Massive MIMO Communications

Transcription

1 Massive MIMO Communications Trinh Van Chien and Emi Björnson Book Chapter N.B.: When citing this work, cite the origina artice. Part of: 5G Mobie Communications, Ed. Wei Xiang, an Zheng, Xuemin Sherman) Shen, 017, pp ISBN: Copyright: Springer DOI: Avaiabe at: Linköping University Eectronic Press

2 Massive MIMO Communications Trinh Van Chien and Emi Björnson Abstract Every new network generation needs to make a eap in area data throughput, to manage the growing wireess data traffic. The Massive MIMO technoogy can bring at east ten-fod improvements in area throughput by increasing the spectra efficiency bit/s/hz/ce), whie using the same bandwidth and density of base stations as in current networks. These extraordinary gains are achieved by equipping the base stations with arrays of a hundred antennas to enabe spatia mutipexing of tens of user terminas. This chapter expains the basic motivations and communication theory behind the Massive MIMO technoogy, and provides impementationreated design guideines. 1 Introduction Much higher area data throughput is required in future ceuar networks, since the goba demand for wireess data traffic is continuousy growing. This goa can be achieved without the need for more bandwidth or additiona base stations if the spectra efficiency measured in bit/s/hz/ce) is improved. This chapter expains why the Massive MIMO mutipe-input mutipe-output) communication technoogy, where muti-antenna base stations spatiay mutipex a mutitude of user terminas over the entire bandwidth, is we-suited for this purpose. The rationae behind the Massive MIMO concept and its transmission protoco is expained from a historica perspective in Sect.. Next, Sect. 3 provides a basic communication theoretic performance anaysis. Cosed-form spectra efficiency expressions are derived and the key properties and performance imitations of Massive MIMO are highighted. The chapter Trinh Van Chien Linköping University, Department of Eectrica Engineering ISY), SE Linköping, Sweden e-mai: trinh.van.chien@iu.se Emi Björnson Linköping University, Department of Eectrica Engineering ISY), SE Linköping, Sweden e-mai: emi.bjornson@iu.se c Springer Internationa Pubishing Switzerand 017. This is the authors manuscript version of the foowing origina pubication: Trinh Van Chien, Emi Björnson, Massive MIMO Communications, in 5G Mobie Communications, W. Xiang et a. eds.), pp , Springer, 017. DOI / The officia pubication is avaiabe here. 1

3 Trinh Van Chien and Emi Björnson is concuded by Sect. 4 where impementation-reated design guideines are given, particuary regarding power aocation and the reuse of piot sequences for efficient channe estimation. Muti-ce simuations are provided to showcase that the Massive MIMO technoogy can provide ten-fod or even 50-fod improvements in spectra efficiency over contemporary technoogy, without the need for advanced signa processing or network coordination. Finay, the fu mathematica detais are provided in Appendix at the end of this chapter. Importance of Improving the Spectra Efficiency The wireess information traffic has doubed every two and a haf years since the beginning of wireess communications, as observed by Martin Cooper at ArrayComm in the nineties. Different technoogies and use cases have dominated in different periods, but the exponentia increase is currenty driven by wireess data traffic in ceuar and oca area networks. There are no indications that this trend wi break anytime soon; in fact, a sighty faster traffic growth is predicted in the we-reputed Cisco Visua Networking Index and Ericsson Mobiity Report. To keep up with the rapid traffic growth, a key goa of the 5G technoogies is to improve the area throughput by orders of magnitude; 100 and even 1000 higher throughput are reguary mentioned as 5G design goas. The area throughput of a wireess network is measured in bit/s/km and can be modeed as foows: Area throughput bit/s/km ) = Bandwidth Hz) Ce density ces/km ) Spectra efficiency bit/s/hz/ce). This simpe formua reveas that there are three main components that can be improved to yied higher area throughput: 1) more bandwidth can be aocated for 5G services; ) the network can be densified by adding more ces with independenty operating access points; and 3) the efficiency of the data transmissions per ce and for a given amount of bandwidth) can be improved. The improvements in area throughput in previous network generations have greaty resuted from ce densification and aocation of more bandwidth. In urban environments, where contemporary networks are facing the highest traffic demands, ceuar networks are nowadays depoyed with a few hundred meters inter-site distances and wireess oca area networks WLANs) are avaiabe amost everywhere. Further ce densification is certainy possibe, but it appears that we are reaching a saturation point. Moreover, the most vauabe frequency bands are beow 6 GHz because these frequencies can provide good network coverage and service quaity, whie higher bands might ony work we under short-range ine-of-sight conditions. In a typica country ike Sweden, the ceuar and WLAN technoogies have in tota been aocated more than 1 GHz of bandwidth in the interva beow 6 GHz and thus we cannot expect any major bandwidth improvements either.

4 Massive MIMO Communications 3 In contrast, the spectra efficiency SE) has not seen any major improvements in previous network generations. Hence, it might be a factor that can be greaty improved in the future and possiby become the primary way to achieve high area throughput in 5G networks. In this chapter, we describe the rationae and background of the physica-ayer technoogy Massive mutipe-input mutipe-output MIMO), which provides the means to improve the SE of future networks by one or two orders of magnitude..1 Muti-User MIMO Communication The SE of a singe-input singe-output SISO) communication channe, from a singe-antenna transmitter to a singe-antenna receiver, is upper bounded by the Shannon capacity, which has the form og 1 + SNR) bit/s/hz for additive white Gaussian noise AWGN) channes. The SISO capacity is thus a ogarithmic function of the signa-to-noise ratio SNR), denoted here as SNR. To improve the SE we need to increase the SNR, which corresponds to increasing the power of the transmitted signa. For exampe, suppose we have a system that operates at bit/s/hz and we woud ike to doube its SE to 4 bit/s/hz, then this corresponds to improving the SNR by a factor 5, from 3 to 15. The next doubing of the SE, from 4 to 8 bit/s/hz, requires another 17 times more power. In other words, the ogarithm of the SE expression forces us to increase the transmit power exponentiay fast to achieve a inear increase in the SE of the SISO channe. This is ceary a very inefficient and non-scaabe way to improve the SE, and the approach aso breaks down when there are interfering transmissions in other ces that scae their transmit powers in the same manner. We therefore need to identify another way to improve the SE of ceuar networks. Each base station BS) in a ceuar network serves a mutitude of user terminas. Traditionay, the time/frequency resources have been divided into resource bocks and ony one of the user terminas was active per bock. This termina can then receive a singe data stream with an SE quantified as og 1 + SNR). The efficient way to increase the SE of a ceuar network is to have mutipe parae transmissions. If there are G parae and independent transmissions, the sum SE becomes Gog 1 + SNR) where G acts as a mutipicative pre-og factor. Parae transmissions can be reaized by having mutipe transmit antennas and mutipe receive antennas. There are two distinct cases: 1. Point-to-point MIMO [39], where a BS with mutipe antennas communicates with a singe user termina having mutipe antennas.. Muti-user MIMO [34], where a BS with mutipe antennas communicates with mutipe user terminas, each having one or mutipe antennas. There are many reasons why muti-user MIMO is the most scaabe and attractive soution [17]. Firsty, the waveength is 5-30 cm in the frequency range of ceuar communication 1-6 GHz). This imits the number of antennas that can be depoyed

5 4 Trinh Van Chien and Emi Björnson in a compact user termina for point-to-point MIMO, whie one can have amost any number of spatiay separated singe-antenna terminas in muti-user MIMO. This is an important distinction since the number of simutaneous data streams that can be separated by MIMO processing equas the minimum of the number of transmit and receive antennas. Secondy, the wireess propagation channe to a user termina is ikey to have ony a few dominating paths, which imits the abiity to convey mutipe parae data streams to a termina in point-to-point MIMO. The corresponding restriction on muti-user MIMO is that the users need to be, say, a few meters apart to have sufficienty different channe characteristics, which is a very oose restriction that is true in most practica scenarios. Thirdy, advanced signa processing is needed at the terminas in point-to-point MIMO to detect the mutipe data streams, whie each termina in muti-user MIMO ony needs to detect a singe data stream. The canonica muti-user MIMO system consists of a BS with M antennas that serves singe-antenna terminas; see Fig. 1 for a schematic iustration. The BS mutipexes one data stream per user in the downink and receives one stream per user in the upink. Simpy speaking, the BS uses its antennas to direct each signa towards its desired receiver in the downink, and to separate the mutipe signas received in the upink. If the termina is equipped with mutipe antennas, it is often beneficia to use these extra antennas to mitigate interference and improve the SNR rather than sending mutipe data streams [6]. For the ease of exposition, this chapter concentrates on singe-antenna terminas. In this case, minm,) represents the maxima number of data streams that can be simutaneousy transmitted in the ce, whie sti being separabe in the spatia domain. The number minm,) is referred to as the mutipexing gain of a muti-user MIMO system.. Lessons Learned The research on muti-user MIMO, particuary with muti-antenna BSs, has been going on for decades. Some notabe eary works are the array processing papers [1, 38, 44, 47], the patent [36] on spatia division mutipe access SDMA), and the semina information-theoretic works [11, 18, 4, 43, 46] that characterized the achievabe muti-user capacity regions, assuming that perfect channe state information CSI) is avaiabe in the system. In this section, we summarize some of the main design insights that have been obtained over the years. Capacity-achieving transmission schemes for muti-user MIMO are based upon non-inear signa processing; for exampe, the dirty-paper coding DPC) scheme that achieves the downink capacity and the successive interference canceation SIC) scheme that achieves the upink capacity. The intuition behind these schemes is that the inter-user interference needs to be suppressed, by interference-aware transmit processing or iterative interference-aware receive processing, to achieve the optima performance. These non-inear schemes naturay require extensive computations and accurate CSI, because otherwise the attempts to subtract interference cause more harm than good.

6 Massive MIMO Communications 5 a) b) Fig. 1 Iustration of the downink and upink transmission in a muti-user MIMO system, where the BS is equipped with M antennas and serves user terminas simutaneousy. This iustration focuses on ine-of-sight propagation where the downink signas can be viewed as anguar beams, but muti-user MIMO works equay we in non-ine-of-sight conditions. a) Downink in mutiuser MIMO. b) Upink in muti-user MIMO How arge are the gains of optima non-inear processing e.g., DPC and SIC) over simpified inear processing schemes where each user termina is treated separatey? To investigate this, et us provide a numerica exampe where = 10 user terminas are simutaneousy served by a BS with M antennas. For simpicity, each user is assumed to have an average SNR of 5 db, there is perfect CSI avaiabe everywhere, and the channes are modeed as uncorreated Rayeigh fading this is defined in detai in Sect. 3). Figure shows the average sum SE, as a function of M, achieved by sum capacity-achieving non-inear processing and a simpified inear processing scheme caed zero-forcing ZF), which attempts to suppress a interference. The resuts are representative for both upink and downink transmissions. This simuation shows that the non-inear processing greaty outperforms inear ZF when M. The operating point M = makes particuar sense from a mutipexing perspective since the mutipexing gain minm, ) does not improve if we et M increase for a fixed. Nevertheess, Fig. shows that there are other rea-

7 6 Trinh Van Chien and Emi Björnson Spectra Efficiency bit/s/hz/ce) No interference Non inear: Sum capacity Linear: ZF Number of BS Antennas M) Fig. Average spectra efficiency in a muti-user MIMO system with = 10 users and varying number of BS antennas. Each user has an average SNR of 5 db and the channes are Rayeigh fading. The sum capacity is compared with the performance of inear ZF processing and the upper bound when negecting a interference. The resuts are representative for both upink and downink sons to consider M > ; the capacity increases and the performance with inear ZF processing approaches the capacity. Aready at M = 0 i.e., M/ = ) there is ony a sma gap between optima non-inear processing and inear ZF. In fact, both schemes aso approach the upper curve in Fig. which represents the upper bound where the interference between the users is negected. This shows that we can basicay serve a the users as if each one of them was aone in the ce. First esson earned: Linear processing, such as ZF, provides a sum spectra efficiency cose to the sum capacity when M. The performance anaysis and optimization of inear processing schemes have received much attention from academic researchers. Whie non-inear schemes are hard to impement but reativey easy to anayze and optimize, inear processing schemes have proved to have the opposite characteristics. In particuar, computing the optima downink inear precoding is an NP-hard probem in many cases [7], which requires monotonic optimization toos to sove; see for exampe [9]. Nevertheess, the suboptima ZF curve in Fig. was generated without any compicated optimization, thus showing that the optima inear processing obtained in [9] can ony bring noticabe gains over simpe ZF for M, which is the regime where we have earnt not to operate. As mentioned earier, the BS needs CSI in muti-user MIMO systems to separate the signas associated with the different users. Perfect CSI can typicay not be achieved in practice, since the channes are changing over time and frequency, and thus must be estimated using imited resources. The channe estimation of a frequency-seective channe can be handed by spitting the frequency resources into

8 Massive MIMO Communications 7 mutipe independent frequency-fat subchannes that can be estimated separatey. A known piot sequence is transmitted over each such subchanne and the received signa is used to estimate the channe response. In order to expore a spatia channe dimensions, this sequence must at east have the same ength as the number of transmit antennas [8]. This means that a piot sequence sent by the BS needs to have the ength M, whie the combined piot sequence sent by the singe-antenna user terminas needs to have the ength. There are two ways of impementing the downink and upink transmission over a given frequency band. In frequency division dupex FDD) mode the bandwidth is spit into two separate parts: one for the upink and one for the downink. Piot sequences are needed in both the downink and the upink due to the frequencyseective fading, giving an average piot ength of M + )/ per subchanne. There is an aternative time-division dupex TDD) mode where the whoe bandwidth is used for both downink and upink transmission, but separated in time. If the system switches between downink and upink faster than the channes are changing, then it is sufficient to earn the channes in ony one of the directions. This eads to an average piot ength of minm,) per subchanne, if we send piots ony in the most efficient direction. In the preferabe operating regime of M, we note that TDD systems shoud send piots ony in the upink and the piot ength becomes minm,) =. We concude that TDD is the preferabe mode since it not ony requires shorter piots than FDD, but is aso highy scaabe since the piot ength is independent of the number of BS antennas. We give a concrete numerica exampe in Fig. 3 for downink transmission with = 10 users, an SNR of 5 db, and uncorreated Rayeigh fading channes. Two inear precoding schemes are considered; a) maximum ratio MR) and b) zeroforcing ZF). These schemes are ater defined mathematicay in Sect. 3. This simuation compares the SE obtained when having perfect CSI with the performance when having CSI estimated with piot sequences of ength τ p. The SE is shown as a function of the number of BS antennas, M, and we compare TDD mode using τ p = = 10 with FDD mode using either τ p = 10, τ p = M, or τ p = minm,50), where the atter modes an arbitrariy chosen maximum piot ength of 50 e.g., motivated by piot overhead constraints). In TDD mode there is a visibe performance oss in Fig. 3 as compared to having perfect CSI. The oss with MR precoding is very sma, which shows that it is robust to estimation errors. The performance oss is arger for ZF precoding, since estimation errors make it harder to suppress interference, but we notice that ZF anyway provide higher performance than MR for a considered M. We notice that the performance osses are substantiay constant irrespective of the number of BS antennas, thus TDD systems aways benefit from adding more antennas. In contrast, FDD systems ony benefits from adding more antennas if the piot sequences are aso made onger, as in the case τ p = M. With τ p = 10 there is no benefit from having more than 10 antennas, whie the performance saturates at 50 antennas when τ p = minm,50). In summary, TDD operation is fuy scaabe with respect to the number of BS antennas, whie FDD operation can ony hande more antennas by aso increasing the piot overhead. It is practicay feasibe to depoy FDD systems

9 8 Trinh Van Chien and Emi Björnson with many antennas, particuary for sowy varying channes where we can accept a arge piot overhead, but TDD is aways the better choice in this respect. Second esson earned: The channe estimation is simpified when operating in TDD mode, since the piot sequences ony need to be of ength irrespective of the number of BS antennas M. Note that the upink works in the same way in the TDD and FDD modes, whie the distinct benefit of TDD in terms of scaabiity appears in the downink...1 Favorabe Propagation Reca from Fig. that by adding more BS antennas, both the sum capacityachieving non-inear processing and the simpified inear ZF processing approached the case without interference. This is not a coincidence but a fundamenta property that is referred to as favorabe propagation. Let h 1,h C M represent the channe responses between a BS and two different user terminas. If these vectors are non-zero and orthogona in the sense that h H 1 h = 0, 1) where ) H denotes the conjugate transpose, then the BS can competey separate the signas s 1,s transmitted by the users when it observes y = h 1 s 1 + h s. By simpy computing the inner product between y and h 1, the BS obtains h H 1 y = h H 1 h 1 s 1 + h H 1 h s = h 1 s 1 ) where the inter-user interference disappeared due to 1). The same thing can be done for the second user: h H y = h s. Note that the BS needs perfect knowedge of h 1 and h to compute these inner products. The channe orthogonaity in 1) is caed favorabe propagation, since the two users can communicate with the BS without affecting each other. Is there a chance that practica channes offer favorabe propagation? Probaby not according to the strict definition that h H 1 h = 0, but an approximate form of favorabe propagation is achieved in non-ine-of-sight scenarios with rich scattering: Lemma 1. Suppose that h 1 C M and h C M have independent random entries with zero mean, identica distribution, and bounded fourth-order moments, then amost surey as M. h H 1 h M 0 3) Proof. This is a consequence of the aw of arge numbers. A direct proof is provided aong with Theorem 3.7 in [14].

10 Massive MIMO Communications Spectra Efficiency bit/s/hz) Perfect CSI TDD τ p=10) or FDD τ p=m) FDD τ p=minm,50)) FDD τ p=10) Number of BS Antennas M) a) Spectra Efficiency bit/s/hz) Perfect CSI TDD τ p=10) or FDD τ p=m) FDD τ p=minm,50)) FDD τ p=10) Number of BS Antennas M) b) Fig. 3 Average downink spectra efficiency, as a function of the number of BS antennas, with different processing schemes and different types of CSI avaiabe at the BS. a) Downink simuation with maximum ratio precoding. b) Downink simuation with zero-forcing precoding

11 10 Trinh Van Chien and Emi Björnson This emma shows that the inner product between h 1 and h, if normaized with the number of BS antennas, goes asymptoticay to zero as M increases. We refer to this as asymptotic favorabe propagation and note that this phenomenon expains the behaviors in Fig. ; the difference between having no inter-user interference and suppressing the interference by ZF becomes smaer and smaer as the number of antennas increases, because the oss in desired signa gain when using ZF reduces when the user channes become more orthogona. One specia case in which Lemma 1 hods is h 1,h CN 0,I M ), where CN, ) denotes a muti-variate circuary symmetric compex Gaussian distribution and I M is the M M identity matrix. This is known as uncorreated Rayeigh fading and in this case one can even prove that the variance of the inner product in 3) is 1/M and thus decreases ineary with the number of antennas [31]. Many academic works on Massive MIMO systems consider Rayeigh fading channes, due to the anaytic tractabiity of Gaussian distributions. Nevertheess, Lemma 1 shows that asymptotic favorabe propagation hods for other random channe distributions as we. This mathematica resut can be extended to aso incude correation between the eements in a channe vector. One can aso derive simiar anaytic resuts for ine-of-sight propagation [31] and behaviors that resembe asymptotic favorabe propagation have been observed aso in the rea-word muti-user MIMO channe measurements presented in [16, 0]. Third esson earned: Most wireess channes seem to provide asymptotic favorabe propagation. This esson is yet another reason to design muti-user MIMO systems with M. It is, however, important to note that h H 1 h )/M 0, as M, does not impy that h H 1 h 0. Strict favorabe propagation is unikey to appear in practica or theoretica channes. In fact, the inner product h H 1 h grows roughy as M for Rayeigh fading channes. The key point is that this correation has a negigibe impact, since the SE depends on h H 1 h )/M which goes to zero roughy as 1/ M. Moreover, the main suppression of inter-user interference appears aready at reativey sma number of antennas due to the square root..3 Massive MIMO Concept The Massive MIMO concept was proposed in the semina paper [8] and described in the patent [9], both of which have received numerous scientific awards. Massive MIMO takes muti-user MIMO communications to a new eve by designing a highy scaabe communication protoco that utiizes the three essons described in Sect... The basic information and communication theoretic imits of this 5G technoogy were estabished in eary works such as [3, 19, 1, 3, 30]. In this chapter we define Massive MIMO as foows:

12 Massive MIMO Communications 11 Frequency T c Upink piots and data Downink data B c Time Fig. 4 Iustration of the basic Massive MIMO transmission protoco, where the time-frequency resources are divided into coherence intervas, each containing τ c = B c T c transmission symbos. Each coherence interva contains upink piot sequences and can be used for both upink and downink payoad data transmission based on TDD operation Massive MIMO is a muti-user MIMO system with M antennas and users per BS. The system is characterized by M and operates in TDD mode using inear upink and downink processing. This definition does not manifest any particuar ratio between M and, or any particuar orders of magnitude that these parameters shoud have. One attractive exampe is a system with M in the range of 100 to 00 antennas, serving between = 1 and = 40 users depending on the data traffic variations. The first pubic reatime impementation of Massive MIMO is the LuMaMi testbed described in [41], which features M = 100 and = 10. We stress that other definitions of Massive MIMO are avaiabe in other works and can both be more restrictive e.g., require certain dimensionaity of M and ) and ooser e.g., aso incude FDD mode), but in this chapter we ony consider the definition above. The BS antenna array typicay consists of M dipoe antennas, each having an effective size λ/ λ/, where λ is the waveength. This means that an array area of 1 m can fit 100 antennas at a 1.5 GHz carrier frequency and 400 antennas at 3 GHz. Each antenna is attached to a separate transceiver chain, so that the system can access the individua received signa at each antenna and seect the individua signas to be transmitted from each antenna. The array can have any geometry; inear, rectanguar, cyindrica, and distributed arrays are described in [5]. It is important to note that no mode of the array geometry is expoited in the Massive MIMO processing, thus the antennas can be depoyed arbitrariy without any geometrica array caibration. The basic Massive MIMO transmission protoco is iustrated in Fig. 4. The timefrequency resources are divided into bocks of size B c Hz and T c s, with the purpose of making each user channe approximatey frequency-fat and static within a bock. Hence, the bandwidth B c is seected to be smaer or equa to the anticipated channe coherence bandwidth among the users, whie T c is smaer or equa to the anticipated channe coherence time of the users. For this particuar reason, each bock is referred to as a coherence interva. The number of transmission symbos that fit into

13 1 Trinh Van Chien and Emi Björnson a coherence interva is given by τ c = B c T c, due to the Nyquist-Shannon samping theorem. The dimensionaity of the coherence interva depends greaty on the anticipated system appication. For exampe, a coherence interva of τ c = 00 symbos can be obtained with B c = 00 khz and T c = 1 ms, which supports highway user veocities in urban environments at GHz carrier frequencies. Much arger coherence intervas e.g., τ c at the order of 10 3 or 10 4 ) can be obtained by imiting the appication to scenarios with ow user mobiity and short deay spread. Each coherence interva is operated in TDD mode and can contain both downink and upink payoad transmissions. To enabe channe estimation at the BS, τ p of the symbos in each coherence interva are aocated for upink transmission of piot sequences where τ p ), whie the remaining τ c τ p symbos can be aocated arbitrariy between upink and downink payoad data transmissions. We et γ UL and γ DL denote the fractions of upink and downink payoad transmission, respectivey. This means that the upink contains γ UL τ c τ p ) data symbos and the downink contains γ DL τ c τ p ) data symbos per coherence interva. Naturay, these fractions satisfies γ UL + γ DL = 1 and γ UL,γ DL 0. Notice that no downink piots are assumed in this protoco, since the effective precoded channes converge to their mean vaues when the BS has many antennas due to the aw of arge numbers). It is certainy possibe to aso send a sma amount of downink piots, particuary for estimating the sma fading variations of the effective precoded channes, but the additiona gains from doing this appears to be sma in many reevant Massive MIMO cases [33]. Based on this definition of Massive MIMO, the next sections anayze how arge SEs that the transmission protoco can offer in 5G ceuar networks. 3 Performance Anaysis In this section, we describe the upink detection and downink precoding of a Massive MIMO network, and anayze the achievabe system performance. We consider a basic Massive MIMO network comprising L ces, each consisting of a BS with M antennas and singe-antenna user terminas. The channe response between the th BS and user k in the ith ce is denoted by h i,k = [h i,k,1... h i,k,m ]T C M, where ) T denotes the transpose. These channe vectors are ergodic random variabes that are assumed to take new independent reaizations in each coherence interva; reca the Massive MIMO protoco described in Sect..3. To show that the genera concept of Massive MIMO is appicabe in any propagation environment, we keep the performance anaysis genera by ony defining the basic statistica channe properties: the mean vaue and variance of each channe coefficient h i,k,m note that m stands for the mth antenna at BS, for m = 1,...,M). We et h i,k = E{h i,k } = [ h i,k,1... h i,k,m ]T 4)

14 Massive MIMO Communications 13 denote the vector of mean vaues. The variance of the mth coefficient of h i,k is denoted by βi,k = V{h i,k,m }, 5) which is independent of the antenna index m assuming that the arge-scae fading is stationary over the BS array). We aso assume that each BS and user can keep perfect track of these ong-term statistica properties, and that the user channes are statisticay independent. Using these channe properties, we now anayze the upink and the downink. 3.1 Upink with Linear Detection For each upink symbo, the received baseband signa y C M at the th BS is modeed as y = L i=1 k=1 h i,k pi,k x i,k + n, 6) where x i,k is the normaized transmission symbo with E{ x i,k } = 1) and p i,k is the transmit power of user k in ce i. The receiver hardware at the BS is contaminated by additive white noise, as modeed by the vector n C M which is zero-mean circuary symmetric compex Gaussian distributed with variance σ UL ; that is, n CN 0,σ UL I M). The matrix notations H i = [h i,1... h i, ] CM, P i = diagp i,1,..., p i, ) C, and x i = [x i,1... x i, ] T C can be used to write the muti-ce muti-user MIMO system mode from 6) in a compact matrix form: y = L i=1 H ip 1/ i x i + n. 7) The channes h i,k need to be estimated at BS to perform good detection and this is done in the upink by etting each user transmit a sequence of τ p piot symbos; see Fig. 4. We et τ p = f for some positive integer f e.g., 1,,...) which is caed the piot reuse factor. This aows for inear independence between a tota of τ p different piot sequences. This is, by design, sufficient to aocate independent piot sequence to the users in each ce and to aso divide the L ces into f disjoint ce groups having fuy independent piot sequences. The benefit of having mutipe ce groups is reduced interference during the piot transmission and the corresponding gains in estimation quaity are quantified beow. The upink received signa Y piot C M τ p at the th BS during piot transmission is Y piot L = i=1 H ip 1/ i Φ H i + N 8)

15 14 Trinh Van Chien and Emi Björnson and coects the received signa from 7) over the τ p piot symbos. Here, Φ i = [φ i,1...φ i, ] C τp denotes the piot matrix used by the users in the ith ce, where φ i,k C τ p is the piot sequence used by the kth user in that ce. The piot matrix satisfies Φ H i Φ i = τ p I. Moreover, Φ H Φ i = τ p I if ce and ce j beong to the same ce group i.e., use the same set of piots), whie Φ H Φ i = 0 if the two ces beong to different ce groups. For notationa convenience, we et P {1,...,L} denote the set of ce indices that beong to the same ce group as ce, incuding itsef. Some particuar exampes are given ater in Fig. 7. By using the channe mean and variances, defined in the beginning of Sect. 3, we can use the inear minimum mean square error LMMSE) estimator to separatey acquire each eement of h i,k from the received piot signa 8), which was proposed in [37] as a ow-compexity estimation scheme. The channe estimate ĥ i,k reated to the true channe response h i,k is given by the foowing emma. Lemma. Suppose that BS estimates each channe coefficient separatey from its received signa 8) using an LMMSE estimator. BS can then estimate the channe to the kth user in the jth ce as ĥ j,k = h j,k + p j,k β j,k i P j p i,k τ p β i,k + σ UL ) Y piot φ j,k pi,k τ p h i,k. 9) i P j Each eement of the uncorreated estimation error e j,k = h j,k ĥ j,k has zero mean and the variance ) MSE j,k = β j,k p j,k τ p β j,k 1 i P j p i,k τ p βi,k + σ UL. 10) Proof. The proof is avaiabe in Appendix at the end of this chapter. It is worth emphasizing that the estimation error variance in 10) is independent of M, thus the estimation quaity per channe coefficient is not affected by adding more antennas at the BS. Note that Lemma hods for any correation between the channe coefficients, since each coefficient is estimated separatey. If the channe coefficients are correated, with a known correation structure and distribution, the estimation quaity woud improve with the number of antennas if the estimator is modified appropriatey [8]. We aso stress that the estimation error is ony affected by noise and interference from the users in the same ce group that are aocated the same piot sequence. In addition, we notice that the estimate in 9) can be computed using eementary inear agebra operations, with ow computationa compexity. Using the channe estimates derived in Lemma, in this chapter, we anayze the performance of a Massive MIMO network with non-cooperative BSs. During upink payoad data transmission this means that the BS in ce ony utiizes its own received signa y in 6) and ony targets to detect the signas sent by its own users. Signas coming from users in other ces are perceived as inter-ce interference and eventuay treated as additiona noise. The BS in ce discriminates the

16 Massive MIMO Communications 15 signa transmitted by its kth user from the interference by mutipying the received signa in 6) with a inear detection vector v,k C M as foows: v H,k y = L i=1 t=1 = v H,k,k h p,k x,k + }{{} Desired signa v H,k h i,t pi,t x i,t + v H,k n t=1 t k v H,k,t h p,t x,t + }{{} Intra-ce interference L i=1 t=1 i v H,k h i,t pi,t x i,t } {{ } Inter-ce interference + v H,k n }{{} Residua noise where x i,t is the transmitted data symbo from user t in ce i. As seen from 11), the processed received signa is the superposition of four parts: the desired signa, intra-ce interference, inter-ce interference, and residua noise. Since the inear detection vector v,k appears in a these terms, it can be used to ampify the desired signa, suppress the interference, and/or suppress the noise. More precisey, by gathering the detection vectors at BS in matrix form as V = [v,1...v, ] C M, there are two main schemes being considered in the Massive MIMO iterature: maximum ratio MR) and zero-forcing ZF). These are given by Ĥ, V = Ĥ Ĥ ) H Ĥ 11) for MR, ) 1 1), for ZF. MR detection expoits the M observations in y to maximize the ratio between the average signa gain in 11) and the norm of the detection vector: E { v H,k h,k v,k } = vh,kĥ,k v,k ĥ,k 13) where the expectation is computed with respect to the zero-mean channe estimation error. The inequaity in 13) is satisfied with equaity by v,k = ĥ,k eading to MR detection with V = Ĥ ). In contrast, the ZF detection matrix utiizes the M observations over the antennas to minimize the average intra-ce interference, whie retaining the desired signas: ) E{V H H P1/ x } = V H Ĥ P 1/ x = Ĥ ) H Ĥ 1 ) Ĥ ) H Ĥ P 1/ x = P 1/ x 14) where the expectation is computed with respect to the zero-mean channe estimation error and the second equaity foows from the ZF detection matrix definition. The average processed signa becomes P 1/ x = [ p,1 x,1... p, x, ] T, which contains no intra-ce interference. Note that the inverse of the matrix Ĥ ) H Ĥ

17 16 Trinh Van Chien and Emi Björnson Ce 1 h, p x 11 1,1 1,1 p x h1 1, 1,, H v, 1 x,1 BS Ce i pi,1 xi,1 h i, 1 p x i, i, h i, n y H v,k H v, x k, x, Ce L pl,1 xl,1 h L, 1 p x L, L, h L, Fig. 5 Bock diagram of the upink transmission with inear detection in a muti-ce muti-user MIMO network, where BS receives a inear combination of the signas transmitted from a users in a L ces ony exists if M. There are aso muti-ce variants of ZF detection that can be used to cance out inter-ce interference; see for exampe [] and [7]. A bock diagram of the upink transmission with inear detection is provided in Fig. 5. The purpose of the detection is to make the detected signa x,k at BS equa to the true signa x,k, at east up to a scaing factor. Due to noise and estimation errors, there is aways a mismatch between the signas which is why the communication ink has a imited capacity. If the true signa x,k originates from a discrete consteation set X e.g., a quadrature ampitude moduation QAM)), x,k is seected based on v H,k y by finding the minimum distance over a the candidates x X : x,k = min x X v H,k y v H,k,kĥ p,k x. 15) This expression can be utiized to compute bit error rates and simiar uncoded performance metrics. Since modern communication systems appy channe coding over reativey ong data bocks, to protect against errors, the ergodic channe capacity is a more appropriate performance metric in 5G networks. It merits to note that the ergodic capacitites of the individua communication inks are hard to characterize exacty, particuary under imperfect channe knowedge, but tractabe ower bounds are obtained by the foowing theorem.

18 Massive MIMO Communications 17 Theorem 1. In the upink, a ower bound on the ergodic capacity of an arbitrary user k in ce is R UL,k = γul 1 τ ) p ) og τ 1 + SINR UL,k, 16) c where the signa-to-interference-and-noise ratio SINR) is SINR UL,k = L i=1 t=1 p i,t E { } p,k E v H,k h,k } { } { v H E,k h i,t p,k v H,k h,k + σ UL E { 17) v,k }. Proof. The proof is avaiabe in Appendix at the end of this chapter. Theorem 1 demonstrates that the achievabe SE of an arbitrary user k in ce in a Massive MIMO network can be described by an SINR term SINR UL,k that contains expectations with respect to the sma-scae channe fading. The numerator contains the gain of the desired signa, whie the denominator contains three different terms. The first term is the average power of a the signas, incuding both muti-user interference and the desired signa, whie the second term subtracts the part of the desired signa power that is usabe for decoding. The third term is the effective noise power. The pre-og factor 1 τ p τ c ) compensates for the fact that τ p /τ c of the transmission symbos contain piots instead of payoad data. The SE is aso mutipied by γ UL, which was defined earier as the fraction of upink data. Ceary, MR detection aims at maximizing the numerator of SINR UL,k, whie ZF detection tries to minimize the intra-ce interference. The expectations in Theorem 1 can be computed numericay for any channe distribution and any detection scheme. In the case of MR detection, the desired signa gain E{v H,k h,k } grows as M for most channe distributions, whie the noise term σul E{ v,k } ony grows as M and thus becomes ess significant the more antennas are depoyed at the BS. This property is known as the array gain from coherent detection. The behavior of the muti-user interference terms greaty depends on the channe distribution, but typicay these terms wi aso have the sower scaing of M [31], except for users that interfered with each other during piot transmission i.e., appeared in each other s expressions 10) for the estimation error variance). The atter is a phenomenon caed piot contamination and is further discussed ater in this chapter. To demonstrate these properties in detai, we now consider the specia case in which the channe between BS and user k in ce i is uncorreated Rayeigh fading: ) h i,k CN 0,βi,k I M. 18) Hence, h i,k = E{h i,k } = 0, which means that there is no ine-of-sight channe component. This specia case is reevant in rich-scattering environments where the channe does not have any statisticay dominating directivity.

19 18 Trinh Van Chien and Emi Björnson Subsequenty, the LMMSE estimate in Lemma simpifies to p ĥ j,k β j,k j,k = p i,k τ p βi,k + σ Y piot φ j,k 19) UL i P j and becomes circuary-symmetric compex Gaussian distributed: ) ĥ j,k CN 0,β j,k MSE j,k )I M. 0) There is an important reationship between the two estimated channes ĥ,k and ĥ i,k for ce indices i and such as i P expressed by ĥ pi,k βi,k i,k = p,k β,k ĥ,k. 1) This equation shows that BS cannot te apart the channes of users that send the same piot sequence; the estimates are the same up to a scaing factor. This fact is the cause of piot contamination and wi have a key impact on the performance, as shown ater. Moreover, the LMMSE estimator in 19) is aso the MMSE estimator in the specia case of Rayeigh fading, since the channes are Gaussian distributed [4]. By using these key properties, the ergodic SE in Theorem 1 can be computed in cosed form for MR and ZF detection, as shown by the foowing coroary. Coroary 1. In the upink, if a channes are uncorreated Rayeigh fading, the ower bound on the ergodic capacity of user k in ce stated in Theorem 1 becomes R UL,k = γul 1 τ ) p ) og τ 1 + SINR UL,k, ) c where the SINR is SINR UL,k = G i P \{} p i,k β Gp,k β i,k,k p,k τ p β,k i P p i,k τ p β i,k +σ UL p i,k τ p βi,k i P p i,k τ p β i,k +σ UL i=1 t=1 + L p i,t z i,t + σ UL 3) and the parameters G and z i,t depend on the choice of detection scheme. MR gives G = M and z i,t = β i,t, whie ZF gives G = M and { z MSE i,t i,t =, for i P, βi,t, otherwise. Proof. The proof is avaiabe in Appendix at the end of this chapter.

20 Massive MIMO Communications 19 The cosed-form achievabe SE expressions in Coroary 1 provide many insights on the advantages of spatia muti-user mutipexing and the effects of channe estimation. Firsty, the desired signa term in the numerator of 3) scaes with the number of BS antennas, proportionay to M and M with MR and ZF, respectivey. This array gain is mutipied with the average received signa power per antenna, p,k β,k, and the reative channe estimation quaity p,k τ p β,k i P p i,kτ p βi,k + σ UL, 4) which is a number between 0 and 1 where 1 is perfect CSI and 0 is no CSI). Secondy, we notice that the first term of the denominator in 3) has a simiar structure as the desired signa and represents the coherent piot contamination interference that is ampified aong with the desired signas due the BS s inabiity to te apart users that use the same piot sequence. The piot contamination degrades the SINR by adding additiona interference that scaes as M or M, depending on the detection scheme. However, since piot contamination ony arises at BS from the interfering user in ce i in P, the network can suppress piot contamination by increasing the piot reuse factor f and by designing the ce groups appropriatey. To understand how to suppress piot contamination, we have a ook at the ratio between the piot contamination term and the signa term in 3): G i P \{} Gp,k β p i,k β,k i,k p i,k τ p βi,k i P p i,k τ p β i,k +σ UL p,k τ p β,k i P p i,k τ p β i,k +σ UL = i P \{} ) pi,k βi,k p,k β,k. 5) This ratio represents the reative strength of the piot contamination and 5) shoud preferaby be sma. The piot contamination caused by UE k in ce i is sma whenever βi,k /β,k is sma, which occurs when either β,k is arge i.e., the desired user is cose to its serving BS) or βi,k is sma i.e., the interfering ce is far away). The ce groups shoud be designed based on these properties, and this issue is further discussed in Sect. 4.. Thirdy, the performance in Coroary 1 is aso affected by cassica noise and interference. Since MR focuses ony on maximizing the SNR, the interference term L i=1 t=1 p i,tβi,t is simpy the average signa power received at any antenna of BS. In contrast, ZF pays attention to the intra-ce interference and takes no notice of the noise. The interference suppression repaces the fu channe variance βi,t in the aforementioned interference summation with the estimation error variance MSE i,t for ces i P. Due to the imperfect CSI i.e., MSE i,t > 0) not a intra-ce interference can be removed by ZF. However, the piot contamination aso has the positive effect that not ony intra-ce interference is suppressed, but aso the interce interference coming from other users in the same ce group which use the same piots as in ce ). The fact that the interference and noise terms are independent

21 0 Trinh Van Chien and Emi Björnson of M, whie the desired signa scaes with M, is a consequence of the asymptotic favorabe propagation that was described in Sect...1. If we imit the scope to a singe-ce network, achievabe SE expressions can be obtained directy from Coroary 1 by simpy setting P = {} and removing the interference from a other ces j {1,...,L} \ {}. For simpicity of exposition, we eave out the ce index in this specia case. Coroary. In the singe-ce upink, if a channes are uncorreated Rayeigh fading, a ower bound on the ergodic SE of an arbitrary user k is given by R UL k = γ UL 1 τ ) p og τ c 1 + Gp k τ pβk pk τ p β k + σul ) ). 6) p t z t + σul t=1 Here, the parameters G and z t depend on the detection scheme. MR gives G = M and z t = β t, whie ZF gives G = M and z t = β tσ UL p t τ p β t +σ UL This coroary shows that the spatia muti-user mutipexing capabiity is even greater in isoated singe-ce networks. The most notabe difference compared to a muti-ce network is the ack of inter-ce interference, both during data and piot transmission. In other words, the interference ony originates from users within the own ce, whie piot contamination vanishes thanks to the orthogonaity of a piot sequences in the ce. The SE per ce is therefore higher in singe-ce networks than in the muti-ce networks at east if the ce geometry is the same and we ony negect inter-ce interference. The motivation of having mutipe ces is, of course, to cover a arger area and thereby achieve much higher tota SE. The scenarios when the interference suppression of ZF is beneficia as compared to MR can be identified from Coroary as the cases when t=1 M p t β t σ UL p t τ p β t +σ UL + σ UL >. M. 7) p t β t + σul t=1 To summarize, we have derived upink SE expressions for Massive MIMO networks, for genera channe distributions in Theorem 1 and for Rayeigh fading in Coroary 1. In the atter case, the expressions are in cosed form and can thus be computed and anayzed directy, without having to simuate any channe fading reaizations. These expressions are used in Sect. 4 to iustrate the anticipated performance of Massive MIMO networks.

22 Massive MIMO Communications 1 3. Downink with Linear Precoding Next, we consider the downink of a Massive MIMO network where the BSs are transmitting signas to their users. For an arbitrary BS, we et x C M denote the transmitted signa vector intended for its users. We consider inear precoding where this vector is computed as x = t=1 ρ,t w,t s,t, 8) where the payoad symbo s,t intended for user t in ce has unit transmit power E{ s,t } = 1 and ρ,t represents the transmit power aocated to this particuar user. Moreover, w,t C M, for t = 1,...,, are the corresponding inear precoding vectors that determine the spatia directivity of the signa sent to each user. The received signa y,k C at user k in ce is modeed as y,k = L i=1 h i,k )H x i + n,k, 9) where n,k CN 0,σDL ) is the additive white noise with variance σ DL. Notice that h i,k is the same channe response as in the upink, due to the reciprocity of physica propagation channes within a coherence interva). 1 A bock diagram of the downink transmission is provided in Fig. 6. Since there are no downink piots in the Massive MIMO protoco described in Sect..3, the users are assumed to ony know the channe statistics. The ack of instantaneous CSI woud greaty reduce the performance of sma MIMO systems, but Massive MIMO works we without it since the effective precoded channes quicky approach their mean as more antennas are added. Hence, coherent downink reception is possibe using ony statistica CSI. This eads to a ow-compexity communication soution where a the inteigence is paced at the BS. Since the ergodic capacity is hard to characterize in this case, the foowing theorem derives a ower bound on the capacity between user k in ce and its serving BS. Theorem. In the downink, a ower bound on the ergodic rate an arbitrary user k in ce is R DL,k = γdl 1 τ ) p ) og τ 1 + SINR DL,k, 30) c where the SINR is 1 In fact, the reciproca channe is h i,k )T, using the reguar transpose instead of the conjugate transpose as in 9), but since the ony difference is a compex conjugation we can characterize the performance using 9) without oss of generaity. The reason to use the conjugate transpose is that the notation becomes easier and the reation to the upink is cearer.

23 Trinh Van Chien and Emi Björnson 1,1 s1,1 w 1,1 Ce 1 1 h ) H,k 1, s1, w 1, i,1 s i,1 w i,1 Ce i i ) h,k H n k, User k, Ce i, s i, w i, y k, L,1 s L,1 w L,1 Ce L L H h, ) L, s L, w L, Fig. 6 Bock diagram of the downink transmission with inear precoding in a muti-ce MIMO system, where BSs equipped with M antennas are transmitting signas that reach user k in ce SINR DL,k = L i=1 t=1 } ρ,k E {h,k )H w,k } { }. 31) ρ i,t E { h i E,k )H w i,t ρ,k h,k )H w,k + σ DL Proof. The proof is avaiabe in Appendix at the end of this chapter. The downink achievabe SE provided in Theorem hods for any channe distributions and choice of precoding vectors. Since the upink and downink channes are reciproca, it woud make sense if the upink and downink performance were aso somehow connected. The downink achievabe SE in Theorem indeed bears much simiarity with the corresponding upink expression in Theorem 1. The desired signa terms are the same, except for the potentiay different transmit power parameters and the fact that the detection vector is repaced by the corresponding precoding vector. The interference terms have a simiar structure, but the indices are swapped between the channe vector and the processing vector. This is because the upink interference arrives through different channes for different users whie a the downink interference from a particuar ce comes through the same channe from the BS. These observations ead to the foowing upink-downink duaity [7, 10]: Lemma 3. Suppose that the downink precoding vectors are seected as

24 Massive MIMO Communications 3 w,k = v,k E { v,k } 3) based on the upink detection vectors v,k, for a and k. For any given upink powers p i,t for i = 1,...,L and t = 1,...,), there exist a corresponding set of downink powers ρ i,t for i = 1,...,L and t = 1,...,) such that SINR UL,k = SINRDL,k 33) for a and k, and L i=1 t=1 p i,t σ UL = L i=1 t=1 ρ i,t σdl. 34) Proof. The proof is avaiabe in Appendix at the end of this chapter. This emma shows that the same performance can be achieved in both the upink and the downink, if the downink power is aocated in a particuar way based on the upink powers and the precoding vectors are seected based on the detection vectors as in 3). The downink powers are computed according to 7), which is given in Appendix at the end of this chapter since the important thing for now is that there exist a coection of downink powers that give exacty the same performance in both directions. If σul = σ DL, then the same tota transmit power is used in both directions of the Massive MIMO network; however, the power wi generay be distributed differenty over the users. Motivated by the upink-downink duaity, it makes sense to consider MR and ZF precoding as the main downink precoding schemes. These are defined as w,k = E E ĥ,k { ĥ,k }, for MR, Ĥ r,k { Ĥ r,k }, for ZF, 35) where r,k denotes the kth coumn of Ĥ ) H Ĥ ) 1. Simiar to the upink performance anaysis, we now compute the downink SE in cosed form for uncorreated Rayeigh fading channes, as defined in 18). Because of the channe reciprocity, the channe estimates obtained at the BSs in the upink can aso be used in the downink. In particuar, the channe estimates ĥ i i,k and ĥ i,k for ce indices i and with P i are sti reated as ĥ i p,k β,k i,k = pi,k βi,k i ĥ i i,k, 36) thus showing that piot contamination exists aso in the downink; that is, BS i cannot precode signas toward its user k without aso precode the signa towards user k in ce i P. The next coroary speciaizes Theorem for Rayeigh fading channes.

25 4 Trinh Van Chien and Emi Björnson Coroary 3. In the downink, if a channes are uncorreated Rayeigh fading, the ower bound on the ergodic capacity of user k stated in Theorem becomes R DL,k = γdl 1 τ ) p ) og τ 1 + SINR DL,k, 37) c where the SINR is SINR DL,k = G i P \{} ρ i,k β i Gρ,k β,k,k p,k τ p β,k i P p i,k τ p β i,k +σ UL p,k τ p β,k i i P p i,k τ p β i i,k +σ UL i=1 t=1 + L ρ i,t z i,k + σ DL. 38) The parameters G and z i,k are specified by the precoding scheme. MR precoding gives G = M and z i,k = β,k i, whie ZF precoding gives G = M and { z i,k = MSE i,k, for i P, β,k i, otherwise. Proof. The proof is avaiabe in Appendix at the end of this chapter. For Rayeigh fading channes, Coroary 3 shows that the array gain, piot contamination, and a other attributes of MR and ZF precoding are very simiar to the upink counterparts. Hence, the same kind of observations can be made from Coroary 3 as previousy done for Coroary 1. In the singe-ce scenario, the SE expression in Coroary 3 simpifies to the foowing resut. Coroary 4. In the singe-ce downink, if a channes are uncorreated Rayeigh fading, a ower bound on the ergodic SE of an arbitrary user k is given by ) R DL k = γ DL 1 τ p Gρ k p k τ p βk )og τ 1 + c pk τ p β k + σul ) zk t=1 ρ t + σdl). 39) The parameters G and z k depend on the precoding scheme. MR gives G = M and z k = β k, whie ZF obtains G = M and z k = β kσ UL p k τ p β k +σ UL We concude the anaytica part of this chapter by recaing that the upink and downink spectra efficiencies with Massive MIMO can be easiy computed from Theorem 1 and Theorem for any channe distributions and processing schemes. In the uncorreated Rayeigh fading case there are even cosed-form expressions. The same SINR performance can be achieved in the upink and downink, based on what is known as upink-downink duaity. The intuition is that the downink precoding and upink detection vectors shoud be the same, but that the power aocation needs to be adapted differenty in the two cases..

26 Massive MIMO Communications 5 4 Design Guideines and Anticipated Spectra Efficiency Gains In this section, we provide some basic design guideines for Massive MIMO networks and showcase the SEs that the technoogy can deiver to 5G networks according to the theory deveoped in Sect. 3. For iustrative purposes, we consider a cassic ceuar network topoogy with hexagona ces, where each ce can be iustrated as in Fig. 1. In other words, the BS is depoyed in the center of the ce, whie the users are distributed over the ce area. When many ces of this type are paced next to each other, the ceuar network has the shape showed in Fig. 7. Whie conventiona ceuar networks use sectorization to spit each ce into, say, three static sectors, this is not assumed here. This is because the spatia transceiver processing at the BS in Massive MIMO basicay creates virtua sectors, adapted dynamicay to the positions of the current set of users. 4.1 Power Aocation The average transmit power of user k in ce j is denoted by p j,k in the upink and by ρ j,k in the downink. These are important design parameters that determine the SEs of the users; see Theorem 1 for the upink) and Theorem for the downink). Since inter-user interference is an important factor in any muti-user MIMO system, each transmit power coefficient affects not ony the strength of the desired signa at the desired user, but aso the amount of interference caused to a the other users in the network athough the interference is most severe within a ce and between neighboring ces). The seection of these transmit power coefficients is referred to as power aocation and needs to be addressed propery. A key property of Massive MIMO is that the sma-scae fading in time and the frequency-seective fading variations are negigibe, since they essentiay average out over the many antennas at each BS. For exampe, the SE expressions for Rayeigh fading channes in Coroaries 1 4 ony depend on the channe variances and not on the instantaneous reaizations of the corresponding channe vectors. Therefore, there is no need to change the power aocation between each coherence interva, but ony over the onger time frame where the channe variances change, due to modifications in the arge-scae propagation behaviors e.g., caused by user mobiity). This is a substantia increase of the time frame in which power aocation decisions are to be made, from miiseconds to seconds. This fact makes it possibe to optimize and coordinate the power aocation across ces, in ways that have not been possibe in the past due computationa or deay imitations. A structured approach to power aocation is to find the transmit powers that jointy maximize the network utiity functions U UL {R UL,k }) and U DL {R DL,k }) in the βi,k h i,k upink and downink, respectivey. These utiities are increasing functions of the users SEs, where {R UL,k } and {RDL,k } denote the sets of a SEs. Some particuar exampes of network utiity functions are [5]

27 6 Trinh Van Chien and Emi Björnson L =1 k=1 R,k, Sum utiity, U{R,k }) = L =1 k=1 R,k, Proportiona fairness, 40) min {1,...,L},k {1,...,} R,k, Max-min fairness, where we have omitted the upink/downink superscripts since the same type of utiity function can be utiized in both cases. These utiities are often maximized with respect to a given power budget per user in the upink) and per BS in the downink). For brevity, we wi not provide any further mathematica detais, but briefy outine what is known around power aocation for Massive MIMO. Maximization of the sum utiity SU) provides high SEs to users with good average channe conditions, at the expense of ow SE for users with bad average channe conditions. In contrast, max-min fairness MMF) enforces that each user shoud get equa SE, which effectivey means that users with good channes reduce their SEs to cause ess interference to the users with bad channes. Proportiona fairness PF) can be shown to ie in between these extremes. The SU achieves the highest sum SE, since this is reay what is optimized by this utiity function, whie MMF trades some of the sum SE to obtain a uniform user experience. The choice of network utiity function is a matter of subjective taste, since there is no objectivey optima utiity function [5]. Nevertheess, there seems to be a trend towards more fairnessemphasizing utiities in the Massive MIMO iterature [7, 3, 45], motivated by the fact that contemporary networks are designed to provide high peak rates, whie the ce edge performance is modest and needs to be improved in 5G. In the upink, another important aspect to consider in the power aocation is the fact that a BS cannot simutaneousy receive desired user signas of very different power eves, since then the weak signas wi then drown in the quantization noise caused by the anaog-to-digita conversion. Hence, even if the channe attenuation might differ by 50 db within a ce, these variations need to be brought down to, say, 10 db by the upink power aocation. From a numerica optimization perspective, the downink power aocation probem for fixed upink power aocation) has the same mathematica structure as the seemingy different scenario of singe-antenna muti-ce communications with perfect CSI. The downink utiity optimization can therefore be soved using the techniques described in [4, 7, 35] and references therein. In genera, the PF and MMF utiities give rise to convex optimization probems that can be soved efficienty with guaranteed convergence to the goba optimum. These agorithms can aso be impemented in a distributed fashion [4]. The SU probem is, in contrast, provabe non-convex and hard to sove [7], which means that the optima soution cannot be found under any practica constraints on compexity. The upink power aocation is more compicated than power aocation in the downink; for exampe, because the SE expression in Coroary 1 contains both p j,k and p j,k whie the downink SE expressions ony contain the inear term ρ j,k). Nevertheess, there are severa efficient agorithms that maximize the MMF utiity [13, 45, 1], and the approach in [1] can aso maximize the SU and PF utiities with MR and ZF detection. The work [6] provides an aternative methodoogy to maximize an approximation of the SU metric for other detection methods.

28 Massive MIMO Communications 7 f=1 f=3 f=4 f=4 Fig. 7 Iustration of potentia symmetric reuse patterns created by three different piot reuse factors, f, in a ceuar network with hexagona ces. In the ower right case, each ce is divided into two sub-ces with different sets of piots. If j is the index of a particuar ce, then P j is the index set of a ces having the same coor. Ony the ces with the same coor use the same piot sequences, and thereby degrade each other s CSI estimation quaity and cause piot contaminated interference In summary, power aocation is used in Massive MIMO to distribute the sum SE over the individua users. There are penty of agorithms that can be used to optimize the power aocation, depending on the utiity function that is used in the system. 4. Non-Universa Piot Reuse An important insight from the theoretica anaysis in Sect. 3 is that the SE of a particuar ce j is infuenced by the piot signaing carried out in other ces. The degradations in CSI estimation quaity and piot contaminated interference are caused ony by the interfering ces in P j that use the same piot sequences as ce j. Since the channe attenuation of the interference increases with distance, one woud ike these interfering ces to be as far away from ce j as possibe and the same is desirabe for a ces in P j. Reca that the piot reuse factor f = τ p / was assumed to be an integer in Sect. 3, which eads to a division of the L ces into f disjoint ce groups. The case f = 1 is known as universa piot reuse and f > 1 is caed non-universa pi-