Performance of Cell-Free Massive MIMO with Rician Fading and Phase Shifts

Özgecan Özdogan, Student Member, IEEE, Emil Björnson, Senior Member, IEEE, and Jiayi Zhang, Member, IEEE

Abstract—In this paper, we study the uplink (UL) and down- In its canonical form, cell-free massive MIMO uses MR link (DL) spectral efficiency (SE) of a cell-free massive multiple- combining because of its low complexity. Due to the fact that input-multiple-output (MIMO) system over Rician fading chan- MR combining cannot suppress the interference well, some nels. The phase of the line-of-sight (LoS) path is modeled as a uniformly distributed random variable to take the phase-shifts APs receive more interference from other UEs than signal due to mobility and phase noise into account. Considering the power from the desired UE. The LSFD method is proposed in availability of prior information at the access points (APs), the [7] and [8] to mitigate the interference for co-located massive phase-aware minimum mean square error (MMSE), non-aware MIMO systems. This method is generalized for more realistic linear MMSE (LMMSE), and least-square (LS) estimators are spatially correlated Rayleigh fading channels with arbitrary derived. The MMSE estimator requires perfectly estimated phase knowledge whereas the LMMSE and LS are derived without it. first-layer decoders in [9], [10]. The two-layer decoding tech- In the UL, a two-layer decoding method is investigated in order nique is first adapted to cell-free massive MIMO networks in to mitigate both coherent and non-coherent interference. Closed- [11] for a Rayleigh fading scenario. In the concurrent paper form UL SE expressions with phase-aware MMSE, LMMSE, and [12], the LSFD method is studied in a setup with Rician fading LS estimators are derived for maximum-ratio (MR) combining channels where the LoS phase is static. in the first layer and optimal large-scale fading decoding (LSFD) in the second layer. In the DL, two different transmission modes Joint transmission from multiple APs can be either coherent are studied: coherent and non-coherent. Closed-form DL SE (same data from all APs) or non-coherent (different data). expressions for both transmission modes with MR precoding Only the former has been considered in cell-free massive are derived for the three estimators. Numerical results show MIMO, but it requires that the APs are phase-synchronized. A that the LSFD improves the UL SE performance and coherent synchronization method is outlined in [5], [13], [14] without transmission mode performs much better than non-coherent transmission in the DL. Besides, the performance loss due to validation. the lack of phase information depends on the pilot length and it In densely deployed systems, like cell-free massive MIMO, is small when the pilot contamination is low. the channels typically consist of a combination of a semi- Index Terms—Cell-free massive MIMO, Rician fading, phase deterministic LoS path and small-scale fading caused by shift, performance analysis. multipath propagation, which can be modeled as Rician fading [12], [15]. A small change in the UE location may result in a significant phase-shift of the LoS component, but no I.INTRODUCTION change in amplitude. For instance, if the UE moves half Cell-free massive MIMO refers to a distributed MIMO a wavelength away from the AP, the phase of the channel system with a large number of APs that jointly serve a response changes by ±π. Similarly, hardware effects such as smaller number of user equipments (UEs) [2]–[5]. The APs phase noise may create severe shift in the phase. These effects cooperate via a fronthaul network [5] to spatially multiplex are usually neglected in the analysis of Rician fading channels arXiv:1903.07335v2 [cs.IT] 1 Nov 2019 the UEs on the same time-frequency resource, using network by assuming a LoS path with static phase. Especially in high MIMO methods that only require locally obtained channel mobility scenarios, the phase shift in LoS path may have a state information (CSI) [6]. large impact on system performance. Recently, [16] studied a cell-free network that supports both unmanned aerial vehicles Manuscript received March 13, 2019; revised June 13, 2019; accepted (UAVs) and ground UEs where the channels between AP-UAV August 3, 2019. This work was supported in part by ELLIIT, in part by pairs have Rician distribution with uniformly distributed phase the Swedish Research Council, in part by the National Natural Science Foundation of China (Grant Nos. 61601020 and U1834210), and in part by on the LoS paths. the Beijing Natural Science Foundation (Grant Nos. 4182049 and L171005). Each AP needs to learn the channel statistics of each A preliminary version of this manuscript was presented at IEEE SPAWC 2019 UE that it serves, as well as the statistics of the combined [1]. The associate editor coordinating the review of this article and approving it for publication was R. Brown. (Corresponding author: Özgecan Özdogan.) interfering signals, if Bayesian channel estimators are to be Ö. Özdogan and E. Björnson are with the Department of Electrical used. The large-scale fading coefficients can be estimated with Engineering (ISY), Linköping University, Linköping SE-581 83, Sweden (e- a negligible overhead since the coefficients are deterministic mail: [email protected]; [email protected]). J. Zhang is with the School of Electronic and Information Engineer- [17]; several practical methods to estimate these coefficients ing, Beijing Jiaotong University, Beijing 100044, China (e-mail: jiay- using uplink pilots are presented in [18, Section IV]. However, [email protected]) the phase-shifts are harder to estimate since they change as Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org. frequently as the small-scale fading. Depending on the avail- Digital Object Identifier 10.1109/TWC.2019.2935434 ability of channel statistics, the phase-aware MMSE estimator 2 which requires all prior information, LMMSE estimator with only the large-scale fading parameters, or LS estimator with no prior information can be utilized. In this paper, the specific technical contributions are as follows: • We consider Rician fading channels between the APs and UEs, where the mean and variance are different for every AP-UE pair. Additionally, the phases of the LoS paths are modeled as independent and identically distributed (i.i.d.) random variables in each coherence block. • We derive the phase-aware MMSE, LMMSE, and LS channel estimators and obtain their statistics. In the UL, using the estimates for MR combining in the first-layer, we compute closed-form UL achievable SE expressions

for two-layer decoding scheme. In the DL, we obtain : AP, : UE, : CPU, : Fronthaul closed-form DL achievable SE expressions for both co- connection herent and non-coherent transmission. Fig. 1: Illustration of a cell-free massive MIMO network. The conference version of this paper [1] only considered fading model since |hm,k | is Rice distributed, but hm,k is not the DL transmission and used the phase-aware MMSE and Gaussian distributed as in many prior works that neglected the LMMSE estimators. phase shift. We assume that hm,k is an independent random Reproducible Research: All the simulation results can be variable for every m = 1,..., M, k = 1,..., K and the channel reproduced using the Matlab code and data files available at: realization hm,k in different coherence blocks are i.i.d. https://github.com/emilbjornson/rician-cell-free All APs are connected to a central processing unit (CPU) Notation: Lower and upper case bold letters are used for via a fronthaul network that is error free. The system operates vectors and matrices. The transpose and Hermitian transpose T H in time division duplex (TDD) mode and the uplink (UL) of a matrix A are written as A and A . The superscript and DL channels are estimated by exploiting only UL pilot ( )∗ × . denotes the complex conjugate operation. The M M- transmission and channel reciprocity. dimensional matrix with the diagonal elements d1, d2,..., dM is denoted as diag (d1, d2,..., dM ). The diagonal elements of III.UPLINK CHANNEL ESTIMATION a matrix D are extracted to a M × 1 vector as diag(D) = T τ [d1, d2,..., dM ] . The notation X = [xi, j : i = 1,..., M, j = In each coherence block, p samples are reserved for UL 1,..., N] denotes the M × N matrix. The expectation of a pilot-based channel estimation, using a set of τp mutually random variable X is denoted by E {X}. The expectations are orthogonal pilot sequences. The pilot sequence of UE k is τp ×1 2 taken with respect to all sources of randomness. denoted√ by φk ∈ C and satisfies kφk k = τp. It is scaled by pˆk , with pˆk being the pilot power, and sent to the APs. p,ap The received signal y ∈ Cτp ×1 at AP m is II.SYSTEM MODEL m K We consider a cell-free Massive MIMO system with M APs Õ yp,ap = ppˆ h φ + np , (2) and K UEs. All APs and UEs are equipped with a single m k m,k k m antenna. The multi-antenna AP case can be straightforwardly k=1 covered by treating each antenna as a separate AP, if it is p ∼ N 2 where nm C 0τp , σulIτp is additive noise. AP m com- assumed that there is no correlation between the small-scale p,ap putes an inner product between ym and φk to get sufficient fading coefficients (or phase-shifts) [19]. However, for a more statistics for estimation of h . This results in realistic analysis, single-antenna results can be generalized m,k K to multiple antenna case by taking the spatial correlations p H p,ap Õ p H H p between antennas into account. It will result in non-diagonal ym,k = φk ym = pˆl hm,lφk φl + φk nm. (3) covariance matrices. The channels are assumed to be constant l=1 and frequency-flat in a coherence block of length τc samples We assume that the number of UEs is large such that τp < K (channel uses). The length of each coherence block is deter- and we define the set Pk of UEs that use same pilot sequence mined by the carrier frequency and external factors such as the as UE k, including itself. Since UEs with different pilots have p propagation environment and UE mobility [17]. The channel orthogonal pilots, ym,k in (3) can be rewritten as hm,k between UE k and the AP m is modeled as p p Õ p H p ym,k = pˆk τp hm,k + pˆlτp hm,l + φk nm, (4) ¯ jϕm, k hm,k = hm,k e + gm,k, (1) l∈Pk \{k } ∼ N ¯ ≥ H p ∼ N ( 2 ) where gm,k C 0, βm,k , the mean hm,k 0 represents where φk nm C 0, σulτp . Based on this signal, we will the LoS component, and ϕm,k ∼ U [−π, π] is the phase-shift. derive three different channel estimators, and characterize their The small-scale fading from non-LoS (NLoS) propagation has statistics. In particular, we will explore the impact of the a variance βm,k that models the large-scale fading, including phase-shift in the LoS paths by considering the analytically geometric pathloss and shadowing. Note that (1) is a Rician tractable cases where the phase-shifts are either perfectly 3

˜ mmse − ˆ mmse known or fully unknown. In co-located MIMO systems, the The estimation error hk = hk hk has zero mean and phase can be estimated by employing classical algorithms such the covariance matrix as estimation of signal parameters via rotational invariance techniques (ESPRIT) [20] and multiple signal classiﬁcation Ck = Rk − pˆk τpΩk, (11) (MUSIC) [21]. However, these methods either require multiple antennas and/or long sample sequences (snapshots) which are where Ωk = Rk Λk Rk . The mean-squared error is MSE = {k − ˆ mmse k2} ( ) ˆ mmse not available in our system setup. In a low mobility scenario E hk hk = tr Ck . The MMSE estimate hk and ˜ mmse in which the APs are equipped with multiple antennas, the the estimation error hk satisfy phase-shifts can be estimated quite well, but the estimates ˆ mmse ¯ ¯ will anyway be imperfect since the estimation resources are E hk Φk hk = Φk hk, (12) practically limited. Nevertheless, we will consider a phase- mmse Cov hˆ Φk h¯ k = Rk − Ck, (13) aware MMSE estimator that serves as an upper bound on k ˜ mmse ∼ N ( ) the achievable performance. The practical performance will hk C 0M, Ck . (14) be somewhere in between the cases with perfectly known and fully unknown phases, which are considered below. We consider two estimators of the latter kind; one Bayesian B. LMMSE Channel Estimator LMMSE and one classical LS estimator. If the channel statistics h¯ m,k , βm,k are available but the phase ϕ is completely unknown at AP m, for every k, we can A. Phase-aware MMSE Channel Estimator m,k derive the LMMSE estimator of hm,k as If the channel statistics h¯ m,k , βm,k are available and the √ p phase ϕm,k is somehow perfectly known at AP m, for every 0 pˆk βm,k y k, we can derive the MMSE estimator of h as [22] hˆ lmmse = m,k , (15) m,k m,k λ0 √ m,k p − p pˆk βm,k ym,k y¯m,k mmse jϕm, k 0 0 hˆ = h¯ m,k e + , (5) where β = β +h¯ 2 and λ = Í pˆ τ (β +h¯ 2 )+ m,k λ m,k m,k m,k m,k l∈Pk l p m,l m,l m,k 2 ˜ lmmse ˆ lmmse p √ σ . The estimation error h = hm,k −h has zero mean where y¯ = Í pˆ τ h¯ ejϕm,l and λ = ul m,k m,k m,k l∈Pk l p m,l m,k and the variance Í 2 ˆ mmse l∈P pˆlτp βm,l + σ . Note that h is a random variable k ul m,k 0 2 and (5) is a single realization for a speciﬁc coherence block. pˆk τp(βm,k ) p p c0 = β0 − . (16) The y , y¯ and ϕm,k change in every coherence block m,k m,k 0 m,k m,k λm,k whereas h¯ m,k and βm,k remain constant for a longer period of time. AP m estimates channels to all the UEs. The estimation lmmse 2 0 The mean-squared error is MSE = E{|hm,k − hˆ | } = c . ˜ mmse − ˆ mmse m,k m,k error hm,k = hm,k hm,k has zero mean and the variance ˆ lmmse ˜ lmmse The LMMSE estimate hm,k and the estimation error hm,k 2 pˆk τp β are uncorrelated random variables and satisfy c = β − m,k . (6) m,k m,k λ m,k n ˆ lmmseo n ˆ lmmseo 0 0 E h = 0, Var h = βm,k − cm,k, (17) The mean-squared error is MSE = E{|h − hˆ mmse|2} = c . m,k m,k m,k m,k m,k n o n o ˆ mmse ˜ mmse E h˜ lmmse = 0, Var h˜ lmmse = c0 . (18) The MMSE estimate hm,k and the estimation error hm,k m,k m,k m,k satisfy n o A detailed derivation is given in Appendix A. We can write E hˆ mmse ϕ = h¯ ejϕm, k , (7) m,k m,k m,k (15) in vector form as n mmse o Var hˆ ϕm,k = βm,k − cm,k, (8) m,k ˆ lmmse p 0 Λ0 p hk = pˆk Rk k yk , (19) n ˜ mmseo n ˜ mmseo E hm,k = 0, Var hm,k = cm,k, (9) where R0 = diag(β0 . . ., β0 ), Λ0 = diag(λ0 , . . ., λ0 )−1. mmse mmse k 1,k M,k k 1,k M,k where hˆ is not Gaussian distributed. Also, hˆ and lmmse lmmse m,k m,k The estimation error h˜ = hk − hˆ has zero mean and ˜ mmse k k hm,k are uncorrelated but not independent random variables. the covariance matrix The APs do not share these channel estimates with each other, or the CPU, but they are used to perform MR processing 0 0 0 C = R − pˆk τpΩ , (20) distributively as in [3], [4], [23]. Nevertheless, the collection k k k of channel estimates of UE k from all APs can be written in 0 0 0 0 where Ωk = Rk Λk Rk . The mean-squared error is MSE = vector form as {k − ˆ lmmse k2} ( 0 ) ˆ lmmse E hk hk = tr Ck . The MMSE estimate hk and ˆ mmse Φ ¯ p Λ ( p − p) the estimation error h˜ lmmse are uncorrelated random variables hk = k hk + pˆk Rk k yk y¯k , (10) k T jϕ jϕ and satisfy where h¯ k = [h¯1,k,..., h¯ M,k ] , Φk = diag e 1,k ,..., e M, k , h iT yp [yp ,..., yp ]T y¯ p yp ,..., yp Λ ˆ lmmse ˆ lmmse 0 0 k = 1,k M,k , k = ¯1,k ¯M,k , k = E hk = 0M, Cov hk = Rk − Ck, (21) −1 ˜ lmmse ˜ lmmse 0 diag λ1,k, . . ., λM,k , and Rk = diag(β1,k, . . ., βM,k ). E hk = 0M, Cov hk = Ck . (22) 4

C. LS Channel Estimator with No Prior Information If AP m has no prior information regarding the phase or statistics, then we can derive the non-Bayesian LS estimator √ | p − |2 that minimizes ym,k pk τp hm,k , which is ˆ ls 1 p hm,k = √ ym,k . (23) pk τp The LS estimator and estimation error satisfy

n ˆ ls o n ˆ ls o 1 0 E hm,k = 0, Var hm,k = λm,k, (24) pˆk τp Fig. 2: Two-layer decoding technique. n ˜ ls o n ˜ ls o 1 0 0 E hm,k = 0, Var hm,k = λm,k − βm,k . (25) pˆk τp p the likely quality of the received signal. For instance, if the In vector form, we have hˆ ls = √ 1 y for each user k. The k pk τp k UE is far away from the AP, potentially the received signal LS estimator and estimation error satisfy will contain a considerable amount of interference and noise ˆ ls ˆ ls 1 0 −1 whereas the desired signal is weak. Then, the corresponding E hk = 0M, Cov hk = (Λk ) , (26) pˆk τp LSFD coefficient for this signal will be small to reduce the risk 1 of adding and amplifying more interference into the detection. ˜ ls ˜ ls (Λ0 )−1 − 0 E hk = 0M, Cov hk = k Rk . (27) Note that assigning α = 1 for all APs and UEs corresponds pˆk τp m,k to the single-layer decoding considered in [3], [4], [24], [25]. A detailed derivation is given in Appendix B. Similarly, setting αm,k = 0 means that AP m is not serving UE k, which can be utilized to handle practical cases where IV. UPLINK DATA TRANSMISSION each UE is served by a subset of all the APs; see [24], [26] Let τu be the number of UL data symbols per coherence for details. The αm,k is deterministic since it only depends on block, where τu ≤ τc − τp. The received signal at AP m is deterministic large-scale fading coefficients. K Based on (30), the UL ergodic channel capacity of UE k can ul Õ ul ym = hm,k sk + nm, (28) be lower bounded using the use-and-then-forget (UatF) bound k=1 [17, Th. 4.4]. The expected values of the desired signal, inter- user interference and noise related terms at all APs for UE k where sk ∼ NC (0, pk ) is the UL signal with power pk = | |2 ul ∼ N ( 2 ) can be computed respectively as E sk and nm C 0, σul is additive receiver noise. AP m selects the receiver combining scalar as v = hˆ ∈ m,k m,k 2 mmse lmmse ls M {hˆ , hˆ , hˆ } to locally detect the signal of UE k. Õ n o 2 m,k m,k m,k ∗ E ∗ E H H Then, the combined received signal at AP m is αm,k vm,k hm,k = vk Ak hk (31) m=1 K 2 ∗ ul ∗ Õ ∗ ∗ ul  M  n o s˜m,k = vm,k ym = vm,k hm,k sk + vm,k hm,l sl + vm,k nm. (29)  Õ ∗ ∗  H H 2 E α v hm,l = E v A hl (32) l=1 m,k m,k k k l,k  m=1    A second layer decoding is performed to mitigate the inter- M 2  Õ  E  ∗ ∗  E k k2 user interference using LSFD coefficients. The single-layer αm,k vm,k = Ak vk (33) decoded signal in (29) is sent to the CPU and it performs  m=1  the second layer decoding by computing the LSFD weighted   T signal as where Ak = diag α1,k, . . ., αM,k and vk = [v1,k,..., vM,k ] . M M Then, a lower bound on the UL ergodic SE with the LSFD Õ ∗ Õ ∗ ∗ sˆk = αm,k s˜m,k = αm,k vm,k hm,k sk receiver is τ m=1 m=1 ul u ( ul) M K M SEk = log2 1 + γk , (34) τc Õ Õ ∗ ∗ Õ ∗ ∗ ul + αm,k vm,k hm,l sl + αm,k vm,k nm. (30) m=1 l=1 m=1 where the effective SINR γul is l,k k where αm,k is the complex LSFD coefficient for AP m and H H 2 UE k. The weighting operation reduces the inter-user inter- pk E v A hk γul = k k . ference by balancing the received signals from all APs. The k K n o Í H H 2 − H H 2 2 k k2 plE vk Ak hl pk E vk Ak hk + σulE Ak vk CPU computes the LSFD weights based on the large-scale l=1 fading coefficients that vary slowly compared to the small- (35) scale coefficients. Recall that the value of large-scale fading and the expectations are with respect to all sources of random- between an AP-UE pair depends mainly on the distance and ness. This bound is referred as UatF bound since the channel shadowing. Thus, it gives us a valuable information about estimates are used for combining locally at the APs and then 5

K they are unknown at the CPU. The effective SINR of UE k in Γ Í p R0Z Í p p p τ2z zH − p L2 where k = l l k + l∈Pk \{k } l ˆk ˆl p k,l k,l k k + (35) with the LSFD receiver can be rewritten as l=1 2 σulZk, bk = diag (Zk ), and zk,l = diag (RlΛk Rk ). The SE in H 2 −1 pk a bk,k (41) with the maximizing LSFD receiver vector ak = Γ bk γul = k , (36) k k K is H Í Γ(1) − H 2 Γ(2) τ ak pl k,l pk bk,k bk,k + σul k ak ul u H −1 l=1 SEk = log2 1 + pk bk Γk bk . (45) τc where Proof: The proof is given in Appendix C. T M×1 ak = α1,k, . . ., αM,k, ∈ C , (37) T h n ∗ o n ∗ oi M×1 bk,l = E v1,k h1,l ,..., E vM,k hM,l ∈ C , (38) B. Uplink Spectral Efficiency with the LMMSE Estimator (1) h n o i ∗ ∗ ∈ M×M ˆ lmmse Γk,l = E vm,k hm,lvn,k hn,l : m, n C , (39) Theorem 2: If MR combining with vm,k = hm,k is used ∀ ul τu (2) based on the LMMSE estimator, then SE = log (1 + Γ = diag E |v |2 ,..., E |v |2 ∈ RM×M . (40) k τc 2 k 1,k M,k ul,lmmse) ul,lmmse γk with γk in (46), at the top of next page, where In order to maximize the SE of UE k, we need to select the lmmse,(1) H 0 0 LSFD vector that maximizes (36) as shown in the following Tk,l = pˆk τptr Ak RlΩk Ak , (47) Lemma. lmmse,(2) h 2 H 2Λ0 Ω0 0Λ0 0 2 Lemma 1: For a given set of pilot and data power coeffi- Tk,l = pˆk pˆlτp tr Ak Rl k k Ak + tr Ak Rl k Rk cients, the SE of UE k is H 0 0 H 0 0 0 2 i +2tr Ak Λk Ωk LlRlAk − tr Ak (RlΛk Rk ) Ak . (48) ul τu SEk = (41) τc The effective SINR in (46) can be reformulated in Rayleigh K !−1 quotient form as given in (49), at the top of next page where © H Õ (1) H 2 (2) ª × log2 1 + pk bk,k pl Γk,l − pk bk,k bk,k + σ Γk bk,k ul ® lmmse,(2) l=1 Γ = pˆ pˆ τ2 (50) « ¬ k,l k l p h i with the LSFD vector 2 0 0 0 0 H 0 0 0 2 × Rl Λk Ωk + 2Λk Ωk LlRl + dk,ldk,l − (RlΛk Rk ) , − K ! 1 K Õ (1) (2) lmmse,(1) Γ − H 2 Γ lmmse Õ ak = pl k,l pk bk,k bk,k + σul k bk,k (42) Γk = pl Γk + l=1 l=1 Õ lmmse,(2) lmmse lmmse H 0 2 Proof: The SINR expression in (36) is a generalized pl Γk,l − pk bk,k (bk,k ) + pˆk τpΩk σul, (51) Rayleigh quotient with respect to ak and we apply [17, Lemma l∈Pk B.10] to obtain the maximizing ak in (42). Inserting this vector lmmse,(1) lmmse ( 0 ) 0 0 into (34) gives the SE in (41). bk = pˆk τpdiag Ωk , Γk,l = pˆk τpRlΩk , and dk,l = β0 β0 β0 β0 The UL SE of two-layer decoding scheme for Rayleigh 0 0 0 1,l 1, k M,l M, k T M×1 diag RlΛk Rk = [ λ0 ,..., λ0 ] ∈ R . The SE fading channels with MMSE estimator when using MR com- 1,k M, k in (41) with the maximizing LSFD receiver vector a = bining is presented in [11]. Lemma 1 generalizes this results k (Γlmmse)−1blmmse is to cover arbitrary channel distributions and beamforming k k schemes. Every choice of channel model and beamforming τ −1 (1) (2) ul u ( lmmse)H Γlmmse lmmse will affect the values of bk,l, Γ , and Γ , but the solution SEk = log2 1 + pk bk k bk . (52) k,l k τc structure in Lemma 1 remains the same. In the following ul subsections, γk is computed for MR combining when using Proof: The proof is given in Appendix D. the phase-aware MMSE, LMMSE, and LS channel estimators derived in Section III. C. Uplink Spectral Efficiency with the LS Estimator ls A. Uplink Spectral Efficiency with the Phase-aware MMSE Theorem 3: If MR combining with vm,k = hˆ is used m,k Estimator based on the LS estimator, then SEul = τu log 1 + γul,ls k τc 2 k ˆ mmse ul,ls Theorem 1: If MR combining with vm,k = hm,k is with γ in (53), at the top of next page, where used based on the phase-aware MMSE estimator, then k τ ul,mmse ul,mmse SEul = u log 1 + γ with γ in (43), at the ls,(1) 1 k τc 2 k k H 0 −1 0 Tk,l = tr Ak (Λk ) RlAk , (54) top of next page, where Zk = pˆk τpΩk + Lk and Lk = pˆk τp ¯ 2 ¯ 2 ls, (2) pˆl diag h1,k,..., hM,k . In Rayleigh quotient form, the SINR of Tk,l = (55) UE k is pˆk H H 2 ul,mmse pk ak bk bk ak H 2 H 0 H 0 2 γ = , (44) × tr A (R + 2LlRl)Ak + tr A R − tr A (R ) Ak . k H Γ k l k l k l ak k ak 6

p |tr (A Z ) |2 γul,mmse = k k k , (43) k K Õ H 0 Õ 2 2 H 2 2 pltr Ak RlZk Ak + pl pˆk pˆlτp |tr (Ak RlΛk Rk ) | + tr Ak (σulZk − pk Lk )Ak l=1 l∈Pk \{k }

2 2 0 2 pk pˆ τp |tr Ak Ω | γul,lmmse = k k , (46) k K Õ lmmse,(1) Õ lmmse,(2) 2 2 0 2 2 H 0 plTk,l + plTk,l − pk pˆk τp |tr Ak Ωk | + σulpˆk τptr Ak Ωk Ak l=1 l∈Pk H lmmse lmmse H pk a b (b ) ak γul,lmmse = k k k , (49) k lmmse,(1) lmmse,(2) aH ÍK p Γ + Í p Γ − p blmmse(blmmse)H + pˆ τ Ω0 σ2 a k l=1 l k l∈Pk l k,l k k k k p k ul k

0 2 pk |tr Ak Rk | γul,ls = , (53) k K σ2 Õ ls,(1) Õ ls, (2) 0 2 ul H 0 −1 plTk,l + plTk,l − pk |tr Ak Rk | + tr Ak Λk Ak pˆk τp l=1 l∈Pk H ls ls H pk a b (b ) ak γul,ls = k k k , (56) k ls,(1) ls,(2) σ2 H ÍK Í ls ls H ul 0 −1 a pl Γ + pl Γ − pk b (b ) + Λ ak k l=1 k,l l∈Pk k,l k k pˆk τp k

The effective SINR in (53) can be reformulated in Rayleigh massive MIMO, thus it is a conservative lower bound on the quotient form as given in (56) where performance.

ls,(2) pˆl 2 ls ls H 0 2 Γk,l = Rl + 2LlRl + bl (bl ) − (Rl) , (57) pˆk V. COHERENT DOWNLINK TRANSMISSION K 2 Λ0 −1 ls Õ ls,(1) Õ ls,(2) ls ls H σul k Γk = pl Γk,l + pl Γk,l − pk bk (bk ) + , pˆk τp Each coherence block contains τd DL data symbols, where l=1 l∈Pk τ = τ − τ − τ . In this section, we assume that each BS (58) d c u p transmits to each UE and sends the same data symbol as ls,(1) bls = diag R0 and Γ = 1 (Λ0 )−1R0. Note that the the other APs. By setting some transmit powers to zero, the l l k,l pˆk τp k l analysis also covers cases where each AP only serves a subset SINR expressions with the LS estimator above are proportional of the UEs. The transmitted signal from AP m is to the ones with LMMSE estimator since the LMMSE and LS estimators equal up to a scaling factor that depends on K Õ the large-scale fading coefﬁcients. The SE in (41) with the x = wcoh ς , (60) −1 m m,k k ls ls k=1 maximizing LSFD receiver ak = Γk bk is − ∼ N τu 1 where ςk C (0, 1) is the DL data signal to UE k which ul ls H ls ls q ρ SEk = log2 1 + pk (bk ) Γk bk (59) coh m, k ˆ is same for all APs. The scalar w = 2 hm,k τc m,k E{ |hˆm,k | } and it is equal to the UL SE with LMMSE estimator in (52). is the coherent beamforming and ρm,k ≥ 0 is chosen to 2 dl Proof: The proof is given in Appendix E. satisfy the DL power constraint E{|xm| } ≤ ρ which implies ÍK ≤ dl Remark 1: Note that the LSFD method only requires the l=1 ρm,l ρ . The received signal at the kth UE is large-scale fading coefﬁcients and not the channel realizations K Õ themselves. Hence, it can be implemented along with the ydl = hH w ς + hH w ς + ndl, (61) LMMSE and MMSE estimators without requiring any addi- k k k k k l l k l=1 tional information. However, the optimal LSFD vector cannot l,k be practically computed when using the LS estimator since k the statistics are not known in that case. We have nevertheless where the receiver noise at UE is denoted as dl ∼ N 2 1/2 ˆ included the optimal LSFD vector in that case to demonstrate nk C 0, σdl and wk = Dk hk with Dk = that it makes the SE expressions with the LMMSE and LS ρ1,k ρM,k 1/2 diag ˆ 2 ,..., ˆ 2 . Note that D is a determin- estimators identical, thus LSFD can compensate for the use E{ |h1, k | } E{ |hM, k | } k of a bad channel estimator. We will demonstrate later that istic matrix whereas hˆ k is a random vector. Based on the the LS estimator (without LSFD) performs poorly in cell-free signal in (61), the ergodic DL capacity of UE k is lower 7

bounded using the UatF bound as SEdl = τd log 1 + γdl,coh C. Coherent Downlink Spectral Efficiency with the LS Esti- k τc 2 k [bit/s/Hz] with mator / H 2 1 2 ˆ ls E w hk Theorem 6: If MR precoding with wk = Dk hk is used γdl,coh = k , (62) k K based on the LS estimator, then the expectations in (62) are Õ n H 2o H 2 2 computed as E wl hk − E wk hk + σdl l=1 E wH h = tr D1/2R0 , (69) where the expectations are with respect to all sources of k k k k dl,coherent n 2o 1 randomness [17, Th. 4.6]. Next, the effective SINR γk H 0 −1 0 E wl hk = tr Dl(Λl) Rk (70) is computed when using the phase-aware MMSE, LMMSE, pˆlτp and LS estimators. 2 tr D R2 + 2L R + tr D1/2R0  l k k k l k pˆk  A. Coherent Downlink Spectral Efficiency with the Phase- + 0 2 −tr Dl(R ) , l ∈ Pk aware MMSE Estimator pˆl  k  1/2 ˆ mmse 0 l < Pk . Theorem 4: If MR precoding with wk = Dk hk is  used based on the phase-aware MMSE estimator, then the Inserting these expressions into (62) gives the DL SINR at expectations in (62) are computed as UE k as H 1/2 E wk hk = tr Dk Zk , (63) dl,ls γk n 2o E wH h p τ (D R Ω ) p τ (D Ω L ) D S 2 l k = ˆl ptr l k l + ˆl ptr l l k + tr l l,k 1/2 0 tr Dk Rk 2 1/2 = ,  2 Λ ∈ P \{ } K pˆk pˆlτp tr Dl Rk lRl , l k k 2  Õ 1 0 −1 0 Õ dl,ls, (2) 1/2 0 2  2 2 tr Dl(Λ ) R + T − tr D R + σ  2 2 1/2 1/2 l k k,l k k dl pˆ τ tr D Ωl + tr D Ll pˆlτp + tr (D R L ) + l p l l l=1 l∈Pk l k l +2p ˆ τ tr D1/2Ω tr D1/2L , l = k  l p l l l l where  0 l < Pk .  dl,ls, (2) pˆk (64) Tk,l = (71) pˆl Inserting these expressions into (62) gives the DL SINR at 2 × tr D R2 + 2L R + tr D1/2R0 − tr D (R0 )2 . UE k in (65), at the top of next page. l k k k l k l k Proof: The proof is given in Appendix C. Proof: The proof is given in Appendix E. B. Coherent Downlink Spectral Efficiency with the LMMSE Estimator 1/2 ˆ lmmse VI.NON-COHERENT DOWNLINK TRANSMISSION Theorem 5: If MR precoding with wk = Dk hk is used based on the LMMSE estimator, then the expectations in (62) In this section, we consider the alternative case in which are computed as each AP is allowed to transmit to each UE but sends a different 1/2 data symbol than the other APs, to alleviate need for phase- E wH h = pˆ τ tr D Ω0 , (66) k k k p k k synchronizing the APs. The transmitted signal from AP m is n H 2o 0 0 E wl hk = pˆlτptr DlRk Ωl (67) K Õ 0 0 2 xm = wm,k ςm,k, (72) tr DlΩ Λ R + 2Lk Rk  l l k k=1  2 + pˆ pˆ τ2 1/2 0 0 0 0 0 0 2 l k p + tr D R Λ R − tr Dl(R Λ R ) , l ∈ Pk  l k l l k l l where ςm,k ∼ NC 0, ρm,k is the DL data signal to UE k. The  ˆ 0 l P . hm,k < k beamforming scalar wm,k = q is a scaled version  ˆ 2 Inserting these expressions into (62) gives the DL SINR at E{ |hm, k | } of the channel estimate and ρm,k is chosen to satisfy the DL UE k as 2 dl ÍK power constraint E{|xm| } ≤ ρ which implies l=1 ρm,l ≤ dl,lmmse dl γk = (68) ρ . The received signal at the kth UE is 2 2 | ( 1/2Ω0 )|2 pˆk τp tr Dk k M K M , dl Õ ∗ Õ Õ ∗ dl K yk = hn,k wn,k ςn,k + hn,k wn,l ςn,l + nk , (73) Õ 0 0 Õ dl,lmmse,(2) 0 2 2 2 2 n=1 l n=1 pˆlτptr DlRk Ωl + Tk,l − pˆk τp |tr Dk Ωk | + σdl =1 l,k l=1 l∈Pk h dl,lmmse,(2) 2 0 0 2 where the receiver noise at UE k is denoted ndl ∼ N 0, σ2 . where Tk,l = pˆk pˆlτp tr DlΩlΛl Rk + 2Lk Rk + k C dl 2 Lemma 2: Based on the signal in (73), if the UE k detects tr D1/2R0 Λ0R0 − tr D (R0 Λ0R0)2 . l k l l l k l l the M signals using successive interference cancellation (with Proof: The proof is given in Appendix D. arbitrary decoding order) a lower bound on the DL sum SE of 8

2 tr D1/2Z k k γdl,mmse = . (65) k K Õ Õ 2 tr D R0 Z + pˆ pˆ τ2 tr D1/2R Λ R − tr D L2 + σ2 l k l k l p l k l l k k dl l=1 l∈Pk \{k } tr (D Z ) γdl,mmse k k , (78) k K Õ 0 Õ 2 2 −1 2 −1 2 tr DlRk + pˆk pˆlτptr Dl(Rk ΛlRl) Zl − tr Dk Lk Zk + σdl l=1 l∈Pk \{k }

UE k is SEdl,nc = τd log (1 + γdl,nc) where the effective SINR B. Non-Coherent Downlink Spectral Efﬁciency with the k τc 2 k is LMMSE and LS Estimator hˆlmmse dl,nc If MR precoding with w = m, k is used based γ = m,k r n o k E |hˆlmmse |2 n o 2 m, k ÍM ρ E h∗ w on the LMMSE estimator without phase information, then the n=1 n,k n,k n,k . expectations in (74) are computed as M K M Õ Õ n o Õ n o 2 ∗ 2 ∗ 2 q ρn,lE |hn,k wn,l | − ρn,k E hn,k wn,k + σdl n ∗ o 2 0 2 0 −1 E h wn,k = E |wn,k | = pˆk τp(β ) (λ ) , (79) n=1 l=1 n=1 n,k n,k n,k (74) n ∗ 2o 0 E |hn,k wn,l | = βn,k Proof: The proof is given in Appendix F. ( 0 −1 2 2 pˆk τp(λ ) β + 2h¯ βn,k , l ∈ Pk From the effective SINR above, we notice that the numerator + n,l n,k n,k (80) contains the sum of the squared contributions from different 0 l < Pk, APs, which is different from the coherent case where the n o | ˆ lmmse|2 ( 0 )2( 0 )−1 summation is inside the square. Hence, the coherent case gives where E hm,k = pˆk τp βm,k λm,k . If MR precoding a larger signal term, but it requires that all APs are synchro- hˆls with w = m, k is used based on the LS estimator nized and co-operate to achieve a coherent beamforming gain m,k r n o E |hˆls |2 that is analogous to co-located massive MIMO. The numerical m, k without any prior information regarding the phase or statistics, comparisons of these DL transmission modes are presented in computing expectations in (74) gives the exactly same results Section VII. In the following subsections, the effective γdl,nc n o k as (79) and (80) where E |hˆ ls |2 = 1 λ0 . Note that this is calculated for MR precoding when using the phase-aware m,k pˆk τp m,k MMSE, LMMSE, and LS estimators. is due to the precoding normalization at each AP: √ ˆ lmmse 0 ( 0 )−1 p hm,k pˆk βm,k λm,k ym,k wm,k = r = q A. Non-coherent Downlink Spectral Efﬁciency with the Phase- n o ( 0 )2( 0 )−1 E |hˆ lmmse|2 pˆk τp βm,k λm,k aware MMSE Estimator m,k hˆmmse 1 p m,k √ ˆ ls If MR precoding with w = is used based p τ ym,k h m,k r n o k p m,k E |hˆmmse |2 = = . (81) m,k q 1 0 r n o pˆ τ λm,k | ˆ ls |2 on the MMSE estimator, then the expectations in (74) are k p E hm,k computed as Substituting (79) and (80) into (74) gives the same DL SINR n o q ∗ 2 −1 ¯ 2 E hn,k wn,k = pˆk τp βn,k λn,k + hn,k, (75) at UE k with the LMMSE and LS estimators n ˆ mmse 2o 2 −1 ¯ 2 dl,lmmse dl,ls E |hn,l | = pˆlτp βn,lλn,l + hn,l, (76) γk = γk = (82) 0 n ∗ 2o 0 pˆk τptr(Dk Ωk ) E |h wn,l | = β . n,k n,k K 2 2 2 −2 Õ 0 Õ 0 2 0 2 pˆk pˆl τp βn,k βn,l λn,l tr(D R ) + pˆ τ tr(D Λ (R + 2R L )) − pˆ τ (D Ω ) + σ  , l ∈ P \{k} l k k p l l k k k k p k k dl  pˆ τ β2 λ−1 +h¯2 k  l p n,l n,l n,l l=1 l∈Pk  pˆ2 τ2 β4 λ−2 +2p ˆ τ β2 λ−1 h¯2 + k p n, k n, k k p n, k n, k n,k , l = k (77) (83) pˆ τ β2 λ−1 +h¯2  k p n, k n,k n,k  Proof: The proof is also given in Appendix D and 0, l < Pk .  Appendix E. Substituting these expressions into (74) gives the DL SINR at In contrast to the coherent DL transmission case, the SE is UE k as given in (78), at the top of this page where Dk = the same for both LMMSE and LS estimator because there diag ρ1,k, . . ., ρM,k is the downlink power matrix. is no cooperation between APs. Thus, the expectations are Proof: The proof is also given in Appendix C. calculated for each antenna individually. 9

1.8 MMSE 1 1.6 LMMSE LS 1.4 Single-layer Monte-Carlo 0.8 decoding 1.2 Two-layer decoding Two-layer 1 0.6 decoding

0.8 0.4 0.6 Single-layer

Average UL SE [bit/s/Hz] 0.4 decoding 0.2 MMSE 0.2 LMMSE

0 Cumulative Distribution Function (CDF) LS 40 50 60 70 80 90 100 0 Number of APs, M 0 1 2 3 4 5 6 SE per UE [bit/s/Hz] Fig. 3: Average UL SE for different number of APs. K = Fig. 4: CDF of UL SE for M = 100, K = 40 and τp = 5. 40, τp = 5.

1 1

= 20 0.8 p 0.8

K = 40 0.6 0.6

0.4 0.4 K=10 = 5 p

0.2 0.2 MMSE MMSE Cumulative Distribution Function (CDF) Cumulative Distribution Function (CDF) LMMSE/LS LMMSE/LS 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 SE per UE [bit/s/Hz] SE per UE [bit/s/Hz]

Fig. 5: CDF of UL SE for M = 100, K = 40 and τp = Fig. 6: CDF of UL SE for M = 100, τp = 5 and K = [5, 20]. Two-layer decoding. [10, 40]. Two-layer decoding.

VII.NUMERICAL RESULTS independent and δ is the shadow fading parameter, 0 ≤ δ ≤ 1. In this section, the closed-form SE expressions are validated The random variables am and bk model the shadow fading and evaluated by simulating a cell-free massive MIMO net- effect from blocking objects in the vicinity of the AP m and M K UE k, respectively. The covariance functions for arbitrary AP work. We have APs and = 40 UEs that are independently −dm, n 2 d and uniformly distributed within a square of size 1×1 km with and UE pairs are given as E {aman} = 2 dc , and E {bk bl } = a wrap-around setup. Maximum ratio precoding/combining is −dk,l 2 ddc , where d denote the distance between AP m and used at UL/DL in all simulations. The pathloss is computed m,n AP n and d is the distance between UE k and UE l. The d based on the COST 321 Walfish-Ikegami model for micro- k,l dc is decorrelation distance which depends on the environment. cells in [27] with AP height 12.5 m and UE height 1.5 m. We set the parameters d = 100 m, σ = 8 and δ = 0.5 in All AP-UE pairs have a LoS path and the path-loss (PL) is dc sf the simulation. With this model, the large scale coefficient of modeled (in dB) as hm,k are d − − m,k r PLm,k = 30.18 26 log10 + Fm,k, (84) κm,k p 1 1 m h¯ m,k = PLm,k, βm,k = PLm,k . (85) κm,k + 1 κm,k + 1 where dm,n is the distance between AP m and UE k and Fm,k is the shadow fading coefficient. The Rician κ-factor is calculated We consider communication over a 20 MHz channel and the 1.3−0.003d as κm,k = 10 m,k [27] . total receiver noise power is −94 dBm. Each coherence block √ We assume√ correlated shadow fading as in [3] with Fm,k = consists of τc = 200 samples and τp pilots. The pilots of first τp − ∼ N( 2 ) ∼ N( 2 ) δam+ 1 δbk where am 0, σsf and bk 0, σsf are UEs are allocated randomly. The rest of UEs sequentially pick 10

1.6 1 Coherent trans. with MMSE Non-coherent Coherent trans. with LMMSE transmission 1.4 Coherent trans. with LS Non-coherent trans. with MMSE 0.8 Non-coherent trans. with LMMSE/LS 1.2 Monte-Carlo

Coherent 1 0.6 transmission

0.8 0.4

0.6 Coherent trans. with MMSE

Average DL SE [bit/s/Hz] Coherent trans. with LMMSE 0.2 0.4 Coherent trans. with LS Non-coherent trans. with MMSE Cumulative Distribution Function (CDF) Non-coherent trans. with LMMSE/LS 0.2 0 40 50 60 70 80 90 100 0 1 2 3 4 5 6 Number of APs, M SE per UE [bit/s/Hz] Fig. 7: Average DL SE versus different number of APs for Fig. 8: CDF of DL SE for coherent and non-coherent coherent and non-coherent transmission. K = 40, τp = 5. transmission with M = 100, K = 40 and τp = 5.

1 1 = 20 p

0.96 0.8 0.8 0.94 = 20 p 0.92 0.9 = 5 0.6 0.6 0.88 p

1.5 1.6 1.7

0.4 0.4 = 5 p

0.2 MMSE 0.2 LMMSE MMSE Cumulative Distribution Function (CDF) Cumulative Distribution Function (CDF) LS LMMSE/LS 0 0 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 SE per UE [bit/s/Hz] SE per UE [bit/s/Hz] Fig. 9: CDF of DL SE with coherent transmission for pilot Fig. 10: CDF of DL SE with non-coherent transmission for lengths where M = 100, K = 40, τp = [5, 20]. pilot lengths where M = 100, K = 40, τp = [5, 20].

≤ ≤ dl their pilots that give least interference to UEs in the current is the power fraction parameter (0 ηm,k 1) and ρ = K × 200 mW. pilot set. In UL transmission, we set τu = τc − τp and τd = τc − τp in DL transmission, which means that each coherence Fig. 3 shows the average UL SE as function of the number block is either used for only UL data or only DL data. of APs with the phase-aware MMSE, LMMSE, and LS estimators. The average is taken over different UE locations and In the UL, all UEs transmit with the same power 200 mW. shadow fading realizations. The “” markers are generated by In the two-layer decoding scheme, we utilized the maximizing Monte Carlo simulations. The fact that the markers overlap LSFD vectors for each estimator that is computed by only with the curves validates of our analytical results. Fig. 4 using large-scale fading coefﬁcients. In single-layer decoding, compares the cumulative distribution functions (CDFs) of the the LSFD vectors are a = [1,..., 1], k which simply corre- k UL SE of estimators with two-layer and single-layer decoding. spond to conventional MR combining.∀ The same DL power is The randomness is due to random UE locations and shadow assigned to UE k for both coherent and non-coherent transmis- fading realizations. From Fig. 3 and Fig. 4, we observe sion. The power is allocated proportional to the channel quality that two-layer decoding gives improvements in all the cases. of UE by using the matrices for the coherent and non-coherent dl Especially, the LS estimator beneﬁts the most from using ρ η1, k ηM, k case respectively as Dk = diag 2 ,..., 2 LSFD since it is more vulnerable to interference. Also, the K E{ |hˆ1,k | } E{ |hˆ M,k | } dl β +h¯2 LS and LMMSE estimators coincide if the maximizing LSFD ρ m, k m, k and Dk = K diag η1,k, . . ., ηM,k where ηm,k = ÍK ¯2 vector is used as discussed in Section IV-C. l=1 βm,l +hm,l 11

each UE gets higher SEs. 1 Fig. 7 shows the average DL SE as a function of the number of APs with coherent and non-coherent transmission 0.8 for the phase-aware MMSE, LMMSE, and LS estimators. The K = 40 average is taken over different UE locations and shadow fading 0.6 realizations. Fig. 8 shows the CDF of DL SE with phase- aware MMSE, LMMSE, and LS estimators for coherent and non-coherent transmission. These ﬁgures demonstrate that the 0.4 coherent transmission provides higher SE in almost all cases except when using the LS estimator. Also, the performance gap K = 10 0.2 MMSE between the phase-aware MMSE and LMMSE/LS estimator is LMMSE higher in the coherent case since it is sensitive to phase errors.

Cumulative Distribution Function (CDF) LS 0 Fig. 9 shows the CDF of DL SE with phase-aware MMSE, 0 1 2 3 4 5 6 7 SE per UE [bit/s/Hz] LMMSE and LS estimators for different pilot lengths with coherent transmission. Similar to Fig. 5, the LMMSE and Fig. 11: CDF of DL SE with coherent transmission for LS estimators beneﬁt from increased pilot length where the M K [ , ] τ different number of UEs where = 100, = 10 40 , p = 5. MMSE estimator does not. Also, we observe that the performance gap due to the lack of phase knowledge between phase- aware MMSE and LMMSE estimators are 42.6% and 13.4% 1 for τp = 5 and τp = 20 respectively. In Fig. 10, we consider the same setting as in Fig. 9 but with non-coherent transmission. 0.8 The effect of phase knowledge is small in the non-coherent K = 40 transmission as expected. Quantitatively, it is 10.9% for τp = 5 and 2.4% for τp = 20. 0.6 Fig. 11 and Fig. 12 show the CDF of DL SE with phase-aware MMSE, LMMSE and LS estimators for different K = 10 0.4 number of UEs with coherent transmission and non-coherent transmission respectively. In both ﬁgures, the performance is better when the number of UEs in the network is lower in all 0.2 cases. MMSE Cumulative Distribution Function (CDF) LMMSE/LS 0 0 0.5 1 1.5 2 2.5 3 3.5 VIII.CONCLUSION SE per UE [bit/s/Hz] This paper studied the SE of a cell-free massive MIMO Fig. 12: CDF of DL SE with non-coherent transmission for system over Rician fading channels. The phase of the LoS different number of UEs where M = 100, K = [10, 40], τp = 5. path is modeled as a uniformly distributed random variable to take phase-shifts due to mobility and phase noise into account. To determine the importance of knowing the phase, Fig. 5 shows the CDFs of the SEs with different estimators the phase-aware MMSE, non-aware LMMSE, and LS estima- with two-layer decoding for different pilot lengths. In case of tors were derived. In the UL, a two-layer decoding method LMMSE and LS estimation, increasing the pilot length reduces was studied in order to mitigate the both coherent and non- the pilot contamination and improves the estimation quality. coherent interference. We observed that the LSFD method Thus, the CDF curve of LMMSE/LS estimators gets closer provides a substantial gain in UL SE for all estimators, and to the phase-aware MMSE estimator with τp = 20. However, should therefore always be used in cell-free massive MIMO. the MMSE estimator can already suppress the interference and Furthermore, the performance losses as a result of unavailable therefore increasing the pilot length reduces the pre-log factor phase knowledge depends on the pilot length. If there is no in the SE expression more than the SINR is improved, thereby strongly interfering user, the LoS and NLoS paths can be reducing the performance slightly. Moreover, the gaps between jointly estimated without having to know the phase. In the phase-aware MMSE and LMMSE estimators for different pilot tested scenarios, we observed 6.9% performance loss for low lengths indicate that the amount of performance loss due pilot contamination and 24.8% for high pilot contamination. to lack of phase knowledge depends on the degree of pilot In the DL part, coherent and non-coherent transmission contamination. The gap becomes larger when the pilot length were studied. We noticed that the losses from the lack of decrease. The loss is 6.9% when τp = 20 is used and 24.8% phase knowledge are 13.4% and 2.4% for coherent and non- when τp = 5. coherent transmission respectively in the tested scenarios with Fig. 6 compares the CDF of the SEs with different estima- low pilot contamination. When we reduced the pilot length, it tors with two-layer decoding for different number of UEs. As loss increased up to 42.6%. Therefore, the pilot length should the number of UEs decrease while the pilot length is kept the be adjusted by taking the phase shifts into account in high same, the interference and pilot contamination decrease and mobility or low-quality hardware scenarios. In order to deal 12

with the cases where there is not enough pilots, methods to APPENDIX C explicitly estimate the phases could be considered in future PROOFOF UL AND DLSE WITH MMSE ESTIMATOR work. Besides, the coherent transmission performs much better The expectations in (35) and (74) are calculated here. than the non-coherent one, which is the reason of its wide use n o n o ∗ (ˆ mmse)∗ ˆ mmse in the cell-free literature. We begin with E vm,k hm,k = E hm,k hm,k = 2 −1 ¯ 2 pˆk τp βm,k λm,k + hm,k which gives APPENDIX A M DERIVATION OF THE LMMSEESTIMATOR Õ n o E vH AH h = α∗ E v∗ h (94) The estimation is performed based on the received pilot k k k m,k m,k m,k m=1 signal in (3) where hm,k is the desired part. The LMMSE M estimator and the mean-square error are respectively given in Õ ∗ 2 −1 ¯ 2 = αm,k pˆk τp βm,k λm,k + hm,k = tr pˆk τpAk Ωk + Ak Lk . [22, Chapter 12] as m=1 n p ∗o E hm,k (y ) Similarly, we compute the second term as ˆ lmmse m,k p hm,k = n o ym,k, (86) | p |2 E kA v k2 = (95) E ym,k k k M n o 2 Õ n o E h (yp )∗ | ∗ |2 | ∗ |2 Ω H H m,k m,k αm,k E vm,k = tr pˆk τpAk k Ak + Ak Lk Ak . 0 | |2 − cm,k = E hm,k n o . (87) m=1 E |yp |2 m,k The last expectation in the denominator of (35) is written Computing the following expectations and substituting them as into (86) gives the LMMSE estimator and the corresponding  M 2 MSE as given in Section III-B: n 2o  Õ  E H H E  ∗ ∗  vk Ak hl = αm,k vm,k hm,l (96) n p ∗o p p ∗ E h (y ) = pˆ τ E |h |2 + E h (φH n )  m=1  m,k m,k k p m,k m,k k m   n o M M ∗ Õ p ∗ p 2 Õ Õ ∗ n ∗ ∗ o + pˆlτpE hm,k h = pˆk τp(βm,k + h¯ ), (88) m,l m,k = αm,k αn,k E vm,k hm,l vn,k hn,l . l∈Pk \{k } m=1 n=1 n p o Õ p | |2 2 | |2 | H |2 ∗ E ym,k = pˆlτpE hm,l + E φk nm n ∗ ∗ o The E v hm,l v hn,l is computed for all possible l∈Pk m,k n,k Õ AP and UE combinations. We utilize the independence of = pˆ τ2(β + h¯ 2 ) + σ2 τ = τ λ0 (89) l p m,l m,k ul p p m,k channel estimates at different APs. The ﬁrst case is m , n l∈P n ∗ o k and l P and E v∗ h v∗ h = 0 since v∗ and n o < k m,k m,l n,k n,l n,k ˆ lmmse The mean and variance of the estimator are E hm,k = hn,l are independent and both have zero mean. For m , n and √ β0 m, k n p o n lmmseo n lmmse 2o l ∈ Pk \{k}, we obtain pˆk 0 E y = 0 and Var hˆ = E khˆ k = λm, k m,k m,k m,k ( 0 )2 n ∗ ∗ ∗ o n mmse ∗ mmseo n mmse ∗ mmseo βm, k n p o 0 0 − ( ) ( ) (ˆ ) ˆ (ˆ ) ˆ E | |2 ( )2( ) 1 E vm,k hm,l vn,k hn,l = E hm,k hm,l E hn,k hn,l , pˆk (λ0 )2 ym,k = pˆk τp βm,k λm,k . m, k (97) APPENDIX B where DERIVATION OF THE LSESTIMATOR n o n ∗ (ˆ mmse)∗ ˆ mmse p −1 ( p − p ) ¯ jϕm, k The mean and variance of the LS estimator can be computed E hm,k hm,l = E pˆk βm,k λm,k ym,k y¯m,k + hm,k e as p − p p o × pˆ β λ 1 (y − y¯ ) + h¯ ejϕm,l n o 1 n o l m,l m,k m,k m,k m,l E hˆ ls E yp , m,k = √ m,k = 0 (90) p −1 pˆk τp = pˆk pˆlτp βm,l βm,k λm,k, (98) n o 1 n o 1 Var hˆ ls = E |yp |2 = λ0 . (91) n √ m,k 2 m,k m,k E h¯ h¯ e−jϕm,k ejϕm,l E ( p β λ−1 pˆk τp pˆk τp since m,k m,l = 0 and ˆk m,k m,k p p o ( − ))∗ ¯ jϕm,l The estimation error mean and variance are ym,k y¯m,k hm,le = 0. Repeating the same calcula- ( ) n o 1 tion for AP n and putting it into (97) gives E h˜ ls = E h − √ yp = 0, (92) m,k m,k m,k n o pˆk τp ∗ ∗ ∗ 2 −1 −1 E (vm,k hm,l) (vn,k hn,l) = pˆk pˆlτp βm,l βm,k λm,k βn,l βn,k λn,k . n o n o 1 n o Var h˜ ls = E h 2 + E |yp |2 (93) (99) m,k m,k 2 m,k pˆk τp 1 n o 1 n o Another case is m , n and l = k and we obtain − E h (yp )∗ − E (h )∗ yp √ m,k m,k √ m,k m,k n ∗ o pˆk τp pˆk τp ∗ ∗ 2 2 2 −1 2 −1 E vm,k hm,k vn,k hn,k = pˆk τp βm,k λm,k βn,k λn,k λ0 λ0 ¯ 2 m,k ¯ 2 m,k 0 ¯ 2 ¯ 2 2 −1 ¯ 2 2 −1 ¯ 2 = βm,k + hm,k + − 2(βm,k + hm,k ) = − βm,k . + hm,k hn,k + pˆk τp βm,k λm,k hn,k + pˆk τp βn,k λn,k hm,k . (100) pˆk τp pˆk τp 13

− / ˆ p 1 2 ¯ jϕm,l Similarly for m = n and l = k, we calculate the following and similarly hm,l = pˆlτp βm,lλm,k w + hm,le where equations ( p − p ) ∼ N ∼ N ( ) ym,k y¯m,k C 0, τpλm,k and w C 0, 1 . After this computation, we directly compute n ∗ 2o 2 −1 2 E |hˆ h˜ m,k | = pˆk τp β λ + h¯ cm,k, (101) 2 n − / m,k m,k m,k m,k E hˆ ∗ hˆ = E ppˆ τ β λ 1 2w∗ + h¯ e−jϕm, k m,k m,l k p m,k m,k m,k n − / E |hˆ |4 = E ppˆ τ β λ 1 2w∗ + h¯ e−jϕm, k m,k k p m,k m,k m,k 2 p −1/2 jϕm,l × pˆlτp βm,lλ w + h¯ m,le − / 2 m,k p 1 2 ¯ jϕm, k 2 2 4 −2 4 × pˆk τp βm,k λ w + hm,k e = pˆ τp β λ E |w| m,k k m,k m,k 2 2 −1 E | |4 2 −1 ¯ 2 2 ¯ 4 2 2 4 −2 = pˆk pˆlτp βm,l βm,k λm,k w + 4p ˆk τp βm,k λm,k hm,k E |w| + hm,k = 2p ˆk τp βm,k λm,k 2 −1 ¯ 2 2 2 −1 ¯ 2 2 2 −1 ¯ 2 ¯ 2 ¯ 4 + pˆk τp βm,k λm,k hm,lE |w| + pˆlτp βm,lλm,k hm,k E |w| + 3p ˆk τp βm,k λm,k hm,k + (βm,k − cm,k )hm,k + hm,k . (102) ¯ 2 ¯ 2 2 2 2 −2 + hm,l hm,k = 2p ˆk pˆlτp βm,l βm,k λm,k where w ∼ N (0, 1). Combining (101) and (102) gives the 2 −1 ¯ 2 2 −1 ¯ 2 ¯ 2 ¯ 2 C + pˆk τp βm,k λm,k hm,l + pˆlτp βm,lλm,k hm,k + hm,l hm,k result for m = n and l = k as 2 −1 2 −1 ¯ 2 = 2p ˆk τp βm,k λm,k (βm,l − cm,l) + pˆk τp βm,k λm,k hm,l n o n o ¯ 2 ¯ 2 ¯ 2 ˆ ∗ 2 ˆ 4 ˆ ∗ ˜ 2 3 −1 + (βm,l − cm,l)hm,k + hm,l hm,k . (108) E |hm,k hm,k | = E |hm,k | + E |hm,k hm,k | = pˆk τp βm,k λm,k ¯ 2 2 −1 ¯ 2 ¯ 4 2 2 4 −2 Then, it yields + βm,k hm,k + 3p ˆk τp βm,k λm,k hm,k + hm,k + pˆk τp βm,k λm,k . n ∗ 2o 2 −1 ¯ 2 ¯ 2 ¯ 2 (103) E |vm,k hm,l | = pˆk τp βm,l βm,k λm,k + βm,l hm,k + hm,k hm,l ( pˆ pˆ τ2 β2 β2 λ−2 , l ∈ P \{k} 2 −1 ¯ 2 k l p m,l m,k m,k k Putting all the equations together for l = k, we obtain + pˆk τp βm,k λm,k hm,l + 0, l < Pk . (109) M M ∗ n H H 2o Õ Õ ∗ n ∗ ∗ o E vk Ak hk = αm,k αn,k E vm,k hm,k vn,k hn,k Finally, arranging all these equations gives m=1 n=1 n H H 2o H H M M E vk Ak hl = pˆk τptr Ak RlAk Ωk + pˆk τptr Ak Ωk Ak Ll Õ Õ ∗ 2 2 2 −1 2 −1 ¯ 2 ¯ 2 = αm,k αn,k pˆk τp βm,k λm,k βn,k λn,k + hm,k hn,k H H m=1 n=1 + tr Ak RlAk Lk + tr Ak Sk,lAk (110) 2 −1 ¯ 2 2 −1 ¯ 2 2 2 + pˆk τp β λ h + pˆk τp β λ h pˆk pˆlτ |tr (Ak RlΛk Rk ) | , l ∈ Pk \{k} m,k m,k n,k n,k n,k m,k  p  2 2 2 2 M pˆk τp |tr (Ak Ωk ) | + tr (Ak Lk ) Õ 2 3 −1 2 2 −1 2 + + |αm,k | pˆk τp β λ + βm,k h¯ + pˆk τp β λ h¯ +2p ˆ τ tr (A Ω ) tr (A L ), l = k m,k m,k m,k m,k m,k m,k  k p k k k k m=1  0, l < Pk . 2 2 4 −2 H 2  +pˆ τp β λ = pˆk τptr A Rk Ak Ωk + tr (Ak Lk ) k m,k m,k k where Sk,l = Lk Ll if l , k and zero matrix otherwise. H 2 2 2 H Inserting the required results into (35) and (74) gives the + tr A Rk Ak Lk + pˆ τ |tr (Ak Ωk ) | + pˆk τptr A Ωk Ak Lk k k p k results for UL and DL SEs respectively. + 2p ˆk τptr (Ak Ωk ) tr (Ak Lk ) . (104) APPENDIX D PROOFOF UL AND DLSE WITH LMMSEESTIMATOR The remaining cases are computed as follows. For l < Pk n o and m = n, E |v∗ h |2 = (pˆ τ β2 λ−1 + h¯ 2 )(β + In this part, we note that the LMMSE estimate is not m,k m,l k p m,k m,k m,k m,l Gaussian distributed. We begin with computing the ﬁrst term 2 h¯ ). For l ∈ Pk \{k} and m = n, we obtain m,l n ∗ o n ˆ lmmse ∗ ˆ lmmse ˜ lmmse o E vm,k hm,k = E (hm,k ) (hm,k + hm,k ) 2 2 2 n o n ∗ o E ∗ E ˆ ∗ ˆ E ˆ ∗ ˜ = E |hˆ lmmse|2 + E (hˆ lmmse) h˜ lmmse vm,k hm,l = hm,k hm,l + hm,k hm,l , (105) m,k m,k m,k (a) n o n o 2 = pˆ τ (β0 )2(λ0 )−1 + E (hˆ lmmse)∗ E h˜ lmmse E ˆ ∗ ˜ 2 −1 ¯ 2 k p m,k m,k m,k m,k hm,k hm,l = pˆk τp βm,k λm,k + hm,k cm,l . (106) (b) 0 2 0 −1 = pˆk τp(βm,k ) (λm,k ) (111) 2 where (a) uses the fact that the estimate and estimation error To calculate E hˆ ∗ hˆ , we rewrite the MMSE estimator m,k m,l are uncorrelated and (b) uses that both estimates have zero in (5) for UEs l ∈ Pk \{k} as mean. Calculating (111) for each AP m gives M Õ n o ˆ p −1 ( p − p ) ¯ jϕm, k H H ∗ ∗ hm,k = pˆk βm,k λm,k ym,k y¯m,k + hm,k e E vk Ak hk = αm,k E vm,k hm,k m=1 p −1 √ 1 1/2 −1/2 p p jϕ ( )( − ) ¯ m, k M = pˆk βm,k λm,k τp √ λm,k λm,k ym,k y¯m,k + hm,k e τp Õ ∗ 0 2 0 −1 0 = α pˆk τp(β ) (λ ) = pˆk τptr Ak Ω . (112) − / m,k m,k m,k k p 1 2 ¯ jϕm, k = pˆk τp βm,k λm,k w + hm,k e (107) m=1 14

Similarly, we obtain case is m = n and l ∈ Pk . In this case the channel estimators hˆ lmmse hˆ lmmse 2 ( m),k and m,l are no longer independent. We obtain  M  M M 2  Õ  Õ ∗ ∗ Õ E kAk vk k = E αm,k vm,k = E αm,k vm,k αn,k vn,k ∗ 2 n ∗ o p 0 0 − p  m=1  m=1 n=1 E |(hˆ lmmse) hlmmse|2 = E pˆ β (λ ) 1 y h   m,k m,l k m,k m,k m,k m,l M ( ) n o a Õ 2 ˆ lmmse 2 0 H 0 2 0 −2 2 n 4o n H p ∗ 2o = |αm,k | E |hm,k | = pˆk τptr Ak Ωk Ak , (113) = pˆk (βm,k ) (λm,k ) pˆlτpE hm,l + E (φk nm) hm,l m=1 2   where (a) utilizes the independence of the zero mean channel  Õ p ∗ ª 0 2 0 −2 +E pˆz τp h hm,l ® = pˆk (β ) (λ ) estimates at different APs. m,z ® m,k m,k  z ∈Pk \{l}  The last expectation in the denominator of (35) is written  ¬ as h 2 2 ¯ 2 0 0 i × pˆlτp βm,l + 2hm,l βm,l + τpλm,k βm,l M M n 2o Õ Õ n ∗ o 0 0 2 0 −1 2 0 2 0 −2 2 H H ∗ ∗ ∗ = pˆk τp β (β ) (λ ) + pˆk pˆlτ (β ) (λ ) β E vk Ak hl = αm,k αn,k E vm,k hm,l vn,k hn,l . m,l m,k m,k p m,k m,k m,l 2 0 2 0 −2 2 m=1 n=1 + 2p ˆk pˆlτ (β ) (λ ) h¯ βm,l (118) (114) p m,k m,k m,l ∗ n 4o n 4o n ∗ ∗ o where E h = E g + h¯ ejϕm,l = Then, we need to compute E v hm,l v hn,l for all m,l m,l m,l m,k n,k possible AP and UE combinations. If m n and l P then 2 ¯ 2 ¯ 4 , < k 2βm,l + 4hm,l βm,l + hm,l . Combining all the cases gives all the terms in the expectation are uncorrelated or independent ∗ (119), at the top of next page. Using matrix notation, we can n ∗ ∗ o and have zero mean which gives E vm,k hm,l vn,k hn,l = reformulate the equation above as 0. However, for m n and l ∈ P , we have ∗ , k n ∗ ∗ o n ∗ o n ∗ o E v hm,l v hn,l = E vm,k h E v hn,l , since n 2o m,k n,k m,l n,k E vH AH h p τ AH R0A Ω0 the pilot contaminating estimators at the same AP are not k k l = ˆk ptr k l k k (120) independent. We calculate tr AH R2Λ0 Ω0 A + 2tr AH Λ0 Ω0 L R A  k l k k k k k k l l k n lmmse ∗ o n lmmse lmmse lmmse ∗o  2 E hˆ h = E hˆ (hˆ + h˜ ) (115) 2  0 0 0 0 0 0 m,k m,l m,k m,l m,l + pˆk pˆlτp tr A R Λ R − tr AH (R Λ R )2A l ∈ P k l k k k l k k k k n o  = E hˆ lmmse(hˆ lmmse)∗ 0 l P m,k m,l  < k n ∗o p 0 ( 0 )−1 p p 0 ( 0 )−1 p = E pˆk βm,k λm,k ym,k pˆl βm,l λm,k ym,k and this finishes the proof. 0 0 n o β β p 0 0 ( 0 )−2 | p |2 p m,k m,l = pˆk pˆl βm,k βm,l λm,k E ym,k = pˆk pˆlτp 0 . λm,k APPENDIX E PROOFOF UL AND DLSE WITHTHE LSESTIMATOR n ∗ o Repeating the same process for E v hn,l gives the final n,k n ∗ o ∗ The first two terms to calculate are E v hm,k = n ∗ ∗ o m,k result for m n and l ∈ Pk as E v hm,l v hn,l = n o n o , m,k n,k ( p )∗ | p |2 0 E ym, k hm,k E ym, k λ 2 0 0 0 −1 0 0 0 −1 √ 0 2 m,k pˆk pˆlτ β β (λ ) β β (λ ) . Another case is m = = β and E |vm,k | = 2 = as p m,k m,l m,k n,k n,l n,k pˆk τp m,k pˆk τp pˆk τp lmmse n and l < Pk in which the channel estimators hˆ and given in Appendix B. Moreover, we calculate n m,k o hˆ lmmse are independent. We directly compute E |v∗ h |2 = m,l m,k m,l E vH AH h n o n o k k k E |(hˆ lmmse)∗hˆ lmmse|2 + E |(hˆ lmmse)∗h˜ lmmse|2 using uncorre- M M m,k m,l m,k m,l Õ ∗ n ∗ o Õ ∗ 0 0 lated zero mean estimators and estimation errors. The first and = αm,k E vm,k hm,k = αm,k βm,k = tr Ak Rk . second terms are derived respectively as m=1 m=1 (121) n lmmse ∗ lmmse 2o E |(hˆ ) hˆ | −1 m,k m,l M H 0 tr A Λk Ak 2 Õ ∗ 2 n ∗ 2o k ∗ 2 E kAk vk k = |α | E |v | = . = E ppˆ β0 (λ0 )−1 yp ppˆ β0 (λ0 )−1 yp m,k m,k pˆ τ k m,k m,k m,k l m,l m,l m,l m=1 k p n o n o (122) ( 0 )2( 0 )2( 0 )−2( 0 )−2 | p |2 | p |2 = pˆk pˆl βm,k βm,l λm,k λm,l E ym,k E ym,l n H H 2o 2 0 2 0 2 0 −1 0 −1 The third term is E v A hl and can be expanded as = pˆk pˆlτp(βm,k ) (βm,l) (λm,k ) (λm,l) , (116) k k n ˆ lmmse ∗ ˜ lmmse 2o n ˆ lmmse 2o n ˜ lmmse 2o E |(hm,k ) hm,l | = E |hm,k | E |hm,l | 2  M  0 0 − 0  Õ ∗ ∗  2 1 E α v hm,l = pˆk τp(βm,k ) (λm,k ) cm,l . (117) m,k m,k  m=1  Rewriting the first term and combining (116) and (117)   n o M M ∗ ∗ 2 0 2 0 −1 0 0 Õ Õ ∗ n ∗ ∗ o leads to E |v hm,l | = pˆk τp(β ) (λ ) (β − c ) + m,k m,k m,k m,l m,l = αm,k αn,k E vm,k hm,l vn,k hn,l . (123) 0 2 0 −1 0 0 0 2 0 −1 m=1 n=1 pˆk τp(βm,k ) (λm,k ) cm,l = pˆk τp βm,l(βm,k ) (λm,k ) . The last 15

M M M n 2o Õ 2 Õ Õ n ∗ o E H H | |2E ∗ ∗ E ∗ ∗ vk Ak hl = αm,k vm,k hm,l + αm,k αn,k vm,k hm,l vn,k hn,l (119) m=1 m=1 n=1 m,n M Õ 2 2 0 2 0 −2 ¯ 2 2 2 0 2 0 −2 2  |αm,k | 2p ˆk pˆlτp(βm,k ) (λm,k ) hm,l βm,l + |αm,k | pˆk pˆlτp(βm,k ) (λm,k ) βm,l  M 0 0 2 m=1 β (β )  M M Õ 2 m,l m,k  Õ Õ = pˆk τp |αm,k | 0 + ∗ 2 0 0 0 −1 0 0 0 −1 λ + αm,k αn,k pˆk pˆlτp βm,k βm,l(λm,k ) βn,k βn,l(λn,k ) , l ∈ Pk m=1 m,k   m=1 n=1   m,n 0, l P .  < k

− M K 2 m 1 2 Õ Õ ∗ Õ n ∗ o 2 © ρn,lE h wn,l − ρn,k E h wn,k + σ ª M M ! M n,k n,k dl ® Õ dl τd Ö dl τd Ö n=1 l=1 n=1 ® SE = log 1 + γ = log ®, (132) n,k τ 2 n,k τ 2 M K m ® n c n c n Õ Õ n o Õ n o 2 =1 =1 =1 E | ∗ |2 − E ∗ 2 ® ρn,l hn,k wn,l ρn,k hn,k wn,k + σdl ® « n=1 l=1 n=1 ¬ M K Õ Õ 2 © ρ E h∗ w + σ2 ª n,l n,k n,l dl ® τ n SEdl = d log =1 l=1 ® . (133) k 2 M K M ® τc Õ Õ n o Õ n o 2 ® E | ∗ |2 − E ∗ 2 ® ρn,l hn,k wn,l ρn,k hn,k wn,k + σdl ® « n=1 l=1 n=1 ¬

Begin with m = n and l ∈ Pk and similar to (118), we obtain In matrix form, we obtain

2 1 2 E v∗ h = E yp h m,k m,l 2 m,k m,l n H H 2o 1 H 0 −1 0 pˆk τp E v A hl = tr A (Λ ) R Ak + k k pˆ τ k k l 1 h i k p = pˆ τ2 β2 + 2h¯ 2 β + τ λ0 β0 . (124) 2 l p m,l m,l m,l p m,k m,l tr AH (R2 + 2L R )A pˆk τp  k l l l k pˆl  2 H 0 H 0 2 (127) p +tr A R − tr A (R ) Ak , l ∈ Pk For m = n and l P , the y and h are independent pˆk k l k l < k m,k m,l  random variables. We can compute directly 0, l P .  < k 2 1 2 E v∗ h = E (yp )∗h m,k m,l 2 m,k m,l pˆk τp 1 2 n o 1 = E yp E h 2 = λ0 β0 . (125) APPENDIX F 2 m,k m,l m,k m,l pˆk τp pˆk τp PROOFOF DLSE WITH NON-COHERENT TRANSMISSION

∗ n ∗ ∗ o For m , n and l ∈ Pk , we have E v hm,l v hn,l = m,k n,k At the beginning of the detection process, UE k does not 1 n p ∗ o n p ∗ o pˆl 0 0 2 E (y ) hm,l E (y ) hn,l = β β and finally pˆk τp m,k n,k pˆk m,l n,l know any of the transmitted signals. It first detects the signal n ∗ o n ∗ o ∗ ∗ from AP 1 by using the average channel E h w1,k only. for m , n and l < Pk E vm,k hm,l vn,k hn,l = 0 since all 1,k the terms are mutually independent and have zero mean. The The received signal can be written as final form is n o n o M 2 M 0 0 ydl = ydl = E h∗ w ς + h∗ w − E h∗ w ς  Õ  Õ λ β 1,k k 1,k 1,k 1,k 1,k 1,k 1,k 1,k 1,k E  α∗ v∗ h  = |α |2 m,k m,l m,k m,k m,l m,k M K M   pˆk τp Õ Õ Õ  m=1  m=1 + h∗ w ς + h∗ w ς + ndl. (128)   n,k n,k n,k n,k n,l n,l k ÍM | |2 2 ¯ 2 n=2 l=1 n=1  m=1 αm,k βm,l + 2hm,l βm,l  l,k  M M pˆl  Õ Õ ∗ 0 0 + + αm,k αn,k βm,l βn,l, l ∈ Pk (126) pˆk  m=1 n=1 where the first term is the desired signal over known deter-   m,n ministic channel while other terms are treated as uncorrelated 0, l P .  < k noise. Sequentially, UE k detects signal from AP m by 16

subtracting the first m − 1 signals: [7] A. Ashikhmin and T. Marzetta, “Pilot contamination precoding in multicell large scale antenna systems,” in IEEE International Symposium on m−1 Information Theory Proceedings, July 2012, pp. 1137–1141. dl dl Õ n ∗ o n ∗ o y = y − E h wn,k ςn,k = E h wm,k ςm,k [8] A. Adhikary, A. Ashikhmin, and T. L. Marzetta, “Uplink interference m,k k n,k m,k reduction in large-scale antenna systems,” IEEE Transactions on Com- n=1 munications, vol. 65, no. 5, pp. 2194–2206, May 2017. K M [9] T. Van Chien, C. Mollén, and E. BjÃ˝urnson,“Large-scale-fading de- ∗ n ∗ o Õ Õ ∗ dl + hm,k wm,k − E hm,k wm,k ςm,k + hn,k wn,l ςn,l + nk coding in cellular massive MIMO systems with spatially correlated l=1 n=1 channels,” IEEE Transactions on Communications, pp. 1–1, December l,k 2018. m−1 M [10] T. Van Chien, C. Mollén, and E. Björnson, “Two layer decoding Õ ∗ n ∗ o Õ ∗ in cellular massive mimo systems with spatial channel correlation,” + hn,k wn,k − E hn,k wn,k ςn,k + hn,k wn,k ςn,k . available on "arxiv.org/abs/1903.07135", March 2019. n=1 n=m+1 [11] E. Nayebi, A. Ashikhmin, T. L. Marzetta, and B. D. Rao, “Performance (129) of cell-free massive MIMO systems with MMSE and LSFD receivers,” in Asilomar Conference on Signals, Systems and Computers, Nov 2016, The first term in (129) is equivalent to having a deterministic pp. 203–207. n ∗ o [12] H. Q. Ngo, H. Tataria, M. Matthaiou, S. Jin, and E. G. Larsson, “On the channel h = E hm,k wm,k and ςm,k is the desired signal. The performance of cell-free massive MIMO in Ricean fading,” in Asilomar other terms are uncorrelated noise υm,k with power Conference on Signals, Systems, and Computers, Oct 2018, pp. 980–984. [13] R. Rogalin, O. Y. Bursalioglu, H. Papadopoulos, G. Caire, A. F. Molisch, 2 n ∗ n ∗ o 2o A. Michaloliakos, V. Balan, and K. Psounis, “Scalable synchronization E |υm,k | = ρm,k E |hm,k wm,k − E hm,k wm,k | (130) and reciprocity calibration for distributed multiuser MIMO,” IEEE m−1 Transactions on Wireless Communications, vol. 13, no. 4, pp. 1815– Õ n ∗ n ∗ o 2o 2 1831, April 2014. + ρn,k E |hn,k wn,k − E hn,k wn,k | + σdl [14] S. Perlman and A. Forenza, “An introduction to pCell,” Feb. 2015, n=1 Artemis Networks LLC, White paper. M K M [15] O. Özdogan, E. Björnson, and J. Zhang, “Cell-free massive MIMO with Õ n o Õ Õ n o + ρ E |h∗ w |2 + ρ E |h∗ w |2 Rician fading: Estimation schemes and spectral efficiency,” in Asilomar n,k n,k n,k n,l n,k n,l Conference on Signals, Systems, and Computers, Oct 2018, pp. 975–979. n=m+1 l=1 n=1 [16] C. D’Andrea, A. Garcia-Rodriguez, G. Geraci, L. G. Giordano, and l,k S. Buzzi, “Cell-free massive MIMO for UAV communications,” avail- M K m Õ Õ n o Õ n o 2 able on "https://arxiv.org/pdf/1902.03578.pdf", February 2019. ∗ 2 ∗ 2 [17] E. Björnson, J. Hoydis, and L. Sanguinetti, “Massive MIMO networks: = ρn,lE |hn,k wn,l | − ρn,k E hn,k wn,k + σdl. Foundations and Trends R n=1 l=1 n=1 Spectral, energy, and hardware efficiency,” in Signal Processing, vol. 11, no. 3-4, pp. 154–655, 2017. dl |h |2 [18] L. Sanguinetti, E. Björnson, and J. Hoydis, “Towards massive MIMO The DL SINR γ = 2 is equal to 2.0: Understanding spatial correlation, interference suppression, and m,k E{ |υm, k | } pilot contamination,” CoRR, vol. abs/1904.03406, 2019. [Online]. n o 2 Available: http://arxiv.org/abs/1904.03406 ∗ [19] T. C. Mai, H. Quoc Ngo, and T. Q. Duong, “Cell-free massive mimo ρm,k E hm,k wm,k dl systems with multi-antenna users,” in 2018 IEEE Global Conference on γm,k = . M K m 2 Signal and Information Processing (GlobalSIP), Nov 2018, pp. 828–832. Õ Õ n ∗ 2o Õ n ∗ o 2 ρn,lE |h wn,l | − ρn,k E h wn,k + σ [20] A. Paulraj, R. Roy, and T. Kailath, “Estimation of signal parameters n,k n,k dl via rotational invariance techniques- esprit,” in Nineteeth Asilomar n=1 l=1 n=1 (131) Conference on Circuits, Systems and Computers, 1985., Nov 1985, pp. dl 83–89. The total SE of UE k is SEk is equal to (132), at the top of [21] R. O. Schmidt, “A signal subspace approach to multiple emitter location previous page. After cancellations of terms that appear in both and spectral estimation,” Ph.D. dissertation, 1981. the numerator and denominator, we obtain (133), at the top of [22] S. M. Kay, Fundamentals of statistical signal processing: Estimation theory. Prentice Hall, 1993. previous page and it is equal to (74). [23] E. Nayebi, A. Ashikhmin, T. L. Marzetta, and H. Yang, “Cell-free massive MIMO systems,” in 2015 49th Asilomar Conference on Signals, Systems and Computers, Nov 2015, pp. 695–699. REFERENCES [24] S. Buzzi and C. D’Andrea, “Cell-free massive MIMO: User-centric approach,” IEEE Wireless Communications Letters, vol. 6, no. 6, pp. [1] O. Özdogan, E. Björnson, and J. Zhang, “Downlink performance of cell- 706–709, December 2017. free massive MIMO with rician fading and phase shifts,” in 2019 IEEE [25] M. Bashar, K. Cumanan, A. G. Burr, H. Q. Ngo, and H. V. Poor, “Mixed 20th International Workshop on Signal Processing Advances in Wireless quality of service in cell-free massive MIMO,” IEEE Communications Communications (SPAWC), 2019. Letters, vol. 22, no. 7, pp. 1494–1497, July 2018. [2] K. T. Truong and R. W. Heath, “The viability of distributed antennas for [26] E. Björnson and L. Sanguinetti, “A new look at cell-free massive MIMO: massive MIMO systems,” in Asilomar Conference on Signals, Systems Making it practical with dynamic cooperation,” in IEEE International and Computers, Nov 2013, pp. 1318–1323. Symposium on Personal, Indoor and Mobile Radio Communications [3] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, (PIMRC), 2019. “Cell-free massive MIMO versus small cells,” IEEE Transactions on [27] 3rd Generation Partnership Project, Technical Specification Group Radio Wireless Communications, vol. 16, no. 3, pp. 1834–1850, March 2017. Access Network; Spatial channel model for Multiple Input Multiple [4] E. Nayebi, A. Ashikhmin, T. L. Marzetta, H. Yang, and B. D. Rao, Output (MIMO) simulations, Mar. 2017, 3GPP TR 25.996 V14.0.0. “Precoding and power optimization in cell-free massive MIMO systems,” IEEE Transactions on Wireless Communications, vol. 16, no. 7, pp. 4445–4459, July 2017. [5] G. Interdonato, E. Björnson, H. Q. Ngo, P. Frenger, and E. G. Larsson, “Ubiquitous cell-free massive MIMO communications,” arXiv preprint arXiv:1804.03421, April 2018. [6] E. Björnson, R. Zakhour, D. Gesbert, and B. Ottersten, “Cooperative multicell precoding: Rate region characterization and distributed strate- gies with instantaneous and statistical CSI,” IEEE Transactions on Signal Processing, vol. 58, no. 8, pp. 4298–4310, Aug 2010. 17

Özgecan Özdogan (SâA˘ Z18)´ received her B.Sc and M.Sc. degrees in Electronics and Communication Engineering from Ärzmirˇ Institute of Technology, Turkey in 2015 and 2017 respectively. She is currently pursuing the Ph.D. degree in communication systems at Linköping University, Sweden.

Emil Björnson Emil Björnson (S’07-M’12-SM’17) received the M.S. degree in engineering mathematics from Lund University, Sweden, in 2007, and the Ph.D. degree in telecommunications from the KTH Royal Institute of Technology, Sweden, in 2011. From 2012 to 2014, he held a joint post-doctoral position at the Alcatel-Lucent Chair on Flexible Radio, SUPELEC, France, and the KTH Royal Insti- tute of Technology. He joined Linköping University, Sweden, in 2014, where he is currently an Associate Professor and a Docent with the Division of Com- munication Systems. He has authored the textbooks Optimal Resource Allocation in Coordinated Multi-Cell Systems (2013) and Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency (2017). He is dedicated to reproducible research and has made a large amount of simulation code publicly available. He performs research on MIMO communications, radio resource allocation, machine learn- ing for communications, and energy efficiency. Since 2017, he has been on the Editorial Board of the IEEE TRANSACTIONS ON COMMUNICATIONS and the IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING since 2016. He has performed MIMO research for over ten years and has filed more than ten MIMO related patent applications. He has received the 2014 Outstanding Young Researcher Award from IEEE ComSoc EMEA, the 2015 Ingvar Carlsson Award, the 2016 Best Ph.D. Award from EURASIP, the 2018 IEEE Marconi Prize Paper Award in Wireless Communications, the 2019 EURASIP Early Career Award, and the 2019 IEEE Communications Society Fred W. Ellersick Prize. He also co-authored papers that received Best Paper Awards at the conferences, including WCSP 2009, the IEEE CAMSAP 2011, the IEEE WCNC 2014, the IEEE ICC 2015, WCSP 2017, and the IEEE SAM 2014.

Jiayi Zhang (S’08–M’14) received the B.Sc. and Ph.D. degree of Communication Engineering from Beijing Jiaotong University, China in 2007 and 2014, respectively. Since 2016, he has been a Professor with School of Electronic and Information Engineer- ing, Beijing Jiaotong University, China. From 2014 to 2016, he was a Postdoctoral Research Associate with the Department of Electronic Engineering, Ts- inghua University, China. From 2014 to 2015, he was also a Humboldt Research Fellow in Institute for Digital Communications, University of Erlangen- Nuermberg, Germany. From 2012 to 2013, he was a visiting Ph.D. student at the Wireless Group, University of Southampton, United Kingdom. His current research interests include massive MIMO, cell-free massive MIMO, and performance analysis of generalized fading channels. He was recognized as an exemplary reviewer of the IEEE COMMUNICA- TIONS LETTERS in 2015 and 2016. He was also recognized as an exemplary reviewer of the IEEE TRANSACTIONS ON COMMUNICATIONS in 2017 and 2018. He is the leading guest editor of IEEE JOURNAL Of SELECTED AREA In COMMUNICATIONS, and serves as an Associate Editor for IEEE TRANSACTIONS ON COMMUNICATIONS, IEEE COMMUNICATIONS LETTERS and IEEE ACCESS. He received the WCSP and IEEE APCC Best Paper Award in 2017.