Throughput Maximization-Based AP Clustering Methods in Downlink Cell-Free MIMO Under Partial CSI Condition

Daisuke ISHII, Takanori HARA, Kenichi HIGUCHI

Summary :

In this paper, we investigate a method for clustering user equipment (UE)-specific transmission access points (APs) in downlink cell-free multiple-input multiple-output (MIMO) assuming that the APs distributed over the system coverage know only part of the instantaneous channel state information (CSI). As a beamforming (BF) method based on partial CSI, we use a layered partially non-orthogonal zero-forcing (ZF) method based on channel matrix muting, which is applicable to the case where different transmitting AP groups are selected for each UE under partial CSI conditions. We propose two AP clustering methods. Both proposed methods first tentatively determine the transmitting APs independently for each UE and then iteratively update the transmitting APs for each UE based on the estimated throughput considering the interference among the UEs. One of the two proposed methods introduces a UE cluster for each UE into the iterative updates of the transmitting APs to balance throughput performance and scalability. Computer simulations show that the proposed methods achieve higher geometric-mean and worst user throughput than those for the conventional methods.

• Publication: IEICE TRANSACTIONS on Communications Vol.E107-B No.10 pp.653-660
• Publication Date: 2024/10/01
• Online ISSN: 1745-1345
• DOI: 10.23919/transcom.2023EBT0008
• Type of Manuscript: PAPER
• Category: Wireless Communication Technologies

1.  Introduction

Cooperative multiple-input multiple-output (MIMO) [1]-[4], which employs MIMO transmission to multiple sets of user equipment (UEs) in coordination among base stations (BSs), eliminates the cell boundaries between coordinated BSs. This approach increases the throughput by utilizing an increased number of transmitter and receiver antennas at the network level. Furthermore, distributed antenna systems (DASs) [5], [6], in which BS antennas are distributed throughout the system coverage area, are effective in reducing the area of insensitivity, reducing the required transmission power, and increasing the total throughput compared to when many antennas are locally deployed. In this context, cell-free MIMO [7]-[9], which is one type of DAS and distributes antennas, such as access points (APs), over the system coverage, has recently been actively researched and developed for application to the 5th generation mobile communication system new radio (NR) and later systems [10]-[12].

Beamforming (BF) is essential to achieve high transmission capacity in MIMO transmission. It takes advantage of the high spatial degrees of freedom of a MIMO channel to increase the received signal power and to suppress interference between spatially multiplexed UEs. A sophisticated BF scheme requires instantaneous channel state information (CSI), which corresponds to the instantaneous complex channel coefficients, at the AP.

However, it is impractical for each AP to obtain ideally the instantaneous CSI for all UEs. For example, in a time division duplex (TDD) system, although the AP estimates the instantaneous CSI using the uplink reference signal, such a CSI estimate for UEs that are far from the AP tends to be inaccurate. Moreover, the CSI estimation and the calculation of sophisticated BF vectors for all UEs require a high level of computational complexity when cell-free MIMO supports a large number of UEs.

To overcome these issues, our research group previously reported a partially non-orthogonal zero-forcing (ZF) method for MIMO with partial CSI knowledge [13]. This method designs BF vectors based on a muting method that inserts zeros into small channel coefficients. The BF enables spatial multiplexing of a group of UEs with different instantaneous CSI knowledge where each AP knows only some instantaneous CSI. Since the degree of freedom of the MIMO channel is used effectively, this method increases user throughput compared to the conventional method [14], [15] that allocates different orthogonal channels to UEs with different instantaneous CSI knowledge.

In addition to the BF design with partial CSI knowledge, approaches to enhance the scalability of cell-free MIMO have been investigated [9], [16], [17]. Although there are many approaches, this paper focuses on a method that determines the transmitting AP groups, which is also known as clustering¹. Methods for AP clustering each UE independently from other UEs have been proposed [9], [18]-[21]. In [19], an AP clustering method was proposed that determines the transmitting AP groups while taking into account the received signal power using the partially non-orthogonal ZF method as a BF method. Moreover, in [20] and [21], this method was extended to consider the interference to other UEs. In contrast, the methods proposed in [9] and [18] rely on path loss to determine the transmitting AP groups. However, independent determination of the transmitting APs for a specific UE induces interference to other UEs, leading to degradation in the system throughput. Therefore, an alternative AP clustering method that takes into account the effects among UEs is desirable to enhance further the performance of downlink cell-free MIMO with partial instantaneous CSI knowledge. In addition to improving the system throughput, the scalability of the AP clustering method is important for accommodating a system with numerous UEs and APs.

In this paper, we propose two AP clustering methods for each UE in downlink cell-free MIMO based on a layered partially non-orthogonal ZF method. Both proposed methods first tentatively determine the transmitting APs independently for each UE and then iteratively update the transmitting APs for each UE using the geometric-mean throughput as a metric. These steps allow the proposed methods to determine the APs for each UE considering the mutual effects among UEs. Moreover, one of the two proposed methods focuses on balancing throughput performance and scalability by introducing a UE cluster into the determination of transmitting APs for each UE. Computer simulations show that the proposed methods achieve higher throughput than that for the conventional methods.

The rest of this paper is organized as follows. Section 2 describes the model of cell-free MIMO with partial CSI knowledge and the layered partially non-orthogonal ZF method. Subsequently, the conventional UE-independent AP clustering methods are reviewed in Sect. 3. Section 4 presents the proposed AP clustering methods. Section 5 shows the simulation results and Sect. 6 concludes the paper.

2.  System Model

2.1  Cell-Free MIMO Model

We consider a downlink cell-free MIMO system that comprises APs with \(N\) transmitter antennas and single-antenna UEs. Let \(\mathcal{L}\) and \(\mathcal{K}\) be the set of APs and UEs in the system coverage, respectively. The set of APs that are candidates for transmitting APs to UE \(k\) is denoted by \(\mathcal{L}_k \subseteq \mathcal{L}\). Throughout the paper, we set the number of APs in \(\mathcal{L}_k\) to \(T\) for all UEs, and \(\mathcal{L}_k\) comprises APs with the \(T\) lowest path loss between it and UE \(k \in \mathcal{K}\).

In this paper, the instantaneous CSI is assumed to be estimated based on the uplink reference signal in a TDD system. To consider the partial CSI condition, we define \(\mathcal{S}_k \subseteq \mathcal{L}_k\) as the set of APs that know the instantaneous CSI of UE \(k\), as shown in Fig. 1. Set \(\mathcal{S}_k\) comprises APs whose path loss between it and UE \(k\) is less than \(G_{k,\min}+\Delta_{\text{CSI}}\) dB, where \(G_{k,\min}\) is the minimum path loss between UE \(k\) and the APs and \(\Delta_{\text{CSI}}\) is a parameter that represents the ease of ascertaining the instantaneous CSI. Namely, \(\mathcal{S}_k=\left\{l\mid\beta_{k,l}\leq G_{k,\min}+\Delta_\mathrm{CSI}\right\}\) with \(\beta_{k,l}\) denoting the path loss between UE \(k\) and AP \(l\).

Fig. 1  System model.

2.2  Layered Partially Non-Orthogonal ZF Method

In this subsection, we review the layered partially non-orthogonal ZF method [19], which is the BF method used in this paper. Although this method utilizes the block diagonalization (BD) method [22] to encompass the case where a UE has multiple antennas, BD is replaced by ZF in the description since this paper assumes that all UEs have a single antenna.

Let \(\mathcal{G}(\mathcal{S}_k)\) be the set of UEs other than UE \(k\) for which at least one of AP \(l \in \mathcal{S}_k\) knows the instantaneous CSI. Thus, \(\mathcal{G}(\mathcal{S}_k)=\{i\colon \mathcal{S}_i\cap \mathcal{S}_k\neq\varnothing, i\neq k\in \mathcal{K}\}\). Moreover, the \(1 \times N\)-dimensional channel matrix between AP \(l\) and UE \(k\) is denoted by \(\mathbf{H}_{k,l}\). Note that AP \(l \notin \mathcal{S}_k\) does not know \(\mathbf{H}_{k,l}\) due to the partial CSI condition.

Let us consider BF used for data transmission to UE \(k\). As a baseline of the layered BD method, it is reasonable that the transmitting APs for UE \(k\) are set to \([\mathcal{S}_k]_1,\ldots,[\mathcal{S}_k]_{|\mathcal{S}_k|}\) where \([\mathcal{S}_k]_l\) represents the \(l\)-th element of \(\mathcal{S}_k\). Term \(\mathbf{H}_{\mathcal{S}_k}\) denotes the \(|\mathcal{G}(\mathcal{S}_k)| \times |\mathcal{S}_k|N\)-dimensional channel matrix between the AP set \(\mathcal{S}_k\) and the UEs in \(\mathcal{G}(\mathcal{S}_k)\). If all the elements of \(\mathcal{G}(\mathcal{S}_k)\) are known, the BF vector to UE \(k\) can be obtained from the Moore-Penrose generalized inverse of \(\mathbf{H}_{\mathcal{S}_k}\) based on the principle of the ZF method. However, due to the partial CSI condition, \(\mathbf{H}_{\mathcal{S}_k}\) may contain unknown elements. Therefore, we use a method to determine the BF vector based on a muted channel matrix, e.g., as in [13]. The muting method [13] first performs a muting operation that sets all \(\mathbf{H}_{i,l}\), i.e., \(i \in \mathcal{G}(\mathcal{S}_k)\), \(l \notin \mathcal{S}_i\), to zero matrix \(\mathbf{O}\), since \(\mathbf{H}_{i,l}\) (\(i \in \mathcal{G}(\mathcal{S}_k)\), \(l \notin \mathcal{S}_i\)) is not known at the AP. Term \(\tilde{\mathbf{H}}_{\mathcal{S}_k}\) is the \(|\mathcal{G}(\mathcal{S}_k)| \times |\mathcal{S}_k|N\)-dimensional matrix obtained after applying the muting operation to \(\mathbf{H}_{\mathcal{S}_k}\). The BF vector to UE \(k\) can be obtained from the Moore-Penrose generalized inverse of \(\tilde{\mathbf{H}}_{\mathcal{S}_k}\). The idea of the muting method is also used in the study of cell-free MIMO in [18].

In this case, the data transmitted by AP \([\mathcal{S}_k]_l\) to UE \(k\) do not interfere with other UEs for which AP \([\mathcal{S}_k]_l\) knows the instantaneous CSI. On the other hand, the transmitted data to UE \(k\) interfere with UEs whose instantaneous CSI is unknown to AP \([\mathcal{S}_k]_l\), resulting in partially non-orthogonal ZF. However, since the path loss of channels with unknown instantaneous CSI is high, the interference power is expected to be largely suppressed by the channel.

Given \(\mathcal{S}_k\), \(F(\mathcal{S}_k)\) is defined as

\[\begin{equation*} F\left(\mathcal{S}_k\right)=\left|\mathcal{S}_k\right|N-\left|\mathcal{G} \left(\mathcal{S}_k\right)\right|. \tag{1} \end{equation*}\]

This indicates the order of the received signal power gain that UE \(k\) can obtain when BF based on ZF is performed. The non-positive value of \(F(\mathcal{S}_k)\) implies that all the degrees of freedom of the MIMO channel are used for interference suppression. Thus, for data transmission to UE \(k\) using AP group \(\mathcal{S}_k\), \(F(\mathcal{S}_k)\) must be greater than one. If \(F(\mathcal{S}_k)\) is less than or close to one, the layered ZF method considers adding to the transmitting APs for UE \(k\) to increase the received signal power gain. Since the instantaneous CSI between the added APs and UE \(k\) is unknown, the use of additional APs does not directly contribute to the transmission quality of UE \(k\). However, the growth in the number of transmitting APs for UE \(k\) increases the channel degrees of freedom that can be used to null out interference in other UEs. As a result, an increase in the received signal power gain can be expected.

Let \(\mathcal{D}_k\) and \(\mathbf{H}_{\mathcal{S}_k\cup \mathcal{D}_k}\) be the AP group used for data transmission to UE \(k\) together with APs in \(\mathcal{S}_k\) (\(\mathcal{S}_k\cap \mathcal{D}_k=\varnothing\)) and let \(\mathbf{H}_{\mathcal{S}_k\cup \mathcal{D}_k}\) be the \(|\mathcal{G}(\mathcal{S}_k\cup\mathcal{D}_k)|\times |\mathcal{S}_k\cup\mathcal{D}_k|N\)-dimensional channel matrix between AP \([\mathcal{S}_k\cup\mathcal{D}_k]_1,\ldots,[\mathcal{S}_k\cup \mathcal{D}_k]_{|\mathcal{S}_k\cup\mathcal{D}_k|}\) and the UEs in \(\mathcal{G}(\mathcal{S}_k\cup \mathcal{D}_k)\), respectively. Matrix \(\tilde{\mathbf{H}}_{\mathcal{S}_k\cup \mathcal{D}_k}\) is obtained by muting operation on \(\mathbf{H}_{\mathcal{S}_k\cup \mathcal{D}_k}\) as described above. Provided that \(F(\mathcal{S}_k\cup \mathcal{D}_k)\) is greater than or equal to one, the BF vector to UE \(k\) can be obtained from the Moore-Penrose generalized inverse matrix of \(\tilde{\mathbf{H}}_{\mathcal{S}_k\cup\mathcal{D}_k}\).

The \(|\mathcal{J}_k|N\)-dimensional BF vector for UE \(k\), whose norm is normalized, is denoted by \(\mathbf{b}_k(\mathcal{J}_k)\) with \(\mathcal{J}_k\) denoting the transmitting AP group used for transmission to UE \(k\). Let \(\tilde{\mathbf{H}}_k\left(\mathcal{J}_k\right)\) be a \(1 \times |\mathcal{J}_k|N\)-dimensional muting channel matrix where the unknown term of instantaneous CSI between UE \(k\) and AP group \(\mathcal{J}_k\) is set to zero. Then, the effective channel gain for UE \(k\) is expressed as

\[\begin{equation*} \lambda_k\left(\mathcal{J}_k\right)=\|\tilde{\mathbf{H}}_k\left(\mathcal{J}_k\right) \mathbf{b}\left(\mathcal{J}_k\right)\|^2. \tag{2} \end{equation*}\]

3.  Conventional AP Clustering Methods

This section describes the conventional AP clustering methods proposed in [18] and [21], which will be evaluated in Sect. 5 along with the proposed methods. In this paper, we refer to them as the path-loss based method and the signal-to-leakage-and-noise ratio (SLNR)-based method, respectively.

Let \(\mathcal{J}_k\) be the set of APs used for data transmission to UE \(k \in \mathcal{K}\). The path-loss based method [18] independently determines \(\mathcal{J}_k\) by selecting APs with the \(U\) lowest path loss between it and UE \(k \in \mathcal{K}\) from the candidate APs. We note that the candidate APs comprise the APs in \(\mathcal{L}_k\), unlike in [18]. Although the process of the path-loss based method is obviously simple, it determines a group of APs for transmission to each UE without considering the mutual effects among other UEs.

On the other hand, the SLNR-based method [21] determines \(\mathcal{J}_k\) based on maximization of the metric below.

\[\begin{equation*} \mathcal{J}_k=\arg\max_{\mathcal{S}_k\cup \mathcal{D}_k} \frac{\lambda_k(\mathcal{S}_k\cup \mathcal{D}_k)}{z_k(\mathcal{S}_k\cup \mathcal{D}_k)}, \tag{3} \end{equation*}\]

where \(z_k(\mathcal{S}_k\cup\mathcal{D}_k)\) is the sum of the estimated interference power that the data transmitted from AP group \(\mathcal{S}_k\cup\mathcal{D}_k\) to UE \(k\) gives to all other UEs and is given below.

\[\begin{equation*} z_k\left(\mathcal{S}_k\cup\mathcal{D}_k\right)=\sum_{i\neq k} \sum_{l\in(\mathcal{S}_k\cup\mathcal{D}_k)\backslash\mathcal{S}_i} \beta_{i,l}\frac{p_k}{\left|\mathcal{S}_k\cup\mathcal{D}_k\right|}, \tag{4} \end{equation*}\]

where \(p_k\) is the transmission power of the data transmitted to UE \(k\). The estimated interference power is calculated using the path loss under the assumption that the selected APs transmit data to the UE using the identical transmission power. The numerator of the metric in (3) is calculated using (2) and corresponds to the desired received signal power of UE \(k\). The SLNR-based method is expected to improve the throughput at the system level by taking into account the amount of interference that the selected AP group may cause to other UEs. However, this method does not attempt to maximize the system throughput directly, rather it maximizes the SLNR for each UE individually.

In light of the above, the conventional AP clustering methods do not fully consider the mutual effect of the AP clustering for one UE on the other UEs. Furthermore, these methods do not select the transmitting AP groups based on the achievable throughput.

4.  Proposed AP Clustering Methods

In this paper, we propose two AP clustering methods to improve the geometric-mean user throughput over the entire system coverage area. The first one achieves the highest throughput since it considers the mutual effects among all UEs. The second one considers scalability in addition to the achievable throughput. Each of the proposed methods is described below.

4.1  Iterative Update Method

We propose an AP clustering method that determines the transmitting AP group for each UE based on an iterative algorithm. Hereafter, this method is referred to as the iterative update method. We note that this method was originally proposed in [23], in which the estimated values of the average user throughput and worst user throughput that is defined by the minimum throughput among all UEs are utilized as a metric. In this paper, the geometric-mean throughput is used as a metric to guarantee user fairness. The iterative update method considers the mutual effects among all UEs by estimating the throughput. The flow of the iterative update method is described as follows.

Let \(\mathcal{J}_k^{(r)}\) be the transmitting AP group for UE \(k\) in the \(r\)-th iteration. The iterative update method initializes \(\mathcal{J}_k^{(r=0)}\) using the SLNR-based method as

\[\begin{equation*} \mathcal{J}_k^{(0)}=\arg\max_{\mathcal{S}_k\cup\mathcal{D}_k} \frac{\lambda_k(\mathcal{S}_k\cup\mathcal{D}_k)}{z_k(\mathcal{S}_k\cup\mathcal{D}_k)}. \tag{5} \end{equation*}\]

After determining \(\mathcal{J}_k^{(0)}\), the transmitting AP groups for all UEs are iteratively updated using the AP groups obtained in the previous iteration. Specifically, in the \(r\)-th iteration, transmitting AP groups \(\mathcal{J}_k^{(r)}\) for each UE \(k\) are updated based on transmitting AP groups \(\{\mathcal{J}_i^{(r-1)}\}\) for all other UEs obtained in the \(r-1\)-th iteration. The updates are performed sequentially for each UE as follows.

\[\begin{equation*} \mathcal{J}_k^{(r)}=\arg\max_{\mathcal{J}_k}\left(\prod\nolimits_{i\in \mathcal{K}} C_i^{(r)}\right)^{\frac{1}{\left|\mathcal{K}\right|}}, \tag{6} \end{equation*}\]

where \(C_k^{(r)}\) is the estimated throughput of UE \(k\) in the AP clustering process for UE \(k\) in the \(r\)-th iteration and is calculated by

\[\begin{equation*} \mbox{$\displaystyle C_k^{(r)}= \log_2\left(1+\frac{\lambda_k\left(\mathcal{J}_k^{(r)}\right)p_k}{\displaystyle \sum\nolimits_{i=1}^{k-1}w_{i,k}\left(\mathcal{J}_i^{(r)}\right)+ \sum\nolimits_{i=k+1}^{|\mathcal{K}|} w_{i,k}\left(\mathcal{J}_i^{(r-1)}\right)+N_0}\right), $} \tag{7} \end{equation*}\]

where \(N_0\) denotes the noise power and

\[\begin{equation*} w_{i,k}\left(\mathcal{J}_i^{(r)}\right)=\sum_{l\in(\mathcal{S}_i\cup\mathcal{D}_i) \backslash\mathcal{S}_k} \beta_{k,l}\frac{p_i}{\left|\mathcal{S}_i\cup\mathcal{D}_i\right|}. \tag{8} \end{equation*}\]

Note that \(w_{i,k}(\mathcal{J}_i^{(r)})\) implies the estimated interference power, which is calculated in the same manner as (4), to UE \(k\) when the transmitting AP group for UE \(i\) is \(\mathcal{J}_i^{(r)}\).

Without loss of generality, it is assumed that the updates are performed in the order of UE number \(1, \ldots, |\mathcal{K}|\) as shown in Fig. 2. The iterative update method calculates (7) for all patterns of the transmitting AP group for each UE. Therefore, its computational complexity depends on \(|\mathcal{K}|\) and the number of candidates for the AP group, which is determined by \(T\).

Fig. 2  Flow of iterative update method.

The above iterative process is repeated until \(r\) reaches \(R\), which is the maximum number of iterations.

4.2  Scalable Iterative Update Method

The iterative update method fully considers the effect from the determination of the transmitting AP group for other UEs. Meanwhile, to calculate the metric in (6), the iterative update method needs to calculate the effective channel gain, which requires generating the BF vector for UE \(k\) using the set of all other UEs whose instantaneous CSI is known at the AP group for UE \(k\). The method also necessitates calculating the estimated interference power from all UEs in the system coverage area, leading to the difficulty in achieving scalability. Moreover, since the iterative update method updates the AP group in order for each UE, the processing time per iteration depends on the number of UEs in the system coverage area. Hence, we also propose the scalable iterative update method to consider throughput and scalability.

The scalable iterative update method determines the UE cluster for each UE and updates the transmitting AP group using the metric based on UEs in the UE cluster. The UE cluster for UE \(k\), which is defined as \(\mathcal{Q}_k\), comprises the UEs that are expected to be nearby UE \(k\). The scalable iterative update method reselects the transmitting AP group for each UE to maximize the metric calculated based on such a UE cluster. The flow of the scalable iterative update method is described as follows.

The scalable iterative update method is constituted by \(R\) iterations the same as in the iterative update method. This method generates a UE cluster for each UE and determines \({\mathcal{J}_k'}^{(0)}\) using the SLNR-based method based on UE cluster \(\mathcal{Q}_k\). Cluster \(\mathcal{Q}_k\) is constituted by any number of other UEs in order of decreasing path loss for UE \(k\). The SLNR-based method using the UE cluster is determined as

\[\begin{equation*} {\mathcal{J}_k'}^{(0)}=\arg\max_{\mathcal{S}_k\cup\mathcal{D}_k} \frac{\lambda_k'\left(\mathcal{S}_k\cup\mathcal{D}_k\right)}{z_k' \left(\mathcal{S}_k\cup\mathcal{D}_k\right)}, \tag{9} \end{equation*}\]

where \(\lambda_k'(\mathcal{S}_k\cup \mathcal{D}_k)\) is the effective channel gain of UE \(k\). It is calculated by (2) using the BF vector generated by considering only the UEs that are both in \(\mathcal{Q}_k\) and \(\mathcal{S}_k\cup \mathcal{D}_k\). Term \(z_k'(\mathcal{S}_k\cup\mathcal{D}_k)\) is the sum of the estimated interference power that the data transmitted from AP group \(\mathcal{S}_k\cup\mathcal{D}_k\) to UE \(k\) given to all UEs in \(\mathcal{Q}_k\) and is given by

\[\begin{equation*} z_k'\left(\mathcal{S}_k\cup\mathcal{D}_k\right)=\sum_{i\in \mathcal{Q}_k} \sum_{l\in (\mathcal{S}_k\cup \mathcal{D}_k)\backslash \mathcal{S}_i}\beta_{i,l} \frac{p_i}{\left|\mathcal{S}_k\cup \mathcal{D}_k\right|}. \tag{10} \end{equation*}\]

After initialization, the transmitting APs for all UEs are iteratively updated. In the \(r\)-th iteration, transmitting APs \({\mathcal{J}_k'}^{(r)}\) for each UE \(k\) are updated based on the transmitting APs of all UEs obtained in the \(r-1\)-th iteration, such as \(\{{\mathcal{J}_i'}^{(r-1)}\}\). The updates are performed independently for each UE as

\[\begin{equation*} {\mathcal{J}_k'}^{(r)}=\arg\max_{\mathcal{J}_k} \left(\prod\nolimits_{i\in \mathcal{Q}_k\cup\{k\}} {C_i'}^{(r)}\right)^{\frac{1}{\left|\mathcal{Q}_k\right|+1}}, \tag{11} \end{equation*}\]

where \({C_k'}^{(r)}\) is the estimated throughput of UE \(k\) in the AP clustering process for UE \(k\) in the \(r\)-th iteration and is calculated by

\[\begin{equation*} {C_k'}^{(r)}=\log_2\left(1+\frac{\lambda_k'\left({\mathcal{J}_k'}^{(r)}\right)p_k}{\displaystyle \sum\nolimits_{i\in \mathcal{Q}_k}w_{i,k}\left({\mathcal{J}_i'}^{(r-1)}\right)+N_0}\right). \tag{12} \end{equation*}\]

The numerator in (12) is the interference power from other UEs that is calculated using \(\{{\mathcal{J}_k'}^{(r-1)}\}\). This indicates that the calculation of (12), namely the estimated throughput, relies only on the results obtained in the previous iteration, unlike (7). Therefore, the scalable iterative method can update the transmitting AP groups of all UEs in parallel, as shown in Fig. 3.

Fig. 3  Flow of scalable iterative update method.

4.3  Computational Complexity

In this subsection, we discuss the computational complexity of the two proposed AP clustering methods. Table 1 gives the computational complexity of the iterative update and scalable iterative update methods. Term \(\mathcal{G}'(\mathcal{S}_k)\) is the set of UEs that AP \(l \in \mathcal{S}_k\) knows the instantaneous CSI in \(\mathcal{Q}_k\), \(\mathcal{G}'(\mathcal{S}_k)=\{i\colon \mathcal{S}_i\cap\mathcal{S}_k\neq\varnothing, i\in \mathcal{Q}_k\}\). In this comparison, it is assumed that the number of elements in set \(\mathcal{G}'(\mathcal{S}_k)\) is larger than \(|\mathcal{S}_k\cup \mathcal{D}_k|N\). As shown in Table 1, the scalable iterative update method especially reduces the computational complexity required for the calculation of the Moore-Penrose general inverse matrix to obtain the BF vector.

Table 1  Comparison of computational complexity.

In the following, we discuss the computational complexity in the path-loss based and SLNR-based methods. The path-loss based method does not need the complex calculations given in Table 1 to determine \(\mathcal{J}_k\) since it independently selects APs with the \(U\) lowest path loss between it and UE \(k\) from the candidate APs. Therefore, the computational complexity level of this method is much lower than that for the proposed methods. On the other hand, the SLNR-based method needs to calculate the estimated interference power that the signal transmitted from the AP group to UE \(k\) gives to other UEs, the Moore-Penrose generalized inverse matrix to obtain the BF vector, and the metric, similar to the iterative update method. However, this method does not calculate the estimated interference power of UE \(k\) and does not iterate the above three calculations. This contrasts with the iterative update method, resulting in a lower computational complexity level. Although the proposed methods require a higher computational complexity level than that for the conventional methods, they increase the user throughput, which will be confirmed in Sect. 5.

5.  Numerical Results

5.1  Simulation Parameters

The user throughput is evaluated by computer simulation according to Table 2. APs and UEs are randomly placed in a wraparound square system coverage with a side of 2 km according to the Poisson point process. The number of AP antennas is set to \(N = 1\), as considered in related cell-free MIMO literature such as [7]-[9], [18]. The transmission bandwidth is set to 100 MHz, and the transmission signal power per UE is set to 40 dBm. The propagation model simulates the distance attenuation, shadowing, and instantaneous fading as given in Table 2, and the noise power density at the UE is set to \(-165\) dBm/Hz. Term \(\Delta_{\text{CSI}}\) in the partial CSI model described in Sect. 2 is parameterized. In the proposed methods, the number of iterations is set to two. In addition to the proposed methods, the scalable iterative update method that only performs initialization (\(r = 0\)), path-loss based method, and SLNR-based method are evaluated. Throughput is calculated based on Shannon’s capacity formula. If \(F(\mathcal{J}_k)\) is less than one, the throughput of UE \(k\) is set to zero.

Table 2  Simulation parameters.

5.2  Simulation Results

Figures 4 and 5 show the geometric-mean and worst user throughput as a function of \(\Delta_{\text{CSI}}\), respectively. In the scalable iterative update method, the number of UEs per UE cluster \(|\mathcal{Q}_k|\) is set to 20, which is half the number of UEs \(|\mathcal{K}|\) in the simulated system coverage area. The geometric-mean and worst user throughput for all the methods tend to decrease when \(\Delta_{\text{CSI}}\) is greater than 10 dB. This is because the number of other UEs that need to be nulled for data transmission to a UE tends to increase when \(\Delta_{\text{CSI}}\) is large. If the number of such UEs is close to the total number of transmitting AP antennas, the received signal power gain by the BF is liable to be small. In addition, data transmission is impossible in the extreme case where there are more UEs that need to be nulled for data transmission to a UE than the number of antennas of all transmitting APs, resulting in a lack of spatial degrees of freedom. As shown in Figs. 4 and 5, most of the methods achieve the highest throughput when \(\Delta_{\text{CSI}}\) is 10 dB. Meanwhile, the large value of \(\Delta_{\text{CSI}}\) necessitates many APs estimating the CSI using the reference signal with a low received signal power. This results in non-negligible CSI estimation errors and is thus infeasible in real systems. It is shown that cell range expansion, which biases the received signal power in the handover criteria and is deployed in 4G and 5G cellular systems, can compensate for the difference in the channel conditions by setting its bias value to 10 dB [24]. Since this bias value is equivalent to \(\Delta_{\text{CSI}}\), it indicates the validity of using \(\Delta_{\text{CSI}} = 10\) dB. Therefore, \(\Delta_{\text{CSI}}\) is set to 10 dB in the subsequent evaluations.

Fig. 4  Geometric-mean user throughput as a function of \(\Delta_{\text{CSI}}\).

Fig. 5  Worst user throughput as a function of \(\Delta_{\text{CSI}}\).

The iterative update method achieves higher geometric-mean and worst user throughput than other methods. This indicates that the iterative update method selects transmitting AP groups considering the effect among all UEs. The throughput of the pass-loss based method with \(U = 15\) is higher than that with \(U = 10\), while it is lower than the SLNR-based method.

The scalable iterative update method outperforms the path-loss based method and improves the user throughput by updating with iterations. Moreover, this method achieves geometric-mean and worst user throughput levels of greater than 60% for each of the SLNR-based and iterative update methods. This implies that the scalable iterative update method can select the appropriate transmitting AP group since updating them has a great impact on the throughput of UEs that neighbor a target UE.

Figures 6 and 7 show geometric-mean and worst user throughput as a function of \(|\mathcal{Q}_k|\), respectively. From the figures, we see that the scalable iterative update method selects the transmitting AP group that improves both geometric-mean and worst user throughput as \(|\mathcal{Q}_k|\) increases. This is because the AP group selection in this method can appropriately take into account the effects among UEs thanks to an increase in UEs that are considered in the calculation of the effective channel gain and interference power. When \(|\mathcal{Q}_k|\) is greater than 30, the throughput of the scalable iterative update method is close to that of the SLNR-based and iterative update methods. In addition, it achieves the same level of throughput at \(|\mathcal{Q}_k|=20\) compared to the path-loss based method with \(U = 15\).

Fig. 6  Geometric-mean user throughput as a function of \(|\mathcal{Q}_k|\).

Fig. 7  Worst user throughput as a function of \(|\mathcal{Q}_k|\).

Finally, Fig. 8 shows the cumulative probability of the number of APs used per UE. Both of the proposed methods achieve comparable throughput to the path-loss based and SLNR-based methods while reducing the number of transmitting APs per UE. Therefore, the proposed methods achieve good throughput while maintaining the same number of APs per UE, which may lead to reduced complexity of the system configuration.

Fig. 8  Cumulative probability of the number of APs per UE.

6.  Conclusion

We considered a downlink cell-free MIMO system with partial instantaneous CSI knowledge and proposed two AP clustering methods that consider the throughput and interference among UEs. The first proposed method, the iterative update method, fully considers the effect among all other UEs in the system coverage and successively updates transmitting APs for each UE with the calculation of the estimated throughput. The other proposed method, the scalable iterative update method, determines the UE cluster for each UE, which comprises the UEs considered in the AP clustering process, and updates the AP groups independently, reducing the computational complexity level. The proposed AP clustering methods improve the geometric-mean and worst user throughput compared to those for the conventional methods while avoiding an excessive increase in the number of transmitting APs per UE. Moreover, the scalable iterative update method achieves comparable throughput to the iterative update and SLNR-based methods while reducing the number of UEs considered in the AP clustering process. In the future, we plan to investigate ways to reduce the load on the coordination process required for the AP clustering in realistic environments where coordination among APs via a central processing unit is limited. Moreover, we will consider improving the layered BF method based on channel-matrix muting and extending the method to cope with the limitation of the transmission power per AP.

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Footnotes

1. To reduce the required computational complexity for the BF, a local BF method in which each AP individually computes the BF vector has been investigated [16]. Besides, the authors of [17] have proposed the method to categorize APs into the ones using a sophisticated BF, such as ZF, and the others using a simple BF.

Authors

Daisuke ISHII
  Tokyo University of Science

Takanori HARA
  Tokyo University of Science

Kenichi HIGUCHI
  Tokyo University of Science

Keyword

MIMO,  cell-free MIMO,  inter-base station cooperation,  transmitter selection,  beamforming,  partial CSI