IEEE ICC 2015 - Workshop on 5G & Beyond - Enabling Technologies and Applications
Uplink Sum-Throughput Evaluation of Sectorized Multi-cell Massive MIMO System Jiahui Li, Qiang He, Limin Xiao, Xibin Xu, and Shidong Zhou State Key Laboratory on Microwave and Digital Communications Tsinghua National Laboratory for Information Science and Technology Department of Electronic Engineering, Tsinghua University, Beijing 100084, China Email:
[email protected]
Abstract—In this paper, a sectorized multi-cell massive multiple-input multiple-output (MIMO) system is considered with a spatially correlated channel model. Since sum-throughput per cell is an important index for evaluating a multi-cell system, we derive the ergodic achievable uplink sum-throughput per cell of the sectorized system and give the deterministic approximation of it based on the large random matrix theory. Numerical results indicate that sum-throughput per cell can be greatly increased compared to the conventional multi-cell massive MIMO system, which validates the effectiveness of the sectorized system. Moreover, it can be seen that the deterministic approximation is consistent with the result of Monte-Carlo simulation. Index Terms—massive MIMO, multi-cell, sectorized, uplink sum-throughput
I. I NTRODUCTION Multiuser multiple-input multiple-output (MIMO) systems, which equip base stations (BSs) with multiple antennas and serve multiple users simultaneously, have been introduced into wireless broadband standards like LTE [1]. Since more BS antennas can improve the system throughput or link reliability [2], very large MIMO or massive MIMO has attracted substantial interest. In massive MIMO time division duplex (TDD) systems, orthogonal pilot sequences are generally used in each cell to guarantee more accurate channel state information (CSI) for each user, thus the length of pilot sequences should be at least equal to the number of users in each cell [3], [4]. As the number of users increases, a great proportion of channel coherence interval (defined as the product of coherence time and coherence bandwidth) will be spent in uplink training phase, which finally limits the system throughput. A typical way to solve the problem mentioned above is to group users and reuse pilot resource among different groups in the same cell [5]-[8]. The method in [5] partitions users into groups, each group with similar channel covariance This work was supported by National Basic Research Program of China (2013CB329002), National Natural Science Foundation of China (61201192), Tsinghua University Initiative Scientific Research Program (2011Z02292), International Science and Technology Cooperation Program (2012DFG12010), National S&T Major Project (2013ZX03001024-004), Key Grant Project of Chinese Ministry of Education (313005), China’s 863 Project (2012AA01A502), the open research fund of National Mobile Communications Research Laboratory, Southeast University (2012D02), TsinghuaQualcomm joint research program.
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eigenvectors, and a two-stage precoding scheme is utilized to serve them. Based on the two-stage precoding framework, [6] proposes an improved K-means user grouping scheme. In [7], a relationship between channel spatial correlations and power angle spectrum is established, and a pilot reuse scheme to reduce the pilot overhead based on this relationship is proposed. However, all these methods need to estimate the channel covariance matrices of all users and execute the user grouping algorithms, which are of high computational complexity. Fortunately, a low complexity sectorization method proposed by [8] can overcome this shortcoming, which groups users based on user locations. The single-cell scenario is considered and the azimuth domain is equally divided into several sectors, each of which corresponds to a user group. The pilot resource is reused among different sectors, which makes more users be simultaneously served and pilot overhead be reduced. It concludes that the sum-throughput can be greatly increased compared to the conventional massive MIMO system. However, the multi-cell scenario is not addressed, which is more practical in real cellular systems. In this paper, we introduce the sectorization method in [8] to the multi-cell scenario, where the BS in each cell is equipped with a large uniform linear antenna array (ULA). Under the assumption that different cells use the same set of orthogonal pilots, we derive the ergodic achievable sum-throughput per cell of the uplink, and give the deterministic approximation of it by using the large random matrix analysis methods introduced in [9]. Based on the Monte-Carlo simulation and the deterministic approximation, the effects of several parameters on the sum-throughput per cell for the sectorized system are evaluated, such as pilot reuse factor among different sectors in each cell, the number of users per cell, uplink signal-to-noise ratio (SNR), coherence interval, signal angle-of-arrival (AOA) spread and inter-cell cross gain. The paper is organized as follows: In Section II, the channel model is described and a benchmark system is given. Section III describes the sectorization method. In Section IV, the ergodic achievable uplink rate and its deterministic approximation for each user in the sectorized system are derived. We present some numerical results and related discussions in Section V and the paper is concluded in Section VI.
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IEEE ICC 2015 - Workshop on 5G & Beyond - Enabling Technologies and Applications
1) Uplink transmission: For each use of the channel in uplink transmission, the signal received by the jth BS is √ X √ Hjl xl + nj (3) yj = pul Hjj xj + pul l6=j N ×K
Fig. 1. A multi-cell system, each cell has one BS equipped with a large ULA with N antennas and K single-antenna users
Notations: Vectors are column vectors and denoted by lower case boldface and italic: x. Matrices are upper case boldface: A. IN is the size-N identity matrix. The trace, transpose and Hermitian transpose are denoted by tr(·), (·)T and (·)H , respectively. The 2-norm of a vector x is denoted by ||x||. CN (0, Σ) stands for the circular symmetric complex Gaussian distribution with mean 0 and covariance matrix Σ. E[·] is the expectation operator and E[·|·] denotes the conditional expectation operator. 1) cells is considered, where the BS deployed at the center of each cell is equipped with a large ULA of N elements and serves K single-antenna users, as schematically shown in Fig. 1. Assume that all the BSs and users are perfectly synchronized and user channels are independent of each other. We consider transmissions over flat fading channels and use a spatial correlation channel model introduced in [5] and [10].
ntrjk
where ∼ CN (0, IN ) is the training noise and ptr is the effective training SNR, which depends on the pilot transmit power and the length of pilot sequences. Assume that ptr is a given parameter. The MMSE estimate of hjjk is given by [9] ˆjjk = Rjjk Qjk ytr h jk
ˆjjk ∼ CN (0, Φjjk ). Here we have which is distributed as h Φjlk = Rjjk Qjk Rjlk , ∀j, l, k !−1 X 1 Qjk = Rjlk + IN , ∀j, k ptr
A. Channel Model In this model, the uplink channel gain vector between the kth (k = 1, . . . , K) user in the lth (l = 1, . . . , L) cell and the jth BS (j = 1, . . . , L) can be modeled as 1 2 vjlk hjlk = Rjlk
(1)
where vjlk ∼ CN (0, IN ) is the fast fading channel vector and Rjlk = E[hjlk hH jlk ] is the channel correlation matrix. For a ULA, we have Rjlk = βjlk E{aN (θjlk )[aN (θjlk )]H } (see [10] for details), where θjlk is a random variable with a certain distribution, describing the signal AOA from the kth user in the lth cell to the jth BS, βjlk is the slow fading coefficient, including path loss and shadow fading, and h iT aN (θ) = 1, e−j2π∆ cos θ , . . . , e−j2π(N −1)∆ cos θ (2) is the steering vector, where ∆ is the antenna spacing normalized by carrier wavelength λ.
(6) (7)
l
˜jjk = hjjk − h ˆjjk . Since MMSE Define estimation error as h estimation is used and Gaussian channel fading coefficients ˜jjk ∼ CN (0, Rjjk − Φjjk ) is independent are considered, h ˆ of hjjk . 3) Achievable uplink rates: Consider MMSE detection of xj . The detector for user k in the jth cell is 1 ˆjjk IN )−1 h (8) pul where Zj is the covariance matrix of channel estimation error and inter-cell interference, given as X ˜ jj H ˜H + Zj = E H Hjl HH jj jl l6=j (9) X XX = (Rjjm − Φjjm ) + Rjlm ˆ jj H ˆ H + Zj + rjk = (H jj
m
B. Benchmark System In the conventional multi-cell massive MIMO system, users in each cell transmit mutually orthogonal pilot sequences to obtain the CSI at the BS, such a system is referred to as the benchmark system in the following analysis.
(5)
l6=j m
Using a standard bound given by [11] based on the worstcase uncorrelated Gaussian noise, the ergodic achievable uplink rate of user k in the jth cell is given by bk bk Cjk = E log2 1 + γjk (10)
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IEEE ICC 2015 - Workshop on 5G & Beyond - Enabling Technologies and Applications
bk where γjk denotes the signal-to-interference-plus-noise ratio (SINR), given by (11) on the bottom of the page.The superscript “bk” refers to the benchmark system. The deterministic bk bk approximation of γjk , denoted by γ¯jk , can be found in [9, eq. (25)].
III. S ECTORIZATION M ETHOD Consider a spatial filter [8] for the ULA with weight vector ω 1 = [ω0 , ω1 , . . . , ωM −1 , 0, . . . , 0]H ∈ CN . The spatial response can be written as u1 (θ) = ω H 1 aN (θ) =
M −1 X
total user number is K = QK 0 . Other parameters such as cell number and BS antenna number are the same as described in Section II. The signal received by sector q of the jth BS can be written as X√ √ pul WqH Hjjp xjp yjq = pul WqH Hjjq xjq + | {z } p6=q desired signal | {z } inter-sector interference X X√ (17) + pul WqH Hjlp xlp +WqH nj l6=j
| ωn e−j2πn∆ cos θ
(12)
n=0
Shift the elements of ω 1 by b times with shift step length m, another weight vector can be obtained: ω b+1 = [ 0, . . . , 0, ω0 , ω1 , . . . , ωM −1 , 0, . . . , 0 ] | {z } | {z }
H
(13)
N −bm−M
bm
{z
inter-cell interference
0
IV. U PLINK R ATE D ERIVATION WITH S ECTORIZATION
ub+1 (θ) = u1 (θ)e−j2πbm∆ cos θ , b ≤ (N − M )/m
(14)
Define weight matrix W = [ω 1 , . . . , ω B ], where B = b(N − M )/mc + 1 and btc denotes a maximum integer not larger than t. The spatial responses corresponding to all these weight vectors can be written as a response vector: u(θ) = WH aN (θ) = u1 (θ)am B (θ)
(15)
where am B (θ) is given by h iT −j2πm∆ cos θ am , . . . , e−j2π(B−1)m∆ cos θ (16) B (θ) = 1, e Elements of u(θ) have the same magnitude for a given θ, which means the beams generated by columns of W have the same directional characteristics, and the phase differences between the elements are the same as those generated by a ULA of B elements with element spacing m∆λ. Therefore, the completely overlapped beams corresponding to each sector can be regarded as multiple sector antennas. Besides, by designing ω 1 , the basic weight vector of W, we can divide the azimuth domain into several sectors. Remark: Since the system using the sectorization method is similar to the current cellular system which has sectors implemented by using directional antennas, many methods used there can be directly applied, e.g., strategies of user scheduling across sectors. In addition, different sectors under the proposed scheme share the same ULA, so it requires smaller size to mount antennas. Assume that the azimuth domain is equally divided into Q sectors. Each sector serves K 0 user simultaneously, and the
In this section, we derive the ergodic achievable uplink rate of each user in the sectorized system. Assume that different cells use the same set of orthogonal pilots (extension to other pilot reuse scheme among different cells can be easily obtained), and the pilot reuse factor among different sectors in one cell is α(α = 1, 2, . . . , Q), which means the Q sectors are divided into α groups, each uses 1/α of the total pilot resource and pilot sequences for each group are orthogonal to each other. Users in each sector use orthogonal pilot sequences, and the same set of pilots are reused by all sectors in the same group. The minimum length of pilot sequences is αK 0 . Let Gg (g = 1, 2, . . . , α) and G represent the index set of the sectors in the gth group and the index set of all the sectors, respectively. Assume that each cell has the same sector pilot reuse scheme. During the training phase, all users will transmit their pilot sequences synchronously. The jth BS estimates the channel vector hjjqk of the kth (k = 1, 2, . . . , K 0 ) user in sector q(q ∈ Gg ) of the jth (j = 1, 2, . . . , L) cell based on the observation ytrjgk ∈ CN , given as X ytrjgk = hjjqk + hjjpk p∈Gg ,p6=q
+
XX l6=j p∈Gg
1 hjlpk + √ ntrjgk ptr
E rH jk
˜jjk h ˜H h jjk
+
(18)
where ntrjgk ∼ CN (0, IN ) denotes the training noise. The MMSE estimate of hjjqk is written as ˆjjqk = Rjjqk Qjgk ytr h jgk
2 ˆ |rH jk hjjk |
h
}
where Hjlp = [hjlp1 , . . . , hjlpK 0 ] ∈ CN ×K is the channel matrix between users in sector p of the lth cell and the jth BS, Wq ∈ CN ×B is the weight matrix for the qth sector, 0 xjp ∈ CK is the transmitted signal vector of users in sector p of the jth cell with i.i.d. zero-mean and unit-variance elements, and nj ∼ CN (0, IN ) is the noise at the receiver.
The corresponding spatial response of ω b+1 is
bk γjk =
p
H H l Hjl Hjl − hjjk hjjk +
P
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1 pul IN
i ˆ rjk H jj
(19)
(11)
IEEE ICC 2015 - Workshop on 5G & Beyond - Enabling Technologies and Applications
ˆjjqk ∼ CN (0, Φjqjqk ). Here we have which is distributed as h Φjqlpk = Rjjqk Qjgk Rjlpk , ∀j, l, g, k, ∀q, p ∈ Gg
Here we have H Θjqlpk = Wq Rjlpk Wq , ∀j, l, q, p, k Ξjqlpk = WqH Φjqlpk Wq , ∀j, l, g, k, ∀q, p ∈ Gg = WqH Rjjqk Qjgk Rjlpk Wq
(20)
H ∀j, l, p, k Rjlpk = E[hjlpk hjlpk ], −1 XX (21) 1 ∀j, g, k Rjlpk + IN , Qjgk = ptr
The ergodic achievable uplink rate of the mth user in sector q of the jth cell is sr sr Cjqk (29) = E log2 1 + γjqk
p∈Gg
l
˜jjqk = hjjqk − h ˆjjqk . Since Define estimation error as h MMSE estimation is used and Gaussian channel fading co˜jjqk ∼ CN (0, Rjjqk − Φjqjqk ) is efficients are considered, h ˆjjqk . independent of h 0 Define Gjqlp = WqH Hjlp = [gjqlp1 , . . . , gjqlpK 0 ] ∈ CB×K ˆ jqlp = WH H ˆ jlp as the as the effective channel matrix, G q ˜ ˜ jlp estimate of the effective channel matrix and Gjqlp = WqH H as the estimation error of the effective channel matrix. Then the receive signal in (17) can be written as yjq =
√
ˆ jqjq xjq + zjq pul G
sr where γjqk denotes the SINR, given by (26) on the bottom of the page. The superscript “sr” refers to the benchmark system. sr sr Denote the deterministic approximation of γjqk by γ¯jqk , which is derived using the large random matrix analysis methods introduced in [9] and given by (27) on the bottom of the page with
νjqlpmk
(22)
p6=q
+
ϑjqlpm
where c 1) Tjq = T Bpqul and δ jq = [δjq1 , . . . , δjqK 0 ]T = c δ Bpqul are given by [9, Th. 1], for S = Zjq /B, D = IB and Rm = Ξjqjqm , cq 0 0 ¯ 2) Tjq = T Bpul is given by [9, Th. 2], for S = Zjq /B, Θ = WqH Wq , D = IB and Rm = Ξjqjqm ,
(23)
pul Gjqlp xlp + WqH nj
p
l6=j
0 2
