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Joint interference alignment and power allocation for NOMAbased multiuser MIMO systems
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 217 (2018)
Abstract
Interference alignment (IA) and nonorthogonal multiple access (NOMA) are key technologies for achieving the capacity scaling required by next generation networks to overcome the unprecedented growth of data network traffic. Each of these technologies was proved to present excellent performance for MIMO systems. In this article, we propose a joint IA and power allocation (PA) framework for NOMAbased multiuser MIMO (MUMIMO) systems. Different approaches for applying IA in downlink NOMAbased MUMIMO systems will be addressed while implementing a PA technique that fully exploits the characteristics of NOMAbased systems. The proposed framework aims to maximize the sumrate of the NOMAbased MUMIMO system through combining IA with PA. The process begins by initially grouping the system users into clusters for optimum implementation of NOMA. The sumrate maximization is carried out under cluster power budget, user qualityofservice (QoS), and robust successive interference cancellation (SIC) constraints. Meanwhile, it uses the power domain multiplexing strategy to allow the users within each cluster to share the data streams without exerting interference to one another. Three iterative joint IA and PA algorithms are proposed for NOMAbased MUMIMO systems. Moreover, these algorithms are compared with orthogonal multiple access (OMA)based MUMIMO counterpart as well as the stateoftheart techniques presented for NOMAbased MUMIMO systems. Numerical simulations verify that the proposed framework can greatly improve the performance of NOMAbased MUMIMO systems in terms of the achievable sumrate when compared with OMAbased MUMIMO and the stateoftheart NOMAbased MUMIMO systems.
Introduction
One of the key challenges facing the fifth generation (5G) mobile networks is the overwhelming growth of data network traffic. Accordingly, nonorthogonal multiple access (NOMA) has recently attracted much attention as a promising radio access technology in 5G mobile networks due to its superior spectral efficiency [1]. The concept behind NOMA is the exploitation of the power domain for implementing a multiple access mechanism in mobile networks. Specifically, the signals of NOMA users are assigned with different power allocation (PA) coefficients according to their channel conditions. Users with poor channel conditions are assigned more power level, and users with better channel conditions are assigned lower power level [2, 3]. One of the major advantages of the NOMA technique is its excellent ability to balance between sumrate and fairness, and accordingly achieves an optimized spectral efficiency for all the served users [2, 4].
The NOMA technique was the core of many research studies in the last few years [1–7]. In [5], a comparison between NOMA and its orthogonal counterparts, in terms of the achievable sumrate, has been accomplished and the results demonstrated the superiority of NOMA as a radio access technology for future 5G cellular networks. In [6], the NOMA technique is used to implement a cooperative transmission strategy for spectrumsharing cognitive radio networks. The user fairness for NOMAbased cellular systems has been addressed in [4]. NOMA technique is also used in cognitive radio networks in order to maintain some predefined qualityofservice (QoS) conditions [7]. The application of MIMONOMA to the downlink mobile communication networks is addressed in [2]. Specifically, they implement MIMONOMA to both cellular and cognitive inspired wireless networks. Additionally, they explored the outage probability for MIMONOMA systems and study the sumrate gap between NOMAbased networks and their orthogonal multiple access (OMA) counterparts.
During the last decade, interference alignment (IA) technique is also proposed as an excellent solution to the interference problem arising in wireless multiuser (MU) communication networks that significantly improve its sumrate [8]. Specifically, implementing IA technique in wireless networks results in sumrate that can scale linearly and without bound with the number of users in the network at high signaltonoise power ratio (SNR) [8, 9]. The key idea behind the IA technique is to align interference signals into a reduced dimensional subspace leaving the remaining subspace for the transmission of useful signal without any interference. Accordingly, maximum degrees of freedom (DoFs) for the whole network can be achieved. The IA scheme is studied for many different networks, e.g., X channel [10], Kuser interference channel (IC) [8, 11], heterogeneous networks [12–14], and cognitive radio networks [15, 16]. Moreover, the importance of the channel state information (CSI) for successful IA implementation is addressed in many works [17, 18]. Additionally, the feasibility conditions for IA implementation were the core of careful research studies [8–10].
Evaluating the capacity of a general IC is still a difficult goal for researchers in wireless communications and information theory [19] communities. However, IA technique is introduced as a DoFs optimal approach to interference management [10, 11]. This means that it can achieve the capacity of the IC at high SNR value. Note that IA approach can be achieved in time, frequency, and space dimensions. However, applying IA approach in the space dimension is the most popular due to the widespread use of MIMO technology. In MIMO networks, IA technique is applied using transmit beamforming matrices that help keeping all undesired received signals at each receiver within the same minimum dimensional subspace, leaving the desired signal subspaces interferencefree. Then, a receive beamforming matrix orthogonal to the interference subspaces at each receiver is used to completely eliminate the undesired interference signals [18, 20].
Recently, maximizing the capacity and accordingly the sumrate for NOMAbased MUMIMO communication networks becomes a target for many research works [21–24]. In [21], the problem of maximizing the sumrate for NOMAbased MIMO communication systems is studied under both total transmit power and minimum rate per user constraints. However, this study gives no attention to the applicability of successive interference cancellation (SIC) technique for networks with large number of users. In this work, they proposed PA scheme based on the CSI corresponding to fullrate transmission condition. The concept of signal alignment for both uplink and downlink transmissions in NOMAbased MIMO systems is addressed in [22]. Specifically, the authors used stochastic geometry to evaluate the performance of the proposed transmission framework with both randomly deployed users and interferers. The authors in [23] proposed a userclustering algorithm for conventional NOMAbased MUMIMO systems. They also investigated the performance of NOMAbased MIMO systems compared to OMAbased MIMO systems and concludes that NOMAbased MIMO systems are offering better capacity than the conventional OMAbased MIMO counterparts. Unlike our work, the method presented in [23] does not consider working with IAbased networks. In [24], the authors proposed a resource allocation scheme based on IA for NOMAbased networks. Specifically, they proposed a PA algorithm for 2users NOMA network that implements the singular value decomposition (SVD)based IA scheme which is not scalable to networks with large number of users. Additionally, they targeted optimizing the sumrate under total power constraint. However, the generalization of the PA to the case where there are K>2 users in the network is done in heuristically nonoptimal manner based on 2users pairing. Moreover, the solutions presented in [24] totally ignores the practical feasibility of SIC technique as well as the QoS requirements at each user, which are all considered in our proposed joint optimization algorithms.
In this article, we propose a joint IA and PA framework for optimizing the sumrate of the NOMAbased MUMIMO systems. The main contributions in our article can be summarized as follows:

We propose a system and signal model for NOMAbased MUMIMO systems that implements both the IA and PA techniques.

We formulate the IA problem for NOMAbased MUMIMO systems as an optimization problem and then find the optimum solutions according to the approach of IA employed for maximizing the system sumrate.

A PA technique for NOMAbased MUMIMO system that employs IA transceivers is introduced, aiming to maximize the sumrate under total power, robust SIC feasibility, and user QoS constraints.

We devise three iterative algorithms for solving the optimization problem in the previous item, through jointly optimizing the IA transceivers and PA coefficients of the system users.

Compare the performance of the joint optimizationbased iterative algorithms in both OMA and NOMAbased MUMIMO systems.
The remaining sections are organized as follow: Section 2 introduces the signal and system model for the considered downlink NOMAbased MUMIMO system. Then, different approaches for realizing the IA conditions for the downlink NOMAbased MUMIMO systems with full CSI available at the basestation (BS) are addressed in Section 3 together with a simple solution for the case of no CSI available at the BS. This is followed by exploiting the role of both the optimum PA and clustering in maximizing the achievable sumrate, in Section 4. Simulation results with their discussion are presented in Section 5, and Section 6 concludes our work.
Notation: Vectors and matrices are written in boldface lowercase and uppercase letters, respectively. The A^{†} and A^{∗} are referred to as the complex Hermitian transpose, and the complex conjugate of matrix A, respectively. The symbols tr (A) and ∥A∥_{2} represent the trace and 2−norm of matrix A, respectively. Moreover, ∥a∥ represents the absolute value for the vector a. The matrix I_{n} stands for the identity matrix of size n×n. The \(\mathbf {x} \sim \mathcal {CN}(\mu, \Sigma)\) means that x is complex Gaussian distributed with mean μ and covariance matrix Σ. The expression \(\sigma _{max}^{2}(\mathbf {H})\) refers to the maximum eigenvalue of the matrix H. The null(A) refers to the null space of the matrix A.
System and signal models
Consider a downlink MUMIMO communication scenario where a BS with M transmit antennas is communicating with multiple UEs, each equipped with N receiving antennas. The served UEs are grouped into M clusters with K UEs in each cluster (Fig. 1). In this work, we are considering scenarios in which the number of antennas at each user, namely N, is greater than the number of antennas equipped at the BS, namely M, that is to say N≥M. This assumption is popular in some 5G scenarios such as ultradense small cells and cloudradio access networks (CRANs) [2]. Through this assumption we are trying to consider some of the realistic scenarios that all 5G communities and mobile communication companies agreed that it will be challenging in the near future. One of the main pillars upon which the next generation mobile networks will based for achieving the 1000 times capacity scaling is the ultradense smallcell networks. In such network design, lowpower lowcost smallcell BSs will be employed for mobile data offloading. So, it is very likely that it owns the same number of antennas as the UE or even less, given the rapid progress in increasing the capabilities of such UEs. Another network design that support our assumption is the CRANs, in which UEs are served by a small number of lowcost remote radio heads (RRHs) to reduce the fronthaul overhead [2].
The powerdomain multiplexed signal of the users’ signals in cluster m is expressed as:
where s_{m,k} is the useful information signal to be transmitted to the k^{th} user in the m^{th} cluster, and α_{m,k} is its corresponding NOMA PA coefficient. The signals to be transmitted by the BS in the downlink direction is firstly precoded using the beamforming filter \(\mathbf {V} \in \mathbb {C}^{\mathrm {M} \times \mathrm {M}}\phantom {\dot {i}\!}\). Accordingly, the BS downlink transmitted signal can be written as:
where \(\phantom {\dot {i}\!}\mathbf {x} = \left [ x_{1} x_{2} \cdots x_{M}\right ]^{T} \in \mathbb {C}^{M \times 1}\) is the combined signal transmitted from the BS to all users in different clusters, with \(\phantom {\dot {i}\!}x_{m} = \sum \limits _{k=1}^{K} \alpha _{m,k} s_{m,k}\) is the data to be transmitted from the BS to the m^{th} cluster (Fig. 1). Let the radio channel over which the BS transmits its signals is denoted as \(\phantom {\dot {i}\!}\mathbf {H} = \left [{\mathbf {H}_{1}}^{T} {\mathbf {H}_{2}}^{T} {\mathbf {H}_{3}}^{T} \cdots {\mathbf {H}_{M}}^{T} \right ]^{T} \in \mathbb {C}^{MKN \times M} \), where \(\mathbf {H}_{m} \in \mathbb {C}^{KN \times M}\) are the channels between the BS and users in cluster m which are all Rayleigh fading channels, and the channel between the BS and the k^{th} user in the m^{th} cluster is denoted as H_{m,k}. The vector \(\tilde {\mathbf {s}}\) represents the powerdomain multiplexed signals for all the M clusters, which can be expressed in matrix form as:
Accordingly, the signal received at the k^{th} user in the m^{th} cluster is decoded using u_{m,k} to give the detected signal as
Assuming that V=[v_{1}v_{2}⋯v_{M}] and \(\mathbf {U}_{i} = \left [\mathbf {u}_{i,1} \mathbf {u}_{i,2} \cdots \mathbf {u}_{i,K}\right ] \in \mathbb {C}^{N \times K}, \forall i \in \{1, 2, \cdots, M \}\). The signal model in (4) can be rewritten as:
The interference signals generated in the assumed scenario can be divided into two parts, namely, the intracluster interference and the intercluster interference. The intracluster interference results from the intentional overlapping/superimposing of signals that are to be transmitted to users belonging to the cluster of the user of interest, namely, the selfinterference generated due to the implementation of the NOMA technique. On the other hand, the interuser interference originated due to the transmission of signals to users who are not belonging to the same cluster of the considered user. Using the notation of intracluster and intercluster interference, the signal model in (5) can be detailed as:
where i,m∈{1,2,⋯,M}, and j,k∈{1,2,⋯,K}. Equation (6) shows how IA and NOMA schemes are integrated together to optimize the sumrate of MUMIMO network. The IA technique is applied through implementing the transmit and receive beamformers, u_{m,k} and v_{m}. On the other hand, NOMA is applied through superimposing the signals of all users in the cluster together using power domain multiplexing, and this is achieved through careful evaluation of the PA coefficients. Our proposed algorithms will jointly optimize the beamforming vectors and the PA coefficients based on different objectives that are all related to the system sumrate. The knowledge of the channel conditions is very critical for the implementation of NOMA systems. Accordingly and without loss of generality, we will assume the channels such that the effective channel gains are ordered as follows:
and according to the principles of NOMA technique, the PA coefficients of the users with in the m^{th} cluster are ordered as follow:
Based on (6), the signaltointerferenceplusnoise power (SINR) ratio for the K^{th} user, the user with smallest effective channel gain in the m^{th} cluster, is given by
where ρ refers to the transmit SNR. According to the principles of the NOMA technique, the k^{th}user,∀ 1<k<K in the m^{th} cluster, needs to decode the messages sent to other users with poorer channel conditions first before decoding its own message. Accordingly, the message s_{m,j}, K≥j≥(k+1), can be detected at k^{th} user in the m^{th} cluster with SINR expressed as:
It is the role of the SIC to remove the message s_{m,j} from the observation of the k^{th} user. This message can be decoded successfully only when meeting the condition \(\text {log}(1+\text {SINR}_{m,k}^{j})> R_{m,j}\), with R_{m,j} denoting the j^{th} user target data rate. The operation of the SIC technique will continue until the k^{th} user decodes its own message with SINR equals \(\text {SINR}_{m,k}^{k}\). The first user in the m^{th} cluster, which is the user with largest effective channel gain, is responsible for decoding all the messages of other users in the cluster. If it is successful, it can decode its own message with SINR equals
IA and PA techniques are both used to improve the sumrate for many different wireless communication scenarios [25]. However, up to the best of our knowledge, the study of combining IA approach with PA technique in NOMAbased MUMIMO environment is not sufficiently conducted. In our proposed framework, the design of the PA scheme depends mainly on the IA strategy to be employed. So, the design of the precoding and decoding filters based on the principles of IA is accomplished first and consequently the PA strategies will be achieved. In the following section, the problem formulation for designing the IAbased precoding and decoding matrices for different IA approaches will be manipulated. This will be followed with the PA design problem which will be addressed in Section 4.
IA solutions for multiuser MIMONOMA
The design of the precoding and decoding filters for all the network nodes depends on the objective of the IA design process and the availability of the CSI. In this section, we consider the case where no global CSI available at the BS followed by the case with full global CSI available at the BS and IA transceivers design in each case. In our designs for NOMAbased MUMIMO system with full global CSI at the BS, the considered objectives of IA technique are SINR maximization approach (maxSINR), interference leakage minimization approach (MIL), and sumrate maximization approach (MaxSR). The derivations and details of each approach are discussed in the below subsections.
The case with no CSI available at the BS
In the proposed framework, the IA technique is responsible for removing the intercluster interference leaving the intracluster interference to the SIC technique implicitly implemented in the NOMA design. Accordingly, the IA conditions that guarantee the removal of intercluster interference are expressed as follows:
The availability of CSI at the BS can be considered as a great system overhead. Accordingly, designing IA precoding and decoding filters with no CSI available at the BS, however, it is nonoptimal, but it is considered in many practical scenarios because of its reduced system overhead of acquiring the global CSI from all the system nodes. One of the possible solutions to (12) is to choose V, as V=I_{M}. Choosing V in this form means that the BS broadcasts user messages without any processing, which reduces the overhead due to handshaking messages in acquiring and forwarding CSI in the network. Accordingly, the decoding filters can be evaluated by substituting in (12) as:
where h_{i,mk} is the i^{th} column of H_{m,k}. As a result of that, the IA constraints at the k^{th} user in the m^{th} cluster can be written as:
The above equation can be solved for u_{m,k} as:
By using the precoding and decoding matrices derived in this section, the intercluster interference will be eliminated, and the SINR for K^{th} user in the m^{th} cluster will be given by:
similarly, messages of users j, K≥j≥k+1≥1 will be successfully detected at the k^{th} user, K>k>1 with SINR:
The SIC scheme implemented with NOMA technique will take care of the remaining intracluster interference. Specifically, for the users k,and j, with K≥j≥k+1≥1, when the message s_{j,1} is successfully detected at the k^{th} user, it will be removed from the k^{th} user’s superimposed received signal, and SIC scheme will continue working until its own message is received with SINR equals \(\text {SINR}_{m,k}^{k}\). The evaluation of the optimum values of the PA coefficients will be discussed in Section 4.
The case with full global CSI available at the BS
In this section, we discuss different approaches for evaluating IAbased precoding and decoding filters according to the objective of the IA optimization problem. For each approach, we will formulate the optimization problem which is used at the BS to determine the optimal IA transceiver design for all users and the BS. Specifically, we explain how to design the optimal IA precoder and decoder for each userBS pair using the MIL, MaxSINR, and MaxSR approaches through solving their respective optimization problems. The general idea of using IA in our scenario is to align the unwanted received signals at the user of interest (the power multiplexed signals assigned to other clusters), into an interference subspace and reducing its projection within the desired signal subspace (the power multiplexed signals of users in its cluster). For example, for the k^{th} user in the m^{th} cluster, the IA constraints are:
IA is used to align the effect of all other clusters by considering them as interference and directing their effects into the interference subspace. This will leave each user only with the effect of its clusterpartners which can be dealt with the SIC technique which is already the core of the NOMA radio access technology.
Interference leakage minimization approach
As the name implies, this approach targets minimizing the other clustersinterference signals deliberated to the desired signal subspace at the user, and the process can be accomplished for the whole cluster at one shot by solving the corresponding optimization problem. The optimization problem corresponding to the m^{th} cluster is formulated as:
where \(l_{m} = \text {tr}\left [\mathbf {U}_{m}^{H} \mathbf {Q}_{m} \mathbf {U}_{m}\right ]\) is the total interference leakage deliberated to the useful subspace of cluster m, and Q_{m} is the interference covariance matrix for the m^{th} cluster, \(\mathbf {Q}_{m} = \sum \limits _{{i=1; \ i\neq m}}^{M} \sum _{k =1}^{K} \alpha _{m,k} \mathbf {H}_{m,k} \mathbf {V} \mathbf {V}^{H} \mathbf {H}_{m,k}^{H}\) [8]. This optimization problem can be solved by fixing a subset of variables (either U_{m} or V), and then optimize for the others, then alternate the roles between the fixed constant variables and the optimization variables. This technique tries to minimize interference leakage by alternatively optimizing the IA beamforming filters. Thus, at the BS, suppose the transmission is carried out in a specific communication direction, the optimization problem (20) is subject to \(\mathbf {U}_{m} \mathbf {U}_{m}^{H} = \mathbf {I}_{N}\), where we optimize for the decoding filters U_{m}. On the other hand, when the communication direction is reversed, the precoding and decoding filters are interchanged, and the optimization problem is now constrained to VV^{H}=I_{M} instead. If we denote d_{m} as d_{m}=min(M,N), the resulted optimization problem in each direction can be solved iteratively using alternative minimization by finding the d_{m} eigenvectors corresponding to smallest d_{m} number of eigenvalues of the interference covariance matrix Q_{m} at each iteration [8]. Therefore, the d_{m} columns of U_{m} are given by:
where ν_{d}[A] refers to the eigenvectors corresponding to the d smallest eigenvalues of A.
MaximumSINR approach
Another criteria that typically used as an objective to the IA design is the SINRmaximization approach. Specifically, the IA transceiver filters can be designed to maximize the SINR instead of only minimizing the interference leakage, where the MIL approach gives no attempt to maximize the desired signal power within the desired signal subspace. In other words, the MIL approach does not depend at all on the channels through which the desired signal arrives at the intended receiver. According to [8], the IA filters are obtained by:
where
The last criteria and optimization problem that is used to design the IA transceiver filters at the BS is the sumrate maximization. The goal is to design the optimum transceivers that maximize the sumrate.
Sumrate maximization approach
With this approach, we want to obtain the optimum transceiver filters that maximize the total system sumrate. Accordingly, the optimization problem that will be implemented to evaluate the IA transceivers have the following form:
where R_{m,k} represents the mutual information rate between the BS and the k^{th} user in the m^{th} cluster, and it is given by:
For solving the optimization problem in (24), we will use an iterative algorithm based on Riemannian optimization method [26], which can be considered as a generalization of the standard euclidean optimization by formulating the optimization problem over smooth manifolds instead of the standard euclidean space [27–29]. The update rule for the IA iterative algorithm that based on maximizing the sumrate is given by the Riemannian optimization over the Grassmann manifold and based on geodesics of a straight line in the euclidean space to the manifold. The update rule of the decoding matrix U_{m}, m∈{1,2,⋯,M} over the Grassmann manifold Gr with gradient descent method is expressed as [27]:
where ϕ_{Gr}(.)is the geodesic emanating from U_{m} on the Grassmann manifolds Gr, U_{m}∈Gr, in the direction of the gradient of the function R_{sum}, expressed as “\(\text {grad}_{\mathbf {U}_{m}} \mathrm {R}_{\text {sum}}\phantom {\dot {i}\!}\)”, and μ is the step size. The gradient, \(\text {grad}_{\mathbf {U}_{m}} \mathrm {R}_{\text {sum}}\phantom {\dot {i}\!}\), on the Grassmann manifold Gr is computed as:
The natural gradient of R_{sum}(U_{m}), expressed as \(\triangledown \mathrm {R}_{\text {sum}}\left (\mathbf {U}_{m}\right)\) is a real valued function. However, U_{m} is a matrix whose components are complex, so according to [30], the gradient can be evaluated as
where \(\frac {\delta (f(x))}{\delta x}\) refers to the partial derivative of the function f(x)with respect to x. The matrices X and Y are expressed as:
The update rule for the iterative algorithm that computes the decoding matrices U_{m} is derived by substituting (28) and (30) in (27).
The proposed joint PA and IA methods
In this section, the applied user clustering models are first explained; then, the concepts and details of the proposed PA algorithms are manipulated. Finally, the proposed joint PA and IA algorithms is proposed based on both the clustering and PA concepts.
User clustering models
User pairing is demonstrated to be very beneficial for the implementation of the NOMA technique in downlink MU scenarios [2]. In a similar fashion, we will illustrate the gain of grouping the users into clusters in the case of downlink NOMAbased MUMIMO system with the implementation of joint IA and PA optimization. The clustering process and its optimization is beyond the scope of this article. However, we will implement two extreme models for user clustering to prove the importance of using it, in terms of sumrate, when dealing with IA and PA for downlink NOMAbased MUMIMO systems. The first model depends on grouping the K users with the best channels together in the first cluster, then the following K best channels’ users are grouped within the second cluster, and so on. On the other hand, the second model depends on distributing the M users with best channel gains, one in each of the M clusters, then the following M best channels is distributed in the same fashion, and so on. Figure 2 illustrates the concepts of the two clustering models. In the following discussion, we will refer to the first and second models as the bestwithbest and the bestwithpoor models, respectively. In both clustering models, the user with the best channel gain can be considered as the cluster head.
Power allocation approach
We consider that the downlink NOMAbased MUMIMO system is divided into clusters and the beamforming process is designed such that a single beam used to send all the data messages to their respective users within a specific cluster. Since each cluster contains the same number of users and their channel gains follow the same random distribution, we assume the power budget of the BS will be divided equally between all the clusters. If we assume that the power budget of the BS is denoted as P_{BS}, subsequently, this power is allocated equally between the M clusters with each cluster allocated an amount equal to (P_{BS}/M). The power budget for each cluster will be allocated among the scheduled users within the cluster according to the principles of the NOMA technique. For the PA among the users within the cluster, the sumrate is maximized under cluster power budget constraints, minimum user sumrate (as quality of service metric) constraints, and constraints related to the implementation of SIC technique, i.e. minimum power differences among NOMA received signals as illustrated in [31]. The PA strategy will be applied separately with each cluster. Without loss of generality, we assume that the effective channel gains of the users within the m^{th} cluster satisfying u_{m,1}H_{m,1}V_{m}>u_{m,2}H_{m,2}V_{m}>⋯>u_{m,k}H_{m,k}V_{m}>⋯>u_{m,K}H_{m,K}V_{m}. Additionally, we refer to the minimum sumrate values that must be guaranteed by all users within the cluster as R_{m,1},R_{m,2},⋯,R_{m,K} where R_{m,k}>0, ∀ m and ∀ k. Since we are choosing the number of clusters equal to the number of transmitting antennas at the BS, each cluster will use the whole system bandwidth BW to serve its users. The PA optimization problem for the users within the m^{th} cluster can be reformulated as:
where P_{th} denotes the threshold minimum received power difference between users’ signals required for carrying out the SIC technique. The previous optimization problem is similar in notation to those mentioned in [31, 32], and accordingly the closed form solution presented in [32] can be applied directly to solve the optimization problem in (31). Let B and C denote the complementary set of users in the m^{th} cluster that meet the minimum sumrate for the users and SIC visibility constraints, respectively. The optimal PA for the first user within the m^{th} cluster is given by:
Additionally, the PA coefficient for the k^{th} user, with k≠1, within the m^{th} cluster can be expressed as:

If k∉B
$$ {}{{\begin{aligned} \alpha_{m,k}^{2} & = \left[\frac{1}{\prod\limits_{\stackrel{j=k}{j\notin {B}}}^{K} 2^{\left[\frac{R_{m,j}}{BW}\right]} \prod\limits_{\stackrel{j=k}{j\in {B}}}^{K} 2}  \sum\limits_{\stackrel{j=k}{j\notin {B}}}^{K} \frac{\left(2^{\left[\frac{R_{m,j}}{BW}\right]}  1\right)}{\left(\frac{\mathrm{P_{BS}}}{M}\right) \left\mathbf{u}_{m,j} \mathbf{H}_{m,j} \mathbf{V}_{m} \right^{2} \prod\limits_{\stackrel{i=k}{i\notin {B}}}^{j} 2^{\left[\frac{R_{m,i}}{BW}\right]} \prod\limits_{\stackrel{i=k}{i\in {B}}}^{j} 2} \right. \\ & \left. \sum\limits_{\stackrel{j=k}{j\notin {C}}}^{K} \frac{P_{th}}{\left(\frac{\mathrm{2P_{BS}}}{M}\right) \left\mathbf{u}_{m,j1} \mathbf{H}_{m,j1} \mathbf{V}_{m} \right^{2} \prod\limits_{\stackrel{i=k}{i\notin {B}}}^{j1} 2^{\left[\frac{R_{m,i}}{BW}\right]} \prod\limits_{\stackrel{i=k}{i\in {B}}}^{j1} 2} \right.\\ & \left. + \frac{1}{\left(\frac{\mathrm{P_{BS}}}{M}\right)\left\mathbf{u}_{m,k} \mathbf{H}_{m,k} \mathbf{V}_{m} \right^{2}}{\vphantom{\frac{1}{\prod\limits_{\stackrel{j=k}{j\notin {B}}}^{K} 2^{\left[\frac{R_{m,j}}{BW}\right]} \prod\limits_{\stackrel{j=k}{j\in {B}}}^{K} 2}}}\right] \times \left(2^{\left[\frac{R_{m,k}}{BW}\right]}  1\right). \end{aligned}}} $$(33) 
If k∈B
$$ {}\begin{aligned} \alpha_{m,k}^{2} & = \frac{1}{\prod\limits_{\stackrel{j=k}{j\notin {B}}}^{K} 2^{\left[\frac{R_{m,j}}{BW}\! \right]} \prod\limits_{\stackrel{j=k}{j\in {B}}}^{K} 2}  \sum\limits_{\stackrel{j=k}{j\notin {B}}}^{K} \frac{\left(2^{\left[\frac{R_{m,j}}{BW}\right]}  1\right)}{\left(\! \frac{\mathrm{P_{BS}}}{M}\right) \left\mathbf{u}_{m,j} \mathbf{H}_{m,j} \mathbf{V}_{m} \right^{2} \prod\limits_{\stackrel{i=k}{i\notin {B}}}^{j} 2^{\left[\frac{R_{m,i}}{BW}\right]}\! \prod\limits_{\stackrel{i=k}{i\in {B}}}^{j} 2} \\ &\quad  \sum\limits_{\stackrel{j=k}{j\notin {C}}}^{K} \frac{P_{th}}{\left(\frac{\mathrm{2P_{BS}}}{M}\right) \left\mathbf{u}_{m,j1} \mathbf{H}_{m,j1} \mathbf{V}_{m} \right^{2} \prod\limits_{\stackrel{i=k}{i\notin {B}}}^{j1} 2^{\left[\frac{R_{m,i}}{BW}\right]} \prod\limits_{\stackrel{i=k}{i\in {B}}}^{j1} 2} \\ &\quad +\frac{P_{th}}{\left(\frac{\mathrm{P_{BS}}}{M}\right)\left\mathbf{u}_{m,k1} \mathbf{H}_{m,k1} \mathbf{V}_{m} \right^{2}}. \end{aligned} $$(34)
The detailed steps for the proposed joint IA and PA framework based on different IA approaches are given in Algorithms 1, 2, and 3.
Note that the PA coefficients are initialized in all the three algorithms as α_{m,k}=1/K. Each of the algorithms is dependent on a specific criteria for achieving the IA conditions as explained in Subsection 3.2. All these algorithms rely on the availability of CSI, and all apply the same power allocation algorithm introduced in Section 4. Since the evaluation of the matrices Q_{m}, and B_{m,k} depends on the PA coefficients, the PA coefficients accordingly affect the design of the transmit and decode beamforming matrices. As a result, the proposed algorithms are jointly optimizing both the PA coefficients and the transceiver filters aiming to eventually optimize the system sumrate. The joint optimization is solved under total power budget, user QoS, and robust SIC constraints as explained in details in Section 4. It is worth noting that the proposed iterative IA techniques are carried out offline as we are assuming stationary block fading channels, which remains constant during the transmission process. Accordingly, the complexity analysis of the algorithms is not as important as the gain in the system sumrate.
Simulation results and discussion
In this section, we introduce simulation results to illustrate the sumrate improvement resulted due to joint optimization of IA transceivers and PA coefficients for downlink NOMAbased MUMIMO systems. Additionally, the simulation results for the proposed NOMAbased MUMIMO system with joint IA and PA optimization is compared with those of the conventional NOMAbased MUMIMO as well as the OMAbased MUMIMO systems. The channels between the BS and the users are all assumed to be Gaussian distributed Rayleigh fading with zero mean and unit variance in addition to the shadowing and the pathloss effects with parameters as mentioned in [32]. Additionally, we assume that perfect full global CSI is available at the BS. In other words, the BS owns a copy of the channel between it and each user in the system [8–18, 32]. The simulation results are obtained through averaging the measurements over 5000 channel realizations. For the proposed NOMAbased MUMIMO network with clustering model employed, we have assumed that the number of clusters in the system equals the number of antennas at the BS, and all clusters have the same size (number of users in the cluster). A list of all the simulation parameters used in evaluating our results are in Table 1. The algorithms that will be involved in the comparison are MILIA implemented in OMAbased MUMIMO system (MILIAMIMOOMA), MILIA implemented in conventional NOMAbased MUMIMO system (MILIAConvMIMONOMA), MILIA implemented in the proposed NOMAbased MUMIMO system (MILIAProposedMIMONOMA), MaxSINRIA implemented in OMAbased MUMIMO system (MaxSINRIAMIMOOMA), MaxSINRIA implemented in conventional NOMAbased MUMIMO system (MaxSINRIAConvMIMONOMA), MaxSINRIA implemented in the proposed NOMAbased MUMIMO system (MaxSINRIAProposedMIMONOMA), MaxSRIA implemented in OMAbased MUMIMO system (MaxSRIAMIMOOMA), MaxSRIA implemented in conventional NOMAbased MUMIMO system (MaxSRIAConvMIMONOMA), MaxSRIA implemented in the proposed NOMAbased MUMIMO (MaxSRIAProposedMIMONOMA), SVDbased IA implemented in OMAbased MUMIMO (SVDIAMIMOOMA), SVDbased IA implemented in conventional NOMAbased MUMIMO system introduced in [24] (SVDIAConvMIMONOMA in [24]), SVDbased IA implemented in the proposed NOMAbased MUMIMO system (SVDIAProposedMIMONOMA), and the conventional NOMAbased MUMIMO system without IA (ConvNOMA without IA [23]).
Figure 3 shows the variation of the sumrate versus the transmitted power, P_{tr}, with different IA approaches for the proposed NOMAbased MUMIMO as well as the conventional MIMONOMA and MIMOOMA systems. In these simulation results, we have assumed that users having the same order within the clusters will be assigned the same minimum sumrate value, which is inserted in (31) as the minimum sumrate constraint, mathematically speaking, we assume that R_{m−1,k}=R_{m,k}=R_{m+1,k}, ∀ m∈{1,2,⋯,M}. It is obvious from the results that, the IA approach that depends on maximizing the system sumrate over the Grassmann manifold outperforms both the MIL and MaxSINR approaches with both the proposed and the conventional systems. The cause behind that fact is that the sumrate maximization approach considers optimizing all the Shanonn’s capacity equation’s parameters, namely the spatial DoFs, desired signal power, and the undesired interference power, while other approaches consider optimizing only one or two of these parameters. Additionally, the proposed NOMAbased MUMIMO system that employs clustering, IA, and optimum PA obtained by solving (31) provides the most higher sumrate performance, followed by the conventional NOMAbased MUMIMO system, and the OMAbased MUMIMO system provides the worst performance in the comparison. Moreover, it is obvious that all the proposed algorithms outperform the stateoftheart algorithms [23, 24].
Figure 4 shows the effect of choosing the clustering model within the proposed NOMAbased MUMIMO system. In our work, we employed the two clustering models shown in Fig. 2. It is very clear from the results that the proposed NOMAbased MUMIMO system performs better with the bestwithpoor clustering model than with the case of bestwithbest clustering model. This is due to the effectiveness of the SIC technique with the bestwithpoor model than with the bestwithbest model. In other words, the clustering model somehow governs and keeps the minimum received power differences among the signals of different users with the NOMAbased MUMIMO system improving the performance of the SIC technique and accordingly provides optimum interference cancellation within the cluster. Another important observation is that the superiority of the bestwithpoor clustering model than the bestwithbest clustering model is guaranteed with any of the IA approaches. At a transmitted power of 35 dB, the proposed MIMONOMA with MaxSR IA approach achieves around 11 (bits\sec\Hz) more sumrate with bestwithpoor clustering model than with bestwithbest model. Similar conclusions can be reported with other IA approaches at different transmitted power levels.
Figure 5 shows the effect of the cluster size, defined as the number of users per cluster, on the system sumrate. The sumrate grows almost linearly with the cluster size for all IA approaches until reaching a specific cluster size, 10 and 12 users at transmission power levels 15dBm and 35dBm, respectively. After that, the sumrate begins to decay due to the lower efficiency of the SIC technique and accordingly the growth of intracluster interference. In other words, the sumrate for the NOMAbased MUMIMO system grows with the cluster size as long as the system meets the constraint of minimum received power differences among the users. Once the system violates this constraint, the SIC technique provides lower efficiency in canceling the intracluster interference and the sumrate begins to decrease. This behavior is common among all IA approaches with different transmission powers.
Conclusions
In this article, we have studied the application of different IA approaches to downlink NOMAbased MUMIMO systems. Specifically, we have proposed a joint IA and PA framework for maximizing the sumrate of the NOMAbased MUMIMO system under different approaches of IA. It turns out from the simulation results that IA is still an excellent scheme for accessing the maximum DoFs of the next generation NOMAbased networks. Additionally, user clustering is proven to be a critical step for the joint optimization of the PA coefficients and IA transceivers in NOMAbased MUMIMO systems. Finally, we concluded that accompanying the NOMA access technology with IA combined with optimum PA algorithm can be considered as a key solution for achieving the capacity scaling targeted by next generation 5G networks. As a future work, the performance of the proposed algorithms would be studied under both instantaneous and statistical CSI. Moreover, the performance of the IAbased transceivers in NOMAbased MUMIMO system will be investigated with partial CSI.
Abbreviations
 BS:

Basestation
 CSI:

Channel state information
 CRAN:

Cloudradio access networks
 DoFs:

Degrees of freedom
 IA:

Interference alignment
 IC:

Interference channel
 MUMIMO:

Multiuser multipleinput multipleoutput
 NOMA:

Nonorthogonal multiple access
 OMA:

Orthogonal multiple access
 PA:

Power allocation
 QoS:

Quality of service
 RRH:

Remote radio heads
 SIC:

Successive interference cancellation
 SINR:

Signal to interference plus noise ratio
 SNR:

Signal to noise ratio
 SVD:

Singular value decomposition
 UE:

User equipment
 5G:

Fifth generation
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Acknowledgements
The authors would like to thank the anonymous reviewers for their insightful comments.
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All authors discussed the experiments; MR performed the experiments and wrote the paper. LH and PZ have made some useful comments on the paper. All authors have read and approved the final manuscript.
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Rihan, M., Huang, L. & Zhang, P. Joint interference alignment and power allocation for NOMAbased multiuser MIMO systems. J Wireless Com Network 2018, 217 (2018). https://doi.org/10.1186/s136380181226y
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Keywords
 Nonorthogonal multiple access (NOMA)
 Interference alignment (IA)
 Power allocation
 Joint optimization
 Grassmann manifold