Cell-Free Massive MIMO versus Small Cells

Transcription

1 Cell-Free Massive MIMO versus Small Cells 1 Hien Quoc Ngo, Alexei Ashikhmin, Hong Yang, Erik G. Larsson, an Thomas L. Marzetta Abstract arxiv: v1 [cs.it] 26 Feb 2016 A Cell-Free Massive MIMO (multiple-input multiple-output) system comprises a very large number of istribute access points (APs) which simultaneously serve a much smaller number of users over the same time/frequency resources base on irectly measure channel characteristics. The APs an users have only one antenna each. The APs acquire channel state information through time-ivision uplex operation an the reception of uplink pilot signals transmitte by the users. The APs perform multiplexing/e-multiplexing through conjugate beamforming on the ownlink an matche filtering on the uplink. Close-form expressions for iniviual user uplink an ownlink throughputs lea to max-min power control algorithms. Max-min power control ensures uniformly goo service throughout the area of coverage. A pilot assignment algorithm helps to mitigate the effects of pilot contamination, but power control is far more important in that regar. Cell-Free Massive MIMO has consierably improve performance with respect to a conventional small-cell scheme, whereby each user is serve by a eicate AP, in terms of both 95%-likely per-user throughput an immunity to shaow faing spatial correlation. Uner uncorrelate shaow faing conitions, the cell-free scheme provies nearly 5-fol improvement in 95%-likely per-user throughput over the small-cell scheme, an 10-fol improvement when shaow faing is correlate. I. INTRODUCTION Massive multiple-input multiple-output (MIMO), where a base station with many antennas simultaneously serves many users in the same time-frequency resource, is a promising 5G wireless access technology that can provie high throughput, reliability, an energy efficiency with simple signal processing [2], [3]. Massive antenna arrays at the base stations can be eploye in collocate or istribute setups. Collocate Massive MIMO architectures, where all service antennas are locate in a compact area, have the avantage of low backhaul requirements. In contrast, in istribute Massive MIMO systems, the service antennas are sprea out over a large area. Owing to their ability to more efficiently exploit iversity against the shaow faing, istribute systems can potentially offer much higher probability of coverage than collocate Massive MIMO [4], at the cost of increase backhaul requirements. H. Q. Ngo an E. G. Larsson are with the Department of Electrical Engineering (ISY), Linköping University, Linköping, Sween ( nqhien@isy.liu.se; egl@isy.liu.se). A. Ashikhmin, H. Yang an T. L. Marzetta are with Bell Laboratories, Alcatel-Lucent, Murray Hill, NJ USA ( aea@research.bell-labs.com; h.yang@research.bell-labs.com; tlm@research.bell-labs.com). The work of H. Q. Ngo an E. G. Larsson was supporte in part by the Sweish Research Council (VR) an ELLIIT. Portions of this work were performe while H. Q. Ngo was with Bell Labs in Part of this work was presente at the 16th IEEE International Workshop on Signal Processing Avances in Wireless Communications (SPAWC) [1].

2 2 In this work, we consier a istribute Massive MIMO system where a large number of service antennas, calle access points (APs), serve a much smaller number of autonomous users istribute over a wie area. All APs cooperate phase-coherently via a backhaul network, an serve all users in the same timefrequency resource via time-ivision uplex (TDD) operation. There are no cells or cell bounaries. We call this system Cell-Free Massive MIMO, an it combines the istribute MIMO an Massive MIMO concepts. It offers two main avantages: 1) a high throughput, energy efficiency an coverage probability; an 2) averaging out of small-scale faing an uncorrelate noise, which results in performance being epenent only on the path loss an shaow faing. Conjugate beamforming/matche filtering techniques, also known as maximum-ratio processing, are use both on uplink an ownlink. These techniques are computationally simple an can be implemente in a istribute manner, that is, with most processing one locally at the APs. In Cell-Free Massive MIMO, there is a central processing unit (CPU), but the information exchange between the APs an this CPU is limite to the payloa ata, an power control coefficients that change slowly. There is no sharing of instantaneous channel state information (CSI) among the APs or the central unit. All channels are estimate at the APs through uplink pilots. The so-obtaine channel estimates are use to precoe the transmitte ata in the ownlink an to perform ata etection in the uplink. Throughout we emphasize per-user throughput rather than sum-throughput. To that en we employ maxmin power control. Relate work: A boy of previous work has suggeste the use of coherent cooperation between base stations, resulting in concepts known as virtual MIMO, network MIMO an cooperative multipoint joint processing (CoMP) [5] [7]. It was inicate in [8] [14], that istribute MIMO may offer higher average rates than collocate MIMO. However, in [8] [14], perfect CSI was assume at both the APs an the users, an the analysis in [14] was asymptotic in the number of antennas an number of users. A realistic analysis of performance of istribute MIMO must account for imperfect CSI, which is an inevitable consequence of the finite channel coherence in a mobile system an which typically limits performance [15]. Distribute Massive MIMO with imperfect CSI was consiere in [6], [16] an in [17] [19] for the special case of orthogonal pilots, an in [6] assuming frequency-ivision uplex (FDD) operation. By contrast, in our work, we assume TDD operation. In [16], the authors exploit the low-rank structure of users channel covariance matrices, an examine performance of uplink transmission with matche filtering etection, uner the assumption that all users use the same pilot sequence. In contrast to this literature, in the current paper, we assume the use of arbitrary pilot sequences in the network an we erive rigorous capacity lower bouns vali for any number of APs an users. The papers cite above compare the performance between istribute an collocate Massive MIMO

3 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 3 systems. An alternative to (istribute) MIMO systems is to eploy small cells, consisting of APs that o not cooperate. Small-cell systems are consierably simpler than Cell-Free Massive MIMO, since only ata an power control coefficients are exchange between the CPU an the APs. It is expecte that Cell-Free Massive MIMO systems perform better than small-cell systems. There has however been little work that compares Massive MIMO an small-cell systems, though the latter systems have receive much interest recently. In [20], the authors show that, when the number of cells is large, a small-cell system is more energy-efficient than a collocate Massive MIMO system. A comparison between small-cell an istribute Massive MIMO systems is reporte in [8], assuming perfect CSI at both the APs an the users. Yet, a comprehensive performance comparison between small-cell an istribute Massive MIMO systems that takes into account the effects of imperfect CSI, pilot assignment, an power control is not available in the existing literature. Specific contributions of the paper: We erive close-form capacity lower bouns for the Cell-Free Massive MIMO ownlink an uplink with finite numbers of APs an users. Our analysis takes into account the effects of channel estimation errors, power control, an non-orthogonality of pilot sequences. We compare two pilot assignment schemes: ranom assignment an greey assignment. We evise max-min power control algorithms that maximize the smallest of all user rates. Globally optimal solutions can be compute by solving a sequence of secon-orer cone programs (SOCPs) for the ownlink, an a sequence of linear programs for the uplink. We quantitatively compare the performance of Cell-Free Massive MIMO to that of small-cell systems, uner uncorrelate an correlate shaow faing moels. The rest of paper is organize as follows. In Section II, we escribe the Cell-Free Massive MIMO system moel. In Section III, we present the achievable ownlink an uplink rates. The pilot assignment an power control schemes are evelope in Section IV. The small-cell system is iscusse in Section V. In Section VI-A2 we escribe the shaow faing correlation moel use in the performance evaluation. We provie numerical results an iscussions in Section VI an finally conclue the paper in Section VII. Notation: Bolface letters enote column vectors. The superscripts (), () T, an () H stan for the conjugate, transpose, an conjugate-transpose, respectively. The Eucliean norm an the expectation operators are enote by ane{ }, respectively. Finally,z CN (0,σ 2 ) enotes a circularly symmetric complex Gaussian ranom variable (RV) z with zero mean an variance σ 2, an z N(0,σ 2 ) enotes a real-value Gaussian RV.

4 4 PSfrag replacements user 1 AP 1 user K AP 2 g CPU AP m user k AP M Fig. 1. Cell-Free Massive MIMO system. II. CELL-FREE MASSIVE MIMO SYSTEM MODEL We consier a Cell-Free Massive MIMO system with M APs an K users. All APs an users are equippe with a single antenna, an they are ranomly locate in a large area. Furthermore, all APs connect to a central processing unit via a backhaul network, see Figure 1. We assume that all M APs simultaneously serve all K users in the same time-frequency resource. The transmission from the APs to the users (ownlink transmission) an the transmission from the users to the APs (uplink transmission) procee by TDD operation. Each coherence interval is ivie into three phases: uplink training, ownlink payloa ata transmission, an uplink payloa ata transmission. In the uplink training phase, the users sen pilot sequences to the APs an each AP estimates the channel to all users. The so-obtaine channel estimates are use to precoe the transmit signals in the ownlink, an to etect the signals transmitte from the users in the uplink. In this work, to avoi sharing of channel state information between the APs, we consier conjugate beamforming in the ownlink an matche filtering in the uplink. No pilots are transmitte in the ownlink of Cell-Free Massive MIMO. The users o not nee to estimate their effective channel gain, but instea rely on channel harening, which makes this gain close to its expecte value, a known eterministic constant. Our capacity bouns account for the error incurre when the terminals use the average effective channel gain instea of the actual effective gain. Channel harening in Massive MIMO is iscusse, for example, in [2]. Notation is aopte an assumptions are mae as follows: The channel moel incorporates the effects of small-scale faing an large-scale faing (that latter inclues path loss an shaowing). The small-scale faing is assume to be static uring each coherence interval, an change inepenently from one coherence interval to the next. The large-scale faing changes much more slowly, an stays constant for several coherence intervals. Depening on

5 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 5 the user mobility, the large-scale faing may stay constant for a uration of at least some 40 smallscale faing coherence intervals [21], [22]. We assume that the channel is reciprocal, i.e., the channel gains on the uplink an on the ownlink are the same. This reciprocity assumption requires TDD operation an perfect calibration of the harware chains. The feasibility of the latter is emonstrate for example in [23]. We let g enote the channel coefficient between the kth user an the mth AP. The channel g is moelle as follows: g = β 1/2 h, (1) where h represents the small-scale faing, an β represents the large-scale faing. We assume that h, m = 1,...,M, K = 1,...K, are inepenent an ientically istribute (i.i..) CN (0,1) RVs. The justification of the assumption of inepenent small-scale faing is that the APs an the users are istribute over a wie area, an hence, the set of scatterers is likely to be ifferent for each AP an each user. We assume that all APs are connecte via perfect backhaul that offers error-free an infinite capacity to the CPU. In practice, backhaul will be subject to significant practical constraints [24], [25]. Future work is neee to quantify the impact of backhaul constraints on performance. In all scenarios, we let q k enote the symbol associate with the kth user. These symbols are mutually inepenent, an inepenent of all noise an channel coefficients. A. Uplink Training The Cell-Free Massive MIMO system employs a wie spectral banwith, an the quantities g an h are epenent on frequency; however β is constant with respect to frequency. The propagation channels are assume to be piece-wise constant over a coherence time interval an a frequency coherence interval. It is necessary to perform training within each such time/frequency coherence block. We assume that β is known, a priori, wherever require. Let τ c be the length of the coherence interval (in samples), which is equal to the prouct of the coherence time an the coherence banwith, an let τ cf be the uplink training uration (in samples) per coherence interval, where the superscript cf stans for cell-free. It is require that τ cf < τ c. During the training phase, all K users simultaneously sen pilot sequences of length τ cf samples to the APs. Let τ cf ϕ k C τcf 1, where ϕ k 2 = 1, be the pilot sequence use by the kth user, k = 1,2,,K. Then, the τ cf 1 receive pilot vector at the mth AP is given by y p,m = τ cf p g ϕ k +w p,m, (2) k=1

6 6 where p is the normalize signal-to-noise ratio (SNR) of each pilot symbol an w p,m is a vector of aitive noise at the mth AP. The elements of w p,m are i.i.. CN (0,1) RVs. Base on the receive pilot signal y p,m, the mth AP estimates the channel g,k = 1,...,K. Denote by ˇy p, the projection of y p,m onto ϕ H k : ˇy p, = ϕ H k y p,m = τ cf pg + τ cf p g ϕ H k ϕ k +ϕ H k w p,m. (3) Although, for arbitrary pilot sequences, ˇy p, is not a sufficient statistic for the estimation of g, one can still use this quantity to obtain suboptimal estimates. In the special case when any two pilot sequences are either ientical or orthogonal, then ˇy p, is a sufficient statistic, an estimates base on ˇy p, are optimal. The MMSE estimate of g given ˇy p, is ĝ = E{ˇy p, g } E { ˇy p, 2} ˇy p, = c ˇy p,, (4) where c τ cf p k k τ cf pβ K k =1 β ϕh k ϕ k Remark 1: If τ cf K, then we can choose ϕ 1,ϕ 2,,ϕ K so that they are pairwisely orthogonal, an hence, the secon term in (3) isappears. Then the channel estimate ĝ is inepenent of g, k k. However, owing to the limite length of the coherence interval, in general, τ cf < K, an mutually nonorthogonal pilot sequences must be use throughout the network. The channel estimate ĝ is egrae by pilot signals transmitte from other users, owing to the secon term in (3). This causes the so-calle pilot contamination effect. Remark 2: The channel estimation is performe in a ecentralize fashion. Each AP autonomously estimates the channels to the K users. The APs o not cooperate on the channel estimation, an no channel estimates are interchange among the APs. B. Downlink Payloa Data Transmission The APs treat the channel estimates as the true channels, an use conjugate beamforming to transmit signals to the K users. The transmitte signal from the mth AP is given by x m = ĝ q k, (5) k=1 where q k, which satisfies E{ q k 2 } = 1, is the symbol intene for the kth user, an η, m = 1,...,M, k = 1,...K, are power control coefficients chosen to satisfy the following power constraint at each AP: E { x m 2}. (6)

7 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 7 With the channel moel in (1), the power constraint E{ x m 2 } can be rewritten as: η γ 1, for all m, (7) where k=1 γ E { ĝ 2} = The receive signal at the kth user is given by r,k = g x m +w,k = τ cf pβ c. (8) k =1 g ĝ q k +w,k, (9) where w,k is aitive CN (0,1) noise at the kth user. Then q k will be etecte from r,k. C. Uplink Payloa Data Transmission In the uplink, all K users simultaneously sen their ata to the APs. Before sening the ata, the kth user weights its symbol q k, E{ q k 2 } = 1, by a power control coefficient η k, 0 η k 1. The receive signal at the mth AP is given by y u,m = u g ηk q k +w u,m, (10) k=1 where u is the normalize uplink SNR an w u,m is aitive noise at the mth AP. We assume that w u,m CN (0,1). To etect the symbol transmitte from the kth user, q k, the mth AP multiplies the receive signal y u,m with the conjugate of its (locally obtaine) channel estimate ĝ. Then the so-obtaine quantity ĝ y u,m is sent to the CPU via a backhaul network. The CPU sees r u,k = ĝ y M u,m = ρ cf u η k ĝ g q k + ĝ w u,m. (11) Then, q k is etecte from r u,k. k =1 III. PERFORMANCE ANALYSIS A. Large-M Analysis In this section, we provie some insights into the performance of Cell-Free Massive MIMO systems when M is very large. We show that, as in the case of Collocate Massive MIMO, when M, the channels between the users an the APs become orthogonal. Therefore, with conjugate beamforming respectively matche filtering, non-coherent interference, small-scale faing, an noise isappear. The only remaining impairment is pilot contamination, which consists of interference from users using same pilot sequences as the user of interest in the training phase.

8 8 On ownlink, from (9), the receive signal at the kth user can be written as: r,k = g ĝ q k+ g ĝ q k +w,k, (12) } {{ } DS k k k } {{ } MUI k where DS k an MUI k represent the esire signal an multiuser interference, respectively. By using the channel estimates in (4), we have M ( g ĝ = c g τ cf p = + τ cf p c g 2 ϕ T k ϕ k + τ cf p g ϕ H k ϕ k + w p, k =1 k k ) c g g ϕt k ϕ k c g w p,, (13) where w p, ϕ H k w p,m. Then by Tchebyshev s theorem [26], 1 we have 1 g M ĝ 1 τ M cf p c β ϕ T k ϕ k Using (14), we obtain the following results: 1 M DS k 1 τ M cf p ρcf 1 M MUI k 1 τ M cf p c β q k k k P M c β ϕ T k ϕ kq k P M 0. (14) 0, (15) P M 0. (16) The above expressions show that when M, the receive signal inclues only the esire signal plus interference originating from the pilot sequence non-orthogonality: ( r τ cf,k M p ρcf M ) M c β q k + c β ϕ T k ϕ kq k k k P 0, as M. (17) If the pilot sequences are pairwisely orthogonal, i.e., ϕ H k ϕ k = 0 for k k, then the receive signal becomes free of interference an noise: r,k M Similar results hol on the uplink. τ cf p ρcf M c β q k P 0, as M. (18) 1 Tchebyshev s theorem: Let X 1,X 2,...X n be inepenent RVs such that E{X i} = µ i an Var{X i} c <, i. Then 1 n (X1 +X Xn) 1 n (µ1 +µ2 +...µn) P 0.

9 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 9 B. Achievable Rate for Finite M In this section, we erive close-form expressions for the ownlink an uplink achievable rates, using the analysis technique from [17], [27]. 1) Achievable Downlink Rate: We assume that each user has knowlege of the channel statistics but not of the channel realizations. The receive signal r,k in (9) can be written as where DS k BU k UI kk r,k = DS k q k +BU k q k + E { M ( M g ĝ UI kk q k +w,k, (19) k k } g ĝ E { M, (20) g ĝ }), (21) g ĝ, (22) represent the strength of esire signal (DS), the beamforming gain uncertainty (BU), an the interference cause by the k th user (UI), respectively. We treat the sum of the secon, thir, an fourth terms in (19) as effective noise. Since q k is inepenent of DS k an BU k, we have E{DS k q k (BU k q k ) } = E{DS k (BU k ) }E { q k 2} = 0. Thus, the first an the secon terms of (19) are uncorrelate. A similar calculation shows that the thir an fourth terms of (19) are uncorrelate with the first term of (19). Therefore, the effective noise an the esire signal are uncorrelate. By using the fact that uncorrelate Gaussian noise represents the worst case, we obtain the following achievable rate of the kth user for Cell-Free (cf) operation: ) R,k cf DS k = log 2 (1+ 2 E{ BU k 2 }+ K k k E{ UI. (23) kk 2 }+1 We next provie a new exact close-form expression for the achievable rate (23), for a finite M. Theorem 1: An achievable ownlink rate of the transmission from the APs to the kth user in the Cell-Free Massive MIMO system with conjugate beamforming, for any finite M an K, is ( M ) 2 R,k cf γ = log 2 1+ ( M ) 2 γ β ϕ H β k ϕ k 2 +. (24) η γ β +1 k k Proof: See Appenix A. k =1

10 10 Remark 3: The main ifferences between the capacity boun expressions for Cell-Free an collocate Massive MIMO systems [3] are: i) in Cell-Free systems, in general β β m k, for m m, whereas in collocate Massive MIMO, β = β m k; an ii) in Cell-Free systems, a power constraint is applie at each AP iniviually, whereas in collocate systems, a total power constraint is applie at each base station. Consier the special case in which all APs are collocate an the power constraint for each AP is replace by a total power constraint over all APs. In this case, we have β = β m k β k, γ = γ m k γ k, an the power control coefficient is η = η k /(Mγ ). If, furthermore, the K pilot sequences are pairwisely orthogonal, then, (24) becomes R cf,k = log 2 ( 1+ ) M γ kη k β K k k =1 η k +1, (25) which is ientical to the rate expression for collocate Massive MIMO systems in [3]. Remark 4: The achievable rate (24) is obtaine uner the assumption that the users only know the channel statistics. However, this achievable rate is close to that in the case where the users know the actual channel realizations. This is a consequence of channel harening, as iscusse in Section II. To see this more quantitatively, we compare the achievable rate (24) with the following expression, R,k cf = E log M η1/2 g ĝ K k k M η1/2 g ĝ 2 +1, (26) which represents an achievable rate for a genie-aie user that knows the instantaneous channel gain. Figure 2 shows a comparison between (24), which assumes that the users only know the channel statistics, an the genie-aie rate (26), which assumes knowlege of the realizations. As seen in the figure, the gap is small, which means that ownlink training is not necessary. 2) Achievable Uplink Rate: The central processing unit etects the esire signal q k from r u,k in (11). We assume that the central processing unit uses only statistical knowlege of the channel when performing the etection. Using a similar methoology as in Section III-B1, we obtain a rigorous closeform expression for the achievable uplink rate as follows. Theorem 2: An achievable uplink rate for the kth user in the Cell-Free Massive MIMO system with matche filtering etection, for any M an K, is given by ( M ) u η 2 k γ Ru,k=log cf 2 1+ ( M ) 2 u η k β ϕ H k ϕ k 2 + u k k β γ η k k =1 γ β + M γ. (27) Remark 5: In the special case that all APs are collocate an all K pilot sequences are pairwisely orthogonal, then β = β m k β k, γ = γ m k γ k, an ϕ H k ϕ k = 0, k k. Equation (27) then

11 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 11 Achievable Rate per User (bits/s/hz) user perfectly knows its effective channel gain user knows only the channel statistics K = 10 K = Number of APs (M) Fig. 2. The achievable rate versus the number of APs for ifferentk. Here,ρ sc = 10 B, p = 0 B,τ cf = K,β = 1,η = 1/(Kγ ), an pilot sequences are pairwisely orthogonal. reuces to R cf u,k = log 2 (1+ u ) M uηkγ k K k =1 η k β k +1, (28) which is precisely the uplink capacity lower boun of a single-cell Massive MIMO system with a collocate array obtaine in [17], an a variation on that in [28]. IV. PILOT ASSIGNMENT AND POWER CONTROL To obtain goo system performance, the available raio resources must be efficiently manage. In this section, we will present methos for pilot sequence assignment an power control. Importantly, pilot assignment an power control can be performe inepenently, because the pilots are not power controlle. A. Greey Pilot Assignment Typically, ifferent users must use non-orthogonal pilot sequences, ue to the limite length of the coherence interval. Since the length of the pilot sequences is τ cf, there exist τ cf orthogonal pilot sequences. Here we focus on the case that τ cf < K. If τ cf K, we simply assign K orthogonal pilot sequences to the K users. A simple baseline metho for assigning pilot sequences of length τ cf samples to the K users is ranom pilot assignment [29]. With ranom pilot assignment, each user will be ranomly assigne one pilot sequence from a preetermine set S ϕ of τ cf orthogonal pilot sequences. Ranom pilot assignment coul alternatively be one by letting each user choose an arbitrary unit-norm vector (i.e. not from a

12 12 preetermine set of pilots). However, it appears from simulations that the latter scheme oes not work well. While ranom pilot assignment is a useful baseline, occasionally two users in close vicinity of each other will use the same pilot sequence, which results in strong pilot contamination. Optimal pilot assignment is a ifficult combinatorial problem. We propose to use a simple greey algorithm, which iteratively refines the pilot assignment. The K users are first ranomly assigne K pilot sequences. Then the user that has the lowest ownlink rate, say user k, upates its pilot sequence so that its pilot contamination effect is minimize. 2 The pilot contamination effect at the k th user is quantifie by the secon term in (3) which has variance 2 E g ϕ H k ϕ k = K k k k k β ϕ H k ϕ k 2. (29) The k th user is assigne a new pilot sequence which minimizes the pilot contamination in (29), summe over all APs: arg min ϕ k β k k ϕ H k ϕ k 2 = argmin ϕ k ( M ) ϕ H K k k k β ϕ k ϕ H k ϕ H k ϕ k ϕ k, (30) where we use the fact that ϕ k 2 = 1. The algorithm then procees iteratively for a preetermine number of iterations. The greey pilot assignment algorithm can be summarize as follows. Algorithm 1 (Greey pilot assignment): 1) Initialization: choose K pilot sequences ϕ 1,,ϕ K using the ranom pilot assignment metho. Choose the number of iterations, N, an set n = 1. 2) Compute R,k cf, using (24). Fin the user with the lowest rate: k = argmin k R cf,k. (31) 3) Upate the pilot sequence for the k th user by choosing ϕ k from S ϕ which minimizes β k k 4) Set n := n+1. Stop if n > N. Otherwise, go to Step 2. ϕ H k ϕ k 2. 2 In principle, this worst user coul be taken to be the user that has either the lowest uplink or the lowest ownlink rate. In our numerical experiments, we reassign the pilot of the user having the lowest ownlink rate, hence giving ownlink performance some priority over uplink performance.

13 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 13 Remark 6: The greey pilot assignment can be performe at the CPU, which connects to all APs via backhaul links. The pilot assignment is recompute on the large-scale faing time scale. 3 This simplifies the signal processing at the central unit significantly. Furthermore, since ϕ k is chosen from S ϕ, to inform the users about their assigne pilots, the CPU only nees to sen an inex to each user. B. Power Control We next show that Cell-Free Massive MIMO can provie uniformly goo service to all users, regarless of their geographical location, by using max-min power control. While power control in general is a well stuie topic, the max-min power control problems that arise when optimizing Cell-Free Massive MIMO are entirely new. The power control is performe at the CPU, an importantly, is one on the large-scale faing time scale. 1) Downlink: In the ownlink, given realizations of the large-scale faing, we fin the power control coefficients η, m = 1,,M,k = 1,,K, that maximize the minimum of the ownlink rates of all users, uner the power constraint (7). At the optimum point, all users get the same rate. Mathematically: max {η } max {η } subject to min k=1,,k min,k K k=1 η γ 1, m = 1,...,M k=1,,k Rcf η 0, k = 1,...,K, m = 1,...,M, where R,k cf is given by (24). Define ς. Then, from (24), (32) is equivalent to ( M ) 2 γ ς subject to ( M ϕ H k ϕ k 2 k k γ β K k=1 η γ 1, m = 1,...,M η 0, k = 1,...,K, m = 1,...,M, ) 2 ς β + M β By introucing slack variables k k an ϑ m, we reformulate (33) as follows: ( M ) 2 γ ς max {ς, k k,ϑ m} subject to min k=1,,k k =1 ϕ H k ϕ k 2 2k k + M β ϑ 2 m + 1 k k K k =1 γ ς2 ϑ2 m, m = 1,...,M M γ β β ς k k, k k 0 ϑ m 1, m = 1,...,M ς 0, k = 1,...,K, m = 1,...,M. γ ς Hence this recomputation is infrequent even in high mobility. For example, at user mobility of v = 100 km/h, an a carrier frequency of f c = 2 GHz, the channel coherence time is on the orer of a millisecon. The large-scale faing changes much more slowly, at least some 40 times slower accoring to [21], [22]. As a result, the greey pilot assignment metho must only be one a few times per secon. (32) (33) (34)

14 14 The equivalence between (33) an (34) follows irectly from the fact that the first an secon constraints in (34) hol with equality at the optimum. Proposition 1: The objective function of (34) is quasi-concave, an the problem (34) is quasi-concave. Proof: See Appenix B. Consequently, (34) can be solve efficiently by a bisection search, in each step solving a sequence of convex feasibility problem [30]. Specifically, the following algorithm solves (34). Algorithm 2 (Bisection algorithm for solving (34)): 1) Initialization: choose the initial values of t min an t max, where t min an t max efine a range of relevant values of the objective function in (34). Choose a tolerance ǫ > 0. 2) Set t := t min+t max 2. Solve the following convex feasibility program: where v k [ v T k1 I k v T k2 v k 1 t γ ς, k = 1,...,K, k =1 γ ς 2 ϑ2 m, m = 1,...,M, γ β β ς k k, k k, 0 ϑ m 1, m = 1,...,M, ς 0, k = 1,...,K, m = 1,...,M, (35) ] T 1, an where v k1 [ ϕ H 1 ϕ k 1k... ϕ H K ϕ ] T K Kk, I k is a K (K 1) matrix obtaine from the K K ientity matrix with the kth column remove, an v k2 [ β1k ϑ 1... β Mk ϑ M ] T. 3) If problem (35) is feasible, then set t min := t, else set t max := t. 4) Stop if t max t min < ǫ. Otherwise, go to Step 2. Remark 7: The max-min power control problem can be irectly extene to a max-min weighte rate problem, where the K users are weighte accoring to priority: maxmin{w k R k }, where w k > 0 is the weighting factor of the kth user. A user with higher priority will be assigne a smaller weighting factor. 2) Uplink: In the uplink, the max-min power control problem can be formulate as follows: max {η k } subject to min k=1,,k Rcf u,k 0 η k 1, k = 1,...,K, (36)

15 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 15 where R cf u,k is given by (27). Problem (36) can be equivalently reformulate as max {η k },t subject to t t R cf u,k, k = 1,...,K 0 η k 1, k = 1,...,K. Proposition 2: The optimization problem (37) is quasi-linear. Proof: From (27), for a given t, all inequalities involve in (37) are linear, an hence, the program (37) is quasi-linear. Consequently, Problem (37) can be efficiently solve by using bisection an solving a sequence of linear feasibility problems. (37) V. SMALL-CELL SYSTEM In this section, we give the system moel, achievable rate expressions, an max-min power control for small-cell systems. These will be use in Section VI where we compare the performance of Cell-Free Massive MIMO an small-cell systems. For small-cell systems, we assume that each user is serve by only one AP. For each user, the available AP with the largest average receive useful signal power is selecte. If an AP has alreay been chosen by another user, this AP becomes unavailable. The AP selection is one user by user in a ranom orer. Let m k be the AP chosen by the kth user. Then, m k argmax β. (38) m {available APs} We consier a short enough time scale that hanovers between APs o not occur. This moeling choice was mae to enable a rigorous performance analysis. While there is preceent for this assumption in other literature [8], [31], future work may aress the issue of hanovers. As a result of this assumption, the performance figures we obtain for small-cell systems may be overoptimistic. In contrast to Cell-Massive MIMO, in the small-cell systems, the channel oes not haren. Specifically, while in Cell-Free Massive MIMO the effective channel is an inner prouct between two M-vectors hence close to its mean, in the small-cell case the effective channel is a single Rayleigh faing scalar coefficient. Consequently, both the users an the APs must estimate their effective channel gain in orer to emoulate the symbols, which requires both uplink an ownlink training. The etaile transmission protocols for the uplink an ownlink of small-cell systems are as follows. A. Downlink Transmission In the ownlink, the users first estimate their channels base on pilots sent from the APs. The soobtaine channel estimates are use to etect the esire signals.

16 16 Let τ sc be the ownlink training uration in samples, τ sc φ k C τsc 1, where φ k 2 = 1, is the pilot sequence transmitte from the m k th AP, an ρ sc,p MMSE estimate of g k can be expresse as is the transmit power per ownlink pilot symbol. The ĝ k = g k ε k, (39) where ε k is the channel estimation error, which is inepenent of the channel estimate ĝ k. Furthermore, we have ĝ k CN (0,µ k) an ε k CN (0,β k µ k), where µ k τ sc ρsc,p τ scρsc,p β2 m k k K k =1 β m k k φ H k φ k (40) After sening the pilots for the channel estimation, the K chosen APs sen the ata. Let α,k q k, E{ q k 2 } = 1, be the symbol transmitte from the m k th AP, estine for the kth user, where α,k is a power control coefficient, 0 α,k 1. The kth user receives y k = ρ sc g k α,k q k +w k k =1 = ρ sc ĝm k k α,k q k + ρ sc ε m k k α,k q k + ρ sc g k α,k q k +w k, (41) where ρ sc is the normalize ownlink transmit SNR an w k CN (0,1) is aitive Gaussian noise. Remark 8: In small-cell systems, since only one single-antenna AP is involve in transmission to a given user, the concept of conjugate beamforming becomes voi. Downlink transmission entails only transmitting the symbol estine for the kth user, appropriately scale to meet the transmit power constraint. Channel estimation at the user is require in orer to emoulate, as there is no channel harening (see iscussion above). 1) Achievable Downlink Rate: Treating the last three terms of (41) as uncorrelate effective noise, we obtain the following achievable ownlink rate for the kth user:,k = E log 2 1+ R sc k k ρ sc α,k ĝ k 2 ρ sc α,k(β k µ k)+ρ sc k k. (42) α,k β k +1 Since the channel oes not haren, applying the bouning techniques in Section III, while not impossible in principle, woul yiel very pessimistic capacity bouns. However, since ĝ k 2 is exponentially istribute with mean µ k, the achievable rate in (42) can be expresse in close form in terms of the exponential integral function Ei( ) [32, Eq. ( )] as: ( R,k sc = (log 2e)e 1/ µ m kk Ei 1 ), (43) µ k

17 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 17 where µ k ρ sc α,kµ k ρ sc α,k(β k µ k)+ρ sc K k k α,k β m k k +1. (44) 2) Max-Min Power Control: As in the Cell-Free Massive MIMO systems, we consier max-min power control which can be formulate as follows: max {α,k } subject to min k=1,,k Rsc,k 0 α,k 1, k = 1,,K. Since R sc,k is a monotonically increasing function of µ m k k, (45) is equivalent to (45) max {α,k } subject to min k=1,,k µ m k k 0 α,k 1, k = 1,,K. Problem (46) is a quasi-linear program, which can be solve by using bisection. (46) B. Uplink Transmission In the uplink, the APs first estimate the channels base on pilots sent from the users. The so-obtaine channel estimates are use to etect the esire signals. Let ρ sc u an 0 α u,k 1 be the normalize SNR an the power control coefficient at the kth user, respectively. Then, following the same methoology as in the erivation of the ownlink transmission, we obtain the following achievable uplink rate for the kth user: where ω k an where ω k is given by ( Ru,k sc = (log 2 e)e 1/ ω m kk Ei 1 ), (47) ω k ρ sc u α u,kω k ρ sc uα u,k (β k ω k)+ρ sc u ω k τ sc u ρsc u,p τu sc ρ sc u,pβm 2 k k K k k α u,k β (48) m k k +1, K k =1 β m k k ψh k ψ k (49) In (49), τ sc u is the uplink training uration in samples, τ sc u ψ k C τsc u 1, where ψ k 2 = 1, is the pilot sequence transmitte from the kth user, an ρ sc u,p is the transmit power per uplink pilot symbol. Similarly to in the ownlink, the max-min power control problem for the uplink can be formulate as a quasi-linear program: max {α u,k } subject to which can be solve by using bisection. min k=1,,k ω m k k 0 α u,k 1, k = 1,,K, (50)

18 18 VI. NUMERICAL RESULTS AND DISCUSSIONS We quantitatively stuy the performance of Cell-Free Massive MIMO, an compare it to that of smallcell systems. We specifically emonstrate the effects of shaow faing correlation. The M APs an K users are uniformly istribute at ranom within a square of size D D km 2. A. Large-Scale Faing Moel We escribe the path loss an shaow faing correlation moels, which are use in the performance evaluation. The large-scale faing coefficient β in (1) moels the path loss an shaow faing, accoring to β = PL 10 σ sh z 10, (51) where PL represents the path loss, an 10 σ sh z 10 represents the shaow faing with the stanar eviation σ sh, an z N(0,1). 1) Path loss Moel: We use a three-slope moel for the path loss [33]: the path loss exponent equals 3.5 if istance between the mth AP an the kth user (enote by ) is greater than 1, equals 2 if 1 > 0, an equals 0 if 0 for some 0 an 1. When > 1, we employ the Hata-COST231 propagation moel. More precisely, the path loss in B is given by L 35log 10 ( ), if > 1 PL = L 15log 10 ( 1 ) 20log 10 ( ), if 0 < 1 (52) L 15log 10 ( 1 ) 20log 10 ( 0 ), if 0 where L log 10 (f) 13.82log 10 (h AP ) (1.1log 10 (f) 0.7)h u +(1.56log 10 (f) 0.8), (53) an where f is the carrier frequency (in MHz), h AP is the AP antenna height (in m), an h u enotes the user antenna height (in m). The path loss PL is a continuous function of. Note that when 1, there is no shaowing. 2) Shaowing Correlation Moel: Most previous work assume that the shaowing coefficients (an therefore z ) are uncorrelate. However, in practice, transmitters/receivers that are in close vicinity of each other may be surroune by common obstacles, an hence, the shaowing coefficients are correlate. This correlation may significantly affect the system performance. For the shaow faing coefficients, we will use a moel with two components [34]: z = δa m + 1 δb k, m = 1,...,M, K = 1,...,K, (54) where a m N(0,1) an b k N(0,1) are inepenent ranom variables, an δ, 0 δ 1, is a parameter. The variable a m moels contributions to the shaow faing that result from obstructing objects in the

19 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 19 vicinity of the mth AP, an which affects the channel from that AP to all users in the same way. The variable b k moels contributions to the shaow faing that result from objects in the vicinity of the kth user, an which affects the channels from that user to all APs in the same way. When δ = 0, the shaow faing from a given user is the same to all APs, but ifferent users are affecte by ifferent shaow faing. Conversely, when δ = 1, the shaow faing from a given AP is the same to all users; however, ifferent APs are affecte by ifferent shaow faing. Varying δ between 0 an 1 traes off between these two extremes. The covariance functions of a m an b k are given by: ) E{a m a m } = 2 a(m,m ecorr, E{b k b k } = 2 u(k,k ) ecorr, (55) where a (m,m ) is the geographical istance between the mth an m th APs, u (k,k ) is the geographical istance between the kth an k th users, an ecorr is a ecorrelation istance which epens on the environment. Typically, the ecorrelation istance is on the orer of m. A shorter ecorrelation istance correspons to an environment with a lower egree of stationarity. This moel for correlation between ifferent geographical locations has been valiate both in theory an by practical experiments [34], [35]. B. Parameters an Setup In all examples, we choose the parameters summarize in Table I. The quantities, ρcf u, an ρcf p in this table are the transmit powers of ownlink ata, uplink ata, an pilot symbols, respectively. The corresponing normalize transmit SNRs, ρcf u, an p can be compute by iviing these powers by the noise power, where the noise power is given by noise power = banwith k B T 0 noise figure (W), TABLE I SYSTEM PARAMETERS FOR THE SIMULATION. Parameter Carrier frequency Banwith Noise figure (uplink an ownlink) AP antenna height User antenna height, u, p σ sh Value 1.9 GHz 20 MHz 9 B 15 m 1.65 m 200, 100, 100 mw 8 B D, 1, , 50, 10 m

20 20 where k B = (Joule per Kelvin) is the Boltzmann constant, an T 0 = 290 (Kelvin) is the noise temperature. To avoi bounary effects, an to imitate a network with an infinite area, the square area is wrappe aroun at the eges, an hence, the simulation area has eight neighbors. We consier the per-user net throughputs which take into account the channel estimation overhea, an are efine as follows: S cf A,k = B 1 τcf /τ c R cf 2 A,k, (56) S sc +τsc A,k = B1 (τsc 2 u )/τ c R sc A,k, (57) where A {, u} correspon to ownlink respectively uplink transmission, B is the spectral banwith, an τ c is again the coherence interval in samples. The terms τ cf /τ c an (τ sc + τsc u )/τ c in (56) an (57) reflect the fact that, for each coherence interval of length τ c samples, in the Cell-Free Massive MIMO systems, we spen τ cf samples for the uplink training, while in the small-cell systems, we spen τ sc+τsc u samples for the uplink an ownlink training. In all examples, we take τ c = 200 samples, corresponing to a coherence banwith of 200 KHz an a coherence time of 1 ms, an choose B = 20 MHz. To ensure a fair comparison between Cell-Free Massive MIMO an small-cell systems, we choose ρ sc = M K ρcf, ρsc u = ρcf u, an ρsc u,p = ρsc,p = ρcf p, which makes the total raiate power equal in all cases. The cumulative istributions of the per-user ownlink/uplink net throughput in our examples are generate as follows: For the case with max-min power control: 1) 200 ranom realizations of the AP/user locations an shaow faing profiles are generate; 2) for each realization, the per-user net throughputs of K users are compute by using max-min power control as iscusse in Section IV-B for Cell-Free Massive MIMO an in Section V for small-cell systems with max-min power control these throughputs are the same for all users; 3) a cumulative istribution is generate over the so-obtaine per-user net throughputs. For the case without power control: same proceure, but in 2) no power control is performe. Without power control, for Cell-Free Massive MIMO, in the ownlink transmission, all APs transmit with full power, an at the mth AP, the power control coefficients η, k = 1,...K, are the same, i.e., η = ( K k =1 γ ) 1, k = 1,...K, (this irectly comes from (7)), while in the uplink, all users transmit with full power, i.e., η k = 1, k = 1,...K. For the small-cell system, in the ownlink, all chosen APs transmit with full power, i.e. α,k = 1, an in the uplink, all users transmit with full power, i.e. α u,k = 1, k = 1,...K. For the correlate shaow faing scenario, we use the shaowing correlation moel iscusse in Section VI-A2, an we choose ecorr = 0.1 km an δ = 0.5.

21 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS 21 For the small-cell systems, the greey pilot assignment works in the same way as the scheme for Cell-Free Massive MIMO iscusse in Section IV-A, except for that in the small-cell systems, since the chosen APs o not cooperate, the worst user will fin a new pilot which minimizes the pilot contamination corresponing to its AP (rather than summe over all APs as in the case of Cell-Free systems). C. Results an Discussions We first compare the performance of Cell-Free Massive MIMO with that of small-cell systems with greey pilot assignment an max-min power control. Figure 3 compares the cumulative istribution of the per-user ownlink net throughput for Cell-Free Massive MIMO an small-cell systems, with M = 100, K = 40, an τ cf = τ sc = τsc u = 20, with an without shaow faing correlation. Cell-Free Massive MIMO significantly outperforms small-cell in both meian an in 95%-likely performance. The net throughput of Cell-Free Massive MIMO is much more concentrate aroun its meian, compare with the small-cell systems. Without shaow faing correlation, the 95%-likely net throughput of the Cell-Free ownlink is about 14 Mbits/s which is 7 times higher than that of the small-cell ownlink (about 2.1 Mbits/s). In particular, we can see that the small-cell systems are much more affecte by shaow faing correlation than Cell-Free Massive MIMO is. This is ue to the fact that when the shaowing coefficients are highly correlate, the gain from choosing the best APs in a small-cell system is reuce. With shaowing correlation, the 95%-likely net throughput of the Cell-Free ownlink is about 10 times higher than that of the small-cell system. The same insights can be obtaine for the uplink, see Figure 4. In aition, owing to the fact that the ownlink uses more power (since M > K an > ρcf u ) an has more power control coefficients to choose than the uplink oes, the ownlink performance is better than the uplink performance. Next we compare Cell-Free Massive MIMO an small-cell systems, assuming that no power control is performe. Figures 5 an 6 show the cumulative istributions of the per-user net throughput for the ownlink an the uplink, respectively, with M = 100, K = 40, an τ cf = τ sc = τsc u = 20, an with the greey pilot assignment metho. In both uncorrelate an correlate shaowing scenarios, Cell-Free Massive MIMO outperforms the small-cell approach in terms of 95%-likely per-user net throughput. In aition, a comparison of Figure 3 (or 4) an Figure 5 (or 6) shows that with power control, the performance of Cell-Free Massive MIMO improves significantly in terms of both meian an 95%-likely throughput. In the uncorrelate shaow faing scenario, the power allocation can improve the 95%-likely Cell-Free throughput by a factor of 2.5 for the ownlink an a factor of 2.3 for the uplink, compare with the case without power control. For the small-cell system, power control improves the 95%-likely

22 correlate shaowing uncorrelate shaowing 0.8 Cumulative Distribution small-cell Cell-Free Massive MIMO Per-User Downlink Net Throughput (Mbits/s) Fig. 3. The cumulative istribution of the per-user ownlink net throughput for correlate an uncorrelate shaow faing, with the greey pilot assignment an max-min power control. Here, M = 100, K = 40, an τ cf = τ sc = Cumulative Distribution small-cell Cell-Free Massive MIMO correlate shaowing uncorrelate shaowing Per-User Uplink Net Throughput (Mbits/s) Fig. 4. The same as Figure 3 but for the uplink, an τ cf = τ sc u = 20. throughput but not the meian throughput (recall that the power control policy explicitly aims at improving the performance of the worst user). In Figures 7 an 8, we consier the same setting as in Figures 3 an 4, but here we use the ranom pilot assignment scheme. These figures provie the same insights as Figures 3 an 4. Furthermore, by comparing these figures with Figures 3 an 4, we can see that with greey pilot assignment, the 95%- likely net throughputs can be improve by about 20% compare with when ranom pilot assignment is use.

23 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS correlate shaowing uncorrelate shaowing Cumulative Distribution small-cell Cell-Free Massive MIMO Per-User Downlink Rate (Mbits/s/Hz) Fig. 5. The cumulative istribution of the per-user ownlink net throughput for correlate an uncorrelate shaow faing, with the greey pilot assignment an without power control. Here, M = 100, K = 40, an τ cf = τ sc = 20. Cumulative Distribution correlate shaowing uncorrelate shaowing Cell-Free Massive MIMO small-cell Per-User Uplink Net Throughput (Mbits/s/Hz) Fig. 6. The same as Figure 5 but for the uplink, an τ cf = τ sc u = 20. In aition, we stuy how the M APs assign powers to a given user in the ownlink of Cell-Free Massive MIMO. From (5), the average transmit power expene by the mth AP on the kth user is η γ. Then p(m,k) η γ M m =1 η m kγ m k (58) is the ratio between the power spent by the mth AP on the kth user an the total power collectively spent by all APs on the kth user. Figure 9 shows the cumulative istribution of the effective number of APs serving each user, for τ cf = 5 an 20, an uncorrelate shaow faing. The effective number of APs

24 correlate shaowing uncorrelate shaowing 0.8 Cumulative Distribution small-cell Cell-Free Massive MIMO Per-User Downlink Net Throughput (Mbits/s) Fig. 7. The cumulative istribution of the per-user ownlink net throughput for correlate an uncorrelate shaow faing, with the ranom pilot assignment an max-min power control. Here, M = 100, K = 40, an τ cf = τ sc = Cumulative Distribution small-cell Cell-Free Massive MIMO correlate shaowing uncorrelate shaowing Per-User Uplink Net Throughput (Mbits/s) Fig. 8. The same as Figure 7 but for the uplink, an τ cf = τ sc u = 20. serving each user is efine as the minimum number of APs that contribute at least 95% of the power allocate to a given user. This plot was generate as follows: 1) 200 ranom realizations of the AP/user locations an shaow faing profiles were generate, each with M = 100 APs an K = 40 users; 2) for each user k in each realization, we foun the minimum number of APs, say n, such that the n largest values of {p(m,k)} sum up to at least 95% (k is arbitrary here, since all users have the same statistics); 3) a cumulative istribution was generate over the 200 realizations. We can see that, on average, only about of the 100 APs really participate in serving a given user. The larger τ cf, the less pilot contamination

25 CELL-FREE MASSIVE MIMO VERSUS SMALL CELLS Cumulative Distribution τ cf = 5 τ cf = Effective Number of APs Serving Each User Fig. 9. Cumulative istribution of the effective number of APs serving each user. Here, M = 100, K = 40, an τ cf = 5 an 20. an the more accurate channel estimates hence, more AP points can usefully serve each user. Finally, we investigate the effect of the number of users K an the training uration τ cf on the performance of Cell-Free Massive MIMO an small-cell systems. Figure 10 shows the average ownlink net throughput versus K for ifferent τ cf. The average is taken over the large-scale faing. We can see that when reucing K or τ cf, the effect of pilot contamination increases, an hence, the performance ecreases. As expecte, Cell-Free Massive MIMO systems outperform small-cell systems. Cell-Free Massive MIMO benefits from favorable propagation, an therefore, it suffers less from interference than the small-cell system oes. As a result, for a fixe τ cf, the performance gap between Cell-Free Massive MIMO an small-cell systems increases with K. Tables II an III summarize the ownlink respectively uplink performances of the Cell-Free Massive MIMO an small-cell systems, uner uncorrelate an correlate shaow faing. TABLE II THE 95%-LIKELY PER-USER NET THROUGHPUT (MBITS/S) OF THE CELL-FREE AND SMALL-CELL DOWNLINK, FOR M = 100, K = 40, AND τ cf = τ sc = 20. greey pilot assignment greey pilot assignment ranom pilot assignment with power control without power control with power control uncorrelate correlate uncorrelate correlate uncorrelate correlate shaowing shaowing shaowing shaowing shaowing shaowing Cell-Free Small-cell