Analyzing Uplink SINR and Rate in Massive. MIMO Systems Using Stochastic Geometry

Transcription

1 Anayzing Upink SINR and Rate in Massive MIMO Systems Using Stochastic Geometry Tianyang Bai and Robert W. Heath, Jr. arxiv:5.2538v2 [cs.it] 2 Apr 26 Abstract This paper proposes a stochastic geometry framework to anayze the SINR and rate performance in a arge-scae upink massive MIMO network. Based on the mode, expressions are derived for spatia average SINR distributions over user and base station distributions with maximum ratio combining MRC) and zero-forcing ZF) receivers. We show that using massive MIMO, the upink SINR in certain urban marco-ce scenarios is imited by interference. In the interference-imited regime, the resuts revea that for MRC receivers, a super-inear poynomia) scaing aw between the number of base station antennas and schedued users per ce preserves the upink SIR distribution, whie a inear scaing appies to ZF receivers. ZF receivers are shown to outperform MRC receivers in the SIR coverage, and the performance gap is quantified in terms of the difference in the number of antennas to achieve the same SIR distribution. Numerica resuts verify the anaysis. It is found that the optima compensation fraction in fractiona power contro to optimize rate is generay different for MRC and ZF receivers. Besides, simuations show that the scaing resuts derived from the proposed framework appy to the networks where base stations are distributed according to a hexagona attice. I. INTRODUCTION Massive mutipe-input and mutipe-output MIMO) is an approach to increase the area spectrum efficiency in 5G ceuar systems [2] [5]. By depoying arge-scae antenna arrays, base stations can use muti-user MIMO to serve a arge number of users and provide high ce throughput [2] [5]. In this paper, we focus on the defacto massive MIMO systems operated beow 6 GHz, where piot-aided channe estimation is performed in the upink, and piots are The authors are with The University of Texas at Austin, Austin, TX, USA. emai: tybai@utexas.edu, rheath@utexas.edu) This work is supported by the Nationa Science Foundation under Grant Nos , 39556, and Parts of the resuts on the anaysis of MRC receivers was presented at the 25 IEEE Goba Conference on Communications [].

2 2 reused across ces to reduce the training overhead [2] [5]. Prior work showed that when the number of base station antennas grows arge, high throughput is achieved 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 [2]. In this paper, we derive the signa-to-interference ratio SIR) distribution for the upink of a massive MIMO network with maximum ratio combining MRC) and zero-forcing ZF) receivers, for a random base station topoogy. The performance with MRC and ZF beamforming in terms of signa-to-noise-and-interference ratio SINR), spectrum efficiency, and energy efficiency was examined in a simpe network topoogy, e.g. in [2], [6] [], where the SIR and rate expressions were conditioned on specific user ocations or equivaenty the received power for each user. The concusions drawn from the conditiona expression, however, need not appy to the spatia average system-eve performance due to the difference in users path osses. For exampe, the inear scaing between the number of users and antennas examined in [8] does not maintain the upink SIR distribution, as wi be shown in our anaysis. This motivates the anaysis of the spatia average performance over different base station and user distributions in arge-scae massive MIMO networks, which was mainy studied using Monte Caro simuations in prior work [], [2]. Stochastic geometry provides a powerfu too to anayze system-eve performance in a argescae network with randomy distributed base stations and users. Assuming a singe antenna at each base station, the spatia average downink SIR and rate distributions were derived for a network with Poisson point process PPP) distributed base stations, and were shown a reasonabe fit with simuations using rea base station data [3]. The stochastic geometry framework in [3] was further extended to anayze the performance of MIMO networks: the downink SIR and rate of muti-user MIMO ceuar system were anayzed, e.g. in [4] [7] assuming perfect channe state information CSI), and in [8] with quantized CSI from imited feedback. For upink anaysis, prior work [9] [2] showed that the upink and downink SIR foows different distributions, due to the difference in network topoogy. In [2], a stochastic geometry upink mode was proposed to take account the pairwise correations in the user ocations, where the SIR distributions derived based on the anaytica mode were shown a good fit with the simuations. The prior resuts in [3] [2], however, do not directy appy to anayze upink massive MIMO networks, as i) they did not take account for the effects of piot contamination, which becomes

3 3 a imiting factor with arge numbers of antennas [2]; ii) the anaysis in [3] [8] was intended for downink performance, which foows different distributions from the upink network; and iii) the resuts in [4] [7] were intended for MIMO networks with a few antennas, where the computationa compexity for the anaytica expressions grows with the number of antennas, and hinders the direct appication to the massive MIMO scenarios. Stochastic geometry was aso appied to study the asymptotic SIR and rate in a massive MIMO networks in [22], [23], where the asymptotic SIR is shown to be approached with impracticay arge number of antennas, e.g. 4 antennas. Reated work in [24] appied stochastic geometry to study the upink interference in a massive MIMO network. A inear scaing between the numbers of base station antennas and schedued users was found to maintain the mean interference, which need not preserve the SIR distribution. In this paper, we propose a stochastic geometry framework to derive the upink SINR and rate distributions in a arge-scae ceuar network using muti-user MIMO. To mode the upink topoogy, we propose an excusion ba mode based on prior work [2], which simpifies the computation. Channe estimation error due to piot contamination is aso considered in the system mode. The proposed framework aso incorporates the fractiona power contro by compensating for a fraction of the path oss as in ong term evoution LTE) systems [25]. Based on the framework, we derive anaytica expressions for the upink SINR distribution for both MRC and ZF receivers in the massive MIMO regime. Unike prior work anayzing asymptotic performance with infinity antennas [22], [23], the SINR coverage is examined as a function of the number of base station antennas and schedued users per ce. We appy the SINR resuts to investigate the interference-imited case, as numerica resuts show that the impact of noise becomes minor in the rban macro-ce scenario with certain typica system parameters. We derive scaing aws between the number of base station antennas and schedued users per ce to maintain the same upink SIR distributions. Unike the inear scaing aw examined in prior work [8], [24], we find that a super-inear scaing is generay required for MRC receivers to maintain the upink SIR distributions, due to the near-far effect from intrace interference. For ZF receivers, we show that a inear scaing aw sti hods, as the intra-ce interference is negigibe. We use the scaing aw resuts to quantify the performance gap between ZF and MRC receivers, in terms of the difference in the number of antennas to provide the same SIR distribution. The resuts show that ZF receivers provides better SIR coverage than MRC

4 4 receivers; the performance gap increases with the number of schedued users in a ce, and is reduced with the fractiona power contro, as it mitigates the near-far effect from intra-ce interference. Simuations verify our anaysis, and indicate that the scaing aws derived from the stochastic geometry framework aso appy to the hexagona mode. Numerica resuts on rate aso show that the average per user rate can be maximized by adjusting the compensation fraction, where the optima fraction is around.5 for MRC receivers, and.2 for ZF receivers. Our prior work in [] focused on the performance of MRC receivers, and provided an expression for the SIR distribution assuming no power contro. In this paper, we incorporate therma noise in the anaysis, extend the resuts to the case of genera fractiona power contro for MRC receivers, and anayze the performance of ZF receivers. This paper is organized as foows. We present the system mode for network topoogy and channe assumptions in Section II. We anayze the performance of MRC receivers in Section III, and that of ZF receivers in Section IV-A, foowed by a performance comparison of two receivers in Section IV-B. We present numerica resuts to verify the anaysis in Section V, and concude the paper in Section VI. Notation: We use the foowing notation throughout this paper: Bod ower-case etters x are used to denote vectors, and bod upper-case etters X are used to denote matrices. We use X[:,k] to denote the k-th row of matrix X, X as the Hermitian transpose of X, and X as the pseudo-inverse of X. We use E to denote expectation, and P to denote probabiity. II. SYSTEM MODEL In this section, we introduce the system mode for an upink massive MIMO ceuar network. We focus on the networks operated in the sub-6 GHz band; the proposed mode can be extended for massive MIMO at miimeter wave mmwave) frequencies by incorporating key differences in propagation and hardware constraints [26]. 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 k-th schedued user in the ce of -th base station, and h k) the channe vector from X to Y k). We consider a ceuar network with perfect synchronization, and assume the foowing piotaided channe estimation in the upink. In the upink channe training stage, the schedued users Y k) send their assigned piots t k, and base stations X estimate the channes by correating the

5 5 corresponding piots and using an minimum mean square error MMSE) estimator; in the upink data transmission, the base stations wi appy either MRC or ZF receivers, based on the channe estimates derived from upink piots. Further, we assume the piots {t k } k K are orthogona and fuy reused in the network. Note that the system mode assumption appies to genera upink muti-user MIMO networks with piot-aided channe estimation in the upink, incuding but not imited to the time-division dupex TDD) massive MIMO [2]. 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 [27]. We express the channe vector h k) n CM as ) /2Φ h k) n = β k) k)/2 n n w k) n, ) where β k) n is the arge-scae path oss, wk) n CM is a Gaussian vector with the distribution CN,I M ) for Rayeigh fading, and Φ k) n CM M is the covariance matrix to account for potentia correations in sma-scae fading. In this paper, we focus on the case of identicay and independenty distributed IID) channe fading channes, i.e., Φ k) n = I M. The incorporation of spatia correations in fading is deferred to future work. The arge-scae path oss gain β k) n is computed as β k) n = C R k) n ) α, 2) where C is a constant determined by the carrier frequency and reference distance, α > 2 is the path oss exponent. Next, we introduce the network topoogy assumptions based on stochastic geometry. We assume the base stations are distributed as a PPP with a density λ b. A user is assumed to be associated with the base station that provides the minimum path oss signa. In this paper, each base station is assumed to serve K schedued users that are independenty and uniformy distributed in its Voronoi ce [28]. The assumption is equivaent to that in prior work [2], where the schedued user process is obtained by i) generating an overa user process as a PPP, and ii) randomy seecting K out of associated users in each ce as its schedued users, under the fu buffer assumption that the overa user process is sufficienty dense, such that each base

6 6 station has at east K candidate users in the ce. Without oss of generaity, a typica schedued user Y ) is fixed at the origin, and its serving base station X is denoted as the tagged base station in this paper. We wi investigate the SINR and rate performance at this typica user. Now we focus on modeing the distribution of schedued user process in a resource bock. For k K, the k-th schedued user Y k) in each ce is assigned with the same piot t k. Let N k) u be the point process formed by the ocations of the k-th schedued users Y k) each ce. Note that the schedued user process N k) u from is non-stationary aso non-ppp), as their ocations are correated with the base station process, and the presence of one schedued user using t k prohibits the others in the same ce [9] [2]. Unfortunatey, the correations in the schedued users ocations make the exact anaysis intractabe. In [2], the authors proposed an upink mode to account for the pairwise correations, where the other-ce schedued users for base station X in N k) u is modeed as an inhomogeneous PPP with a density function of λ u r) = λ b e λ br 2), 3) where r is the distance to base station X. To further simpify the anaysis, e.g., the computation in 23) and 26), we propose an excusion ba approximation, as a first-order approximation of the mode in [2], on the distribution of the schedued user process N k) u as foows. Assumption : The foowing assumptions are made to approximate the exact schedued users process N k) u. ) The distances R k) from a user to their associated base stations are assumed to be IID Rayeigh random variabes with mean.5 /λ b [9]. 2) The other-ce schedued user process N k) u is modeed by a homogenous PPP of density λ b outside an excusion ba centered at the tagged base station X with a radius R e. 3) The schedued users processes using different piots N u k) and N k ) u independent for k k. are assumed to be Note that in the excusion ba mode, we equivaenty use a step function λ b Ir < R e )) to approximate the density function in 3), where I ) is the indicator function. In this paper, we et R e = /πλ b ) by matching the average number of the excuded points from a homogenous PPP of density λ b in the step function and in 3), i.e., by etting λ b πre 2 = 2πλ b e λ bπr 2 rdr =. An aternative expanation for our choice of R e is to et the size of the excusion ba πr 2 e equa the average ce size /λ b [29]. In Section V, we

7 7 show that the SINR distributions derived based on the excusion ba assumption, as we as the approximations made in our subsequent derivation, match we with the simuation using the exact user distribution. Fractiona power contro, as used in the LTE systems [25], is assumed in both the upink training and upink data stages: the user Y k) transmits with power ) ǫ, P k) = P t β k) 4) where β k) is the path oss in the corresponding signa ink, ǫ [,] is the fraction of the path oss compensation, and P t is the open oop transmit power with no power contro. We omit the constraint on the maximum upink transmit power for simpicity; the constraint can be incorporated by appying the truncated channe inversion power contro mode [2] to determine the transmit power. We note that ignoring the maximum transmit power constraint increases the average transmit power, and reduces the impact of noise. The incorporation of more compicated power contro agorithms is deferred to future work. The noise power is denoted as σ 2. In the upink training stage, after correating the received training signa with the corresponding piot, base station X has an observation of the channe h ) as u ) = P ) h ) + P ) h ) +n t, > where n t is the noise vector in the training stage foowing the distribution CN ), σ2i K M. We assume for >, the arge-scae path osses β ) are perfecty known to base station X. Since the channes are assumed to be IID Rayeigh fading, the channe h ) MMSE estimator as where h ) can be decomposed as h ) = P ) β ) P) β ) + σ2 K is estimated by an u ), 5) is the estimation of h). Due to the orthogonaity principe, the channe vector h) h ) = h ) +ĥ) whereĥ) is the estimation error foowing the distribution CN Let s k) be the upink data symbo for user Y k), 6) ) ),β ) P ) β ) I. P) β [ ] ) +σ2 K with E s k) 2 = P k). In upink data transmission, base station X is assumed to use the combiner vector g k) to decode s k) from

8 8 Y k), based on the channe estimate the typica user X ) is = g ) ) h s k) +g ) ĥ) s k) + ŝ ) h k). Then, at base station X, the decoded symbo ŝ ) for,k),) g ) h k) sk) +g ) n u } {{ } unknown at base station, 7) where n u C M is the therma noise vector in the upink data transmission. Treating the unknown terms at base station X as uncorreated additive noise, the upink SINR for the typica user Y ) is SINR = P ) g ) ) h 2 P ) E g ) ĥ) 2 +,k),) Pk) E g ) h k) 2 + g ) 2 σ 2, 8) where the expectation operator is taken over the channe estimation error and sma-scae fading in the interference inks. We wi investigate the SINR distributions for MRC and ZF receivers in the foowing sections. The proposed system mode represents a simpe muti-user MIMO systems in which the SINR expression can be anayzed using stochastic geometry. In the foowing sections, we wi study the upink SINR and rate distributions for MRC and ZF receivers, when the number of base station antennas is arge. III. PERFORMANCE ANALYSIS FOR MRC RECEIVERS In this section, we derive an approximate SINR distribution in an upink muti-user MIMO network, where the approximation becomes tight in the massive MIMO regime, e.g. when M > 64. Then, we focus on the interference imited case, as numerica resuts show that the upink SINR is dominated by the interference in certain urban macro-ce scenarios. We derive a scaing aw between the number of users and antennas that maintains the upink SIR distribution at the typica user. Finay, we present a method to compute the per-user achievabe rate and ce throughput, based on the SINR distribution. A. SIR Coverage Anaysis Now we investigate the upink SINR coverage based on the system mode. With MRC receivers, we assume that base station X appies the combining vector g k) as a scaed version

9 9 of the channe estimate h ) to decode the signa from Y ) : g ) = P) β ) + σ2 K P ) β ) h ) = u). 9) Note the scaing on the combining vector is intended to simpify expressions, and wi not change the SINR distribution. Then, using the combining vector in 9), the SINR expression can be simpified in ) as SINR = M ) 2 + where k) = > β k) β ) ) ǫ ) K + k=2 ) ǫβ k) M +) β k) β ) ) 2 ǫ) ) ǫ K + k= k) + σ2 KP t, and k) 2 = > β k) ) β ) ) 2ǫ β k) ) ) ǫ ) ) + ) 2. The derivation to obtain ) is given in Appendix A. Note that k) and k) 2 correspond to the sum of certain interference terms from other-ce users. Next, we denote the exact SINR distribution for ) using the exact schedued user distribution defined in Section II but not the excusion ba assumption) as PSINR > T). Due to piot contamination, the combining vector g k) is correated with certain interference channe vectors as shown in 9). As a resut, the denominator in ) contains cross-products of the path osses from different interferers. Moreover, different cross-product terms in the denominator of ) can be correated, as they may contains common path oss terms, which renders the exact derivation ofpsinr > T) intractabe. Therefore, we compute an approximate SINR distribution PSINR > T), which we argue in Section V is a good match for PSINR > T), in Theorem. Theorem MRC SINR): In the proposed massive MIMO networks, an approximate upink SINR distribution with MRC receivers can be computed as N N PSINR > T) = ) ) n+ e t TηC t α ǫ) TηC 2 t α 2 ǫ) C 3 t)dt, ) n n= wheren is the number of terms used in the cacuation,η = NN!) σ N,C σ 2 = 2 ) C = K+ 2Γ α ǫ 2 +) +C M+ α 2) σ 2, C 2 = MΓα ǫ+) + K 2Γ α 2, ǫ 2 +) +C M+)α ) M+ α 2) σ 2) C 3 t) = e u u α 2 ǫ) TηC 4 t) du ηtc 4 t) ) K ) e u K TηC 4 t)+u ǫ)du, α 2 KP tc ǫ λ b π) α ǫ) 2,

10 ) C 4 t) = 2Γ α ǫ 2 +) +C M+ α 2 σ 2 )t α ǫ) +t α2 ǫ), and Γα) = e t t α dt is the gamma function. Proof: See Appendix B. Besides the excusion ba approximation, the main approximation in Theorem is to repace certain out-of-ce interference terms by their means in 23) and 26). The approximation resuts in a minor error in the SINR distribution, as i) with K users in a ce, the intra-ce interference dominates the out-of-ce interference with high probabiity; ii) with arge antenna arrays, the ratio of the signa power to certain out-of-ce interference power terms, e.g. the terms in k), decays as M. In Section V, using N 5 terms, the distribution PSINR > T) computed in Theorem is shown to be a good match with the SINR distribution PSINR > T) from Monte Caro simuations. In addition, the error of the approximation becomes more prominent with a smaer noise power, as a the approximations are made with respect to the interference distribution. The expression is intended for the massive MIMO regime when M, as the error of the approximations decays with. In simuations, we find that the resuts in the M theorem generay appies to the muti-user MIMO networks with not-so-arge M, e.g. the case of M,K) =,2). In Theorem, the noise power is taken account by the parameter C σ 2 = σ 2 KP tc ǫ λ b π) α ǫ) 2 which shows that the impact of noise on the SINR is reduced with a arger number of schedued users per ce K, a higher base station density λ b, a smaer path oss α, and a arger power contro parameter ǫ. Besides, the impact of noise goes down with arger M, as in the expressions for C and C 2, the noise parameter C σ 2 is divided by M +). Next, we focus on the performance of interference-imited networks. We wi show in Section V that the impact of noise is negigibe in certain urban macro-ce cases with M = 64 antennas at base stations. Then, the genera expression in Theorem can be further simpified in the foowing specia cases. Y k) Case Fu power contro, ǫ = ): In this case, the transmitting power at schedued user is adjusted to compensate for the fu path oss, i.e., P k) = P t β k), such that a base station receives equa signa powers from a of its associated users. When ǫ =, the SIR distribution can be simpified as in the foowing coroary.,

11 Coroary.: With ǫ = and σ 2 =, the approximate SIR distribution can be computed as N ) N PSIR > T) = ) n+ e Tη C5 K+Γ α ).5) + M+ α, 2) n where C 5 = 4Γ2α.5)+α 2 4)Γ α.5) α 2) 2. n= Based on Coroary., a inear scaing aw between the number of users and antennas is observed as foows. Coroary.2: With ǫ = and σ 2 =, to maintain the upink SIR distribution unchanged, the scaing aw between the number of base station antennasm and users per cek is approximatey ) M +) K + Γα.5) K. 3) C 5 Note that when ǫ =, the inear scaing aw matches prior resuts in [8, Sec. IV], where the path oss to a associated users in the typica ce was assumed to be identica. The inear scaing aw, however, does not appy to other cases with ǫ <, e.g. in the foowing case without power contro. Case 2 No power contro, ǫ = ): In this case, the fraction of the path oss compensation is ǫ =. Then, the upink SIR can be evauated as foows. as Theorem 2: With ǫ = and σ 2 =, an approximate upink SIR distribution can be cacuated PSIR > T) = N n= N ) ) n+ e µγ 2/α)nηT)2/α +)t nηt α tα dt, 4) n where N is the number of terms used in the computation, and µ = Proof: The proof is simiar to that in [, Appendix A]. K. M+) 2/α We wi show in Section V that Theorem 2 provides a tight approximation of the exact SIR distribution PSIR > T), when N 5 terms are used. Moreover, note that in 4), the number of antennas M and the number of schedued users per ce K ony affect the vaue of µ. Therefore, by Theorem 2, in the no power contro case, we observe the foowing scaing aw to maintain SIR. Coroary 2.: Assuming no power contro, the approximate scaing aw to maintain the same upink SIR distribution is M + ) K α/2, which is a superinear poynomia scaing when α > 2.

12 2 In the case of no power contro, the difference in the path osses between the typica user and the intra-ce interferers affect the SIR distribution, and thus the scaing aw to maintain the SIR becomes a function of the path oss exponent. The super-inearity in the scaing aw can be expained by the near-far effect of the intra-ce interference from mutipe users in a ce. With no power contro, the ce edge users wi receive weaker signas than the ce center user. With a uniform user distribution in a ce, the typica user wi be more ikey to be ocated at the ce edge. When increasing the number of schedued users K in a ce, the probabiity that the interference from a ce-center interferer dominates the signa from the typica user increases. Therefore, compared with the inear scaing aw with fu power contro ǫ = ) where such near-far effect is mitigated, more antennas wi be needed in the no power contro case to reduce the intra-ce interference, and preserve the SIR distribution, when increasing K. Next, we focus on the scaing aw in the genera fractiona power contro case with ǫ,). It is difficut to derive the exact scaing aw directy from the expression ), due to the integra form. Since with the fractiona power contro, the equivaent path oss exponent in the signa ink ineary scaes with ǫ, we propose the foowing approximate scaing aw by ineary fitting the exponent s of the scaing aw M +) K s, based on two specia cases of ǫ: by Coroary.2, when ǫ =, s = ; and by Theorem 2, when ǫ =, s = α. Therefore, for genera < ǫ <, 2 the ineary fitted exponent of the scaing aw s is given as foows. Scaing aw : With fractiona power contro, the scaing aw between M and K is approximatey M +) K s, where the exponent of the scaing aw is s = α 2 ǫ)+ǫ. Scaing aw reveas that a superinear) poynomia scaing aw between K and M is required to maintain upink SIR distribution, for a genera ǫ <. The resuts in Scaing aw are verified by numerica simuations in Section V. B. Rate Anaysis In this section, we appy the SINR resuts to compute the achievabe rate. First, we define the average achievabe spectrum efficiency at a typica user as τ = E[og 2 +min{sinr,t max })], 5) where T max is a SINR distortion threshod determined by imiting factors ike distortion in the radio frequency front-end. By [3, Section III-C], given the SINR distribution PSINR > T), the

13 3 average achievabe spectrum efficiency can be computed as τ = Tmax PSINR>x) dx. To take n2) +x account for the overhead, et ψ be the fraction time for overhead. In this paper, for simpicity, we ony consider the overhead due to upink channe training, and compute the overhead fraction ψ as ψ = Tt T c = K T c, where T t and T c are the ength of channe training period and coherent time, in terms of the number of symbo time. The ength of channe training is assumed to be equa to the number of schedued users in a ce, as we assumed fu reuse of orthogona piots throughout the network. Then the average achievabe rate with the overhead penaty τ equas τ = K ψ)τ, 6) Note that when ignoring therma noise, the scaing aw to maintain SINR distribution aso maintains the average achievabe rate τ. When taking account for the training overhead penaty ψ, however, the scaing aw wi not keep τ unchanged, as ψ ineary decreases with K, uness ψ is negigibe, e.g. when the coherence time T c K. Next, we define the average ce throughput τ ce, in terms of spectrum efficiency, as τ ce = K ψ)τ. 7) We wi examine the average ce throughput as a function of M and K in Section V. Before that, we continue to present the resuts for ZF receivers in the next section. IV. PERFORMANCE ANALYSIS WITH ZF RECEIVERS In this section, we wi investigate the performance of ZF receivers in IID fading channes. First, we derive the SINR and rate distributions with ZF receivers. Then, we appy the anaytica resuts to compare the performance of MRC and ZF receivers in an interference-imited network. In particuar, we aim to answer the question: compared with MRC receivers, how many antennas can be saved by appying ZF receivers, whie keeping the same upink SIR distribution. A. SINR Anaysis of ZF Receivers Now we begin to investigate the performance of ZF receivers in an upink massive MIMO network. For ZF receivers, we sti focus on the case of IID fading. To cance the intra-ce interference, base station X wi appy the combining vector g k) for user Y k) g k) = H [:,k], 8) as

14 where H = [ ] u ),u2),...,uk) associated users in ce X, and u k) = C M K is the matrix of a estimated channes to the P) β ) +σ2 K P k) β k) 4 h k) is a scaed version of the channe estimate. The scaing in the channe estimates wi not change the upink SINR distribution, as it wi ony cause certain scaing in the corresponding combining vector. Simiar to the case of MRC receivers, the exact upink SINR distribution is difficut to derive, as due to piot contamination, the combining vector is correated with certain interference channe vectors. Therefore, appying the same approximations in 23) and 26), we derive an approximate distribution for the upink SINR expression in 8) for the typica user X ) in the foowing theorem. Theorem 3: With M K and ZF receivers, an approximate upink SINR distribution for the typica user can be cacuated by PSINR > T) = where the constant C 6 = C 9 N n= N ) ) n+ e nηtc 6t α 2 ǫ) +C 7 t α ǫ) ) t dt, 9) n ) + + MK ) + MK )C 8, M K+ M+ M K+) 2 M K+) 2 C 7 = M Γ α ǫ+) K )M + + M + α M + M K +) 2 ) C C 8 = 2Γα ǫ 2 +)+α 2)C σ 2 α 2)+C σ 2)+2Γ α ǫ 2 +), C 9 = 2Γα ǫ 2 +) α 2 +C σ 2, C σ 2 = σ 2 terms used in the computation, and η = NN!) N. Proof: See Appendix C. KP tc ǫ λ b π) α ǫ) 2 K )M M K +) 2C 8C 9,, N is the number of Note that when K =, the SINR distribution in 9) for ZF receivers is the same as that for MRC receivers in 2). We wi verify the tightness of the approximation PSINR > T) PSINR > T) by numerica simuation in Section V. We have the foowing remark on the appicabe regime for Theorem 3. Remark : We need the condition M K in the proof, as the error in the approximation in 28) decays as. In numerica simuations, we find that the approximate SINR distribution M K+ in Theorem 3 shows a good match with the simuations when M K comment appies to Scaing aw 2 beow. 3 with M. The same Next, we focus on the interference-imited case. Based on Theorem 3, we can derive an approximate scaing aw between M and K to maintain the SIR distribution in the region of M K as foows.

15 5 Scaing aw 2: With ZF receivers and σ 2 =, the upink SIR distribution of the typica user remains approximatey unchanged when the number of antennas M ineary scaes with the number of users per ce K as M +) K. Proof: Note that when σ 2 =, C σ =. The dependence on M and K in 9) ony occurs in the constants C 6 and C 7. Therefore, it is sufficient to show that a inear scaing between M and K approximatey) maintains the vaues of C 6 and C 7. Note that when M, and M K, the foowing imits hod: M+, M K+, and M M+. Therefore, it K foows that when keeping = t, im M+ M C 6 = C 8+C 9 )t, and im t M C 7 = Γα ǫ+) + ) α t 4Γ 2α ǫ 2 +) + 2C 8Γ α ǫ t α 2) 2 2 +), which are invariant when M +) ineary scaes with K. α 2 Compared with MRC receivers, the near-far effect for users in a ce becomes minor with ZF receivers, as the intra-ce interference is argey suppressed. Therefore, a inear scaing aw appies for ZF receivers even without power contro. Based on the SINR coverage resuts, the achievabe rate per user and sum throughput can be computed foowing the same ine as in Section III-B. In the next section, we wi use the derived resuts to compare the SIR coverage performance between MRC and ZF receivers in an interference-imited network. B. Comparison of SIR Coverage Performance Now assuming the network is interference-imited, we compare the SIR coverage between ZF and MRC receivers. Prior work [8] showed that ZF and MRC receivers have the same asymptotic performance, both which are imited by the piot contamination. The anaysis in [24] showed that by suppressing intra-ce interference, which turns to be more dominant than the out-of-ce interference, the ZF receivers suffers from ess interference than MRC receivers. In this section, we make a quantitative comparison by answering the foowing question: in IID fading channes, how many base station antennas M ZF is needed for ZF receivers to provide the same upink SIR distribution as MRC receivers with M MRC antennas? Based on Scaing aw and Scaing aw 2, we have the foowing proposition to determine M ZF to match the SIR coverage with MRC receivers. Proposition : Assuming M ZF K, ZF receivers with M ZF +) = ξm MRC +) antennas approximatey provide the same upink SIR distribution as MRC receivers with M MRC antennas in a massive MIMO networks, where the scaing factor ξ = K α 2 ) ǫ), and K is the number of schedued users in a ce.

16 6 Proof: For the ease of notation, et ZFM,K) and MRCM,K) denote the upink SIR distributions with ZF and MRC receivers of M antennas, when serving K users in a ce. By Scaing aw 2, when M ZF K, ZFM ZF,K) ZF M ZF+,). Next, note that when K =, i.e., K with a singe schedued user in a ce, MRC and ZF receivers provide the same SIR coverage. Thus, it foows that ZFM ZF,K) = ZF M ZF+ K,) = MRCM ZF+,). Last, by Scaing aw, K ZFM ZF,K) MRC M ZF+,) MRCM K ZF +)K α 2 ) ǫ),k). The condition M ZF K in the proposition is required to ensure the appicabiity of Scaing aw 2. In numerica simuations, the resut is found to be a good approximation with M ZF K > 3. Note that the exponent of the scaing factor α ) ǫ) is non-positive, which indicates we 2 need M MRC M ZF to provide the same SIR coverage. Further, the scaing factor ξ increases with the number of the schedued user K, which reveas that the performance gap between MRC and ZF receivers grows with K. When K increases, the mitigation of the intra-ce interference from K ) users by ZF receivers becomes more prominent to improve SIR coverage. In addition, Proposition aso shows that in terms of the SIR distribution, the performance gap reduces with arger ǫ in the power contro scheme, as the scaing factor ξ is a decreasing function of ǫ. Simuations show that with ǫ =, ony a minor gap exists between the SIR coverage curves for ZF and MRC receivers. Last, we note that Proposition, which is drawn based on the SIR distribution, need not extend to a genera SINR distribution that is not dominated by interference; prior work [6] showed that when the noise is not negigibe, MRC receivers woud provide a comparabe or even better SINR, compared with the ZF receivers. In the foowing section, we wi present numerica resuts to vaidate our anaytica resuts. V. NUMERICAL RESULTS In this section, we verify our anaytica resuts with numerica simuations, which foow the procedure as: ) generating the base station process as a PPP of density λ b ; 2) generating the overa user process as a PPP of density λ u,o, where we use λ u,o = 6λ b, uness otherwise specified; 3) associating the points in the overa user process to base stations, based on the minima path oss rue, and then randomy scheduing K out of the associated users in each ce as their schedued users; 4) picking the base station cosest to the origin as the tagged base station X, and its first schedued user Y ) as the typica user; 5) generating channe vectors

17 7 as IID Gaussian vectors, and computing the SINR for the iteration; 6) repeating the step )-5) for, iterations, and computing the empirica distribution of the SINR at Y ). For the simuations using hexagona grids, we foow the same procedure except that the base station process is generated as a 9-ce hexagona grid, and the tagged base station is the center ce. In addition, we wi use N = 5 terms when evauating the anaytica expressions ZF: M= SINR CCDF.5.4 SINR CCDF.5.4 ZF: M= MRC: M= MRC: M= SINR threshod in db SINR threshod in db a) ISD=5 meters. b) ISD= meters. Fig.. Comparison of SINR and SIR distributions. In the figures, we use markers to represent SINR curves, soid ines for SIR. We assume K = users per ce, ǫ =, and α = 4 in a cases. The gap between the SIR and SINR distributions becomes minor when ISD=5 meters, which is the typica size for the urban macro ces [3]. SIR CCDF Anay: M,K,ε)=64,,) Simu: M,K,ε)=64,,) Anay: M,K,ε)=64,,.5) Simu: M,K,ε)=64,,.5) Anay: M,K,ε)=64,,) Simu: M,K,ε)=64,,) Anay: M,K,ε)=28,2,) Simu: M,K,ε)=28,2,) SIR threshod in db Fig. 2. SIR coverage for MRC receivers. In the simuations, we assume α = 4. The anaytica curves are drawn based on Theorem, which are shown a good fit with simuation. The difference in the curves for M,K,ǫ) = 64,,) and M,K,ǫ) = 28,2,) indicates that inear scaing between M and K does not generay preserve SIR for MRC receivers.

18 8 SIR CCDF Anay: M,K,ε)=64,,) Simu: M,K,ε)=64,,) Anay: M,K,ε)=64,,.5) Simu: M,K,ε)=64,,.5) Anay: M,K,ε)=64,,) Anay: M,K,ε)=64,,) Anay: M,K,ε)=28,2,) SIR threshod in db Fig. 3. SIR distributions with ZF receivers. We assume α = 4, and IID fading channe. The anaytica curves are potted based on Theorem 3. Simuations verify the anaytica resuts, and show that when both M and K doube, the SIR curves remain amost unchanged. SIR CCDF ZF: M,K,ε)=64,4,) MRC: M,K,ε)=64,4,) ZF: M,K,ε)=64,2,) MRC: M,K,ε)=64,2,) ZF: M,K,ε)=64,2,) MRC:M,K,ε)=64,2,) SIR threshod in db Fig. 4. Comparison of SIR coverage with MRC and ZF receivers. We assume α = 4. As the doube arrays dispay, when fixing ǫ =, the performance gap in SIR coverage is shown to increase with K; when fixing K = 2, the gap diminishes when ǫ. Impact of the noise: To begin with, we examine the impact of noise by comparing the SINR and SIR distributions for both MRC and ZF receivers in different scenarios in Fig.. In the simuations, we assume P t =23 dbm, and the bandwidth is 2 MHz as in the current LTE standards [3]. We examine the case of ǫ =, which maximizes the impact of the noise. We simuate with two inter-site distances ISDs): an average ISD of 5 meters in Fig. a), and meters in Fig. b). Note that a typica ISD of 5 meters is assumed for urban macro-

19 9 CCDF of SIR MRC: M,K,ε)=64,5,) ZF: M,K,ε)=3,5,) MRC: M,K,ε)=64,5,.5) ZF: M,K,ε)=28,5,.5) MRC: M,K,ε)=64,5,.8) ZF: M,K,ε)=46,5,.8) SIR threshod in db Fig. 5. Verification of Proposition. In the simuation, α = 4. In the simuation, we use the SIR curve of MRC64,5) as a baseine for comparison. We use Proposition to compute the required number of antennas for ZF receivers, to have the SIR distribution of the baseine curve M,K,α,ε)=32,5,3,.5) M,K,α,ε)=77,,3,.5) M,K,α,ε)=32,5,4,) M,K,α,ε)=3,,4,) M,K,α,ε)=32,5,5,.9) M,K,α,ε)=72,,5,.9) M,K,α,ε)=32,5,3,.5) M,K,α,ε)=64,,3,.5) M,K,α,ε)=32,5,4,) M,K,α,ε)=64,,4,) M,K,α,ε)=32,5,5,.9) M,K,α,ε)=64,,3,.9) SIR CCDF SIR CCDF SIR threshod in db SIR threshod in db a) SIR for MRC receivers in the hexagona grid mode. b) SIR for ZF receivers in the hexagona grid mode. Fig. 6. Verification of the scaing aws in the hexagona mode. We use M,K) = 32,5) as the baseine curves. When increasing the number of users to K=, we compute the required M to preserve the SIR distribution as baseine curves, according to Scaing aw and Scaing aw 2. Simuations indicates that the scaing aw resuts appy to the hexagona mode. ces in the 3GPP standards [3]. In Fig. a), the network with ISD=5 meters is shown to be interference-imited with M = 64 antennas for both MRC and ZF receivers, as the SIR curves amost coincide with the SINR curves, which justifies the interference-imited assumption in urban marco ces. In the sparse network with ISD= meters, however, simuations show that even with M=64 antennas, notabe gaps exist between the SINR and SIR distributions, especiay

20 2 2 MRC:ρ=,ε= Number of antennas M MRC:ρ=,ε=.5 MRC:ρ=,ε= ZF: ρ=, ε [,] MRC:ε= MRC:ε=.5 MRC:ε= 2 ZF Number of schedued users per ce K Fig. 7. Comparison of different scaing aws. We pot the required number of antennas to provide the same SIR as that of the case M,K) = 6,5) as a function of K with different system parameters. Spectrum efficiency per user in bps/hz MRC: M=28 ZF: M=28 MRC:M=64 ZF:M= Compensation fraction ε Fig. 8. Average spectrum efficiency per user in an interference-imited network. In the simuation, we assume T max = 2 db, which sets the maximum spectrum efficiency per data stream as 7 bps/hz. Training overhead is not taken account in this figure. for ZF receivers. In addition, the resuts in Fig. b) shows that when the noise power is high, even with no power contro, ZF and MRC receivers have a simiar SINR coverage performance, which indicates that the SIR comparison resuts in Proposition need not extends to genera SINR comparisons. SIR coverage for MRC receivers: In Fig. 2, we verify the anaytica resuts for the SIR distribution with MRC receivers. Numerica resuts show that the SIR coverage is sensitive to the compensation fraction ǫ in the fractiona power contro: a arge compensation fraction ǫ improves

21 Ce throughput in bps/hz 2 5 ZF: M=64 ZF: M=28 MRC: M=64 MRC: M=28 Ce throughput in bps/hz ZF: M=64 ZF: M=28 MRC: M=64 MRC: M= Number of schedued users per ce K Number of schedued users per ce K a) Ce throughput when T c=4 symbos. b) Ce throughput when T c=2 symbos. Fig. 9. Upink ce throughput as a function of K. The overhead due to channe training is taken account in the simuations. We simuate an interference-imited network with ISD=5 meters. We use ǫ =.5 for MRC receivers, and ǫ =.2 for ZF receivers, which are shown to optimize the per user rate. the SIR coverage in the ow SIR regime at the expense of sacrificing the coverage in the high SIR regime. Besides, a comparison of the curves for M,K) = 64,) and M,K) = 28,2) shows that the inear scaing aw does not maintain the SIR distribution when ǫ =. SIR coverage for ZF receivers: We verify the anaysis for ZF receivers in Fig. 3. The anaytica curves generay match we with numerica simuations. A comparison of the curves for M,K,ǫ) = 64,,) and M,K,ǫ) = 28,2,) shows that unike the case of MRC receivers, a inear scaing aw between M and K maintains the SIR distribution, even when there is no fractiona power contro impemented. SIR comparison between MRC and ZF receivers: We compare the upink SIR distributions for MRC and ZF receivers in Fig. 4. Simuations show that ZF receivers provide better SIR coverage, due to the suppression of intra-ce interference. Moreover, for the same ǫ, the performance gap between MRC and ZF receivers increases with the number of schedued users K, as the strength of tota intra-ce interference aso increases with K. When fixing M and K, the performance gap decreases with ǫ; when ǫ =, the SIR coverage gap becomes minima between MRC and ZF receivers. With fu compensation of path oss in power contro, inear scaing aws between M and K appy to both MRC and ZF receivers, as the near-far effect for users in a ce is mitigated. When M K, the difference in the average residue) intra-ce

22 22 TABLE I COHERENCE TIMET c IN THE EXAMPLES Mobiity Max. veocity Max. Dopper f D Coherence time T c High 5 Km/h 92.6 Hz 4 Symbos Low Km/h 8.5 Hz 2 Symbos interference between MRC and ZF receivers becomes minor, as it decays with M. In Fig. 5, we verify our theoretica resuts in Proposition. In the simuation, we fix the number of antennas for the MRC receivers to be M MRC = 64, and use Proposition to cacuate the required M ZF, to maintain the same SIR distribution. Numerica resuts show a good match with our anaysis; the minor mismatch in the case ǫ = is because Proposition theoreticay requires M ZF K, whie we use M ZF K = 3 5 in the simuation. Verification with hexagona grid mode: We verify the scaing aws derived from stochastic geometry with the hexagona grid mode in Fig. 6. In the simuations, 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. In Fig. 6a), for MRC receivers, we use a M,K) = 32,5) as the baseine curve for comparison. When doubing the number of schedued users to K =, we use Scaing aw to compute the required M to maintain the same SIR distribution, which is shown to be amost accurate with extensive combinations of the system parameters in the hexagona grid mode. Simiary, resuts in Fig. 6b) verifies the inear scaing aw for ZF receivers in Scaing aw 2. This indicates that the stochastic geometry mode provides reasonabe predictions even for the hexagona mode. Comparison of scaing aws: We compare scaing aws to maintain the upink SIR distribution in different scenarios in Fig. 7. We pot the required number of antennas to maintain the same SIR distribution as that in the case ofm,k) = 6,5), as a function of K. As shown in the pot, for MRC receivers, given the path oss exponent α, the sope of the scaing aw is determined by the fraction of path oss compensation ǫ: the inear scaing aw proposed in prior work [8], [24] is ony achieved when ǫ =. Athough the choice of ǫ = makes the system with MRC receivers ineary scaabe, it need not maximize the per-user rate, as wi be shown in Fig. 8. On the contrary, for ZF receivers, the inear scaing appies for a ǫ [,].

23 23 Rate performance: We iustrate the resuts on the average spectrum efficiency per user in Fig. 8. In the simuation, the average ISD is 5 meters, and K =, which is shown to be interference-imited in Fig.. Consistent with the SIR resuts, in a interference-imited network, ZF receivers provide a higher spectrum efficiency per user. Numerica resuts aso show that the average spectrum efficiency is sensitive to the fraction of the path oss compensation ǫ; the optimum ǫ for per user rate is generay around.5 for MRC receiver, and.2 for ZF receivers. In addition, we aso observe that there is a minor performance gap in rate between ZF and MRC receivers under fu channe compensation power contro, as predicted by Proposition. Last, we examine the ce throughput in a system operated at 2 GHz in Fig. 9. As an exampe, we consider an OFDM system, where the symbo time is 66.7 µs. We consider two cases with different mobiities as isted in Tabe I; the coherence time T c is computed as T c = 4f D [32], where f D is the maximum dopper frequency. In this simuation, we assume the density of overa users to be times the base station density, to simuate the case with arge K. In Fig. 9a), in the high mobiity case, when T c < M, the optima K for ce throughput is imited by the duration of T c, and the optima vaue generay is K Tc. In the exampe of ow mobiity 2 case, when T c > M, the resuts in Fig. 9b) show that the optima K depends much on M: for ZF receivers, the ce throughput drops fast when K M approaches to, and the optima K is around M 2 for maximum throughput; for MRC receivers, the ce throughput becomes saturated approximatey when K > M. In addition, ZF receivers generay achieve better ce throughput 3 than MRC receivers; the ony exception is the case of M K, where the ce throughput of ZF receivers drops beow that of MRC. In addition, compared with the singe user per ce case K = ), the resuts confirm that massive MIMO improves the ce throughput by serving mutipe users simutaneousy. VI. CONCLUSIONS In this paper, we proposed a stochastic geometry framework to anayze the spatia average SINR coverage and rate in massive MIMO networks. We appied the anaysis and numerica resuts to draw severa important system design insights about the SINR coverage and rate in upink massive MIMO networks. The upink massive MIMO networks can be interference-imited in urban marco ces ISD=5 meters) with M = 64 antennas at base stations.

24 24 With MRC receivers, the number of antennas M shoud scae super-ineary with the number of schedued users per ce K as M + ) K α 2 ǫ)+ǫ, to maintain the upink SIR distribution; a inear scaing aw ony appies to the case of fu path oss compensation in the power contro, i.e., when ǫ =. With ZF receivers, a inear scaing between the number of antennas M and users per ce K maintains the upink SIR distribution in massive MIMO. When noise is negigibe, ZF receivers provide better SIR coverage rate than MRC receivers. The performance gap increases with K, and decreases with path oss compensation faction ǫ. The gap becomes minor when ǫ =. The SIR coverage and rate are sensitive to the fraction ǫ of path oss compensation in power contro. Larger ǫ improves coverage in the ow SIR regime whie reducing coverage probabiity at high SIR. Numerica resuts show that the optima ǫ for rate is around.5 for MRC, and.2 for ZF receivers in certain cases. A. Derivation of ): APPENDIX With the combining vector in 9), the SINR expression equas 2) as SIR = P ) E u ) ĥ) P ) 2 +,k),) Pk) E u ) n th k) In the numerator, the signa power can be computed as ) 2 P ) β ) P ) u ) ) h 2 a) = P) β ) + σ2 K c) = P ) β ) h ) ) 2 u ) 4 b) ) 2M 2 +M) = P 2 t 2 + P ) h ) ) 2 P ) β ) 2 h k) P) β ) + σ2 K β ) ) 2 ǫ)m 2 +M), + u ) 2 σ 2. ) 2 E u ) 4 where a) foows from the MMSE estimator in 5), b) foows from the fact that u ) 4 M E u ) 4, and the approximation error decays as [33], and c) foows from the fact that M 2. 2 E u ) 4 = M 2 +M) P) β ) + K) σ2 Next, we compute the first term in the denominator of 2) as P ) E u ) ĥ) 2 = M P) E u ) 2 E ĥ) 2 2)

25 a) = M P) M = MP 2 t β ) ) P ) β ) + σ2 Mβ ) K ) ) ǫ > β ) ) ǫβ ) + σ2 P t K, P ) β ) P) β ) + σ2 K ) 25 where a) foows from the fact that) the channe estimation error ĥ) foows the distribution CN,β ) P ) β ) )I M. P) β ) +σ2 K Next, we simpify the second term in the denominator. Note that uness,,k) = n,n,m), h ) hk) and h) n hm) n are uncorreated zero-mean random variabes. Therefore, we can simpify the second term in the denominator of 2) as,k),) =,k),) E n th k) + σ 2 K MP t P ) h ) hk) 2 2) β k) ) ǫβ k) + > Note that for k =, =, the expression is simpified as [ ] [ ] P k) P ) E h ) hk) 2 = P )2 E h ) 4 = M 2 +M)P )2 for k > or k =, >, it foows that [ ] P k) P ) E h ) hk) 2 Therefore, we can express 22) as σ 2,k),) = MP 2 t K MP t k,),) β k) β k) ) ǫβ k) ) ǫβ k) + > β ) [ ] ) P k) P ) E h ) hk) 2. 22) ) 2 = P 2 t M 2 +M) = MP ) P k) β ) β k) = MP 2 t β ) β k) [ ] ) P k) P ) E h ) hk) 2 σ 2 KP t + ) ) ǫβ β ) ) ) ǫβ ) β k). +M 2 P 2 t Last, the therma noise term in the denominator can be simpified as ) ) ǫβ u ) 2 σ 2 = P t Mσ 2 β ) ) + σ2. P t K β ) > ) 2ǫ β ) β ) ) 2; ) 2ǫ Then the expression in ) is obtained through agebraic manipuation in the denominator. β ) ) 2.