Channel estimation is crucial for massive massive multiple-input multiple-output (MIMO) systems to scale up multi-user (MU) MIMO, providing great improvement in spectral and energy efficiency. This paper presents a simple and practical channel estimator for multi-cell MU massive MIMO time division duplex (TDD) systems with pilot contamination in flat Rayleigh fading channels, i.e., the gains of the channels follow the Rayleigh distribution. We also assume uncorrelated antennas. The proposed estimator addresses performance under moderate to strong pilot contamination without previous knowledge of the cross-cell large-scale channel coefficients. This estimator performs asymptotically as well as the minimum mean square error (MMSE) estimator with respect to the number of antennas. An approximate analytical mean square error (MSE) expression is also derived for the proposed estimator. Keywords: Massive MU-MIMO, Channel estimation, Flat fading, Pilot contamination, Maximum likelihood 1 Introduction consequence, the pilot sequences have to be reused in Massive multiple-input multiple-output (MIMO) antenna neighbor cells of cellular systems. Therefore, channel esti- systems potentially allow base stations (BSs) to operate mates obtained in a given cell get contaminated by the with huge improvements in spectral and radiated energy pilots transmitted by the users in other cells [2]. This efficiency, using relatively low-complexity linear process- coherent interference is known in the literature as pilot ing. The higher spectral efficiency is attained by serving contamination, i.e., the channel estimate at the base sta- several terminals in the same time-frequency resource tion in one cell becomes contaminated by the pilots of through spatial multiplexing, and the increase in energy the users from other cells [3]. The contamination not only efficiency is mostly due to the array gain provided by the reduces the quality of the channel estimates, i.e., increases large set of antennas [1]. the MSE, but also makes the channel estimates statistically The expected massive MIMO improvements assume dependent, even though the true channels are statisti- that accurate channel estimations are available at both cally independent. Moreover, pilot contamination does the receiver and transmitter for detection and precod- not disappear with the addition of more antennas [4]. ing, respectively. Additionally, the reuse of frequencies Massive MIMO systems operating in TDD assume and pilot reference sequences in cellular communica- channel reciprocity between uplink and downlink in order tion systems causes interferences in channel estimation, to minimize pilot overhead, transmitting pilot reference degrading its performance. Since both the time-frequency signals only in the uplink. In this scenario, pilot over- resources allocated for pilot transmission and the chan- head cost is proportional to the number of terminals nel coherence time are limited, the number of possi- and improved estimation quality can be achieved due to ble orthogonal pilot sequences is also limited, and as a the large number of antennas [5, 6]. Base stations esti- mate channels usually based on least squares (LS) [3]or minimum mean square error (MMSE) [7–9]methods. *Correspondence: felipe.pereira@ugent.be Besides, inter and intra-cell large-scale fading coefficients Department of Information Technology, Ghent University, Technologiepark-Zwijnaarde, 15, 9052 Gent, Belgium are assumed to be perfectly known when applying the Full list of author information is available at the end of the article MMSE method in the great majority of works [5, 9–13]. © The Authors 2018, corrected publication March 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Figueiredo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:14 Page 2 of 10 In a real-world network deployment, although chang- results confirm that the performance of the proposed ing slowly, the large-scale fading coefficients must be channel estimator approaches that of the ideal MMSE estimated and updated from time to time. Additionally, estimator asymptotically with the number M of anten- the estimation error of the large-scale fading coefficients nas, i.e., M →∞. Additionally, in contrast with [21], we impacts significantly on the performance of uplink data derive an approximate analytical MSE expression for the decoding and downlink transmission (e.g., precoding and proposed channel estimator that is more mathematically beamforming) [14–16]. Approaches on how to estimate tractable and not susceptible to numerical issues. the large-scale fading coefficients are presented in the following pieces of work [10, 14, 17]. 1.1 Related work The most commonly used analytical massive MIMO In this section, we survey previous work on channel esti- channel is the spatially i.i.d. frequency non-selective (flat) mation and pilot contamination mitigation. fading channel model. Flat fading channels are also known A TDD cellular system employing BSs equipped with as amplitude varying channels and narrowband channels large numbers of antennas that communicate simulta- as the signal’s bandwidth is narrow compared to channel’s neously with smaller numbers of cheap, single-antenna bandwidth [18]. In this narrowband channel model, the terminals through MU MIMO techniques is proposed in channel gain between any pair of transmit-receive anten- [3]. The author employs LS channel estimation in order nas is modeled as a complex Gaussian random variable. to study and evaluate the problems caused by pilot con- This model relies on two assumptions: (i) the antenna ele- tamination to such systems. He concludes that even when ments in the transmitter and receiver being spatially well different sets of orthogonal pilots are used in different separated once the more widely spaced (in wavelengths) cells, it makes little difference to the resulting signal- the antenna elements, the smaller the spatial channel cor- to-interference ratio (SIR). This work is the first one to relation [19, 20], and(ii) thepresenceofalargenumberof present the massive MIMO concept and identify its intrin- temporally but narrowly separated multipaths (common sic issues, however, it fails to suggest ways to mitigate the in a rich-scattering environment), whose combined gain, pilot contamination problem. by the central-limit theorem, can be approximated by a The impact of pilot contamination on multi-cell sys- Gaussian random variable [20]. tems is studied in [5]. The authors adopt MMSE channel Flat fading channels present a channel response that estimation for the analysis of pilot contamination and the exhibits flat gain and linear phase over a bandwidth achievable rates in a massive MIMO system suffering from (coherence bandwidth) that is greater than the signal’s such problem. They propose a multi-cell MMSE-based bandwidth. Therefore, all frequency components of the precoding method that mitigates the pilot contamina- signal will experience the same magnitude of fading, tion problem by considering the set of training sequences resulting in a scalar channel response. The gain applied assigned to the users in the solution of an optimization to the signal varies over time according to a fading dis- problem that minimizes the error seen by users in the tribution. In this work, we consider that the gain applied serving cell and the interference seen by the users in all to the signal passing through this channel will vary ran- other cells. Simulation results show that the proposed domly, according to a Rayleigh distribution. We addition- approach has significant gains over certain single-cell pre- ally assume that the antenna spacing is sufficiently large coding methods such as zero-forcing. In summary, the so that the antennas are uncorrelated. authors address the pilot contamination problem through In this paper, we deal with the channel estimation a precoding technique and assume that the large-scale and pilot contamination problems associated with uplink fading coefficients are known to all BSs. training in flat Rayleigh fading channels and understand MMSE channel estimation is used in [7]toderive its impact on the operation of multi-cell MU massive approximations of the achievable uplink and downlink MIMO TDD cellular systems. We propose and evaluate rates with several linear precoders and detectors for real- an efficient and practical channel estimator that does not istic system dimensions, i.e., systems where the number of require previous knowledge of inter/intra-cell large-scale antennas is not extremely large compared to the number fading coefficients (i.e., interference) and noise power. Dif- of users. Simulation results show that the approximations ferently from [21], we employ the maximum likelihood are asymptotically tight, but accurate for realistic systems. (ML) method to find an estimator for the interference The authors do not propose any approach to mitigate plus noise power term in the MMSE channel estimator. the pilot contamination problem, however, they study and We show that this estimator is not only unbiased but also evaluate its impact on the achievable rates. achieves the Crámer-Rao lower bound. We replace this The impact of pilot contamination effect on the achiev- estimator back into the MMSE estimator and prove that able uplink ergodic rate when using linear detection in the performance of the new channel estimator asymptoti- multi-cell MU massive MIMO systems under a more cally approaches that of the MMSE estimator. Simulation realistic physical channel model is assessed in [8]. The Figueiredo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:14 Page 3 of 10 authors assume that the channel vectors for different users show that in the ideal case, where the desired and the are correlated, or not asymptotically orthogonal due to interference covariance matrices span distinct subspaces, the antennas not being sufficiently well separated and/or the pilot contamination effect tends to vanish in the large the propagation environment not offering rich enough antenna array case. As a consequence, users with mutu- scattering. Moreover, they assume that the BS performs ally non-overlapping angle of arrival (AoA) hardly con- MMSE channel estimation based on training sequences taminate each other. Based on the results, the authors received on the uplink and a priori knowledge of the propose a coordinated pilot assignment strategy which large-scale fading coefficients. assigns carefully selected groups of users to identical pilot In [9], the polynomial expansion (PE) technique is sequences. applied to channel estimation of massive MIMO sys- A semi-blind iterative space-alternating generalized tems in order to approximate the MMSE estima- expectation maximization (SAGE) based channel estima- tor and thereby obtain a new set of low-complexity tion algorithm for massive MIMO systems with pilot channel estimators. Conventional MMSE estimators contamination is proposed in [13]. The proposed method present cubic complexity due to an inversion opera- does not assume a priori knowledge on the large-scale tion while the estimator proposed in [9]reduces this fading coefficients of the interfering cells, employing an to square complexity by approximating the inverse by estimate obtained from the received signal. The method a L-degree matrix polynomial. The proposed estima- updates the pilot based MMSE channel estimates iter- tor achieves near-optimal MSE with low polynomial atively with the help of the SAGE algorithm, which degrees. However, statistical knowledge of channel and improves the initial estimate with the help of pilot disturbance parameters at the receiver is assumed in symbols and soft information of the transmitted data. this paper. However, as it refines the channel estimates over some Outer multi-cellular precoding is employed in [10]to iterations starting from an initial MMSE channel esti- devise a method used to eliminate pilot contamination mation, it presents a computational complexity that is in massive MIMO systems. Each BS performs two lev- higher than the one presented by pure blind and linear els of precoding, firstly it estimates and shares only the estimators. large-scale fading coefficients with a central entity (net- After surveying the literature on channel estimation and work controller) which computes the precoding matrices pilot contamination mentioned above, it is clear that, for and sends them back to the BSs, i.e, outer precoding. clarity, in the great majority of studies the authors always Next, each BS performs local precoding using estimates assume complete knowledge on large-scale fading coeffi- of the fast-fading vectors, i.e., inner precoding. The pro- cients, i.e., path-loss and shadow fading, of the interfering posed approach is shown to completely mitigate the pilot cells, which is not the case in practical deployments of MU contamination problem, making it possible to construct Massive MIMO systems. Furthermore, several studies interference and noise free multi-cell massive MIMO sys- propose solutions that present additional computational tems with frequency reuse one and infinite downlink and complexity in order to mitigate the pilot contamination uplink signal-to-interference-plus-noise ratios (SINRs). problem. The proposed method employs MMSE channel estima- The main contribution of our work is the proposal and tion, however, the effectiveness of this method lies in assessment of a simple and practical channel estimator the estimation accuracy of the shared large-scale fading used to mitigate the pilot contamination problem. The coefficients from each BS. The authors also propose a proposed estimator does not assume a priori knowledge of method to estimate the large-scale fading coefficients. As the large-scale fading coefficients of the interfering cells. this approach needs to share the large-scale coefficients Moreover, it does not require the heavy overhead created with the network controller for outer precoding computa- by their estimation once it obtains them from the received tion, it presents a higher computational complexity than signal. non-cooperative approaches. The authors in [11], adopt a massive MIMO system 1.2 Organization model that is based on spatially correlated channels. They The remainder of this work is divided into four parts: devise a covariance aided channel estimation method First, we present the problem structure, signal model which exploits the covariance information of both desired adopted for this study and briefly discuss two well-known and interfering user channels. The Bayesian method is channel estimators, namely, LS and MMSE linear estima- used to derive two different channel estimators (it is tors. Then, we introduce the proposed channel estimator also shown that the Bayesian estimators coincide with for flat Rayleigh fading channels. Later, some numerical the MMSE estimators), one for all channels from users results are presented in order to support the effectiveness in all cells to the target cell and the other one for of the proposed estimator against the well-known linear the channels from users within the target cell. Results estimators. Finally, we present our conclusions. Figueiredo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:14 Page 4 of 10 2 Problem structure Coherence Time Let us assume as illustrated in Fig. 1 a multi-cell system with L cells, where each cell has a BS at its center with Uplink Training M co-located antenna elements and K randomly located Uplink Data single antenna users. Let us also assume Rayleigh fading channels being independent across users and antennas. Downlink Data Let g represent the complex gain of the channel from ilkm Fig. 2 TDD transmission protocol the kth user in the lth cell to the mth BS antenna in the √ √ ith cell. We can write g = β h where β is ilkm ilk ilkm ilk ⎡ ⎤ the large-scale coefficient encompassing both path loss g g ··· g il11 il21 ilK1 and log-normal shadowing. We assume the same large- ⎢ ⎥ g g ··· g il12 il22 ilK2 ⎢ ⎥ scale coefficient value for all BS co-located antennas, and ⎢ ⎥ g g ··· g il13 il23 ilK3 G = ⎢ ⎥.(1) il h is the small-scale coefficient with a circularly sym- ilkm ⎢ ⎥ . . . . . . . ⎣ ⎦ . . . metric complex normal distribution CN (0, 1). We assume g g ··· g that the large-scale fading coefficients do not depend il1M il2M ilKM on the frequency as well as on the antenna index m Based on the assumption of channel reciprocity, we of a given BS because typically, the distance between a adopt the TDD protocol depicted in Fig. 2 and proposed user and a BS is significantly larger than the distance in [22]. Due to the reciprocity principle, only the uplink between the BS antennas [10]. Therefore, between a BS channels need to be estimated while the downlink chan- and a user, there is only one large-scale fading coeffi- nels are equal to the transpose of the uplink channels. It cient. Moreover, these coefficients only change when a is important to note that the length of the TDD frames is user considerably change its geographical location. The limited by the channel coherence time [22, 23]. Accord- wireless channels are considered static during the channel ing to the TDD protocol, first, all users in all cells send coherence time (i.e., channel estimates are effective only their uplink training sequences synchronously. After that, in this time interval) and independent across users and the BSs use the training sequences to estimate the uplink antennas. channels. Next, the users send uplink data signals. Then, The M × 1 channel vector from the kth user in the lth the BSs use the estimated channels to detect uplink data cell to the M antennas at the ith BS is defined by g = ilk and generate precoding matrices used to transmit down- g , g , ··· , g . The overall M × K channel matrix ilk1 ilk2 ilkM link data. G is obtained by column concatenating vectors g for il ilk all cell users, that is, G = g , g ··· g .For detec- il il1 il2 ilK 2.1 Uplink training tion and precoding, BS i needs to know the channels of Each user transmits an uplink training sequence so that the users in cell i,namely {g , ∀k}.The same wayasin iik the user serving BS can estimate the channels per antenna the literature, we treat {β } as being deterministic during ilk and subsequently detect the transmitted user data. We the channel estimation [1, 6, 8, 13]. As described, the over- assume that users in different cells transmit data at the all channel matrix G can also be defined directly by the il same time-frequency resource (a typical scenario in mas- channel coefficients, sive MIMO) and that the pilot reuse factor is one, the worst possible use case scenario [3]. As all BSs reuse the same set of pilots and transmit at the same time-frequency resource, the pilot contamination problem arises, conse- quently, all the other BSs will also receive the pilots sent by users being served by other BSs, limiting the quality of the channel estimation [24]. The pilot signals of K users are represented by a N × K matrix S of the form S = [s , s , ··· , s ],where N is the 1 2 K length of the pilot sequences. Each pilot sequence is of N −1 0 1 the form s = s , s , ··· , s .The pilotmatrix, S, k k k exhibits orthogonal property S S = NI . The pilots are created by applying cyclic shifts to Zadoff-Chu (ZC) root sequences with length N,where N is a prime number. These sequences exhibit some useful properties: (i) cyclically shifted versions of themselves are Fig. 1 Problem definition orthogonal to each other, (ii) constant amplitude, (iii) zero Figueiredo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:14 Page 5 of 10 auto-correlation, (iv) flat frequency domain response, and The MSE per antenna of the LS estimator is given by (v) cross-correlation between two ZC sequences is low LS LS [25]. Some of the reasons why they are adopted in com- η = E g ˆ − g = ζ − β.(7) iik ik iik ik iik munication systems like long-term evolution (LTE) are As known, the LS estimator has larger MSE than the (i) channel estimation at receiver is made simpler due to MMSE estimator; however, it does not need prior knowl- their small variation in frequency, (ii) inter-cell interfer- ence is reduced as they present low cross-correlation, (iii) edge of the large-scale fading coefficients, {β }. ilk high peak to average power ratio (PAPR) is reduced due to their small variation in time. ZC sequences are used in this Remark 1 Due to pilot contamination, as q →∞, LS work due to the properties mentioned above [25]; how- η → β . ilk ik l=1,l=i ever, any other sequences could be used as long as they exhibit the required orthogonal property. Additionally, we 2.3 MMSE channel estimator assume that N ≥ K in order to avoid underdetermined A great number of massive MIMO works adopt the systems. MMSE estimation method to obtain channel knowledge The received uplink training sequences at the ith BS can [5, 8]. Those works assume that all large-scale fading coef- be represented as a M × N matrix defined as ficients, i.e., {β , i ≥ 1, l ≤ L,1 ≤ k ≤ K },are ilk perfectly known. In practice, this assumption might not be reasonable. In case we consider the coefficients {β } ilk Y = q G S + N,(2) i i il perfectly known at the BS, the ideal MMSE estimator is l=1 given by [26] where q is the uplink power or transmit signal to noise iik ratio (TX SNR) and N is a M × N noise matrix with inde- MMSE g = z,(8) ik iik pendent and identically distributed elements following ik CN (0, 1). MMSE iik where g ˆ ∼ CN 0 , I and the MSE of MMSE Equation (2) can also be written as showed below, M M iik ζ ik which clearly highlights the coherent inter-cell interfer- estimator is given by ence caused by users employing the same pilot sequences 1 β iik in other BSs. MMSE MMSE η = E g ˆ − g = β 1 − .(9) iik iik ik iik M ζ L ik √ √ H H Y = qG S + q G S + N .(3) i ii il i Remark 2 Due to pilot contamination, as q →∞, l=1,l=i Noise Desired pilot signals mmse iik η → β 1 − . iik L Undesired pilot signals ik ilk l=1 2.2 LS channel estimator 3 Proposed channel estimator For estimation of the channel g at BS i, a sufficient ilk In this work, we employ the ML method to estimate statistic [26–28]isgiven by the parameter ζ [26]. Applying the ML method to ik f (z ; ζ ) ∼ CN (0 , ζ I ), we find the following estima- ik ik M ik M 1 N s i k z = √ Y s = g + √ ik i k tor for ζ given the observation z ilk ik ik qN qN l=1 ik (4) ζ = . (10) ik N s i k = g + g + . iik ilk qN l=1,l=i This estimator has E ζ = ζ , which shows that the Desired channel ik ik Noise Inter-cell interference ML estimator is unbiased, and var ζ = ζ /M.Inorder ik ik where z is a column vector with a CN (0 , ζ I ) distri- ik M ik M to assess the efficiency of the estimator we derive the bution and Cramér-Rao bound as [26] ζ = β +.(5) ik ilk ik var ζ ≥ . (11) qN ik l=1 M Therefore, the ML estimator derived for ζ is the min- Additionally, the term corresponding to noise in (4)has ik imum variance unbiased estimator (MVUE), i.e., it is a CN 0 , I distribution. M M qN an unbiased estimator that has lower variance than any Therefore, the least square estimator is given by [26] other unbiased estimator for all possible values of the LS g ˆ = z.(6) parameter [26]. ik iik Figueiredo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:14 Page 6 of 10 This simple and effective estimator is derived based where on the observation that the MMSE estimator does not 1 1 k (1 − t) + k w t(1 − t) ik ik θ = need to know the individual large-scale fading coeffi- ik k (1 − t) + 2k w t(1 − t) + t (16) 0 −1 ik ik cients, {β }, as assumed in the existing literature, but ilk .f (t)f (w)dwdt T W just ζ suffices. The proposed estimator for ζ makes ik ik the acquisition of inter-cell large-scale fading coefficients iik with k = ,and f (t) and f (w) are given by T W ik unnecessary. The task of gaining knowledge of those coef- ζ −β ik iik ficients may be unjustifiable in practice due to the exces- (2M) M−1 f (t) = (t(1 − t)) ,0 < t < 1 (17) sive, e.g., in case there are L cells serving K users in each T ((M)) oneofthem, each BS needstoacquire (L − 1)K inter-cell large-scale coefficients. M 1 1 2 M− f (w) = B , M (1 − w ) , |w| < 1. (18) ˆ W Swapping ζ with ζ in (8) produces the proposed ik ik π 2 channel estimator, which is defined by The difference between the closed-form, given by Eq. (9) in [21], and the approximated MSE expressions are prop ik g ˆ = Mβ . (12) iik defined by iik ik ! " prop (closed-form) prop (approx.) iik η −η = 2β − θ , (19) iik ik ik ik This estimator approaches the ideal MMSE estima- ik tor asymptotically with respect to M. The estimator has where θ is defined in [21]. prop ik E g ˆ = 0 and variance given by iik iik Remark 4 As both q and M →∞, θ → and then, ik ik 2 prop(closed−form) prop(approx.) prop prop prop iik η − η → 0. ik ik Var g ˆ = E g ˆ g ˆ = I . iik iik iik M − 1 ζ ik We find Remark 4 by using Remark 3 and equaling (13) the closed-form and approximated MSE expressions. This remark shows that the difference between the closed-form As can be seen by analyzing equation (13), as M → 2 and the approximated MSE expressions decreases, tend- prop iik ∞,Var g ˆ → . An approximation to the MSE per ing to 0, as both uplink power, q, and number of receiving iik ζ ik antenna of this estimator is given by antennas, M,increase. 1 (M − 2)β Remark 5 The average normalized squared Euclidean prop prop iik η = E g ˆ − g ≈ β 1 − . prop iik iik MMSE ik iik distance between g ˆ and g ˆ is given by M (M − 1)ζ ik iik iik (14) 1 1 prop MMSE iik E g ˆ − g ˆ = . (20) iik iik M M − 1 ζ ik The approximate MSE in (14)for theproposedesti- The proof of (20) is given in Appendix B.From(5) mator decreases with increasing transmitting power q, and (20), it is easily noticeable that the average distance increasing M or decreasing β , which means smaller iik decreases with increasing M,decreasing q,increasing interference level from other cells, i.e., smaller pilot β , i = l, and decreasing β . contamination. ilk iik 4 Numerical results and discussion Remark 3 Due to pilot contamination, as q →∞ and In this section, we compare the performance of the pro- prop iik M →∞, η → β 1 − . iik L ik β posed channel estimator with that of the MMSE and LS ilk l=1 estimators. We adopt a typical multi-cell structure as the Remark 3 clearly shows that the MSE of the proposed one shown in Fig. 1 with L = 7 cells (one central cell estimator tends to that of the MMSE estimator when both surrounded by 6 other cells), K = 10 users in each cell, q and M →∞. The proof for the approximation of the frequency reuse factor of 1 and N = K pilot symbols. We MSEisgiven in Appendix A. consider two different types of setups for {β },one with ilk For the sake of clarity, we reproduce below the closed- fixed values and other with random values. For the fixed form MSE equation (9) presented in [21]. case, we set β = 1and β = a, ∀ l = i,where a repre- iik ilk sents the cross-cell interference level. The value selected M β prop(closed-form) for a in the fixed case is 0.05, and it is chosen so that there iik η = + β − 2β θ (15) iik iik ik ik M − 1 ζ is moderate cross-cell interference level from users being ik Figueiredo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:14 Page 7 of 10 served by other BSs, i.e., not being served by the central cell. For the random case, users in each cell are uniformly distributed within a ring with radii d = 100 m and d = 0 1 1000 m respectively. The large-scale fading coefficients ilk {β } are independently generated by β = ψ/ , ilk ilk where v = 3.8, 10 log (ψ ) ∼ N 0, σ with shadow, dB σ = 8, and d is the distance of the kth user in shadow, dB ilk the lth cell to the ith BS. Both, the path loss exponent, v, and the standard deviation of the log-normal shadow- ing, σ , are common values for outdoor shadowed shadow, dB urban cellular radio environments [18, 29]. The results in Fig. 3 show MSE versus SNR (uplink pilot power q) performances for a = 0.05 and M = 70. As can be seen, analytical, approximated, and simula- tion MSEs match for all estimators. With the increase of Fig. 4 MSE performance versus number of BS collocated antennas, M SNR, MSEs of all the estimation methods decrease. There areMSE floors forall thethree estimators duetopilot contamination (see Remarks 1, 2,and 3). At low SNR, the finite floating-point number represented by the IEEE dou- MSEofthe proposed estimatorisveryclose to thatof ble precision format, i.e., 1.7977e+308 [30], for values of the ideal MMSE estimator. On the other hand, as can be M greater than 85. A double precision variable goes to noticed, with theincreaseoftheSNR, thegap between +Inf after the largest possible number [30]. On the other the ideal MMSE estimator and the proposed one increases hand, as can be seen in Fig. 4, the approximate ana- lytical MSE expression (14) does not present the same (see Remark 5). problem and, therefore, can be used to evaluate the MSE In Fig. 4, we compare MSE versus the number of BS for any number of antennas, M without any numerical antennas M under the setting of a = 0.05 and TX SNR issue. q = 10 dB. With the increase of M, the MSE of the In Fig. 5, we compare MSE performance with respect to proposed estimator approaches that of the ideal MMSE, various levels of cross-cell interference, a,with q = 10 dB while the MSE of LS estimator does not change. Due to and two different number of antennas, M = 30 and numerical issues, the closed-form MSE expression pre- M = 90. We can see that when a increases (the effect sented in [21] does not produce values for M > 85. During of pilot contamination increases), the estimation perfor- our simulations, comparing the closed-form expression mance degrades. At a low cross-cell interference level, givenbyequation(15) and the approximated MSE expres- LS presents a slightly better MSE when compared to sion given by (14), we noticed that the (2M) function the proposed estimator. This difference disappears as M in the numerator of equation (16) grows without bound, increases, as can be noticed in the plot with M = 90. As reaching values that are greater than the largest possible MMSE (analytical) MMSE (simulated) LS (analytical) LS (simulated) Prop. (approximated) Prop. (simulated) M = 70, a = 0.05 -10 -5 0 5 10 15 20 25 30 SNR [dB] Fig. 3 Channel estimation MSE versus uplink pilot power Fig. 5 Channel estimation MSE versus cross-cell interference level MSE Figueiredo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:14 Page 8 of 10 the interference level increases, the proposed method out- performs the LS estimator substantially and approaches the ideal MMSE performance (see Remark 5). In Fig. 6, we evaluate the MSE performance under ran- dom large-scale fading coefficients {β } with M = 30. ilk The results are obtained by averaging MSEs over 10000 realizations of {β }. As can be observed, simulation MSE ilk matches with the analytical MSE. Additionally, the sensi- tivity of the proposed estimator against inaccuracy of β iik by using an estimate β = β 1 + N 0, σ is inves- iik iik tigated. The performance degradation for σ = 0.1 is noticeable at high SNR but for σ = 0.01, it is insignif- icant. The proposed estimator still outperforms the LS estimator significantly. In Fig. 7,wecompare thedistancebetween theproposed and MMSE channel estimators for different number of Fig. 7 Distance between proposed and MMSE estimators (Remark 5) antennas, M,with a = 0.05. As the Remark 5 states, the distance is small at low SNR, increasing with SNR until a ceiling is reached. As can be also noticed, the ceiling value decreases with the number of antennas, M. with pilot contamination in a flat channel environment. In Fig. 8, we compare the absolute distance between The proposed estimator replaces the combined interfer- the approximated MSE expression presented in (13)and ence plus noise power term in the ideal MMSE estimator the analytical (closed form) MSE expression presented in with a maximum likelihood estimator for that term. More- [21] for various SNR and M values with a = 0.05. over, the proposed estimator presents MSE results that are Thedistancebetween theMSE expressionsissmall at very close to that of the ideal MMSE estimator without low SNR, increasing with SNR until a ceiling value is requiring previous knowledge of noise and interference reached. As can be noticed, the ceiling value decreases statistics. Additionally, we have derived an approximate with the number of antennas, M.For M = 50, the analytical MSE expression for the proposed estimator ceiling distance is smaller than 1e − 4, showing that which can be useful in system design and performance the approximated MSE expression can replace the one evaluation. We have also shown that the MSE expres- presented in [21]. sion presented here asymptotically approaches that of the MMSE estimator. Finally, the simpler approximate analyt- 5Conclusions ical MSE expression presented here can be used instead of In this work, we have introduced a simple and practi- the more complex and susceptible to numerical issues one cal channel estimator for massive MIMO TDD systems presented in [21]. -2 -4 -6 M = 10 -8 M = 50 M = 100 M = 200 M = 500 -50 -40 -30 -20 -10 0 10 20 30 40 50 SNR [dB] Fig. 8 Absolute distance between closed form and approximated Fig. 6 Average channel estimation MSE under random {β } MSE expressions ilk Distance between analytical and approximated errors Figueiredo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:14 Page 9 of 10 Appendix A From Lemma 1,weknowthat ∼ (MP, ζ ). ik ik For the proof of the approximate MSE of the proposed Then, applying Lemma 2 to (25), we figure out that # $ estimator, we need to present a few Lemmas. E 1/ = 1/ζ (M − 1) and consequently, the first ik ik expectation term is defined as Lemma 1 If X ∼ CN 0, σ ∀m are independent, then m 2 Mβ prop 2 iik 2 2 g ˆ = . (26) ∼ M, σ . m=1 iik M ζ (M − 1) ik 1 The second expectation term is defined as −1 Lemma 2 If X ∼ (k, θ) and ∼ (k, θ),i.e.,the # $ 1 1 M inverse-gamma distribution, then E = . 1 1 X θ(k−1) 2 2 = E = β . (27) iik iikm iik M M m=1 Lemma 3 Let μ and μ be the expectations of X and Y, X Y Finally, in order to find the expected value of the third σ be thevarianceofY,and σ be their covariance. Then, XY term, first, we use (4)and (12)torewrite it as the expectation, E{X/Y }, can be approximated by ! " ! " ! H H X μ σ μ X XY X z g g g 2 iik iik ik l=1 ilk E ≈ − + σ . (21) −2β E R =−2β E R iik iik Y 2 2 2 3 ik ik Y μ μ μ Y Y w g iik ik +E R Proof For a function that depends on two variables, ik x and y, the second order Taylor expansion series about (28) the point (a, b) is given by where w = N s / qN ∼ CN 0 , I . i M M ik k qN g(x, y) = g(a, b) + g (a, b)(x − a) + g (a, b)(y − b) x y In order to avoid the numerical issues mentioned ear- 1 2 + g (a, b)(x − a) + 2g (a, b)(x − a)(y − b) xx xy 2! lier in this work and find a simpler and more tractable +g (a, b)(y − b) , yy equation for the MSE of the proposed channel estimator, (22) we find approximations to the two ratios of random vari- ables in (28). It is possible to approximate the moments where the subscripts denote the respective partial deriva- 2 of a function g(X, Y ) using Taylor series expansions, pro- tives. The partial derivatives are defined by g =−X/Y , 3 2 vided g is sufficiently differentiable and that the moments g = 2X/Y , g = 1/Y , g = 0, and g =−1/Y . yy x xx xy of X and Y are finite. Therefore, applying Lemma 3 sep- Applying the derivatives into (22), the second order Tay- arately to each one of the terms in the second and third lor expansion of g(X, Y ) = X/Y around the mean point lines of (28), we are able to find an approximation to the (μ , μ ), the following is obtained X Y third expectation, which is defined as μ μ X 1 x x ≈ − (Y − μ ) + (X − μ ) y x Y μ μ L y y prop H β 2 β ilk 1 iik l=1 − E R g ˆ g ≈−2β 1− + iik iik M iik ζ Mζ M 2μ ik ik 1 x 2 2 + (Y − μ ) − (Y − μ )(X − μ ) . y y x 3 2 2! μ μ " y y 2β −β iik iik + =− . (23) 2 Mζ qN ik ik Finally, applying the expectation operator, E {.},to(23) (29) concludes the proof. After finding the three expectations, (26), (27), and (29), prop by substituting them back in the expansion of η ,we prop ik Proof of the approximate MSE, η ik complete the proof. Forthe proofofthe approximateMSE,weexpanditas 1 1 Appendix B prop prop 2 2 η = E g ˆ + E iik ik iik Here, we present proof for (20). First, we expand the nor- M M (24) prop MMSE malized Euclidean distance between g ˆ and g ˆ as prop H iik iik − E R g ˆ g , iik iik prop 2 2 1 1 MMSE E g ˆ + E g ˆ andfindthese threeexpectations. M iik M iik (30) From (12), the first expectation can be written as prop H 2 MMSE ˆ ˆ − E R g g . ! " M iik iik prop ik 2 2 g ˆ = Mβ E Then, we compute these three different expectations. iik iik 2 2 M [ ik prop 2 ! " (25) The first one is given by (26), E g ˆ = 1 M iik = Mβ E . 2 MMSE iik Mβ /ζ (M − 1). Next, by recalling that g ˆ ∼ ik ik iik iik Figueiredo et al. EURASIP Journal on Wireless Communications and Networking (2018) 2018:14 Page 10 of 10 β 2 1 MMSE 6. E Bjornson, EG Larsson, TL Marzetta, Massive MIMO: ten myths and one iik CN 0 , I ,wehavethat E g ˆ = M M ζ M iik ik critical question. IEEE Commun. Mag. 54(2) (2016) 2 7. J Hoydis, S ten Brink, M Debbah, Massive MIMO in the UL/DL of cellular β /ζ . For the last expectation term, using (8)and (12), ik iik networks: how many antennas do we need? IEEE J.Sel. Areas Commun. we can write it as 31(2), 160–171 (2013) 8. HQ Ngo, EG Larsson, TL Marzetta, The multicell multiuser MIMO uplink prop H 2β with very large antenna arrays and a finite-dimensional channel. IEEE 2 MMSE iik ik − E R g ˆ g ˆ =− E R M iik iik ζ Trans. Commun. 61(6), 2350–2361 (2013) ik ik 2β 9. N Shariati, E Bjornson, M Bengtsson, M Debb, Low-complexity polynomial iik =− . ζ channel estimation in large-scale MIMO with arbitrary statistics. IEEE J. Sel. ik Top. Signal Process. 8(5), 815–830 (2014) (31) 10. 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EURASIP Journal on Wireless Communications and Networking – Springer Journals
Published: Jan 15, 2018
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