Pilot allocation scheme based on coalition game for TDD massive MIMO systems

With the appearance of smartphones and tablets, wireless communication requires increasingly higher communication rates. Therefore, many technologies have been proposed to improve the communication rate of 5G, and massive multiple-input multiple-output (MIMO) is one of these technologies. Compared with traditional wireless networks, massive MIMO significantly enhances the spectral and power efficiency [1].

However, the excellent performance of massive MIMO depends on the accuracy of channel estimation. For conventional channel estimation in time division duplex (TDD) massive MIMO systems, users transmit pilot signals in the uplink, and base stations (BSs) receive the pilot signal and estimate channel state information (CSI). Then, BSs can use the CSI for uplinking received signal detection and downlinking transmission precoding. Therefore, the accuracy of channel estimation is one of the key factors that affect the system performance of uplink and downlink. In pilot-based channel estimation, the number of pilots is finite and limited by the channel coherence time, so mobile users in different cells might have to reuse the same pilot. Pilot reuse seriously affects the channel estimation accuracy at BSs and further influences the system performance of uplink and downlink, where the influence is known as pilot contamination [2]. Recent studies have shown that pilot contamination is one of the key factors that affect the performance of TDD massive MIMO systems [1‐4]. Therefore, many methods have been proposed to reduce pilot contamination, such as pilot contamination precoding [4], time-shifted pilots [5, 6], spatial domain-based pilot allocation [7], design and transmit power control of pilot sequences [8, 9], and pilot reuse [10, 11].

For instance, reference [4] shows that the performance of TDD massive MIMO system is limited by pilot contamination. Furthermore, reference [4] considers the uplink precoding and downlink precoding (called pilot contamination precoding) based on slow fading coefficients to reduce pilot contamination. Papers [5, 6] discuss the time-shifted pilot method in hexagonal cells, which means cells are divided into different groups, and cells in different groups transmit pilots in different time slots; analyses show that this method can effectively reduce pilot contamination. However, this method assumes that the number of users in each cell is fixed and equal. In addition, it requires strict system synchronization, making implementation complex and difficult. Paper [7] proposes a spatial domain method with the main idea that users with high channel orthogonality can use the same pilot. This method also assumes that cells have a regular shape and that the number of users in each cell is fixed and equal. Furthermore, this method is highly complex because BSs need to know the CSIs between all users and all BSs to carry out the search for highly orthogonal channels. Paper [8] gives a design for non-orthogonal pilot signals, and paper [9] proposes a pilot power allocation method based on cell grouping to alleviate pilot contamination. The methods in [8, 9] both assume that cells have a regular shape and the number of users per cell is fixed. Paper [10] presents a pilot reuse method based on cell grouping and analyzes the system performance of different pilot reuse factors. In paper [11], a fractional pilot reuse method is proposed in which users are classified into cell-center users and cell-edge users, where the cell-center users reuse the same pilot subset and the cell-edge users use the remaining pilots according to cell grouping.

Studies [12, 13] try to introduce game theory into pilot allocation to reduce pilot contamination. However, the pilot allocation methods in [12, 13] both consider the games between cells. Simply because of the games between cells, the suitable application scene of [12, 13] is still fixed and has an equal number of users in each cell, although reference [13] attempts to discuss irregularly shaped cells.

Reviewing the studies in [4‐13], reasonable pilot allocation can effectively reduce pilot contamination and improve system performance. However, in order to facilitate theoretical analysis and simulation, most of these pilot allocation methods assume that cells have a regular shape (most are hexagons), the number of users per cell is equal, and/or channels are independent. However, in reality, the cell shapes are affected by various factors, such as landforms and buildings, and do not always have a regular shape. Moreover, mobile users are randomly distributed in reality, so it is impossible for the number of users in each cell to be equal or fixed. When the number of base station antennas is very large, it is difficult to ensure the distance between the antennas, and it is difficult to ensure that all channels are independent.

Most of these pilot allocation methods (in references [4‐13]) assume that the number of users per cell is equal or fixed for two main reasons. First, when the number of users in each cell is equal or fixed, it is easy to allocate pilots. For example, in references [4‐13], in order to avoid the use of the same pilot in adjacent cells (pilot contamination is large in this case), these pilot allocation methods [4‐13] always reserve a certain number of pilots for each cell. If the number of users in each cell is equal or fixed, it is convenient to estimate the number of pilots per cell, so that when pilots are allocated, the pilots of adjacent cells can be differentiated. Second, when the number of users in each cell is equal, the dimension of the channel matrix in each cell is equal, which is convenient for merging in mathematical operation in theoretical analysis.

If we modify these methods [4‐13] to the actual scenes in which all users are randomly distributed and the number of users per cell is unequal, the system performance will not be guaranteed. For example, if the number of users is large in a cell, using these methods [4‐13], pre-allocated pilots are insufficient, and pilot reuse will cause large pilot contamination. On the other hand, if the number of users in a cell is small, using these methods [4‐13], the pilot resource cannot be fully utilized, and this will cause the waste of pilot resources and then affect overall performance. Obviously, this point reflects that these methods [4‐13] lack flexibility.

The assumption of a regular cell shape is also beneficial to pilot allocation. When the cell shape is regular, cells can be grouped first. For example, as shown in Fig. 1a, taking pilot reuse factor 3 as an example, the cells in the picture have three types of background pattern. The cells with the same background pattern are grouped together and reuse the same pilot subset (the total pilot set has been divided into three orthogonal pilot subsets). Cells with different background patterns use different orthogonal pilot subsets. References [9‐11] use similar cell grouping pilot allocation methods. However, the actual situation is that the cell shape is irregular. The pilot multiplexing in 11 cells shown in Fig. 1b is based on the pilot allocation method in Fig. 1a. It can be seen that in Fig. 1b, because of the irregular cell shape, the same pilot subset is used in adjacent cells, which will cause serious pilot contamination; thus, the system performance cannot be guaranteed.

So we can conclude that if we modify these methods [4‐13] to the actual scenes (in which cells have irregular shapes, users are randomly distributed), the system performance will not be guaranteed.

Papers [14, 15] propose different greedy algorithms to allocate pilots in the scene of correlated fading channels. When compared with the methods in [4‐13], greedy algorithms in papers [14, 15] are flexible and can be easily generalized or modified to the actual scenes, but they still cannot guarantee the target performance requirement. Paper [14] presents a statistical greedy pilot scheduling algorithm, which focuses on users in a single cell and searches users with small channel correlations to use the same pilot. This algorithm does not consider the application scenario of multi-cells and the correlation of the channels between different users and different BSs, so it cannot reduce the interference among different cells and thus cannot guarantee system performance in multi-cell scenarios. That is to say, although this algorithm can be generalized to multi-cell scenarios of arbitrary BS locations and an unequal number of users per cell, it cannot guarantee target performance satisfaction. Thus, it is essentially different from our work; the proposed pilot allocation scheme in our work can be used in arbitrary scenarios as well as guarantee system performance.

Paper [15] proposes a greedy approach; its main idea is selecting one user from each cell (there are P users per cell) to form a user set during each search, and users in the same set use the same pilot. The principle of user selection is to minimize the sum mean square error of channel estimation for the users using the same pilot. The greedy approach assumes the number of users per cell is equal; it cannot be used in the case of different numbers of users per cell, so it is different from our work and we have to make modifications in simulations. In addition, it only considers the sum-mean-square error of users with the same pilot and does not consider the total performance of all users; furthermore, it is a one-time search for all users and cannot achieve global optimization, whereas the proposed pilot allocation scheme in our work considers the total performance of all users and performs a cyclic search for all users. Therefore, in comparison, the greedy approach in [15] cannot meet the needs of total performance, whereas our scheme can meet overall performance requirements through cyclic search.

In order to find the flexible pilot allocation method that can be used in any actual scenes and provide performance guarantee, we introduce the idea of a coalition game into pilot allocation and propose a pilot allocation scheme based on the coalition game in this paper. We consider the game between users and divide users into different sub-coalitions. Because this coalition game is among users, the proposed pilot allocation scheme is more flexible and can be used in any actual scene. Because the purpose of each coalition adjustment is to improve the target performance (average utility function), the target performance can be satisfied through the cyclic search of coalition adjustment. The contributions of this paper are as follows.

The remainder of this paper is organized as follows. The system model for the massive MIMO system is described in Section 2. Different utility functions are analyzed under correlated fading channels in Section 3, including mean square error of channel estimation (MSE-CE), mean square error of signal detection (MSE-SD), received signal-to-interference-plus-noise ratio (SINR), and spectrum efficiency (SE). In Section 4, the definition of the coalition game is given, and then the coalition structure, the adjustment principle of coalition structure, and the condition of final stability are defined. After that, the coalition formation algorithm is given, a pilot allocation scheme based on the coalition game is proposed, and the implementation method of the proposed scheme is discussed. In Section 5, simulations are developed with respect to different utility functions, and the proposed pilot allocation scheme is compared with the greedy approach in reference [15] and the random coalition scheme. Section 6 concludes this paper. The proof of theorem is given in the Appendices 1, 2, and 3.

MSE-CE means the mean square error of channel estimation. During the uplink pilot transmission, all users transmit pilots at the same time, thus the received pilot signal at the BS l iswhere w_jk ∈ ℂ^P × 1 is the pilot sequence used by user (j, k), and \( \mathbb{E}\left\{{\boldsymbol{w}}_{jk}^H{\boldsymbol{w}}_{jk}\right\}=1 \). t_jk is the pilot transmit power of user (j, k). In order to avoid the near-far issue in the uplink, here we consider the transmit power control and let t_jk = T/β_jjk, where Tis a design parameter of transmit power. \( {\boldsymbol{N}}_l^{\mathrm{pilot}}\in {\mathrm{\mathbb{C}}}^{M\times P} \) is the equivalent additive white Gaussian noise (AWGN) at the receiver with independent and identically distributed (i.i.d.) elements of \( \mathcal{CN}\left(0,{\sigma}^2\right) \), where σ²is the variance of each element in noise .

After decorrelation and power normalization of \( {\boldsymbol{y}}_l^{\mathrm{pilot}} \), BS l can obtain the channel observation of h_ljk, which iswhere \( {\Lambda}_{\mathcal{S}}\left(j,k\right) \) is one of the sub-coalitions in coalition structure \( \mathcal{S} \), and user (j, k) belongs to the sub-coalition \( {\Lambda}_{\mathcal{S}}\left(j,k\right) \).

According to the definition of minimum mean square error (MMSE) estimation and expressions (14), (15) in reference [14], the MMSE estimation of channel h_ljk at BS l, which is based on the channel observation \( {\overline{\overline{\boldsymbol{h}}}}_{ljk} \), is given by

where \( {\mathbf{C}}_{ljk}=\sum \limits_{\left(m,n\right)\in {\Lambda}_{\mathcal{S}}\left(j,k\right)}\kern0em \frac{t_{mn}}{t_{jk}}{\mathbf{R}}_{lmn}+\frac{\sigma^2}{t_{jk}}{\mathbf{I}}_M \) is the covariance matrix of \( {\overline{\overline{\boldsymbol{h}}}}_{ljk} \).

Similar to expression (7), we can obtain the channel estimation of h_jjk (the channel between user (j, k) and its own BS j), which is

with covariance \( {\mathbf{R}}_{{\widehat{h}}_{jjk}}={\mathbf{R}}_{jjk}{\mathbf{C}}_{jjk}^{-1}{\mathbf{R}}_{jjk} \). Thus, we can determine that the channel estimation error is \( {\tilde{\boldsymbol{h}}}_{jjk}={\boldsymbol{h}}_{jjk}-{\widehat{\boldsymbol{h}}}_{jjk} \). From the orthogonality principle of MMSE estimation [14, 18], channel estimation error \( {\tilde{\boldsymbol{h}}}_{jjk} \) is independent of \( {\widehat{\boldsymbol{h}}}_{jjk} \) and the covariance of \( {\tilde{\boldsymbol{h}}}_{jjk} \) is \( {\mathbf{R}}_{{\tilde{h}}_{jjk}}={\mathbf{R}}_{jjk}-{\mathbf{R}}_{{\widehat{h}}_{jjk}}={\mathbf{R}}_{jjk}-{\mathbf{R}}_{jjk}{\mathbf{C}}_{jjk}^{-1}{\mathbf{R}}_{jjk} \). Note that \( {\widehat{\boldsymbol{h}}}_{jjk} \) and \( {\mathbf{R}}_{{\tilde{h}}_{jjk}} \) are also mean and covariance of h_jjk conditioned on \( {\overline{\overline{\boldsymbol{h}}}}_{ljk} \), respectively [18].

The mean square error of channel estimation (MSE-CE) \( {\widehat{\boldsymbol{h}}}_{jjk} \) means the sum of squares estimation errors for each element of h_jjk (you can refer to expression (18) in [14]); so the MSE-CE of \( {\widehat{\boldsymbol{h}}}_{jjk} \) is \( \mathrm{MSE}\hbox{-} {\mathrm{CE}}_{jk}={\varepsilon}_{{\tilde{h}}_{jjk}}^{\mathrm{MSE}\hbox{-} \mathrm{CE}}=\mathbb{E}\left\{{\left({\tilde{\boldsymbol{h}}}_{jjk}\right)}^H{\tilde{\boldsymbol{h}}}_{jjk}\right\}=\mathbb{E}\left\{\mathrm{tr}\left\{{\tilde{\boldsymbol{h}}}_{jjk}{\tilde{\boldsymbol{h}}}_{jjk}^H\right\}\right\}=\mathrm{tr}\left\{{\mathbf{R}}_{{\tilde{h}}_{jjk}}\right\} \). We define the sum MSE-CE of all users as the sum of the estimation mean square error (MSE) of the channels between each user and its own BS. Thus, the average MSE-CE per user is

MSE-SD means the mean square error of signal detection. During the uplink data transmission, all users send their data to all BSs at the same time, and the received signal at BS j is

where x_mn ∈ ℂ^1 × 1 represents the symbol transmitted from user (m, n), \( \mathbb{E}\left\{{\left|{x}_{mn}\right|}^2\right\}=1 \), \( {\boldsymbol{y}}_j^{\mathrm{data}}={\left[{y}_{j1},\cdots, {y}_{jM}\right]}^T\in {\mathrm{\mathbb{C}}}^{M\times 1} \), \( {\boldsymbol{n}}_j^{\mathrm{data}} \) is AWGN, and \( {\boldsymbol{n}}_j^{\mathrm{data}}\sim \mathcal{CN}\left(0,{\sigma}^2{\mathbf{I}}_M\right) \), \( {\boldsymbol{g}}_{jmn}=\sqrt{t_{mn}^d}{\boldsymbol{h}}_{jmn} \).

Let \( {\boldsymbol{H}}_{jm}=\left[{\boldsymbol{h}}_{jm1},{\boldsymbol{h}}_{jm2},\cdots, {\boldsymbol{h}}_{{jm K}_m}\right]\in {\mathrm{\mathbb{C}}}^{M\times {K}_m} \), and \( {\boldsymbol{T}}_m^d=\operatorname{diag}\left(\sqrt{t_{m1}^d},\cdots, \sqrt{t_{mK_m}^d}\right) \), \( {\boldsymbol{x}}_m={\left[{x}_{m1},\cdots, {x}_{mK_m}\right]}^T \), \( {\boldsymbol{G}}_{jm}={\boldsymbol{H}}_{jm}{\boldsymbol{T}}_m^d=\left[{\boldsymbol{g}}_{jm1},\cdots, {\boldsymbol{g}}_{{jm K}_m}\right] \); thus, the received signal at BS j can be rewritten as \( {\boldsymbol{y}}_j^{\mathrm{data}}=\sum \limits_{m=1}^L{\boldsymbol{H}}_{jm}\kern0.5em {\boldsymbol{T}}_m^d{\boldsymbol{x}}_m+{\boldsymbol{n}}_j^{\mathrm{data}}=\sum \limits_{m=1}^L{\boldsymbol{G}}_{jm}{\boldsymbol{x}}_m+{\boldsymbol{n}}_j^{\mathrm{data}} \).

To distinguish the symbol of each user, the BS j selects the combination matrix \( {\boldsymbol{W}}_{jj}^{\mathrm{scheme}}\in {\mathrm{\mathbb{C}}}^{M\times {K}_j} \), which is directly multiplied with the received signal, and the processed data of all K_j users in the jth cell can be written as

Considering the maximum ratio combining (MRC), zero-forcing combining (ZFC), and MMSE schemes of the received signal \( {\boldsymbol{y}}_j^{\mathrm{data}} \), thuswhere \( {\widehat{\boldsymbol{H}}}_{jj}=\left[{\widehat{\boldsymbol{h}}}_{jj1},{\widehat{\boldsymbol{h}}}_{jj2},\cdots, {\widehat{\boldsymbol{h}}}_{{jj K}_j}\right]\in {\mathrm{\mathbb{C}}}^{M\times {K}_j} \), and \( {\boldsymbol{T}}_j^d=\operatorname{diag}\left(\sqrt{t_{j1}^d},\cdots, \sqrt{t_{jK_j}^d}\right) \), \( {\boldsymbol{D}}_{jj}=\operatorname{diag}\left(\frac{1}{\delta_{jj1}},\cdots, \frac{1}{\delta_{{jj K}_j}}\right) \) is the diagonal matrix parameter that is set for \( \mathbb{E}\left\{{\left({\boldsymbol{w}}_{jjk}^{MRC}\right)}^H{\boldsymbol{h}}_{jjk}\right\}=1 \), and \( {\delta}_{jjk}=\mathrm{tr}\left\{{\mathbf{R}}_{{\widehat{h}}_{jjk}}\right\} \).

Theorem 1 For a given coalition structure \( \mathcal{S} \), the mean square error of signal detection (MSE-SD) for user (j, k) at BS j for MRC, ZFC, and MMSE schemes are given by the following expressions (13), (14) and (15) respectively,where \( {\boldsymbol{Q}}_1={\mathbb{E}}_{\boldsymbol{h}}\left\{{\boldsymbol{w}}_{jjk}^{zfc}{\left({\boldsymbol{w}}_{jjk}^{zfc}\right)}^H\right\} \), \( {\boldsymbol{Q}}_2={\mathbb{E}}_{\boldsymbol{h}}\left\{{\widehat{\boldsymbol{h}}}_{jjk}{\left({\boldsymbol{w}}_{jjk}^{zfc}\right)}^H\right\} \), \( {\boldsymbol{Q}}_3={\mathbb{E}}_{\boldsymbol{h}}\left\{{\left({\widehat{\boldsymbol{H}}}_{jj}^H{\widehat{\boldsymbol{H}}}_{jj}\right)}^{-1}\right\} \), \( {\boldsymbol{Q}}_4={\mathbb{E}}_{\boldsymbol{h}}\left\{{\sigma}^2{\left({\widehat{\boldsymbol{H}}}_{jj}^H{\widehat{\boldsymbol{H}}}_{jj}+{\sigma}^2{\left({\boldsymbol{T}}_j^d\right)}^{\hbox{-} 2}\right)}^{-1}\right\} \), \( {\boldsymbol{Q}}_5={\mathbb{E}}_{\boldsymbol{h}}\left\{{\boldsymbol{w}}_{jjk}^{\mathrm{mmse}}{\left({\boldsymbol{w}}_{jjk}^{\mathrm{mmse}}\right)}^H\right\} \), \( {\boldsymbol{Q}}_6={\mathbb{E}}_{\boldsymbol{h}}\left\{{\widehat{\boldsymbol{h}}}_{jjk}{\left({\boldsymbol{w}}_{jjk}^{\mathrm{mmse}}\right)}^H\right\} \), \( {\boldsymbol{Q}}_7={\mathbb{E}}_{\boldsymbol{h}}\left\{{\left({\boldsymbol{W}}_{jj}^{\mathrm{mmse}}\right)}^H{\boldsymbol{W}}_{jj}^{\mathrm{mmse}}\right\} \), \( {\boldsymbol{w}}_{jjk}^{\mathrm{zfc}} \) is the kth column of the matrix \( {\boldsymbol{W}}_{jj}^{\mathrm{zfc}}={\widehat{\boldsymbol{H}}}_{jj}{\left({\widehat{\boldsymbol{H}}}_{jj}^H{\widehat{\boldsymbol{H}}}_{jj}\right)}^{-1} \), and \( {\boldsymbol{w}}_{jjk}^{\mathrm{mmse}} \) is the kth column of the matrix \( {\boldsymbol{W}}_{jj}^{\mathrm{mmse}}={\widehat{\boldsymbol{H}}}_{jj}{\left({\widehat{\boldsymbol{H}}}_{jj}^H{\widehat{\boldsymbol{H}}}_{jj}+{\sigma}^2{\left({\boldsymbol{T}}_j^d\right)}^{\hbox{-} 2}\right)}^{-1} \).

For a given coalition structure \( \mathcal{S} \), the uplink received SINR for user (j, k) at BS j for MRC, ZFC, and MMSE schemes are given by the expressions (17), (18) and (19) respectively.

Proof The proofs of Theorem 1 for MRC, ZFC, and MMSE schemes are given in Appendices 1, 2, and 3, respectively.

Based on previous theoretical analyses, according to Shannon capacity and referring to expression (11) in [19], the uplink spectrum efficiency (SE) of user (j,k) iswhere the \( {\mathrm{SINR}}_{jk}^{\mathrm{scheme}} \) for MRC, ZFC, and MMSE schemes are given by expressions (17), (18), and (19), respectively.