ransmi anenna is acive a a ime. Therefore, he opimal receiver in (5) can be simplified o, [ˆl(ML),ŝ (ML) ] = arg min l 1,,...N } s s 1,s,...s M} = arg min l 1,,...N } s s 1,s,...s M} y h l s F Nr } } y r h l,r s r=1 where y r andh l,r are he r h enries of y andh l respecively. III. SPHERE DECODERS FOR SM In his secion we inroduce wo SDs ailored for SM, SM Rx and SM Tx. SM Rx aims a reducing he number of summaions over N r required by he ML receiver in (6). SM Tx aims a reducing he number of poins (l,s) he ML receiver searches over. Firs, for ease of derivaion, we inroduce he real valued equivalen of he complex valued model in (4) following [3], where, ȳ = H x l,s + n (5) (6) = h l s + n (7) ȳ = [ Re y T},Im y T}] T [ ] ReH} ImH} H = ImH} ReH} (8) (9) x l,s = [ Re x T l,s},im x T l,s }] T (1) n = [ Re n T},Im n T}] T h l = [ Hl, H ] l+n [ ] Res} s = Ims} (11) (1) (13) where Re } and Im } denoe real and imaginary pars respecively, and H l is he l h column of H. A. SM Rx Deecor The SM Rx is a reduced complexiy and close o opimum BER achieving decoder, which aims a reducing he receive search space. The deecor can formally be wrien as follows: [ˆl(Rx),ŝ (Rx) ] = arg min l 1,,...N } r=1 s s 1,s,...s M} Ñ r(l,s) ȳr h l,r s (14)

4 4 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION Ri+1 = z D xl,s F = N (R i R ( )+ zν D (ν,l) Res} D ) (ν,l+n)ims} ν=1 () where h l,r is he r h row of h l, and, } n Ñ r (l,s) = max n y r h l,r s R n 1,,...N r} r=1 (15) The idea behind SM Rx is ha i keeps combining he received signals as long as he Euclidean disance in (14) is less or equal o he radius R. Whenever a poin is found o be inside he sphere, he radius, R, is updaed wih he Euclidean disance of ha poin. The poin wih he minimum Euclidean disance and Ñr(, ) = N r is considered o be he soluion. B. SM Tx Deecor The convenional SD is designed for SMX, where all anennas are acive a each ime insance [7], [31] [33]. However, in SM only one anenna is acive a a ime. Therefore, a modified SD algorihm ailored for SM, named SM Tx, is presened. More specifically, similar o convenional SDs, he SM Tx scheme reduces he number of poins (l,s) for l 1,,...N } and s s 1,s,...s M } o be searched hrough in (6), i.e., he ransmi search space, by compuing he Euclidean disances only for hose poins ha lie inside a sphere wih radius R and are cenred around he received signal. However, unlike convenional SDs, in our scheme he se of poins inside he sphere are much simpler o compue, as here is only a single acive anenna in SM. In his secion, we show how o compue he se of poins. The Cholesky facorisaion of he (N N ) posiive definie marix Ḡ = H T H+ϕĪ N is Ḡ = D D T, where σ ϕ = n N > N r (16) N N r Then he SM Tx can be formally wrien as follow, [ˆl(Tx SD),ŝ (Tx SD) ] } = argmin z D x l,s F (l,s) Θ R (17) where Θ R is he subse of poins (l,s) for l 1,,...N } and s s 1,s,...s M } in he ransmi search space ha lie inside a sphere wih radius R and cenred around he received signal z, z = D ρ and ρ = Ḡ 1 H T ȳ. Unlike SM Rx, SM Tx reduces he compuaional complexiy of he ML receiver by reducing he ransmi search space, which is done by he efficien compuaion of he subse Θ R. Afer some algebraic manipulaions as shown in Appendix A, he subse of poins Θ R lie in he inervals: R i + z l+n Ims} R i + z l+n D (l+n,l+n ) D (l+n,l+n ) R + z l l+n D l,l (18) Res} R + z l l+n D l,l (19) where z a b = z a D (a,b) Ims} () R = R N ν=n +1 z ν l+n (1) Noe, every ime a poin is found inside he sphere, he radius R is updaed as shown in (), wih he Euclidean disance of ha poin. Moreover, (19) needs o be compued only for hose poins ha lie inside he inerval in (18), for he reason ha (19) depends implicily on (18). Because of he unique properies of SM he inervals in (18) and (19) needs o be calculaed only once for each possible ransmi poin, unlike convenional SDs where he inervals have o be calculae N imes for each ransmi poin. Furhermore, we noe ha SM Tx works for boh underdeermined MIMO seup wihn > N r, and overdeermined MIMO seup wih N N r. As opposed o he SM Rx scheme, he SM Tx scheme uses some pre compuaions o esimae he poins ha lie inside he sphere of radius R. These addiional compuaions are carefully aken ino accoun in he analysis of he compuaional complexiy of he SM Tx scheme and is comparison wih he ML opimum deecor in secion IV. IV. COMPUTATIONAL COMPLEXITY OF SM RX AND SM TX In his secion, we analyse he compuaional complexiy of SM ML, SM Rx and SM Tx. The complexiy is compued as he number of real muliplicaion and division operaions needed by each algorihm [34]. A. SM ML The compuaional complexiy of SM ML receiver in (6), yields, C SM ML = 8N r m, (3) as he ML deecor searches hrough he whole ransmi and receive ( search spaces. Noe, evaluaing he Euclidean disance y r h l,r s ) requires 8 real muliplicaions. The compuaional complexiy of SMX ML receiver in (5) is equal o C SMX ML = 4(N +1)N r m. (4) ( Noe, y Hx ) in (5) requires (N +1) complex muliplicaions. From (3) and (4), he complexiy of SM does no depend on he number of ransmi anennas, and i is equal o he complexiy of SIMO sysems. However, he complexiy of SMX increases linearly wih he number of ransmi anennas.

5 YOUNIS e al.: GENERALISED SPHERE DECODING FOR SPATIAL MODULATION 5 Thus, he reducion of SM ML receiver complexiy relaive o he complexiy of he SMX ML decoder for he same specral efficiency is given by, C ML rel = 1 ( 1 N +1 ). (5) From (5), he reducion in complexiy offered by SM increases wih he increase in he number of ransmi anennas. For example for N = 4 SM offers a 6% reducion in complexiy compared o SMX, and as he number of ransmi anennas increases he reducion increases. B. SM Rx The complexiy of he SM Rx receiver is given by: N M C Rx SD = 3 Ñ r (l,s) (6) l=1 s=1 I is easy o show ha C Rx SD lies in he inerval 3 m C Rx SD 6N r m, where he lower bound corresponds o he scenario where Ñ r (l,s) = 1, and he upper bound corresponds o he scenario where Ñ r (l,s) = N r for l 1,,...N } and s s 1,s,...s M }. An ineresing observaion is ha SM Rx offers a reducion in complexiy even for he case of N r = 1, where he complexiy lies in he inerval 3 m C Rx SD 6 m. We noe ha he SM Rx soluion requires no pre compuaions wih respec o he ML opimum deecor. In fac, Ñ r (l,s) for l 1,,...N } and s s 1,s,...s M } in (15) are implicily compued when solving he deecion problem in (14). C. SM Tx The compuaional complexiy of SM Tx can be upper bounded by, C Tx SD C ΘR +3N cardθ R } (7) where card } denoes he cardinaliy of a se, and C ΘR is he complexiy of finding he poins in he subse Θ R, C ΘR = C Pre-Comp +C Inerval (8) where, 1) C Pre-Comp is he number of operaions needed o compue he Cholesky decomposiion. Calculaing he upper riangular marix D using Cholesky decomposiion has he complexiy [34], C CH = 4N 3 /3 (9) I can be easily shown ha he calculaion of Ḡ, ρ and z requires N r N (N +1), N (N +N r +1) and N (N +1) real operaions respecively, where back subsiuion algorihm was used for calculaing ρ [34]. Hence, C Pre-Comp = C CH +N (4N r N +6N r +6N +3) (3) ) C Inerval is he number of operaions needed o compue he inervals in (18),(19), where, C inerval = N +(N +3)N (19) (31) For (18): N real divisions are needed. For (19): (N +3)N (19) real muliplicaions are needed, where (N +3) is he number of real compuaions needed o compue (19), and N (19) is he number of imes (19) is compued, which is calculaed by simulaions. Noe, i) he inerval in (19) depends on he anenna index l and only he imaginary par of he symbol s, ii) some symbols share he same imaginary par. Therefore, (19) is only calculaed for hose poins (l,s) which lie in he inerval in (18) and does no have he same l and Ims} as a previously calculaed poin. V. ERROR PROBABILITY OF SM SDS AND INITIAL RADIUS SELECTION METHOD In his secion, we derive an analyical expresion for he BER performance of SM SD, and we show ha SM SD offers a near opimum performance. The BER for SM SD is esimaed using he union bound [35], which can be expressed as follows, BER SM SD l,s l,s N ( x l,s, x l,s ) E H Pr e,sm SD } m m (3) where N ( x l,s, x l,s ) is he number of bis in error beween x l,s and x l,s, and, ) ) Pr (( lsm SD = Pr, s SM SD (l,s ) (33) e,sm SD is he pairwise) error probabiliy of deciding on he poin ( lsm SD, s SM SD given ha he poin (l,s ) is ransmied. The probabiliy of error Pr e,sm SD can be hough of as wo muually exclusive evens depending on wheher he ransmied poin (l,s ) is inside he sphere. In oher words, he probabiliy of error for SM SD can be separaed in wo pars, as shown in (34) [36]: ( ) Pr ( l SM ML, s SM ML ) (l,s ) : The probabiliy of deciding on he incorrec ransmied symbol and/or used anenna combinaion, given ha he ransmied poin (l,s ) is inside he sphere. Pr((l,s ) / Θ R ): The probabiliy ha he ransmied poin (l,s ) is ouside he se of poins Θ R considered by he SD algorihm. ( ( ) ) Pr Pr ( l SM ML, s SM ML ) (l,s ) +Pr((l,s ) / Θ R ) e,sm SD (34)

6 6 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION E H Pr e,sm SD } = [ ( σ ζ s 4σn )] Nr N r 1 r= ( Nr 1+r r )[ 1 ζ ( σ s 4σ n )] r (46) However, he probabiliy of error for he ML decoder is, ( ) Pr Pr ( l, s) (l,s ) (35) e,sm ML Thus, SM SD will have a near opimum performance when, ( ) Pr((l,s ) / Θ R ) << Pr ( l, s) (l,s ) (36) The probabiliy of no having he ransmied poin (l,s ) inside Θ R can be wrien as, ( Nr ) Pr((l,s ) / Θ R ) = Pr ȳr h l,r s > R where, r=1 ( ( ) ) R = Pr κ > σ n / ( ( ) ) R γ N r, σ n = 1 Γ(N r ) N r κ = n r σ n / r=1 (37) (38) is a cenral chi-squared random variable wih N r degree of freedom having a cumulaive disribuion funcion (CDF) equal o [35], F κ (a,b) = γ(b/,a/) (39) Γ(b/) where γ(c, d) is he lower incomplee gamma funcion given by, γ(c,d) = d and Γ(c) is he gamma funcion given by, Γ(c) = c 1 e d (4) c 1 e d (41) The iniial sphere radius considered in SM SD is a funcion of he noise variance as given in [37], R = αn r σ n (4) where α is a consan chosen o saisfy (36). This can be done by seing Pr((l,s ) / Θ R ) = 1 6 and back solving (37). For N r = 1,,4, α = 13.8,8.3,5.3 respecively. Finally, Pr e,sm SD can be formulaed as, ( Pr ȳ hl s = Pr > ȳ hl s ) e,sm SD where, ξ = Re ( hl s h ) } T l s n = Pr ( ξ > h l s h l s ) (43) N ( (,σn hl s h l s )) (44) Thus, hl s Pr = Q h l s (45) e,sm SD σ n where Q(x) = (1/ π) + e / d. x In he case of Rayleigh fading, we can derive he closed form soluion for E H Pr e,sm SD } in (3) by employing he soluion from [38, eq. (6)]. Noe ha he argumen of he Q-funcion in (45) can be represened as he summaion of N r squared Gaussian random variables, wih zero mean and variance equal o 1. This means ha he argumen in he Q funcion can be described by a cenral chi squared disribuion wih N r degrees of freedom. The resul for E H Pr e,sm SD } is as given in (46), where σs = x l,s x l,s F and, ζ(c) = 1 ( ) c 1 (47) 1+c Plugging (46) ino (3) gives a closed form expression for he BER of SM SD. In he nex secion, we show ha (3) gives a igh approximaion of he BER of SM SD, and ha SM SD offers a near opimum performance. VI. RESULTS In he following, Mone Carlo simulaion resuls for a leas 1 6 Rayleigh fading channel realisaions are shown o compare he performance and compuaional complexiy of large scale MIMO, SM ML, SM SD and SMX SD. A. Analyical performance of SM SD Figs. 1- show he BER simulaion resuls for SM ML, SM Rx and SM Tx compared wih he analyical bound Bi Error Raio SM ML, SM ML, SM ML, SM Rx, SM Rx, SM Rx, SM Tx, SM Tx, SM Tx, N =, 3 QAM N = 8, 8 QAM N = 3, BPSK N =, 3 QAM N = 8, 8 QAM N = 3, BPSK N =, 3 QAM N = 8, 8 QAM N = 3, BPSK Analyical, N =, 3 QAM Analyical, N = 8, 8 QAM Analyical, N = 3, BPSK Fig. 1. BER agains SNR. m = 6, and N r = 4.

7 YOUNIS e al.: GENERALISED SPHERE DECODING FOR SPATIAL MODULATION 7 Bi Error Raio SM ML, N =, 18 QAM SM ML, N = 8, 3 QAM SM ML, SM ML, N = 3, 8 QAM N = 18, BPSK SM Rx, N =, 18 QAM SM Rx, N = 8, 3 QAM SM Rx, SM Rx, N = 3, 8 QAM N = 18, BPSK SM Tx, N =, 18 QAM SM Tx, N = 8, 3 QAM SM Tx, SM Tx, N = 3, 8 QAM N = 18, BPSK Analyical, N =, 18 QAM Analyical, N = 8, 3 QAM Analyical, N = 3, 8 QAM Analyical, N = 18, BPSK Fig.. BER agains SNR. m = 8, and N r = 4. derived in secion V, where m = 6,8 and N r = 4. From he figures we can see ha boh SM Tx and SM Rx offer a near opimum performance, where he resuls overlap wih SM ML. Furhermore, Figs. 1- validae our analyical bound as for BER < 1 all graphs closely mach he analyical resuls. Noe, i is will known ha he union bound is loose for low SNR [35]. B. Comparison of he BER performance of SM and SMX Figs. 3 and 4 show a BER comparison beween all possible combinaions of SM and SMX for m = 6 and N r =,4. In Fig. 3, we can observe ha he BER performance depends on he he number of ransmi anennas used and, consequenly, he consellaion size. The smaller he consellaion size, he beer he performance. Anoher observaion ha can be made is ha SM and SMX offer nearly he same performance when using he same consellaion size. In Fig. 4, where he number of receive anennas is increased, we noice ha SM performs Bi Error Raio 1 SM SD, N =, 3 QAM SM SD, N = 8, 8 QAM SM SD, N = 3, BPSK 1 1 SMX SD, N =, 8 QAM SMX SD, N = 6, BPSK Fig. 4. BER agains SNR. m = 6, and N r = 4. 1 Bi Error Raio SM SD, N =, 18 QAM SM SD, N = 8, 3 QAM SM SD, N = 3, 8 QAM SM SD, N = 18, BPSK SMX SD, N =, 16 QAM SMX SD, N = 8, BPSK Fig. 5. BER agains SNR. m = 8, and N r = SM SD, N =, 3 QAM SM SD, N = 8, 8 QAM SM SD, N = 3, BPSK SMX SD, N =, 8 QAM SMX SD, N = 6, BPSK SM SD, N =, 18 QAM SM SD, N = 8, 3 QAM SM SD, N = 3, 8 QAM SM SD, N = 18, BPSK SMX SD, N =, 16 QAM SMX SD, N = 8, BPSK Bi Error Raio 1 Bi Error Raio Fig. 3. BER agains SNR. m = 6, and N r = Fig. 6. BER agains SNR. m = 8, and N r = 4.

8 8 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION C rel (%) SM Tx, N=, 3 QAM SM Rx, N= 4, 16 QAM SM Rx, N= 3, BPSK SMX SD, N=, 8 QAM SMX SD, N= 3= QPSK C rel (%) SM Tx, N=, 18 QAM SM Rx, N= 4, 64 QAM SM Rx, N= 18, BPSK SMX SD, N=, 16 QAM SMX SD, N= 4, QPSK Fig. 7. Compuaional complexiy agains SNR. m = 6, and N r =. Fig. 9. Compuaional complexiy agains SNR. m = 8, and N r = SM Tx, N=, 3 QAM SM Rx, N= 4, 16 QAM SM Rx, N= 3, BPSK SMX SD, N=, 8 QAM SMX SD, N= 3, QPSK C rel (%) C rel (%) SM Tx, N=,18 QAM SM Tx, N= 4, 64 QAM SM Rx, N= 8, 3 QAM SM Rx, N= 18, BPSK SMX SD, N=, 16 QAM SMX SD, N= 4, QPSK SMX SD, N= 8, BPSK Fig. 8. Compuaional complexiy agains SNR. m = 6, and N r = 4. Fig. 1. Compuaional complexiy agains SNR. m = 8, and N r = 4. beer han SMX. In paricular, BPSK SM provides a 1 db beer performance han BPSK SMX. Also 8 QAM SM offers a slighly beer performance (.5 db) han 8-QAM SMX. In Figs. 5 and 6, he BER comparisons for m = 8 and N r =,4 are shown. In Fig. 5, SM and SMX offer similar performance for he same consellaion size. However, SM offers a beer performance when he number of receive anennas increases as shown in Fig. 6. In summary, SM offers a similar or beer performance han SMX, where he performance of boh sysems depends on he size of he consellaion diagram and he number of receive anennas. We also noe ha he BER performance of SM can be improved by increasing he number of receive anennas. C. Complexiy Analysis In Figs. 7-1, he compuaional complexiy of SM Rx and SM Tx provided in (6) and (7) respecively is compared wih he compuaional complexiy of SMX SD, where he iniial radius is chosen according o (4). In paricular, he figures show he relaive compuaional complexiy of he SDs wih respec o he SM ML deecor, i.e C rel (%) = 1 (C SD /C ML ). Noe, for SM he SD wih he lowes complexiy is chosen. In Figs. 7 and 8, he relaive compuaional complexiies for m = 6 and N r =,4 are shown. Fig. 7, shows ha for large consellaion sizes he lowes relaive compuaional complexiy is offered by SM Tx N =. The relaive compuaional complexiy ranges beween 4% for low SNR and 16% for high SNR. However, for lower consellaion sizes SM Rx provides he lowes relaive compuaional complexiy, which is beween 56% for low SNR and 6% for high SNR. As SM Rx reduces he receive search space, he reducion in he compuaional complexiy offered by SM Rx does no depend on he number of ransmi anennas. Therefore, only SM Rx wihn = 4,3 are shown, where boh scenarios offer nearly he same relaive compuaional complexiy. Finally, from Fig. 7 we can see ha SMX SD N = and N = 3

9 YOUNIS e al.: GENERALISED SPHERE DECODING FOR SPATIAL MODULATION 9 are less complex han SM ML wih a relaive compuaional complexiy 48% and 79% 8% respecively. However, comparing SM SD o SMX SD N =, for 3 QAM SM SD is 3% less complex han SMX SD, and for BPSK SM SD is % less complex han SMX SD. In Fig. 8, i can be seen ha for large consellaion sizes SM Tx is sill he bes choice wih a relaive complexiy ha ranges beween % for low SNR and 1% for high SNR, which is 15% less han SMX SD N =. For smaller consellaion sizes SM Rx is he bes choice wih relaive complexiy ha ranges beween 55% for low SNR and 14% for high SNR, offering a 3% exra reducion in complexiy when compared o SMX SD N =. Noe, i) SMX SD N = 6 is no shown in he figure, because his scenario has a complexiy higher han he complexiy of SM ML, ii) he complexiy of SMX SD N = 3 increased wih he increase of SNR, for he reason ha, in he underdeermined case ϕ depends on he SNR (5). The relaive complexiy for m = 8 and N r =,4 is shown in Fig. 9 and 1. Since SM Tx reduces he ransmi search space, he reducion in complexiy increased by more han 1% wih he increase in he wordsize and consequenly he consellaion size. In Fig. 9 for high consellaion sizes SM Tx N = is he bes choice wih a relaive complexiy ha reaches 4% for high SNR,. In Fig. 1 for high consellaion sizes SM Tx N = and N = 4 are he bes choice wih a relaive complexiy ha reaches 3% and 1% respecively. On he oher hand, SM Rx reduces he receive search space, herefore, i sill offers nearly he same relaive complexiy. However, he complexiy reduces wih he increase of N r, where SM Rx N r = 4 is ( 1%) less complex han SM Rx N r =. Finally, from boh figures i can be seen ha alhough SM ML is much less complex han SMX ML, SMX SD is less complex han SM ML. For ha reason, SM SD has o be developed, where SM SD is ( %) less complex han SMX SD for N r =, and ( 1%) less complex han SMX SD for N r = 4. Noe, he complexiy of boh SM Tx and SMX SD decreases wih he increase of N r, because for he case of N r < N, he less under-deermined he sysem, he fewer pre compuaions are needed. To summarize, wo SDs for SM are inroduced: SM Tx which reduces he ransmi search space, and SM Rx which reduces he receive search space. Boh deecion algorihms are shown o offer a significan reducion in compuaional complexiy while mainaining a near opimum BER performance. For sysems wih few ransmi anennas, SM Tx is shown o be he beer choice. For sysems wih wih a larger number of receive anennas, SM Rx is shown o be he beer candidae in erms of complexiy reducion. The decision for he mos appropriae SD depends on he paricular deploymen scenario. VII. CONCLUSION In his paper we have inroduced and analysed he performance/complexiy rade off of wo SDs designed specifically for SM. The proposed SDs provide a subsanial reducion in he compuaional complexiy while reaining he same BER as he ML opimum deecor. The closed form analyical performance of SM in i.i.d. Rayleigh fla fading channels has been derived, and analyical and simulaion resuls were shown o closely agree. Furhermore, numerical resuls have highlighed ha no SD is superior o he ohers, and ha he bes soluion o use depends on he MIMO seup, and he SNR a he receiver. In general, SM Rx is he bes choice for lower specral efficiencies, and SM Tx is he bes opion for higher specral efficiencies. Finally, simulaion resuls showed ha SM using SD offers a significan reducion in compuaional complexiy and nearly he same BER performance as SMX using ML decoder or SD. Overall, SM SD offers i) hardware complexiy and power consumpion ha does no depend on he number of ransmi anennas, ii) BER performance ha increases wih he increase of he number of ransmi anennas, and iii) a large reducion in compuaional complexiy compared o SMX. Thus, we believe ha SM SD is an ideal candidae for large scale MIMO sysems. APPENDIX PROOF OF THE INTERVALS (18), (19) Proof: 1) Firs (17) can be hough of as an inequaliy, R ȳ H x l,s } F (48) Then add ϕ x H l,s x l,s o boh sides of (48) o ge (49), where Ḡ = H H H + ϕī N is a (N N ) posiive definie marix, wih a Cholesky facorisaion defined as Ḡ = D H D, where D is a (N N ) upper riangular marix. Now by defining ρ = Ḡ 1 H H ȳ, and adding ρ D H D ρ o boh sides of (49), i can be re wrien as, R ϕ z D x l,s } F N N z i D i,j x l,s (j) i=1 where, z = D ρ and, j=i (5) Rϕ = R +ϕ x T l,s x l,s +ȳ T H ρ ȳ T ȳ (51) σ ϕ = n N > N r (5) N N r For simpliciy, in his paper we assume R ϕ = R. R +ϕ x H l,s x l,s ȳ H x l,s F +ϕ xh l,s x } l,s ȳ H ȳ ȳ H H xl,s + x H l,s H H ȳ+ x H l,sḡ x } l,s (49)

10 1 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION ) Second, we noe a necessary condiion ha he poins of he ransmi search space need o saisfy o belong o he subse Θ R is (for all i = 1,,...,N ): N R z i D i,j x l,s (j) (53) j=i which is a condiion similar o convenional SD algorihms [31]. 3) We need o ake ino accoun ha in SM only a single anenna is acive a any ime insance. In he equivalen real valued signal model in (1), his is equivalen o having only wo, ou of N, non zero enries in he signal vecors x l,s and x l,s, respecively. By aking his remark ino accoun, i follows ha: a) if i = N + 1,N +,...,N, hen only he imaginary par of x l,s plays a role in (53), and, hus, only one enry x l,s (l+n ) can be non zero; and b) if i = 1,,...,N, hen boh real and imaginary pars of x l,s play a role in (53), and, hus, only wo enries x l,s (l), x l,s (l+n ) can be non zero. The consideraions in a) and b) lead o he inervals in (18) and (19), respecively, which are direcly obained by solving he inequaliy in (53). REFERENCES [1] E. Telaar, Capaciy of muli-anenna Gaussian channels, European Trans. Telecommun., vol. 1, no. 6, pp , Nov. / Dec [] G. J. Foschini, Layered space-ime archiecure for wireless communicaion in a fading environmen when using muli-elemen anennas, Bell Labs Tech. J., vol. 1, no., pp , [3] J. Hoydis, S. en Brink, and M. Debbah, Massive MIMO: How many anennas do we need? in 11 49h Annual Alleron Conf. Commun., Conrol, Compu. (Alleron), Sep. 11, pp [4] S. Mohammed, A. Zaki, A. Chockalingam, and B. Rajan, High-rae space-ime coded large-mimo sysems: Low-complexiy deecion and channel esimaion, IEEE J. Sel. Topics Signal Process., vol. 3, no. 6, pp , Dec. 9. [5] B. Cerao and E. Vierbo, Hardware implemenaion of a lowcomplexiy deecor for large MIMO, in IEEE In. Symp. Circuis Sys. (ISCAS 9), May 9, pp [6] S. Mohammed, A. Chockalingam, and B. 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11 YOUNIS e al.: GENERALISED SPHERE DECODING FOR SPATIAL MODULATION 11 Abdelhamid Younis received a BSc in Elecrical and Elecronic Engineering wih honours in 7 from he Universiy of Benghazi, Libya and an MSc wih disincion in Signal Processing and Communicaion Engineering in 9 from he Universiy of Edinburgh, UK. He is currenly compleing a Ph.D. in Communicaion Engineering a he Insiue of Digial Communicaions (IDCOM) a he Universiy of Edinburgh where in 1 he was awarded he Overseas Research Suden Award (ORS) in recogniion of his work. His main research ineress lie in he area of wireless communicaion and digial signal processing wih a paricular focus on spaial modulaion, MIMO wireless communicaions, reduced complexiy MIMO design and opical wireless communicaions. Raed Mesleh (S -M 8-SM 13) Dr Mesleh holds a Ph.D. in Elecrical Engineering from Jacobs Universiy in Bremen, Germany and several years of pos-docoral wireless communicaion and opical wireless communicaion research experience in Germany. In Ocober 1, he joined Universiy of Tabuk in Saudi Arabia where he is now an assisan professor and he direcor of research excellence uni. His main research ineress are in spaial modulaion, MIMO cooperaive wireless communicaion echniques and opical wireless communicaion. Dr Mesleh publicaions received more han 8 ciaions since 7. He has published more han 5 publicaions in op-ier journals and conferences, and he holds 7 graned paens. He also serves as on he TPC for academic conferences and is a regular reviewer for mos of IEEE/OSA Communicaion Sociey s journals and IEEE/OSA Phoonics Sociey s journals. Sinan Sinanović (S 98-M 7) is a lecurer a Glasgow Caledonian Universiy. He has obained his Ph.D. in elecrical and compuer engineering from Rice Universiy, Houson, Texas, in 6. In he same year, he joined Jacobs Universiy Bremen in Germany as a pos docoral fellow. In 7, he joined he Universiy of Edinburgh in he UK where he has worked as a research fellow in he Insiue for Digial Communicaions. While working wih Halliburon Energy Services, he has developed acousic elemery receiver which was paened. He has also worked for Texas Insrumens on developmen of ASIC esing. He is a member of he Tau Bea Pi engineering honor sociey and a member of Ea Kappa Nu elecrical engineering honor sociey. He won an honorable menion a he Inernaional Mah Olympiad in Marco Di Renzo (S 5 AM 7 M 9) was born in L Aquila, Ialy, in He received he Laurea (cum laude) and he Ph.D. degrees in Elecrical and Informaion Engineering from he Deparmen of Elecrical and Informaion Engineering, Universiy of L Aquila, Ialy, in April 3 and in January 7, respecively. From Augus o January 8, he was wih he Cener of Excellence for Research DEWS, Universiy of L Aquila, Ialy. From February 8 o April 9, he was a Research Associae wih he Telecommunicaions Technological Cener of Caalonia (CTTC), Barcelona, Spain. From May 9 o December 9, he was an EPSRC Research Fellow wih he Insiue for Digial Communicaions (IDCOM), The Universiy of Edinburgh, Edinburgh, Unied Kingdom (UK). Since January 1, he has been a Tenured Researcher ( Chargé de Recherche Tiulaire ) wih he French Naional Cener for Scienific Research (CNRS), as well as a faculy member of he Laboraory of Signals and Sysems (LS), a join research laboraory of he CNRS, he École Supérieure d Élecricié (SUPÉLEC), and he Universiy of Paris Sud XI, Paris, France. His main research ineress are in he area of wireless communicaions heory. He is a Principal Invesigaor of hree European funded research projecs (Marie Curie ITN GREENET, Marie Curie IAPP WSN4QoL, and Marie Curie ITN CROSSFIRE). Dr. Di Renzo is he recipien of he special menion for he ousanding five year (1997 3) academic career, Universiy of L Aquila, Ialy; he THALES Communicaions fellowship for docoral sudies (3 6), Universiy of L Aquila, Ialy; he Bes Spin Off Company Award (4), Abruzzo Region, Ialy; he Torres Quevedo award for research on ulra wide band sysems and cooperaive localizaion for wireless neworks (8 9), Minisry of Science and Innovaion, Spain; he Dérogaion pour l Encadremen de Thèse (1), Universiy of Paris Sud XI, France; he 1 IEEE CAMAD Bes Paper Award from he IEEE Communicaions Sociey; and he 1 Exemplary Reviewer Award from he IEEE WIRELESS COMMUNICATIONS LETTERS of he IEEE Communicaions Sociey. He currenly serves as an Edior of he IEEE COMMUNICATIONS LETTERS. Professor Harald Haas (S98-A-M3) holds he Chair of Mobile Communicaions in he Insiue for Digial Communicaions (IDCOM) a he Universiy of Edinburgh, and he currenly is he CTO of a universiy spin-ou company purevlc Ld. His main research ineress are in inerference coordinaion in wireless neworks, spaial modulaion and opical wireless communicaion. Prof. Haas holds 3 paens. He has published more han 6 journal papers including a Science Aricle and more han 16 peer-reviewed conference papers. Nine of his papers are invied papers. Prof. Haas has co-auhored a book eniled Nex Generaion Mobile Access Technologies: Implemening TDD wih Cambridge Universiy Press. Since 7 Prof. Haas has been a Regular High Level Visiing Scienis suppored by he Chinese 111-program a Beijing Universiy of Poss and Telecommunicaions (BUPT). He was an invied speaker a he TED Global conference 11, and his work on opical wireless communicaion was lised among he 5 bes invenions in 11 in Time Magazine. He recenly has been awarded a presigious Fellowship of he Engineering and Physical Sciences Research Council (EPSRC) in he UK.