Upink Massive MIMO SIR Anaysis: How do Antennas Scae with Users? Tianyang Bai and Robert W. Heath, Jr. Wireess Networking and Communication Group The University of Texas at Austin 66 Guadaupe Street, C83, Austin, TX 787 {tybai, rheath}@utexas.edu Abstract Massive mutipe-input mutipe-output MIMO is a potentia physica ayer technoogy for 5G ceuar networks. This paper everages stochastic geometry to derive the upink signato-interference SIR distribution in massive MIMO networks. Based on the derived expressions, a scaing aw between the number of base station antennas and schedued users per ce is provided to preserve the upink SIR distribution, where the impacts of correation in sma-scae fading and power contro in the form of fractiona path oss compensation are taken account. Numerica resuts verify the anaysis, and show that fractiona power contro with a compensation fraction of.5 is neary optima for the average achievabe rate in certain cases. I. INTRODUCTION Massive mutipe-input and mutipe-output MIMO is an approach to increase the area spectrum efficiency in 5G ceuar systems [] [4]. It works by using many antennas in a arge-scae antenna array to simutaneousy serve a arge number of users and provide high sum throughput [] [4]. In this paper, we focus on massive MIMO operated in timedivision dupex TDD mode beow 6 GHz, where reciprocity in the channe is expoited to avoid feedback, and piots are reused to reduce the training overhead [] [4]. Prior work showed that due to asymptotic orthogonaity between channes, high throughput coud be achieved with arge-scae antenna arrays through simpe signa processing, and that the asymptotic performance of massive MIMO in the imit of the number of base station antennas is imited by piot contamination []. Most prior work studied the performance of massive MIMO using a simpified network topoogy, e.g. considering ony a few base stations in a hexagona grid [], [5] [7]. With the densification of ceuar networks, it is of interest to consider ess reguar network topoogies with a arge number of base stations. Fortunatey, simpe characterizations of the signato-interference ratio SIR and rate in arge networks were enabed by stochastic geometry [8]. The work in [8], however, does not directy extend to massive MIMO networks, as it focused on the singe user per ce scenario with perfect channe state information CSI. Stochastic geometry was appied to study the asymptotic SIR and rate in a massive MIMO networks in [9], [], where the asymptotic SIR is shown to be approached with impracticay arge number of antennas, e.g. 4 antennas. Reated work in [] appied stochastic geometry to study the upink performance under the identicay and independenty distributed IID fading assumption. A inear scaing between the numbers of base station antennas and schedued users was found to maintain the same mean interference, which need not preserve the SIR distribution. In this paper, we appy stochastic geometry to study the upink SIR performance in a arge-scae massive MIMO network using maximum ratio combining MRC. The proposed mode incorporates correated sma-scaing fading with exponentia correation [2], and fractiona power contro by compensating for a fraction of the path oss as in ong term evoution LTE systems [3]. Compared with prior stochastic geometry network modes [8], the mode characterizes the distribution of both inter-ce and intra-ce users in the upink, and accounts for the effect of imperfect CSI due to piot contamination. Unike prior work that focused on the asymptotic performance [9], [], we derive anaytica expressions for the non-asymptotic upink SIR distribution, as a function of the number of base station antennas and schedued users per ce. Based on the anaytica expressions, the scaing aw between the base station antennas and schedued users per ce is obtained to maintain the same upink SIR distribution. Our anaysis shows that i to maintain the same upink SIR, the number of antennas shoud generay scae superineary with the number of schedued users per ce; ii the inear scaing aw in [] preserves the SIR distribution ony in the case of fu compensation of the path oss in power contro; and iii correations in sma-scae fading reduce the upink SIR coverage. Numerica resuts aso indicate that the average per user rate can be maximized by adjusting the compensation fraction in the fractiona power contro, and the optima fraction is around.5 in certain cases. II. SYSTEM MODEL In this section, we introduce the upink system mode for a massive MIMO ceuar network operated in the sub-6 GHz band. The mode can be extended for massive MIMO at miimeter wave mmwave frequencies by incorporating certain differences in propagation and hardware constraints [4]. Each base station is assumed to have M antennas. In each time-frequency resource bock, a base station can simutaneousy schedue K users in its ce. Let X be the ocation of the -th base station, Y k be the ocation of the U.S. Government work not protected by U.S. copyright
k-th schedued user in the ce of -th base station, and h k the channe vector from X to Y k. As in [], we consider a network operated in the foowing TDD mode with perfect synchronization: in the upink channe training stage, the schedued users send their assigned piots T k, and base stations estimate the channes using the orthogonaity of the piots; in the upink data transmission, the base stations appy MRC to receive the upink data, based on the channe estimates derived from upink piots. Further, we assume fu reuse of the orthogona piots {T k } k K in the network. Now, we introduce the channe mode assumptions. The channe is assumed to be constant during one resource bock and fades independenty from bock to bock. Moreover, we appy a narrowband channe mode, as frequency seectivity in fading can be minimized by techniques ike orthogona frequency-division mutipexing OFDM and frequency domain equaization [5]. To mode the correated sma-scae fading, we express the channe vector as /2 h k = β k k/2 Φ w k, where β k is the arge-scae path oss, wk is a Gaussian vector with the distribution CN, I M for Rayeigh fading, and Φ k is the covariance matrix to account for correations in sma-scae fading. Let λ k [m] be the eigenvaues of the covariance matrix Φ k. We assume that for a channes, the trace of the covariance matrix is normaized to M, i.e, M m= λk [m] =M, and the average square of the eigenvaues is upper bounded by a constant γ: im sup M M M m= λ k2 [m] =γ. 2 Note that the constraint in 2 is satisfied by many common channe modes, incuding the IID fading mode, the exponentia correation matrix mode [2], and the case of uniform inear arrays with certain continuous anguar spread [6]. is computed as α β k max = C R k,δ, 3 The arge-scae path oss gain β k where C is a constant determined by the carrier frequency and reference distance, R k is the distance from X to Y n k, α>2 is the path oss exponent, and δ is a sma distance, e.g. the reference distance of meter, intended to address the near fied effect. Simiar path oss modes have been used in prior work on ceuar network anaysis [7, Page 69]. The condition that δ > is essentia to prove the convergence of SIR in the asymptotic regime with infinity antennas [4], which requires the finiteness of the path oss. For the anaysis in the non-asymptotic regime, we wi assume δ =as in [8], [8] for tractabiity of the anaysis. Next, we introduce the network topoogy assumptions based on stochastic geometry. We assume the base stations are distributed as a Poisson point process PPP with density λ b. A user is assumed to be associated with the base station that provides the minimum path oss signa. The users, either schedued or not in a resource bock, are uniformy distributed on the pane with sufficienty high density, such that in any resource bock, a base station has at east K candidate users in its ce for potentia scheduing. Without oss of generaity, a typica schedued user Y is fixed at the origin. We wi investigate the SIR and rate performance at this typica user. Now we focus on the distribution of schedued users in a resource bock. For k K, etn u k be the point process formed by the ocations of the schedued users Y k, i.e., a the schedued users assigned with the k-th piot sequence T k. Note that even though the users are distributed as a PPP on the pane, the schedued users do not form a PPP, as their ocations are correated due to the unique assignment of the piot T k within a ce. Unfortunatey, the correations in the schedued users ocations make the exact anaysis intractabe [8]. Therefore, we make the foowing approximations on the distribution of the schedued user process N u k. Assumption : The foowing approximations are assumed to mode the schedued users process N u k : The distances R k from a user to their associated base stations are assumed to be IID. Note that R k is a Rayeigh random variabe with mean.5 /λ b [8]. 2 Schedued users assigned with different piots are independenty distributed, i.e., for k k, the processes N u k and N k u are independent. 3 The other-ce schedued users of N u k are modeed by the excusion ba mode [4]: they form a homogeneous PPP with density λ b outside an excusion ba centered at the base station X. The radius of the excusion ba is / πλ b. The assumptions aow for tractabe anaysis of the SIR and ead to tight approximations as reveaed by the simuations in Section IV. In addition, the excusion ba mode can be viewed as the first-order equivaence of the upink topoogy mode in [8], where the pairwise correations in users ocations are taken account. Next, the fractiona power contro, as used in the LTE systems [3], is assumed in both the upink training and upink ɛ, data stages: the user Y k transmits with power P t β k where β k is the path oss in the desired ink, ɛ [, ] is the fraction of the path oss compensation, and P t is the open oop transmit power with no power contro. Further, we omit the constraint on the maximum upink transmit power for simpicity. To maintain tractabiity of the anaysis using stochastic geometry, we assume that the base stations estimate the channe by correating the received training signa with the corresponding piot without using the minimum mean squared error estimation as empoyed in [6]. Therma noise is ignored in our anaysis, as ceuar networks beow 6 GHz are mosty interference-imited, and the impact of noise vanishes when the number of antennas M grows arge []. Hence,
in the channe estimation stage, the estimate of the channe ɛ/2 h k k at the base station X is h = β k k h ɛ/2 β k k h, where ɛ/2 β k k h is the estimation error caused by piot contamination. In the upink, to decode the upink data from Y, the base station X is assumed to use the channe estimate h to perform MRC. Therefore, the upink SIR for the user Y is SIR = β,k, ɛ h h 2 ɛ β k h hk 2. 4 The proposed system mode represents a simpe massive MIMO systems in which the SIR expression can be anayzed using stochastic geometry. In the foowing sections, we wi study the upink SIR distribution. III. PERFORMANCE ANALYSIS In this section, we first introduce the asymptotic SIR resuts, when the number of antennas goes to infinity. Then we focus on the non-asymptotic case with finite base station antennas, where we propose to sove the probem of how the number of antennas M shoud scae with the number of schedued users per ce K, to maintain the same upink SIR. A. Asymptotic SIR Anaysis Now we study the asymptotic upink SIR when the number of base station antennas goes to infinity. Simiar to the anaysis in finite-size networks [], when M, the upink SIR converges to an asymptotic equivaence in the proposed argescae networks. The convergence resuts are summarized in the foowing theorem. The proof of the resuts can be found in [4] Theorem : With δ>, the upink SIR in the proposed network converges to its asymptotic equivaence in probabiity as SIR UL p. β 2 ɛ β 2 β 2ɛ, when the number of antennas M. Further, for ɛ <, the distribution of the asymptotic SIR can be approximatey as PSIR >T e α T Γɛ α α ɛ, 5 where Γ is the Gamma function. Though the asymptotic resuts provide an upper performance bound for the upink SIR with finite antennas, however, the tightness of the bound is not guaranteed. As in [], [4], simuations show that it may require more than 4 antennas to approach the asymptotic bound in certain cases. Consequenty, we continue to study the non-asymptotic SIR in the subsequent section. B. SIR Anaysis with Finite Antennas The non-asymptotic SIR for finite base station antennas is generay difficut to anayze, due to the correation between the interference terms that do not vanish in the non-asymptotic SIR expression. To obtain insights on the upink performance, we begin the anaysis with the foowing simpe case. Case IID fading with no power contro: The smascae fading in a inks is assumed to be IID Rayeigh, and the fraction of the path oss compensation is ɛ =.Inthis case, the upink SIR can be evauated as foows. Theorem 2: With IID Rayeigh fading and no fractiona power contro, the upink SIR distribution can be approximated as N N PSIR >T n e μat a2tα dt, 6 n K n= where μ =, a M 2/α =Γ 2/αnηT 2/α, a 2 = nηt α, N is the number of terms used in the approximation, η = NN! N, and Γ is the Gamma function. Proof: See Appendix A. We wi show in Section IV that Theorem 2 provides a tight approximation, when N 5 terms are used. Moreover, note that in 6, the number of antennas M and the number of schedued users per ce K ony affect the vaue of μ. Therefore, by Theorem 2, assuming IID fading channe and no power contro, the scaing aw to maintain the same upink SIR distribution is M K α/2, which is superinear when α>2. Next, we extend the resuts to correated fading. Case 2 Correated fading with no power contro: For the ease of iustration, we use the exponentia antenna correation mode [2] as an exampe to account for the fading correations, whie the resuts appy to other correation modes satisfying the constraint in 2. In the exponentia correation mode [2], the i, j-th entry of correation matrix Φ k is Φ k [i, j] =ρ i j, where ρ [, is the correation coefficient of the channes between neighbouring antennas. Measurements showed high correations, e.g. ρ>.5, between antennas haf-waveength apart in massive MIMO systems [9]. For simpicity, we assume ρ remains the same for a channes in the anaysis. In the correated fading case, we can compute the upink SIR in the foowing coroary. Coroary 2.: With correated fading, the upink SIR distribution can be appropriated via 6 by repacing the scaing Kγ constant μ as μ = 2/α, where γ is the constant M2γ 2/α defined in 2. Moreover, in the exponentia correation mode, γ is computed as γ = 2π 2 ρ 2 2π ρ 2 2ρ cosu du. Proof: See Appendix B. Note that with correations in fading, the scaing aw to preserve the SIR distribution is M 2γ K α/2. Compared with the IID fading case, the correations in fading resut in a shift of 2γ in the scaing aw, which is approximatey twice the difference in the average square of the eigenvaues of the fading covariance matrices. Moreover, given the fact that γ increases with ρ, it foows that the higher the correation, the more antennas are needed to maintain the
same SIR distribution. Next, we continue to study the impact of fractiona power contro. Case 3 Fractiona power contro: The case with fractiona power contro is generay difficut to anayze due to the compicated SIR expression. In the case of fu path oss compensation ɛ =, where a the schedued users in a ce have the identica effective path oss after the compensation in power contro, the anaysis in [6, Coroary 2] show that a inear scaing between M and K is sufficient to keep the SIR unchanged. For genera ɛ,, we can approximate the scaing aw between M and K in the foowing coroary. Coroary 2.2: With fractiona power contro, the scaing aw between M and K is approximatey M 2γ K s, where the exponent of the scaing aw is s = α 2 ɛɛ. Proof: Note that by [6], when ɛ =, s =, and by Theorem 2, when ɛ =, s = α 2. Therefore, for genera < ɛ<, the exponent of the scaing aw s can be approximated as the inear fitting from the two extreme cases. Though an approximation, the proposed scaing aw in Coroary 2.2 matches the simuation resuts. Last, we can appy the SIR resuts to compute the achievabe rate. Let the average achievabe spectrum efficiency at a typica user be ξ = og 2 min{sir,t max }, where T max is a SINR distortion threshod determined by the imiting factors ike non-inearity in the radio frequency front-end. Given the SIR distribution PSIR >T, the average achievabe spectrum efficiency can be computed as in [2, Section III-C]. IV. NUMERICAL RESULTS In this section, we verify our anaytica resuts with numerica simuations. As a genera setup of Monte Caro simuations, we assume the user density is 6 times the base station density, and the base stations randomy pick K out of the associated users to serve in a resource bock. In the simuation, the average inter-site distance between base stations is 3 meters. CCDF of SIR.7.6.5.4.3.2. Simu: K,M,ρ=4,6,.8 Anay: K,M,ρ=4,6,.8 Simu: K,M,ρ=8,88,.8 Simu: K,M,ρ=4,6, Anay: K,M,ρ=4,6, Simu: K,M,ρ=8,67, 5 5 5 2 25 3 35 4 SIR threshod in db Fig. 2. SIR distributions with correated fading. We assume ɛ =, and α = 4. The anaytica curves are drawn using N = 5 terms, based on Theorem 2 and Coroary 2.. CCDF of SIR.9.8.7.6.5.4.3.2. K,M,ε=5,32, K,M,ε=,3, K,M,ε=5,32,.5 K,M,ε=,92,.5 K,M,ε=5,32, K,M,ε=,64, 5 5 5 2 25 3 35 4 SIR threhod in db Fig. 3. SIR distributions with fractiona power contro. We assume α =4, and IID fading channe. Note that ɛ =is for fu path oss compensation, and ɛ =for no compensation..9 CCDF of SINR.8.7.6.5.4.3.2. M= 2 M= 3 M= 4 M= 5 5 5 2 25 3 35 4 SINR threshod in db Fig.. Convergence to the asymptotic SIR. In the simuations, we assume α = 4, K =,andɛ =. The asymptotic curve is drawn based on Theorem. Asymptotic SIR distribution: In Fig., we show the convergence of upink SIR to its asymptotic equivaence. We assume IID fading channe in the simuations. Numerica resuts show that more than 4 antennas are required to approach the performance in the asymptotic regime. Impact of fading correations: We pot the SIR distributions with IID fading and correated fading in Fig. 2. Note that ρ =represents IID fading, and ρ =.8 indicates high fading correations. In the simuations, we doube the number of schedued users K from 4 to 8, and scae the number of antennas M according to the proposed scaing aws, which are shown to preserve the SIR distributions in Fig. 2. The impact of correation can be summarized as foows: i correations in fading degrade the SIR coverage in massive MIMO networks, as fixing M and K, the CCDF of SIR decreases with ρ; ii the correations requires more antennas than the IID fading case to maintain the upink SIR when increasing the number
CCDF of SIR.9.8.7.6.5.4.3.2. K,M,ρ,ε,α=5,32,,,4 K,M,ρ,ε,α=,3,,,4 K,M,ρ,ε,α=5,32,,.5,5 K,M,ρ,ε,α=,,,.5,5 K,M,ρ,ε,α=5,32,.5,.7,4 K,M,ρ,ε,α=,82,.5,.7,4 5 5 5 2 25 3 35 4 SIR threshod in db Fig. 4. SIR distribution in the hexagona grid mode. When K=, we compute the required M to preserve the SIR distribution as that of K =5, according to Coroary 2.2. Simuations confirm the accuracy of the proposed scaing aw. Average spectrum efficiency per user bps/hz 2.8.6.4.2.8.6 ρ= ρ=.2 ρ=.5 ρ=.8.4..2.3.4.5.6.7.8.9 fraction of channe inversion ε Fig. 5. Average spectrum efficiency per user. In the simuation, we assume T max =2dB, which sets the maximum spectrum efficiency per data stream as 7 bps/hz. Correations in fading reduce the average spectrum efficiency. of schedued users per ce. Impact of fractiona power contro: We examine the impact of fractiona power contro in Fig. 3. We compute the required number of antennas M by Coroary 2.2, when increasing the number of schedued user K from 5 to, and check if the scaing aw preserves the SIR distribution by simuations: the proposed scaing aw is shown to be accurate in Fig. 3. Numerica resuts aso show that a arge compensation fraction ɛ improves the SIR coverage in the ow SIR regime at the expense of sacrificing the coverage in the high SIR regime. Intuitivey speaking, fractiona power contro improves the ce-edge user SIR by trading off the performance of the non-ce-edge users. Verification with hexagona grid mode: In Fig. 4, we use a ayout of 9 hexagona ces with inter-site distance of 3 meters; ony the schedued users in the centra ce are counted for the SIR statistics, to avoid edge effect. With K =schedued users per ce, the required M to maintain the same SIR as that of K, M = 5, 32 is computed by Coroary 2.2, which is shown to be amost accurate with extensive combinations of the system parameters in the hexagona grid mode. This shows that the stochastic geometry mode provides reasonabe predictions even for the hexagona mode. Rate performance: We iustrate the resuts on the average spectrum efficiency per user in Fig. 5. In the simuation, we assume M =64base station antennas, and K =schedued users per ce. Consistent with the SIR resuts, higher eves of correations in fading resut in ower per user rates. Numerica resuts aso show that the average spectrum efficiency is sensitive to the fraction of the path oss compensation, whie the range of the optimum ɛ in the simuations is generay between.5-.6. V. CONCLUSIONS In this paper, the upink SIR in massive MIMO networks with MRC receivers was studied using a stochastic geometry mode. Approximate expressions to compute the SIR distributions were derived in both asymptotic and non-asymptotic cases. Based on the SIR anaysis, the scaing aw between the numbers of antennas and schedued users per ce was provided to maintain the same upink SIR distribution. Our anaysis showed that compared with the scaing aw of IID fading, the spatia correation resuted in a constant offset in the number of antennas. The exponent of the scaing aw is affected by the fractiona power contro. The anaytica resuts were verified with numerica simuations. Besides, numerica resuts indicated that the optimum fraction of the fractiona power contro is roughy.5 to maximize the average achievabe rate. For future work, the framework can be extended to study the performance of more advanced combining schemes, e.g. with zero-forcing receivers. APPENDIX A. Proof of Theorem 2: First, with M base station antennas, the upink SIR expression in 4 can be approximated as SIR a = b c = E h h 2,k, E h h 2 M β2 M β 2 β β β β k β βk M β 2 M β2 β M β 2 M C 2 πλ b α /α β,k, βk,,k, βk where in a the expectations are taken with respect to the sma-scae fading in h k ; in b we drop certain terms that do not scae with M in both the numerator and denominator;
in c we approximate β2 by its mean, which is computed by Campbe s Theorem [2]. Next, conditioning on R = x, we compute the conditiona upink SIR distribution as P SIR >T R = x = P >T πλ b α x 2α x α,k, βk α CM a P g >T πλ b α x 2α x α,k, βk α CM ηt b πλ b α x 2α x α N,k, βk α CM E e c = N N n e nηt πλ b n n= α α x 2α πkλ b Γ 2/α nηt M 2/α x 2 where in a we use a dummy gamma variabe g with unit mean and shape parameter N to approximate the constant number one, and the approximation foows from the fact that g converges to one when N goes to infinity, n i.e., im n x n e nx n Γn = δx [22], where δx is the Dirac deta function; in b, the approximation foows from Azer s inequaity [2], [23, Appendix A], where η = NN! N ; in c we approximate the point process invoved with,k, βk as a PPP with density Kλ b, and it foows from computing the Lapacian functiona of the homogeneous PPP [2]. Last, noting that R is a Rayeigh random variabe with mean.5 /λ b [8], we obtain the upink SIR distribution by de-conditioning on R = x and changing the variabe as t = πλ b x 2 B. Proof of Coroary 2.: First, simiar to the proof in Appendix A, the upink SIR expression with fading correations can be approximated as SIR a b E h h 2,k, E h h 2 M2γ M γ M β 2 M2γ M γ M M2γ γ β2 β M2γ γ β 2 β2 β M m= λk2 M,k, βk,k, βk where in a γ M = is the average square of the eigenvaues for the covariance matrix Φ k in the case of M antennas; in b we approximate γ M with its superior imit γ as defined in 2. Moreover, when assuming the exponentia antenna correation mode, the covariance matrices of the fading correation Φ k are Toepitz matrices: by [24], the asymptotic imit of γ M can be computed as, 2π ρ 2 γ = 2π foows the same ine as in Appendix A. ρ 2 2ρ cosx 2 dx. 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