A lowcomplexity algorithm for the joint antenna selection and user scheduling in multicell multiuser downlink massive MIMO systems
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Abstract
The massive MIMO (multipleinput multipleoutput) technology plays a key role in the nextgeneration (5G) wireless communication systems, which are equipped with a large number of antennas at the base station (BS) of a network to improve cell capacity for network communication systems. However, activating a large number of BS antennas needs a large number of radiofrequency (RF) chains that introduce the high cost of the hardware and high power consumption. Our objective is to achieve the optimal combination subset of BS antennas and users to approach the maximum cell capacity, simultaneously. However, the optimal solution to this problem can be achieved by using an exhaustive search (ES) algorithm by considering all possible combinations of BS antennas and users, which leads to the exponential growth of the combinatorial complexity with the increasing of the number of BS antennas and active users. Thus, the ES algorithm cannot be used in massive MIMO systems because of its high computational complexity. Hence, considering the tradeoff between network performance and computational complexity, we proposed a lowcomplexity joint antenna selection and user scheduling (JASUS) method based on Adaptive Markov Chain Monte Carlo (AMCMC) algorithm for multicell multiuser massive MIMO downlink systems. AMCMC algorithm is helpful for selecting combination subset of antennas and users to approach the maximum cell capacity with consideration of the multicell interference. Performance analysis and simulation results show that AMCMC algorithm performs extremely closely to ESbased JASUS algorithm. Compared with other algorithms in our experiments, the higher cell capacity and nearoptimal system performance can be obtained by using the AMCMC algorithm. At the same time, the computational complexity is reduced significantly by combining with AMCMC.
Keywords
5G Massive MIMO systems Antenna selection User scheduling Adaptive Markov chain Monte Carlo algorithmAbbreviations
 AMCMC
Adaptive Markov chain Monte Carlo
 BS
Base station
 CSI
Channel state information
 ES
Exhaustive search
 JASUS
Joint antenna selection and user scheduling
 MIMO
Multipleinput multipleoutput
 MU
Multiuser
 RF
Radio frequency
 SINR
Signaltointerferenceplusnoise ratio
 TDD
Time division duplexing
 ZF
Zeroforcing
1 Introduction
In order to satisfy the rapidly increasing requirements for high data rate in current wireless communication systems, a new massive MIMO (multipleinput multipleoutput) technology was introduced in [1, 2, 3]. Massive MIMO technique plays a key role to enhance the cell capacity without increasing system bandwidth or base station (BS) transmission power for the 5G network systems [4]. The key idea of the massive MIMO technique is to install a large amount of transmit antennas at the BS of a cellular and provide services for several users sharing the same spectrum resources. However, as the number of BS antennas and users increases, the combination complexity and hardware cost also increase dramatically. Therefore, when the numbers of BS transmit antenna and active users are extremely large, the joint antenna selection and user scheduling (JASUS) algorithm [5, 6, 7, 8] can be adopted as an approach to decide the radio frequency (RF) chain configuration to improve the cell capacity in multicell massive multiuser MIMO systems.
In a practical network, one of the key challenges in multicell multiuser massive MIMO systems is the hardware cost and power consumption because the element of each antenna needs a complete RF chain that consists of RF amplifiers and analogtodigital converters, which are very pricey and are the main elements of the power consumption at the BS [9]. Different schemes were used in many types of research, such as hybrid precoding and spatial modulation, to reduce the cost of the hardware and the power consumption of the system [10]. One of the best schemes to solve this problem is to applying antenna selection [11, 12, 13] to decide optimal subset of BS transmit antennas for decreasing the required number of high pricey RF chains while decreasing the resulting network performance loss.
However, in multicell multiuser massive MIMO systems, only a limited number of transmit antennas are selected to provide services for active users scheduled. Hence, if the number of users exceeds, the number of selected transmit antennas, user scheduling must be performed because different wireless channels have different properties. High cell capacity can be obtained by scheduling users with the high channel quality. Therefore, the research of JASUS method for multicell multiuser massive MIMO systems is necessary.
Recently, only a few types of research have studied a low complexity JASUS for downlink massive multiuser MIMO systems. Benmimoune et al. [14] proposed a twostep JASUS scheme for downlink multiuser massive MIMO systems. It successively closed unnecessary antennas and removes undesired users which contribute little to system performance. However, due to the high computational complexity, this algorithm can only be employed to the scenarios with a smaller number of candidate antennas and user sets. Thus, using this algorithm in practical multicell multiuser massive MIMO system scenarios is difficult. Olyaee et al. [15] proposed a JASUS method based on zeroforcing (ZF) precoding algorithm for singlecell multiuser massive MIMO downlink systems. Though the ZF precoding method has a high system performance, it also has a very high computational complexity. For distributed downlink multiuser massive MIMO system, a JASUS method was proposed in [16] by Xu. et al. It successively obtains the majority of gain with limited backhaul capacity. Lee et al. [17] proposed a random antenna selection algorithm, the algorithm can provide significant capacity efficiency gain, but it is difficult to use for multicell multiuser massive MIMO systems. However, the above researches focused on singlecell multiuser massive MIMO systems. Thus, the research of JASUS method for multicell multiuser massive MIMO systems remains a largely open area. Therefore, the novel JASUS algorithm with considered multicell interference, which causes no or only a few decreases of system performance, represents a new promising research topic.
 1.
A low complexity JASUS method based on AMCMC algorithm is proposed for downlink multicell multiuser massive MIMO systems. AMCMC algorithm is helpful for selecting combination subset of antennas and users to approach the maximum cell capacity while decreasing the resulting network performance loss.
 2.
In this paper, we proposed updating rules for the selection probability of each base station transmit antenna and the scheduling probability for each user. In addition, we also proposed a new projection strategy to satisfy the constraints of antenna and user selection.
 3.
Performance analysis and simulation results show that our proposed algorithm produced promising results. Compared with ESbased JASUS algorithm, the proposed algorithm achieved comparable performance with low complexity. In addition, the AMCMCbased JASUS algorithm outperforms greedybased JASUS and normbased JASUS methods in terms of cell capacity and SER (symbol error rate) performance.
Notation: Symbol ℂ denotes the set of complex numbers, vectors are denoted by using lowercase bold letters, matrices are denoted by using bold letters, . denotes the absolute value of a scalar, ‖·‖_{F} denotes the Frobenius norm function, and (.) represents the binomial coefficient.
The remaining content is organized as follows. In Section 2, the system model and capacity maximize problem formulation are described. In Section 3, we formulate the problem of JASUS method based on AMCMC in multicell multiuser massive MIMO systems. Section 4 presents the simulation setup and assumption. In Section 5, we discuss the simulation results and analyze the complexity; finally, this work is concluded in Section 6.
2 System model and problem formulation
In this part, we simply give the system model for multicell massive multiuser
MIMO downlink systems with the system capacity formulation model with consideration of the multicell interference.
2.1 System model
Our objective is to find optimum combinations subset of BS antennas and users to approach the maximum cell capacity while decreasing the resulting network performance loss. Furthermore, we will decrease the number of expensive RF chains and avoid the uneconomic costs of the hardware and decrease power consumption caused by selecting undesired antennas to provide the requirement of service.
2.2 Problem formulation
2.3 Capacity of massive MIMO
Addressing the aforementioned problem by employing an ES method needs to evaluating the cell capacity of \( \varphi \triangleq {C}_M^N\times {C}_U^K \) joint antenna and user combinations, where \( {C}_M^N \) and \( {C}_U^K \) are the binomial coefficient. This fact indicates that the ES method cannot be used in current the massive MIMO systems where U and M are very numerous, it leads to high computational complexity. Thus, a low complexity algorithm for JASUS is needed in order to obtain the best network performance with low computational complexity.
 1.
Intercell interference: the first problem is the intercell interference coming from other cells. In order to solve this problem, cells can adjust their precoding matrices, thus can eliminate or decrease the interference from all users. In order to make better use of the coordinated massive MIMO technology, a group of users in the cells should be scheduled so that each group of users in the cell has the biggest spatial separation with the interference channels of users in neighboring cells.
 2.
Computational complexity: the second problem is how to obtain the best combination subset of antenna and user in each cell with lower computational complexity so as to decrease or eliminate the intercell interference and maximize the sum capacity of all cells. We know that in multicell massive MIMO systems, the various path loss between the antennas coming from neighboring cells and users coming from target cell also bring to much computational complexity same to the antenna selection and user scheduling. Thus, the computational complexity of the ES algorithm becomes very large than of a singlecell JASUS.
 3.
CSI feedback cost: for massive MIMO systems, the perfect CSI feedback depends mainly on the number of active antennas and the users they support. Hence, in order to centralize processing for selecting antennas and scheduling users across cells, the BS needs to exchange the overall CSI of overall combined subset antennas and users at each scheduling period, it brings more burden for BS. In addition, when the number of BS antennas and users in each cell increases, the cost of CSI increases accordingly.
On the base of the aforementioned discussion, a low complexity scheme is needed from the practical point of view for JASUS in multicell multiuser massive MIMO scenarios to reduce the complexity of function (11) while decreasing the cost of the CSI feedback.
3 Joint antenna selection and user scheduling algorithm
In this part, we presented two suboptimal iterative algorithms for JASUS before a discussion of the proposed AMCMC method.
3.1 Normbased JASUS algorithm
3.2 Greedybased JASUS algorithm
3.3 AMCMCbased JASUS algorithm
Although the greedybased JASUS method improved the cell capacity, it ignores the computational complexity of all system. For the actual network communication system, it will not have commercial value or attraction. Hence, considering the tradeoff between cell capacity and complexity, we proposed a lowcomplexity JASUS method based on AMCMC algorithm and its description is as follows.
MCMC [21] is a method of generating random samples, which is often used to calculate statistical estimation, marginal probability, and conditional probability. MCMC algorithms depend on (Markov) sequences with limit distributions corresponding to interest distributions. In the past decades, it has been widely used in many fields such as engineering and statistics [22]. The key idea of the MCMC method is that Markov chains are simulated in state space X, and the stable distribution of the chains is the target distribution π [23].
In order to address our objection problem in (11), we must tackle tow essential issues when applying AMCMC algorithm. The first problem is how to provide a proposal distribution of candidate samples L_{MCMC}. The second problem is how to design the most suitable updating rule for the proposal distribution.
3.3.1 Derivation of the candidate sampling distribution for the MCMC method

A candidate sample Ω_{[new], t} is drawn from proposal distribution Ψ(Ω_{ℓ, t}; R_{t − 1}).

Simulate u ∼ uniform[0, 1], and according to the accepting probability α(Ω_{[ℓ], t}, Ω_{[new], t}), let
After L_{MCMC} iterations, we can achieve a set of samples \( \left\{{\boldsymbol{\Omega}}_{\left[0\right],t},{\boldsymbol{\Omega}}_{\left[1\right],t},\dots, {\boldsymbol{\Omega}}_{\left[{L}_{\mathrm{MCMC}}\right],t}\right\} \), which is subjected to distribution π(Ω_{ℓ, t}).
3.3.2 Derivation of updating rule for the AMCMC algorithm
The updated proposal distribution Eqs. (32) and (33) are iteratively used with the objective to close to the target distribution.
3.4 Constraints for the AMCMCbased JASUS problem
For the JASUS problem, the vectors \( \overline{\boldsymbol{\upomega}} \) and ω are subject to constraint functions (12) and (13), respectively. However, employing functions (32) and (33) with MIS to generate samples, but we cannot ensure that the samples meet the constraints (12) and (13). In order to ensure that the samples drawn from (32) and (33) meet the constraints (12) and (13), we propose a new projection strategy. For convenience, we only introduce the proposed projection strategy for ω because the similar projection strategy can be used to \( \overline{\boldsymbol{\upomega}} \).

If \( {\sum}_{m=1}^M{I}_m\left({\boldsymbol{\upomega}}_{\mathrm{\ell},t}\right)<N \), then the proposed projection strategy sequentially selects the BS antenna with the biggest probability from the set ϕ_{0} to the set ϕ_{1} until ϕ_{1} = N, where ϕ_{1} is the number of elements of the set ϕ_{1}.

If \( {\sum}_{m=1}^M{I}_m\left({\boldsymbol{\upomega}}_{\mathrm{\ell},t}\right)>N \), then the BS antennas with the smallest probability in the set ϕ_{1} are closed sequentially according to the proposed projection strategy until ϕ_{1} = N.
3.5 Convergence analysis for the AMCMCbased JASUS algorithm
In order to obtain a higher convergence rate, the probability parameters R_{t − 1} of the proposal distribution Ψ(Ω_{ℓ, t}; R_{t − 1}) are adjusted. In this paper, we use literature [28] to explain the convergence problem, because our proposed method gives a similar description of the convergence problem and proves the effectiveness of the method through analysis. Besides, the complexity of the MCMC algorithm has been proved to be only related to sample size L_{MCMC} in [22]. The proposed adaptive strategy requires less sample size and iteration times, which can significantly improve the convergence speed of the MCMC algorithm.
3.6 Constrained AMCMCbased JASUS algorithm
4 Simulation configuration
Simulation parameters setting
Parameters  Values 

No. of cells in the simulation Β  7 
Average no. of UEs in one cell  2 ≤ U ≤ 50 
Number of BS antennas M  10 ≤ M ≤ 64 
Intersite distance  500 m 
Cell radius  295 m 
Path loss  128.1 + 37.6×log_{10}(distance(km))[dB] 
Each BS transmission power  10 dB 
Shadowing standard derivation  7 dB 
Noise spectral density  − 174 dBm/Hz 
Users’ speed  0 
System bandwidth  20 MHz 
5 Simulation results and analysis
In this section, we provide numerical results and computational complexity analysis of the proposed algorithm by simulative evaluation.
5.1 Performance evaluation
5.2 Computational complexity analysis
Computational complexity analysis
Algorithms  General case  M = 64, N = 16, U = 50, k = 10 

ESbased JASUS  \( \mathrm{O}\left({\mathrm{C}}_M^N{\mathrm{C}}_U^K{\mathrm{N}}^3\right) \)  2.1 × 10^{28} 
AMCMCbased JASUS  O(N^{3}tL_{MCMC})  2.8 × 10^{7} 
Greedybased JASUS  tO(MUN^{3})  3.9 × 10^{8} 
Normbased JASUS  O(MU) + tO(N^{3})  1.3 × 10^{5} 
Number of iterations t  1 ≤ t ≤ 50  t = 30 
Constant δ  0.5/1/1.5/2/2.5  δ = 2 
Number of promising samples L_{MCMC}  L_{MCMC} = δ × (M + U)  L_{MCMC} = 228 
6 Conclusion
In this paper, we studied the problem of JASUS in a multicell multiuser massive MIMO downlink system operating with TDD mode. Considering the tradeoff between network performance and computational complexity, we proposed a lowcomplexity algorithm for JASUS method based on AMCMC algorithm in the downlink multicell multiuser massive MIMO systems. AMCMC algorithm has been proven helpful for selecting combination subset of antennas and users to approach the maximum cell capacity with consideration of the intercell interference. In our algorithm, the updating rules of the selection probability of each base station antenna and scheduling probability for each user are proposed. In addition, we proposed a new projection strategy to satisfy the constraints of selection. Performance analysis and simulation results show that our proposed algorithm can produce promising results and achieve a good tradeoff between complexity and performance. Compared with ESbased JASUS algorithm, the proposed algorithm achieved comparable performance with very low complexity. In addition, we demonstrate that our proposed algorithm outperforms greedybased JASUS and normbased JASUS methods in terms of cell capacity and SER performance with under poorly conditioned channels. At the same time, the computational complexity is reduced significantly by combining with the proposed algorithm.
Notes
Acknowledgements
This work described in this paper was supported by the National Science and Technology Major Project: No. 2018ZX03001029004.
Authors’ contributions
WG conceived and designed the study. SM and KZ performed the simulation experiments. SM and KZ wrote the paper. XL and ZS reviewed and edited the manuscript. All authors read and approved the final manuscript.
Funding
The funding for the research reported is provided by the National Science and Technology Major Project: No. 2018ZX03001029004. The funds are mainly used for simulation hardware support.
Competing interests
The authors declare that they have no competing interests.
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