Optimizing Large-Scale MIMO Cellular Downlink: Multiplexing, Diversity, or Interference Nulling?

Transcription

1 Optimizing Large-Sale MIMO Cellular Downlink: Multiplexing, Diversity, or Interferene Nulling? Kianoush Hosseini, Wei Yu, and Raviraj S. Adve The Edward S. Rogers Sr. Department of Eletrial and Computer Engineering University of Toronto, Toronto, ON, M5S 3G4, Canada s: {kianoush, weiyu, Abstrat A base-station BS) equipped with multiple antennas an use its spatial dimensions in three ways: 1) serve multiple users to ahieve a multiplexing gain, 2) provide diversity to its users, and/or 3) null interferene at a hosen subset of out-of-ell users. The main ontribution of this paper is to answer the following question: what is the optimal balane between the three ompeting benefits of multiplexing, diversity and interferene nulling? We answer this question in the ontext of the downlink of a ellular network in whih eah user hooses its best serving BS, and requests nearby interfering BSs for interferene nulling. BSs are equipped with a large number of antennas, serve multiple single-antenna users using zero-foring beamforming and equal power assignment, and null interferene at a subset of out-of-ell users. The remaining spatial dimensions provide transmit diversity. We assume perfet hannel state information at the BSs and users. Utilizing tools from stohasti geometry, we show that, surprisingly, to maximize the per-bs ergodi sum rate, at the optimal alloation of spatial resoures, interferene nulling does not bring tangible benefit. A loseto-optimal strategy is to use none of the spatial resoures for interferene nulling, while reserving 60% of spatial resoures for ahieving multiplexing and the rest for providing diversity. I. INTRODUCTION The advent of the massive multiple-input multiple-output MIMO) onept has brought myriapportunities for optimizing transmissions in wireless ellular networks. Having a large number of base-station BS) antennas at our disposal allows us to ahieve multiple system objetives simultaneously: 1) serve multiple users in the same time-frequeny slot to ahieve a multiplexing gain, 2) provide diversity for the sheduled users, and 3) as often stated as a major advantage of massive MIMO systems, null interferene at out-of-ell users. In fat, in the limit as the number of antennas goes to infinity, interell interferene an be ompletely eliminated, leaving pilot ontamination as the only limiting fator [1]. This paper fouses on the design of large-sale MIMO LS-MIMO) downlink systems [2], where eah BS is equipped with a large but finite number of antennas. In this regime, it is ruial to understand the tradeoffs between multiplexing, diversity and interferene nulling. This paper attempts to answer the following key question in suh a design: between providing spatial multiplexing, diversity and mitigating interell interferene, whih takes priority? Equivalently, what is the optimal tradeoff among the three? Toward this end, we onsider an LS-MIMO system where BSs are distributed aording to an independent Poisson point proess PPP). Further, this paper advoates a user-entri lustering strategy. Eah user reeives the intended signal from the losest BS, while requesting interferene nulling from a luster of nearby BSs. We assume that perfet hannel state information CSI) is available without ost, at both the BSs and users. Eah BS serves multiple single-antenna users simultaneously using zero-foring ZF) beamforming with equal power alloation aross users. In addition, the remaining spatial dimensions at eah BS an potentially be used to provide diversity for the sheduled users and/or to null interferene at a subset of out-of-ell users. This paper arrives at the following surprising onlusion: For a single-tier wireless ellular network, devoting spatial dimensions to interferene nulling does not provide tangible benefit. In fat, it is fairly lose to optimum, in terms of maximizing the per-bs ergodi sum rate, to use about 60% of the spatial dimensions for spatial multiplexing i.e., with M antennas, serve K 0.6M users), leaving the remaining dimensions for spatial diversity, while devoting none of the spatial dimensions for interell interferene nulling. This is despite the fat that a) the BSs are densely deployed and the overall network is interferene limited and b) the analysis does not aount for the ost of hannel estimation. Thus, even with perfet CSI available for all diret and interferene hannels at all BSs without any ost, one may still wish to utilize spatial dimensions for providing spatial multiplexing and diversity for the intended users, rather than interferene nulling for the neighboring users. A. Related Work The tradeoff between multiplexing and diversity for the single-user MIMO hannel [3] and for the uplink single-ell multiuser system [4] are well known. However, the optimal alloation of spatial resoures between multiplexing and diversity in a multiell setting has not been studied rigorously. The multiell setting is of partiular interest beause of the possibility of mitigating interell interferene through oordinating transmission strategies aross multiple BSs, also known as interferene oordination [5]. For example, the problems of joint design of beamformers aross multiple BSs in order to minimize transmit power [6], [7] and to maximize SINR [8], [9] have been studied. The LS-MIMO system onsidered in this paper is the finite-dimensional version of non-ooperative massive MIMO

2 systems, wherein eah BS is equipped with an asymptotially large number of antennas and serves its sheduled users independently. In this asymptoti regime, the unorrelated interell interferene ompletely vanishes; interell oordination is therefore not required [1], [10]. However, with a finite number of antennas, it is not lear as to what extent this desirable feature of massive MIMO systems remains optimal. This paper deviates from the massive MIMO literature in that it expliitly aounts for interferene anellation using beamforming and aims to understand whether it is benefiial to use spatial dimensions for interferene anellation. This paper onsiders a user-entri lustering strategy for interferene nulling. The joint design of user-entri lusters and downlink beamformers for a fixed network topology has been studied in [11], [12]. To aount for the randomness in BS and user loations, we use available tools from stohasti geometry [13]. Stohasti models for the analysis of userentri lustering in wireless systems, where eah BS serves a single user, have been introdued in [14], [15]. This paper arries out a stohasti analysis of an LS-MIMO system under user-entri lustering, where eah BS serves multiple users, and aims at understanding the optimal alloation of spatial resoures to multiplexing, diversity and interferene nulling. B. Summary of Contributions The main ontributions of this paper are as follows: 1) Stohasti analysis of LS-MIMO systems with userentri lustering: We present a stohasti analysis of an LS-MIMO system under user-entri lustering. Using tools from stohasti geometry, we obtain a tratable expression for the per-bs ergodi sum rate as a funtion of average luster size. This expression enables us to effiiently explore the performane of LS-MIMO systems with user-entri lustering under different system parameters. 2) Optimizing alloation of spatial resoures for multiplexing, diversity and interferene nulling: The analysis above allows us to answer our entral question as to the optimal alloation of spatial dimensions. Our analysis shows that a lose-to-optimal strategy is to use 60% of spatial dimensions for multiplexing, the remaining dimensions for providing diversity, while reserving none of the spatial dimensions for interferene nulling. II. SYSTEM MODEL We onsider a time-division duplex TDD) LS-MIMO system, where BSs are distributed aording to a homogeneous PPP Φ with density λ over the entire R 2 plane. Eah BS is equipped with M antennas, and is onstrained to use a maximum power of P T. The single-antenna users are distributed aording to an independent point proess with density onsiderably larger than that of the BSs. Users form ells by assoiating with their losest BSs. Perfet CSI is assumed to be available at the BSs and the users. The wireless fading hannel is modeled as follows. Let g ij = β ij h ij C M 1 denote the hannel between BS j and user i, where h ij CN 0, I M ) indiates a small-sale ) α Rayleigh fading omponent, and β ij = 1 + rij is the path-loss omponent; here, r ij is the distane between user i and BS j, α > 2 denotes the path-loss exponent, and is the referene distane. In a given time-frequeny slot, BS b shedules a set of K users, denoted by S b, hosen from its assoiated users. The BS uses ZF beamforming with equal power alloated aross the K users. Further, BS b uses its available antennas to anel interferene at a subset of out-of-ell users denoted by O b. The normalized ZF beam assigned to user i assoiated with BS b, while nulling the interferene at the other K 1 + O b users, is given by ) I M G ib G ib ĝ ib w ib = ) I M G ib G 2, i S b ib ĝ ib where G ib = [ ĝ 1b,..., ĝ i 1)b, ĝ i+1)b,..., ĝ K+ Ob )b], denotes the ardinality of a set, and ĝ jb = g jb / g jb 2. A. User-Centri Clustering in LS-MIMO Systems The hoie of out-of-ell users for interferene anellation depends on the BS lustering strategy. This paper onsiders user-entri lustering, where eah user identifies the interfering BSs with strong average hannel magnitudes. These BSs form the luster from whih the user requests interferene nulling. Ideally, interferene from the BSs within the luster would all be anelled; only the interferene from outside this luster remains, and is treated as additional noise. However, in reality, beause eah BS has finite number of available spatial dimensions, it an only selet a subset of out-of-ell users for interferene nulling. In partiular, eah BS reserves suffiient antennas to serve its K sheduled users with diversity order ζ. Aounting for the orthogonal property of ZF beamforming, BS b must reserve ζ + K 1 dimensions for transmission to its own users, leaving a maximum of O max = M K ζ + 1 dimensions for interferene nulling at out-of-ell users. If the number of users requesting interferene nulling at BS b is larger than O max, then BS b selets the O max users with the strongest hannel magnitudes within the andidate set, i.e., some out-of-ell users requests would have to be ignored. On the other hand, if the number of requesting users for interferene anellation from BS b is smaller than O max, then the extra antennas are used to provide users with diversity order larger than ζ. In the system desribed above, let Φ Si Φ be the set of BSs loated in the luster of user i, and Φ Ii = Φ \ Φ Si be the set of interfering BSs loateutside the luster. Further, let Φ Si,Intf Φ Si indiate the BSs in Φ Si that do not have suffiient antennas to anel interferene at user i. The reeived signal at user i assoiated with BS b is the sum of the intended signal, residual intra-luster interferene, interluster interferene, and reeiver noise; the reeived signal is

3 given by PT y ib = K gh ibw ib s ib + K PT K gh ijw kj s kj }{{} j Φ Si,Intf k=1 intended signal }{{} + K m Φ Ii u=1 intra-luster interferene PT K gh imw um s um } {{ } inter-luster interferene + n ib }{{} noise where s ib denotes the omplex symbol intended for user i assoiated with BS b suh that E [ s ib 2] = 1, and n ib denotes the additive white Gaussian noise with variane σ 2. B. Analytial Model of User-Centri Clustering Strategy This paper aims to provide a stohasti performane analysis of a user-entri LS-MIMO system. For tratability, the following simplified model is proposed for theoretial analysis: A1 : All BSs from whih a user requests interferene nulling are within a irle of radius R entered at the user. A2 : The number of BSs in the irular luster is the same as the average number of BSs B = λπr 2. A3 : Eah BS has enough antennas not only to serve its K sheduled users with a fixed diversity order of ζ, but also to grant all interferene nulling requests. Together with A1, this means that eah BS must anel interferene at all users loated within distane R from itself. Note that to satisfy A3, eah BS should anel interferene at O max = O = M K ζ + 1 out-of-ell users, while serving its K sheduled users. To allow this, given A2, i.e., assuming eah luster inludes exatly B BSs, the luster radius must be R = B/λπ, where B = O + K) /K. Under these assumptions, the signal-to-interferene-plusnoise ratio SINR) of user i sheduled by BS b is given by ρβ ib h H ib γ ib = w ib 2 K m Φ Ii k=1 ρβ im h H im w km P T where ρ = Kσ indiates the per-user signal-to-noise ratio 2 SNR). The instantaneous rate of user i denoted by R ib is therefore given by R ib = log γ ib ) Γ, where Γ is the SNR gap to apaity. III. A STOCHASTIC GEOMETRY ANALYSIS OF LS-MIMO In this setion, we derive the per-bs ergodi sum rate expression for an LS-MIMO system with user-entri lustering. Theorem 1. Under Assumptions A1-A3, for an LS-MIMO system with user-entri lustering employing ZF beamforming, providing a fixed diversity ζ for eah sheduled user, and distributing power equally aross the downlink beams, 1) TABLE I SYSTEM DESIGN PARAMETERS BS density λ = 1/π500 2 m 2 Total bandwidth W = 20 MHz BS max. available power 43 dbm Bakground noise PSD N o = 174 dbm/hz Noise figure N f = 9 db SNR gap Γ = 3 db Path-loss exponent α = 3.76 Referene distane = m the per-bs ergodi sum rate in nats/se/hz is given by e zγ R ell = K exp 2πλ Ψ I zγ) Ψ II zγ))) 0 z ) ) α ζ df rmin zρ 1 + rdo dr dz 2) dr 0 where F rmin r) = 1 exp λπr 2) for r > 0 is the umulative distribution funtion CDF) of distane between a user and its losest BS [13], and Ψ I s) and Ψ II s) are given as Ψ I s) = d2 o 1 + R ) F 1 K, 2α ; 1 2α ; sρ 1 + R ) )) α 3) Ψ II s) = d 2 o 1 + R ) 1 2 F 1 K, 1α ; 1 1α ; sρ 1 + R ) )) α. 4) Proof: Sine eah BS serves K users, the per-bs ergodi sum rate is given by R ell = KE Φ,h R ib ). The proof of Theorem 1 involves expressing the rate funtion in an integral form using log 1 + x) = e t 0 t 1 e xt ) dt and substituting in the Laplae transforms of the interferene and signal powers. The proof uses the same approah as that used in [16], [17]. The details are omitted here. Remark 1. Theorem 1 ompletely haraterizes the per-bs ergodi sum rate of an LS-MIMO system with user-entri lustering as a funtion of important system parameters, suh as BS deployment density λ), number of users sheduled by eah BS K), diversity order per user ζ), and luster radius R ). Although the expression involves a double integral, using the transformation introdued in Remark 1 of [18], it an be omputed effiiently. IV. OPTIMIZATION OF LS-MIMO SYSTEMS In this setion, we use the per-bs ergodi sum rate expression obtained in the preeding setion to answer the following entral question of this paper: given a fixed number

4 of antennas M at eah BS, what is the optimal number of sheduled users K, the optimal diversity order provided to eah user ζ, and the optimal number of interferene nulling diretions O in terms of maximizing the per-bs ergodi sum rate? Note that K [1,, M], ζ [1,, M K + 1] and O = M K ζ +1, i.e., only two of the triple K, ζ, O) are free variables. The result provided by Theorem 1 allows us to obtain the optimal operating point in an effiient manner. The hosen system parameters are listed in Table I. Numerial results are averagever network topologies and small-sale hannel fading realizations. Figs. 1 and 2 plot the per-bs ergodi sum rate of an LS- MIMO system with M = 15 for various ombinations of K, ζ, O) obtained numerially via simulations and analytially using 2). As the results show, when a substantial number of dimensions are devoted to nulling interferene large O), the numerial and analyti results are in good agreement. However, at low values of O, the two results diverge. The main reason for the disrepany is that in the simulations, one O is speified, the exat number of BSs nulling interferene at eah user, i.e., its exat luster size, is known for any given network topology. However, in the analysis, the values of K and O speify the average number of BSs in a luster, i.e., B = K + O) /K, as explained in Setion II-B. Despite the disrepany, the analysis does apture the general behavior of the system. When both O and K are small, inreasing K improves the per-bs ergodi sum rate, while for large values of O and K, inreasing K degrades system performane. Most importantly, the analysis helps identify the region of O with the maximum sum-rate, leading to the main, and surprising, result of this paper. A remarkable impliation of these analytial and simulation results is that the per-bs ergodi sum rate is largely a dereasing funtion of the number of spatial dimensions assigned to null interferene. The analyti expression in 2) is, in fat, a stritly dereasing funtion of O, while the numerial urves are approximately so. It is nearly optimum for eah BS to operate independently of the other BSs and not to spend spatial resoures on nulling interferene. More speifially, the sum-rate optimal operating point obtained numerially is K, ζ, O ) = 8, 6, 2), and the optimal operating point obtained analytially is K, ζ, O ) = 10, 6, 0). The differene between the ahievable sum rates at these two operating points obtained numerially) is only about 4%. Thus, when K and ζ are properly hosen, an LS- MIMO system without interferene nulling, i.e., with eah BS operating independently, performs lose to optimum. Fig. 3 plots the CDF of downlink user rates evaluated at various hoies of K, ζ, O). As an be seen, the gap between the CDF of the downlink user rates obtained at 10, 6, 0) and 8, 6, 2) is negligible. Clearly, operating at either of these two points is signifiantly superior to operating at any of 1, 15, 0), 15, 1, 0) and 7, 3, 6), but there are other hoies e.g., 9, 5, 2), that also perform reasonably well. These results indiate that it is important to properly balane K and ζ, and to keep O small. Finally, Table II lists the optimal K, ζ, O ) obtained Per-BS ergodi sum rate bits/se/hz) K = 8 K = 7 Numerial Analytial K = Number of dimensions for interferene nulling Fig. 1. Per-BS ergodi sum rate of an LS-MIMO system with M = 15, K = 6, 7, 8, and various hoies of ζ and O obtained both numerially and analytially. Per-BS ergodi sum rate bits/se/hz) K = 10 K = 9 Numerial Analytial K = Number of dimensions for interferene nulling Fig. 2. Per-BS ergodi sum rate of an LS-MIMO system with M = 15, K = 8, 9, 10, and various hoies of ζ and O obtained both numerially and analytially. both numerially and analytially for various values of M. The table also lists the performane gap between these operating points evaluated numerially). In all the senarios, the analysis suggests that it is optimal to ignore interell interferene and retain all spatial dimensions for multiplexing and diversity. The simulation results, in good agreement with the analysis, suggest that only a very few of the spatial dimensions should be used for interferene anellation. Moreover, the simulation results indiate that the performane gap between these points for various values of M is negligible. An interesting observation from the table is that the optimal loading fator η = K + O) /M appears to be always lose to 0.6. This implies that, at a lose-to-optimum operating point, a MIMO BS should devote about 60% of its resoures to multiplexing K/M = 0.6) and the remaining to providing diversity to eah user ζ = M K + 1, O = 0).

5 Cumulative distribution funtion Fig , 15, 0) 15, 1, 0) 7, 3, 6) 10, 6, 0) 9, 5, 2) 8, 6, 2) Downlink user rate Mbits/se) CDF of the downlink user rates evaluated at different K, ζ, O). TABLE II OPTIMAL OPERATING POINTS FOR MAXIMIZING PER-BS ERGODIC SUM RATE OF LS-MIMO SYSTEMS M analytial numerial performane gap η 10 6, 5, 0) 5, 5, 1) 4% , 6, 0) 8, 6, 2) 4% , 16, 0) 24, 15, 2) 3% 0.62 It is worth noting that these results are obtained without aounting for the overhead needed for hannel training. To implement interferene nulling as suggested by the optimal solutions obtained numerially, CSI aquisition overhead would inrease. As an example, with M = 40, at the optimal operating point obtained numerially, in order to serve 24 sheduled users and null interferene at 2 out-ofell users, 26 hannels are needed to be estimated, whereas at the optimal point suggested by analysis, i.e., with no interferene nulling, 25 hannels should be estimated at eah BS. Furthermore, with user-entri lustering, lusters may overlap. Therefore, pilot alloation for CSI aquisition must be organized by some entral entity aross the entire network. This, in fat, represents another level of overhead. As a result, in pratial senarios, enabling interferene nulling may diminish the 3-4% apparent gain of operating under the optimal point obtained numerially. Consequently, it is highly likely that in ellular LS-MIMO systems, ignoring interferene nulling, while properly seleting K and ζ, is, in fat, the best operating ondition. V. CONCLUSION This paper onsiders an LS-MIMO system with userentri lustering operating under ZF beamforming and equal power assignment. For suh a system, we derive omputationally effiient expression for the per-bs ergodi sum rate as a funtion of average luster size using tools from stohasti geometry. We then optimize the per-bs ergodi sum rate of the system as a funtion of the number of users to shedule, the diversity order of eah user and the number of interferene diretions to anel. This paper shows that at the optimal operating point, only a very few of spatial dimensions should be reserved for interferene nulling. In fat, our analysis shows that it is near optimal to alloate none of the spatial resoures for interferene nulling, while using 60% of spatial dimensions for ahieving multiplexing and the rest for providing diversity. Reduing multiplexing and diversity dimensions in order to perform interferene nulling does not appear to bring muh benefit to the overall system sum rate. REFERENCES [1] T. L. Marzetta, Nonooperative ellular wireless with unlimited numbers of base station antennas, IEEE Trans. Wireless Commun., vol. 9, no. 11, pp , Nov [2] K. Hosseini, W. Yu, and R. S. Adve, Large-sale MIMO versus network MIMO for multiell interferene mitigation, IEEE J. Sel. 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