# Joint Resource Allocation Optimization for the Wireless Backhaul Link of User-Centric Cell-Free NOMA Networks

Authors: Qiang Wang; Qingxiu Ma
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

In this paper, we investigate the downlink resource allocation problem for the user-centric ultra-dense networks (UUDN) which can provide high area throughput density. UUDN can effectively improve the user experience since one user can be served by more than one access points (APs) simultaneously. However, the ultra-dense node deployment will also result in serious co-channel interference which will degrade the performance of the networks. Nonorthogonal multiple access (NOMA) is introduced to reduce the interference caused by the network densification. We consider the resource allocation for both the wireless backhaul downlink and the user access downlink at the same time in UUDN with NOMA, which involves the APs grouping and power allocation. The resource allocation problem is formulated as a mixed integer nonconvex optimization problem due to the combinatorial nature of APs grouping. We first decompose the problem into the APs grouping subproblem and power allocation subproblem to reduce the computational complexity. For the APs grouping subproblem, we model it as a bipartite graph maximum weight matching problem, and the Hungarian algorithm is used to solve it. When the APs grouping is fixed, the original problem is still difficult to tackle since the power allocation is a max-min optimization problem. The KKT conditions are applied to optimize the power allocation subproblem. Simulation results illustrate that the proposed resource allocation algorithm can effectively improve the system throughput of UUDN.


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## Body

## 1. Introduction

Recently, the new generation of the wireless communication technology is developed explosively [1, 2]. Densification is one of the main features of the new generation of the wireless network, since the number of wireless network nodes has to tend to be massive in order to meet the explosive growth of capacity and traffic data. Nevertheless, the ultra-dense deployment of the wireless nodes will inevitably cause serious co-channel interference and power consumption which will significantly degrade the performance of the network. In this underground, the user-centric ultra-dense network (UUDN) has been presented as one of the promising solutions to improve the user experience in complex electromagnetic environment.In UUDN, substantial low-power and low-cost access points (APs) are deployed, and each user can be served by multiple APs in the same time-frequency resource. Since there are no cell boundaries, each user can obtain excellent experience. Although UUDN has advantages over conventional wireless networks, the ultradense node deployment of UUDN will result in high cost if the traditional wired connectivity backhauling is adopted. In addition, the locations of some APs may be hard to reach for the wired connectivity backhauling. In contrast, wireless backhauling has better cost-competitive and deployment flexibility compared with wired connectivity backhauling. Due to the limitation of the radio resource, the transmit rate of the wireless backhauling link will be a bottleneck of the system performance. Thus, both the transmit rate of the wireless backhauling link and that of the user access link are necessary to be optimized at the same time to improve the performance of the system. In addition, another tricky problem of UUDN is the APs grouping. That is, which APs are grouped as a cluster, and this cluster connects with which users. Furthermore, in each APG, an enhanced AP (EAP) with high capability is selected as a manager of the other APs to reduce the signaling overhead. Obviously, the APs grouping problem is a combinatorial optimization problem with high computational complexity.Nonorthogonal multiple access (NOMA) is another promising wireless access technology since it can significantly improve the spectrum efficiency [3, 4]. In NOMA, multiple users can share the same frequency band and time slot that benefited from the superposition coding and successive interference cancelation (SIC). Different users can be split in the power domain, and the user with lower channel condition will be allocated more power, and so on. Therefore, it is appropriate for UUDN to employ NOMA to reduce the co-channel interference.


### 1.1. Related Works

Seriously, UUDN is one of the cell-free networks, wherein a very large number of distributed APs simultaneously and jointly serve a much smaller number of user equipments (UEs) [5]. Indeed, in accordance with whether one user is served by all APs, the cell-free networks can be divided into two classes: full cooperation cell-free networks and UUDN. In the following, we will introduce the related works about the cell-free networks and provide the motivation of this paper.The full cooperation cell-free network allows all the APs to serve all the users simultaneously. Therefore, it can be regarded as a virtual massive multiple-input and multiple-output (MIMO) system. Liu et al. [6] investigate the achievable uplink rate performance of cell-free massive MIMO systems employing zero-forcing (ZF) detectors and provides a asymptotic lower bound of the achievable uplink rate. Meanwhile, the authors in [6] show that this lower bound is better than that of the conventional MIMO system and demonstrates the potential of the cell-free massive MIMO systems. Liu et al. [7] consider the pilot allocation problem for cell-free massive MIMO system. In [7], a pilot assignment scheme based on graph coloring is proposed to mitigate the severe pilot contamination. In the first, an interference graph is constructed by an AP selection algorithm, and then, the optimal pilot assignment can be achieved by updating the interference graph. Specifically, if there are any two users are served by at least one AP, these two users will not be assigned the same pilot; otherwise, they will share the same pilot. Thus, the pilot allocation problem can be formulated as a graph coloring problem. Jiang et al. and Chen et al. [8, 9] investigate the resource allocation problem for the cell-free visible light communication networks. In [8], the joint user association and power allocation problem are considered to improve the system performance. The joint resource allocation problem is formulated as a nonconvex network utility maximization problem involving of the user fairness, load balancing, and power control. To solve this problem, the authors in [8] divide it into the user association subproblem and the power allocation subproblem. The dual projected gradient algorithm and successive convex approximation algorithm are adopted to solve these two subproblems, respectively. Ngo et al. [10] analyze the performance of cell-free massive MIMO. In [10], the authors derive rigorous closed-form capacity lower bounds for the cell-free massive MIMO downlink and uplink in consideration of the effects of channel estimation errors, power control, and nonorthogonality of pilot sequences. Nayebi et al. [11] compare the performance of the ZF precoder with that of the conjugate beamforming and show that the performance of the zero-forcing precoder outperforms that of the conjugate beamforming. Then, a suboptimal power allocation algorithm and low complexity heuristic power allocation algorithm are proposed for cell-free system with ZF precoder and conjugate beamforming, respectively.As aforementioned, UUDN is another form of the cell-free networks, and some current works have illustrated that UUDN has better performance than full cooperation cell-free networks [12, 13]. Buzzi et al. [12] extend the cell-free approach to the case in which both the APs and the UEs are equipped with multiple antennas and compare the performance of the UUDN with that of the cell-free approach. The results in [12] show that UUDN is more scalable than the full cooperation cell-free deployment and achieves generally better performance than the full cooperation cell-free approach for the vast majority of the users. He et al. [14] consider the joint problem of APs clustering and beamforming in UUDN, which is formulated as a NP-hard nonconvex optimization problem. In [14], a deep residual learning architecture is proposed to solve this problem. The user-centric unmanned aerial vehicle (UAV) swarm network is one of the important components of the next wireless networks. Huang et al. [15] investigate the coverage probability and average achievable rate for the user-centric UAV swarm networks to characterize the performance gain obtained from the increased UAV swarm diversity. NOMA can significantly reduce the cochannel interference caused by the ultradense deployment of the wireless nodes. Liu et al. [16] study the joint resource allocation problem both for the wireless backhauling link and user access link in UUDN with NOMA. The joint resource allocation problem is formulated as a mixed integer nonlinear programming (MINP) problem, in which the transmit rate in the wireless backhauling link is regarded as a constraint. The authors in [16] propose a multiple-to-one two-side matching algorithm and an iterative algorithm based on differ of convex programming (DCP) to solve this problem. Zaidi et al. [17] consider the APs clustering problem for the user-centric wireless networks. In [17], the authors propose three clustering approaches which can provide dynamic, adaptive, and overlapping APs clusters. Zhang et al. [18] investigate the full duplex in UUDN and propose a joint resource allocation scheme in terms of the user access, subchannel allocation, and power control for full duplex UUDN. In [18], the joint resource allocation problem is formulated as a fractional programming, and the multiplicative linear fractional programming algorithm and particle swarm optimization algorithm are applied to solve this problem. Zhu and Yu [19] investigate the performance analysis for the user-centric MIMO networks using stochastic model. In [19], the authors show that base station (BS) cooperation achieves better gain as compared to single-cell processing. Millimeter (mmWave) communication is another promising technology for the next-generation wireless communication network. In [20], the authors propose a dynamic APs clustering model for the mmWave networks and then investigate the coverage probability and average spectral efficiency performance using stochastic geometry tools. Ammar et al. [21] investigate the downlink resource allocation for the user-centric MIMO networks involving the user scheduling, power allocation, and beamforming. In [21], the resource allocation problem is formulated as a fractional programming, and the block coordinate descent and compressive sensing are adopted to solve it.


### 1.2. Contributions

In this paper, we investigate the resource allocation problem in NOMA-enhanced UUDN involving the APs grouping and power allocation. The sum rate maximization problems both for the wireless backhauling link and user access link are considered simultaneously to close to the practical application of the wireless network. The NOMA is applied both in the wireless backhauling link as well as the user access link. The resource allocation of the UUDN is formulated as a mixed integer nonlinear programming problem due to the combinatorial nature of the APs grouping which is extremely difficult to tackle. In order to reduce the computation complexity, we decompose the original optimization problem into the APs grouping subproblem and power allocation subproblem, and we optimize these two subproblems, respectively. For the former one, we first choose one EAP from the APs for each user which is called as the EAP matching problem, and the bipartite graph is adopted as a tool to solve the EAP matching problem. Then, we propose a heuristic matching scheme based on the NOMA principle to match the remaining APs to the users. For the power allocation subproblem, the sum rate maximization problem of the wireless backhauling link and user access link is formulated as a max-min optimization problem. The KKT conditions are adopted to tackle the power allocation subproblem.The main contributions of this paper are summarized as follows:(1)
We investigate the joint resource allocation problem in NOMA-enhanced UUDN involving of the APs grouping and power allocation. Different from of the most current works, we consider the sum rate maximization problem of the wireless backhauling link which is crucial for the whole system performance, and regard it as one of the optimization objectives as well as the sum rate maximization of the user access link instead of the constraints.(2)
Due to the ultradense deployment of the APs and UEs, the APs grouping problem has extremely high computational complexity. To reduce the computational complexity, we divide the APs grouping into two steps. Firstly, we model the EAP matching problem as a bipartite graph maximum weight matching problem, and the Hungarian algorithm is used to solve it. Then, a heuristic matching scheme based on the NOMA principle is proposed to allocate the remaining APs to the users. Indeed, in NOMA-enhanced UUDN, when an AP is assigned to one certain APG, it will cause interference to the other APs in the APG. Accordingly, in the proposed matching scheme, whether an AP is assigned to one certain APG depending on that if the profit of its participation is greater than the loss caused by its participation(3)
The power allocation problem of the NOMA-enhanced UUDN both for the wireless backhauling downlink and user access downlink is formulated as a max-min optimization problem which is difficult to solve. We propose an algorithm based on the KKT conditions to solve it.The rest of the paper is organized as follows. We describe the system model in Section2. In Section 3, we model the EAP matching as a bipartite graph maximum weight matching problem and propose a heuristic matching scheme based on the NOMA principle to allocate the remaining APs to the users. In Section 4, we optimize the power allocation problem using KKT conditions. In Section 5, simulation results are provided to evaluate the performance of the proposed resource allocation algorithm for the NOMA-enhanced UUDN. Finally, we conclude this paper in Section 6.


## 1.1. Related Works

Seriously, UUDN is one of the cell-free networks, wherein a very large number of distributed APs simultaneously and jointly serve a much smaller number of user equipments (UEs) [5]. Indeed, in accordance with whether one user is served by all APs, the cell-free networks can be divided into two classes: full cooperation cell-free networks and UUDN. In the following, we will introduce the related works about the cell-free networks and provide the motivation of this paper.The full cooperation cell-free network allows all the APs to serve all the users simultaneously. Therefore, it can be regarded as a virtual massive multiple-input and multiple-output (MIMO) system. Liu et al. [6] investigate the achievable uplink rate performance of cell-free massive MIMO systems employing zero-forcing (ZF) detectors and provides a asymptotic lower bound of the achievable uplink rate. Meanwhile, the authors in [6] show that this lower bound is better than that of the conventional MIMO system and demonstrates the potential of the cell-free massive MIMO systems. Liu et al. [7] consider the pilot allocation problem for cell-free massive MIMO system. In [7], a pilot assignment scheme based on graph coloring is proposed to mitigate the severe pilot contamination. In the first, an interference graph is constructed by an AP selection algorithm, and then, the optimal pilot assignment can be achieved by updating the interference graph. Specifically, if there are any two users are served by at least one AP, these two users will not be assigned the same pilot; otherwise, they will share the same pilot. Thus, the pilot allocation problem can be formulated as a graph coloring problem. Jiang et al. and Chen et al. [8, 9] investigate the resource allocation problem for the cell-free visible light communication networks. In [8], the joint user association and power allocation problem are considered to improve the system performance. The joint resource allocation problem is formulated as a nonconvex network utility maximization problem involving of the user fairness, load balancing, and power control. To solve this problem, the authors in [8] divide it into the user association subproblem and the power allocation subproblem. The dual projected gradient algorithm and successive convex approximation algorithm are adopted to solve these two subproblems, respectively. Ngo et al. [10] analyze the performance of cell-free massive MIMO. In [10], the authors derive rigorous closed-form capacity lower bounds for the cell-free massive MIMO downlink and uplink in consideration of the effects of channel estimation errors, power control, and nonorthogonality of pilot sequences. Nayebi et al. [11] compare the performance of the ZF precoder with that of the conjugate beamforming and show that the performance of the zero-forcing precoder outperforms that of the conjugate beamforming. Then, a suboptimal power allocation algorithm and low complexity heuristic power allocation algorithm are proposed for cell-free system with ZF precoder and conjugate beamforming, respectively.As aforementioned, UUDN is another form of the cell-free networks, and some current works have illustrated that UUDN has better performance than full cooperation cell-free networks [12, 13]. Buzzi et al. [12] extend the cell-free approach to the case in which both the APs and the UEs are equipped with multiple antennas and compare the performance of the UUDN with that of the cell-free approach. The results in [12] show that UUDN is more scalable than the full cooperation cell-free deployment and achieves generally better performance than the full cooperation cell-free approach for the vast majority of the users. He et al. [14] consider the joint problem of APs clustering and beamforming in UUDN, which is formulated as a NP-hard nonconvex optimization problem. In [14], a deep residual learning architecture is proposed to solve this problem. The user-centric unmanned aerial vehicle (UAV) swarm network is one of the important components of the next wireless networks. Huang et al. [15] investigate the coverage probability and average achievable rate for the user-centric UAV swarm networks to characterize the performance gain obtained from the increased UAV swarm diversity. NOMA can significantly reduce the cochannel interference caused by the ultradense deployment of the wireless nodes. Liu et al. [16] study the joint resource allocation problem both for the wireless backhauling link and user access link in UUDN with NOMA. The joint resource allocation problem is formulated as a mixed integer nonlinear programming (MINP) problem, in which the transmit rate in the wireless backhauling link is regarded as a constraint. The authors in [16] propose a multiple-to-one two-side matching algorithm and an iterative algorithm based on differ of convex programming (DCP) to solve this problem. Zaidi et al. [17] consider the APs clustering problem for the user-centric wireless networks. In [17], the authors propose three clustering approaches which can provide dynamic, adaptive, and overlapping APs clusters. Zhang et al. [18] investigate the full duplex in UUDN and propose a joint resource allocation scheme in terms of the user access, subchannel allocation, and power control for full duplex UUDN. In [18], the joint resource allocation problem is formulated as a fractional programming, and the multiplicative linear fractional programming algorithm and particle swarm optimization algorithm are applied to solve this problem. Zhu and Yu [19] investigate the performance analysis for the user-centric MIMO networks using stochastic model. In [19], the authors show that base station (BS) cooperation achieves better gain as compared to single-cell processing. Millimeter (mmWave) communication is another promising technology for the next-generation wireless communication network. In [20], the authors propose a dynamic APs clustering model for the mmWave networks and then investigate the coverage probability and average spectral efficiency performance using stochastic geometry tools. Ammar et al. [21] investigate the downlink resource allocation for the user-centric MIMO networks involving the user scheduling, power allocation, and beamforming. In [21], the resource allocation problem is formulated as a fractional programming, and the block coordinate descent and compressive sensing are adopted to solve it.


## 1.2. Contributions

In this paper, we investigate the resource allocation problem in NOMA-enhanced UUDN involving the APs grouping and power allocation. The sum rate maximization problems both for the wireless backhauling link and user access link are considered simultaneously to close to the practical application of the wireless network. The NOMA is applied both in the wireless backhauling link as well as the user access link. The resource allocation of the UUDN is formulated as a mixed integer nonlinear programming problem due to the combinatorial nature of the APs grouping which is extremely difficult to tackle. In order to reduce the computation complexity, we decompose the original optimization problem into the APs grouping subproblem and power allocation subproblem, and we optimize these two subproblems, respectively. For the former one, we first choose one EAP from the APs for each user which is called as the EAP matching problem, and the bipartite graph is adopted as a tool to solve the EAP matching problem. Then, we propose a heuristic matching scheme based on the NOMA principle to match the remaining APs to the users. For the power allocation subproblem, the sum rate maximization problem of the wireless backhauling link and user access link is formulated as a max-min optimization problem. The KKT conditions are adopted to tackle the power allocation subproblem.The main contributions of this paper are summarized as follows:(1)
We investigate the joint resource allocation problem in NOMA-enhanced UUDN involving of the APs grouping and power allocation. Different from of the most current works, we consider the sum rate maximization problem of the wireless backhauling link which is crucial for the whole system performance, and regard it as one of the optimization objectives as well as the sum rate maximization of the user access link instead of the constraints.(2)
Due to the ultradense deployment of the APs and UEs, the APs grouping problem has extremely high computational complexity. To reduce the computational complexity, we divide the APs grouping into two steps. Firstly, we model the EAP matching problem as a bipartite graph maximum weight matching problem, and the Hungarian algorithm is used to solve it. Then, a heuristic matching scheme based on the NOMA principle is proposed to allocate the remaining APs to the users. Indeed, in NOMA-enhanced UUDN, when an AP is assigned to one certain APG, it will cause interference to the other APs in the APG. Accordingly, in the proposed matching scheme, whether an AP is assigned to one certain APG depending on that if the profit of its participation is greater than the loss caused by its participation(3)
The power allocation problem of the NOMA-enhanced UUDN both for the wireless backhauling downlink and user access downlink is formulated as a max-min optimization problem which is difficult to solve. We propose an algorithm based on the KKT conditions to solve it.The rest of the paper is organized as follows. We describe the system model in Section2. In Section 3, we model the EAP matching as a bipartite graph maximum weight matching problem and propose a heuristic matching scheme based on the NOMA principle to allocate the remaining APs to the users. In Section 4, we optimize the power allocation problem using KKT conditions. In Section 5, simulation results are provided to evaluate the performance of the proposed resource allocation algorithm for the NOMA-enhanced UUDN. Finally, we conclude this paper in Section 6.


## 2. System Model

We consider a downlink of NOMA-enhanced UUDN consisting of one macro BS (MBS) confirming the complete coverage forM APs and N UEs (M>N) as shown in Figure 1. It is assumed that all the APs are connected to the MBS with a wireless backhaul. For the sake of simplicity, we only consider the access of UEs to the APs, and the the access of UEs to the MBS is not considered in this paper. The downlink data transmission is divided into two processes. In the first process, the MBS transmits signals to APs via the wireless backhaul downlinks. In the second process, the APs transmit signals to UEs via the user access downlinks. In order to reduce the interference, the NOMA is adopted both in the wireless backhaul downlinks and user access downlinks, and all the APs and UEs share the same spectrum resource. In UUDN, each UE can be served by more than one AP. Therefore, all the APs are grouped as N AP groups (APGs) corresponding to the N UEs.Figure 1
System model of the user-centric wireless network.


### 2.1. Problem Formulation

In this subsection, we introduce the signaling model for the wireless backhaul downlinks and the user access downlinks, respectively.


#### 2.1.1. Wireless Backhaul Downlink Throughput

In wireless backhaul downlinks, the MBS transmits signals to different APGs through different spectrum resources, and the APs in the same APG share the same spectrum resource through NOMA. Letpmn denote the transmit power from the MBS to the mnth AP in the nth APG. Denote Dmn by the path loss factor from the MBS to the mnth AP in the nth APG. The Rayleigh fading channel gain from the MBS to the mnth AP in the nth APG is denoted by gmn. Let zmn represents the additive white Gaussian noise with variance σ2. It is assumed that the MBS has the full knowledge of the channel state information (CSI) gmn, and let hmn=Dmngmn denote the channel gain from the MBS to the mnth AP in the nth APG.Without loss of generality, we assume that the channels in each APG are sorted ash1≤h2≤⋯≤hMn, ∀mn, ∀n, where Mn represents the number of APs of the nth APG. In NOMA system, SIC is adopted to mitigate the cochannel interference, and the APs with better channel conditions will decode the signals of the APs with weaker channel conditions. The SINR at the mnth AP in the nth APG can be given as
(1)γm=∑n=1Numnhmpm∑j=m+1Mujnhmpj+σ2,where umn is the AP assignment variable. If umn=1, the mth AP is assigned to the nth UE; otherwise, umn=0. Since each AP can only be assigned to one UE at most, there is ∑n=1Numn≤1, ∀m.The achievable backhaul rate of themth AP in the wireless backhauling link can be denoted as
(2)Cm=log21+γm.


#### 2.1.2. User Access Downlink Throughput

All the APs in the same APG transmit signals to the corresponding user using the same spectrum resource with NOMA. We denotepmn by the transmit power from the mth AP in the nth APG to UE n. Let Dmn denote the path loss factor from the mth AP to UE n, and the Rayleigh fading channel gain from the mth AP to UE n is denoted as gmn. Here, we also assume that all the APs have the full knowledge of the channel state information (CSI) gmn, and let hmn=Dmngmn denote the channel gain from the mth AP in the nth APG to the nth UE. We denote zmn as the additive white Gaussian noise with variance σ2. It is also assumed that all the APs have the full knowledge of CSI, and we denote the hmn=Dmngmn by the channel gain from the mth AP to the nth UE.It is assumed that the channels from the APs to thenth UE can be sorted as h1n≤h2n≤⋯≤hMn. In the user access downlink, each UE receives the signals from multiple APs at the same time. Different from the wireless backhaul downlink, in the user access downlink, the signals with better channel conditions will be interfered by the signals with worse channel conditions. Indeed, the signal with the best channel condition will be decoded in the first. However, the signals with worse signals are still useful for the corresponding user, and they cannot be subtracted.Thus, the SINR of thenth user served by the mth AP is given by
(3)γmn=umnhmnpmn∑j=1m−1ujnhjnpjn+σ2.The achievable access rate from themth AP to the nth UE can be denoted as
(4)Cmn=log21+γmn.


#### 2.1.3. Optimization Problem

In the above subsections, we have given the achievable backhaul rate and access rate of thenth UE served by the mth AP. The actual achievable rate of the nth user can be denoted as
(5)Rmn=minCm,Cmn.The system throughput can be obtained(6)R=∑n=1N∑m=1MminCm,Cmn.The optimization problem can be formulated as(7)P1maxp,u∑n=1N∑m=1MminCm,Cmn,(8)s.t.∑m=1Mpm≤Pb,(9)pmn≤P,∀m,(10)∑n=1Numn≤1,∀m,(11)pm≥0,pmn≥0,∀n,∀m,(12)umn∈0,1,∀n,∀m,where constraints (8) and (9) mean the peak power constraints of the MBS and APs, respectively, and (10) is imposed to guarantee that each AP can only be assigned to one UE at most.ProblemP1 is difficult to solve since this problem is both a mixed integer programming problem and a max-min optimization problem. To reduce the computational complexity, we decompose problem P1 into two subproblems in terms of the APs grouping and power allocation. Due to the ultradense deployment of APs and UEs, the computational complexity of the APs grouping subproblem is also extremely high. We model the EAP matching problem as a bipartite graph maximum weight matching problem, and the Hungarian algorithm is used to solve it. Then, a heuristic matching scheme based on the NOMA principle is proposed to allocate the remaining APs to the UEs. A list of important variables is provided in Table 1.Table 1
Summary of the main variables and their definitions.
VariableDefinitionDmPath loss factor from the MBS to themth APgmRayleigh fading gain from the MBS to themth APhmChannel gain from the MBS to themth APpmTransmit power from the MBS to themth APγmSINR at themth APCmAchievable backhaul rate of themth APDmnPath loss factor from themth AP to the nth UEgmnRayleigh fading gain from themth AP to UE nhmnChannel gain from themth AP to UE npmnTransmit power from themth AP to UE nγmnSINR of thenth UE served by the mth APCmnAchievable access rate from themth AP to UE n


## 2.1. Problem Formulation

In this subsection, we introduce the signaling model for the wireless backhaul downlinks and the user access downlinks, respectively.


### 2.1.1. Wireless Backhaul Downlink Throughput

In wireless backhaul downlinks, the MBS transmits signals to different APGs through different spectrum resources, and the APs in the same APG share the same spectrum resource through NOMA. Letpmn denote the transmit power from the MBS to the mnth AP in the nth APG. Denote Dmn by the path loss factor from the MBS to the mnth AP in the nth APG. The Rayleigh fading channel gain from the MBS to the mnth AP in the nth APG is denoted by gmn. Let zmn represents the additive white Gaussian noise with variance σ2. It is assumed that the MBS has the full knowledge of the channel state information (CSI) gmn, and let hmn=Dmngmn denote the channel gain from the MBS to the mnth AP in the nth APG.Without loss of generality, we assume that the channels in each APG are sorted ash1≤h2≤⋯≤hMn, ∀mn, ∀n, where Mn represents the number of APs of the nth APG. In NOMA system, SIC is adopted to mitigate the cochannel interference, and the APs with better channel conditions will decode the signals of the APs with weaker channel conditions. The SINR at the mnth AP in the nth APG can be given as
(1)γm=∑n=1Numnhmpm∑j=m+1Mujnhmpj+σ2,where umn is the AP assignment variable. If umn=1, the mth AP is assigned to the nth UE; otherwise, umn=0. Since each AP can only be assigned to one UE at most, there is ∑n=1Numn≤1, ∀m.The achievable backhaul rate of themth AP in the wireless backhauling link can be denoted as
(2)Cm=log21+γm.


### 2.1.2. User Access Downlink Throughput

All the APs in the same APG transmit signals to the corresponding user using the same spectrum resource with NOMA. We denotepmn by the transmit power from the mth AP in the nth APG to UE n. Let Dmn denote the path loss factor from the mth AP to UE n, and the Rayleigh fading channel gain from the mth AP to UE n is denoted as gmn. Here, we also assume that all the APs have the full knowledge of the channel state information (CSI) gmn, and let hmn=Dmngmn denote the channel gain from the mth AP in the nth APG to the nth UE. We denote zmn as the additive white Gaussian noise with variance σ2. It is also assumed that all the APs have the full knowledge of CSI, and we denote the hmn=Dmngmn by the channel gain from the mth AP to the nth UE.It is assumed that the channels from the APs to thenth UE can be sorted as h1n≤h2n≤⋯≤hMn. In the user access downlink, each UE receives the signals from multiple APs at the same time. Different from the wireless backhaul downlink, in the user access downlink, the signals with better channel conditions will be interfered by the signals with worse channel conditions. Indeed, the signal with the best channel condition will be decoded in the first. However, the signals with worse signals are still useful for the corresponding user, and they cannot be subtracted.Thus, the SINR of thenth user served by the mth AP is given by
(3)γmn=umnhmnpmn∑j=1m−1ujnhjnpjn+σ2.The achievable access rate from themth AP to the nth UE can be denoted as
(4)Cmn=log21+γmn.


### 2.1.3. Optimization Problem

In the above subsections, we have given the achievable backhaul rate and access rate of thenth UE served by the mth AP. The actual achievable rate of the nth user can be denoted as
(5)Rmn=minCm,Cmn.The system throughput can be obtained(6)R=∑n=1N∑m=1MminCm,Cmn.The optimization problem can be formulated as(7)P1maxp,u∑n=1N∑m=1MminCm,Cmn,(8)s.t.∑m=1Mpm≤Pb,(9)pmn≤P,∀m,(10)∑n=1Numn≤1,∀m,(11)pm≥0,pmn≥0,∀n,∀m,(12)umn∈0,1,∀n,∀m,where constraints (8) and (9) mean the peak power constraints of the MBS and APs, respectively, and (10) is imposed to guarantee that each AP can only be assigned to one UE at most.ProblemP1 is difficult to solve since this problem is both a mixed integer programming problem and a max-min optimization problem. To reduce the computational complexity, we decompose problem P1 into two subproblems in terms of the APs grouping and power allocation. Due to the ultradense deployment of APs and UEs, the computational complexity of the APs grouping subproblem is also extremely high. We model the EAP matching problem as a bipartite graph maximum weight matching problem, and the Hungarian algorithm is used to solve it. Then, a heuristic matching scheme based on the NOMA principle is proposed to allocate the remaining APs to the UEs. A list of important variables is provided in Table 1.Table 1
Summary of the main variables and their definitions.
VariableDefinitionDmPath loss factor from the MBS to themth APgmRayleigh fading gain from the MBS to themth APhmChannel gain from the MBS to themth APpmTransmit power from the MBS to themth APγmSINR at themth APCmAchievable backhaul rate of themth APDmnPath loss factor from themth AP to the nth UEgmnRayleigh fading gain from themth AP to UE nhmnChannel gain from themth AP to UE npmnTransmit power from themth AP to UE nγmnSINR of thenth UE served by the mth APCmnAchievable access rate from themth AP to UE n


## 2.1.1. Wireless Backhaul Downlink Throughput

In wireless backhaul downlinks, the MBS transmits signals to different APGs through different spectrum resources, and the APs in the same APG share the same spectrum resource through NOMA. Letpmn denote the transmit power from the MBS to the mnth AP in the nth APG. Denote Dmn by the path loss factor from the MBS to the mnth AP in the nth APG. The Rayleigh fading channel gain from the MBS to the mnth AP in the nth APG is denoted by gmn. Let zmn represents the additive white Gaussian noise with variance σ2. It is assumed that the MBS has the full knowledge of the channel state information (CSI) gmn, and let hmn=Dmngmn denote the channel gain from the MBS to the mnth AP in the nth APG.Without loss of generality, we assume that the channels in each APG are sorted ash1≤h2≤⋯≤hMn, ∀mn, ∀n, where Mn represents the number of APs of the nth APG. In NOMA system, SIC is adopted to mitigate the cochannel interference, and the APs with better channel conditions will decode the signals of the APs with weaker channel conditions. The SINR at the mnth AP in the nth APG can be given as
(1)γm=∑n=1Numnhmpm∑j=m+1Mujnhmpj+σ2,where umn is the AP assignment variable. If umn=1, the mth AP is assigned to the nth UE; otherwise, umn=0. Since each AP can only be assigned to one UE at most, there is ∑n=1Numn≤1, ∀m.The achievable backhaul rate of themth AP in the wireless backhauling link can be denoted as
(2)Cm=log21+γm.


## 2.1.2. User Access Downlink Throughput

All the APs in the same APG transmit signals to the corresponding user using the same spectrum resource with NOMA. We denotepmn by the transmit power from the mth AP in the nth APG to UE n. Let Dmn denote the path loss factor from the mth AP to UE n, and the Rayleigh fading channel gain from the mth AP to UE n is denoted as gmn. Here, we also assume that all the APs have the full knowledge of the channel state information (CSI) gmn, and let hmn=Dmngmn denote the channel gain from the mth AP in the nth APG to the nth UE. We denote zmn as the additive white Gaussian noise with variance σ2. It is also assumed that all the APs have the full knowledge of CSI, and we denote the hmn=Dmngmn by the channel gain from the mth AP to the nth UE.It is assumed that the channels from the APs to thenth UE can be sorted as h1n≤h2n≤⋯≤hMn. In the user access downlink, each UE receives the signals from multiple APs at the same time. Different from the wireless backhaul downlink, in the user access downlink, the signals with better channel conditions will be interfered by the signals with worse channel conditions. Indeed, the signal with the best channel condition will be decoded in the first. However, the signals with worse signals are still useful for the corresponding user, and they cannot be subtracted.Thus, the SINR of thenth user served by the mth AP is given by
(3)γmn=umnhmnpmn∑j=1m−1ujnhjnpjn+σ2.The achievable access rate from themth AP to the nth UE can be denoted as
(4)Cmn=log21+γmn.


## 2.1.3. Optimization Problem

In the above subsections, we have given the achievable backhaul rate and access rate of thenth UE served by the mth AP. The actual achievable rate of the nth user can be denoted as
(5)Rmn=minCm,Cmn.The system throughput can be obtained(6)R=∑n=1N∑m=1MminCm,Cmn.The optimization problem can be formulated as(7)P1maxp,u∑n=1N∑m=1MminCm,Cmn,(8)s.t.∑m=1Mpm≤Pb,(9)pmn≤P,∀m,(10)∑n=1Numn≤1,∀m,(11)pm≥0,pmn≥0,∀n,∀m,(12)umn∈0,1,∀n,∀m,where constraints (8) and (9) mean the peak power constraints of the MBS and APs, respectively, and (10) is imposed to guarantee that each AP can only be assigned to one UE at most.ProblemP1 is difficult to solve since this problem is both a mixed integer programming problem and a max-min optimization problem. To reduce the computational complexity, we decompose problem P1 into two subproblems in terms of the APs grouping and power allocation. Due to the ultradense deployment of APs and UEs, the computational complexity of the APs grouping subproblem is also extremely high. We model the EAP matching problem as a bipartite graph maximum weight matching problem, and the Hungarian algorithm is used to solve it. Then, a heuristic matching scheme based on the NOMA principle is proposed to allocate the remaining APs to the UEs. A list of important variables is provided in Table 1.Table 1
Summary of the main variables and their definitions.
VariableDefinitionDmPath loss factor from the MBS to themth APgmRayleigh fading gain from the MBS to themth APhmChannel gain from the MBS to themth APpmTransmit power from the MBS to themth APγmSINR at themth APCmAchievable backhaul rate of themth APDmnPath loss factor from themth AP to the nth UEgmnRayleigh fading gain from themth AP to UE nhmnChannel gain from themth AP to UE npmnTransmit power from themth AP to UE nγmnSINR of thenth UE served by the mth APCmnAchievable access rate from themth AP to UE n


## 3. Optimization of APs Grouping

In this section, we optimize the APs grouping subproblem with fixed power allocation. We only consider the throughput optimization problem of user access downlink as follows:(13)P2maxu∑n=1N∑m=1Mlog21+umnhmnpmn∑j=1m−1ujnhjnpjn+σ2,(14)s.t.∑n=1Numn≤1∀m,(15)umn∈0,1,∀n,∀m.ProblemP2 is a combinatorial optimization problem with high complexity. Next, we first assign an EAP to each UE using the bipartite graph model.


### 3.1. EAP Assignment Using Bipartite Graph

In this subsection, we first assign an EAP to each UE. That is, each UE is assigned only one AP with best channel condition in this step. We model the EAP assignment problem as a bipartite graphA = VAP,VUE,E, in which VAP=vAPm and VUE=vUEn mean the sets of APs and UEs, respectively. vAPm and vUEn denote the mth AP and the nth UE, respectively. E = em,n is the set of edges connecting to the vertices in different sets, where em,n denotes the edge that connect to vAPm and vUEn. Matching set M is defined as a set of pairwise nonadjacent edges. Let wm,n=log21+hmnpmn/σ2 denote the weight of em,n. Then, we can obtain a maximum weighted bipartite matching (MWBM) problem of bipartite graph. Indeed, umn is equal to the selection of edge in E. If em,n is selected, um,n =1. Otherwise, um,n =0. In this paper, The Hungarian algorithm is adopted to solve the MWBM problem.


### 3.2. Heuristic Algorithm for the Remaining APs Assignment

In the above subsection, we model the EAP assignment as an MWBM problem of bipartite graph, and each UE has been assigned one AP. In this subsection, we propose a heuristic algorithm based on NOMA principle to assign the remaining APs to UEs.Assuming themth AP is waiting to be assigned, the nth UE has been assigned Mn APs. It is assumed that the channels from these Mn APs to UE n are sorted as: h1n≤⋯≤hjn≤⋯≤hMn. Let RMn denote the sum rate of UE n served by Mn APs. RMn can be written as
(16)RMn=∑jn=1nMnlog21+hjnnpjnn∑in=1njn−1hinnpinn+σ2.If themth AP is assigned to the nth APG, let RMnm denote the new sum rate of the nth UE. Without loss of generality, we assume the current channel order is h1n≤⋯≤hmn≤⋯≤hM+1n. Then, RMnm can be given as
(17)RMnm=∑jn=1nM+1nlog21+hjnnpjnn∑in=1njn−1hinnpinn+σ2.Then, we define the utility of themth AP for the nth APG:
(18)Umn=RMnm−RMn.For themth AP, we will compare all the utilities with each other to find the APG with the best utility for AP m. Let
(19)Umn∗=maxUm1,⋯,UmN.Then, APm will be assigned to the nth APG.In accordance with the proposed heuristic matching algorithm, one AP will be assigned to the APG with the best utility. Indeed, according to Equation (3), if one AP is far from one UE, this AP will achieve less transmit rate since the channel condition of this AP is weaker. Furthermore, it will cause serious interference to the APs with better channel conditions. Therefore, according to the principle of NOMA, one AP will be assigned to the UE with closer distance to it since the APs with the better channel conditions will achieve larger transmit rate meanwhile they will cause weaker interference. The details of the proposed heuristic algorithm for the remaining APs assignment are shown in Algorithm 1.Algorithm 1: Heuristic algorithm for the remaining APs assignment.
Initialize the number of the APGs denoted byN, and the number of the APs in the nth (n=1,⋯,N) APG denoted by Mn. Let M denote the set of all APs, and M denote the cardinality of M, and mn denote the set of the APs in the nth APG, and mn is initialized as 1.form =1:Mdoforn =1:NdoComputeUmn according to (18)end forFind the APG with the best utility for themth AP denoted by n∗. The mth AP is assigned to the nth APG, and let mn∗=mn∗⋃m.end for


## 3.1. EAP Assignment Using Bipartite Graph

In this subsection, we first assign an EAP to each UE. That is, each UE is assigned only one AP with best channel condition in this step. We model the EAP assignment problem as a bipartite graphA = VAP,VUE,E, in which VAP=vAPm and VUE=vUEn mean the sets of APs and UEs, respectively. vAPm and vUEn denote the mth AP and the nth UE, respectively. E = em,n is the set of edges connecting to the vertices in different sets, where em,n denotes the edge that connect to vAPm and vUEn. Matching set M is defined as a set of pairwise nonadjacent edges. Let wm,n=log21+hmnpmn/σ2 denote the weight of em,n. Then, we can obtain a maximum weighted bipartite matching (MWBM) problem of bipartite graph. Indeed, umn is equal to the selection of edge in E. If em,n is selected, um,n =1. Otherwise, um,n =0. In this paper, The Hungarian algorithm is adopted to solve the MWBM problem.


## 3.2. Heuristic Algorithm for the Remaining APs Assignment

In the above subsection, we model the EAP assignment as an MWBM problem of bipartite graph, and each UE has been assigned one AP. In this subsection, we propose a heuristic algorithm based on NOMA principle to assign the remaining APs to UEs.Assuming themth AP is waiting to be assigned, the nth UE has been assigned Mn APs. It is assumed that the channels from these Mn APs to UE n are sorted as: h1n≤⋯≤hjn≤⋯≤hMn. Let RMn denote the sum rate of UE n served by Mn APs. RMn can be written as
(16)RMn=∑jn=1nMnlog21+hjnnpjnn∑in=1njn−1hinnpinn+σ2.If themth AP is assigned to the nth APG, let RMnm denote the new sum rate of the nth UE. Without loss of generality, we assume the current channel order is h1n≤⋯≤hmn≤⋯≤hM+1n. Then, RMnm can be given as
(17)RMnm=∑jn=1nM+1nlog21+hjnnpjnn∑in=1njn−1hinnpinn+σ2.Then, we define the utility of themth AP for the nth APG:
(18)Umn=RMnm−RMn.For themth AP, we will compare all the utilities with each other to find the APG with the best utility for AP m. Let
(19)Umn∗=maxUm1,⋯,UmN.Then, APm will be assigned to the nth APG.In accordance with the proposed heuristic matching algorithm, one AP will be assigned to the APG with the best utility. Indeed, according to Equation (3), if one AP is far from one UE, this AP will achieve less transmit rate since the channel condition of this AP is weaker. Furthermore, it will cause serious interference to the APs with better channel conditions. Therefore, according to the principle of NOMA, one AP will be assigned to the UE with closer distance to it since the APs with the better channel conditions will achieve larger transmit rate meanwhile they will cause weaker interference. The details of the proposed heuristic algorithm for the remaining APs assignment are shown in Algorithm 1.Algorithm 1: Heuristic algorithm for the remaining APs assignment.
Initialize the number of the APGs denoted byN, and the number of the APs in the nth (n=1,⋯,N) APG denoted by Mn. Let M denote the set of all APs, and M denote the cardinality of M, and mn denote the set of the APs in the nth APG, and mn is initialized as 1.form =1:Mdoforn =1:NdoComputeUmn according to (18)end forFind the APG with the best utility for themth AP denoted by n∗. The mth AP is assigned to the nth APG, and let mn∗=mn∗⋃m.end for


## 4. Sum Rate Maximization

In the above section, we have optimized the APs grouping subproblem with fixed power allocation. In this section, we solve the power allocation subproblem based on the APs grouping result in the above section. For the sake of simplicity, we letpmn and pmnn denote the transmit power from the MBS to the mnth AP of the nth APG and the transmit power from the mnth AP of the nth APG to the nth UE, respectively. Denote hmn and hmnn by the channel gain from the MBS to the mnth AP of the nth APG and the channel gain from the mnth AP of the nth APG to the nth UE, respectively. Then, the power allocation subproblem can be rewritten as
(20)P3maxp∑n=1N∑m=1MminC^mn,C^mnn,(21)s.t.∑n=1N∑mn=1Mnpmn≤Pb,(22)pmn≤P,∀m,(23)pmnn≤P,∀n,∀mn,where C^mn=log1+pmn/∑jn=mn+1Mnpjn+Hmn and C^mnn=log1+pmnn/∑jn=1mn−1HjnnPjnn+Hmnn, and Hmn=σ2/hmn, and Hjn=hjnn/hmnn, and Hmnn=σ2/hmnn.ProblemP3 is a max-min optimization problem. The solution of P3 is achieved when C^mn=C^mnn. Then, P3 can be rewritten as
(24)P4maxp∑n=1N∑mn=1Mnlogpmn+∑jn=mn+1Mnpjn+Hmn−log∑jn=mn+1Mnpjn+Hmn,(25)s.t.∑n=1N∑mn=1Mnpmn≤Pb,(26)pmnn≤P,∀n,∀mn,(27)pmn∑jn=mn+1MnPj+Hmn=pmnn∑jn=1mn−1HjnnPjnn+Hmnn,(28)pmn≥0,pmnn≥0,∀n,∀mn.The Lagrangian function of problemP4 can be written as
(29)Lp,μ,λ=∑n=1N∑mn=1Mnlogpmn+∑jn=mn+1Mnpjn+Hmn−log∑jn=mn+1Mnpjn+Hmn+μPb−∑n=1N∑mn=1Mnpmn+∑n=1N∑mn=1MnλmnnPa∑j=mn+1MnPj+Hmn−pmn∑jn=1mn−1HjnnPjnn+Hmnn,where μ and λ=λ1n,⋯,λMnnT are the Lagrange multipliers associated to the corresponding constraints of problem P4. The KKT conditions of problem P4 can be given as
(30)∂L∂pmn=1pmn+∑j=mn+1Mnpj+Hmn+∑l=1mn−11pl+∑jn=l+1Mnpjn+Hl−1∑j=l+1Mnpj+Hl−μ+Pa∑l=1mn−1λmnn−λmnn∑jn=1mn−1HjnnPjnn+Hmnn=0,(31)μPb−∑n=1N∑mn=1Mnpmn=0,(32)λmnnPa∑j=mn+1MnPj+Hmn−pmn∑jn=1mn−1HjnnPjnn+Hmnn=0,∀n,∀mn,(33)μ≥0,λmnn≥0,∀n,∀mn.In the following discussion, we first assume thatH1nn>H2nn>⋯>HMnn. The special case of Hmnn=Hm+1nn, mn=1n,⋯,M−1n will be discussed later.From (30), we can obtain
(34)∂L∂pmn−∂L∂pm−1n=1pmn+∑jn=mn+1Mnpjn+Hmn−1∑jn=mnMnpjn+Hm−1n−λmnn∑jn=1mn−1HjnnPjnn+Hmnn+λm−1nn∑jn=1mn−2HjnnPjnn+Hm−1nn+λm−1nPa=0.SinceHmnn<Hm−1nn, it can be obtained that
(35)λmnn∑jn=1mn−1HjnnPjnn+Hmnn−λm−1nn∑jn=1mn−2HjnnPjnn+Hm−1nn+Pa≥0.Then, we have(36)λmnnλm−1nn≥∑jn=1mn−2HjnnPjnn+Hm−1nn+Pa∑jn=1mn−1HjnnPjnn+Hmnn.Owning toHmn−1n≤1 and pmn−1n≤Pa,
(37)λmnnλm−1nn>1.Then, we have(38)λMnn>⋯>λmnn>λm−1nn>⋯>λ1nn≥0.According to the KKT conditions, sinceλMnn>⋯>λmnn>λm−1nn>⋯>λ2nn>0, we can obtain pmn=Pa, mn=2n,⋯,Mn. Then, there is only one Lagrangian multiplier λ1nn that should be considered. There are two cases for λ1nn, namely, λ1nn>0, or λ1nn=0. If λ1nn>0, it can be observed that constraint (32) hold with equality for all the UEs, and we have p1nn=Pa. For the case of λ1nn=0, constraint (32) for p1nn is strictly satisfied. In this case, it is difficult to optimize p1nn since the channel condition of the 1nth UE is the worst, and it will interfere all the other UEs in the same APG. In this work, we only consider the case of λ1nn>0, and the case of λ1nn=0 will be considered in the future works.Next, we optimize the transmit power from the MBS to APs. From (27), we can obtain
(39)pMn=PaHMnM−1Pa+HM−1nn,(40)pM−kn=Pa∑l=M−k+1nMnpl+HM−knM−k−1Pa+HM−k−1Pa+HM−knnk=1,⋯,M−1.In the following, we consider the special case that there are at least two APs which have the same channel conditions both for the wireless backhaul link and user access link. We regard themnth AP and the m+1nth AP in the nth APG as one virtual AP denoted by mn′. Firstly, we consider the special case for the user access link with hmnn=hm+1nn and illustrate that the channel conditions of the nth APG in which the mnth AP and m+1nth AP are regarded as one virtual AP have the order: h1nn<⋯<hm−1nn<hmn′n<hm+2nn<⋯<hMnn.Theorem 1.
The channel conditions of thenth APG in which the mnth AP and m+1nth AP are regarded as one virtual AP have the order: h1nn<⋯<hm−1nn<hmn′n<hm+2nn<⋯<hMnn.Proof.
Let the achievable access rate of the APm′ is denoted by Cmn′n=Cmnn+Cm+1nn. There is
(41)Cmnn+Cm+1nn=log1+hmnnpmnn∑jn=1nm−1nhjnnpjnn+σ2+log1+hm+1nnpm+1nn∑jn=1nmnhjnnpjnn+σ2=log1+hmnnpmnn+hm+1nnpm+1nn∑jn=1nm−1nhjnnpjnn+σ2.
Lethmn′n denote the channel condition from the virtual AP m′ to the nth UE. From (41), it can be obtained
(42)hmnnpmnn+hm+1nnpm+1nn=hmn′npmnn+pm+1nn.
Then, there is(43)hmn′n=hmnnpmnnpmnn+pm+1nn+hm+1nnpm+1nnpmnn+pm+1nn.
Thus,hmn′n can be regarded as the weighted sum of hmnn and hm+1nn. Therefore, the channel conditions of the nth APG in which the mnth AP and m+1nth AP are regarded as one virtual AP have the order: h1nn<⋯<hMnn.
According to Theorem1, the optimal pmn′n can be obtained as Pa, and pmnn and pm+1nn can be arbitrarily determined by pmnn∗+pm+1nn∗=Pa.
Next, we consider the special case for the wireless backhaul link with ofHmnn=Hm+1nn, mn=1n,⋯,M−1n. Then, it is obvious that the channel conditions of the APs in the nth APG have the order as: H1n>⋯>Hm−1n>Hmn′>Hm+2n. Therefore, the optimal power allocation pmn′∗ for the virtual AP mn′ in the nth APG can be obtained using the above method. The transmit rate of the mn′th AP is denoted by Cmn′=Cmn+Cm+1n. We have
(44)Cmn+Cm+1n=log1+pmn+pm+1n∑jn=1nM−1npjn+Hmn.
It can be observed from (44) that the sum rate of UE mn′ is determined by the sum of the mnth UE’s transmit power and the m+1nth UE’s transmit power. Therefore, pmn and pm+1n can be arbitrarily determined by pmn∗+pm+1n∗=pmn′∗.


## 5. Simulation Results

In this section, we evaluate the performance of the proposed APs grouping and power allocation algorithm for the UUDNs. In simulation, we assume that the UUDN is constructed by one MBS, some APs and UEs, and it is assumed that the coverage region of the MBS is a square with area of 0.1 KM×0.1 KM. The transmit power of the MBS and APs are set as 46 dBm and 30 dBm, respectively. In the network considered by the simulation, the MBS is located in the center, and the APs and UEs are distributed in the coverage region of the MBS uniformly. The WINNER model [22] is adopted as the large-scale path loss model, and the small-scale fading is modeled as Rayleigh fading. The system bandwidth is set to 20 MHz, and the spectrum density of the noise is set to -174 dBm/Hz.


### 5.1. Comparison of the APs Grouping

Figures2 and 3 compare the performance of the proposed APs grouping algorithm with that of the exhaustive search and random APs grouping. It can be seen from Figure 2 that the system throughput increases with the increase of the number of the APs which is intuitive. In addition, it can also be observed that the exhaustive search outperforms the proposed APs grouping algorithm and random APs grouping. Indeed, the exhaustive search provides a top bound of the system throughput; however, the computational complexity of exhaustive search is too high to be adopted. Figure 2 shows that the performance of the proposed APs grouping algorithm is close to that of the exhaustive search, and it has better performance comparing with the random APs grouping.Figure 2
Sum rate for the exhaustive search, proposed APs grouping algorithm and randomly APs grouping with the increase of the number of APs.Figure 3
Sum rate for the exhaustive search, proposed APs grouping algorithm and randomly APs grouping with the increase of the number of UEs.Figure3 shows the performance of the proposed APs grouping algorithm and the other two compared APs grouping methods with the increase of the number of the UEs. From Figure 3, it can be observed that the system throughput increases with the number of the UEs, which can be explained as the multiuser diversity. Indeed, with the increase of the UEs, the proportion of the channels with good channel conditions can also be improved. Figure 3 also shows that the performance of the proposed APs grouping algorithm is close to that of the exhaustive search, and the proposed APs grouping algorithm outperforms the random APs grouping.


### 5.2. Comparing with the Benchmarks

In this subsection, we evaluate the proposed joint APs grouping and power allocation algorithm through comparing it with some benchmarks. These benchmarks are as follows:(i)
RandomAPsGrouping: In this benchmark, the random APs grouping is adopted, and the power allocation uses the proposed power allocation algorithm.(ii)
NoPowerAllocation: In this benchmark, the proposed APs grouping algorithm is adopted, and the transmit power of both the MBS and APs are fixed.(iii)
RandomAPsGrouping&NoPowerAllocation: In this benchmark, the random APs grouping is adopted, and the transmit power of both the MBS and APs is fixed.(iv)
User-centricAccessSchemewithOMA: In this benchmark, the NOMA will not be applied in both the wireless backhaul link and user access link.(v)
Cell-centricAccessSchemewithNOMA: In this benchmark, each UE can be served by at most one AP in the access link, and the NOMA is applied in each cell.(vi)
Cell-centricAccessSchemewithOMA: In this benchmark, each UE can be served by at most one AP in the access link, and the NOMA is not applied in each cell.Figures4 and 5 show the performance of the proposed joint APs grouping and power allocation algorithm and these three benchmarks involving the APs grouping and power allocation with the increase of the number of APs and UEs, respectively. From Figure 4, it can be seen that the system throughput increases with the increase of the number of APs. The results of Figure 4 show that the proposed joint APs grouping and power allocation algorithm outperform the corresponding benchmarks and illustrate the effectiveness of the proposed joint APs grouping and power allocation algorithm. Furthermore, it can also be observed that the NoPowerAllocation benchmark has better performance comparing with the RandomAPsGrouping benchmark and RandomAPsGrouping&NoPowerAllocation benchmark. This result demonstrates that the APs grouping in UUDN is more important than power allocation. The results of Figure 5 is similar with that of Figure 4 which illustrate the effectiveness of the proposed joint APs grouping and power allocation algorithm, and it can also be seen from Figure 5 that the system throughput increases with the increase of the number of UEs due to the multiuser diversity.Figure 4
Sum rate for the proposed joint resource allocation algorithm and the benchmarks involving the APs grouping and power allocation with the increase of the number of APs.Figure 5
Sum rate for the proposed joint resource allocation algorithm and the benchmarks involving the APs grouping and power allocation with the increase of the number of UEs.Then, we compare the performance of the proposed joint APs grouping and power allocation algorithm with that of the UU-OMA benchmark, CC-NOMA benchmark, and CC-OMA benchmark to evaluate the effectiveness of the user-centric network and NOMA, and the results are depicted in Figures6 and 7. From Figures 6 and 7, it can be seen that NOMA can effectively improve the system performance. Indeed, the NOMA is adopted both in the wireless backhaul link and user access link, which can significantly decrease the cochannel interference by exploiting the power domain. In addition, the results of Figures 6 and 7 also show that the user-centric network outperforms the traditional cell-centric network since each UE can be served by multiple APs simultaneously.Figure 6
Sum rate for the proposed joint resource allocation algorithm and the benchmarks involving the OMA and cell-centric network with the increase of the number of APs.Figure 7
Sum rate for the proposed joint resource allocation algorithm and the benchmarks involving the OMA and cell-centric network with the increase of the number of UEs.


### 5.3. Convergence Analysis

In this subsection, we evaluate the convergence of the proposed joint APs grouping and power allocation algorithm, and the results is depicted in Figure8. It can be observed from Figure 8 that the proposed algorithm converges within 20 iterations.Figure 8
Sum rate with the increase of the number of iterations.


## 5.1. Comparison of the APs Grouping

Figures2 and 3 compare the performance of the proposed APs grouping algorithm with that of the exhaustive search and random APs grouping. It can be seen from Figure 2 that the system throughput increases with the increase of the number of the APs which is intuitive. In addition, it can also be observed that the exhaustive search outperforms the proposed APs grouping algorithm and random APs grouping. Indeed, the exhaustive search provides a top bound of the system throughput; however, the computational complexity of exhaustive search is too high to be adopted. Figure 2 shows that the performance of the proposed APs grouping algorithm is close to that of the exhaustive search, and it has better performance comparing with the random APs grouping.Figure 2
Sum rate for the exhaustive search, proposed APs grouping algorithm and randomly APs grouping with the increase of the number of APs.Figure 3
Sum rate for the exhaustive search, proposed APs grouping algorithm and randomly APs grouping with the increase of the number of UEs.Figure3 shows the performance of the proposed APs grouping algorithm and the other two compared APs grouping methods with the increase of the number of the UEs. From Figure 3, it can be observed that the system throughput increases with the number of the UEs, which can be explained as the multiuser diversity. Indeed, with the increase of the UEs, the proportion of the channels with good channel conditions can also be improved. Figure 3 also shows that the performance of the proposed APs grouping algorithm is close to that of the exhaustive search, and the proposed APs grouping algorithm outperforms the random APs grouping.


## 5.2. Comparing with the Benchmarks

In this subsection, we evaluate the proposed joint APs grouping and power allocation algorithm through comparing it with some benchmarks. These benchmarks are as follows:(i)
RandomAPsGrouping: In this benchmark, the random APs grouping is adopted, and the power allocation uses the proposed power allocation algorithm.(ii)
NoPowerAllocation: In this benchmark, the proposed APs grouping algorithm is adopted, and the transmit power of both the MBS and APs are fixed.(iii)
RandomAPsGrouping&NoPowerAllocation: In this benchmark, the random APs grouping is adopted, and the transmit power of both the MBS and APs is fixed.(iv)
User-centricAccessSchemewithOMA: In this benchmark, the NOMA will not be applied in both the wireless backhaul link and user access link.(v)
Cell-centricAccessSchemewithNOMA: In this benchmark, each UE can be served by at most one AP in the access link, and the NOMA is applied in each cell.(vi)
Cell-centricAccessSchemewithOMA: In this benchmark, each UE can be served by at most one AP in the access link, and the NOMA is not applied in each cell.Figures4 and 5 show the performance of the proposed joint APs grouping and power allocation algorithm and these three benchmarks involving the APs grouping and power allocation with the increase of the number of APs and UEs, respectively. From Figure 4, it can be seen that the system throughput increases with the increase of the number of APs. The results of Figure 4 show that the proposed joint APs grouping and power allocation algorithm outperform the corresponding benchmarks and illustrate the effectiveness of the proposed joint APs grouping and power allocation algorithm. Furthermore, it can also be observed that the NoPowerAllocation benchmark has better performance comparing with the RandomAPsGrouping benchmark and RandomAPsGrouping&NoPowerAllocation benchmark. This result demonstrates that the APs grouping in UUDN is more important than power allocation. The results of Figure 5 is similar with that of Figure 4 which illustrate the effectiveness of the proposed joint APs grouping and power allocation algorithm, and it can also be seen from Figure 5 that the system throughput increases with the increase of the number of UEs due to the multiuser diversity.Figure 4
Sum rate for the proposed joint resource allocation algorithm and the benchmarks involving the APs grouping and power allocation with the increase of the number of APs.Figure 5
Sum rate for the proposed joint resource allocation algorithm and the benchmarks involving the APs grouping and power allocation with the increase of the number of UEs.Then, we compare the performance of the proposed joint APs grouping and power allocation algorithm with that of the UU-OMA benchmark, CC-NOMA benchmark, and CC-OMA benchmark to evaluate the effectiveness of the user-centric network and NOMA, and the results are depicted in Figures6 and 7. From Figures 6 and 7, it can be seen that NOMA can effectively improve the system performance. Indeed, the NOMA is adopted both in the wireless backhaul link and user access link, which can significantly decrease the cochannel interference by exploiting the power domain. In addition, the results of Figures 6 and 7 also show that the user-centric network outperforms the traditional cell-centric network since each UE can be served by multiple APs simultaneously.Figure 6
Sum rate for the proposed joint resource allocation algorithm and the benchmarks involving the OMA and cell-centric network with the increase of the number of APs.Figure 7
Sum rate for the proposed joint resource allocation algorithm and the benchmarks involving the OMA and cell-centric network with the increase of the number of UEs.


## 5.3. Convergence Analysis

In this subsection, we evaluate the convergence of the proposed joint APs grouping and power allocation algorithm, and the results is depicted in Figure8. It can be observed from Figure 8 that the proposed algorithm converges within 20 iterations.Figure 8
Sum rate with the increase of the number of iterations.


## 6. Conclusion

In this paper, we investigate the resource allocation problem for both the wireless backhaul link and user access link in NOMA-enhanced UUDN. The NOMA technique is adopted both in the wireless backhaul link and user access link to improve the spectrum efficiency. The resource allocation problem involves the APs grouping problem and power allocation problem, and it is formulated as a mixed integer nonconvex optimization problem due to the combinatorial nature of APs grouping. In order to reduce the computational complexity of the original optimization problem, we decompose it into two subproblems in terms of the APs grouping and power allocation. For the APs grouping subproblem, we propose a two-stage heuristic algorithm. In the first, we model the EAP assignment problem as a bipartite graph and obtain an MWBM problem, and the Hungarian algorithm is adopted to solve the MWBM problem. Then, we propose a heuristic algorithm based on the NOMA principle to assign the remaining APs. The power allocation subproblem is formulated as max-min optimization problem since the wireless backhaul link and user access link are considered simultaneously. We propose a suboptimal algorithm based on the KKT conditions to tackle it.



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*Source: 3599520-2022-08-25.xml*