Complex Quadrature Spatial Modulation

Transcription

1 Complex Quadrature Spatial Modulatio Maar Moaise ad Saetbyeol Lee I tis paper, we propose a spatial modulatio (SM) sceme referred to as complex quadrature spatial modulatio (CQSM). I cotrast to quadrature spatial modulatio (QSM), CQSM trasmits two complex sigal costellatio symbols o te real ad quadrature spatial dimesios at eac cael use, icreasig te spectral efficiecy. o tis ed, sigal symbols trasmitted at ay give time istat are draw from two differet modulatio sets. e first modulatio set is ay of te covetioal QAM/PSK alpabets, wile te secod is a rotated versio of it. e optimal rotatio agle is obtaied troug simulatios for several modulatio scemes ad aalytically prove for te case of QPSK, were bot results coicide. Simulatio results sowed tat CQSM outperformed QSM ad geeralized SM (GSM) by approximately 5 ad 4.5 db, respectively, for te same trasmissio rate. Its performace was similar to tat of QSM; owever, it acieved iger trasmissio rates. It was additioally sow umerically ad aalytically tat CQSM outperformed QSM for a relatively large umber of trasmit ateas. Keywords: Mikowski sum, multiple-iput multipleoutput (MIMO) system, quadrature spatial modulatio, spatial modulatio, uitary rotatio. I. Itroductio Spatial modulatio (SM) as emerged i te last decade as a multiple-iput multiple-output (MIMO) tecique tat utilizes bot sigal costellatio symbols, suc as te quadrature amplitude ad pase sift keyig modulatio (QAM/PSK) alpabet, ad te spatial dimesio, wic is te idex of a sigle or multiple trasmit ateas, to covey iformatio from trasmitter to receiver [1]. e fudametal cocept of SM is tat a trasmitter equipped wit a sigle radio frequecy (RF) cai ad several pysical ateas ca ave a relatively ig capacity troug usig te spatial dimesio, represeted by te idices of te trasmit ateas, to covey iformatio to te receiver. e receiver te recovers te desigated iformatio troug demodulatig bot te sigal symbol ad trasmit atea idex. A special case of te SM is referred to as space sift keyig (SSK), wereby te covetioal modulated symbols, QAM/PSK, are replaced by te presece or absece of eergy assiged to a particular atea []. Bot SM ad SSK scemes were geeralized i [3] ad [4], were more ta oe trasmit atea ca be simultaeously used to icrease te system s spectral efficiecy or reduce te umber of required pysical ateas to acieve a target performace criterio. A geeralized SM sceme for large-scale MIMO systems was proposed i [5]. A improved SM (ISM) was proposed i [6] [7], were a simple mappig from te iput bits to te trasmissio vectors is performed for ay umber of trasmit ateas. Hece, te log-two umber of te trasmit atea coditio imposed i te covetioal SM ad SSK scemes is relaxed. Moreover, a combiatio of space-time block codig ad te SM sceme is itroduced i [8], were te proposed tecique outperformed te covetioal SM i

2 terms of error performace by 3 to 5 db, wile acievig te same spectral efficiecy. I [9], a precodig-aided spatial modulatio (PSM) approac is proposed i wic te trasmitted symbol is precoded usig a cael matrix-based criterio so tat a sigle receive atea is activated, tereby coveyig additioal iformatio to te receiver. I additio, atea selectio teciques for bot SM ad PSK scemes ave bee proposed to acieve furter diversity ad power gais (see [10] ad [11] ad teir refereces). Detailed comparisos amog spatial modulatio scemes are itroduced i [1] ad [13]. Recetly, a quadrature spatial modulatio (QSM) sceme was itroduced i [14]. I QSM, te spatial costellatio symbols are expaded ito i-pase ad quadrature dimesios. e real part of te sigal costellatio symbol is trasmitted o te first spatial costellatio dimesio; te imagiary part is trasmitted o te secod. Sice real ad imagiary parts of te sigal costellatio symbol are trasmitted over ortogoal carriers, QSM does ot suffer iter-cael iterferece (ICI). Based o its structure, QSM icreases te spectral efficiecy by log ( ) bits/s/hz compared to te covetioal SM, wic acieves q + log ( ) bits/s/hz, were q ad deote te umber of bits per sigal costellatio symbol ad umber of pysical ateas, respectively. e QSM sceme is used i [15] to efficietly mitigate eavesdroppig. Additioally, a precodig-aided QSM is itroduced i [16], were te idexes of te desigated receive ateas are used to covey iformatio. I tis paper, we advace te covetioal SM tecique to acieve a spectral efficiecy of (q + log ( )) bits/s/hz. e proposed sceme is called complex quadrature spatial modulatio (CQSM). Istead of trasmittig te real ad imagiary parts of a sigal costellatio symbol o a desigated spatial costellatio dimesio, CQSM trasmits two complex sigal modulatio symbols, draw from two differet modulatio sets, at eac cael use. e first ad secod symbols are draw from a covetioal PSK/QAM modulatio set ad a rotated versio of it, respectively. e rotatio agle as a direct impact o te bit-error-rate (BER) performace of te CQSM sceme. erefore, after itroducig CQSM, te rotatio agle is optimized troug extesive Mote Carlo simulatios. e optimal value is aalytically obtaied i Appedix I for te case of QPSK modulatio. It is erei sow, bot umerically ad aalytically, tat CQSM outperforms QSM for a ig umber of trasmit ateas, wile acievig a iger spectral efficiecy. For te same spectral efficiecy, CQSM outperforms SM, geeralized SM (GSM), ad QSM by at least 4 db. e remaider of tis paper is orgaized as follows. I Sectio II, we itroduce te system model ad QSM. I Sectio III, a detailed descriptio of CQSM is give ad te rotatio agle is umerically optimized ad aalytically derived for te case of QPSK. A performace evaluatio ad te computatioal complexity of CQSM are addressed i Sectios IV ad V, respectively. I Sectio VI, simulatio results are provided ad te covergece of te optimal rotatio agle i te case of relatively large-scale systems is addressed. I Sectio VII, te coclusios are preseted. II. System Model ad Related Work 1. System Model We cosider a commuicatio system i wic a base statio (BS) equipped wit trasmit ateas commuicates o te dowlik wit a mobile statio (MS) equipped wit R receive ateas. At eac cael use, te BS seds M bits to te MS o bot te sigal costellatio symbols ad te spatial costellatio dimesios. A sigal costellatio set is deoted by wit cardiality of = q, were q deotes te umber of bits per sigal costellatio symbol. e elemets s ave a average power of oe; tat is, E[s*s] = 1. Cael matrix H couples te R receive ad trasmit ateas, were its elemet i, j Cis a circularly symmetric complex Gaussia variable wit a mea ad variace of zero ad uity, respectively. e R -dimesioal additive oise vector at te receiver is deoted by, wose elemet i is circularly symmetric complex Gaussia wit a mea ad variace of zero ad, respectively.. Quadrature Spatial Modulatio (QSM) e operatio of te QSM sceme is explaied troug te followig umeric example. Assume tat te followig message m = [ ], were m = M, is to be trasmitted at a particular cael usig QPSK modulatio ad four trasmit ateas. e first two bits [1 1] modulate a complex symbol, s = s R + js I = 1 j, were s R ad s I are te real ad imagiary parts of s, respectively. e followig log ( ) bits [0 1] modulate te atea idex l R of te atea used to trasmit te real part. e last log ( ) bits [0 0] modulate te idex l I of te atea used to trasmit te imagiary part. erefore, te vectors obtaied for real ad imagiary parts of te symbol s are give by s R = [ ] ad s I = [ ], resultig i a trasmitted vector give by s = s R + js I = [ j 1 0 0]. e trasmitted vector is te ormalized so tat te average power per sigal costellatio symbol is equal to oe. e operator [.] deotes a vector/matrix traspose. Based o tis descriptio, te spectral efficiecy (quatified i bits/s/hz)

3 of te QSM sceme is give by: c q (1) qsm log ( ). were ad are te idices of te spatial costellatio symbols, tat is, te ateas, from wic te real ad imagiary parts of s are trasmitted, respectively. I additio, is te -t colum of cael matrix H. Based o Fig. 1, te o-zero elemets of s are draw from eiter a BPSK alpabet we or from a QPSK alpabet we =. At te receiver side, te ML detector of te QSM sceme fids,, s R, ad s I as follows: ˆ ( ˆ,, sˆ, sˆ ) argmi s j s.,, s, s y (3) III. Complex Quadrature Spatial Modulatio (CQSM) e CQSM sceme trasmits two complex sigal costellatio symbols at eac cael use. O te oter ad, QSM trasmits a sigle complex symbol per cael use. e CQSM trasmissio leads to a spectral efficiecy of Fig. 1. Example of te QSM sceme usig QPSK ad four trasmit ateas. Fig.. Example of te CQSM sceme usig QPSK ad four trasmit ateas. is example is depicted i Fig. 1. e received vector y C R1 is give by: yhs s j s,, 1,, () c q (4) cqsm log ( ). e CQSM sceme is explaied troug te followig umeric example. Assume tat message m = [ ] is to be trasmitted at a particular cael use by employig QPSK ad four trasmit ateas. We iitially assume tat te two symbols are draw from te same costellatio set. It will be later sow tat tis trivial coice is ot optimal. e first ad secod pair of bits, tat is [1 1] ad [0 1], modulate te complex symbols, s a = (1 j) ad s b = ( 1 j), respectively. e tird ad fourt pair of bits, [0 0] ad [1 0], modulate te idices of te ateas from wic s a ad s b, respectively, are trasmitted. e obtaied vectors, avig te oly o-zero elemets s a ad s b, are give by s a = [s a 0 0 0] ad s b = [0 0 s b 0]. Fially, te trasmitted vector is give by s = s a + s b. is example is depicted i Fig.. Based o te CQSM sceme, te received vector y C R1 is terefore give by yhs s s,, 1,, (5) a b ad te ML receiver for te CQSM is give by ( ˆ, ˆ, sˆ, sˆ ) argmi y ( s s ) a b a b,, s, s a a b b,, s, s a a b b H argmi g Re{ y g} were g = (s a + s b ). For te example depicted i Fig., all ypoteses of s a +s b for, {1,, }, s a a ad s b b are geerated. e ypotesis tat miimizes te square Euclidea orm i (6) is te maximum-likeliood (ML) solutio. ad of te ML solutio are coverted to biary, wit 1 correspodig to 00, to 01, ad so o. e symbols s a (6)

4 ad s b correspodig to te ML solutio are demodulated to te equivalet biary bits, as sow Fig.. Accordigly, all te bits of trasmitted message m are recovered. of π/4. is rotatio does ot cage te power of te sigal symbols or te agle betwee tem. erefore, b remais a valid QPSK costellatio set. I CQSM, rotatig te symbols i b is required to make te symbols i set a b uique. Cosequetly, te sigal detectio at te receiver side becomes possible. e rotatio agle is te optimized so tat te biterror rate is miimized. O te oter ad, i te secod case, tat is = = k, te system modeled i (5) is rewritte as y ( s s ) s, k 1,, (8) k a b k c were s c c, wic is defied as { s s s, s, a, b 1,, 4}. (9) c a b a a b b Fig. 3. Example of te received sigal costellatio set for (a) = 0, ad (b) = π / 4, usig QPSK modulatio set a ad its rotated versio b at te trasmitter side. By defiitio, c is te Mikowski sum of te sets a ad b, wic is referred to as c = a b [17]. Furter exploratio of c i te case of QPSK modulatio is give i Appedix I. O oe ad, te performace of te ML receiver depeds, amog oter factors, o te miimum distace amog te fiite lattice poits wose basis is matrix H workig o te trasmitted vector s, wose o-zero elemet s i d = a b c. O te oter ad, te BER performace also crucially depeds o te miimum Euclidea distace amog te sigal costellatio poits, s i d. At tis poit, we defie te two rotatio agles to be optimized i te sequel: Fig. 4. Miimum Euclidea distace amog te members of set d versus rotatio agle for BPSK, QPSK, 8PSK, ad 16QAM modulatio sets of a. I te sequel, ad for te sake of simplicity, we assume te case of QPSK modulatio, were a straigtforward extesio of our cojecture is possible for oter cases. Cosiderig (5), tere are two distict cases impacted by te values of ad :, ad =. I te first case, were te sigal costellatio symbols are trasmitted from differet ateas, te elemets of s belog to te set d = a b. Based o (6), te performace of te ML receiver depeds, amog oter factors, o te miimum Euclidea distace betwee eac pair of symbols i te resultig modulatio set, d, were a larger miimum Euclidea distace results i better performace. o maximize te miimum Euclidea distace betwee eac pair of symbols i te resultig set d, it suffices to defie b as: (7) /4 { j s s e s, i 1,, }, b b i i a a were e iπ/4 is a uitary rotatio, or, simply, a rotatio, wit agle 1. (s) is rotatio agle, suc tat s a a ad s b b =s a e j. at is, te costellatio set b is a rotated versio of a. e optimal value of tis agle, referred to as opt (s), maximizes te miimum Euclidea distace amog te symbols i d.. (s, H) is te same as (s), except its optimal value, referred to as opt (s, H), miimizes te BER of te wole system. First, we address te optimizatio of (s). e obvious coice of te rotatio agle = π/4 made i te case of is o loger valid. Figures 3(a) ad (b) depict a example of te resultig costellatio set d for te rotatio agles =0 ad π/4, respectively, were te alpabet of eac set is idicated wit a uique marker. I Fig. 3(a), te rotatio agle = 0, wic implies tat a = b. I te sigal costellatio set d, te symbols 0+j0, (1+j1), ( 1+j), ( 1 j), (1 j) are repeated four, two, two, two, ad two times, respectively, ad eac elemet of a is repeated twice. e demodulatio of te received sigal is impossible because te mappig from a ad b to d does ot result i uique sigal modulatio symbols. Figure 3(b), o te oter ad, depicts te resultig costellatio set d for = π/4, were te symbols i te set d are uique, makig it possible to demodulate te received sigal usig te ML detector by usig (6). Based o tis discussio, te coice of (s) sould satisfy te followig two coditios: 1. e elemets of set d must be uique. It is terefore

5 ituitively cocluded tat, i te case of QPSK modulatio, = 0 ad = π/ are to be excluded because tese coices result i idetical a ad b. e optimizatio problem is terefore give by ˆ ( s ) arg max d ( ) arg max mi s s (10) opt mi d i k [0, / ] [0, / ] s, s i k d ik Figure 4 depicts te miimum Euclidea distace amog te symbols i te costellatio set d, deoted d mi ( d ), for BPSK, QPSK, 8PSK, ad 16QAM versus te rotatio agle i te rage from 0 to π/ rad. e results are obtaied troug simulatios. I Appedix I, te optimal rotatio agle is aalytically obtaied i te case of QPSK modulatio, were te aalytical results ad results sow i Fig. 4 coicide. For BPSK, rotatio agles greater ta or equal to 60 o will maximize d mi ( d ). I te case of te oter modulatio scemes, te curves are eve symmetric at approximately = 45 o. For istace, te case of QPSK wit = π/6 ad π/3 results i a maximum d mi ( d ). able 1. Optimal rotatio agle(s) opt(s) ad correspodig max-mi Euclidea distace amog te symbols i sigal costellatio set d for differet modulatio scemes. Modulatio opt(s) (degree) dmi(d) BPSK 60 1 QPSK 30, PSK 17.3, 7.7, 6.3, QAM 30, able. Optimal rotatio agle(s) opt(s, H) tat miimize(s) te BER rate of te ML receiver for several system cofiguratios. (, R) Modulatio opt(s,h) (, R) Modulatio opt(s,h) (, ) (, 4) (, 8) (4, 4) QPSK 30 (4, 5) QPSK 35 16QAM 15 16QAM 15 QPSK 3.5 (4, 6) QPSK QAM 15 16QAM 30.5 QPSK 34.5 (4, 8) QPSK QAM 15 16QAM 30.5 QPSK 35 (8, 8) QPSK 38 16QAM 15 (16,16) QPSK 41 Fig. 5. Miimum Euclidea distace amog te members of set d versus rotatio agle for BPSK, QPSK, 8PSK, ad 16QAM modulatio sets a.. e miimum Euclidea distace betwee eac pair of symbols i set d must be maximized. is is motivated by te fact tat te performace of te ML receiver depeds o te miimum distace betwee te received sigal costellatio poits, amog oter factors. able 1 summarizes te optimal rotatio agle opt (s) ad te correspodig d mi ( d ). Cosiderig Fig. 4 ad able 1, it is wort clarifyig tat, i te case of 16QAM, d mi ( d ) at = 30 o is sligtly greater ta tat at = 14.7 o. For 8PSK, te distace d mi ( d ) i te rage [0, π/4] is eve symmetric aroud π/8 rad. As stated earlier, tere are several variables tat affect te BER performace of te ML receiver. Amog tese variables, rotatio agle plays a importat role. It is terefore iterestig to optimize te rotatio agle, takig ito

6 cosideratio all oter system parameters, icludig te cael matrix ad system cofiguratio, to metio few. I te sequel, te optimal rotatio agle tat miimizes te BER performace of te wole system, referred to as opt (s, H), is optimized. e optimal rotatio agle for te ML receiver is obtaied troug Mote Carlo simulatios for QPSK ad 16QAM scemes ad several R scearios. e results are depicted i Fig. 5. able summarizes tese results. e curves depicted i Fig. 5 are simulated for differet values of ad R. e SNR value i eac sceario is cose suc tat te miimum BER is aroud We te rotatio agle leads to a small Euclidea distace amog te symbols of d, for example = 3 o ad 37 o i Fig. 5(c), te effect of te rotatio agle will domiate tat of SNR, leadig to almost te same BER performace regardless of te system cofiguratio. However, we te rotatio agle leads to a large Euclidea distace amog te symbols of d, for example = 30 o i Fig. 5(c), te effect of SNR becomes more obvious. By comparig Figs. 4 ad 5, we coclude tat te curves of d mi ( d ) versus (s), ad BER versus (s, H), ave te same sape; a icrease i d mi ( d ) leads to improvemet i te BER performace. However, te values of opt (s) ad opt (s, H) are differet for te same system settigs. is is maily due to te umber of used trasmit ad receive ateas. Furter explaatio is give i ligt of te simulatio results i Sectio VI. IV. Performace Aalysis of CQSM Let g s s ad gˆ ˆ sˆ ˆ sˆ be two received a b a b vector sigals. e, te pairwise error probability (PEP) is give as were g gˆ Pr[ g gˆ H ] Q Q (11) 1 H ( g gˆ ) ( g g ˆ), (1) ad its expected value is give i (15), were is te variace of te cael gai. e average PEP assumig R receive ateas is give by [14], [19] were 1 R 1 k R k R P ( g g ˆ) [1 ] e (13) k 0 k 1 / 1 1 /. e BER of te CQSM is upper-bouded by te followig average bit-error probability (ABEP): M M 1 1 P P ( g g ˆ ) e (14) b M e i k i, k i1 k M were e i,k is te umber of bit errors associated wit P ( g g ˆ ). e i k ( x x xˆ xˆ a b a b if,,, if,,, ( x xˆ x xˆ b b a a if,,, ( x xˆ x xˆ a a b b if,,, ( x x xˆ xˆ a b a b if,,, ( x x xˆ xˆ a b a b if,,, ( x x xˆ xˆ a b b a if,,, ( xˆ xˆ x x a b a b if,,, ( x xˆ xˆ x a a b b if,,, ( x xˆ xˆ x b a b a if,,, ( x x xˆ xˆ a b a b if,,, ( x x xˆ xˆ a b a b if,,, ( x xˆ x xˆ a a b b (15)

7 V. Computatioal Complexity e evaluatio of te secod lie of (6) requires (4 R +1) real multiplicatios ad (4 R 1) real additios. Sice te ML searc is performed over a M-dimesioal space, CQSM requires te followig umber of real multiplicatios ad additios: mul add M (4 1), R M (4 1). R (11) Sice te ML detectors for bot CQSM ad QSM ave te same form of optimizatio fuctio, give above i (6) ad (3) i [14], respectively, bot scemes ave equivalet computatioal complexity for te same spectral efficiecy of M bits/s/hz. e computatioal complexity required to evaluate (6) is listed i able system. ese eacemets of CQSM icur o additioal computatioal costs, as detailed i Sectio V. able 3. Receiver computatioal complexity. erm Real multiplicatios Real additios g R R 1 Re{y H g} R +1 R 1 g -Re{y H g} 0 1 otal 4R +1 4R 1 Fig. 6. BER performaces of SM, GSM, QSM ad CQSM scemes for te same spectral efficiecy of 8 bits/s/hz. VI. Simulatio Results ad Discussio I tis sectio, te receiver is cosidered to ave perfect kowledge of te cael state iformatio. I additio, data bits are cosidered to be radom suc tat te sigal ad spatial symbols are uiformly distributed. e optimal rotatio agles used to obtai te simulatio results are depicted i able. Figure 6 sows te BER performace of SM, GSM, QSM ad CQSM for te same spectral efficiecy of 8 bits/s/hz. e GSM trasmitter is equipped wit = 7 ad employs a combiatio of U = ateas to trasmit oe sigal symbol at eac cael use. Assumig bot SM ad GSM use 16QAM, SM requires 16 trasmit ateas to acieve te same spectral efficiecy, compared to oly seve used by GSM. e reductio i comes at a moderate computatioal cost [18]. We SM uses 64QAM ad four trasmit ateas, it lags te performace of GSM by approximately 3 db at a target BER of erefore, a tradeoff betwee performace ad te umber of trasmit ateas ca be acieved. O te oter ad, CQSM outperforms SM ad QSM by 7.1 db ad 5.1 db, respectively, at a target BER of Moreover, CQSM outperforms GSM by 4.5 db wile requirig four trasmit ateas, compared to seve required by GSM. Fially, CQSM outperforms QSM by 5.1 db i te case of te Fig. 7. BER performace of QSM ad CQSM for te same trasmissio rate of 1 bits/s/hz. Figure 7 depicts te performace of CQSM ad QSM scemes usig 16QAM ad 56QAM, respectively, were bot scemes acieve te same trasmissio rate. e upper boud of te ABEP of CQSM give i (14) is also sow for te cosidered scearios. CQSM still outperforms te QSM sceme by 4.1, 4.5, ad 5. db i te case of 48, 46, ad 44 systems, respectively. e outperformace of CQSM is sligtly reduced as te umber of receive ateas is icreased. Figure 8 depicts te performace of CQSM ad QSM scemes for data rates of 8 ad 6 bits/s/hz, respectively, usig QPSK modulatio. e aalytical ABEP of CQSM give i (14) is also sow for te cosidered scearios. QSM outperforms CQSM by 0.5, 0.57, ad 1 db for 44, 46, ad 48 systems, respectively. is degradatio is tolerable as te CQSM sceme icreases te acieved spectral efficiecy by

8 33.33%. Fially, te performace of CQSM ad QSM scemes for several = R scearios usig QPSK modulatio are depicted i Fig. 9. e proposed system acieves a icrease of 50%, 33.33%, 5% ad 0% i spectral efficiecy usig, 4, 8, ad 16 trasmit ateas, respectively. As te umber of trasmit ateas icreases, te performace gap betwee CQSM ad QSM decreases. I te case of = 16, te proposed sceme outperforms QSM for most of te simulated values of SNR. is performace tred is explaied i te sequel. Witout loss of geerality, we cosider te case of QPSK modulatio because te followig cojecture ca be simply exteded to ay oter modulatio set. te followig two probabilities old true. 1 Pr[ s ], i c 1 Pr[ s s ]. i a i b As grows large, te followig two limits are satisfied: lim Pr[ s ] 0, i lim Pr[ s s ] 1. i a i b is implies tat, at a very large, d a b. I tis case, te optimum rotatio agle is ituitively give by opt (s) = opt (s, H) = 45 o ad d mi ( d ) = si(/) = is value is larger ta 0.518; te Euclidea distace at = 30 o we te symbols i d are cosidered to be equally probable. is cojecture coicides wit te optimal values of te rotatio agle listed i able : opt (s, H) icreases from 30 to 41 we te umber of trasmit ateas icreases from to 16. Usig Mote Carlo simulatios, it is additioally determied tat te optimal rotatio agle opt (s, H) obtaied for te 33 system is 41.5 o. is result coicides wit our cojecture o te covergece of te optimal rotatio agle preseted i Sectio III. c VII. Coclusios Fig. 8. BER performace of QSM ad CQSM wit trasmissio rates of 6 ad 8 bits/s/hz, respectively, usig QPSK modulatio for several system cofiguratios. I tis paper, we proposed a CQSM sceme, were two complex costellatio symbols draw from two differet modulatio sets are trasmitted at eac cael use, leadig to a iger trasmissio rate compared to QSM. e first symbol is draw from a covetioal QAM/PSK modulatio set; te secod is draw from a rotated versio of te former set. Sice te rotatio agle affects te system BER performace, it was optimized usig Mote Carlo simulatios as well as aalytically. Simulatio results sowed tat, for te same trasmissio rate, CQSM outperformed te GSM ad QSM by at least 4 to 5 db i several system settigs. It was also umerically ad aalytically sow tat, as te umber of trasmit ateas became large, CQSM outperformed QSM wile acievig a iger trasmissio rate. Appedix I Fig. 9. BER performace of QSM ad CQSM usig QPSK modulatio. CQSM acieves 6, 8, 10, ad 1 bits/s/hz versus 4, 6, 8, 10 bits/s/hz for QSM, usig, 4, 8, ad 16 trasmit ateas, respectively. Let s i be a o-zero elemet of te trasmitted vector s. e, Let a be give by a ( 1) / { j i s e i 1,, 4} i e, te rotated costellatio set is give by

9 i Fig. 10. e, d s s 3 si( ) cos( ) 1 1 Fig. 10. Example of te resultig sigal modulatio sets, a, te rotated set b, ad teir Mikowski sum c for QPSK modulatio ad rotatio agle = π/4. b (( 1) / ) { j i s e i 1,, 4} i were is te rotatio agle of te sigal costellatio set. Based o CQSM, c is defied as { s s s, s, i, k 1,, 4} c i k i a k b were c = a b is te Mikowski sum of te sets a ad b. From a computatioal geometry perspective, te Mikowski sum is represeted as te uio of te followig subsets: were c 4 i1 { s s s, s, k 1,, 4} i i k i b k a at is, sice te symbols i a are located at te corers of a square cetered at te origi, te subset i is a sifted versio of a ad is cetered at s i b. Figure 10 depicts a example of te resultig sigal modulatio set. It is wort metioig tat: 1. Owig to te structure of te QPSK modulatio set, te resultig d is eve symmetric aroud π/4. at is, te values of d mi ( d ) are idetical for te agles ad (π/ ). erefore, te optimizatio of is carried out i te iterval [0, π/4]. Figure 11 depicts d for rotatio agles [0, π/4].. O accout of te symmetry of te resultig sigal modulatio set d, te searc of te optimum rotatio agle reduces to: opt [0, / 4] ( s ) arg max mi( d, d ). i 1 Let s 1 a = 1, s c = (j-je j ), ad s 3 b = e j, as idicated Fig. 11. Sigal costellatio symbols i d for QPSK modulatio ad rotatio agle [0, π/]. d s s cos( ) 1 3 Sice d 1 is a strictly decreasig fuctio ad d is a strictly icreasig fuctio, te optimal rotatio agle terefore satisfies te coditio d 1 = d. From te above defiitios of d 1 ad d, te optimal rotatio agle satisfies si() = 0.5, wic implies tat () s opt 6 i te case of QPSK modulatio. A secod geometrical aalysis of c tat egeders furter isigt o te resultig receiver performace is give i te sequel. Let c be rewritte as a uio of four disjoit subsets i, i = 1,, 4. e symbols of eac subset are located at te four corers of a square for ay rotatio agle. Additioally, te symbols belogig to te same subset ave te same power ad a esemble mea of zero. Let te polar represetatio of te first elemet of te subset i be writte as r i e ji. e, wit te cotributio of trigoometric aalysis, it is sow tat For istace, r cos( ), 1 1 r si( ), 4 r si( ), r cos( ), 4 4

10 j s 1 e 1 cos( ) j si( ). 1 1 erefore, r 1 is simply give as above, ad agle 1 is give by ta ta 1 si( ) si( / ) cos( / ) ta 1 cos( ) cos ( / ) si( / ) cos( / ) Figure 11 depicts te sets a, b, 1,, 3, ad 4 for [0, π/]. e values of several rotatio agles are also added to idicate te directio of rotatio. ese results ca be used for te derivatio of te optimal rotatio agle. Furtermore, tey ca foster furter isigt i aalyzig te CQSM sceme i te case of ig-order PSK modulatio. Refereces [1] R. Mesle, H. Haas, S. Siaovic, C.-W. A, ad S. Yu, Spatial modulatio, IEEE rasactios o Veicular ecology, vol. 57, o. 4, July 008, pp [] J. Jegaata, A. Grayeb, L. Szczeciski, ad A. Cero, Space sift keyig modulatio for MIMO caels, IEEE rasactios o Wireless Commuicatios, vol. 8, o. 7, July 009, [3] J. Jegaata, A. Grayeb, ad L. Szczeciski, Geeralized space sift keyig modulatio for MIMO caels, i Proc. PIMRC, 008, pp [4] A. Youis, S. Siaovic, M. Di Rezo, R. Mesle, ad H. Haas, Geeralized spere decodig for spatial modulatio, IEEE rasactios o Commuicatios, vol. 61, o. 7, July 013, pp [5] A. Youis, R. Mesle, M. Di Rezo, ad H. Haas, Geeralized spatial modulatio for large-scale MIMO, i Proc. Eusipco, Sep. 014, pp [6] J. M. Lua-Rivera ad M. G. Gozalez-Perez, A improved spatial modulatio sceme for MIMO caels, i Proc. EUCAP, 01, pp [7] J. M. Lua-Rivera, M. G. Gozalez-Perez, ad D. U. Campos- Delgado, Improvig te performace of spatial modulatio scemes for MIMO caels, Wireless Persoal Commuicatios, vol. 77, o. 3, August 014, pp [8] E. Basar, U. Aygolu, E. Paayirci, ad H. V. Poor, Space-time block codig for spatial modulatio, i Proc. IEEE PIMRC, 010, pp [9] L.L. Yag, rasmitter preprocessig aided modulatio for multiple-iput multiple-output systems, i Proc. IEEE VC, Sprig 011, pp [10] J. Zeg, Fast receive atea subset selectio for pre-codig aided spatial modulatio, IEEE Wireless Commuicatios Letters, vol. 4, o. 3, Jue. 015, pp [11] S.B. Lee ad M. Moaise, Gram-Scmidt ortogoalizatiobased atea selectio for pre-codig aided spatial modulatio, Joural of elecomm., Electroic ad Computer Egieerig, vol. 8, o. 9, Dec. 016, pp [1] M. We, X. Ceg, ad L. Yag, Idex Modulatio for 5G Wireless Commuicatios. Spriger, 017. [13] E. Basar, Idex modulatio teciques for 5G wireless etworks, IEEE Commuicatios Magazie, vol. 54, o. 7, Jul. 016, pp [14] R. Mesle, S. Ikki, H. M. Aggoue, Quadrature spatial modulatio, IEEE rasactios o Veicular ecology, vol. 64, o. 6, Jue 015, pp [15] Z. Huag, Z. Gao, ad L. Su, Ati-eavesdroppig sceme based o quadrature spatial modulatio, IEEE Commuicatios Letters, vol. PP, o. 99, November 016, pp [16] J. Li, M. We, X. Ceg, Y. Ya, S. Sog, ad M. Lee, Geeralized precodig-aided quadrature spatial modulatio, IEEE rasactios o Veicular ecology, vol. 66. o., Feb. 017, pp [17] M. de Berg, O. Ceog, M. va Kreveld, ad M. Overmars, Computatioal Geometry: Algoritms ad Applicatios, 3rd ed. Berli, Germay: Spriger, 008. [18] A. Youis, N. Serafimovski, R. Mesle, ad H. Haas, Geeralized spatial modulatio, i Proc. Asilomar, 010, pp [19] G. Gritsc, H., Weiricter, ad M. Rupp, A uio boud of te bit error ratio for data trasmissio over correlated wireless MIMO caels, i Proc. ICASSP, 004, pp

11 Maar Moaise received a MSc degree i commuicatios ad sigal processig from te Uiversity of Nice-Sopia Atiplois, Frace, i 005, ad a PD from Ia Uiversity, Rep. of Korea, i 010, bot i commuicatios egieerig. From 001 to 004, e was a cell plaig egieer at te Palestiia elecommuicatios Compay. Sice 010, e as bee a assistat professor at te Departmet of EEC Egieerig, Koreaec, Rep. of Korea. His researc iterests iclude MIMO systems, commuicatio systems, ad social etwork aalysis. Saetbyeol Lee received a BEg degree i electroics egieerig from Korea ec, Rep. of Korea i 015. Se is curretly pursuig a MSc degree at te Departmet of Electrical, Electroics, ad Commuicatios Egieerig, Korea ec, Rep. of Korea. Her researc iterests iclude MIMO systems wit a empasis o spatial modulatio ad iterferece aligmet.