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A lowcomplexity algorithm for the joint antenna selection and user scheduling in multicell multiuser downlink massive MIMO systems
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 208 (2019)
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.
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.
In this paper, we consider the problem of JASUS in multicell multiuser massive MIMO downlink system operating with TDD mode. Considering the tradeoff between cell capacity and complexity, we proposed a lowcomplexity algorithm for JASUS method based on AMCMC. In our proposed method, only a small subset of BS transmit antennas is selected to serve predetermined active users, thus reduces the number of RF chains, avoids uneconomical hardware costs, and reduces power consumption caused by the selection of unnecessary transmit antennas to provide the required services. The main contributions of our work are as follows.

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.
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.
System model
As shown in Fig. 1, the considered scenario is a multicell multiuser massive MIMO downlink system operating in TDD mode, and the cell capacity maximizing problem is studied with consideration of the intercell interference. The system is composed of Β hexagonal cells. All B BSs, where B = {1, 2, …, Β} are installed with M antennas and serves U (M ≥ U ≥ 1) singleantenna users in each cell. The blockfading channel model is assumed. We assume that BS can select N transmit antennas among the M transmit antennas and schedule K(K ≤ N) users among the U users within the cell to be served simultaneously. The channel vector g_{iju} ∈ ℂ^{M} from the jth BS and user u in cell i can be expressed as
where β_{iju} denotes the large scale channel fading between jth BS and user u in cell i, including shadowing and path loss. h_{iju} is the smallscale fading vector, and h_{iju} = [h_{iju1}, h_{iju2}, …, h_{ijuM}]^{T} ∈ ℂ^{M}. Then, the overall downlink transmission matrix G_{ij} ∈ ℂ^{M × U} between the BS in cell j and all users in cell i can be expressed as
where H_{ij} = [h_{ij1}, h_{ij2}, …, h_{ijU}] ∈ ℂ^{M × U} is the overall smallscale fading matrix and D_{ij} = diag(β_{ij1}, β_{ij2}, …, β_{ijU}).
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.
Problem formulation
In the downlink system, the signal received by users in cell i can be written as
where p_{t} denotes the transmitted power, s_{j} = [s_{j1}, s_{j2}, …, s_{jU}]^{T} ∈ ℂ^{U} is the transmit signals for users in cell j, W_{j} ∈ ℂ^{M × U} is the precoding matrix of BS in cell j, and n_{i} = [n_{i1}, n_{i2}, …, n_{iu}]^{T} ∈ ℂ^{U} is the noise vector at the uth user in cell i. The downlink signal received by user u in cell i can be written as
where w_{jb} is the bth column of precoding matrix W_{j}. Formula (4) including the desired signal for the user u in cell i, intracell interference signal (comes from other users in the same cell), and intercell interference signal (comes from other cells), respectively.
Since we used the zeroforcing (ZF) methods and supposing that the channel state information (CSI) is perfectly known at the BS, the intracell interference signal (second term) in function (4) drops to zero according to prevenient works [14, 18,19,20]. Therefore, function (4) becomes
We assume that BS can choose N transmit antennas among the M transmit antennas, and schedule K(K ≤ N) users among the U users within the cell to be served simultaneously. For convenience, we give the selected subset of the antenna and scheduled subset of user indicator functions are ω and \( \overline{\boldsymbol{\upomega}} \),
where ω and \( \overline{\boldsymbol{\upomega}} \) are binary vectors that include two values 0 and 1 to indicate if a given antenna or a given user is selected. (e.g., 1 → selected, 0 → unselected).
For making an easy description, we will define to two indicator functions, which are I_{m}(ω) ≜ ω_{m} ∈ {0, 1} and \( {I}_u\left(\overline{\boldsymbol{\upomega}}\right)\triangleq {\varpi}_u\in \left\{0,1\right\} \), respectively. We use these to indicate whether the mth BS antenna and the uth user are selected or not, respectively. \( {\mathbf{s}}_{i\left\{\overline{\boldsymbol{\upomega}}\right\}}\in {\mathrm{\mathbb{C}}}^K \) denotes transmit signal vector, subblock channel matrix of corresponding denotes by \( {\mathbf{G}}_{ij\left\{\overline{\boldsymbol{\upomega}},\boldsymbol{\upomega} \right\}}\in {\mathrm{\mathbb{C}}}^{N\times K} \) and \( {\mathbf{n}}_{i\left\{\overline{\boldsymbol{\upomega}}\right\}}\in {\mathrm{\mathbb{C}}}^K \) is noise vector, respectively. Finally, in order to denote the joint antenna and user selection, we employ the 2tuple \( \boldsymbol{\Omega} \triangleq \left(\overline{\boldsymbol{\upomega}};\boldsymbol{\upomega} \right) \). In order to have an easy explanation, we will interchangeably use Ω and \( \left(\overline{\boldsymbol{\upomega}};\boldsymbol{\upomega} \right) \) for the following part. After using antenna selection and user scheduling method, function (5) becomes
where \( {\mathbf{w}}_{j\left\{{\varpi}_b\right\}} \) is bth column of precoding matrix \( {\mathbf{W}}_{j\left\{\overline{\boldsymbol{\upomega}},\boldsymbol{\upomega} \right\}}\in {\mathrm{\mathbb{C}}}^{N\times K} \) and \( {n}_{i\left\{{\varpi}_u\right\}} \) is the noise at the uth user in cell i.
Capacity of massive MIMO
According to the aforementioned discussion, the received signaltointerferenceplusnoise ratio (SINR) for the user u ∈ U (which is connected to cell i) with a selected channel vector \( {\mathbf{g}}_{ij\left\{\overline{\boldsymbol{\upomega}},\boldsymbol{\upomega} \right\}} \) can be written as
Considering the intercell interference, the formula of sum capacity for cell i can be expressed as
Our target is to jointly select the optimal combination sets of BS transmit antenna and active user to approach the maximum cell capacity while decreasing the computational complexity. Hence, the problem of JASUS can be written as
subject to
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.
Obviously, formula (11) serves as the target function in this study. Therefore, address the problem (11) in multicell multiuser massive MIMO systems, we need to solve the following three main problems:

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.
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.
Normbased JASUS algorithm
Firstly, we presented a normbased JASUS method for addressing the objective function (10). The normbased JASUS scheme maximizes \( {\left\Vert {\mathbf{G}}_{\left\{\overline{\boldsymbol{\upomega}},\boldsymbol{\upomega} \right\}}\right\Vert}_F \), where ‖.‖_{F} is the Frobenius norm function. Let \( {C}_{\mathrm{sum}}^{\mathrm{NB}}\left({\mathbf{G}}_{\left\{\overline{\boldsymbol{\upomega}},\boldsymbol{\upomega} \right\}}\right)={\left\Vert {\mathbf{G}}_{\left\{\overline{\boldsymbol{\upomega}},\boldsymbol{\upomega} \right\}}\right\Vert}_F \). This scheme including initialization step and iterative updating step, respectively. This both steps use the vector norm as criteria which considerably decrease each iterative computation complexity. The normbased JASUS problem is modeled as
where ω_{NB} is the combination selection indicator. Nevertheless, it has still a problem that when performing transmit antenna selection and user scheduling simply based on the Fnorm criteria, it would sacrifice some cell capacity. In sum, the normbased JASUS method has extremely low complexity, but it cannot guarantee a high sum capacity performance.
Greedybased JASUS algorithm
In order to enhance the cell capacity over normbased JASUS, we presented a greedybased JASUS algorithm. Unlike to the normbased JASUS algorithm, the greedybased JASUS algorithm maximizes the cell capacity in each step. This method also includes initialization step and iterative updating step. The greedybased JASUS problem is modeled as
where ω_{GR} is the combination selection indicator. Compared with the normbased JASUS algorithm, this method has a good capacity performance, but it has high computational complexity.
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.
Derivation of the candidate sampling distribution for the MCMC method
The biggest advantage of the MCMC method is that it can search the “elite samples” instead of exhaustively searching the whole samples. At iteration t, the samples \( {\left\{{\boldsymbol{\Omega}}_{\mathrm{\ell},t}=\left\{{\overline{\boldsymbol{\upomega}}}_{\mathrm{\ell},t},{\boldsymbol{\upomega}}_{\mathrm{\ell},t}\right\}\right\}}_{\mathrm{\ell}=1}^{L_{\mathrm{MCMC}}} \) from an MCMC method can be employed to estimate the maximum value of the target function \( {C}_{\mathrm{sum}}^i\left({\mathbf{G}}_{\left\{\overline{\boldsymbol{\upomega}},\boldsymbol{\upomega} \right\}}\right) \)
where L_{MCMC} denotes the total number of samples, and \( {\Phi}_C^{\ast } \) is the estimated value of formula (11).
Given that scheduled of per user ϖ_{u} and selection of each antenna ω_{m} are binary variables, we use the Boltzmann distribution of the objective function \( {C}_{\mathrm{sum}}^i\left({\mathbf{G}}_{\left\{{\overline{\boldsymbol{\upomega}}}_{\mathrm{\ell},t},{\boldsymbol{\upomega}}_{\mathrm{\ell},t}\right\}}\right) \) with a suitable temperature τ
where \( \Gamma ={\sum}_{{\boldsymbol{\Omega}}_{\mathrm{\ell},t}\triangleq \left({\overline{\boldsymbol{\upomega}}}_{\mathrm{\ell},t},{\boldsymbol{\upomega}}_{\mathrm{\ell},t}\right)}\exp \left\{\left({C}_{\mathrm{sum}}^i\left({\boldsymbol{\Omega}}_{\mathrm{\ell},t}\right)/\tau \right)\right\} \) is a normalization constant in the MCMC method that can be neglected. Thus, maximizing \( {C}_{\mathrm{sum}}^i\left({\boldsymbol{\Omega}}_{\mathrm{\ell},t}\right) \) is equivalent to maximizing π(Ω_{ℓ, t}), and π(Ω_{ℓ, t}) is the target distribution.
In order to prove the MCMC method for searching the distribution π(Ω_{ℓ, t}), we use a MIS (metropolized independence sampler) [23], which is a generic MCMC method. The step is as follow. An initial value Ω_{[0], t} is chosen randomly. Given the current sample Ω_{[ℓ], t},

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
Where
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}).
Derivation of updating rule for the AMCMC algorithm
In this part, we provide updating rule for the proposal distribution. For the AMCMC method, the joint proposal distribution is proportional to the product of Bernoulli distributions, namely
where p_{u} denotes the probability of the uth user being selected for communicating with the BS. That is, ϖ_{u} ∼ Ber(p_{u}) for u = 1, 2, …, U, and g_{m} is the probability of the mth BS antenna being selected. That is, ω_{m} ∼ Ber(g_{m}) for m = 1, 2, …, M. We use the indicator functions I_{m}(ω_{ℓ, t}) and \( {I}_u\left({\overline{\boldsymbol{\upomega}}}_{\mathrm{\ell},t}\right) \) to indicate whether the mth BS antenna and the uth user are selected or not, respectively. R_{t − 1} ≜ {P_{t − 1}, g_{t − 1}}, where \( {\mathbf{P}}_t={\left\{{p}_{u,t}\right\}}_{u=1}^U \), \( {\mathbf{g}}_t={\left\{{\mathrm{g}}_{m,t}\right\}}_{m=1}^M \), and Γ^{′} is a normalization constant that can be ignored in the AMCMC. The adaptation scheme is employed to adjust the parameterized proposal distribution Ψ(Ω_{ℓ, t}; R_{t − 1}) and minimize the KullbackLeibler divergence [24, 25] between the target distribution π(Ω_{ℓ, t}) and the proposal distribution Ψ(Ω_{ℓ, t}; R_{t − 1}), namely
It is observed that \( \tilde{D}=\pi \left({\boldsymbol{\Omega}}_{\mathrm{\ell},t}\right)\times \log \pi \left({\boldsymbol{\Omega}}_{\mathrm{\ell},t}\right)D\left[\pi \left({\boldsymbol{\Omega}}_{\mathrm{\ell},t}\right)\left\Vert \Psi \left({\boldsymbol{\Omega}}_{\mathrm{\ell},t};{\mathbf{R}}_{t1}\right)\right.\right] \) is a convex function [26]. Hence, the minimization of the KullbackLeibler divergence D[π(Ω_{ℓ,t})‖Ψ(Ω_{ℓ,t}; R_{t − 1})] w.r.t. R can be achieved when \( \partial \tilde{D}/\partial \mathbf{R}=0 \). Thus, \( \tilde{D} \) can be written as
We set the partial derivative of (22) to zero with respect to R. Then, the Eq. (22) can be written as
where the partial derivatives of logΨ(Ω_{ℓ,t}; R_{t − 1}) with respect to p_{u} and g_{m} are respectively given by
By substituting (24) and (25) into (23), we obtain
Given a number of samples \( {\left\{{\boldsymbol{\Omega}}_{\mathrm{\ell},t}=\left\{{\overline{\boldsymbol{\upomega}}}_{\mathrm{\ell},t},{\boldsymbol{\upomega}}_{\mathrm{\ell},t}\right\}\right\}}_{\mathrm{\ell}=1}^{L_{\mathrm{MCMC}}} \) drawn from target distribution π(Ω_{ℓ, t}), the Monte Carlo estimate of \( \partial \tilde{D}/\partial {p}_u \) and \( \partial \tilde{D}/\partial {g}_m \) are
Applying the RobbinsMonro stochastic approximation scheme [26], we can achieve the recursive update function to close to the root of \( \partial \tilde{D}/\partial {p}_u=0 \) and \( \partial \tilde{D}/\partial {g}_m=0 \), namely
where r_{t} denotes the sequence of decreasing step sizes [27]. In addition, we can simplify formulas (30) and (31), because (1 − p_{u, t − 1})p_{u, t − 1} and (1 − g_{m, t − 1})g_{m, t − 1} has no significant impact on the convergence of (30) and (31). Hence, Eqs. (30) and (31) becomes
The updated proposal distribution Eqs. (32) and (33) are iteratively used with the objective to close to the target distribution.
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 \( \overline{\boldsymbol{\upomega}} \).
Assume the sample ω_{ℓ, t} drawn from Ψ(·; g_{t − 1}) at the tth iteration. We define the two sets, which are ϕ_{0} = {m : I_{m}(ω_{ℓ, t}) = 0} and ϕ_{1} = {m : I_{m}(ω_{ℓ, t}) = 1}, respectively. We use these to collect the indices for the unselected and selected BS antennas, respectively. The following projection strategy is applied if \( {\sum}_{m=1}^M{I}_m\left({\boldsymbol{\upomega}}_{\mathrm{\ell},t}\right)\ne N \):

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.
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.
Constrained AMCMCbased JASUS algorithm
On the base of the aforementioned discussion, we can be written the proposed AMCMCbased JASUS algorithm by the following steps. At iteration t, L_{MCMC} samples \( {\left\{{\boldsymbol{\Omega}}_{\mathrm{\ell},t}=\left\{{\overline{\boldsymbol{\upomega}}}_{\mathrm{\ell},t},{\boldsymbol{\upomega}}_{\mathrm{\ell},t}\right\}\right\}}_{\mathrm{\ell}=1}^{L_{\mathrm{MCMC}}} \) from the MCMC method are can be generated by employing MIS according to proposal distribution Ψ(Ω_{ℓ, t}; R_{t − 1}). Then, the new proposal distribution Ψ(Ω_{ℓ, t}; R_{t}) will be updated by the KullbackLeibler divergence until it approach the target distribution π(Ω_{ℓ, t}). The detailed AMCMCbased JASUS algorithm is described as follows.
Simulation configuration
In this section, the simulation configuration and simulation parameters are described. The considered scenario is a multicell multiuser massive MIMO downlink system operating in TDD mode with Β = 7, as shown in Fig. 2. The simulation is done with a static network simulator. The key simulation parameters are summarized in Table 1. In this simulation, we assume that the CSI is perfectly known at the transmitter, the total power is uniformly allocated among the transmit antennas. The system composed of Β hexagonal cells. All B BSs, where B = {1, 2, …, Β} are installed with M antennas and serve U singleantenna users in each cell. Each BS is located at the cell center while U singleantenna users are randomly located in the cell. There is no user movement and handover during the simulation process.
Simulation results and analysis
In this section, we provide numerical results and computational complexity analysis of the proposed algorithm by simulative evaluation.
Performance evaluation
As can be seen the result from Fig. 3, the multicell multiuser massive MIMO downlink system using different JASUS method at various SINR. We find that the sum capacity of the AMCMCbased JASUS algorithm is very close to the maximum capacity result obtained by the ESbased JASUS algorithm with a wide range of SINRs. For example, when the SINR is 20 dB, the achieved values of cell capacity by using ES and AMCMC algorithms are 50.7 and 49.9 b/s/Hz, respectively. Ninetyeight percent of the optimal capacity is obtained by our proposed method. The result shows that the AMCMCbased JASUS method has a good performance compared to the greedybased JASUS method with the wide range of SINRs, both AMCMCbased JASUS and greedybased JASUS methods have better capacity performances compared to the normbased JASUS algorithm. This simulation result shows that, when the SINR is 30 dB, ES, AMCMC, and greedy algorithms enhanced cell capacity of approximately 9.1, 8.2, and 5.8 b/s/Hz, respectively.
Figure 4 shows the increase of sum cell capacity during each iteration for AMCMCbased JASUS algorithm with SINR = 20 dB. As can be seen from Fig. 4, we find that the AMCMCbased JASUS converges after about t = 30 iterations. As expected, the sum cell capacity obtained by AMCMCbased JASUS strictly monotonically increased with a number of iteration.
Now, the assumed scenario is a multicell multiuser massive MIMO downlink system with 50 active users (U = 50) at SINR = 20 dB, and we assume a different number of transmit antenna, M, from 16 to 60. Sixteen antennas (N = 16) were selected to be used by the transmitter and ten users (K = 10) were served. As can be seen in the result from Fig. 5, the cell capacity difference between the ESbased JASUS and AMCMCbased JASUS scheme is relatively small, and the cell capacity achieved by the aforementioned algorithms slightly grows with M. In summary, when numbers of the selected BS antennas (N) and scheduled users (K) are confirmed, the increase of the number of transmitting antennas (M) has little effect on the system capacity performance. Thus, it can be proven from the result that activation of more transmit antennas at the BS side is unnecessary.
A cell capacity performance comparison of each JASUS algorithms with various numbers of the selected antennas at SINR = 20 dB is shown in Fig. 6. The various numbers of the selected antennas (12 ≤ N ≤ 24) correspond to the different maximum cell capacity of the networks. Compared with the normbased JASUS, the JASUS algorithms, which are ES, AMCMC, and greedy, had more significant enhancement for the system performance. For example, the maximum cell capacity enhancements, which are approximately 2.5, 2.2, and 1.7 b/s/Hz at SINR = 20 dB, are achieved when 18 (N = 18) BS antennas are selected. From the figure, it can be observed, when N > 18, the cell capacity achieved by the JASUS algorithms is slightly growing when the number of selected antenna goes large. Thus, it can be proven from the result that when numbers of the scheduled users K are confirmed, the system capacity sequentially increases until the numbers of selected BS antenna close to the N = 18, when N > 18, the increasing number of selected antennas has no significant effect on the system capacity performance. Therefore, the results show that more antenna selection is unnecessary at the base station. Thus, the proposed algorithm is demonstrated to be effective. In addition, we considerably decreased system cost and power consumption while approach the maximum cell capacity by selected suitable transmit antennas at BS side.
Figure 7 shows that the cell capacity different K at SINR = 20 dB, for user scheduling and with transmit antenna selection. It can be observed that the cell sum capacity increases with increasing of user K. The different numbers of the scheduled users (2 ≤ K ≤ 16) correspond to the different maximum cell capacity of the networks. Compared with the normbased result, ES, AMCMC, and greedy algorithms enhanced the system performance. For example, the maximum cell capacity enhancement, which are approximately 6.2, 5.5, and 4.0 b/s/Hz at SINR = 20 dB, are achieved when ten (K = 10) users are scheduled. From the figure, it can be observed that when K > 10, the cell capacity achieved by the aforementioned algorithms slightly grows when the number of the scheduled user goes large. This result proves that the behavior of JASUS algorithm does not change drastically when the scheduled user number becomes large.
Finally, we discuss on SER of different JASUS algorithms. A 16QAM scheme is used with a ZF receiver. The SER performance of the linear ZF receiver system is shown in Fig. 8. Compared with the results of normbased JASUS algorithm, ES, AMMC, and greedy algorithms improved the SER of the system. Same to the case of the cell capacity performance, the system SER of the AMCMCbased JASUS algorithm is close to that of the ESbased JASUS algorithm. At the same time, we find that ESbased JASUS and AMMCbased JASUS have better SER performance than greedybased JASUS and normbased JASUS, especially the SINR to high. When the SINR is 20 dB, the SER performances of the ESbased JASUS, AMCMCbased JASUS, greedybased JASUS, and normbased JASUS algorithms are approximately 3.5 × 10^{−2}, 3.9 × 10^{−2}, 4.9 × 10^{−2}, and 6.8 × 10^{−2}, respectively.
Computational complexity analysis
The computational complexities of the introduced different JASUS algorithm are analyzed in this section. Table 2 summarizes the computational complexity of our proposed algorithm along with the complexity of another algorithm. The asymptotic notations, which reflect the computational complexity, was used to evaluation how the scheme responds to changes of parameters which are M, N, U, and K. From Table 2, we can easily observe the computational complexity gap between the four methods. Note that \( {C}_M^N \) denotes the binomial coefficient, and the matrix inverse operation [29,30,31,32] makes the computational complexity of per sample up to O(N^{3}). Thus, the overall complexity of our proposed Algorithm 1 for the problem of (11) is O(N^{3}tL_{MCMC}), where t × L_{MCMC} is the total number of target function evaluations. Ultimately, we can be observed from Table 2 that our proposed algorithm has a very low computational complexity compared to ESbased JASUS and greedybased JASUS algorithms. However, the normbased JASUS algorithm has a very low computational complexity compared to our proposed algorithm, but it also has a very low cell capacity. This result shows that our proposed algorithm is suitable for practical multicell multiuser massive MIMO system.
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.
Abbreviations
 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
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Acknowledgements
This work described in this paper was supported by the National Science and Technology Major Project: No. 2018ZX03001029004.
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.
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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.
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Saidiwaerdi Maimaiti received the M.Sc. degree in signal and information processing from Southwest Jiaotong University, Chengdu, China, in 2014. He is currently working towards the Ph.D. degree in Information and Communications Engineering, Key Laboratory of Universal Wireless Communications, Ministry of Education, Beijing University of Posts and Telecommunications Beijing, China. His research interests include massive MIMO, interference management, radio network planning, resource management, and intelligent network optimization in 5G network systems.
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Correspondence to Saidiwaerdi Maimaiti.
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Maimaiti, S., Chuai, G., Gao, W. et al. A lowcomplexity algorithm for the joint antenna selection and user scheduling in multicell multiuser downlink massive MIMO systems. J Wireless Com Network 2019, 208 (2019) doi:10.1186/s1363801915297
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Keywords
 5G
 Massive MIMO systems
 Antenna selection
 User scheduling
 Adaptive Markov chain Monte Carlo algorithm