Scaling Laws for Noncoherent Energy-based Communications in the SIMO MAC

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1 Scalig Laws for Nocoheret Eergy-based Commuicatios i the SIMO MAC Maiak Chowdhury, Alexadros Maolakos ad Adrea Goldsmith Fellow, IEEE Abstract We cosider a oe-shot commuicatio settig i which several sigle atea trasmitters commuicate with a receiver with a large umber of ateas, i.e., the receiver decodes trasmitted iformatio at the ed of every symbol time. Motivated by the optimal ocoheret detector i a Rayleigh fadig chael, we cosider a ocoheret eergy-based commuicatio scheme that does ot require ay kowledge of istataeous chael state iformatio at either the trasmitter or the receiver; it uses oly the statistics of the chael ad oise. We show that, for geeral chael fadig statistics, the performace of the cosidered oe-shot multiuser ocoheret scheme is the same, i a scalig law sese, as that of the optimal coheret scheme exploitig perfect chael kowledge ad codig across time. Furthermore, we preset a umerical evaluatio of the performace of this scheme i represetative fadig ad oise statistics. Idex Terms Massive MIMO, Nocoheret Commuicatio, Eergy Receiver, Multiuser Commuicatio I. INTRODUCTION Systems with a large umber of ateas at either the trasmitters or receivers ejoy promisig asymptotic properties. As show i [], the effects of fast fadig ad oise vaish ad eve zero forcig receiver architectures perform well i the asymptotic limit. I practice, however, may of these beefits are tied to the assumptio of reasoably good chael state iformatio (CSI) at the trasmitter ad the receiver. Obtaiig perfect CSI, however, is ofte a bottleeck i systems with large atea arrays operatig either at high speeds or at high carrier frequecies. The authors are with the Departmet of Electrical Egieerig, Staford Uiversity, Staford, CA Questios or commets ca be addressed to {maiakch, amaolak, adreag}@staford.edu. Parts of this work were preseted at CISS, 204 ad at ISIT, 204. This work is supported by a 3Com Corporatio Staford Graduate Fellowship, a Alcatel-Lucet Staford Graduate Fellowship, a A.G. Levetis Foudatio scholarship, the NSF Ceter for Sciece of Iformatio (CSoI): NSF-CCF , NSF grat , ONR grat N ad by a research grat from CableLabs. Copyright (c) 204 IEEE. Persoal use of this material is permitted. However, permissio to use this material for ay other purposes must be obtaied from the IEEE by sedig a request to pubs-permissios@ieee.org. Acquisitio of CSI i cellular systems requires the use of orthogoal pilot sequeces i the idividual cells. I fact, i a surprisig result i [], it was show that the pilot cotamiatio problem, due to the reuse of pilot sequeces i adjacet cells, is the oly factor limitig the rate of a cellular base statio servig a fixed umber of users with a asymptotically large umber of ateas. With a fiite but large umber of ateas, the chael estimatio overhead maifests itself i additioal performace degradatio due to icorrect CSI [2], [3]. Moreover, i large atea arrays, the RF chai desig with perfect CSI acquisitio quickly becomes prohibitive i terms of cost/eergy, thereby motivatig aalog or hybrid atea array desigs [4]. Give the difficulty of chael state iformatio acquisitio i multiple-iput multiple-output (MIMO) systems with a large umber of ateas, i this work, we ask how much of the potetial performace gais from a large atea deploymet ca be achieved without ay istataeous CSI either at the receiver (CSIR) or at the trasmitter (CSIT). We cosider oe-shot achievability schemes, i.e., we do ot use codig across differet symbol times or coherece itervals ad we assume that the receiver decodes iformatio at the ed of each symbol time. We show that this oe-shot ocoheret scheme achieves the same scalig behavior of achievable rates as perfectly coheret schemes whe the umber of receive ateas approaches ifiity, eve for large coherece times ad large decodig blocklegths (i.e., multi-shot schemes). The reaso why we ca commuicate reliably i this settig eve without kowig the istataeous CSI is that there is a lot of spatial diversity i a large atea array, which, as we show later i the paper, may obviate the eed for ay chael state iformatio. I this mauscript, we cosider a average-eergy-based ocoheret commuicatio system with multiple (but fixed) sigle atea trasmitters ad a receiver with a large umber of ateas. Due to the large umber of receive ateas, we refer to this system as a massive sigle-iput multiple-output (SIMO) multiple access chael (MAC). The receiver performs oly a

2 average eergy measuremet ad decodes based o the estimate of the average received eergy at the ed of each symbol time. For this scheme, we characterize the rate of decay of the achievable symbol error probability as a fuctio of the umber of the ateas at the receiver. Our aalysis suggests that the proposed average eergybased commuicatio system ca achieve the same performace as a commuicatio system with perfect CSIR i terms of the scalig law of achievable rates with a icreasig umber of receive ateas. We also cosider a costellatio desig problem for the proposed commuicatio scheme, based o the miimum distace criterio used to establish the scalig law result. We provide umerical results which demostrate the performace of this scheme for represetative fadig ad oise statistics. Note that while our achievable scheme uses symbol-by-symbol trasmissio ad decodig, our results o the optimal scalig laws are applicable eve to multi-shot schemes. I particular, multi-shot schemes ca icrease the error expoets associated with the probability of error or improve the pre-log factor i the optimal achievable rates, but do ot chage the rate scalig. Lastly, we poit out that while our aalysis of achievable rates does ot cosider codig across time, icorporatig codig does help i improvig the error expoet associated with the achievable bit error rates. We also compare the performace of a joit multiuser costellatio desig with a sigle user desig multiplexed i time. We observe that while a joit multiuser desig ca outperform time-divisio schemes for certai chaels, it is ot uiformly better. I particular, we preset a example of a chael for which this is ot the case. The rest of the paper is orgaized as follows. I Sectio II we preset the system model ad the average eergybased ocoheret detector, together with the optimal ocoheret maximum likelihood (ML) decoder. Sectio III summarizes related work. The, Sectio IV presets a upper boud o the symbol error probability as a fuctio of the umber of receive ateas. Sectio V shows, usig simple costellatio desigs, that for a geeral multiuser system, the performace of the eergybased decoder is the same, i a scalig law sese, as the performace of the optimal coheret detector. Sectio VI presets ad solves a costellatio desig problem ivolvig a miimum distace criterio (a more exhaustive treatmet of costellatio desig is preseted i [5], icludig compariso with the miimum distace criterio described here). Sectio VII presets umerical evaluatios of the proposed schemes, demostratig their symbol error performace i typical scearios with a icreasig umber of receive ateas. We preset coclusios ad future research directios i Sectio VIII. A. Notatio We use [] to deote the set {, 2,, } where is a iteger. C refers to the set of complex umbers ad C m is the set of all complex-valued matrices of size m. For a matrix H C m, the (i, j)-th elemet is deoted by H i,j. Re( ) ad Im( ) represet the real ad imagiary terms, respectively. E [U] deotes the expectatio of the radom variable U. CN (µ, R) represets the distributio of circularly symmetric complex Gaussia (CSCG) radom vectors with mea vector µ ad a covariace matrix R. The symbol is used to deote a defiitio, Θ( ) to idicate equality up to a costat, ad. = to deote equality to the first order i( the) expoet,. i.e., a = b meas that lim log a b = 0. For x, y R, x > y x i > y i, i []. II. SYSTEM MODEL We cosider m sigle atea trasmitters ad oe receiver with ateas. The system is represeted as y = Hx + ν, () with y C, x C m, ν C, H = [H, H 2,..., H m ] C m, H j C ad each ν i CN (0, σ 2 ). The i th elemet of H j is distributed as H i,j f(h), such that E[H i,j ] = µ j, E [ H i,j µ j 2] = σj 2, ad µ j 2 + σj 2 = for all j [m]. We further assume that the desity fuctio f(h) is such that, for ay fixed x C, the momet geeratig fuctio of y i 2, i.e., M(θ) E[e θ yi 2 ], exists ad is twice differetiable i a iterval aroud θ = 0. May fadig distributios fall withi this model, e.g., Rayleigh ad Ricia fadig [6], i which case H i,j CN (µ j, σj 2 ). We assume that the istataeous chael realizatio is ukow to both the trasmitters ad the receiver ad ivestigate schemes for recoverig trasmitter data, based oly o the kowledge of the statistics of the system. As already described, we focus oly o symbol-bysymbol ecodig ad decodig schemes for achievability. We will show that a particular ocoheret oeshot scheme suffices to achieve a scalig behavior that 2

3 is optimal eve uder more geeral settigs of larger coherece times ad larger decodig blocklegths. I this work, motivated by the simplicity of a averageeergy-based receiver desig, we focus oly o schemes which use the average eergy of the received sigal to reliably trasfer the iformatio; that is, the iformatio is modulated o the eergy of the trasmitted symbols ad the receiver estimates the eergy of the received sigals. A atural questio that arises is how much is lost i performace by usig the average-eergy-based decoder. As explaied i the ext sectio, we show that, i terms of the scalig law of achievable rates, there is o loss i performace with this decoder with respect to the optimal decoder usig perfect chael state iformatio as the umber of ateas grows asymptotically large. We ext describe the ecodig ad decodig procedure. A. Ecoder For j [m], the j th trasmitter trasmits symbols from the costellatios C j = {c,j, c 2,j,, c Lj,j} C, subject to a idividual average power costrait L j c i,j 2, j. (2) L j Here c i,j C j is the i th costellatio poit of the j th trasmitter ad the total size of the costellatio for trasmitter j is L j. The set of all possible trasmitted poits is deoted by C = {x C m : x j C j, j [m]}. By defiitio, we ca verify that the cardiality of C is give by C = m j= L j L. We ow describe our decoders. B. Decoders We describe two differet decoders. The first oe, the average eergy-based decoder, does ot require phase iformatio at ay of the receiver ateas ad is suboptimal for geeral fadig statistics, i terms of miimizig the symbol error probability. The secod decoder, the ML decoder, utilizes phase iformatio at all the receiver ateas ad is the optimal decoder with respect to miimizig the symbol error probability. This work aalyses the performace of the former i the case of a massive SIMO MAC system. ) Eergy-based decoder: Based o its kowledge of the statistics of the chaels ad oise ad of the costellatios {C j } m j=, the decoder divides the positive real lie ito o-itersectig itervals or decodig regios {I x } x C. I order to detect the trasmitted x C, the decoder first computes y 2 R +, (3) i.e., it computes the average received power across all the ateas. It the returs } ˆx {x C : y 2 I x. The, the probability of error give that x is trasmitted is give by P e (x ) Pr{x ˆx}. 2) Nocoheret ML decoder: While the eergy-based decoder is the optimal choice for some chael statistics ad has good asymptotic properties, for geeral statistics, it may ot be optimal. The optimal detector with respect to miimizig the probability of symbol error, assumig equiprobable sigalig ad oly the kowledge of the chael distributio at the receiver, is give by the followig maximum likelihood decoder: ˆx argmax x f(y), where f( ) is the likelihood fuctio of the output y give the chael ad oise statistics, ad the fact that x was trasmitted. It turs out that the two decoders are the same for some represetative chael statistics. We summarize this observatio i the followig lemma. Lemma. For zero mea Gaussia statistics o both the chael oise ad the chael coefficiets, (i.e., i i.i.d. Rayleigh fadig with i.i.d. AWGN), the ocoheret ML decoder reduces to the average eergy-based decoder. Proof: The proof is preseted i Appedix B. III. RELATED WORK The use of ocoheret schemes i wireless systems is ot ew. The earliest icaratios of wireless cellular systems (from G systems which used differetial modulatio to 2G systems which used frequecy shift keyig [6]) were ocoheret due to limitatios i device maufacturig ad less cocer about spectral efficiecy for voice-oly systems. As maufacturig improved, demad for high data-rates grew, ad spectrum became more expesive, coheret systems became prevalet due to their spectral efficiecy (e.g. i almost all 3G ad 4G 3

4 systems). With may emergig applicatios, however, spectrum is o loger a bottleeck ad processig complexity/power cosumptio are equally importat, if ot more importat, cosideratios. A importat ad power hugry compoet of may high speed systems is the RF frot ed (the clock recovery ad phase acquisitio circuits), ad the aalog to digital coverter (ADC). This is icreasigly a bottleeck ot oly i low power applicatios but also i applicatios with larger atea arrays [7] ad higher carrier frequecies. This makes the simplicity of ocoheret desigs especially attractive i such systems. The resurgece i iterest i ocoheret desigs for moder commuicatio systems led to a lot of fudametal work i differet aspects of ocoheret system desig i the late 990s ad early 2000s. A ocoheret commuicatio system that uses the Geeralized Likelihood Ratio Test (GLRT) is described i [8]. I this scheme, the chael ad the trasmitted symbols are joitly estimated at the receiver. The authors propose a miimum distace criterio for sigal ad code desig by characterizig the performace of the GLRT decoder i the AWGN chael for the high SNR regime. They subsequetly apply this to DPSK ad QAM alphabets. As described i [8], the GLRT decoder is idetical to the ocoheret ML decoder i geeral settigs (as log as the phase of the chael distributio is uiformly distributed over [0, 2π] ad the sigals are of equal eergy). I [9] a alterate sigal costellatio desig is proposed based o a metric motivated by a uio boud o the probability of error uder ocoheret ML decodig. Similar metrics, motivated by a high SNR aalysis, are preseted i [0] where the worst-case chordal distace is employed to space the codewords as far apart as possible. The capacity of the ocoheret chael uder a high SNR assumptio has also bee well studied uder differet settigs. Fudametal isights for the high SNR capacity i a block fadig chael model were derived i [], [2], which also derived capacity optimal sigalig distributios for the high SNR ocoheret chael. Capacity optimal high SNR iput distributios with coherece times shorter tha the total umber of ateas have bee derived i [3]. The capacity expressios i some of these works suggest i particular that small coherece times or a small rak of the chael matrix ca be very detrimetal. Multiuser couterparts of these ideas ca be foud i [4], [5]. I particular, [4] shows that time-divisio strategies may be capacity achievig i ocoheret high SNR settigs. Aother high SNR metric related to ocoheret fadig chaels is the cocept of a fadig umber [6], [7]. This is the gap betwee the high SNR asymptote of log(log(snr)) ad the ocoheret chael capacity. As discussed further i [6], [7], the fadig umber ca be computed uder very geeral assumptios o the fadig statistics of a sigle-iput sigle-output (SISO) chael ad bouded for geeral MIMO chaels. There is also a lie of work o characterizig the achievable rates i block fadig chaels (possibly takig ito accout the chael estimatio overhead) i the regime of fiite SNR ad coherece times. I [8], the chael capacity for a sigle-user MIMO chael is studied uder differet assumptios o the chael state iformatio at the receiver. I particular, uder Gaussia assumptios o the chael matrix, it is show that the ocoheret capacity scales with the umber of receivers as the logarithm of the umber of receiver ateas. This is show to be true ot oly for very geeral (possibly o-gaussia) statistics, but also for geeral multi-user chaels ad oe-shot (eergy-detectio-based) commuicatios. The use of eergy detectio i the cotext of large atea arrays also appears i [9] where the authors propose trasmissio ad detectio schemes based o Gaussia approximatios of the average eergy statistic i the limit of a large umber of ateas. There is also prior work focused o the impact of imperfect chael estimatio or chael coherece time i coheret systems. Specifically, i [20], the authors cosider the effect of pilot-based chael estimatio o achievable rates ad idetify the capacity loss due to imperfect chael estimatio as beig proportioal to the square root of the Doppler of the chael. I [2] the authors determie the mutual iformatio for Gaussia codebooks o block fadig chaels for geeral SNR, coherece time ad umber of trasmit/receive ateas. I that work, however, it is ot immediately clear how the achievable rates scale with the umber of ateas. Moreover, while Gaussia codebooks are capacityachievig uder geeral coditios for additive Gaussia oise chaels, uder multiplicative oise they may o loger be optimal. For example, [22] ad [23] show that the capacity-achievig distributio for a SISO chael with Rayleigh ad Ricia fadig ad a coherece time of is achieved by a discrete iput distributio. While the structure of the capacity-achievig distributio for a geeral ocoheret SIMO chael with a coherece time greater tha oe is ot kow to the best of our kowledge, our aalysis i this paper is applicable to characterizig the achievable rates for geeral discrete iput distributios i SIMO chaels with a large um- 4

5 ber of receiver ateas. Our work differs from the prior work described above i that we cosider a fiite trasmit power ad a asymptotic aalysis i the umber of receiver ateas. To the best of our kowledge, we are the first to cosider a average-eergy-based ocoheret massive MIMO system with a asymptotically large umber of ateas, which we first proposed i [24]. IV. ERROR PROBABILITY UPPER BOUNDS AND THE RATE FUNCTION I this sectio we aalyse the performace of the proposed eergy detector i the massive SIMO MAC. From our system model, we have y 2 Hx + ν 2 = = Hx 2 + ν 2 + ν) 2Re((Hx) m ( µi 2 + σ i 2) x i 2 + i<j 2 Re((µ i x i ) (µ j x j )) + σ 2 a.s. as. Thus, i the asymptotic regime of large, the received eergy statistic coverges almost surely to a determiistic quatity depedig o the statistics of the chael ad the costellatio poits trasmitted. For a fiite the value may deviate from this limitig value by a oise term. To characterize the oise appearig i the system, we use the followig lemma [25]: Lemma 2. For ay a > 0 ad zero mea, i.i.d. radom variables u,..., u, we have that ( Pr u i.= a) e I(a), where I(a) = sup θ>0 ( θa log(e[e θu ]) ). For completeess, a short proof of this fact is preseted i Appedix A. We use Lemma 2 to characterize the deviatio of y 2 / from its limitig value r(x), where m ( r(x) µi 2 + σ i 2) x i 2 + i<j 2 Re ((µ i x i ) (µ j x j )) + σ 2 (4) is the expectatio of the received eergy statistic y 2 /. To do so, choose u i i Lemma 2 to be 2 u i = H i,j x j + ν i r(x), (5) j thereby makig u i a zero mea radom variable. Note that u i depeds o c ad the statistics of H i,j ad ν i. With this defiitio, the followig holds y 2 = r(x) + i u i. Usig Lemma 2, we get that ( ) y 2.= Pr r(x) > a e I,x(a), where ( ( [ I,x (a) sup θa log E e θu ])). θ>0 Similarly, by ivokig the same lemma for the radom variable u i, we get the followig ( ) y 2.= Pr r(x) < a e I 2,x(a), where ( ( I 2,x (a) sup θa log E[e θu ] )). θ>0 Combiig the above two results ad usig a uio boud, we get that ( y 2 ) Pr r(x).= > a e I x(a), where I x (a) mi (I,x (a), I 2,x (a)) is defied as the rate fuctio of the (possibly multiuser) costellatio poit x. Observe that I x (a) is the error expoet associated with P e (x), i.e., I x (a) = lim log(p e(x)), whe the trasmitters sed x, ad the receiver uses a decodig iterval of I x = (r(x) a, r(x) + a]. We ow show how these results may be used to desig good costellatios for ifiite ad/or fiite systems. Before that, we collect some observatios about I x (a) i the followig lemma, which have bee proved i Appedix C. Lemma 3. The rate fuctio I x (d) ejoys the followig properties. ) The rate fuctio is o-egative ad mootoically icreasig for positive a. 2) For a fixed d ad a scalar x C, the rate fuctio is mootoically decreasig with icreasig x. 3) For ay fiite δ 0 > 0, there exists a large eough σ 2 (oise variace) such that I x+δ (a) I x (a) ɛ, δ < δ 0. (6) 5

6 4) The rate fuctio for ay zero mea U with a twice differetiable momet geeratig fuctio aroud 0 satisfies I x (a) lim a 0 a 2 = 2 E [U 2 ]. (7) The third poit implies that i the limitig case of low SNR, the rate fuctio I x (a) does ot deped o the trasmitted poit x. Recall that the rate fuctio depeds o σ 2 through the radom variable u i defied i (5). Moreover, for a small a, the rate fuctio is quadratic with a weight determied by the secod order statistics of u i defied earlier. V. ASYMPTOTIC THROUGHPUT CHARACTERIZATION I this sectio, we compare the achievable throughput of our eergy based decodig scheme with a coheret scheme as a fuctio of the umber of receiver ateas. We first defie the symmetric ergodic capacity m C E [ log( + σ i (H) 2 /σ 2 ) ] of the coheret chael where σ i (H) is the i th largest sigular value of H ad σ i+ σ i 0 for all i i [m]. I the limit of large, it ca be show that 0 < ξ σ2 i ξ 2, almost surely for all i i [m] ad for some ξ, ξ 2 > 0 idepedet of. This gives us that C = Θ(log ) for coheret systems. The followig lemma shows that this same scalig behavior ca be achieved by our ocoheret scheme. Lemma 4. There is a m-user costellatio with bouded average power per user which achieves a rate of K log2 bits per trasmissio per trasmittig user for some K > 0, with vaishig probability of error with a icreasig umber of receiver ateas. Proof. We show this through explicitly costructig a costellatio which has the above scalig behavior. We fix M to be a iteger ad choose the followig costellatios: { } 2(i )M j C j = : i [M], M m for all j [m]. Note that the eergy i each costellatio poit of C j is bouded above by 2. Note also that this costellatio satisfies the average power costrait (2) o all users, the j th user uses the costellatio C j ad that each trasmitter experieces a rate log 2 (M). Usig these costellatios, we ote that mi r(c ) r(c 2 ) = 2 c,c 2 C M m, where we use that fact that µ j 2 +σ 2 j =, Re(µ j 2 µ j ) for j j 2. Choosig M = K for some K > 0, we get that each user achieves a rate of log M = K log. We ow proceed to boud the probability of error. For a small a, we ote that usig Lemma 3, we have where a 2 mi I x (a) mi x x 2E[U 2 ] = a2, (8) 2α 0 α 0 max x (E[U 2 ]). It follows that (8) represets a lower boud o the worst case error expoet across all costellatio poits. We ow ote that E[U 2 ] is fiite sice it may oly deped o the chael ad oise statistics through the first four momets (which are bouded due to the existece of momet geeratig fuctios) ad through the first four powers of the trasmitted symbols. Moreover, sice c i,j 2 2 for all i, j, E[U 2 ] ca be bouded above by a costat depedig oly o m ad idepedet of. It may also be show that i the limit of low SNR, E[U 2 ] is the same regardless of the trasmitted costellatio poits. Thus a sufficiet criterio to get a vaishig error probability while achievig a K log() rate is to guaratee a high miimum distace betwee r(c ) ad r(c 2 ) for c, c 2 C, c c 2. By the small a characterizatio of the rate fuctio i Lemma 2, we ca choose decisio regios for which the symbol error rate P e satisfies P e e t/(0.5m m ) 2 = e t/((0.5 K) m ) 2 e t 2 Km, where t is a costat idepedet of. By choosig K sufficietly small, i.e. K < 2m, we see that the error rate vaishes as. This establishes Lemma 4. We ow metio a outcome of Lemma 4 which is also the mai theorem of our paper. Theorem. For ay fadig distributio idepedet across receiver atea elemets, oe-shot eergydetectio-based ocoheret commuicatios achieve the same scalig behavior of achievable rates i a massive SIMO system as perfectly coheret systems (Θ(log )) i the limit of a large umber of receiver ateas. Proof. The proof follows directly from Lemma 4 ad the observatio that the oe-shot scheme used to establish Lemma 4 is a special case of more geeral ocoheret schemes over larger coherece times ad decodig blocklegths. 6

7 Note that the proof of Lemma 4 above uses a sufficiet coditio to guaratee a vaishig probability of error while at the same time givig a K log() scalig of achievable rates. This coditio ivolves focusig o just the miimum distace betwee (possibly multiuser) costellatio poits at the receiver. We explore differet aspects of this costellatio desig i the followig two sectios. VI. MINIMUM DISTANCE CONSTELLATION DESIGN We first cosider sigle user costellatio desigs ad the preset how multiuser desigs are differet. A. Desig problem for sigle user We have the followig optimizatio problem maximize C L L ci, 2 mi r(x ) r(x 2 ), x,x 2 C,x x 2 (9) where r(x) evaluates to r(x) = µ 2 x 2 + σ 2. The solutio to this is give by c i, 2 = 2(i ) L, i {, 2,, L }, (0) where L = L. Note that the phase of the particular costellatio poits we choose does ot matter, sice we use a average eergy decoder ad the oise is complex Gaussia with uiform phase. So we choose the phase to be zero, i.e., c i, R +. A achievable scheme follows by settig the decisio regios I x = (r(x) a, r(x) + a], with a = L, which leads to the followig upper boud o the probability of symbol error P e = P e (x) U L L x C. = max x C e Ix(a), () where U L ( e I,c, (a) + e I2,c 2, (a) + e I,c 2, (a) L + + e I,c L, (a) + e I2,c (a)) L,. Numerical evaluatio of the symbol error rates achieved usig this scheme is preseted i Sectio VII. B. Desig problem for 2 users Similar to the oe user case, to guaratee vaishig error probability as icreases, we treat the costellatio desig as a joit codebook desig problem ad we maximize the miimum distace betwee the received costellatio poits as maximize C L j L j ci,j 2, j [2] mi r(x ) r(x 2 ). x,x 2 C,x x 2 (2) This, i geeral, is a o-covex problem due to the absolute value i the objective fuctio. I order to simplify the problem, oe ca cosider a total orderig o the received costellatio poits {r(x)} x C. With the total orderig costrait, (2) becomes efficietly solvable i some special cases. Depedig o the chael statistics, the problem ca be either a liear program, covex quadratic program or a o covex quadratic program. The details are preseted i Appedix D. The performace obtaied from these schemes is preseted i Sectio VII. C. Joit multiuser codebook desig versus sigle user desig with time-divisio A atural way to exted the sigle user desigs to multiple users is to employ a time-divisio strategy ad use the sigle user codes for each user i a roud robi fashio. I this sectio, we ivestigate how this time-divisio strategy compares with a joit multiuser codebook desig. ) Rayleigh fadig: We show that the suggested average-eergy-based system caot achieve a higher rate fuctio (or error expoet) tha that of the correspodig sigle user desigs whe used i a timedivisio fashio. Specifically, we show i Appedix F that the joit codebook desig always yields a smaller error expoet compared to a time-divisio strategy whe the chaels experiece symmetric Rayleigh fadig. 2) AWGN chael ad Ricia fadig: I this sectio we show that if the multiuser chaels have o-zero correlatio (e.g. both of them are Ricia chaels with the same o zero K factor), the a joit multiuser codebook desig ca outperform a time-divisio strategy. The mai ituitio behid why this is the case is that the possible eergy levels of the received sigal for joit multiuser trasmissios i such a case are more separated (ad hece more distiguishable) due to the costructive iterferece of trasmissios from differet users at the receiver. 7

8 Cosider 2 users with the same AWGN chael with oise variace σ 2. I time-divisio, each user trasmits i a roud robi fashio for oe slot from a costellatio of size L 2. I the suggested multiuser desig, both users trasmit cocurretly for two slots usig a joit costellatio of size L, such that the average trasmit power across two cosecutive slots at each user is the same. I the time-divisio scheme, the best sigle user costellatio for low SNR is give by (0), that is C tdma = { 0, 2 4 3, 3, } 2, (3) with ( decisio regios I x = x 2 + σ 2 3, ] x2 + σ 2 + 3, x Ctdma. The above costellatio satisfies the average power costrait ad, i the limit of low SNR (i.e. high σ 2 ), it leads to the best rate fuctio for the time-divisio scheme, which is mi Ix tdma x ( ) ( ) I 0. (4) 3 3 I the above, I 0 ( ) is the rate fuctio associated with the trasmissio of the zero symbol. Observe that, as the SNR becomes lower ad lower, the best rate fuctio is more ad more idepedet of the trasmitted symbol (Lemma 3). For the joit multiuser codebook desig, cosider the followig costellatios: C = { } { } ɛ, ɛ, C 2 = 0, 3 3ɛ, (5) for ay 0 ɛ < 2/5. The above costellatios satisfy the average power costrait for each user across two slots, assumig the users iterchage the costellatios, i.e., user trasmits from C durig trasmissio slot, ad from C 2 durig the secod time slot; user 2 trasmits C 2 whe user trasmits from C ad from C otherwise. Sice c 2 =, 2 c C C 2 ad the trasmitters employ equiprobable sigallig, each user trasmits o average sigals of power. ay Ricia chael with o-zero K could be used. The simplest example is to use a AWGN chael which correspods to ifiite K. I the limit of ifiite receive ateas, the received costellatio poits are {ɛ, 23 + ɛ, 43 ( ) 4 2ɛ + 2 ɛ 3 3ɛ, (2 ) ( ) } ɛ 3 3ɛ ɛ, (6) ad the decisio regios are such that the boudary poits are located i the midpoit betwee cosecutive received costellatio poits. It is easy to verify that for 0 ɛ < 2/5, the above costellatio ad decisio regios lead to a miimum distace betwee received poits which is always more tha 3, i.e., i the limit of low SNR, the rate expoet is mi Ic 2-user c ( 3 + ɛ ) ( ) 3 I 0 + ɛ, (7) where ɛ > 0. Based o Lemma 3 ad equatios (5) ad (7) it follows that the joit multiuser codebook desig ca achieve a higher rate fuctio tha the sigle user desig. Ituitively, the reaso behid this is the fact that the multiuser desig ca make use of the costructive iterferece betwee the chaels of the two trasmitters. I situatios where this costructive iterferece is small (or abset altogether), the gais are small or eve oexistet (e.g., same or symmetric Rayleigh fadig for all users). Numerical examples of this are preseted i Subsectio VII-C. VII. NUMERICAL RESULTS We ow preset differet aspects of the proposed commuicatio system desig through umerical simulatios. Subsectio VII-A demostrates the sigle user performace for the costellatio desig of Subsectio VI-A for Ricia fadig with differet values of the K-factor. The correspodig results for two users are preseted i Subsectio VII-B, i which case we use the costellatios geerated as explaied i Subsectio VI-B. Compariso of the joit multiuser codebook desig for two users with the correspodig time-divisio scheme is preseted i Subsectio VII-C. Subsectio VII-D compares the eergy-based detector with the ocoheret ML detector for both Rayleigh ad Ricia fadig. Fially, Subsectio VII-E demostrates through a example that time codig across multiple chael realizatios could provide additioal gais over the oeshot system. 8

9 P e, L =4 U L, L =4 P e, L =8 U L, L = Number of ateas P e, L =4 U L, L =4 P e, L =8 (a) σ 2 = (low SNR) U L, L = Number of ateas (b) σ 2 = 0. (high SNR) Fig. : SER performace of the sigle user desig i Rayleigh fadig (K = 0) at low ad high SNR P e, L =4 U L, L =4 P e, L =8 U L, L = Number of ateas P e, L =4 U L, L =4 P e, L =8 U L, L =8 (a) σ 2 = (low SNR) Number of ateas (b) σ 2 = 0. (high SNR) Fig. 2: SER performace of the sigle user desig i Ricia fadig (K = 0) at low ad high SNR. A. Sigle user performace We plot the umerical estimate of the probability of symbol error rate (SER, obtaied through Mote Carlo simulatios) with the correspodig aalytical boud U L for the case of a sigle user with a icreasig umber of receive ateas ad a costellatio with equidistat power levels for a Ricia fadig chael with K-factor K = 0 (Fig. ) ad K = 0 (Fig. 2). We cosider additive white Gaussia oise with variace σ 2 = {0., }, ad 2 bit (L = 4) ad 3 bit (L = 8) costellatios. We observe that, as the LOS factor icreases, the performace of the system improves sigificatly. Furthermore, we see that for the curret costellatio desig, eve if it is asymptotically the best as σ 2 icreases, a impractically large umber of receive ateas is eeded i order to get a reasoable performace for values of σ 2 of practical iterest. I [26], we demostrate that by optimizig the costellatios, it is possible to get good performace with a umber of ateas that is commesurate with today s commercial offerigs of large atea systems. B. Two users performace We ow cosider two users, whose costellatio is desiged as described i Subsectio VI-B for the same chaels as above, i.e., Ricia fadig with K = 0 (Fig. 3) K = 0 (Fig. 4) ad σ 2 {0., }, for a bit ad 2 bit costellatio per user with the same average power per user as before. Observe that the SER for L =, i.e., O-Off Keyig, is sigificatly better tha L = 2 as the umber of ateas icrease. Furthermore, the gap betwee the upper boud ad the umerical estimate o the probability of symbol error appears to be costat, which is a good idicatio that the former serves well as a good approximatio of the latter, at least i Ricia fadig chaels. C. Joit multiuser codebook versus time-divisio scheme I this sceario, we umerically compare the joit twouser codebook desigs of Subsectio VI-B, with the case of time-divisio betwee the two users whe average 9

10 P e, L =2 U L, L =2 P e, L =4 U L, L = Number of ateas (a) σ 2 = 0 4 P e, L =2 U L, L =2 P e, L =4 U L, L = Number of ateas (b) σ 2 = 0. Fig. 3: SER performace of the 2 user desig i Rayleigh fadig (K = 0) at low ad high SNR Pe, L =2 UL, L =2 Pe, L =4 UL, L = Number of ateas (a) σ 2 = Pe, L =2 UL, L =2 Pe, L =4 UL, L =4 power does ot exceed oe. We assume that, for the joit multiuser codebook, the average trasmitted power per user over two cosecutive slots does ot exceed oe ad that each user trasmits oe bit per symbol. The umerical results are for Ricia fadig (Fig. 5) with K = 00 ad σ 2 = 2. Observe that the joit multiuser desig achieves better performace compared to the desig with time-divisio across the users Multiuser Pe Multiuser UL TDM UL TDM Pe Number of ateas Fig. 5: 2 user desig vs. time-divisio codig for Ricia fadig with K = 00 ad σ 2 = at a rate of bit per user. The 2 user multiuser scheme achieves better performace compared to the TDM sigle user desig. D. Compariso with the ocoheret ML detector As show i Appedix B, usig the ocoheret ML detector (described i Sectio II-B2; this, i geeral, requires usig phase iformatio i additio to eergy measuremets) does ot improve performace if the chael ad oise statistics are zero-mea Gaussia. I most other cases, phase detectors give sigificat performace improvemets. We preset umerical results for the case of a sigle user usig the optimal ML decoder ad compare it with the average-eergybased decoder with equidistat costellatio poits. We cosider a low SNR (σ 2 = ), ad Ricia fadig with both low (K = 0.) ad high (K = 00) K-factors (Fig. 6). Recall that havig equidistat costellatio poits is optimal for the average eergy-based decoder for low SNR. E. Codig Number of ateas (b) σ 2 = 0. Fig. 4: SER performace of the 2 user desig i Ricia fadig (K = 0) at low ad high SNR I this sectio, we cosider the case of a sigle trasmitter which uses codewords that spa multiple (deoted as T for some T > ) symbol duratios. We also assume that the chael chages betwee the symbols. We show through a explicit example that it is possible 2 Cosiderig eve higher σ 2 leads to the same coclusios. 0

11 0 0 0 ML, K =20dB Eergy-based, K =20dB ML, K = 0 db Eergy-based, K = 0 db Number of ateas Fig. 6: Performace compariso of the ocoheret ML ad eergy-based decoders for a 2 bit costellatio i a Ricia fadig sceario with low SNR (σ 2 = ) ad low (K = 0.) ad high (K = 00) K-factor. Observe that at a low K-factor, the differece betwee the ML ad the eergy-based detector is egligible. to get a strictly better symbol error expoet for the same average rate by trasmittig ad joitly decodig codewords over T = 3 symbols eve though the asymptotic scalig law is the same for both cases with a icreasig umber of receive ateas. Specifically, cosider a receiver with ateas which measures the eergies of three cosecutive symbols, i.e., { y () 2, y(2) 2, y(3) 2 } (8) where y (l) C is the received sigal at the l slot such that y (l) = H (l) p l + ν with H (l) C ad p l is the trasmitted power, such that {p, p 2, p 3 } comes from some codebook C (3). Cosider the case of σ 2, i.e., the oise power is much greater tha the average sigal power P =. I this case, usig Lemma 3 it follows that the error expoet of each costellatio is approximately idepedet of the trasmitted power, i.e., the rate fuctio would deped oly o the miimum distace betwee the eighborig costellatio poits, which are ow poits i the R T space, ad thus a miimum-distace-based costellatio desig would be asymptotically optimal. For o space time codig where the user trasmits oe bit of iformatio, d mi = 2, sice the costellatio poits (p, p 2, p 3 ) ca take oly 2 N = 8 possible values, i.e., C = { {0, 0, 0}, {0, 0, 2}, {0, 2, 0}, {2, 0, 0}, {0, 2, 2}, {2, 0, 2}, {2, 2, 0}, {2, 2, 2} }, for which, the miimum Euclidea distace is d mi = 2. Usig a differet code, however, oe ca achieve a miimum distace of d mi = 2.8 with the same average power. This costellatio is as follows. C ={{0, α, α}, {0, 0, 0}, {0, 0, α}, {α, 0, α}, {0, 0, 2α}, {0, α, 0}, {α, 0, 0}, {α, α, 0}}, where α = 2.8. Thus, for σ, codig helps to icrease the Euclidea distace betwee costellatio poits ad icrease the rate fuctio. Note that the latter may ot be the best performace that ca be achieved with a block code of legth three ad may be improved upo further by searchig more exhaustively over costellatio poits. The empirical SER performace of the above codes are show i Fig C C Number of ateas x 0 4 Fig. 7: Compariso of probability of symbol error betwee C ad C. I [27] prelimiary results are preseted which demostrate i more detail such gais by costructig either a specific codebook or applyig radom codig bouds. VIII. CONCLUSIONS AND FUTURE WORK We cosider a ocoheret average-eergy-based commuicatio scheme for the massive SIMO MAC. Usig a oe-shot trasmissio ad decodig scheme, we characterize the symbol error rate performace of the proposed system for geeral fadig ad oise statistics. Our aalysis shows that i terms of the scalig law of achievable rates with a icreasig umber of receive ateas, the performace of this scheme comes arbitrarily close to that of a coheret system exploitig istataeous chael state iformatio ad codig over large coherece times ad blocklegths. Moreover, we preset a simple costellatio desig scheme based o maximizig the miimum distace - a aalytically-tractable sufficiet coditio to guaratee vaishig probability of error with a icreasig umber of ateas.

12 The achievable schemes preseted i this work suggest that the spatial diversity already preset i a multiple atea system ca ot oly help us desig simple systems, but ca also help us achieve close to optimal performace. I particular, we show that this holds eve for a symbol-by-symbol commuicatio system. However, i geeral, takig ito accout multiple time slots, fiite SNR, or detectors beyod average eergy detectors will help the ocoheret performace. A comprehesive theory aalyzig whe to use ocoheret commuicatio over coheret commuicatio is the ultimate goal of this lie of research. While coheret commuicatio uderlies may moder commuicatio system desigs, the icreased overhead/difficulty of chael state acquisitio together with the icreased spatial diversity i future systems may make ocoheret commuicatio a competitive paradigm i the ot-too-distat future. IX. ACKNOWLEDGEMENTS The authors would like to thak the editor ad the aoymous reviewers for their thoughtful commets which helped to greatly improve the presetatio i the paper. The first author would like to thak Yair Yoa for helpful discussios. The authors would also like to thak Jiyua Che for isightful discussios cocerig the use of time-divisio schemes i multiuser systems. for large eough. Cosider the θ such that I(a) = θ a log ( E [ e θ U ]). We ca the defie a chage of measure dµ (u) = eθ u E [e θ U ] dµ(u), where µ(u) is the origial measure (e.g., iduced by the defiitio (5)). Uder this chage of measure, ote that udµ (u) = a (this follows from the sufficiet coditio for optimality of θ ). Thus if µ (ad µ ) represets the distributio of ui uder µ (ad µ respectively), the we have that µ (A) = x A ( [ ] ) E e θ U e θ x dµ (x), where A = {x x a}. A lower boud o µ (A) ca the be had by cosiderig, for some δ > 0, A δ = {x a x a + δ}. We get that µ (A) µ (A δ ) ( E Notig that by the CLT, µ (A δ ) 2 get that [ e θ U ] e θ (a+δ) ) µ (A δ ). log(µ (A)) lim I(a) θ δ. By choosig a δ small eough we get that ( Pr u ) i a e (I(a)), for large eough. as, we APPENDIX A I this appedix, we outlie a proof for the large deviatio bouds stated i Lemma 2. By a applicatio of the Markov iequality, we have that ( Pr u ) ( ) i a = Pr e θ u i e θa = [ E for some θ > 0 ] e θ u i e θa E [e θ u i e θa e I(a), ( [ ])) where I(a) sup θ >0 θ (E a log e θu. We ow establish that, for all ɛ > 0, ( Pr u ) i a e (I(a)+ɛ), ] APPENDIX B I this appedix, we show how the ML decoder metioed i Subsectio II-B2 reduces to the eergy-based decoder i Subsectio II-B uder the stated assumptios o the Gaussia statistics of the chael. We first ote that give a poit x C m, the output of the chael is distributed as y = Hx + ν, ad has a desity fuctio of y 2 (π( x 2 + σ 2 )) e y /( x 2 +σ 2). Thus the log-likelihood fuctio L(x) for zero mea Gaussia chael ad oise is as follows: L(x) = w (x) y 2 + w 2 (x), for appropriate fuctios w (x) ad w 2 (x). Thus the ocoheret ML decoder correspods to specifyig decisio regios for the sufficiet statistic y 2 ; which is precisely the average eergy-based decoder. 2

13 APPENDIX C Low a asymptotics of I x (a) I this appedix, we prove the properties of the rate fuctio metioed i Lemma 3. We first show mootoicity ad positivity for all a > 0. Mootoicity with a We ote that for a > a 2, θa log ( E [ e θu ]) > θa 2 log(e [ e θu ] ). Thus ( sup θa log ( E [ e θu ])) ( > sup θa2 log ( E [ e θu ])), θ θ or I x (a ) > I x (a 2 ). Note also that for all a > 0, I x (a) I x (0) = 0. Thus strict mootoicity ad positivity of I x (a) for a 0 is established. Mootoicity with x for x C We first observe that I x (a) is a strictly covex fuctio. This follows from the fact that supremum (over θ) of covex fuctios g(θ) = θa log ( E [ e ]) θu is strictly covex ad that I x (a) is strictly mootoic. Moreover, by the existece of E [ e ] θu, ote that g(θ) is differetiable, at least where E[e θu ] exists, ad is fiite. By differetiatig, we write dow a ecessary coditio for the θ which maximizes g(θ). a E [ e θ U ] = E[Ue θ U ]. We ow characterize the depedece of θ o a. By takig partial derivatives with respect to a o both sides we get that [ θ a = E ] e θ U E [U 2 e θ U ] E [Ue θ U ]. (2) We ow ote that at a = 0, θ = 0, ad that the deomiator i the right had side of (2) is a fiite costat ad positive. This suggests that as a 0, θ a = s, for some costat s > 0. We ca fid out the costat s by otig that For x, x 2 C such that x 2 > x 2 2, we show that E [ e θu ] is larger for x tha for x 2. This establishes the mootoicity claimed i the theorem. We start by writig out U i terms of x, as was doe i (5), ad simplifyig for m = : u i = H i, x + ν i 2 x 2 σ 2. We coditio o the distributio of H i, ad observe that E [ e θui] σ 2 H i, 2 x 2 θ = E Hi, e x 2 e 2θσ 2 2θσ 2 (9) for k 2θσ 2 >. (k H i, 2 ) x 2 θ 2θσ 2 = E Hi, e e σ2 2θσ 2, (20) We ow observe [ that ] for ay radom ] variable y, with ] E [y] > 0, E e x 2 y = E [e x 2 y + + E [e x 2 y is a icreasig fuctio of x 2, where y + = max(y, 0), y = max(0, y). This may be see by computig the derivative with respect to x 2 ad observig that the derivative is positive, i.e., ] ] E [y + e x 2 y + E [y e x 2 y > E[y] > 0. Usig this together with the fact that y = k H i, 2, we get our result. θ a a=0 = s = E [U 2 ]. (22) From this we observe that θ a log ( E [ e ]) θ U lim a 0 a 2 = 2 E [U 2 ]. Low SNR behavior of the rate fuctio We start from the defiitio of the rate fuctio ( ( [ I x (a) = sup θa log E e θu ])), θ>0 where 2 u i = H i,j c j + ν i r(c), j Note that if ν i N (0, σ 2 ), we have the followig equivalet defiitio for I x : I x (a) = sup (θ a ( ])) θ>0 σ 2 log E [e θ Uσ 2 The latter ca be thought of as a rate fuctio Ĩc for the radom variable Ũ = U/σ2 evaluated at a/σ 2. By usig the results of the previous subsectio, we observe that lim Ĩc(a/σ 2 ) σ σ4 a 2 = lim σ ] = k, E [Ũ 2 3

14 for some 0 < k < idepedet of c if c <. This establishes the claim that for a large eough σ 2, i.e. i the low SNR regime, the rate fuctios are idetical, i.e. idepedet of c. APPENDIX D I this appedix, we show how the optimizatio problem (2) for 2 users ca be solved efficietly for certai chael statistics by suitably restrictig the search space of possible costellatio poits i C. To be specific, without loss of geerality, assume that c i,j c k,j, k > i, j [m], that is, there exists a partial orderig of the trasmit power levels of each user. This leads to r(c i,j ) r(c k,j ), k > i, j [m]. (23) Comig up with a total orderig of {r(x)} x C such that it is cosistet with (23), would lead to the best solutio of (2) which satisfies the chose orderig. Let s reparametrize (2) as follows: p,i = c i,, p i,2 = c i,2, s k,l = r([p 2 k,, p 2 l,2]) = p 2 k, + p 2 l,2 + 2p k, p l,2 µ µ 2. (24) By expadig the partial order to a total orderig we formulate the followig (possibly o covex) quadratic problem, where the total orderig iformatio is used i the first costrait. maximize {p i,j} i [Lj ],j [2] {s k,l } k [L ],l [L 2 ] t (25) s.t. s k,l + t < s k2,l 2, if s k,l < s k2,l 2, s k,l = p 2 k, + p 2 l,2 + 2p k, p l,2 µ µ 2, L j p 2 i,j, ad p i,j 0 i, j. L j ) Special Cases: No LOS, Oly LOS: I the case of zero LOS i ay chael, i.e., µ = 0 or µ 2 = 0, (25) becomes the followig liear program maximize {p 2 i,j } i [L j ],j [2] t (26) s.t. p 2 k, + p 2 l,2 + t < p 2 k 2, + p 2 l 2,2, if p 2 k, + p 2 l,2 < p 2 k 2, + p 2 l 2,2, (27) L j p 2 i,j, i, j [2]. L j I the case of oly LOS i both chaels, i.e., µ = µ 2 =, (25) reduces to the covex quadratic program maximize {p i,j} i [Lj ],j [2] t s.t. (p k, + p l,2) 2 + t < (p k2, + p l2,2) 2, if (p k, + p l,2) 2 < (p k2, + p l2,2) 2, (28) L j p 2 i,j, ad p i,j 0, i, j [2]. L j Therefore, i these special cases a way to approach the problem is to eumerate all possible total orderigs that agree with the iitial partial orderig (23) (referred to as liear extesios [28]) ad keep the solutio that gives the largest objective fuctio. However, sice the problem of geeratig the set of liear extesios is a NP-hard problem, we were able to idetify the optimal total orderig oly for small costellatio sizes. For istace, cosider the case of µ = µ 2 = 0 ad L = 3, L 2 = 4. After eumeratig all possible total orderigs that agree with (23), two total orderigs which lead to the best value for the objective fuctio of (26) are depicted i Fig. 8. Optimal orderigs have bee idetified through exhaustive search for all L 4, L 2 4. Fig. 9 depicts the miimum distace achieved for the total orderigs depicted i Fig. 8, where we compare it with a upper boud o the miimum 4 distace, dmi = L L 2, obtaied by cosiderig a sigle super-user with L L 2 costellatio poits ad the same total power costrait. s,$ s,2$ s,3$ s,4$ s 2,$ s 2,2$ s 2,3$ s 2,4$ s 3,$ s 3,2$ s 3,3$ s 3,4$ s,$ s,2$ s,3$ s,4$ s 2,$ s 2,2$ s 2,3$ s 2,4$ s 3,$ s 3,2$ s 3,3$ s 3,4$ Fig. 8: A arrow from s k,l to s k2,l 2 implies s k,l s k2,l 2. The two total orderigs i gree lead to the same optimal costellatio desig for L = 3, L 2 = 4, µ = µ 2 = 0. Iitial partial orderig is i red. 2) Geeral Case: I the geeral case, imposig a total orderig does ot covexify the problem. Yet, it is still possible to solve the o covex quadratic program (25) with a startig poit that results from the o LOS or oly LOS solutio to fid a local maximum. This approach has bee used i the umerical results i Sectio VII. 4

15 log 0 (dmi) 0 2 L =6 L = L 2 Achievable d mi Fig. 9: The miimum distace achieved by the two orderigs show i Fig. 8 for differet costellatio sizes. APPENDIX E I this appedix, we show how to solve a istace of the desig problem outlied i Subsectio VII-D for the case of Ricia statistics. We show i particular that collectig phase iformatio of the received sigals ca play a major role i improvig the SER performace or brigig dow the umber of ateas required to achieved a certai SER. The critical observatio is the fact that for Rayleigh fadig ad AWGN chaels, y 2 is a sufficiet statistic for decodig x from y. For geeral statistics with equiprobable sigallig, a likelihood detector will outperform this detector. This likelihood detector for Ricia fadig looks like y j argmax µ jx j 2 x σ 2 + j σ2 j x j 2 + log π(σ 2 + j σj 2 x j 2 ) We plot the performace with this detector i Subsectio VII-D. APPENDIX F I this appedix, we show how a time-divisio scheme will always outperform a joit multiuser codebook desig for symmetric Rayleigh fadig, i.e. Rayleigh fadig with the same average power for all users. We cosider the followig two desig problems: A joit multiuser codebook desig with a symmetric rate of log 2 (L) m per user. A sigle user costellatio desig with a rate of log 2 (L). The idea is that the sigle user costellatio ca be used i a TDMA scheme over m time slots, ad we ivestigate whether it is advatageous to perform a joit multiuser desig. As described i (9), the sigle user desig problem is maximize C L L ci, 2 mi r(x ) r(x 2 ). x,x 2 C,x x 2 The joit multiuser desig problem (similar to the problem (2)) is maximize C,{p j} j [m] L /m L /m c i,j 2 p j, j [m] j pj p j 0 mi r(x ) r(x 2 ). x,x 2 C,x x 2 Note that the desig is over C which is a collectio of L m dimesioal costellatio poits. For ay feasible C, we ca defie a C = { x x C}. We observe that for such a costellatio, x 2. x C Thus the set of feasible r( x) i the multiuser desig problem is a subset of the achievable r(x) for the sigle user case. Thus the objective attaiable usig a sigle user desig with TDMA ca be o worse tha a joit multiuser desig with the same costraits. I other words, i Rayleigh fadig with symmetric average power, for the same average rate, the time-divisio scheme will always have a error expoet which is at least the same as that of the joit multiuser desig. Ituitively, this is due to the fact that, from a error expoet poit of view, the joit desig problem is more costraied tha the sigle user desig problem for symmetric Rayleigh fadig chaels. REFERENCES [] T. L. Marzetta, Nocooperative cellular wireless with ulimited umbers of base statio ateas, IEEE Trasactios o Wireless Commuicatios, vol. 9, o., pp , 200. [2] K. T. Truog ad R. W. Heath, Effects of chael agig i massive MIMO systems, Joural of Commuicatios ad Networks, vol. 5, o. 4, pp , 203. [3] A. Pitarokoilis, S. K. Mohammed, ad E. G. Larsso, Effect of oscillator phase oise o uplik performace of large MU-MIMO systems, i 50th Aual Allerto Coferece o Commuicatio, Cotrol, ad Computig (Allerto), 202. IEEE, 202, pp [4] V. I. Barousis, M. A. Sedaghat, R. R. Müller, ad C. B. Papadias, Massive atea arrays with low frot-ed hardware complexity: A eablig techology for the emergig small cell ad distributed etwork architectures, arxiv preprit arxiv: ,

16 [5] A. Maolakos, M. Chowdhury, ad A. J. Goldsmith, Costellatio desig i a eergy-based ocoheret massive SIMO system, Submitted to IEEE Trasactios o Wireless Commuicatios, 205. [6] A. Goldsmith, Wireless commuicatios. Cambridge Uiversity Press, [7] E. G. Larsso, O. Edfors, F. Tufvesso, ad T. L. Marzetta, Massive MIMO for ext geeratio wireless systems, arxiv preprit arxiv: , 203. [8] D. Warrier ad U. Madhow, Nocoheret commuicatio i space ad time, 999. [9] M. L. McCloud, M. Brehler, ad M. K. Varaasi, Sigal costellatios for ocoheret space-time commuicatios. [0] A. Barg ad D. Y. Nogi, Bouds o packigs of spheres i the Grassma maifold, IEEE Trasactios o Iformatio Theory, vol. 48, o. 9, pp , [] L. Zheg ad D. N. C. Tse, Commuicatio o the Grassma maifold: A geometric approach to the ocoheret multipleatea chael, IEEE Trasactios o Iformatio Theory, vol. 48, o. 2, pp , [2] B. M. Hochwald ad T. L. Marzetta, Uitary space-time modulatio for multiple-atea commuicatios i rayleigh flat fadig, IEEE Trasactios o Iformatio Theory, vol. 46, o. 2, pp , [3] W. Yag, G. Durisi, ad E. Riegler, O the capacity of large- MIMO block-fadig chaels, arxiv preprit arxiv: , 202. [4] S. Shamai ad T. L. Marzetta, Multiuser capacity i block fadig with o chael state iformatio, IEEE Trasactios o Iformatio Theory, vol. 48, o. 4, pp , [5] S. Murugesa, E. Uysal-Biyikoglu, ad P. Schiter, Optimizatio of traiig ad schedulig i the o-coheret SIMO multiple access chael, IEEE Joural o Selected Areas i Commuicatios, vol. 25, o. 7, pp , [6] A. Lapidoth ad S. M. Moser, The fadig umber of SIMO fadig chaels with memory, i Proceedigs of IEEE Iteratioal Symposium o Iformatio Theory ad its Applicatios (ISITA), Parma, Italy, 2004, pp [7] T. Koch ad A. Lapidoth, The fadig umber ad degrees of freedom i o-coheret mimo fadig chaels: A peace pipe, i Proceedigs of Iteratioal Symposium o Iformatio Theory, IEEE, 2005, pp [8] A. M. Segupta ad P. P. Mitra, Capacity of multivariate chaels with multiplicative oise: I. radom matrix techiques ad large- expasios for full trasfer matrices, arxiv preprit physics/00008, [9] À. Oliveras Martíez, E. De Carvalho, P. Popovski, ad G. F. Pederse, Eergy detectio usig very large atea array receivers, i th Asilomar Coferece o Sigals, Systems ad Computers, pp [20] N. Jidal ad A. Lozao, A uified treatmet of optimum pilot overhead i multipath fadig chaels, IEEE Trasactios o Commuicatios, vol. 58, o. 0, pp , 200. [2] F. Rusek, A. Lozao, ad N. Jidal, Mutual iformatio of IID complex gaussia sigals o block Rayleigh-faded chaels, IEEE Trasactios o Iformatio Theory, vol. 58, o., pp , 202. [22] I. C. Abou-Faycal, M. D. Trott, ad S. Shamai, The capacity of discrete-time memoryless rayleigh-fadig chaels, IEEE Trasactios o Iformatio Theory, vol. 47, o. 4, pp , 200. [23] M. C. Gursoy, H. V. Poor, ad S. Verdú, The ocoheret Ricia fadig chael-part I: structure of the capacity-achievig iput, IEEE Trasactios o Wireless Commuicatios, vol. 4, o. 5, pp , [24] M. Chowdhury, A. Maolakos, ad A. J. Goldsmith, Desig ad performace of ocoheret massive SIMO systems, i th Aual Coferece o Iformatio Scieces ad Systems (CISS). IEEE, 204, pp. 6. [25] A. Dembo et al., Large deviatios techiques ad applicatios. Spriger, 200, vol. 38. [26] A. Maolakos, M. Chowdhury, ad A. J. Goldsmith, Costellatio desig i ocoheret massive SIMO systems, i IEEE Global Telecommuicatios Coferece (GLOBECOM), 204. [27] B. Kott, A. Maolakos, M. Chowdhury, ad A. J. Goldsmith, Beefits of codig i a ocoheret massive SIMO system, To appear i Proceedigs of IEEE ICC, 205. [28] G. Pruesse ad F. Ruskey, Geeratig Liear Extesios Fast, SIAM Joural o Computig, vol. 23, o. 2, pp , 994. MAINAK CHOWDHURY is pursuig a Ph.D. degree i the Wireless Systems Laboratory at Staford Uiversity. He received a Bachelor of Techology i Electrical Egieerig from the Idia Istitute of Techology, Kapur, Idia (20), ad a Master of Sciece i Electrical Egieerig from Staford Uiversity (203). His research focuses o commuicatio systems with a large umber of trasmitters or receivers. ALEXANDROS MANOLAKOS received i 205 a Ph.D. degree i the Electrical Egieerig departmet at Staford Uiversity uder the supervisio of Prof. Adrea Goldsmith. He was the recipiet of the Arvaitidis Staford Graduate fellowship (SGF) i Memory of William K. Livill. I 200, he received with Hoors a Diploma degree i Electrical ad Computer Egieerig from the Natioal Techical Uiversity of Athes i Greece ad i 202 a Master of Sciece i Electrical Egieerig from Staford Uiversity. I 202, he was awarded the Roberto Padovai Scholarship from Corporate R&D, Qualcomm, Sa Diego. He is curretly a seior egieer at Qualcomm Corporate R&D i Sa Diego workig with the fifth geeratio (5G) cellular team. His research iterests iclude massive MIMO wireless systems, iterferece mitigatio i wireless commuicatios ad statistical sigal processig. 6

17 ANDREA GOLDSMITH (S 90-M 93- SM 99-F 05) is the Stephe Harris professor i the School of Egieerig ad a professor of Electrical Egieerig at Staford Uiversity. She was previously o the faculty of Electrical Egieerig at Caltech. She co-fouded ad serves as CTO of Accelera, Ic., which develops softwaredefied wireless etwork techology, ad previously co-fouded ad served as CTO of Quatea Commuicatios Ic., which develops high-performace WiFi chipsets. She has previously held idustry positios at Maxim Techologies, Memorylik Corporatio, ad AT&T Bell Laboratories. Dr. Goldsmith is a Fellow of the IEEE ad of Staford, ad she has received several awards for her work, icludig the IEEE Commuicatios Society ad Iformatio Theory Society joit paper award, the IEEE Commuicatios Society Best Tutorial Paper Award, the Natioal Academy of Egieerig Gilbreth Lecture Award, the IEEE Wireless Commuicatios Techical Committee Recogitio Award, the Alfred P. Sloa Fellowship, ad the Silico Valley/Sa Jose Busiess Jourals Wome of Ifluece Award. She is author of the book Wireless Commuicatios ad co-author of the books MIMO Wireless Commuicatios ad Priciples of Cogitive Radio, all published by Cambridge Uiversity Press. She received the B.S., M.S. ad Ph.D. degrees i Electrical Egieerig from U.C. Berkeley. Dr. Goldsmith is curretly o the Steerig Committee for the IEEE Trasactios o Wireless Commuicatios, ad has previously served as editor for the IEEE Trasactios o Iformatio Theory, the Joural o Foudatios ad Treds i Commuicatios ad Iformatio Theory ad i Networks, the IEEE Trasactios o Commuicatios, ad the IEEE Wireless Commuicatios Magazie. Dr. Goldsmith participates actively i committees ad coferece orgaizatio for the IEEE Iformatio Theory ad Commuicatios Societies ad has served o the Board of Goverors for both societies. She has bee a Distiguished Lecturer for both societies, served as the Presidet of the IEEE Iformatio Theory Society i 2009, fouded ad chaired the studet committee of the IEEE Iformatio Theory society, ad curretly chairs the Emergig Techology Committee ad is a member of the Strategic Plaig Committee i the IEEE Commuicatios Society. At Staford she received the iaugural Uiversity Postdoc Metorig Award, served as Chair of its Faculty Seate, ad curretly serves o its Faculty Seate ad o its Budget Group. 7