Throughput Analysis of Massive MIMO Uplink with Low-Resolution ADCs

Transcription

1 Throughput Aalysis of Massive MIMO Uplik with Low-Resolutio ADCs Sve Jacobsso, Studet Member, IEEE, Giuseppe Durisi, Seior Member, IEEE, Mikael Coldrey, Member, IEEE, Ulf Gustavsso, Christoph Studer, Seior Member, IEEE Abstract We ivestigate the iformatio-theoretic throughput that is achievable over a frequecy flat fadig commuicatio lik whe the receiver is equipped with low-resolutio aalog-to-digital coverters ADCs). We focus 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 ivestigate the uplik throughput achievable by a massive multiple-iput multiple-output system i which the base statio is equipped with a large umber of low-resolutio ADCs. We propose a ovel high- SNR approximatio to the rate achievable with oe-bit ADCs that is accurate for a broad rage of system parameters. We show that for the oe-bit quatized case, LS estimatio together with maximalratio combig or zero-forcig detectio eables reliable multi-user commuicatio with high-order costellatios i spite of the severe oliearity itroduced by the ADCs. We demostrate that the rate S. Jacobsso is with Ericsso Research 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 Research, Gotheburg, Swede 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 Research uder grats SM3-28 ad ID4-22, ad by the Swedish Govermet Agecy for Iovatio Systems VINNOVA) withi the VINN Excellece ceter Chase. The work of C. Studer was supported i part by Xilix Ic., ad by the US Natioal Sciece Foudatio NSF) uder grats ECCS-486 ad 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 25 [].

2 2 achievable i the ifiite-precisio o quatizatio) case ca be approached usig ADCs with oly a few bits of resolutio. We fially ivestigate the robustess of the studied low-resolutio ADC system agaist receive power imbalaces betwee the differet users, caused for example by imperfect power cotrol. Idex Terms Aalog-to-digital coverter ADC), chael capacity, joit pilot-data JPD) processig, least squares LS) chael estimatio, low-resolutio quatizatio, multi-user massive multiple-iput multiple-output MIMO). I. INTRODUCTION Massive multiple-iput multiple-output MIMO) is a promisig multi-user MIMO techology for ext geeratio cellular commuicatio systems 5G) [2]. With massive MIMO, the umber of ateas at the base statio BS) is scaled up by several orders of magitude compared to traditioal multi-atea systems with the goals of eablig sigificat gais i capacity ad eergy efficiecy [2], [3]. Icreasig the umber of 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, cosiderably icreases the hardware cost ad the radio-frequecy RF) circuit power cosumptio [4]. 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] [7], where it is foud that massive MIMO provides some degrees of robustess towards sigal distortios caused by low-cost RF compoets. A. Quatized Massive MIMO I this paper, we cosider a uplik massive MIMO system i.e., UEs commuicate to a BS) ad focus o a particular source of sigal distortio, amely the quatizatio oise caused by the use of low-resolutio aalog-to-digital coverters 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, cotiuousamplitude basebad received sigal to oe out of 2 b quatizatio labels, by operatig f s 2 b

3 3 coversio steps per secod. I moder high-speed ADCs e.g., with samplig rates larger tha GS/s), the dissipated power scales expoetially i the umber of bits ad liearly i the samplig rate [8], [9]. 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 RF compoets ad the basebad-processig 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 usig a pair of -bit ADCs operatig at GS/s. Such a system would produce Tbit/s of data. This exceeds by far the rate supported by the commo public radio iterface CPRI) used over today s fiber-optical frothaul liks []. Alleviatig this capacity bottleeck is of particular importace i a cloud radio access etwork C-RAN) architecture [], 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 radiofrequecy circuitry ca be relaxed. The reaso is that the quatizatio oise may be domiatig 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 RF circuitry. The oe-bit resolutio case, where the i-phase ad quadrature compoets of the cotiuousvalued received samples are quatized separately usig oe-bit quatizers, is particularly attractive because of the resultig low hardware complexity [2], [3]. Ideed, a oe-bit quatizer ca be realized usig oly a simple comparator. Furthermore, i a oe-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 Receivers employig low-resolutio ADCs eed to cope with the severe oliearity that is itroduced by the coarse quatizatio, which may reder sigalig schemes ad receiver algorithms developed for the case of high-resolutio ADCs suboptimal.

4 4 The impact of the oe-bit ADC oliearity o the performace of commuicatio systems has bee previously studied i the literature uder various chael-model assumptios. I [4], 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 oe-bit quatizer is a zero-threshold comparator. It turs out that i the low-snr regime, a zero-threshold comparator is ot optimal [5]. The optimal strategy ivolves the use of flash-sigalig [6, Def. 2] 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. Therefore, we shall focus exclusively o the zero-threshold comparator architecture i the remaider of the paper. For the Rayleigh-fadig case, uder the assumptio that the receiver has access to perfect chael state iformatio CSI), it is show i [7] that QPSK is capacity achievig agai for the SISO case). The assumptio that perfect CSI is available may, however, be urealistic i the oe-bit quatized case, sice the oliear distortio caused by the oe-bit quatizers 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 SNR exceeds a certai threshold that depeds o the coherece time of the fadig process [8]. For SNR values that are below this threshold, o-off QPSK is capacity achievig [8]. For the oe-bit quatized MIMO case, the capacity-achievig distributio is ukow. I [9], it is show that QPSK is optimal at low SNR, agai uder the assumptio of perfect CSI at the receiver. Mo ad Heath Jr. [2] derived high-snr bouds o capacity for the case whe also the trasmitter has access to perfect CSI. The chael-estimatio overhead i massive MIMO ca be reduced usig reciprocity-based time-divisio duplexig TDD), where the chael estimates that have bee obtaied i the uplik are used for dowlik beamformig [2]. Chael estimatio o the basis of quatized observatios is cosidered i, e.g., [2], [22] see also [23] for a compressive-sesig versio of this problem). A closed-form solutio for the maximum likelihood ML) estimate i the oe-bit case is derived i [22], uder the assumptio of time-multiplexed pilots. The use of oe-bit ADCs i massive MIMO was cosidered i [24]. There, the authors examied the achievable uplik throughput for the sceario where the UEs trasmit QPSK

5 5 symbols, ad the BS employs a least squares LS) chael estimator, followed by a maximal ratio combiig MRC) or zero-forcig ZF) detector. Their results show that large sum-rate throughputs ca be achieved despite the coarse quatizatio. The results i [24] were exteded to high-order modulatios e.g., 6-QAM) by the authors of this paper i []. 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 ad that the SNR is ot too high. Choi et al. [25] recetly developed a detector ad a chael estimator capable of supportig high-order costellatios such as 6-QAM. A mixed-adc architecture, where may oe-bit ADCs are complemeted with few highprecisio ADCs o some of the ateas is proposed i [26]. It is foud that the additio of a relatively small umber of high-resolutio ADCs icrease the system performace sigificatly. 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 [27]. The proposed receiver employs LS estimatio followed by a message-passig-based detector. The performace of a low-resolutio quatized massive MIMO system usig orthogoal frequecy divisio multiplexig OFDM) ad operatig over a widebad chael was ivestigated i [28]. There, it is foud that usig ADCs with oly four to six bits resolutio is sufficiet to achieve performace close to the ifiite-precisio i.e., o quatizatio) case, at o additioal cost i terms of digital sigal processig complexity. All of the results reviewed so far hold uder the assumptio of Nyquist-rate samplig at the receiver. However, it is worth poitig out that Nyquist-rate samplig is ot optimal i the presece of quatizatio at the receiver [29], [3]. For example, for the oe-bit quatized complex AWGN chael, high-order costellatios such as 6-QAM ca be supported eve i the SISO case, if oe allows for oversamplig at the receiver [3]. 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 Rayleigh block-fadig massive MIMO chael, whe the receiver is equipped with low-resolutio ADCs. Our cotributios are summarized as follows:

6 6 Focusig o the oe-bit ADC architecture, we geeralize the aalysis preseted i [24] to iclude high-order modulatios. We show that MRC/ZF detectio combied with LS estimatio at the BS allows for both multi-user operatio ad the use of high-order costellatios such as 6-QAM, if the umber of ateas is sufficietly large. Furthermore, the rates achievable with 6-QAM tur out to exceed the oes reported i [24] for QPSK, for SNR values as low as 5 db per atea, ad for atea arrays of elemets or more. Our results also suggest that there exists a trade-off betwee the umber of BS ateas ad the resolutio of the ADCs used at each atea port. Through a umerical study, we determie the miimum ADC resolutio eeded to make the performace gap to the ifiite-precisio case egligible. Our simulatios suggest that oly few bits e.g., three to four) are required to achieve a performace close to the ifiiteprecisio case for a large rage of system parameters. For example, cosider users commuicatig with a BS that is equipped with 2 ateas. Furthermore, assume that the SNR is db, that 64-QAM is selected, ad that ZF is used at the BS. The ADCs with three-bit resolutio are sufficiet to attai 97% of the ifiite-precisio per-user rate. This holds provided that all UEs are received with the same average power at the BS. I other words, whe perfect power cotrol is assumed. Fially, we assess the impact o performace of imperfect power cotrol. Specifically, we characterize through umerical simulatios the ADC resolutio eeded to separate the iteded user from the iterferers as a fuctio of the sigal-to-iterferece ratio SIR). For example, whe the umber of ateas is 2 ad the SIR is 2 db, the achievable per-user rate with three-bit ADCs is 89% of the ifiite-precisio rate. However, whe the SIR is 4 db, three-bit ADCs achieve oly 5% of the ifiite-precisio per-user rate. With oe-bit ADCs, we attai oly about 3% of the ifiite-precisio per-user rate at the same iterferece level. This paper complemets the aalysis previously reported i [] by geeralizig it to ZF receivers, to multi-bit quatizatio, ad to the case of imperfect power cotrol. Furthermore, we develop a accurate ad easy-to-evaluate high-snr approximatio to the rate achievable with QAM costellatios ad oe-bit ADCs.

7 7 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{ } to deote the trace of a matrix, ad to deote the l 2 -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 mea µ ad covariace Σ is deoted by CN µ, 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 R{s} ad I{s}. The superscripts ad H deote 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. I Sectio III, we focus o the oe-bit-adc case ad aalyze the rate achievable with fiite-cardiality costellatios. We also derive a high-snr approximatio of the rate, which turs out to be accurate for a broad rage of system parameters. I Sectio IV, we cosider the case of multi-bit quatizatio ad determie the ADC resolutio required to approach the rate achievable i the ifiite-precisio case. We coclude i Sectio V. II. SYSTEM MODEL We cosider the sigle-cell uplik system depicted i Fig.. 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 Rayleigh block-fadig chael see, e.g., [32]), i.e., a chael that stays costat for 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 a arbitrary coherece iterval ad before quatizatio is modeled as y t = Hx t + w t, t =, 2,..., T. ) 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

8 8 UE RF Re Im Low-resolutio ADC Low-resolutio ADC UE 2. RF. Re Im Low-resolutio ADC Low-resolutio ADC. UE K RF RF Re Im Re Im Low-resolutio ADC Low-resolutio ADC Low-resolutio ADC Low-resolutio ADC Fig.. Quatized massive MIMO uplik system model. ad CN, )-distributed. Furthermore, the vector w t C N, whose etries are idepedet ad CN, )-distributed, stads for the AWGN. 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 2 b + quatizatio thresholds T b = {τ, τ,..., τ 2 b}, such that = τ < τ < < τ 2 b =, ad a set of 2 b quatizatio labels Q b = {q, q,..., q 2 } where q b i τ i, τ i+ ]. Let R b = Q b Q b. We shall describe the joit operatio of the 2N b-bit ADCs at the BS by the fuctio Q b ) : C N R N b that maps the received sigal y t with etries {y,t } ito the quatized output r t with etries {r,t } i the followig way: if R{y,t } τ k, τ k+ ] ad I{y,t } τ l, τ l+ ], the r,t = q k + jq 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 =, 2,..., T. 2) I the oe-bit case, uder the assumptio that τ = zero-threshold comparator) ad that Q = {, }, the quatizatio outcomes at each atea belog to the set R = { + j, + j, j, j}. Furthermore, we ca write the quatized received sigal at the th atea, at discrete time t, as follows: r,t = Q y,t ) = sgr{y,t }) + jsgi{y,t }) 3)

9 9 where sg ) deotes the sig fuctio defied as, if x < sgx) =, if x. 4) 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 [33] [35], a operatio that has to be performed usig quatized observatios. We further assume that codig ca be performed over may coherece itervals. Let X = [x, x 2,..., x T ] be the K T matrix of trasmitted sigals withi a coherece iterval, ad let R = [r, r 2,..., r T ] be the correspodig N T matrix of received quatized samples. For a give quatizatio fuctio, the ergodic sum-rate capacity is [32] Cρ) = T sup IX; R). 5) 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 [ tr{xx H } ] KT ρ. 6) Sice the oise variace is ormalized to oe, we ca thik of ρ as the SNR. The sum-rate capacity i 5) is, i geeral, ot kow i closed form, eve i the ifiite-precisio case for which tight capacity bouds have bee reported recetly i [36]). 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 ). Because of the large dimesioality of the fadig matrix H, simple, low-complexity chael-estimatio methods are favorable for massive MIMO [2]. Therefore, as i [24], we shall focus o LS chael estimatio. For the oe-bit SISO case, oe ca actually show that LS estimatio combied with joit pilot ad data processig JPD) achieves the capacity [37], []. However, this result does ot exted to MIMO systems. Although JPD processig is advatageous for low-resolutio quatized massive MIMO [38], we will cosider oly the versio of LS estimatio that relies exclusively o pilot symbols ad does ot exploit JPD processig. Ideed, JPD processig is more computatioally demadig ad may ot be suitable for massive MIMO.

10 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. Furthermore, we assume that all users trasmit the same umber of pilots. Accordig to the LS priciple, a estimate of H is obtaied as P Ĥ = argmi r t Hx t 2 7) H t= P ) P ) = r t x H t x t x H t. 8) t= The use of time-iterleaved pilots esures that the matrix P t= x tx H t i 8) is ideed ivertible. 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 6). To limit the complexity further, we will focus o the 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 [39]. We shall cosider two types of liear receivers, amely MRC ad ZF. With MRC, we maximize the stregth of a UEs sigal, by usig the chael estimates to combie the received sigal coheretly. I the ifiite-precisio case ad if perfect CSI is available at the receiver, this results i a array gai proportioal to N. With ZF, we additioally try to suppress the iterferece from other UEs at the cost of reducig the array gai to N K + see, e.g., [39]). 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 +, P + 2,..., T is obtaied as follows: t= ˆx k,t = a H k r t. 9) Here, a k C N deotes the receive filter for the kth user, which is give by ĥ k / ĥk 2, for MRC a k = Ĥ ) k, for ZF. ) With ĥk we deote the kth colum of the matrix Ĥ, which is obtaied through 8). Furthermore, Ĥ ) k is the kth colum of the pseudo-iverse of the chael estimate matrix Ĥ = ĤĤH Ĥ). This pilot-trasmissio method is chose for coveiece; it may be suboptimal.

11 Quadrature Quadrature Quadrature I-phase a) N = 2 ateas, ρ = db. I-phase b) N = 2 ateas, ρ = db. I-phase c) N = 2 ateas, ρ = 2 db. Quadrature Quadrature Quadrature I-phase I-phase I-phase d) N = 2 ateas, ρ = 2 db, cor- e) N = 2 ateas, ρ = 2 db, f) N = 2 ateas, ρ = 2 db, o- related fadig chael vector h = h). ofadig chael vector h = ). fadig chael vector h = ), Gaussia ditherig ). Fig. 2. Sigle-user MRC outputs for 6-QAM iputs as a fuctio of the umber of receive ateas N ad the SNR ρ. The LS chael estimates are based o P = 2 pilot symbols. III. MASSIVE MIMO WITH ONE-BIT ADCS I this sectio, we focus o the oe-bit-adc case ad assess the rates achievable with QAM costellatios of varyig size. A. High-order Modulatio Formats with Oe-bit ADCs: Why Does it Work? ) The role of additive oise: Although QPSK is optimal i the SISO case, 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. 2 where we plot the MRC receiver output for 3 differet chael fadig realizatios) correspodig to 6-QAM data symbols for the case whe a sigle user trasmits also P = 2 pilots to let the BS acquire LS chael estimates. As the size of the BS atea array icreases, the 6-QAM costellatio becomes

12 2 progressively distiguishable see Fig. 2b), provided that ρ is ot too high. Ideed, additive oise is oe of the factors that eables the detectio of the 6-QAM costellatio; the other is the differet phase of the fadig coefficiets associated with each receive atea. The explaatio is as follows: due to the oe-bit ADCs, the quatized received output at each atea belogs to the set R of cardiality 4. These 4 possible outputs are the averaged by the MRC 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 6-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 MRC tha for the outer costellatio poits. To highlight the importace of the additive oise, we cosider i Fig. 2c the case whe ρ = 2 db. Sice the additive oise is egligible, the output of the MRC 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 6-QAM symbols ca still be detected. Ideed, because of the idepedet fadig, the phase distortio caused by the coarse quatizatio, which is sigificat at each atea, is zero-mea ad will therefore be averaged out with MRC. 2) Impact of spatial correlatio: If the fadig coefficiets are correlated over the atea array, the ability to recover the phase of the trasmitted sigal may be lost at high SNR. To demostrate this, let h = h with h CN, ), deote the chael fadig vector. Here, is the all-oe vector. For this case, the phase distortio due to fadig is equal o all ateas ad it ca o loger be averaged out with MRC see Fig. 2d). Whe the oise is egligible ad whe the chael is ofadig i.e., whe h = ) the costellatio collapses to a oisy QPSK diagram see Fig. 2e). For both of these ufavorable cases, high-order modulatios are ot supported by the chael. 3) Ditherig: A possible remedy to this problem is to radomize the quatizatio error amog observatios by itetioally addig oise to the sigal prior to the ADC. This approach is commoly referred to as ditherig, ad its advatages are well documeted see, e.g., [4] [42]). We ca write the quatized dithered sigal at time t as r t = Q hx t + w t + d t ), t =, 2,..., T ) where d t C N is the dither sigal at time t.

13 3 To highlight the beefits of ditherig, we cosider agai the case whe h = ad ρ = 2 db, ad show i Fig. 2f the received 6-QAM costellatio after we have applied ditherig. Similarly to [26], we have used a CN, ρ )I N )-distributed dither sigal. We ote that it is ow possible to detect the 6-QAM symbols. Ditherig ca also be used to recover the 6-QAM costellatios i Fig. 2c ad 2d, for the idepedet ad correlated Rayleigh-fadig case, respectively. Ditherig which requires kowledge about the SNR) ca be implemeted by, for example, addig DC biases to the comparators i the ADCs. Sice we strive to keep the receiver hardware complexity at a miimum, we will oly cosider odithered quatizatio i the remaider of this paper. For the idepedet Rayleigh-fadig case cosidered i this paper, ditherig is useful at high SNR. However, i massive MIMO, the SNR per atea is typically low, as we rely o the massive umber of ateas to provide large array gai. Furthermore, i a multi-user sceario, the iterferece from other UEs will also perturb the sigal, causig beeficial radomizatio i the quatizatio error. B. Sum-rate Capacity Lower-Boud It follows from, e.g., [43], that the achievable rate R k) ρ) for user k =, 2,..., K with LS estimatio ad MRC or ZF detectio is R k) ρ) = T P T Ix k ; ˆx k Ĥ) 2) 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: Cρ) K R k) ρ). 3) k= I order to compute the achievable rate, we expad the mutual iformatio i 2) as follows [ Ix k ; ˆx k Ĥ) = E Pˆxk x x k,ˆx k,ĥ log,ĥˆx k k x k, Ĥ) ] 2. 4) Pˆxk Ĥˆx k Ĥ) Computig 4) requires oe to obtai the coditioal probability mass fuctios Pˆxk x,ĥˆx k k x k, [ Ĥ) ad Pˆxk Ĥˆx k Ĥ) = E x k Pˆxk x,ĥˆx k k x k, ]. Ĥ) Sice o closed-form expressios are available, oe eeds to estimate them by Mote-Carlo samplig, i.e., by simulatig may oise ad iterferece realizatios, ad by mappig the resultig ˆx k to poits over a rectagular grid i the complex plae as described i [24]. With this techique, we obtai a lower boud o R k) ρ) [44, p. 353]

14 4 that becomes icreasigly tight as the grid spacig is made smaller. 2 Note that 4) holds for every choice of iput distributio ad for ADCs with arbitrary resolutio. I the remaider of this sectio, we will however focus o oe-bit ADCs ad QAM costellatios. C. High-SNR Approximatio of the Achievable Rate Evaluatig the coditioal probabilities i 4) is tedious as it ivolves the simulatio of a large umber of oise ad iterferece trials for each realizatio of the chael H. We ext provide a accurate high-snr approximatio of 2) that is easier to compute i practice. Our high-snr approximatio relies o the followig assumptios: a sigle pilot per user suffices to accurately estimate the sig of the real ad imagiary part of each etry of the chael matrix; the real part ˆx R k = R{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). These assumptios result i R k) ρ) T K T h ) ˆx R k, ˆx I k Ĥ )] 2 E x k,ĥ [log ) 2 2πe) 2 det Σx k, Ĥ). 5) Here, h ) deotes the differetial etropy of a radom vector [45]. I Appedix A, we provide closed-form expressios for the mea µx k, Ĥ) ad the variace Σx k, Ĥ) for the MRC case see 29) 33)). For the ZF case, the mea is provided i 34) whereas we resort to Mote- Carlo simulatios to compute its covariace matrix. As we will illustrate i Sectio III-D, the approximatio 5) turs out to be accurate over a large rage of SNR values. D. Numerical Evaluatio of the Achievable Rate We ow assess the rates achievable with LS estimatio o a multi-user massive MIMO uplik chael whe the receiver is equipped with oe-bit ADCs. 2 The umerical routies used to evaluate 2) ca be dowloaded at massive MIMO.

15 5 ) Sigle-user case: I Fig. 3 we compare for the sigle-user case, the rates achievable with QPSK, 6-QAM, ad 64-QAM as a fuctio of ρ. 3 We depict both the rates achievable with oe-bit ADCs ad the oes for the ifiite-precisio case. The rates with oe-bit ADCs, which are computed usig 2) ad 4), are compared with the approximatio provided i 5) to verify its accuracy. The ifiite-precisio rates are computed usig 2) ad 4); ideed the evaluatio of these expressio ca be effected efficietly i the ifiite precisio case. The umber of receive ateas is N = 2, 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 oe-bit ADCs, higher-order modulatios outperform QPSK already at SNR values as low as ρ = 5 db. Note that the achievable rate does ot icrease mootoically with ρ i the 6-QAM ad 64-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. 2c). Note also that the high-snr approximatio 5) closely tracks the simulatio results for SNR values as low as db. 5 The proposed approximatio 5) eables us to accurately predict the SNR 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 SNR value. We ote that, whe QPSK is used, the differece i the achievable rates betwee the oe-bit quatized case ad the ifiite-precisio case is margial a observatio that was already reported i [24]. I cotrast, the rate loss is more proouced for higher-order costellatios. 2) Multi-user case: I Fig. 4, we plot the rates achievable with MRC ad ZF for both the oe-bit-adc ad the ifiite-precisio case whe K = users are active. 6 Motivated by the results i Fig. 3, we oly compare the rates achievable with 6-QAM ad 64-QAM. Note agai that the high-snr approximatio 5) turs out to be accurate for a large rage of SNR values. 3 To evaluate the mutual iformatio 4), we have simulated 3 radom fadig chael realizatios. For each chael realizatio we have cosidered 3 radom oise realizatios for each symbol i the trasmitted costellatio. 4 For a LTE-like system operatig at 2 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 = 42 symbols. 5 The accuracy of the approximatio 5) at low SNR values depeds critically o T ; the approximatio teds to overestimate the rate for low values of T ad to uderestimate the rate for large value of T. 6 To evaluate the mutual iformatio 4), we have simulated 3 radom fadig chael realizatios. For each chael realizatio we have cosidered 3 radom oise ad iterferece realizatios for each user ad each symbol i the trasmitted costellatio.

16 6 Rate [bits/chael use] QAM 6-QAM QPSK if. prec. oe-bit ADCs SNR, ρ [db] Fig. 3. Sigle-user achievable rate with LS estimatio ad MRC as a fuctio of the SNR ρ; N = 2, K =, T = 42; the umber of pilots P is optimized for each value of ρ. The dashed lies correspod to the high-snr approximatio 5), ad the marks correspod to the rates computed via 2) ad 4). For the oe-bit-adc case, idepedetly of whether MRC or ZF is used, the rate per user is sigificatly reduced compared to the sigle-user case cf. Fig. 3). This suggests that, with high-order modulatios, the system becomes iterferece limited because the oe-bit ADCs partially destroy the orthogoality betwee the fadig chaels associated with differet users. I fact, there is virtually o differece betwee the sigle-user ad the multi-user rate i the ifiite-precisio case if ZF is used cf. Fig. 3 ad Fig. 4b). I cotrast, whe MRC is employed, the system is iterferece limited also i the ifiite-precisio case. More pilots are required i the oe-bit-adc case compared to the ifiite-precisio case, as it is more challegig to perform chael estimatio based o the coarsely quatized observatios. For example, whe ρ = db, ad oe uses 6-QAM i combiatio with ZF, the optimal umber of pilots i the ifiite-precisio case is oe per user, whereas i the oe-bit-adc case it is five per user. 3) Depedece o the umber of BS ateas: I Fig. 5, we plot the per-user achievable rates as a fuctio of the umber of BS ateas. Here, ρ = db, K =, ad T = 42. As i the previous cases, the umber of pilot symbols is optimized, this time for each value of N.

17 7 6 6 Rate per user [bits/chael use] if. prec. oe-bit ADCs 64-QAM 6-QAM Rate per user [bits/chael use] if. prec. oe-bit ADCs 64-QAM 6-QAM SNR, ρ [db] SNR, ρ [db] a) MRC receiver. b) ZF receiver. Fig. 4. Per-user achievable rate with LS estimatio as a fuctio of the SNR ρ; N = 2, K =, T = 42; the umber of pilots P is optimized for each value of ρ. The dashed lies correspod to the high-snr approximatio 5), ad the marks correspod to the rates computed via 2) ad 4). The high-snr approximatio is agai show to be accurate for all values of N, despite the low value of ρ. We ote that higher-order costellatios outperform QPSK also whe the umber of receive ateas is much smaller tha 2. Furthermore, whe QPSK is used, the achievable rate saturates rapidly as the umber of receive ateas is icreased. 4) 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 = db, N = 2, ad K =. Here, the rates are computed via 2) ad 4). The umber of pilot symbols is umerically optimized for each value of T. We also depict the achievable rates for the perfect-csi case. Similarly to the SISO case see []), as T icreases the per-user achievable rate approaches the perfect receiver-csi rate. However, this covergece occurs at a slower pace tha for the ifiite-precisio case. This suggests that the oe-bit ADC architecture is less suitable for high-mobility scearios. Note also that the achievable rate is zero whe T. I fact, whe orthogoal pilot sequeces are trasmitted, at least pilot symbols are required whe K =.

18 8 Rate per user [bits/chael use] QAM 6-QAM QPSK if. prec. oe-bit ADCs Number of ateas, N Fig. 5. Per-user achievable rate with LS estimatio ad ZF as a fuctio of N; ρ = db, K =, T = 42; the umber of pilots P is optimized for each value of N. The dashed lies correspod to the high-snr approximatio 5), ad the marks correspod to the rates computed via 2) ad 4). Rate per user [bits/chael use] if. prec. Perfect receiver-csi oe-bit ADCs 64-QAM 6-QAM Coherece iterval, T Fig. 6. Per-user achievable rate with LS estimatio ad ZF as a fuctio of T ; ρ = db, N = 2, K = ; the umber of pilots P is optimized for each value of T.

19 9 6 Rate per user [bits/chael use] b =, 2, 3, 4, SNR, ρ [db] Fig. 7. Per-user achievable rate with LS estimatio, 64-QAM, ad ZF as a fuctio of the SNR ρ; N = 2, K =, T = 42; the umber of pilots P is optimized for each value of ρ. IV. MASSIVE MIMO WITH MULTI-BIT ADCS We ow tur our attetio to the multi-bit-adc case. To determie the quatizatio labels ad levels we approximate the chael output by a Gaussia radom variable ad use the Lloyd-Max algorithm [46], [47]. 7 The motivatio behid the Gaussia approximatio is that the per-atea received sigal coverges to a Gaussia radom variable with zero mea ad variace Kρ + as the umber of users grows. A. Depedece o ADC resolutio Focusig o 64-QAM ad ZF, we compare i Fig. 7 the achievable rate computed usig 2) ad 4) as a fuctio of the ADC resolutio ad the SNR. 8 We observe that with two-bit ADCs, the achievable rate icreases sigificatly compared to the oe-bit-adc case. For example, at ρ = db, we achieve 9% of the ifiite-precisio rate, compared to 7% with oe-bit ADCs. 7 For the system-parameter values chose i this sectio, a uiform quatizer yields similar performace. 8 Also i the multi-bit-adc case, oe ca derive a Gaussia approximatio to the achievable rate similar to 5), this approximatio is ot detailed for space costraits.

20 2 6 Rate per user [bits/chael use] b =, 2, 3, 4, Number of ateas, N Fig. 8. Per-user achievable rate with LS estimatio, 64-QAM, ad ZF as a fuctio of the N; ρ = db, K =, T = 42; the umber of pilots P is optimized for each value of ρ. Icreasig the ADC resolutio beyod four bits seems uecessary for the system parameters cosidered i Fig. 7. I Fig. 8, we compare the achievable rate as a fuctio of the ADC resolutio ad the umber of BS ateas. We observe that the use of ADCs with oly three-bit resolutio etails virtually o loss i terms of achievable rate compared to the ifiite-precisio case, for the etire rage of N show i the figure. Furthermore, to achieve 3 bits per chael use i the oe-bit-adc case, about 36 ateas are required. I cotrast, for the two-bit-adc case oly 8 ateas are required to meet the same target rate, ad for the three-bit-adc case, 6 ateas suffice. We thus observe that there exists a trade-off betwee the umber of BS ateas ad the resolutio of the ADCs required at each atea port. We ote that whe decidig o whether to equip a BS with a large umber of ateas ad low-resolutio ADCs, or to equip it with fewer ateas but with high-precisio ADCs, the power cosumptio of the ADCs alog with that of other hardware compoets i the trasceivers has to be take ito accout.

21 2 4 4 b = Rate [bits/chael use] 3 2 b =, 2, 3, 4, Rate [bits/chael use] 3 2 b =, 2, 3, SIR, ξ [db] a) MRC receiver SIR, ξ [db] b) ZF receiver. Fig. 9. Achievable rate with LS estimatio ad 6-QAM as a fuctio of the SIR ξ; ρ = db, N = 2, K =, T = 42, ad P =. B. 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 SNR. 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. To ivestigate this issue, we cosider first the case whe there are oly two users i the cell. Both users trasmit 6-QAM symbols, the first oe with SNR ρ = db, the secod oe with a varyig trasmit power. Specifically, its SNR ρ 2 rages from db to 5 db. Hece, the SIR for the first user before receiver filterig is ξ = ρ /ρ 2 i liear scale). I Fig. 9, we plot the achievable rates for the first user, for varyig ADC resolutio, as a fuctio of the SIR. As before, N = 2 ad T = 42. We set the umber of pilots per user to P/K =. Note that the iterferig sigal is i-bad ad ca ot be removed by RF filterig. With MRC, the system is sesitive to iterferece eve i the ifiite-precisio case. Cosequetly, icreasig the ADC resolutio beyod three bits provides o gai i terms of

22 22 TABLE I SUMMARY OF SIMULATION PARAMETERS Descriptio 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 Assumptio Circular cell 335 meters 35 meters log d) db 2 ateas users 42 chael uses pilots per user 2 GHz 2 MHz 8.5 dbm 74.2 dbm/hz 5 db iterferece mitigatio. I cotrast, with ZF the system ca hadle substatially more iterferece. Ideed, i the ifiite-precisio case, the achievable rate is uaffected by the iterferece. With oe-bit ADCs o the other had, the iterferece caot be mitigated. For example, at SIR ξ = 2 db the rate drops to 43% of the ifiite-precisio case. For SIR ξ = 4 db, less tha 3% is attaied. By icreasig the resolutio beyod oe bit, the system ca tolerate more iterferece. For three-bit ADCs, we achieve 89% ad 5% of the ifiite-precisio rate whe the SIR is 2 db ad 4 db, respectively. 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 [48]. 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 = 85 meters from the BS, results i a SNR of approximately ρ = db. The remaiig K users i the cell are radomly dropped accordig to a uiform distributio o the circular rig of ier radius d d meters ad outer radius d + d meters, for a

23 % worst throughput [bits/chael use] 3 2 b =, 2, 3, 4, % worst throughput [bits/chael use] 3 2 b =, 2, 3, 4, d [meters] a) MRC receiver d [meters] b) ZF receiver. Fig.. The % worst throughput with LS estimatio ad 6-QAM for a user located d = 85 meters away from the BS as a fuctio of d for the parameters specified i Table I. distace spread < d < 5 meters. The case d = correspods to the sceario whe power cotrol is executed perfectly. The case d = 5 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 SNR for each iterferig user lies i the rage [ 9. db, 5.3 db]. I Fig., we plot the % worst throughput i.e., the throughput correspodig to the % poit of the CDF of throughputs), for the iteded user located d = 85 meters away from the BS, as a fuctio of d. We focus o 6-QAM ad assume that the received sigal power level for each user is kow to the BS. To attai the curves, we have cosidered radom iterferig user drops for each d value. 9 As expected, the gap to the ifiite-precisio rate grows as d icreases. I the ucoordiated case, with oe-bit ADCs ad ZF, we attai 57% of the rate achievable with perfect power cotrol. The correspodig umber for the three-bit-adc case is 79%. This shows that high rates are achievable with low-resolutio ADCs eve i absece of power cotrol. 9 To compute the rate i 2), we simulate 3 radom fadig chael realizatios for each user drop.

24 24 V. CONCLUSIONS We have aalyzed the performace of a low-resolutio quatized uplik massive MIMO system operatig over a frequecy flat Rayleigh block-fadig chael whose realizatios are ot kow a priori to trasmitter ad receiver. We have show that for the oe-bit massive MIMO case, high-order costellatios, such as 6-QAM, ca be used to covey iformatio at higher rates tha with QPSK. This holds i spite of the oliearity itroduced by the oe-bit quatizers. Furthermore, reliable commuicatio ca be achieved by usig simple sigal processig techiques at the receiver, i.e., LS chael estimatio ad MRC detectio. By icreasig the resolutio of the ADCs by oly a few bits, e.g., to three or four bits, we ca achieve ear ifiite-precisio performace for a large 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 [28] i which the authors reported that, with a specific modulatio ad codig scheme take from IEEE 82., four to six bits are required to achieve a packet error rate below 2 at a SNR close to the oe eeded i the ifiite-precisio 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 RF ad basebad processig circuitry. APPENDIX A DERIVATION OF 5) To attai the high-snr approximatio 5), we shall assume that a sigle pilot per user is sufficiet to correctly estimate the quadrat i which each etry of the chael matrix H lies. I this case, the chael estimate i 8) reduces to Ĥ = sg{r{h}} + jsg{i{h}}. 6) I what follows, we will derive a closed-form approximatio to the achievable rate 2) uder the assumptio that the estimate i 6) is kow to the receiver. To this ed, we first eed to determie the real ad imagiary parts of the received sigal. To keep the otatio compact, we set

25 25 ˆx R k = R{ˆx k} ad ˆx I k = I{ˆx k}. Furthermore, by writig a,k = a R,k + jai,k, h,k = h R,k + jhi,k, ad r = 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 N ) ˆx R k = R{a,kr } = a R a R,k r R + a I,k r) I,k a I,k = sg{h R,k }cr,r + sg{h I,k } ci,i. 7) = = Here, we have defied c R,R = sg{h R,k }rr ad c I,I = sg{h I,k }ri. Similarly, for the imagiary part we ca write ˆx I k = I{a,kr } = = = ) N a R a R,k r I a I,k r R,k = = Here, we have defied c R,I = sg{h R,k }cr,i = sg{h R,k }ri ad c I,R = sg{h I,k }rr. Now, we collect the real ad imagiary compoets i a vector [ˆx R k, ˆxI k ]T a I,k sg{h I,k } ci,r ). 8) 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 2 2 covariace matrix Σx k, Ĥ). It follows that the mutual iformatio i 4) ca be approximated as ) Ix k ; ˆx k ˆx Ĥ) h R k, ˆx I k Ĥ )] 2 E x k,ĥ [log 2 2πe) 2 det Σx k, Ĥ). 9) Here, the differetial etropy hˆx R k, ˆxI k Ĥ) is evaluated by assumig that [ˆxR k, ˆxI k ]T is coditioally Gaussia give x k ad Ĥ, which implies that for iputs draw from a QAM costellatio, the coditioal probability of [ˆx R k, ˆxI k ]T give Ĥ is a Gaussia mixture. The achievable rate i 5) follows from 9) by takig ito accout the rate loss due to the trasmissio of P = K pilot symbols oe per user) 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 biary radom variables c R,R Pr { c R,R, c I,I, c R,I ad c I,R = } = Pr { sg{h R,k}r R = } = Pr { sg{h R,k} sg {, whose support is {, +}. For c R,R, it holds that R{h,k x k } + R{w } + j k R{h,j x j } } = } Φζ RR, ), 2) where i the last step, we have approximated the iterferece term j k R{h,jx j } as a zero-mea Gaussia radom variable with variace ρ j k h,j 2 ad defied h ζ, RR R =,k x R k 2ρ + ρ j k h,j 2 hi,k xi k sg{h R,k } ). 2)

26 26 For the sigle-user case, the approximatio 2) is exact sice there is o iterferece. Proceedig i a aalogous way, we ca show that Pr { c I,I ad that Pr { c I,R = } Φζ I,R 2, ), where ζ II 2, = 2ρ + ρ j k h,j 2 = } Φζ2,), II that Pr { c R,I = } Φζ R,I, ) h I,k ) x R k + hr,k xi k sg{h I,k } 22) ζ R,I, = 2ρ + ρ j k h,j 2 h R,k ) x I k + hi,k xr k sg{h R,k } 23) ad ζ I,R 2, = 2ρ + ρ j k h,j 2 h I,k ) x I k hr,k xr k sg{h I,k }. 24) Next, we use these approximatios to derive µx k, Ĥ) ad Σx k, Ĥ). ) MRC receiver: It follows from ) ad 6) that the receive filter for the kth user ca be writte as a k = 2N sg{r{h k}} + jsg{i{h k }}). Cosequetly, the real ad imagiary compoets of ˆx k reduce to ad ˆx R k = 2N ˆx I k = 2N = = For the real compoet, we ote that c R,R sg{h R,k } r R + sg{h I,k} r I ) = 2N sg{h R,k } r I sg{h I,k} r R ) = 2N c I,I ) = 2} = Φ ζ, RR )Φ ζ2,) II ad Pr{c R,R coditioal mea of ˆx R k give x k ad Ĥ, ca be writte as [ ] E ˆx R k x k, Ĥ = E [ ] c R,R + c I,I 2N = N = N = = = Similarly, for the imagiary part, it holds that [ ] E ˆx I k x k, Ĥ = N = = c R,R c R,I ) + c I,I 25) ) c I,R. 26) + c I,I ) is supported o { 2,, 2} ad that Pr{c R,R + + c I,I ) = 2} = Φζ, RR )Φζ2,). II Thus, the Φζ RR, )Φζ II 2,) Φ ζ RR, )Φ ζ II 2,) ) Φζ RR, ) Φ ζ II 2,) ). 27) = ) Φζ R,I, ) Φ ζ I,R 2, ). 28)

27 27 The sought-after mea vector is the µx k, H) = N ΦζRR, ) Φ ζ2,) II. 29) Φζ R,I, ) Φ ζ I,R 2, ) = Movig to Σx k, H), its first etry ca be obtaied as follows [ ) ] [ ] [Σx k, H)], = E ˆx R 2 2 k xk, Ĥ E ˆx R k x k, Ĥ = N 2 = N 2 = = Aalogously, it holds that [ ] [Σx k, H)] 2,2 = E ˆx I 2 k) xk, Ĥ Φζ RR, )Φζ II 2,) + Φ ζ RR, )Φ ζ II 2,) Φζ RR, ) Φ ζ II 2,) ) 2 ) Φζ RR, )Φ ζ RR, ) + Φζ II 2,)Φ ζ II 2,) ). 3) = N 2 = Furthermore, it ca be verified that c R,R [ ] [Σx k, H)],2 = E ˆx R k ˆx I k x k, Ĥ E = N 2 = N 2 = N 2 = E [ c R,R = = Fially, because of symmetry, E [ c R,R [ ] 2 E ˆx I k x k, Ĥ ) Φζ R,I, )Φ ζ R,I, ) + Φζ I,R 2, )Φ ζ I,R 2, ). 3) ) + c I,I c R,I [ ] ˆx R k x k, Ĥ E ) + c I,I c R,I ] [ + c I,I E c R,I ) c I,R =, which meas that we ca write [ ] ˆx I k x k, Ĥ )] [ c I,R E c R,R ] + c I,R ] [ + c I,I E c R,I ]) + c I,R Φζ RR, ) Φ ζ2,) ) ) II Φζ R,I, ) Φ ζ I,R 2, ). 32) [Σx k, H)] 2, = [Σx k, H)],2. 33) 2) ZF receiver: For the ZF receiver, we resort to 7) ad 8) to derive the required mea vector. I this case, c R,R + c I,I ) ad c R,I c I,R ) are quaterary radom variables. Followig the same steps as for the MRC receiver, the mea ca be writte as follows: µx k, H) = 2 ar ),k sg{h R,k }ΦζRR, ) ai,k sg{h 2,) ar,k I,k }Φ ζii sg{h R,k } a R ) = 2,k sg{h R,k }ΦζR,I, ) ai,k sg{h I,k }Φ ζi,r 2, ) ar,k sg{h R,k } ai,k sg{h I,k } ai,k sg{h I,k }. 34)