Accepted SCIENCE CHINA. Information Sciences. Article. Just Accepted. Multi-cell massive MIMO transmission with coordinated pilot reuse

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1 SCIENCE CHINA Information Sciences Article Multi-cell massive MIMO transmission with coordinated pilot reuse ZHONG Wen, YOU Li, LIAN TengTeng, GAO XiQi Sci China Inf Sci, Just Manuscript DOI: /s Publication Date Web: Just Just manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The Science China Press SCP provides Just as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. Just manuscripts appear in full in PDF format. Just manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Obect Identifier DOI. Just is an optional service offered to authors. Therefore, the Just Web site may not include all articles that will be published in the ournal. After a manuscript is technically edited and formatted, it will be removed from the Just Web site and published as an Online First article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the ournal pertain. SCP cannot be held responsible for errors or consequences arising from the use of information contained in these Just manuscripts. *Corresponding author wzhong@seu.edu.cn Science China Press and Springer-Verlag Berlin Heidelberg 2015 info.scichina.com link.springerlink.com

2 . RESEARCH PAPER. SCIENCE CHINA Information Sciences Manuscript January 2015, Vol. 58 xxxxxx:1 xxxxxx:14 doi: xxxxxxxxxxxxxx Multi-cell massive MIMO transmission with coordinated pilot reuse ZHONG Wen *, YOU Li, LIAN TengTeng & GAO XiQi National Mobile Communications Research Laboratory, Southeast University, Naning , China Received January 14, 2015; accepted May 27, 2015; published online xxx xx, 2015 Abstract In this paper, we propose a coordinated pilot reuse CPR approach to reduce the pilot overhead for multi-cell massive multi-input multi-output transmission. Unlike the conventional multi-cell pilot reuse approach in which pilots can only be reused among different cells, the proposed CPR approach allows pilots to be reused among both inter-cell and intra-cell user equipments, and thus, pilot overhead can be efficiently reduced. For spatially correlated Rayleigh fading channels, we first present a CPR-based channel estimation method and a low complexity pilot allocation algorithm. Because CPR might lead to additional pilot interference, we develop a statistically robust uplink receiver and downlink precoder that takes channel estimation errors into account. The proposed uplink receiver and downlink precoder are robust to channel state information inaccuracy, and thus, can guarantee a certain transmission performance. Monte-Carlo simulations illustrate the significant performance improvement in net spectral efficiency offered by the proposed CPR approach. Keywords efficiency pilot overhead, multi-cell massive MIMO, coordinated pilot reuse, robust transceiver, net spectral Citation Zhong W, You L, Lian T T, et al. Multi-cell massive MIMO transmission with coordinated pilot reuse. Sci China Inf Sci, 2015, 58: xxxxxx14, doi: xxxxxxxxxxxxxx 1 Introduction Massive multi-input multi-output MIMO systems, which employ a large number of antennas at base stations BSs to simultaneously serve multiple user equipments UEs, has recently drawn considerable research attention. With the number of BS antennas growing without limit, all of the effects of uncorrelated noise and fast fading vanish, and the data rate, link reliability, and spectral- and energy-efficiency are substantially improved. Consequently, massive MIMO is considered as one of the key technologies for next-generation wireless communications [1 3]. Performance of massive MIMO communication depends on the degree of knowledge of the channel state information CSI. In practical wireless communication systems, CSI is not genie provided : it must be acquired through channel training. For the case of frequency division duplex systems, the overhead of channel training and CSI feedback scales with the number of antennas at the BS, which become prohibitive in massive MIMO. On the other hand, in time division duplex TDD massive MIMO, using uplink pilots, BSs are capable of estimating the forward- and reverse-link channels by exploiting the * Corresponding author wzhong@seu.edu.cn

3 Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:2 channel reciprocity [3, 4]; the pilot overhead is proportional to the number of UEs, which is far smaller than the number of antennas at the BS in massive MIMO. In practical cellular networks, the resource that can be utilized for channel training is limited. Thus, pilot reuse in multi-cell scenarios is inevitable. Previous works such as [3,5,6] assume a pilot allocation scheme in which UEs in the same cell use a set of mutually orthogonal pilots, and the same set of pilots is reused among cells. For this pilot approach, pilot contamination is thought to be one of the limiting factors of massive MIMO. Several approaches have been proposed to mitigate pilot contamination. For instance, a covariance-based coordinated pilot assignment scheme was proposed in [5]. However, this approach is still considered to be precious-resource-consuming, especially when the UE density in the same cell becomes relatively high. In [7], an intra-cell pilot reuse approach that leverages channel sparsity in the spatial domain was proposed. With this approach, the number of orthogonal pilots can be much smaller than that of UEs. It was shown in [7] that the sum mean square error MSE of the channel estimation can be minimized when the channel angle of arrival intervals of the intra-cell UEs reusing the pilots are non-overlapping. Thus, pilot reuse among UEs in the same cell is feasible and beneficial for UEs located in almost orthogonal spatial directions. Motivated by this, we propose the coordinated pilot reuse CPR approach for multicell massive MIMO transmission. Note that extension of the work in [7] to the multi-cell setting is not straightforward, because full cooperation among cells is usually difficult to implement in practice. However, in our proposed CPR scheme, only coordination of the channel covariance information among cells is required, and the overhead of information exchange among cells can be significantly reduced. The proposed CPR approach allows pilots to be reused among both intra-cell and inter-cell UEs via appropriate pilot scheduling with the channel spatial property taken into consideration and thus pilot overhead can be significantly reduced. With the proposed CPR approach, multi-cell massive MIMO transmission consists of four phases as follows. Coordinated pilot allocation phase: Exchange channel covariance information of all UEs among BSs and allocate pilots using the CPR approach. Channel estimation phase: All UEs send allocated pilot sequences. Each BS estimates CSI based on the received training signals. Uplink data transmission phase: UEs send message-bearing symbols. Each BS uses the obtained CSI for coherent detection of the uplink data. Downlink data transmission phase: Each BS precodes downlink data using the obtained CSI. The remainder of this paper is organized as follows. Section 2 describes the spatially correlated Rayleigh fading massive MIMO channel model. Section 3 presents the CPR-based channel estimation method and a low complexity pilot allocation algorithm. In Section 4, we develop a statistically robust uplink receiver and downlink precoder that takes CSI inaccuracy into account. We present numerical results in Section 5 and conclude the paper in Section 6. The following notations are used throughout this paper. Uppercase and lowercase boldface letters denote matrices and vectors, respectively. Superscripts th, t, u, d, and opt denote expressions related to the index, channel training, uplink data transmission, downlink data transmission, and optimum values, respectively. The notations H, T,, 1, tr { }, and 2 represent the conugate transpose, transpose, conugate, inverse, trace, and l 2 -norm operations, respectively. E A {B} denotes the expectation of B taken over A. The operator diag { } denotes a diagonal matrix with the vector argument as diagonal elements. [ ] xy and [ ] z represent the x, y th element of the matrix argument and the z th element of the vector argument, respectively. min { } denotes the minimum value. We use C CN m, R to indicate that the matrix C exhibits a circular symmetric complex Gaussian distribution with mean m and covariance R. ȷ is the imaginary unit. The notation indicates is equivalent to. 2 Massive MIMO channel model Our scenario includes N hexagonal cells, each consisting of a central M-antenna BS and K M singleantenna UEs, as shown in Figure 1. A TDD protocol and universal frequency reuse are employed. We assume the discrete-time block-fading law that the channel remains constant over an interval with a

4 Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:3 The th cell g kn k,n g nkn Then th cell Figure 1 An N-cell system, with one BS equipped with M antennas and K single-antenna UEs in each cell. Uplink channels from UE k, n to the th BS and the n th BS are denoted by g kn and g nkn, respectively. length T, and independently evolves from interval to interval. We denote the k th UE in the th cell as UE k,, and let R kn be the covariance matrix corresponding to the channel vector g kn from UE k, n to the th BS. We assume a Gaussian channel distribution, i.e., g kn CN 0, R kn. When the number of BS antennas is sufficiently large, R kn can be well approximated by [7] R kn β kn Fdiag {r kn } F H, 1 where β kn is the large-scale fading coefficient between UE k, n and the th BS, and r kn depends on the respective channel power angle spectrum, which models the channel power distribution in the angular domain [8]. We assume that a uniform linear array is equipped at the BS, with M antennas spaced at one-half wavelengths, and that the channel angle of arrival intervals of UEs is equal to [ π/2, π/2]. Then, the following approximations can be obtained [7] [F] xy = 1 e M ȷ2π x 1y 1 M/2 M, 1 x, y M, 2 [r kn ] m = M θ m θ m 1 S kn θ m 1, 1 m M, 3 θ m = arcsin2m /M 1, 0 < m M, 4 where S kn θ denotes the power angle spectrum of the channel between UE k, n and the th BS. We assume that + S kn θ dθ = 1, and that the channel covariance matrix in 1 satisfies the channel power normalization as tr {R kn } = β kn M π 2 π 2 S kn θ dθ. 5 By exploiting resource blocks in which the desired UE and interfering UEs are known to be assigned to different time slots, the individual covariance matrices can be estimated separately [5]. The proposed CPR approach requires the exchange of channel covariance information among BSs; this can be achieved using low-rate backhaul links because the second-order channel statistics vary much more slowly than the instantaneous CSI. In this work, we assume that the channel covariance matrices of all UEs can be obtained and shared by the BSs. 3 CPR-based channel training In this section, we investigate how to assign a set of pilots with length τ among inter- and intra-cell UEs for channel estimation. In our approach, the number of available orthogonal pilots is smaller than the

5 Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:4 number of UEs in each cell, i.e., τ < K, and some of the intra-cell UEs and inter-cell UEs will reuse the same pilots. We detail CPR for channel estimation and propose a low complexity pilot allocation algorithm below. 3.1 CPR-based channel estimation We denote an arbitrary CPR pattern with a given UE set and a pilot set by PK, τ {[k, n, π kn ] : k, n K, π kn T } where K {1, 1,..., K, 1, 1, 2,..., K, N} is the set of UEs, T {1, 2,..., τ} is the index set of available orthogonal pilots, and π kn is the index of the pilot assigned to UE k, n. K π {k, n : π kn = π, π T } denotes the set of UEs sharing the pilot with index π. During the uplink channel training phase, UEs in each cell send their respective pilots for the BS to estimate the channel parameters. The training signal received by the th BS can be expressed as Y = N G n X n + N = N l=1 K g ln x T π ln + N, 6 where G n = [g 1n, g 2n,..., g Kn ] is the matrix consisting of channels from UEs in the n th cell to the th BS, N C M τ is the independent additive Gaussian noise matrix with elements independently and identically distributed as CN 0, σz, t and X n = [x π1n, x π2n,..., x πkn ] T is the uplink pilot matrix sent from UEs in the n th cell. We assume that x H π k x πln = τσx t δ π k π ln. After decorrelation and power normalization of the received pilot signals [3], the th BS gets the corresponding channel observation. Specifically, with pilot x πk, the th BS gets the channel observation of UE k, as yπ t k = 1 σxτ t Y x π k = 1 N K σxτ t g ln x T π ln + N x π k 7 l=1 N K = g ln δ 1 x πln x πk + σ t l=1 xτ N x π k 8 = g ln + 1 σ t l,n K πk xτ N x π k. 9 With the property of unitary transformation, it is not hard to show that the noise term 1 σx t τ N x π k in 9 is still Gaussian with elements independently and identically distributed as CN 0, σz/σ t x. t Let ρ t = σx/σ t z t be the uplink channel training signal-to-noise ratio SNR, then 9 can be rewritten as yπ t k = g ln + 1 ρt τ nt π k l,n K πk = g k + g ln + 1 ρt τ nt π k, 10 l,n K πk \{k,} }{{}}{{} Pilot Interference Pilot Noise where K πk is the set of UEs using the same pilot as UE k,, n t π k CN 0, I M is the independent additive noise vector, and the notation \ denotes the set subtraction operation. The covariance matrix of yπ t k is given by } H C k E {y y tπk tπk = R ln + 1 ρ t I. 11 τ l,n K πk We obtain the minimum MSE-based channel estimate of g k as ĝ k = R k C 1 k yt π k. 12

6 Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:5 Because of the orthogonality principle of minimum MSE MMSE estimation [9], the channel estimation error g kn = g kn ĝ kn is independent of ĝ kn. We define the sum MSE of channel estimation with CPR applied MSE-CPR as N K ϵ t tr { } R gk, 13 =1 k=1 where R gk = R k R k C 1 k R k is the covariance matrix of g k. From 13, we can obtain a condition under which MSE-CPR can be minimized in the following theorem. Theorem 1. The minimum value of the MSE-CPR can be achieved provided that R k R ln = 0 tr {R k R ln } = 0, 14 for k,, l, n K, k, l, n, and π k = π ln. The minimum value is given by ε t = N =1 k=1 Proof. See Appendix A. Using 1, 2, and 3, we can reduce the condition in 14 to K tr {R k R k R k ρ t τ I R k}. 15 S k θ m S ln θ m = 0 0 m M 1, 16 which shows that under the condition of non-overlapping power angle spectrums of UEs using the same pilot, the MSE-CPR can be minimized. This result stems from the fact that channels of UEs reusing the pilots are completely separated in the angular domain; thus, pilot interference does not occur. In realistic outdoor propagation environments, the BSs are typically mounted at elevated positions; thus, the scattering around the BSs is limited. As a result, most of the channel power concentrates in a narrow angle range, which makes CPR feasible, especially for UEs with channels lying in nearly orthogonal spatial directions. 3.2 Coordinated pilot allocation algorithm A straightforward method for determining the optimal CPR pattern that minimizes MSE-CPR is to use an exhaustive search. However, the exponential implementation complexity of exhaustive search is prohibitively high, especially when the number of UEs is large. In the previous subsection, we demonstrated that pilot interference will vanish in the ideal case, namely, where the desired and the interference covariance matrices are orthogonal. Motivated by this result, we propose a low complexity pilot allocation algorithm, which we refer to as the coordinated pilot allocation algorithm, as summarized in Algorithm 1. The idea is to exploit the covariance information of all UEs so as to assign orthogonal pilots to UEs with similar covariance matrices and assign the same pilots to UEs with almost orthogonal covariance matrices. We define the degree of relevance in Definition 1 as follows. Definition 1. The degree of relevance of any set U composed of different UEs is defined as DRU tr {R k R ln } + tr {R nln R nk }. 17 k,,l,n U k, l,n Specifically, when U = K i, it represents the degree of relevance of the UE set that reuses the pilot with index i. Herein, we evaluate the complexity of the proposed coordinated pilot allocation algorithm and exhaustive search method in terms of the required number of complex-valued scalar multiplications. For the proposed algorithm, NK 1 m=1 m NK m =NK 1 NK NK + 1/6 calculations of degree of relevance defined in 17 are required, and the number of scalar multiplications is O M 2 in each calculation.

7 Algorithm 1 Coordinated pilot allocation algorithm Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:6 Require: N, K, T, τ, R kn 1 N, k, n K; Ensure: PK, T ; Initialization: K un = K, T un = T, U temp = ; Assign the 1 st pilot to 1, 1: π 11 = 1; Update: K 1 = {1, 1}, K un = K un \ K 1, T un = T un \ {π 11 }, U temp = K 1 ; while K un and T un do For p T un, select the UE s, t = arg max DR{l} U temp ; l K un Assign the p th pilot to s, t: π st = p; Update: K p = {s, t}, K un = K un \ K p, T un = T un \ {π st}, U temp = U temp K p; end while while K un do For u, v K un, select the pilot q = arg min DR{u, v} K c; c T Assign the q th pilot to u, v: π uv = q; Update: K q = K q {u, v}, K un = K un \ {u, v}; end while Therefore, the computational complexity of the proposed algorithm is O M 2 N 3 K 3. For an exhaustive search, the pilot assignment problem is combinatorial and can be expressed as minimize PK,T ϵ t. 18 Recalling 13, we can derive that the number of scalar multiplications in 18 is O NM 3 K. Because there exist τ NK patterns for P K, T, the computational complexity of the exhaustive search is O τ NK NM 3 K. Thus, the complexity of the proposed CPR algorithm can be significantly reduced relative to the exhaustive search, while substantial performance gains in terms of achievable net spectral efficiency over the conventional orthogonal pilot reuse OPR scheme can still be achieved, as one shall see in the simulation results. 4 Robust data transmission In practical propagation environments, pilot interference cannot be eliminated and might degrade the channel estimation performance. Thus, it is reasonable to design uplink receivers and downlink precoders that are robust to imperfect CSI so as to guarantee a certain transmission performance. After channel training, each BS obtains the channel estimates of the corresponding UEs. When the number of antennas at the BS tends to infinity, the simple linear receivers and precoders are optimal [3, 10]. Consequently, we will investigate linear robust uplink receivers and downlink precoders for massive MIMO data transmission. 4.1 Robust uplink data transmission During the uplink data transmission phase, signals received at the th BS can be written as y u = N G n a u n + 1 ρ u n u, 19 where G n = {g 1n,..., g Kn } is the channel matrix from UEs in the n th cell to the th BS, a u n C K 1 is the data vector transmitted from UEs in the n th cell to the n th BS with mean 0 and covariance I K, ρ u is the average SNR per UE in the uplink data transmission phase, and n u CN 0, I M is the independent additive noise vector. We consider the linear receiver matrix W at the th BS; the estimate of the data transmitted by the UEs in the th cell is â u = W T y u. 20 The MSE of uplink signal detection MSE-USD of the th BS can be defined as { â } ϵ u E a u,n u, G u n a u,

8 Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:7 where G n = [ g 1n,..., g Kn ] denotes the channel estimation errors. The optimal receiver, which minimizes ϵ u, can be obtained in the following theorem. Theorem 2. The optimal robust receiver which minimizes the MSE-USD is given by [ N K ] 1 W opt = R gkn + ĜnĜH n + 1 ρ u I M Ĝ. 22 k=1 Proof. See Appendix B. Using a standard bound based on worst-case uncorrelated additive noise, the ergodic achievable uplink sum rate [10] with the robust receiver given in 22 is given by R u,sum = N =1 k=1 where the equivalent signal-to-interference-noise ratio SINR is given by SINR u k = w T k [ N K Derivations can be found in Appendix C. 4.2 Robust downlink data transmission K E { } log SINR u k, 23 wkĝk T 2 k=1 R g kn + ĜnĜH n + 1 ρ u I ĝ k ĝ H k ] w k. 24 During the downlink transmission phase, the signals received by the UEs in the th cell can be written as y d = N G T nb n a d n + 1 ρ d n d, 25 where B n is the linear precoder matrix of the n th BS satisfying the power constraint tr { } B H n B n K, a d n C K 1 is the downlink data signal vector with mean 0 and covariance I K, ρ d is the average SNR per UE, and n d CN 0, I K is the independent additive noise vector. We define the MSE of the downlink signal detection MSE-DSD in the th cell as { } ϵ d E a d,n d, G n α y d a d 2 2 = E a d,n d, G n α G T B a d + 1 ρ d n d a d N α G T nb n a d 2 n 2, 26 where α is the power scaling factor of the UEs in the th cell. The minimization of ϵ d in 26 requires oint optimization of the precoders in all cells, which has high complexity. Herein, we consider the leakage-based precoding scheme motivated by [11]; the corresponding MSE can be defined as 2 ϵ d = E a d,n d, G n α G T B a d + 1 N n d a d + α G T nb a d ρ d We formulate the following optimization problem to obtain the optimal precoder B minimize ϵ d, B,α subect to tr { B H } 28 B K. The optimal solution is given in the following theorem. 2 n n

9 Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx: Figure 2 Schematic diagram of the UE distributions considered in the simulations unit: m. Theorem 3. where The optimal solution to problem 28 is given by [ N K F = B opt = k= Figure 3 Simulation scenarios for a comparison between robust and non-robust transceivers unit: m. K tr { } F F H F, 29 R gnk + ĜnĜH n ] ρ d I M Ĝ. 30 Proof. See Appendix D. Note that when ρ u = ρd, we have Fopt = W opt, i.e., the robust downlink precoder can be obtained from the uplink receiver with appropriate power scaling; thus, the implementation complexity of the leakage-based robust downlink precoder can be reduced. With the robust precoder given in 29, we can obtain the sum rate of downlink data transmission phase as N K } R d,sum = E {log SINR d k, 31 where the equivalent SINR is given by SINR d k = N =1 k=1 Derivations can be found in Appendix E. 5 Numerical results { } E α gk T b 2 k { α K m=1 E gnk T b mn 2} { } + E{α2 }. 32 E α gk T b 2 k To evaluate the performance of the proposed CPR approach, we adopt a cluster model composed of N hexagonal cells with a radius r c from center to vertex. K UEs are uniformly distributed in a 120 sector in each cell, excluding a central disk of radius r h. Each BS is equipped with a uniform linear array. The simulation scenario is depicted in Figure 2. The channel path loss from UE k, n to the th BS is given by [5] ρ d β kn = r c /d kn υ, 33

10 MSE USD db Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:9 Table 1 Simulation parameters Parameter Value Parameter Value N 3 M 128 K 6 υ 3.8 r h 100 m r c 800 m φ [2,5 ] T [12, 14,..., 24] ρ Conventional Robust, True Robust, Estimated Pattern A Pattern B [ 10, 5,..., 30] db SNR db Figure 4 MSE-USD versus SNR for the robust receiver and conventional receiver using the true and the estimated channel covariance matrices, respectively for two specific UE distribution patterns N = 3, K = 9, φ = 5. MSE DSD db Conventional Robust, True Robust, Estimated Pattern A Pattern B SNR db Figure 5 MSE-DSD versus SNR for the robust precoder using the true and the estimated channel covariance matrices, respectively and conventional precoder for two specific UE distribution patterns N = 3, K = 9, φ = 5. where υ is the path-loss exponent and d kn is the geographical distance from UE k, n to the th BS. Here, we assume the angular spreads ASs of different UEs to be φ for simplicity. In typical urban wireless propagation environments, the channel power angle spectrum can be modeled by [8] 1 S kn θ = e 2φ 1 e 2π/φ 2 θ θ kn φ, θ [θ kn π, θ kn + π 34 where θ kn, which is the mean angle of arrival of the channel between UE k, n and the th BS, is assumed to be uniformly distributed in the angle interval [ π/3, π/3]. The channel training SNR and the uplink/downlink data transmission SNR are assumed to be equal, i.e., ρ t = ρ u = ρ d = ρ. The channel covariance matrices of the UEs are generated using Eq. 1. We assume that both uplink and downlink data transmissions occupy half of the data transmission interval. Then, the average net spectral efficiency is given by R ave,net = 1 τ/t R u,sum + R d,sum, 35 2 where τ and T are the lengths of the pilot sequences and the entire coherence block, respectively. The maor simulation parameters are listed in Table 1, and will be used in the following simulations, unless otherwise stated. We adopt Algorithm 1 to allocate pilots and determine τ by choosing pilots of length τ T that maximize the net spectral efficiency with given values of φ and SNR. We compare the proposed CPR approach to the pilot allocation approach in [5], in which the pilot length is equal to the number of UEs in each cell. First, we employ the MSE-USD and MSE-DSD metrics to evaluate the performance of the proposed robust uplink receiver and downlink precoder, respectively. We consider two specific UE distribution patterns as depicted in Figure 3. For patterns A and B, the UEs are randomly distributed over a circle with a radius of m and m, respectively. In Figures 4 and 5, we plot the performance

11 Average Pilot Overhead % Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx: Block Length OPR CPR, σ = 5 CPR, σ = 2 Figure 6 Average pilot overhead versus block length with different ASs for CPR and OPR N = 3, K = 6, ρ = 20 db. Net Spectral Efficiency bits/s/hz CPR, σ = 2 CPR, σ = 5 OPR, σ = 2 OPR, σ = Block Length Figure 8 Net spectral efficiency versus block length with different ASs for CPR and OPR N = 3, K = 6, ρ = 20 db. Average Pilot Overhead % OPR 10 CPR, σ = 5 CPR, σ = SNRdB Figure 7 Average pilot overhead versus SNR with different ASs for CPR and OPR N = 3, K = 6, T = 12. Net Spectral Efficiency bits/s/hz CPR, σ = 2 CPR, σ = 5 OPR, σ = 2 OPR, σ = SNR db Figure 9 Net spectral efficiency versus SNR with different ASs for CPR and OPR N = 3, K = 6, T = 12. of MSE-USD and MSE-DSD, respectively, for the proposed robust transceiver using the true and the estimated channel covariance matrices that are obtained via averaging 100 samples, respectively and the conventional MMSE transceiver for the uplink and downlink cases. We can observe that the robust transceiver outperforms the conventional one, especially in the high SNR regime where pilot interference dominates. Additionally, the performance of robust transceivers with estimated channel covariance matrices can approach that of robust transceivers with true channel covariance matrices. Figures 6 and 7 show the average pilot overhead for CPR and OPR. We observe that CPR outperforms OPR, especially when the block length is short and/or AS is small. Figures 8 and 9 show the net spectral efficiency performance of CPR and OPR. It can be observed that the performance gains of CPR over OPR become larger with smaller ASs or shorter block lengths. The pilot overhead dominates when the coherence block length is short. In addition, with small ASs, the channel power becomes more concentrated in the angle domain; thus, the desired and the interference covariance matrices can be well separated and some of the UEs can share pilots. The gains of CPR over OPR are substantial, especially in the case of K = 6, φ = 2, ρ = 20 db, and T = 12, in which CPR provides approximately 34.8% net spectral efficiency gain relative to OPR.

12 6 Conclusion Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:11 In this paper, we proposed a coordinated pilot reuse CPR approach for massive MIMO systems in which both inter- and intra-cell UEs can reuse pilots. For spatially correlated Rayleigh fading channels, we first presented the CPR-based channel estimation method and a low complexity pilot allocation algorithm. Because pilot interference in realistic propagation environments cannot be eliminated and the obtained channel state information might be inaccurate, we further developed a statistically robust uplink receiver and downlink precoder that takes channel estimation errors into account. The simulation results confirmed our analyses and demonstrated the significant performance improvement in net spectral efficiency offered by the proposed CPR approach relative to the conventional orthogonal pilot reuse approach. Acknowledgements Part of this paper was presented at the IEEE PIMRC, Washington DC, USA, Sep [12]. This work was supported by the National Natural Science Foundation of China under Grants , , and ; the China High-Tech 863 Plan under Grants 2015AA and 2014AA01A704; the National Science and Technology Maor Proect of China under Grant 2014ZX ; and the Program for Jiangsu Innovation Team. The work of L. You was supported in part by the China Scholarship Council CSC. References 1 Rusek F, Persson D, Lau B K, et al. Scaling up MIMO: Opportunities and challenges with very large arrays. IEEE Signal Process Mag, 2013, 30: Larsson E G, Edfors O, Tufvesson F, et al. Massive MIMO for next generation wireless systems. IEEE Commun Mag, 2014, 52: Marzetta T L. Noncooperative cellular wireless with unlimited numbers of base station antennas. IEEE Trans Wireless Commun, 2010, 9: Marzetta T L. How much training is required for multiuser MIMO?. In: Proceedings of IEEE Fortieth Asilomar Conference on Signals, Systems and Computers, Pacific Grove, USA, Yin H, Gesbert D, Filippou M, et al. A coordinated approach to channel estimation in large-scale multiple-antenna systems. IEEE J Sel Areas Commun, 2012, 31: Li M, Jin S, Gao X Q. Spatial orthogonality-based pilot reuse for multi-cell massive MIMO transmission. In: Proceedings of IEEE International Conference on Wireless Communications and Signal Processing, Hangzhou, China, You L, Gao X Q, Xia X G, et al. Pilot reuse for massive MIMO transmission over spatially correlated Rayleigh fading channels. IEEE Trans Wireless Commun, 2015, 14: Pedersen K I, Mogensen P E, Fleury B H. A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments. IEEE Trans Veh Technol, 2000, 49: Kailath T, Sayed A H, Hassibi B. Linear Estimation. Upper Saddle River: Prentice Hall, Hoydis J, ten Brink S, Debbah M. Massive MIMO in the UL/DL of cellular networks: How many antennas do we need?. IEEE J Sel Area Comm, 2013, 31: Jose J, Ashikhmin A, Marzetta T L, et al. Pilot contamination and precoding in multi-cell TDD systems. IEEE Trans Wireless Commun, 2011, 10: Lian T T, You L, Zhong W, et al. Coordinated pilot reuse for multi-cell massive MIMO transmission. In: Proceedings of IEEE Symposium on Personal Indoor and Mobile Radio Communications, Washington DC, USA, Horn R A, Johnson C R. Matrix Analysis. 2nd ed. New York: Cambridge University Press, Appendix A { Define the notation ϵ t k tr R gk }. As the channel covariance matrices are Hermitian positive semidefinite, we can obtain { ϵ t k tr R k R k R k ρ t τ I R k}. A1 With the condition that R k R ln = 0 tr { R k R ln } = 0, A2

13 Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:12 for k,, l, n K, k, l, n, we can obtain C k R k = R k + 1 ρ t τ I R k = R k R k + 1 ρ t τ I. A3 Then, we obtain the following expression C 1 k R k = R k R k ρ t τ I. A4 Using A3, we can obtain C 1 k R k = R k ρ t τ I R k, A5 which achieves the equality in A1. Thus, we can conclude that under the condition of A2, the minimum value of the MSE-CPR is given by K K { } K K ε t = min ϵ t k = tr {R k R k R k ρ t τ I R k}. A6 =1 k=1 =1 k=1 Appendix B Substituting 19 and 20 into 21, we can obtain H ϵ u = tr E W T N G n a u n + 1 n u ρ u a u W T N G n a u n + 1 n u ρ u a u tr {Q }. B1 Because a u n CN 0, I K, n u CN 0, I M, data signals and additive noise are independent; thus, Q can be simplified as { N } Q = E W T G n G H n W + 1 ρ u W T W + I K W T G G H W. B2 Using the fact that G n = Ĝn + G n and G n CN 0, K k=1 R g kn, Q can be further simplified as [ N N ] Q = W T Ĝ n Ĝ H n + K R gkn + 1 ρ u I M W + I K W T Ĝ ĜH W. B3 k=1 Thus, we obtain ϵ u, which is given by [ ϵ u {W = tr N N ] } T Ĝ n Ĝ H n + K R gkn + 1 ρ u I M W + I K W T Ĝ ĜH W. B4 k=1 ϵ u is convex with respect to W. Thus, we can obtain the optimal value of W opt as follows [ N W ϵ u = N ] T Ĝ n Ĝ H n + K R gkn + 1 ρ u I M W Ĝ = 0, B5 k=1 which leads to [ N N ] Ĝ n Ĝ n + K R gkn + 1 ρ u I M W Ĝ = 0. B6 k=1 Thus, the optimal robust receiver can be obtained as [ N K ] 1 W = R gkn + ĜnĜH n + 1 ρ u I M Ĝ. B7 k=1 Appendix C Take â u k and au kn as the kth element of vectors of â u and au n, respectively, and w k as the k th column vector of the matrix W. Substituting 19 into 20 and then expanding, we have â u k = N K wt k g mn a u mn + 1 n u. C1 m=1 ρ u BS has obtained the channel estimate ĝ k through uplink training; therefore, we have â u k = wt kĝka u k + w T N K k ĝ mn a u mn + g qp a u qp + 1 n u. }{{} m,n k, p=1 q=1 ρ u C2 Signal } {{ } Effective Noise Thus, the uplink SINR can be expressed as SINR u k = wk T wkĝk T 2 [ N K k=1 R g + ĜnĜH kn n ]. C3 + 1 ρ u I ĝ k ĝk H wk

14 Appendix D Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:13 Eq. 27 can be simplified as { N }} ϵ d {tr = E α 2 α 2 BH G n GT n B α G T B α B H G + ρ d + 1 K. D1 Because G n = Ĝn + G n and G n Gn CN 0, K k=1 R g kn, D1 can be further simplified to N K } ϵ d {α = tr α 2 2 BH R g kn + Ĝ nĝt n B α Ĝ T B α B H Ĝ + ρ d + 1 K. D2 k=1 The above obective function is non-convex with respect to B, α. Note that 28 is equivalent to minimize B minimize α B ϵ d B, α, { } subect to tr B H B K. The optimal value of α for the inner unconstrained optimization problem can be readily obtained through N K } ϵ d {2α α = 0 tr B H R g kn + Ĝ nĝt n B ĜT B B H Ĝ + 2α ρ d K = 0, k=1 which yields } tr {ĜT B + B H Ĝ α = { { K 2 + tr B H N K } ρ d k=1 n }. D5 R gkn + Ĝ nĝt B Substituting D5 into D2, we have 2 tr {ĜT ϵ d = K B + B H } Ĝ { { K 4 + tr B H N K } ρ d k=1 n }. D6 R gkn + Ĝ nĝt B Then, the problem in D3 can be simplified to minimize B, { } subect to tr B H B K. B ϵ d The feasible set of D7 is compact and the obective function is continuous over the feasible set. Thus, there exists a global minimum for D7 according to the Weierstrass extreme value theorem [13]. In the following analysis, we will derive B by introducing the Lagrangian function and the Karush-Kuhn-Tucker KKT necessary conditions. The Lagrangian associated with the problem in 28 is { } LB, α, λ = ϵ d + λ tr B B H K, D8 where λ is the Lagrange multiplier associated with the inequality constraint. The corresponding KKT conditions can be obtained as { N K } LB, α, λ = tr 2α B H R g α kn + Ĝ nĝt n B ĜT B B H Ĝ + 2α ρ d K = 0, D9 k=1 N K B LB, α, λ = α 2 R g kn + Ĝ nĝt n B α Ĝ + λ B = 0, D10 k=1 { } λ 0, tr B B H K, D11 { } λ tr B B H K = 0. D12 From D9, we have { } N K } tr {ĜT B + B H Ĝ = 2α tr B H R g kn + Ĝ nĝt n B + 2α ρ d K. D13 k=1 With D10, we have Thus, we arrive at D3 D4 D7 [ N Ĝ = α K ] R g kn + Ĝ nĝt n + λ α 2 I M B. D14 k=1 B H Ĝ = ĜT B = α B H [ N K R g kn + Ĝ nĝt n k=1 ] + λ α 2 I M B. D15

15 Zhong W, et al. Sci China Inf Sci January 2015 Vol. 58 xxxxxx:14 Substituting D15 into D13, we can obtain When combining the previous result withd12, we obtain Substituting λ into D14, we can obtain B as { } λ tr B H B = α2 ρ d K > 0. D16 { } tr B H B = K, λ = α2 ρ d. D17 [ B = 1 N K ] 1 R g α kn + Ĝ nĝt n + 1 ρ d I M Ĝ D18 k=1 [ = 1 N K ] 1 R gkn + α ĜnĜH n + 1 ρ d I M Ĝ, D19 k=1 { } where α is chosen so as to satisfy the power constraint tr B H B = K. Appendix E Take yk d, ad kn, and nd k as the kth element of y d, ad n, and nd, respectively, and b kn as the k th column vector of matrix B n. From 25, we have N yk d = K gnk T b mna d mn + 1 n d m=1 ρ d k. E1 { } The average effective channel perfectly learned by UE k, is E α gk T b k. After potentially power scaling, signal x d k = α yk d received by UE k, can be expressed as } x d k {α = E gk T b N K k a d k + α gnk T bmnad mn + α } n d }{{} m=1 ρ d k {α E gk T b k a d k. E2 Signal }{{} Effective Noise Thus, the downlink SINR can be expressed as { } E α g T SINR d k = k b 2 k { N α K m=1 E gnk mn T b 2} { } + E α 2 { }. E3 E α gk T b 2 k ρ d