Throughput Analysis of Massive MIMO Uplink with Low-Resolution ADCs

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1 TO APPEA IN IEEE TANSACTIONS ON WIELESS COMMUNICATIONS 1 Throughput Aalysis of Massive MIMO Uplik with Low-esolutio ADCs Sve Jacobsso, Studet Member, IEEE, Giuseppe Durisi, Seior Member, IEEE, Mikael Coldrey, Member, IEEE, Ulf Gustavsso, ad Christoph Studer, Seior Member, IEEE arxiv: v3 [cs.it] 3 Apr 17 Abstract We ivestigate the uplik throughput achievable by a multiple-user MU massive multiple-iput multiple-output MIMO system i which the base statio is equipped with a large umber of low-resolutio aalog-to-digital coverters ADCs. Our focus is o the case where either the trasmitter or the receiver have ay a priori chael state iformatio. This implies that the fadig realizatios have to be leared through pilot trasmissio followed by chael estimatio at the receiver, based o coarsely quatized observatios. We propose a ovel chael estimator, based o Bussgag s decompositio, ad a ovel approximatio to the rate achievable with fiite-resolutio ADCs, both for the case of fiite-cardiality costellatios ad of Gaussia iputs, that is accurate for a broad rage of system parameters. Through umerical results, we illustrate that, for the 1-bit quatized case, pilot-based chael estimatio together with maximal-ratio combig or zero-forcig detectio eables reliable multi-user commuicatio with high-order costellatios i spite of the severe oliearity itroduced by the ADCs. Furthermore, we show that the rate achievable i the ifiite-resolutio o quatizatio case ca be approached usig ADCs with oly a few bits of resolutio. We fially ivestigate the robustess of low-adc-resolutio MU-MIMO uplik agaist receive power imbalaces betwee the differet users, caused for example by imperfect power cotrol. Idex Terms Aalog-to-digital coverter ADC, chael capacity, liear miimum mea square error LMMSE chael estimatio, low-resolutio quatizatio, multi-user massive multiple-iput multiple-output MIMO. I. INTODUCTION Massive multiple-iput multiple-output MIMO is a promisig multi-user MU MIMO techology for ext geeratio cellular commuicatio systems 5G []. With massive MIMO, the umber of ateas at the base statio BS is scaled up by several orders of magitude compared to traditioal multiatea systems with the goals of eablig sigificat gais i capacity ad eergy efficiecy [], [3]. Icreasig the umber of S. Jacobsso is with Ericsso esearch ad Chalmers Uiversity of Techology, Gotheburg, Swede sve.jacobsso@ericsso.com G. Durisi is with Chalmers Uiversity of Techology, Gotheburg, Swede durisi@chalmers.se M. Coldrey ad U. Gustavsso are with Ericsso esearch, Gotheburg, Swede {mikael.coldrey,ulf.gustavsso}@ericsso.com C. Studer is with Corell Uiversity, Ithaca, NY studer@corell.edu The work of S. Jacobsso ad G. Durisi was supported i part by the Swedish Foudatio for Strategic esearch uder grat ID1-, ad by the Swedish Govermet Agecy for Iovatio Systems VINNOVA withi the competece ceter ChaseO. The work of C. Studer was supported i part by Xilix Ic., ad by the US Natioal Sciece Foudatio NSF uder grats ECCS-186, CCF , ad CAEE CCF The material i this paper was preseted i part at the IEEE Iteratioal Coferece o Commuicatios ICC Workshop o 5G ad Beyod: Eablig Techologies ad Applicatios, Lodo, U.K., Jue 15 [1]. BS atea elemets leads to high spatial resolutio; this makes it possible to simultaeously serve several user equipmets UEs i the same time-frequecy resource, which brigs large capacity gais. The improvemets i terms of radiated eergy efficiecy are eabled by the array gai that is provided by the large umber of ateas. Equippig the BS with a large umber of atea elemets, however, icreases cosiderably the hardware cost ad the power cosumptio of the radio-frequecy F circuits []. This calls for the use of low-cost ad power-efficiet hardware compoets, which, however, reduce the sigal quality due to a icreased level of impairmets. The aggregate impact of hardware impairmets o massive MIMO systems has bee ivestigated i, e.g., [5] [8], where it is foud that massive MIMO provides to a certai extet robustess agaist sigal distortios caused by low-cost F compoets. However, most of these aalyses rely o the assumptio that the distortio caused by the hardware imperfectios ca be modeled as a additive Gaussia radom variable that is idepedet of the trasmit sigal. It is prima facie uclear how accurate such modelig assumptio is, especially for the distortio caused by low-resolutio aalog-to-digital coverters ADCs. This has bee oted i [8, Sec. IV.A] where it is poited out that such modelig assumptio targets ADCs with high resolutio. A. Quatized Massive MIMO I this paper, we cosider a uplik massive MU-MIMO system ad focus o the sigal distortio caused by the use of low-resolutio ADCs at the BS. A ADC with samplig rate f s Hz ad a resolutio of b bits maps each sample of the cotiuous-time, cotiuous-amplitude basebad received sigal to oe out of b quatizatio labels, by operatig f s b coversio steps per secod. I moder high-speed ADCs e.g., with samplig rates larger tha 1 GS/s, the dissipated power scales expoetially i the umber of bits ad liearly i the samplig rate [9], [1]. This implies that for widebad massive MIMO systems where hudreds of high-speed coverters are required, the resolutio of the ADCs may have to be kept low i order to maitai the power cosumed at the BS withi acceptable levels. A additioal motivatio for reducig the ADC resolutio is to limit the amout of data that has to be trasferred over the lik that coects the F compoets ad the basebadprocessig uit. For example, cosider a BS that is equipped with a atea array of 5 elemets. At each atea elemet, the i-phase ad quadrature samples are quatized separately

2 TO APPEA IN IEEE TANSACTIONS ON WIELESS COMMUNICATIONS usig a pair of 1-bit ADCs operatig at 1 GS/s. Such a system would produce 1 Tbit/s of data. This exceeds by far the rate supported by the commo public radio iterface CPI used over today s fiber-optical frothaul liks [11]. Alleviatig this capacity bottleeck is of particular importace i a cloud radio access etwork C-AN architecture [1], where the basebad processig is migrated from the BSs to a cetralized uit, which may be placed at a sigificat distace from the BS atea array. A implicatio of lowerig the ADC resolutio is that the requiremet o accurate radio-frequecy circuitry ca be relaxed. The reaso is that the quatizatio oise may domiate the oise itroduced by other compoets such as mixers, oscillators, filters, ad low-oise amplifiers. Hece, further power-cosumptio reductios may be achieved by relaxig the quality requiremets o the F circuitry. The 1-bit resolutio case, where the i-phase ad quadrature compoets of the cotiuous-valued received samples are quatized separately usig a pair of 1-bit ADCs, is particularly attractive because of the resultig low hardware complexity [13], [1]. Ideed, a 1-bit ADC ca be realized usig oly a simple comparator. Furthermore, i a 1-bit architecture, there is o eed for automatic gai cotrol circuitry, which is otherwise eeded to match the dyamic rage of the ADCs. B. Previous Work eceivers employig low-resolutio ADCs eed to cope with the severe oliearity itroduced by the coarse quatizatio, which may reder sigalig schemes ad receiver algorithms developed for the case of high-resolutio ADCs suboptimal. The impact of the 1-bit ADC oliearity o the performace of commuicatio systems has bee previously studied i the literature uder various chael-model assumptios. I [15], it is prove that BPSK is capacity achievig over a real-valued ofadig sigle-iput sigle-output SISO Gaussia chael. For the complex-valued Gaussia chael, QPSK is optimal. These results hold uder the assumptio that the 1-bit quatizer is a zero-threshold comparator. It turs out that i the low-sn regime, a zero-threshold comparator is ot optimal [16]. The optimal strategy ivolves the use of flashsigalig [17, Def. ] ad requires a optimizatio over the threshold value. Ufortuately, the power gai obtaiable usig this optimal strategy maifests itself oly at extremely low values of spectral efficiecy. For the ayleigh-fadig case, uder the assumptio that the receiver has access to perfect chael state iformatio CSI, it is show i [18] that QPSK is capacity achievig agai for the SISO case. The assumptio that perfect CSI is available may, however, be urealistic i the 1-bit quatized case, sice the oliear distortio caused by the 1-bit ADCs makes chael estimatio challegig. I particular, if the fadig process evolves rapidly, the cost of trasmittig traiig symbols caot be eglected. For the more practically relevat case whe the chael is ot kow a priori to the receiver, but must be leared for example, via pilot symbols, QPSK is optimal whe the SN exceeds a certai threshold that depeds o the coherece time of the fadig process [19]. For SN values that are below this threshold, o-off QPSK is capacity achievig [19]. For the 1-bit quatized MIMO case, the capacity-achievig distributio is ukow. I [], it is show that QPSK is optimal at low SN, agai uder the assumptio of perfect CSI at the receiver. Mo ad Heath Jr. [1] derived high-sn bouds o capacity, ad showed that high-order modulatios are supported. However, their aalysis relies o the assumptio that the trasmitter has access to perfect CSI, which is urealistic i low-resolutio architectures. Their cotributio leaves ope the questio o whether high-order modulatios are supported i traiig based schemes where the receiver has partial kowledge of the chael ad the trasmitter i our case, the UE has o chael kowledge. Chael estimatio o the basis of quatized observatios is cosidered i, e.g., [], [3] see also [] for a compressivesesig versio of this problem. A closed-form solutio for the maximum likelihood ML estimate i the 1-bit case is derived i [3], uder the assumptio of time-multiplexed pilots. The use of 1-bit ADCs i massive MIMO was cosidered i [5]. There, the authors examied the achievable uplik throughput for the sceario where the UEs trasmit QPSK symbols, ad the BS employs a least squares LS chael estimator, followed by a maximal ratio combiig MC or zero-forcig ZF detector. Their results show that large sum-rate throughputs ca be achieved despite the coarse quatizatio. The results i [5] were exteded to high-order modulatios e.g., 16-QAM by the authors of this paper i [1]. There, we showed that oe ca detect ot oly the phase, but also the amplitude of the trasmitted sigal, provided that the umber of BS ateas is sufficietly large, hece, aswerig positively the questio left ope i [1]. Choi et al. [6] recetly developed a detector ad a chael estimator capable of supportig high-order costellatios such as 16-QAM. Agai for the case of 1-bit ADCs, Li et al. [7], [8] proposed a liear miimum mea square error LMMSE chael estimator based o Bussgag s decompositio that was show to be superior to the oe proposed i [6]. Furthermore, they derive a approximatio o the rates achievable with Gaussia iputs. The accuracy of this approximatio is ot fully validated i [8], sice o compariso with actual achievable rates is provided. We et al. [9] proposed a joit chael- ad data-estimatio algorithm that offers sigificat improvemet compared to the case whe chael estimatio ad data detectio are treated separately. However, as oted i [9], the complexity of the proposed algorithm is too high for practical implemetatios. A mixed-adc architecture where may 1-bit ADCs are complemeted with few high-resolutio ADCs is proposed i [3]. It is foud that the additio of a relatively small umber of high-resolutio ADCs icreases the system performace sigificatly. Specifically, the authors of [3] preset a achievability boud uder Gaussia sigalig ad miimum distace decodig that holds for the setup where chael estimates are acquired through the high-resolutio ADCs. This relies o the assumptio that each high-resolutio ADC ca be liked to several F chais through a switch. The disadvatage of such architecture is that ADC switches icrease hardware complexity. Furthermore, the time eeded to acquire chael

3 S. JACOBSSON, G. DUISI, M. COLDEY, U. GUSTAVSSON, AND C. STUDE 3 estimates icreases dramatically. I all of the cotributios reviewed so far, low-resolutio quatized massive MIMO systems have bee ivestigated solely for commuicatio over frequecy-flat, arrowbad, chaels. A spatial-modulatio-based massive MIMO system over a frequecy-selective chael was studied i [31]. The proposed receiver employs LS estimatio followed by a message-passig-based detector. The performace of a lowresolutio quatized massive MIMO system usig orthogoal frequecy divisio multiplexig OFDM ad operatig over a widebad chael was ivestigated i [3]. There, it is foud that usig ADCs with oly to 6 bits resolutio is sufficiet to achieve performace close to the ifiite-resolutio i.e., o quatizatio case, at o additioal cost i terms of digital sigal processig complexity. A capacity lower boud for widebad chaels ad 1-bit ADCs has bee recetly reported i [33]. The aalysis i [33] relies o the same sigal decompositio used i [7], [8] for the frequecyflat case. However, differetly from [7], [8], the temporal correlatio of the quatizatio oise i the chael-estimatio phase is igored. All the results reviewed so far hold uder the assumptio of Nyquist-rate samplig at the receiver. It is worth poitig out that Nyquist-rate samplig is ot optimal i the presece of quatizatio at the receiver [3] [36]. For example, for the 1-bit quatized complex AWGN chael, high-order costellatios such as 16-QAM ca be supported eve i the SISO case, if oe allows for oversamplig at the receiver [37]. C. Cotributios Focusig o Nyquist-rate samplig, ad o the sceario where either the trasmitter or the receiver have a priori CSI, we ivestigate the rates achievable over a frequecy-flat ayleigh block-fadig MU-MIMO chael, whe the receiver is equipped with low-resolutio ADCs. Our cotributios are summarized as follows: We propose a ovel chael estimator for the case of multibit ADCs ad ouiform quatizatio regios usig Bussgag s decompositio. This estimator recovers the LMMSE estimator proposed i [7], [8], [33] for the case of 1-bit ADCs. We preset a easy-to-evaluate approximatio o the rates achievable with fiite-cardiality costellatios uder the assumptio of traiig-based chael estimatio. The approximatio is explicit i the umber of pilots used to estimate the chael ad i the resolutio of the ADCs; by comparig it with a umerically computed lower boud o the achievable rates, we show that this approximatio is accurate for a large rage of SN values. We also obtai a closed-form approximatio o the rates achievable with Gaussia iputs that is derived usig Bussgag s decompositio. This approximatio recovers for the 1-bit case the approximatio recetly preseted i [7], [8]. A compariso with a umerically computed lower boud o the achievable rates reveals that, i the 1-bit case, this Gaussia approximatio is accurate at low SN, but overestimates the achievable rate at high SN i the multiuser sceario. Through a umerical study, we determie the miimum ADC resolutio eeded to make the performace gap to the ifiite-resolutio case egligible. Our simulatios suggest that oly few bits e.g., 3 bits are required to achieve a performace close to the ifiite-resolutio case for a large rage of system parameters. This holds also whe the users are received at vastly differet power levels imperfect power cotrol. This paper complemets the aalysis previously reported i [1] by geeralizig it to ZF receivers, to multi-bit quatizatio, ad to the case of imperfect power cotrol. Furthermore, the proposed chael estimator ad the rate approximatios are ovel. D. Notatio Lowercase ad uppercase boldface letters deote colum vectors ad matrices, respectively. The idetity matrix of size N N is deoted by I N. We use tr ad diag to deote the trace ad the mai diagoal of a matrix, ad to deote the l -orm of a vector. The multivariate ormal distributio with mea µ ad covariace Σ is deoted by N µ, Σ. Furthermore, the multivariate complex-valued circularly-symmetric Gaussia probability desity fuctio with zero mea ad covariace Σ is deoted by CN, Σ. The operator E x [ ] stads for the expectatio over the radom variable x. The mutual iformatio betwee two radom variables x ad y is idicated by Ix; y. The real ad imagiary parts of a complex scalar s are {s} ad I{s}. The superscripts T,, ad H deote traspose, complex cojugate, ad Hermitia traspose, respectively. The fuctio Φx is the cumulative distributio fuctio CDF of a stadard ormal radom variable. E. Paper Outlie The rest of the paper is orgaized as follows. I Sectio II, we itroduce the massive MIMO system model ad the chaelestimatio ad data-detectio problems. I Sectio III, we derive a approximatio o the rate achievable with fiitecardiality costellatios ad Gaussia iputs. I Sectio IV, we validate the accuracy of our approximatios for differet scearios ad determie the ADC resolutio required to approach the rate achievable i the ifiite-resolutio case. We coclude i Sectio V. II. CHANNEL ESTIMATION AND DATA DETECTION WITH LOW-ESOLUTION ADCS A. System Model ad Sum-ate Capacity We cosider the sigle-cell uplik system depicted i Fig. 1. Here, K sigle-atea users are served by a BS that is equipped with a array of N > K ateas. We model the subchaels betwee each trasmit-receive atea pair as a ayleigh block-fadig chael see, e.g., [38], i.e., a chael that stays costat for a block of T chael uses, ad chages idepedetly from block to block. We shall refer to T as the chael coherece iterval. We further assume that the subchaels are mutually idepedet. The discrete-time complex basebad received sigal over all ateas withi

4 TO APPEA IN IEEE TANSACTIONS ON WIELESS COMMUNICATIONS UE 1 UE UE K. F F. F F e Im e Im e Im e Im Low-resolutio ADC Low-resolutio ADC Low-resolutio ADC Low-resolutio ADC. Low-resolutio ADC Low-resolutio ADC Low-resolutio ADC Fig. 1. Quatized massive MIMO uplik system model. Low-resolutio ADC a arbitrary coherece iterval ad before quatizatio is modeled as y t = Hx t + w t, t = 1,,..., T. 1 Here, x t C K deotes the chael iput from all users at time t, ad H C N K is the chael matrix coectig the K users to the N BS ateas. The etries of H are idepedet ad CN, 1-distributed. Furthermore, the vector w t C N, whose etries are idepedet ad CN, 1-distributed, stads for the AWGN. Throughout the paper, we cosider the case where CSI is ot available a priori to the trasmitter or to the receiver, i.e., they are both ot aware of the realizatio of H. This sceario captures the cost of learig the fadig chael [39] [1], a operatio that has to be performed usig quatized observatios ad may yield sigificat performace loss i the case of low-resolutio ADCs. We further assume that codig ca be performed over may coherece itervals. Let X = [x 1, x,..., x T ] be the K T matrix of trasmitted sigals withi a coherece iterval, ad let = [r 1, r,..., r T ] be the correspodig N T matrix of received quatized samples. For a give quatizatio fuctio, the ergodic sumrate capacity is [38] Cρ = 1 sup IX;. T Here, the supremum is over all probability distributios o X for which X has idepedet rows ad the followig average power costrait is satisfied: E [ trxx H ] KT ρ. 3 Sice the oise variace is ormalized to oe, we ca thik of ρ as the SN. The sum-rate capacity i is, i geeral, ot kow i closed form, eve i the ifiite-resolutio case, for which tight capacity bouds have bee reported recetly i []. B. Quatizatio of a Complex-Valued Vector The i-phase ad quadrature compoets of the received sigal at each atea are quatized separately by a ADC of b-bit resolutio. We characterize the ADC by a set of b + 1 quatizatio thresholds T b = {τ, τ 1,..., τ b}, such that = τ < τ 1 < < τ b =, ad a set of b quatizatio labels L b = {l, l 1,..., l 1} where l b i τ i, τ i+1 ]. Let b = L b L b. We shall describe the joit operatio of the N b-bit ADCs at the BS by the fuctio Q b : C N N b that maps the received sigal y t with etries {y,t } to the quatized output r t with etries {r,t } i the followig way: if {y,t } τ k, τ k+1 ] ad I{y,t } τ l, τ l+1 ], the r,t = l k + jl l. Usig this covetio, the quatized received sigal ca be writte as r t = Q b y t = Q b Hx t + w t, t = 1,,..., T. Fidig the optimal quatizatio labels, i.e., the oes that miimize the mea square error MSE betwee the oquatized received vector y t ad the quatized vector r t, requires oe to determie the probability desity fuctio PDF of y t. Note that such PDF depeds o the choice of the iput costellatio x t. Sice adaptig the quatizatio labels to the choice of the iput costellatio appears to be impractical from a implemetatio poit of view, i this paper we shall cosider the followig suboptimal choice of the set of quatizatio labels L b ad thresholds T b : we first approximate each etry of the oquatized chael output vector y t by a complex Gaussia radom variable with variace Kρ + 1 ad the use the Lloyd-Max algorithm [3], [] to fid a set of labels L b = { l, l 1,..., l } that miimize the mea square error betwee the oquatized ad the quatized sigal. The, we rescale the labels such that the variace of each etry of r t is Kρ + 1. Specifically, we multiply each label i the set Lb by the factor α = b 1 i= l i Φ τ Kρ + 1 i+1 Kρ+1 Φ τ i Kρ+1 5 to produce the set of labels L b = α L b. Some commets o our choice are i order. The Gaussia approximatio is accurate at low SN or whe the umber of UE is sufficietly large. Whe such coditios are ot fulfilled, it may result i a suboptimal choice of the quatizatio labels. The rescalig of the labels by α i 5 turs out to simplify the performace aalysis see Sectios II-D ad III-C. I the 1-bit case, we ca write the quatized received sigal at the th atea, at discrete time t, as follows: Kρ + 1 Q 1 y,t = sg{y,t } + jsgi{y,t }. 6 Here, sg deotes the sigum fuctio defied as { 1, if x < sgx = +1, if x. C. Sigal Decompositio usig Bussgag s Theorem The quatizatio of a vector usig fiite-resolutio ADCs causes a distortio that is correlated with the iput to the quatizer. Whe the iput to the quatizer is Gaussia, we ca use Bussgag s theorem [5] to decompose the quatized sigal i the coveiet form detailed i the followig theorem. Theorem 1: Let r = Q b y deote the output from a set of ADCs described by the set of labels L b ad the set of 7

5 S. JACOBSSON, G. DUISI, M. COLDEY, U. GUSTAVSSON, AND C. STUDE 5 thresholds T b. Assume that y CN N, K where K C N N. The, the quatized vector r ca be decomposed as r = G b y + d 8 where the quatizatio distortio d ad y are ucorrelated. Furthermore, G b N N is the followig diagoal matrix: 1 G b = diagk 1/ l i exp τ π i diagk 1 i= exp τi+1 diagk 1. 9 Here, l i correspods to the ith elemet of the set of labels L b ad τ i to the ith elemet of the set of thresholds T b. Proof: See Appedix A. Bussgag s theorem has bee used previously i the literature to decompose the quatized sigal i the 1-bit-ADC case see, e.g., [7], [8]. A geeralizatio of this result to the case of multi-bit uiform ADCs has bee recetly proposed i [6] i the cotext of dowlik precodig. The more geeral result i Theorem 1 allows oe to hadle the case of ouiform quatizers as well. For the special case whe diagk = Kρ + 1I N, which will tur out relevat for our aalysis i Sectio II-D, the matrix G b i 9 reduces to with G b = i= G b = G b I N 1 l i πkρ + 1 e τ i Kρ+1 e τ i+1 Kρ Note that i the ifiite-resolutio case b =, it follows from 8 that G = I N ad, hece, G = 1 see 1. For the 1-bit-ADCs case, we have that G 1 = /π, a well-kow result used recetly i [8], [33] to aalyze the throughput achievable with 1-bit ADCs. We shall use the Bussgag decompositio to develop a chael estimator i the ext sectio as well as a approximatio o the rates achievable with Gaussia iputs i Sectio III-C. D. Chael Estimatio A commo approach to trasmittig iformatio over fadig chaels whose realizatios are ot kow a priori to the receiver is to reserve a certai umber of time slots i each coherece iterval for the trasmissio of pilot symbols. These pilots are the used at the receiver to estimate the fadig chael. Assume that P pilot symbols are used i each coherece iterval K P T. We shall assume that the pilot sequeces used by differet UEs are pairwise orthogoal, i.e., that P x t x H t = P ρ I K. 1 t=1 Let h deote the chael vector whose etries cotai the chael gai betwee the kth UE, k = 1,..., K, ad 1 We use the covetio that the fuctio exp applied to a diagoal matrix acts elemet-wise o its diagoal etries. the th BS atea. Furthermore, let X p = [x 1,..., x P ] T deote the matrix cotaiig the P pilot symbols trasmitted by the K UEs. Fially, let y p = X p h + w p ad r p = Q b y p deote the oquatized ad quatized pilot sequeces received at the th atea durig the traiig phase. Here, w p = [w,1,..., w,p ] T is the additive oise. For the 1-bit case, the LMMSE estimator of h was obtaied i [8]. Proceedig similarly to [8], we geeralize the LMMSE estimator [8] to the multi-bit case. Specifically, let C p y ad C p r be the covariace matrices of y p ad r p, respectively. Usig Bussgag s decompositio 8 recall that both additive oise ad fadig are Gaussia ad the fact that diagc p y = Kρ + 1I P, which follows from 1 ad implies that G b = G b I P see 9, we coclude that the LMMSE estimator for the multi-bit case is ĥ = G b X H p C 1 r p r p. 13 The computatio of 13 requires kowledge of the covariace matrix C p r. For the case of 1-bit ADCs, oe ca compute C p r i closed form, as show i [8]. For the multi-bit case, however, C p r is ot kow i closed form. To overcome this issue, we shall ext preset a alterative chael estimator, which is a approximatio of 13, but admits a simple closed-form expressio. Usig Bussgag s decompositio 8, we write C p r as C r p = G bc p y + C p d = G bx p X H p + G bi P + C p d. 1 Here, C p d deotes the covariace matrix of the quatizatio distortio. To simplify 1, we shall ext assume that the offdiagoal elemets of C p d are zero, i.e., we shall igore the temporal correlatio of the quatizato distortio. Specifically, we assume that C d p = 1 G bkρ + 1I P. 15 The assumptio i 15 is accurate i the low-sn regime or whe the umber of UEs K is large, ad it is actually exact if the umber of pilots P coicides with the umber of UEs K. The costat o the right-had side of 15 follows from the power ormalizatio 5. Substitutig 15 ito 1 ad 1 ito 13, we obtai ĥ = G b X H p G b X p X H p + G b + 1 G bkρ + 1 I P 1 r p 16 = G b G b X H p X p + G b + 1 G bkρ + 1 I K 1 X H p r p 17 = G b G b P ρ + G b + 1 G bkρ X H p r p. 18 ewritig 18 i matrix form, we obtai the followig simplified estimator, which we shall use i the remaider of the paper: P G b t=1 Ĥ = r tx H t G b P ρ + G b G b Kρ + 1. Some commets o 19 are i order. For the case of 1-

6 6 TO APPEA IN IEEE TANSACTIONS ON WIELESS COMMUNICATIONS bit ADCs, the estimator 19 coicides with the oe derived i [33]. Uder the assumptio that the umber of pilots P is equal to the umber of UEs K, the covariace matrix C p d is ideed diagoal, ad the estimator 19 is actually the LMMSE estimator 13. This fact has bee observed i [8] for the 1-bit case. For the ifiite-resolutio case G = 1, 19 coicides with the classic miimum mea square error MMSE estimator see, e.g., [7]. Let H = Ĥ + H where H deotes the estimatio error. Uder the assumptio that 15 holds, the variace of the chael estimate ad of the estimatio error take the followig forms: ad ˆσ = 1 NK E [ trĥĥ H] = G b P ρ G b P ρ + G b + 1 G b Kρ σ = 1 [ tr ] NK E H HH G b = + 1 G b Kρ + 1 G b P ρ + G b G b Kρ + 1. For the case P = K, 1 ad 3 are exact. Note that i the ifiite-resolutio case G = 1, 1 ad 3 recover wellkow results for MMSE estimatio see, e.g., [7, Eq. 19]. E. Data Detectio We shall focus o the practically relevat case whe the BS employs a liear receiver. Liear receiver processig although iferior to oliear processig techiques such as successive iterferece cacellatio is less computatioally demadig ad has bee show to yield good performace if the umber of ateas exceeds sigificatly the umber of active users [8]. We shall cosider two types of liear receivers, amely MC ad ZF. Usig either of the two methods, a soft estimate ˆx k,t of the trasmitted symbol x k,t from the kth user at time t = P + 1, P +,..., T is obtaied as follows: ˆx k,t = a H k r t. Here, a k C N deotes the receive filter for the kth user, which is give by {ĥk / ĥk, for MC a k = 5 Ĥ k, for ZF where Ĥ k is the kth colum of the pseudo-iverse of the chael estimate matrix Ĥ = ĤĤH Ĥ 1. F. High-Order Modulatio Formats with 1-bit ADCs: Why Does it Work? Although for 1-bit ADCs, QPSK is optimal i the SISO case [19], the use of multiple ateas at the receiver opes up the possibility of usig higher-order modulatio schemes to support higher rates. This observatio is demostrated i Fig. where we plot the MC receiver output for 3 differet chael fadig realizatios for the case whe a sigle user trasmits 16-QAM data symbols. The chael Quadrature I-phase a N = ateas, ρ = db. Quadrature 1 1 Quadrature I-phase b N = ateas, ρ = db. 1 1 I-phase c N = ateas, ρ = db. Fig.. Sigle-user MC outputs for 16-QAM iputs as a fuctio of the umber of receive ateas N ad the SN ρ. The chael estimates are based o P = pilot symbols. estimate is acquired usig P = pilots. As the size of the BS atea array icreases, the 16-QAM costellatio becomes distiguishable see Fig. b, provided that ρ is ot too high. Ideed, additive oise is oe of the factors that eables the detectio of the 16-QAM costellatio. The other is the differet phase of the fadig coefficiets associated with each receive atea. The explaatio is as follows: i the 1- bit ADCs case, the quatized received output at each atea belogs to the set 1 of cardiality. These possible outputs are the averaged by the MC filter to produce a output a scalar that belogs to a alphabet with much higher cardiality. The cardiality depeds o the umber of pilots ad o the umber of receive ateas. The key observatio is that the ier poits of the 16-QAM costellatio, which are more susceptible to oise, are more likely to be erroeously detected at each atea. This results i a smaller averaged value after MC tha for the outer costellatio poits. To highlight the importace of the additive oise, we cosider i Fig. c the case whe ρ = db. Sice the additive oise is egligible, the output of the MC filter lies approximately o a circle, which suggests that the amplitude of the trasmitted sigal caot be used to covey iformatio. However, the phase of the 16-QAM symbols ca still be detected. Ideed, cosider the followig argumet. At high SN ad i the sigle-user case, the sigal received at the th atea ca be well-approximated by r = Q 1 h x + w Q 1 h x = Q 1 e jφ+θ. 6 Here, φ ad θ deote the phase of h ad of x, respectively. Furthermore, agai at high SN, the th etry a of the MC I the remaider of this sectio, we shall drop the time idex t ad the user idex k because they are superfluous.

7 S. JACOBSSON, G. DUISI, M. COLDEY, U. GUSTAVSSON, AND C. STUDE 7 filter a i 5 is well-approximated by a 1 N Q 1h = 1 N Q 1 e jφ. 7 Usig 6 ad 7, we ca approximate the MC output at high SN by ˆx 1 N N Q 1 e jφ Q1 e jφ +θ. 8 =1 To aalyze 8, let us assume without loss of geerality that < θ < π/. Sice φ is uiformly distributed o [, π] recall that we assumed h to be ayleigh distributed, oe ca show that the phase of the radom variable Q 1 e jφ Q1 e jφ +θ is equal to with probability 1 θ/π ad is equal π/ with probability θ/π. Hece, its mea is θ. Sice the fadig coefficiets {h }, ad, hece, also their phases {φ }, are idepedet, the phase of ˆx i 8 coverges to θ as N grows large, due to the cetral limit theorem. As show i Fig. c, N = ateas are sufficiet to distiguish the phase of 16-QAM costellatio poits at db of SN. Note that idepedece betwee the {h } is crucial for the cetral limit theorem to hold ad for the phases to be distiguishable. III. ACHIEVABLE ATE ANALYSIS I this sectio, we shall characterize the rate achievable i a low-resolutio quatized massive MIMO uplik system. I cotrast to [7], [8], [9], [5] we shall maily focus o fiitecardiality costellatios. Usig Bussgag s decompositio, we also provide a closed-form approximatio of the achievable rate with Gaussia iputs, which turs out accurate at low SN. A. Sum-ate Lower-Boud for Fiite-Cardiality Iputs It follows from, e.g., [51], that the achievable rate k ρ for user k = 1,,..., K with pilot-based chael estimatio ad MC or ZF detectio is k ρ = T P Ix k ; ˆx k Ĥ 9 T where x k ad ˆx k are distributed as x k,t ad ˆx k,t respectively. It follows that the sum-rate capacity ca be lower-bouded as follows: K Cρ k ρ. 3 k=1 I order to compute the achievable rate, we expad the mutual iformatio i 9 as follows: [ Ix k ; ˆx k Ĥ = E Pˆxk x x k,ˆx k,ĥ log,ĥˆx k k x k, Ĥ ]. 31 Pˆxk Ĥˆx k Ĥ To compute 31, oe eeds the coditioal probability mass fuctios Pˆxk x,ĥˆx k k x k, Ĥ ad Pˆx k Ĥˆx k Ĥ = [ E xk Pˆxk x,ĥˆx k k x k, ]. Ĥ Sice o closed-form expressios are available for these quatities, we estimate them by Mote- Carlo samplig. Specifically, we simulate may oise ad iterferece realizatios, ad map the resultig ˆx k to poits over a rectagular grid i the complex plae. With this techique, oe obtais a lower boud o k ρ [5, p. 353] that becomes icreasigly tight as the grid spacig is made smaller. 3 Note that 31 holds for every choice of iput distributio ad for ADCs with arbitrary resolutio. B. Sum-ate Approximatio for Fiite-Cardiality Iputs The evaluatio of 31 usig the method just described is extremely time cosumig. We ext provide a accurate approximatio of 31 for fiite-cardiality costellatios that is easier to evaluate, although still ot i closed form ote that eve for the ifiite-resolutio case, o closed-form expressio for the rate achievable with fiite-cardiality costellatios is available. The approximatio relies o the followig assumptio: the real part ˆx k = {ˆx k} ad the imagiary part ˆx I k = I{ˆx k} of the soft estimate ˆx k of the trasmitted symbol x k are coditioally joitly Gaussia give x k ad Ĥ, with coditioal mea µx k, Ĥ ad coditioal covariace Σx k, Ĥ. We use this assumptio to approximate 9 as follows see Appedix B: k ρ T P h ˆx k, ˆx I k T Ĥ 1 ] E x k,ĥ [log πe det Σx k, Ĥ. 3 Here, h deotes the coditioal differetial etropy [53]. Note that uder the above Gaussia approximatio, the coditioal probability of [ˆx k, ˆxI k ]T give Ĥ is a Gaussia mixture, for which the differetial etropy is ot kow i closed form but ca be computed efficietly. The accuracy of the approximatio i 3 depeds crucially o the choice of µx k, Ĥ ad Σx k, Ĥ. I Appedix B, we provide suitable choices for µx k, Ĥ ad Σx k, Ĥ for the MC case see For the ZF case, µx k, Ĥ is provided i 56 whereas, to improve the accuracy of the approximatio, we resort to the umerical method described i Appedix B to compute Σx k, Ĥ. As we shall illustrate i Sectio IV, the resultig approximatio 3 turs out to be accurate for all system parameters cosidered i this paper. C. Sum-ate Approximatio for Gaussia Iputs Next, we preset a approximatio o the achievable rate 9 assumig Gaussia iputs. I cotrast to [8], [33], where a similar approximatio is derived for the 1-bit case, we shall cosider the case of multi-bit ADCs. The approximatio relies o Bussgag s decompositio ad o the assumptio that the quatizer iput y ca be modeled as a Gaussia radom vector ad that its covariace matrix satisfies C y = Kρ + 1I N. 33 Both the Gaussia assumptio ad 33 are accurate at low SN or whe the umber of UEs is large. Uder these assumptios, 3 The umerical routies used to evaluate 9 ca be dowloaded at https: //github.com/ifotheorychalmers/1-bit massive MIMO. To keep otatio compact, we drop i this sectio the time idex t sice it is superfluous.

8 8 TO APPEA IN IEEE TANSACTIONS ON WIELESS COMMUNICATIONS we ca use Bussgag s theorem to decompose the received sigal as r = Q b y = G b y + d where d is the quatizatio distortio. Here, we have used that G b = G b I N, which follows from 33. Furthermore, due to the power ormalizatio 5 ad due to 33, the covariace matrix C r of r satisfies C r = Kρ + 1I N. Hece, the covariace matrix C d of the quatizatio distortio d must be equal to MSE [db] 1 15 b = 1,, 3, MSE [db] 1 15 b = 1,, 3, C d = C r G bc y = 1 G b Kρ + 1 IN. 35 Substitutig 3 ito, we obtai ˆx k = a H k G b y + d = G b a H k Ĥx + ah k 36 where Ĥ is the chael estimate 19, ad H is the correspodig estimatio error. Here, we have defied = G b Hx + Gb w + d. Note that the oise ad the iput vector x are ucorrelated provided that both 33 ad 15 hold. Assumig that this is ideed the case, we ca approximate the mutual iformatio 9 by usig the auxiliary chael lower boud [5, p. 353] ad treatig the additive oise a H k i 36 as a Gaussia radom variable. Specifically, let ρ = G b ˆσ ρ G b K σ ρ + G b + 1 G b Kρ where ˆσ ad σ are give by 1 ad 3, respectively. I 37, the three terms i the deomiator correspod to the estimatio error, the additive oise, ad the quatizatio distortio, respectively. Sice the chael iput x is Gaussia, usig [5, p. 353] we obtai the followig approximatio: Ix k ; ˆx k Ĥ E log 1 + ρ a H k ĥk ρ a H k ĥj + a k j k. 38 Uder the additioal assumptio that Ĥ is Gaussia, we ca use [5, Eq. 16 ad ] to further lower-boud 38 ad obtai the followig closed-form Gaussia approximatios for the rates achievable with MC ad ZF, respectively: MC ρ T P N 1 ρ log T K 1 ρ + 1 ad ZF ρ T P log T 1 + N K ρ. Here, we have multiplied the log terms by T P /T to take ito accout the pilot overhead. Note that for the ifiiteresolutio case G = 1, we recover from 39 ad the achievable rate with imperfect CSI reported i [5, Eq. 39] ad [5, Eq. ] for the MC ad ZF receiver, respectively. For the case of 1-bit ADCs G 1 = /π, we recover from the achievable rate approximatio with ZF recetly reported i [8]. As we shall demostrate i Sectio IV, despite the several assumptios ivoked to obtai 39 ad, these approximatios tur out to be accurate i the low-sn regime SN, ρ [db] a P = K pilots SN, ρ [db] b P = 3K pilots. Fig. 3. MSE of the chael estimator 19 as a fuctio of the SN ρ; QPSK pilots, N =, K = 1. The solid lies correspod to the MSE approximatio 3 ad the marks correspod to the exact MSE, which was computed umerically. IV. NUMEICAL ESULTS We ow assess the rates achievable with the above detailed chael estimatio ad data-detectio schemes detailed i the previous sectio o a massive MU-MIMO uplik system where the receiver is equipped with low-resolutio ADCs. We assume that the users are able to coordiate the trasmissio of their pilots: whe oe of the UEs trasmits pilots, the other UEs remai idle. I other words, pilots are trasmitted i a roud robi fashio. 5 The use of time-iterleaved pilots esures that P t=1 x tx H t = P ρ I K. Also, because of the idle time, each user ca trasmit its pilots at a power level that is K times higher tha the power level for the data symbols, while still satisfyig the average-power costrait 3. A. Chael Estimatio We start by validatig the accuracy of the approximatio for the MSE of the simplified chael estimator 19 give i 3. Specifically, we compare i Fig. 3 the exact MSE of the estimator 19, which is evaluated umerically, with the approximatio 3, for differet values of SN ρ, umber of pilots P, ad ADC resolutio b. We ote that if P = K Fig. 3a, the the MSE approximatio 3 is ideed exact as we claimed i Sectio II-D. For the case of P = 3K Fig. 3b, the approximatio 3 turs out to be accurate at low SN. Furthermore, the accuracy of 3 icreases with the resolutio of the ADCs. Ideed, 3 relies o the assumptio that the off-diagoal elemets of the covariace matrix i 15 are zero ad these etries vaish as the ADC resolutio icreases see, e.g., [55, p. 51] for more details. B. Achievable ate 1 Sigle-User Case, 1-bit ADCs: I Fig. we compare for the sigle-user 1-bit ADC case, the rates achievable with QPSK, 16-QAM, ad 6-QAM as a fuctio of ρ for the 5 This pilot-trasmissio method is chose for coveiece; it may be suboptimal.

9 S. JACOBSSON, G. DUISI, M. COLDEY, U. GUSTAVSSON, AND C. STUDE 9 ate [bits/chael use] 6 6-QAM 16-QAM QPSK if. res. 1-bit ADCs SN, ρ [db] Fig.. Sigle-user achievable rate with MC as a fuctio of the SN ρ; N =, K = 1, T = 11; the umber of pilots P is optimized for each value of ρ. The solid lies correspod to the fiite-cardiality approximatio 3, the dashed lies correspods to the Gaussia approximatio 39, ad the marks correspod to the rates computed via 9 ad 31. ate per user [bits/chael use] 6 if. res. 1-bit ADCs 6-QAM 16-QAM SN, ρ [db] a MC receiver. ate per user [bits/chael use] 6 if. res. 1-bit ADCs 6-QAM 16-QAM SN, ρ [db] b ZF receiver. Fig. 5. Per-user achievable rate as a fuctio of the SN ρ; N =, K = 1, T = 11; the umber of pilots P is optimized for each value of ρ. The solid lies correspod to the fiite-cardiality approximatio 3, the dashed lies correspods to the Gaussia approximatios 39,, ad the marks correspod to the rates computed via 9 ad 31. MC receiver. 6 We depict both the rates achievable with 1-bit ADCs ad the oes for the ifiite-resolutio case. The rates with 1-bit ADCs, which are computed usig 9 ad 31, are compared with the approximatio for fiitecardiality costellatios provided i 3 ad the Gaussia approximatio 39 to verify their accuracy. The ifiiteresolutio rates are computed usig 9 ad 31 ad are also compared with the Gaussia approximatio 39 The umber of receive ateas is N = ad the coherece iterval is T = The umber of trasmitted pilots P is umerically optimized for every value of ρ. We see that, despite usig 1-bit ADCs, higher-order modulatios outperform QPSK already at SN values as low as ρ = 15 db. Note that the achievable rate does ot icrease mootoically with ρ i the 16-QAM ad 6-QAM case. Ideed, as ρ gets large the costellatio gets projected oto the uit circle ad the umber of distiguishable costellatio poits becomes smaller see Fig. c. Note also that the approximatio 3 for fiite-cardiality costellatios closely tracks the simulatio results for all SN values. This approximatio eables us to accurately predict the SN value beyod which the rates achievable with a give costellatio saturates. This, i tur, allows us to idetify the most appropriate costellatio for a give SN value. The Gaussia approximatio 39 tracks the rates achievable with fiite-cardiality costellatios accurately i the low- SN regime. We ote that, whe QPSK is used, the differece i the achievable rates betwee the 1-bit quatized case ad the ifiite-resolutio case is margial a observatio that was already reported i [5]. I cotrast, the rate loss is more proouced for higher-order costellatios. 6 To evaluate the mutual iformatio 31, we have simulated 3 radom fadig chael realizatios. For each chael realizatio we have cosidered 3 radom oise realizatios for each symbol i the costellatio. 7 For a LTE-like system operatig at GHz, with symbol time equal to 66.7 µs, ad with UEs movig at a speed of 3 km/h, the duratio of the coherece iterval accordig to Jake s model is approximately T = 11 symbols. ate per user [bits/chael use] 6 if. res. 1-bit ADCs Coherece iterval, T Fig. 6. Per-user achievable rate with 6-QAM ad ZF as a fuctio of T ; ρ = 1 db, N =, K = 1; the umber of pilots P is optimized for each value of T. The solid lies correspod to the fiite-cardiality approximatio 3, the dashed lies correspods to the Gaussia approximatio, ad the marks correspod to the rates computed via 9 ad 31. Multi-User Case, 1-bit ADCs: I Fig. 5, we plot the rates achievable with MC ad ZF for both the 1-bit-ADC ad the ifiite-resolutio case whe K = 1 users are active. Motivated by the results i Fig., we oly compare the rates achievable with 16-QAM ad 6-QAM. Note agai that the approximatio 3 turs out to be accurate for a all SN values, whereas the Gaussia approximatio is accurate oly at low SN. Note also that rates with 16-QAM ad 6-QAM saturate at the same level at high SN for both MC ad ZF. This implies that the system is effectively distortio ad iterferece limited, ad that the Gaussia approximatios 39, overestimate the rate for high SN values. 3 Depedece o the Coherece Iterval: I Fig. 6, we plot the per-user achievable rates with ZF, as a fuctio of the coherece iterval T for ρ = 1 db, N =, K = 1, ad 6-QAM costellatio. We observe that the reductio i the achievable rate whe T is made smaller is similar for both the 1-bit ad ifiite-resolutio case. Hece, operatig i a high-

10 1 TO APPEA IN IEEE TANSACTIONS ON WIELESS COMMUNICATIONS ate per user [bits/chael use] 6 b = 1,, 3, SN, ρ [db] Fig. 7. Per-user achievable rate with 6-QAM ad ZF as a fuctio of the SN ρ; N =, K = 1, T = 11; the umber of pilots P is optimized for each value of ρ. The solid lies correspod to the fiite-cardiality approximatio 3, the dashed lies correspods to the Gaussia approximatio, ad the marks correspod to the rates computed via 9 ad 31. mobility sceario leads to similar performace losses i both cases. Note also that the achievable rate is zero whe T 1. I fact, whe orthogoal pilot sequeces are trasmitted, at least 1 pilot symbols are required whe K = 1. Depedece o ADC esolutio: Focusig o 6-QAM ad ZF, we compare i Fig. 7 the achievable rate as a fuctio of the ADC resolutio ad the SN. We observe that with -bit ADCs, the achievable rate icreases sigificatly compared to the 1-bit-ADC case. For example, at ρ = 1 db, we achieve 9% of the ifiite-resolutio rate, compared to 71% with 1-bit ADCs. Icreasig the ADC resolutio beyod 3 bits seems uecessary for the system parameters cosidered i Fig. 7. This coclusio is supported by both the approximatio for fiite-cardiality costellatios ad the oe for Gaussia iputs. We ote that the Gaussia approximatio is agai accurate at low SN. Furthermore, as expected its accuracy icreases with the ADC resolutio. C. Impact of Large-Scale Fadig ad Imperfect Power Cotrol So far, we have cosidered oly the case whe all users operate at the same average SN. This correspods to the sceario where perfect power cotrol ca be performed i the uplik, which is clearly favorable for low-resolutio ADC architectures. If, however, the received sigal powers are vastly differet, low-power sigals may ot be distiguishable from high-power iterferers for cases i which the ADCs resolutio is too low. I practical systems, large spreads i the received power is typically avoided through power cotrol. However, perfect power cotrol may be impossible to achieve i practice due to limitatios o the UE trasmit power, for example. We ext ivestigate how relaxig the accuracy of the UE trasmit power cotrol will impact the system performace. We cosider a sigle-cell sceario ad adapt the urba-macro path loss model i [56]. The simulatio parameters for this study are summarized i Table I. The trasmit power for all UEs is set to 8.5 dbm, which for the first user that is located d 1 = Descriptio TABLE I SUMMAY OF SIMULATION PAAMETES Cell layout Cell radius Miimum distace betwee UE ad BS Path loss Number of BS ateas N Number of sigle-atea users K Coherece iterval T Number of pilots per user P/K Carrier frequecy System badwidth UE trasmit power Noise spectral efficiecy Noise figure 1% worst throughput [bits/chael use] 3 1 b = 1,, 3, d [meters] a MC receiver. 1% worst throughput [bits/chael use] 3 1 Assumptio Circular cell 335 meters 35 meters log 1 d db ateas 1 users 11 chael uses 1 pilots per user GHz MHz 8.5 dbm 17. dbm/hz 5 db b = 1,, 3, d [meters] b ZF receiver. Fig. 8. The 1% worst throughput with 16-QAM for a user located d 1 = 185 meters away from the BS as a fuctio of d for the parameters specified i Table I. 185 meters from the BS, results i a SN of approximately ρ 1 = 1 db. The remaiig K 1 users i the cell are radomly dropped accordig to a uiform distributio o the circular rig of ier radius d 1 d meters ad outer radius d 1 + d meters, for a distace spread < d < 15 meters. The case d = correspods to the sceario whe power cotrol is executed perfectly. The case d = 15 meters correspods to the worst-case sceario of ucoordiated uplik trasmissio, where o power cotrol is performed by the UEs. I the latter case, the SN for each iterferig user lies i the rage [ 19. db, 15.3 db]. I Fig. 8, we plot the 1% worst throughput i.e., the throughput correspodig to the 1% poit of the CDF of throughputs, for the iteded user located d 1 = 185 meters away from the BS, as a fuctio of d. We focus o 16- QAM ad assume that the received sigal power level for each user is kow to the BS. To attai the curves, we have cosidered 1 3 radom iterferig user drops for each d value. As expected, the gap to the ifiite-resolutio rate grows as d icreases. I the ucoordiated case, with 1-bit ADCs

11 S. JACOBSSON, G. DUISI, M. COLDEY, U. GUSTAVSSON, AND C. STUDE 11 ad ZF, we attai 57% of the rate achievable with perfect power cotrol. The correspodig umber for the 3-bit-ADC case is 79%. This shows that high rates are achievable with low-resolutio ADCs eve i absece of power cotrol. V. CONCLUSIONS We have aalyzed the performace of a low-resolutio quatized uplik massive MU-MIMO system operatig over a frequecy flat ayleigh block-fadig chael whose realizatios are ot kow a priori to trasmitter ad receiver. I particular, we have show that for the 1-bit massive MIMO case, high-order costellatios, such as 16-QAM, ca be used to covey iformatio at higher rates tha with QPSK; this holds i spite of the oliearity itroduced by the 1-bit ADCs. Furthermore, reliable commuicatio ca be achieved by usig simple sigal processig techiques at the receiver, i.e., pilot-based chael estimatio based o the Bussgag decompositio 19 ad MC detectio. By icreasig the resolutio of the ADCs by oly a few bits, e.g., to 3 bits, we ca achieve ear ifiite-resolutio performace for a broad rage of system parameters; furthermore, the system becomes robust agaist differeces i the received sigal power from the differet users, due for example, to large-scale fadig or imperfect power cotrol. A extesio of our aalysis to a OFDM based setup for trasmissio over frequecy-selective chaels is curretly uder ivestigatio. Such a extesio could be used to bechmark the results recetly reported i [3] i which the authors reported that, with a specific modulatio ad codig scheme take from IEEE 8.11, to 6 bits are required to achieve a packet error rate below 1 at a SN close to the oe eeded i the ifiite-resolutio case. We coclude that for a fair compariso betwee the performace attaiable usig low-resolutio versus high-resolutio ADCs, oe should take ito accout the overall power cosumptio, icludig the power cosumed by F ad basebad processig circuitry. APPENDIX A POOF OF THEOEM 1 It follows from Bussgag s theorem [5] that E [ ry H] = G b E [ yy H] 1 where G b is a N N diagoal matrix with [G b ]. = 1 σ E[Q b y y]. Here, y deotes [ the th etry of the vector y, = 1,..., N, ad σ = E y ] = [K],. It follows from 1 that we ca write the quatized sigal as r = G b y + d, where d ad y are ucorrelated. Note ow that the quatizer output Q b y is equal to l i + jl i if ad oly if {y } [τ i, τ i+1 ad I{y } [τ j, τ j+1. Thus, E y [Q b y y] = = L 1 i= Q b zz σ π l i σ π exp e τ i σ z σ e τ dz 3 i+1 σ. We obtai 9 by substitutig i ad by usig that σ = [K],. APPENDIX B DEIVATION OF 3 To keep the otatio compact, we set ˆx k = {ˆx k} ad ˆx I k = I{ˆx k }. By lettig a,k = a,k + jai,k, h,k = h,k + jhi,k, ad r = r + jr I, where a,k deotes the th etry of the receive filter a k ad r the th etry of the received vector r, we ca express the real compoets of the received sigal as ˆx k = N {a,kr } = =1 N a,k r + a I,k r I. 5 =1 Similarly, for the imagiary part we ca write ˆx I k = N I{a,kr } = =1 N a,k r I a I,k r. 6 =1 Now, we collect the real ad imagiary compoets i a vector [ˆx k, ˆxI k ]T ad approximate their coditioal distributio give the chael iput ad the chael estimate as a bivariate Gaussia radom vector with mea µx k, Ĥ ad covariace matrix Σx k, Ĥ. Uder this assumptio, we have that Ix k ; ˆx k ˆx Ĥ = h k, ˆx I k Ĥ 1 ] E x k,ĥ [log πe det Σx k, Ĥ. 7 It is worth emphasizig that, differetly from 31, the differetial etropy hˆx k, ˆxI k Ĥ i 7 is evaluated uder the assumptio that [ˆx k, ˆxI k ]T is coditioally Gaussia give x k ad Ĥ. The coditioal probability of [ˆx k, ˆxI k ]T give Ĥ is a Gaussia mixture. The achievable rate i 3 follows from 7 by takig ito accout the rate loss due to the trasmissio of P pilot symbols to estimate the chael. We shall ext discuss how to choose µx k, Ĥ ad Σx k, Ĥ. We start by fidig a suitable approximatio for the probability mass fuctios of the radom variables r ad r. I For r, it holds that p,i = Pr { r } = l i 8 = Pr { r } { } < τ i+1 Pr r τ i 9 Φζ i+1 Φζ i 5 where i the last step we have approximated the iterferece term j k {h,jx j } by a zero-mea Gaussia radom variable with variace ρ j k h,j ad defied ζi = τ i h,k x k + hi,k xi k 1 + ρ j k h,j. 51 For the sigle-user case, the approximatio 5 is exact sice there is o iterferece. Proceedig i a aalogous way, we ca show that p I,i Φζ I i+1 Φζ I i 5