1 An Improved Coalition Game Approach for MIMO-NOMA Clustering Integrating Beamforming and Power Allocation Jiefei Ding, Jun Cai, Senior Member, IEEE, and Changyan Yi, Member, IEEE Abstract—The multiple-input multiple-output non-orthogonal multiple access (MIMO-NOMA) has been considered as a promising multiple access technology for the fifth generation (5G) networks to improve the system capacity and the spectral efficiency. In this paper, we propose a cluster beamforming strategy to jointly optimize beamforming vectors and power allocation coefficients for mobile users (MUs) in MIMO-NOMA clustering with the aim of reducing the total power consumption. This approach avoids the peer effect during the process of beamforming matrix calculation and can obtain a closed-form solution of beamforming strategy for multi-MU MIMO-NOMA clusters. To minimize the total power consumption, we further propose an improved coalition game approach to effectively optimize MU clustering for the large-scale MIMO networks, in which the size of a cluster is flexible. Furthermore, we discuss two different MIMO-NOMA scenarios and show that employing a NOMA power coefficient set can achieve a better performance than employing a single NOMA power coefficient for each MU in a cluster. Simulation results include the performance analysis for a single MIMO-NOMA cluster and the clustering result for a large-scale MIMO system with many MUs, which show that the proposed approach is superior than counterparts in finding the power efficient MIMO-NOMA clusters. Index Terms—MIMO-NOMA, mobile user clustering, coalition game, beamforming, power allocation. I. I NTRODUCTION Due to ever-increasing wireless transmission demands and limited spectrum resources, the traditional orthogonal multiple access (OMA) approaches its bottleneck in service provi- sioning especially during transmission rush hours. To further improve the network capacity, non-orthogonal multiple access (NOMA) has been recently proposed as a spectral efficient multiple access technique for 5G mobile networks, which allows multiple users to share the same spectrum in a non- orthogonal way [1], [2]. In NOMA, signals for different mobile users (MUs) are superposed based on the selected power coefficients and will be transmitted through the same channel. MUs detect the desired signals by exploring the successive interference cancellation (SIC) method [3], [4]. It has been proved in [5] that a hybrid way with NOMA and OMA can greatly improve the system performance over OMA in terms of spectrum efficiency and power efficiency. Recently,combining NOMA with multiple-input multiple- output (MIMO) technique attracts more and more research in- Jiefei Ding, Jun Cai and Changyan Yi are with the Network Intelligence and Innovation Lab (NI 2 L), ECE, University of Manitoba, Winnipeg, MB, Canada, R3T 5V6. (Corresponding authors: Jun Cai and Changyan Yi) E-mail: [email protected], {jun.cai, changyan.yi}@umanitoba.ca. terests. The combination of MIMO and NOMA (called MIMO- NOMA) introduces advantages of both technologies and can further improve spectrum reuse efficiency [6], transmission throughputs [7], [8] and power efficiency [9]. However, MIMO-NOMA also brings some new challenges as it has to effectively coordinate the resource utilization in both space and power domains, involving MU clustering (MU group formation and decoding order selection), resource manage- ment (beamforming calculation, NOMA power allocation), and potential peer effect (i.e., the interaction of MUs in beamforming calculation). The key is how to solve a mixed integer programing problem and simplify the computation through appropriate transformations or approximations so as to design a joint optimization approach. MU clustering is generally accompanied by resource man- agement. In literature, most of existing researches on this problem employed either multi-stage optimization approaches or simplified models to reduce computational complexity. Specifically, in [10] and [11], MU clustering optimization and the system optimization (through resource management) were decomposed into two independent subproblems, while ignoring their inherent interactions. Moreover, the objective function of MU clustering (such as the correlation coefficient maximization or the channel gain-difference maximization) was commonly set to be different from that of the system optimization (such as sum-rate maximization or power con- sumption minimization), so that the overall solutions may be far from optimum. The simplified models included simplified MIMO beamforming models (e.g., considering an identity beamforming matrix [12] or selecting the beamforming vector of the cell-edge MU as the cluster beamforming vector [13]) or simplified NOMA models (e.g., fixing the cluster size and power allocation coefficient [14] or a 2-MU system [15]). However, these approaches may be ineffective for large-scale MIMO systems. To further maximize the benefits of MIMO- NOMA, a joint optimization approach should be proposed. Unfortunately, the joint MU clustering and system optimiza- tion is NP hard, and the complexity increases incredibly with the number of MUs. By further considering the mobility and a large quantity of MUs, an optimal effective and computational efficient solution is preferred [16], [17]. Coalition game is an approach to explore players’ cooperative behaviors, which has been widely employed in clustering problems [18], [19]. The framework of a coalition game to determine a stable coalition structure is based on the merge-and-split rule that no player has incentive to break away. This approach is easy to be applied Copyright c 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].
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1
An Improved Coalition Game Approach forMIMO-NOMA Clustering Integrating Beamforming
and Power AllocationJiefei Ding, Jun Cai, Senior Member, IEEE, and Changyan Yi, Member, IEEE
Abstract—The multiple-input multiple-output non-orthogonalmultiple access (MIMO-NOMA) has been considered as apromising multiple access technology for the fifth generation(5G) networks to improve the system capacity and the spectralefficiency. In this paper, we propose a cluster beamformingstrategy to jointly optimize beamforming vectors and powerallocation coefficients for mobile users (MUs) in MIMO-NOMAclustering with the aim of reducing the total power consumption.This approach avoids the peer effect during the process ofbeamforming matrix calculation and can obtain a closed-formsolution of beamforming strategy for multi-MU MIMO-NOMAclusters. To minimize the total power consumption, we furtherpropose an improved coalition game approach to effectivelyoptimize MU clustering for the large-scale MIMO networks, inwhich the size of a cluster is flexible. Furthermore, we discusstwo different MIMO-NOMA scenarios and show that employinga NOMA power coefficient set can achieve a better performancethan employing a single NOMA power coefficient for each MUin a cluster. Simulation results include the performance analysisfor a single MIMO-NOMA cluster and the clustering result fora large-scale MIMO system with many MUs, which show thatthe proposed approach is superior than counterparts in findingthe power efficient MIMO-NOMA clusters.
Index Terms—MIMO-NOMA, mobile user clustering, coalitiongame, beamforming, power allocation.
I. INTRODUCTION
Due to ever-increasing wireless transmission demands andlimited spectrum resources, the traditional orthogonal multipleaccess (OMA) approaches its bottleneck in service provi-sioning especially during transmission rush hours. To furtherimprove the network capacity, non-orthogonal multiple access(NOMA) has been recently proposed as a spectral efficientmultiple access technique for 5G mobile networks, whichallows multiple users to share the same spectrum in a non-orthogonal way [1], [2]. In NOMA, signals for different mobileusers (MUs) are superposed based on the selected powercoefficients and will be transmitted through the same channel.MUs detect the desired signals by exploring the successiveinterference cancellation (SIC) method [3], [4]. It has beenproved in [5] that a hybrid way with NOMA and OMA cangreatly improve the system performance over OMA in termsof spectrum efficiency and power efficiency.
Recently, combining NOMA with multiple-input multiple-output (MIMO) technique attracts more and more research in-
Jiefei Ding, Jun Cai and Changyan Yi are with the Network Intelligenceand Innovation Lab (NI2L), ECE, University of Manitoba, Winnipeg, MB,Canada, R3T 5V6. (Corresponding authors: Jun Cai and Changyan Yi)E-mail: [email protected], jun.cai, [email protected].
terests. The combination of MIMO and NOMA(called MIMO-NOMA) introduces advantages of both technologies and canfurther improve spectrum reuse efficiency [6], transmissionthroughputs [7], [8] and power efficiency [9]. However,MIMO-NOMA also brings some new challenges as it has toeffectively coordinate the resource utilization in both spaceand power domains, involving MU clustering (MU groupformation and decoding order selection), resource manage-ment (beamforming calculation, NOMA power allocation),and potential peer effect (i.e., the interaction of MUs inbeamforming calculation). The key is how to solve a mixedinteger programing problem and simplify the computationthrough appropriate transformations or approximations so asto design a joint optimization approach.
MU clustering is generally accompanied by resource man-agement. In literature, most of existing researches on thisproblem employed either multi-stage optimization approachesor simplified models to reduce computational complexity.Specifically, in [10] and [11], MU clustering optimizationand the system optimization (through resource management)were decomposed into two independent subproblems, whileignoring their inherent interactions. Moreover, the objectivefunction of MU clustering (such as the correlation coefficientmaximization or the channel gain-difference maximization)was commonly set to be different from that of the systemoptimization (such as sum-rate maximization or power con-sumption minimization), so that the overall solutions may befar from optimum. The simplified models included simplifiedMIMO beamforming models (e.g., considering an identitybeamforming matrix [12] or selecting the beamforming vectorof the cell-edge MU as the cluster beamforming vector [13])or simplified NOMA models (e.g., fixing the cluster size andpower allocation coefficient [14] or a 2-MU system [15]).However, these approaches may be ineffective for large-scaleMIMO systems. To further maximize the benefits of MIMO-NOMA, a joint optimization approach should be proposed.
Unfortunately, the joint MU clustering and system optimiza-tion is NP hard, and the complexity increases incredibly withthe number of MUs. By further considering the mobility and alarge quantity of MUs, an optimal effective and computationalefficient solution is preferred [16], [17]. Coalition game is anapproach to explore players’ cooperative behaviors, which hasbeen widely employed in clustering problems [18], [19]. Theframework of a coalition game to determine a stable coalitionstructure is based on the merge-and-split rule that no player hasincentive to break away. This approach is easy to be applied
and has a fast convergence rate as a distributed algorithm.Moreover, coalition game allows the size of a NOMA clusterto be flexible in MU clustering if a large quantity of MUs isconsidered. However, the strategy of each MU is to maximizeits own utility in the traditional coalition rather than improvingthe system performance (the global optimization), so that thesolution may not be the global optimum. By considering theseissues, an improved coalition game is required to make thePareto optimal solution close to the global optimal one.
In this paper, we consider a more general MIMO-NOMAclustering model and propose a new cluster beamformingstrategy for MU clustering to minimize the total power con-sumption. Our main focus includes two aspects: to constructa joint optimization problem and to resolve a multi-userclustering problem. In the first aspect, we carefully designthe cluster beamforming strategy to let MIMO beamformingand NOMA power allocation to be jointly optimized. Then, aclosed-form optimal solution is obtained and employed on MUclustering so as to make the resource management and MUclustering to be jointly considered. For the second aspect, animproved coalition game is proposed to manage MU clusteringin the large-scale system for system power minimization whichis compared with the traditional one. Moreover, we considertwo alternative scenarios, i.e., a power coefficient set (MIMO-NOMA1) or a single power coefficient (MIMO-NOMA2) forpower allocation, and analyze their properties. By furthercomparing with two existing clustering approaches in literature(i.e., the channel gain-correlation based and the channel gain-difference based approaches), we show that the proposedscheme is more effective in reducing the total number offormed coalitions and the total power consumption. The maincontributions of this paper are summarized as follows.
• We propose a novel cluster beamforming strategy forMIMO-NOMA and employ it under two scenarios, i.e.,MIMO-NOMA1 and MIMO-NOMA2. We employ a gen-eral beamforming approach (zero-forcing beamforming)to simplify the computation of our proposed cluster beam-forming strategy, so that the peer effect during MU clus-tering is canceled. Through equivalent transformation, theNOMA power coefficient allocation can be integratedwith the beamforming calculation. Moreover, we derivean optimal decoding order which can perform better thanthe existing works in terms of power reduction.
• We formulate MU clustering as a coalition game due toits advantages of a distributed optimization, so that it hasa fast convergence speed and the flexible cluster size. Tofind the optimal result for the total power consumptionminimization, we further make some improvements onthe traditional coalition game by following the particleswarm optimization (POS) method which adjusts theutility function for each MU towards a global optimalsolution.
• We analyze three major factors that may affect the per-formance of a MIMO-NOMA cluster: the radius of MUs,the radius difference between two MUs, and the channelcorrelation coefficient. In simulations, we observe that theradius of the cell-edge MU and the real part of channel
correlation coefficient are the key factors influencingpower consumption. Based on this fact, we find the radiusthresholds under the considered simulation environment.
The rest of this paper is organized as follows. In SectionII, we review the existing researches on MIMO-NOMA clus-tering, which are compared with our work. In Section III,we describe MIMO-NOMA and MIMO-OMA scenarios in asingle base station MIMO system and introduce the systemmodel. The results of the cluster beamforming strategy fordifferent scenarios and cluster sizes are derived in SectionIV. MIMO-NOMA clustering approaches which include atraditional coalition game and an improved coalition game areproposed in Section V. In Section VI, simulation results arepresented and discussed. Section VII concludes the paper.
Notations: ‖.‖ and |.| denote the 2-norm and the absolutevalue, respectively. (.)∗, (.)† and (.)H stand for the conjugatetranspose, the pseudo-inverse, and the Hermitian, respectively.<(.) and =(.) denote the real part and the imaginary part ofa complex value, respectively. CN(, ) represents the complexnormal distribution.
II. RELATED WORK ON MIMO-NOMA MU CLUSTERING
In literature, there are a few researches focusing on theMIMO-NOMA mobile user (MU) clustering and power allo-cation. Due to high complexity, existing works in this areacan be classified into two categories: simplified models ormulti-stage optimization approaches. The simplified modelsin [13]–[15] are either heuristic or ineffective for the large-scale MIMO system, so that they cannot be applied in ourconsidered problem. In multi-stage optimization approaches,MU clustering and system optimization are usually managedin different stages. There are two different ways to decomposea complex problem into a hierarchical structure.
In the first way, MUs were grouped in the first stage basedon their channel gains, i.e., the gain-correlation based [20]or the gain-difference based [21] clustering approaches. AfterMU clustering, the MU group set was fixed based on anassumption that NOMA was always better than OMA. Bythis way, the system optimization can be conducted throughadjusting power allocation coefficients or calculating beam-forming vectors. However, according to our analyses, NOMAmay not be always better than OMA in a MIMO system, andits performance depends on both the correlation and differenceof MU’s channel gains. In addition, decoupling MU clusteringand the system optimization may result in the solution deviatedfrom the optimum.
The second way is to directly optimize MU clustering toimprove system performance. In the first stage, an optimizationwas conducted on a specific NOMA cluster to get a closed-form solution for power allocation and beamforming strategy[22]. Then, it can be applied to search for an optimal clusterset among a large quantity of MUs to maximize system perfor-mance. This approach is only applied for a fixed cluster sizescenario. In [23], matching game approach was employed inMU clustering optimization for a single antenna BS network,where users and sub-channel were considered as two setsof players. This algorithm grouped MUs into limited sub-channels and maximized the total sum-rate without limitations
3
Channel gain vector
Beamforming VectorWireless BS
MU 1 MU 2
MU 3
1h2h
3 3 1,Hh w =
1w v= 2w v=
MIMO-NOMA cluster
Wirellllless BS
Antennas
1 M...
3
Hh
3w
3 0( 3).H
ih w i= ¹
Fig. 1: A MIMO-NOMA scenario
on MU cluster size. However, in the MIMO system, all sub-channels can be allocated to all MUs simultaneously, so thatthe two-side matching does not applicable.
Different from all these existing works, in this paper, weconsider a MIMO-NOMA optimization problem and formulateit to be a joint optimization of beamforming calculation,NOMA power allocation and MU clustering. Moreover, thecluster size is flexible and MU clustering is managed by acoalition game. Furthermore, for each stable coalition solution,we do not need to recalculate the coalition utility. To achievethe total power consumption minimum, we improve the utilityfunction based on the result of coalition.
III. SYSTEM MODEL
In this section, we first present the considered system model,which includes beamforming model, signal model and powermodel in both MIMO-OMA and MIMO-NOMA scenarios.Since these models for the different cluster sizes are onlydifferent in the dimension of vectors, we present a case of2 MUs (mobile users) for the explanation purpose. Then, weformulate an optimization problem which includes resourcemanagement and NOMA clustering.
A. Cluster Beamforming model
Considering a single cell downlink MIMO system, it has onebase station (BS) (equipped with M antennas) and N MUs(each equipped with a single antenna), in which M ≥N . Lethu∈RM×1andwu∈RM×1denote the channel gain vector andbeamforming vector from the BS to MU u, respectively. W =[w1, ...,wN ] and H=[hT1 , ...,h
TN ]T denote the beamforming
matrix and the channel gain matrix for all MUs, respectively.The difference between MIMO-OMA and MIMO-NOMA isthat the orthogonal condition does not exist among MUs inthe same cluster. As the example in Fig. 1, for a MU, itmay have two states: served by MIMO-OMA (MO); or servedMIMO-NOMA in a cluster (MN) as a cell-center user or acell-edge user. Towards a MU in the MO state, e.g., MU3, itsbeamforming vector should satisfy the orthogonal conditions(i.e., hHi w3 = 0, i ∈ N, i 6= 3), and the normalized condition(hH3 w3 = 1). However, for MUs in a MIMO-NOMA cluster(e.g., MU1 and MU2), they share a same beamforming vectorv which satisfies hH1 v 6= 0, hH2 v 6= 0 and hH3 v = 0.
If all MUs are served by MIMO-OMA, and the beam-forming matrix is calculated by the ZF-beamforming (zero-forcing beamforming) strategy [24], [25]. Because of the zero-interference condition (hHj wk = 0 for j 6= k), the interferenceamong MUs access to the same BS can be cancelled. Thus,we have
W = H† = H∗(HH∗)−1. (1)Note that beamforming vectors for all MUs are determined
at the same time. Thus, a change to one MU’s beamformingvector (e.g., it is grouped in a different cluster), all MUs’beamforming vectors need to be recalculated. This effect isnamed as the peer effect and has been ignored in most existingworks. In this paper, we will show later that our proposedapproach can effectively avoid such peer effects.
B. Signal model
Let G = g1, g2, ..., gα denote a set of clusters, and thereare a total of α MIMO-NOMA clusters. In this set, a clusteris denoted as gk = gk,1, gk,2, ..., gk,nk, where there are nkMUs sharing the same beamforming vector and being alignedfrom the cell-center to the cell-edge, i.e., gk,1 and gk,nk denotea cell-center user and a cell-edge user, respectively.
For a specific MU i in the MO state, the received signal yican be expressed as
yi = hHi w′
i
√pixi + ni, (2)
where w′
i∈RM×1is the beamforming vector allocated to MUi, pi is the transmit power (Since ‖wi‖2 is unnormalized, theactual BS’s power consumption is calculated as in (4)), xiis the data symbol transmitting to MU i, and ni denotes theadditive white Gaussian noise with zero mean and variance σ2
i .We assume ni/x2i = σ2 for all MUs. For a guaranteed qualityof service, it is required that the signal-to-interference-plus-noise ratio (SINR) at MU i is larger than a pre-determinedthreshold δ as
γi =
∣∣∣hHi w′i
√pi
∣∣∣2σ2 ≥ δ. (3)
To meet the minimal SINR requirement, the BS’ powerconsumption can be expressed as
Pi =∥∥∥w′i∥∥∥2pi =
∥∥∥w′i∥∥∥2|hHi w
′i|2σ2δ =
∥∥∥w′i∥∥∥2σ2δ. (4)
If MU i and MU j are grouped as a MIMO-NOMA clusterk, i.e., gk = gk,1 = i, gk,2 = j, the signal vector to betransmitted by the BS is given by
sk =√pk
vk,1(õi,1xi +
õj,1xj)
...vk,M (
õi,Mxi +
õj,Mxj)
, (5)
where µε =√µε,1, ...,√µε,Mis the NOMA power coefficient
set for signal xε, ε = i, j. On each antenna, let µi,n+µj,n = 1.Besides, the beamforming vector of the cluster k is denoted byvk=vk,1, ..., vk,M, and the transmit power is indicated by pk.Note that in [10], MIMO-NOMA is ordinarily considered asan extension of single-antenna NOMA, so that a single powercoefficient was employed to each MU on all antennas, i.e.,µi,1 = µi,2 = ... = µi,M . However, in this paper, we considera more general case by relaxing the power coefficients on
4
different antennas to be different [22], [26]. We term thesetwo cases as MIMO-NOMA1 (with a power coefficient set)and MIMO-NOMA2 (with a single power coefficient).
For MIMO-NOMA1, note that NOMA employs the powercoefficient set to distinguish different signals on the powerdomain while MIMO beamforming working the space domain.To simplify calculations, we combine the beamforming vectorvk and the power coefficient õj,m to transform the originalsuperposed transmit signal formation into a new form as shownin (6). Note that the beamforming vectors for cluster membersare different and non-orthogonal. Let
vk =
√piw2
i,1 + pjw2j,1...√
piw2i,M + pjw2
j,M
,√pk√µε,m =
√pεwε,mvk,m
for m ∈M, and ε = i, j.
The transmit signal vector can be transformed into
sk = wi√pixi +wj
√pjxj , (6)
where the transformed beamforming vectors for MU i andMU j are denoted by wi and wj , respectively, and thecorresponding allocated power parameters are indicated by piand pj . Note that wi and wj are different and non-orthogonal.
For MIMO-NOMA2, the transmit signal vector can bereformulated as
sk = vk√pk(√µi,1xi +
õj,1xj). (7)
For both cases in MIMO-NOMA, the received signal at MUε is yε =hHε sk+nε (ε= i, j), and the SINR condition (3) isapplied.
C. Problem formulation
Our objective is to minimize the total power consumption,and the system optimization problem can be formulated as
minW ,µ,p
N∑i=1
Pi(wi, µi, pi), (8a)
s.t. (3) and (8b)
hiwj
6=0 If MUs i and j are in the same cluster,=0 Otherwise.
(8c)
γji ≥ ζδ, (8d)
where Pi is the power consumption for MU i (as calculated in(10) and (20)). Note that Pi is determined by the beamformingvector wi, the transmit power pi and the power allocationcoefficient µi (in the MN state 0<µi<1, and in the MO stateµi = 1). Constraint (8d) indicates that the achievable rate (orSINR) of decoding signal should be larger than a predefinedthreshold ζδ (ζ is a predefined parameter within (0,1]) in orderto ensure successful SIC decoding (refer to appendix D) [33].
IV. BEAMFORMING STRATEGY FOR A MIMO-NOMACLUSTER
In this section, we formulate a partial ZF-beamformingproblem for MIMO-NOMA clusters and introduce our clusterbeamforming strategy for both MIMO-NOMA1 and MIMO-NOMA2 scenarios. In this strategy, we try to minimize the
group power consumption and get the closed-form solutions.For explanation purpose, we start our discussion from the caseof the 2-MU cluster, and then discuss its extension to thegeneral multi-MU case.
A. MIMO-NOMA1 for a 2-MU cluster
We consider MUs i and j (located from the cell-center to thecell-edge), and the pre-determined decoding order for them isin the reverse order. After decoding the superposed message,the received signals for MU i and MU j can be respectivelyrepresented as
yi = hHi sk + ni = hHi wi√pixi + ni,
yj = hHj sk + nj = hHj wj√pjxj + hHj wi
√pixi + nj .
(9)
To meet the minimal SINR requirement, the power con-sumption for MUi and MUj can be respectively expressed as
Pi = ‖wi‖2pi = ‖wi‖2
|hHi wi|2σ2δ = ‖wi‖2σ2δ,
Pj = ‖wj‖2pj =‖wj‖2
|hHj wj|2(∣∣hHj wi∣∣2pi + σ2)δ
= ‖wj‖2(∣∣hHj wi∣∣2pi + σ2)δ.
(10)
A basic condition for forming a MIMO-NOMA cluster isthat its total power consumption is lower than that beforeclustering. According to (4), if MU i and MU j are both inthe MO state, the total power consumption is
P′
i + P′
j =∥∥∥w′i∥∥∥2σ2δ +
∥∥∥w′j∥∥∥2σ2δ, (11)
where w′
i and w′
j are the beamforming vectors in the MO stateand can be directly obtained by (1). The power reduction byadopting MIMO-NOMA can be calculated by
∆Pk = P′
i + P′
j − (Pi + Pj)
=∥∥∥w′i∥∥∥2σ2δ +
∥∥∥w′j∥∥∥2σ2δ−(‖wi‖2σ2δ + ‖wj‖2(
∣∣hHj wi∣∣2pi + σ2)δ).
(12)
By considering the possible computing cost for NOMA asΩ, we set ∆Pk > Ω (e.g., Ω = 0.01w) as the condition fora beneficial MIMO-NOMA cluster. Obviously, if ∆Pk < Ω,a NOMA cluster k will not be formed. To maximize ∆Pk,the optimal beamforming vectors can be derived based on thefollowing optimization problem
min f(wi,wj) = ‖wi‖2 + ‖wj‖2(∣∣hHj wi∣∣2δ + 1). (13)
According to ZF-beamforming strategy, the beamformingvectors of MUs in a cluster should be orthogonal with all out-of-cluster MUs’ channel gains, but non-orthogonal with thoseof MUs inside this cluster. Let H = [hTi ,h
Tj ,h
T1 , ...h
TN ]T ∈
RN×M be a reorganized channel gain matrix and W =[wi,wj ,w1, ...wN ] ∈ RM×N be the beamforming matrix.Then, we have
HW =[hTi hTj hT1 . . .h
TN
]T[wi wj w1 . . .wN
]=
1 λ 02×(N−2)β 1
0(N−2)×2 I(N−2)×(N−2)
N×N
= T ,(14)
where β and λ are two non-zero parameters denoting thenon-orthogonal relationship within a cluster [26]. From the
5
definition of beamforming matrix in the MO state W′
asdetermined in (1), we have
W = H∗(HH∗)−1HW = W′T . (15)
From (15), we notice that beamforming vectors for a NOMAcluster is actually in a linear space expanded by the relatedbeamforming vectors in the MO state (w
′
i and w′
j), i.e., wi=w′
i+βw′
j andwj=w′
j+λw′
i. Thus, one of our most importantobservations is that the cluster beamforming vectors in the MNstate are only determined by coefficients β and λ and have noeffects on other MUs’ beamforming vectors. Therefore, wecan avoid the peer effect.
From (15), the beamforming vector wj of MU j in the MNstate can be derived by
‖wj‖2 =∥∥∥w′j∥∥∥2β2 + 2βw
′
iw′
j +∥∥∥w′i∥∥∥2. (16)
For simplifying notations, in the following, we define no-tation w
′
iw′
j which equals to w′Ti w
′
j in the real-valued caseor <(w
′
i)<(w′
j) + =(w′
i)=(w′
j) in the complex-valued case.Thus, problem (13) can be transformed into an expression withonly two variables (i.e., β and λ) as
min f(wi,wj)=g(λ, β)=(∥∥∥w′j∥∥∥2β2+2βw
′
iw′
j+∥∥∥w′i∥∥∥2)
+(∥∥∥w′i∥∥∥2λ2 + 2λw
′
jw′
i +∥∥∥w′j∥∥∥2)
(δ(∣∣∣hHj w′j∣∣∣2β2 + 2β
∣∣∣hHj w′i∣∣∣∣∣∣hHj w′j∣∣∣+∣∣∣hHj w′i∣∣∣2) + 1).
(17)
We notice that, after NOMA decoding, the impact of MUj on MU i is eliminated. Thus, the power consumption ofMU j only depends on β, and λ only appears in the secondterm of (17). Moreover, ‖wi‖2 and (
∣∣hHj wi∣∣2δ+1) are alwayslarger than zero in any value of β. Therefore, min‖wj‖2 isthe necessary condition of minf(wi,wj), and ‖wj‖2 can beregarded as an independent sub-optimization problem. Wecalculate the 1st and the 2nd derivatives of (17) with respect to
λ. Since ∂2g(λ,β)∂λ2 =2
∥∥∥w′i∥∥∥2>0, g(λ, β) will reach the minimal
point when ∂g(λ,β)∂λ = 0, which results in λ=− w
′jw′i
‖w′i‖2. Then,
g(λ, β) can be simplified asg(λ, β) = g(β)
= (∥∥∥w′j∥∥∥2β2+2βw
′
iw′
j+∥∥∥w′i∥∥∥2)+‖wj‖2(δβ2+1).
(18)
Similarly, since ∂2g(β)∂β2 =2(
∥∥∥w′j∥∥∥2+δ‖wj‖2)>0, by letting∂g(β)∂β = 0, we have
β= −B2A =−w′iw′j
‖w′j‖2+δ‖wj‖2=
−∥∥∥w′i∥∥∥2w′iw′j
‖w′i‖2‖w′j‖2(1+δ)−δ|w′iw′j|2. (19)
Note that, for a beneficial NOMA cluster, the decoding orderis the descending order of the Euclidian 2-norm of beamform-ing vectors in the MO state (as shown in appendix A). For
example, if∥∥∥w′j∥∥∥2<∥∥∥w′i∥∥∥2, MUi is decoded firstly. Otherwise,
we should first decode MUj . Note that this decoding order hasnot been shown in any existing works and can ensure that thepower reduction is maximized.
B. MIMO-NOMA1 for a cluster with the size larger than 2We now extend our analysis to a general case, where there
are more than 2 MUs in a MIMO-NOMA cluster. For analysis
purpose, we consider that there are n MUs in a cluster locatingfrom the cell-center to the cell-edge, indexed from 1 to n. Inaddition, the pre-determined decoding order is in the reverseorder. According to (14), the beamforming vector of MU ican be calculated by wi = λi1w
′
1 +λi2w′
2 + ...+λinw′
n, i =1, 2, ...n and λii = 1. After decoding the superposed message,the received signals and corresponding power consumptionsfor MU i can be derived by
yi = hHi wi√pixi +
∑i−1ε=1(hHi wε
√pεxε) + ni,
Pi = ‖wi‖2(∑i−1ε=1
∣∣hHi wε∣∣2pε + σ2)δ.(20)
Specifically, the power consumptions for MUs n and n − 1are respectively equal to
To determine λij , i, j = 1, 2, ...n, the objective function isto minimize the overall power consumption, i.e., The objectivefunction is denoted by
minP1 + ...+ Pn.
Based on the similar observations in (17), we propose arecursive process to determine the beamforming vectors fol-lowing the order from MU n to MU 1. Specifically, since theparameters from λn1 to λnn−1 are only related with ‖wn‖2as power allocation pn is positive, the first subproblem fordetermining MU n’s beamforming vector is denoted as
min g(λn1, ..., λnn−1) = ‖wn‖2
=λ2n1w′21 +...+w
′2n+2λ2n1λ
2n2w
′21 w
′22 +...+2λ2n1w
′21 w
′2n .
The solution can be derived by letting the 1st derivatives ofthe objective function be zero, i.e.,
[λn1, λn2, ..., λnn−1] = BA−1,
A =
∥∥∥w′1∥∥∥2 w′
1w′
2 ... w′
1w′
n−1
w′
2w′
1
∥∥∥w2′∥∥∥2 ... w
′
2w′
n−1......
...w′
n−1w′1 w
′
n−1w′
2 ...∥∥∥w′n−1∥∥∥2
,B = −[w
′
1w′
n,w′
2w′
n, ...,w′
n−1w′
n].
(22)
After that, we substitute this solution back to the objectivefunction, and formulate the second subproblem to determinethe beamforming vector of MU n − 1 as (based on the factthat pn−1 > 0)
min g(λn−11, ..., λn−1n) = ‖wn−1‖2 + ‖wn‖2λ2n−1n.Following the similar procedure as in (22), we can deriveλn−1j, j = 1, 2, ..., n. By substituting the solution back tothe objective function, we can derive the subproblem for MUn−2. This process will be continued till all MUs have beenconsidered.
To better illustrate this recursive solution process, we usea 3-MU cluster gk = i, j, l as an example to explainthe analysis details as follows. To meet the minimal SINRrequirement, the received signal and the power consumptionfor MU l (which for MUs i and j are the same as in (9) and(10)) can be expressed as
6
yl = hHl wl√plxl +
∑ε=i,j
(hHl wε√pεxε) + nl,
Pl = ‖wl‖2(∣∣hHl wi∣∣2pi +
∣∣hHl wj∣∣2pj + σ2)δ.(23)
where pi = σ2δ, pj = (∣∣hHj wi∣∣2pi + σ2)δ.
Similar to (14), beamforming vectors for a 3-MU NOMAcluster are determined by
[wi,wj ,wl] = [w′
i,w′
j ,w′
l ]
1 λ1 λ2λ3 1 λ4λ5 λ6 1
, (24)
and the coefficients λε (ε = 1, 2, ..., 6) can be derived basedon the following optimization problem to minimize the totalpower consumption as
min g(λ1, ..., λ6) = Pi + Pj + Pl. (25)Starting from MUk beamforming vector ‖wl‖2, we have
min ‖wl‖2 =∥∥∥w′i∥∥∥2λ22 +
∥∥∥w′j∥∥∥2λ24 + 2w′
iw′
jλ2λ4+
2w′
iw′
lλ2 + 2w′
jw′
lλ4 +∥∥∥w′l∥∥∥2. (26)
The solution is
λ2 =w′jw′lw′iw′j−w
′iw′l
∥∥∥w′j∥∥∥2‖w′j‖2‖w′i‖2−|w′iw′j |2
, λ4 =w′iw′lw′iw′j−w
′jw′l
∥∥∥w′i∥∥∥2‖w′j‖2‖w′i‖2−|w′iw′j |2
.
Thus, we have wl = w′
iλ2 + w′
jλ4 + w′
l , and (25) can betransformed intomin g(λ1, ..., λ6) = (‖wj‖2(
∣∣hHj wi∣∣2δ + 1)+
θδ∣∣hHl wj∣∣2(∣∣hHj wi∣∣2δ+1))+(‖wi‖2+θ(δ
∣∣hHl wi∣∣2+1)),(27)
where θ = ‖wl‖2. Next, we focus on wj . Since (∣∣hHj wi∣∣2δ+
1) > 0, a second sub-optimization is formulated as
min ‖wj‖2α+∣∣hHl wj∣∣2β=α
∥∥∥w′i∥∥∥2λ21+(α∥∥∥w′l∥∥∥2+β)λ26
+2αw′
iw′
lλ1λ6 + 2αw′
iw′
jλ1 + 2αw′
jw′
lλ6 + α∥∥∥w′j∥∥∥2,(28)
where α = (∣∣hHj wi∣∣2δ + 1) and β = αθδ. We obtain
λ1 =w′iw′j(∥∥∥w′l∥∥∥2+θδ)−w′iw′lw′jw′l
|w′iw′l |2−‖w′i‖2(‖w′l‖2+θδ)
,λ6 =
∥∥∥w′i∥∥∥2w′jw′l−w′iw′jw′iw′l|w′iw
′l |2−‖w′i‖2(‖w′l‖2+θδ)
,
and wj = w′
iλ1 + w′
j + w′
lλ6. With these results, problem(25) can be further simplified as
min g(λ1, ..., λ6)=‖wi‖2+∣∣hHj wi∣∣2(‖wj‖2δ+θδ2
∣∣hHl wj∣∣2)
+θδ∣∣hHl wi∣∣2 =(
∥∥∥w′j∥∥∥2+α′)λ23+(∥∥∥wl′∥∥∥2 + β
′)λ25
+2w′
jw′
lλ3λ5 + 2w′
iw′
jλ3 + 2w′
iw′
lλ5 +∥∥∥w′i∥∥∥2,
(29)
where α′
= (∥∥∥w′j∥∥∥2δ +
∣∣hHl wj∣∣2δ2θ), β′
= θδ. We have
λ3 =w′iw′j(∥∥∥wl′∥∥∥2+β′ )−w′iw′lw′jw′l
|w′jw′l |2−(‖w′j‖2+α′ )(‖wl′‖2+β′ )
,
λ5 =w′iw′l (∥∥∥w′j∥∥∥2+α′ )−w′iw′jw′jw′l
|w′jw′l |2−(‖w′j‖2+α′ )(‖wl′‖2+β′ )
,
and wi=w′
i+w′
jλ3+w′
lλ5. In summary, the closed-form solu-tion of beamforming strategy for a 3-MU cluster is obtainedby three steps, and in each step, the optimization is linear.
C. MIMO-NOMA2 for a cluster with a size of 2In MIMO-NOMA2, a same power allocation coefficient is
employed for different antennas. Thus, the received signals for
MU i and MU j can be represented asyi=h
Hi sk+ni=h
Hi vk√pk√µi,1xi+ni,
yj=hHj sk+nj=h
Hj vk√pk√µj,1xj+h
Hj vk√pk√µi,1xi+nj .
(30)
With the minimal SINR requirement satisfied, the powerconsumption for them can be respectively expressed as
Pi = ‖vk‖2pkµi,1 = ‖vk‖2
|hHi vk|2σ2δ,
Pj = ‖vk‖2pkµj,1 = ‖vk‖2
|hHj vk|2 (∣∣hHj vk∣∣2pkµi,1 + σ2)δ
= ‖vk‖2( δ
|hHi vk|2 + 1
|hHj vk|2 )σ2δ.
(31)
To minimize the total power consumption, we formulate anoptimization problem as
min f(vk) = ‖vk‖2(δ + 1∣∣hHi vk∣∣2 +
1∣∣hHj vk∣∣2 )σ2δ. (32)
Since the cluster beamforming vector vk should be or-thogonal to channel gain vectors of MU l (i.e., hlHvk =0, l ∈ M, l 6= i, j), according to (14), vk should be inthe linear space determined by w
′
i and w′
j . We rewrite vk asvk = βw
′
i+λw′
j . Then, problem (32) can also be transformedwith respect to β and λ asmin g(β, λ)
= (∥∥∥w′i∥∥∥2β2 + 2βλw
′
iw′
j +∥∥∥w′j∥∥∥2λ2)( δ+1
β2 + 1λ2 ).
(33)
From (33), we notice that g(β, λ) is only determined by theratio of β
λ . Thus, without loss of generality, we assume thatβ = χλ, and let λ = 1, so that β = χ. Then, the optimizationproblem can be reformulated as
f(vk)=g(χ)=(∥∥∥w′i∥∥∥2χ2+2χw
′
iw′
j+∥∥∥w′j∥∥∥2)( δ+1χ2 +1). (34)
Although the optimal result of χ can be obtained by letting∂g(χ)∂χ = 0, the closed-form solution cannot be derived directly.
If the distance between two MUs in a cluster is large, the cell-edge MU will consume much more energy than the cell-center
MU, i.e.,∥∥∥w′i∥∥∥2 ∥∥∥w′j∥∥∥2 and β λ, and we have δ+1
χ2 1.
By letting δ+1χ2 = 0, problem (32) can be approximated as
min g(χ) = (∥∥∥w′i∥∥∥2χ2 + 2χw
′
iw′
j +∥∥∥w′j∥∥∥2). (35)
Since ∂2g(χ)∂χ2 = 2
∥∥∥w′i∥∥∥2> 0, by letting ∂g(χ)∂χ = 0, we have
the closed-form solution as χ = − w′iw′j
‖w′i‖2. The accuracy of
the approximation will be decreased as two MUs get closer.However, if two MUs are too close, condition ∆Pk>θ maynot be satisfied. In the simulation part, we will show that thisapproximate solution is in high-accuracy. Moreover, the resultof χ for δ+1
χ2 ≥ 1 does not exist as shown in appendix B.In summary, for MIMO-NOMA2 scenario, we can simplify
the objective function based on the fact that∥∥∥w′i∥∥∥2∥∥∥w′j∥∥∥2.
Moreover, from (31), we have µi,1µj,1
=|hHj vk|2
δ|hHj vk|2+|hHi vk|2 , then
we can get the allocated power coefficients (µi,1 and µj,1).
D. MIMO-NOMA2 for a cluster with the size large than 2
For the cluster with n (n > 2) MUs,, the beamformingvector of MU i can be calculated by vk = λ1w
′
1 + λ2w′
2 +
7
... + λnw′
n. After decoding the superposed message andapproximation, the received signals and corresponding powerconsumptions for MU i can be derived by
yi =hHi vk√pk(√µi,1xi+
i−1∑j=1
õj,1xj) + ni.
Pi = ‖vk‖2( 1
|hHi vk|2 +i−1∑j=1
δ(1+δ)(i−1−j)
|hHj vk|2 )σ2δ.
The objective function can be denoted as
min g(λ1, ..., λn) = ‖vk‖2(
i−1∑j=1
(δ + 1)(n−j)
λ2j+
1
λ2n)σ2δ. (36)
If (δ+1)(n−j)
λ2j
is the largest one, after approximation, theobjective function can be transformed as
min g(λ1, ..., λn) = ‖vk‖2((δ + 1)(n−j)
λ2j)σ2δ.
Following the same way as in (22), we can obtain the closedform solution by
[λ1λj, ...,
λj−1λj
,λj+1
λj, ...,
λnλj
] = −[w′
1w′
j , ...,w′
nw′
j ]
∥∥∥w′1∥∥∥2 w′
1w′
2 ... w′
1w′
n
w′
1w′
2
∥∥∥w′2∥∥∥2 ... w′
2w′
n......
...w′
1w′
n w′
2w′
n ...∥∥∥w′n∥∥∥2
−1
.
For example, in a 3-MU cluster gk = i, j, l, the receivedsignal and the power consumption for MU l (which for MUsi and j are shown in (30) and (31)) can be written asyl = hHl vk
√pk√µl,1xl + hHl vk
√pk√µi,1xi
+hHl vk√pk√µj,1xj + nl,
Pl=‖vk‖2pkµl,1=‖vk‖2( δ(δ+1)
|hHi vk|2 + δ
|hHj vk|2 + 1
|hHl vk|2 )σ2δ.(37)
Similarly, we assume that vk = λ1wi + λ2wj + λ3wl andλ2 = αλ1, λ3 = βλ1. In order to minimize the total powerconsumption, the optimization function is given by
min g(α, β)=(∥∥∥w′i∥∥∥2+α2
∥∥∥w′j∥∥∥2+β2∥∥∥w′l∥∥∥2+
2(αw′
iw′
j+βw′
iw′
l+αβw′
jw′
l))((δ+1)2+ δ+1α2 + 1
β2).(38)
In this case, problem (38) is determined by the value ofα and β. Since i is the cell-center MU and l is the cell-edge
MU, we have∥∥∥w′i∥∥∥2 ∥∥∥w′l∥∥∥2, λ1 λ3 and (δ + 1)2 1
β2 .
If MU j is close to MU i, we have (δ+1)α2 1
β2 . Thus, theobjective function can be approximated as
min g(α, β) = (∥∥∥w′i∥∥∥2 + α2
∥∥∥w′j∥∥∥2 + β2∥∥∥w′l∥∥∥2
+2αw′
iw′
j + 2βw′
iw′
l + 2αβw′
jw′
l)1β2 .
(39)
We first consider β as a constant, and let ∂(g(α,β))∂α =0 be-
cause
∥∥∥w′j∥∥∥2β2 >0. As ∂2(g(α,β))
∂α2 =
∥∥∥w′j∥∥∥2β2 >0, the objective func-
tion will reach the minimum point when α = −w′iw′j+w
′jw′lβ
‖w′j‖2.
Then, substituting α to (38), it can be further simplified as
According to [27], the SIC approach is based on the evalua-tion of received signal strength which is used to determine theweight (or precoders) on each decoding layer. Such receivedsignal strength depends on channel gains, beamforming vec-tors, and power distribution. In MIMO-NOMA1, the receivedsignal strengths for different signals are distinguished by theproduct of the channel gain vector, the transformed beam-forming vector and the power coefficient. While in MIMO-NOMA2, since the channel gains, the beamforming vectorsand the power allocations are same for different users, the re-ceived signal strength for different signals can be distinguishedby the power coefficients. Take a 2-MU cluster for example. Inthe case of MIMO-NOMA1, for MU j in (9), the precoders ofxj and xi are determined by √pj and λ
√pi. However, in the
case of MIMO-NOMA2, for MU j in (30), the precoders of xjand xi are determined byõj,1 andõi,1. Therefore, althoughdifferent precoders are used in MIMO-NOMA1, there is noextra overhead introduced. This observation can be easilyextended to a cluster with any size.
V. MIMO-NOMA CLUSTERING APPROACH
Based on the aforementioned NOMA cluster beamformingdesign, we have two observations: i) power reduction canbe achieved through MU clustering; ii) the maximum powerreduction for a cluster is only related to MUs in this clusterbut independent with other out-of-cluster MUs. Based on thesetwo observations, MU clustering problem becomes a groupingproblem for exploring an optimal cluster set. In this paper,a new coalition game approach is proposed to solve suchgrouping problem by exploring players’ cooperative behaviors.
The conditions for grouping MUs together include: thepotential cluster is beneficial, and MUs can be successfullydecoded (refer to appendix D) [33]. We assume that allMUs are willing to join this clustering process and try tomaximize their utilities which are assigned by the BS. Theutility function of MU i is evaluated by the average value ofcluster power reduction ∆Pk ,i.e.,
8
Ui =
∆Pk/nk,
0,i ∈ gk,
Otherwise.(42)
In a traditional coalition game, the cluster with a higherpower reduction will be more potentially formed. Thus, thefinal clustering result may be the best choice for each player(leading to Pareto optimality) but may not be the global opti-mal solution in terms of minimizing the total power consump-tion. For example, considering four MUs (A, B, C and D) in acoalition game, the potential clusters are g1=MUA,MUB,g2=MUB ,MUC and g3=MUC ,MUD, and the achiev-able power reductions for them are ∆P1 =∆P3 =2, ∆P2 =3,respectively. The Pareto optimal solution is G=g2 becauseMUB and MUC will obtain the maximum utility 1.5. How-ever, the global optimal solution is G= g1, g3 as the totalpower reduction is 4. Therefore, some improvement on thetraditional coalition game should be proposed. By consideringthe fact that a global optimal solution may be obtained whenboth the utility of each MU and the number of formed clustersare considered. We introduce a random variable into the designof utility function as in particle swarm optimization (PSO)approach [28] to achieve a balance between the number offormed clusters and the power reduction. The newly designedutility function U∗i (t) is defined as
U∗i (t) = Ui − κ(t)∑nkj=1 Ugj , (43)
where Ui is the average group utility in (42), Ugj is the averagegroup utility of cluster member MU j before a new clusteris formed, and κ(t) denotes an update rate. This update rateis worked for MU i only when other cluster members (suchas j ∈ gk and j 6= i) are already in different clusters andwith non-zero utilities. Therefore, to make MU j split from itsformer cluster and join a newly cluster with MU i, utility of thenewly cluster should be large enough to overcome the penaltyof splitting. Note that κ(t) is critical to the optimal solution,and the traditional coalition game is a special case whenκ(t) = 0. For the stability of a coalition game, κ(t) is onlyupdated after all MUs converge to a Pareto optimal solution.We define t as the time to update κ(t), which follows
κ(t) =
κ(t− 1) + θ1rand(1)∆U(t),
κ(t− 1) + θ2rand(1),∆U(t) > 0,Otherwise.
∆U(t) =∑Nj=1 Pj(t)−
∑Nj=1 P
∗j ,
(44)
where rand(1) is a random variable within 0 to 1, Pj(t) isthe current result of power consumption, P ∗j is the minimumpower consumption resulted from the history, and θ1 and θ2are two parameters related to speed. Note that the update ratedepends on the difference between the optimal result and thecurrent result, and it can adjust the utility function to escapefrom the local optimal result and toward the global optimalsolution.
After defining the utility functions of all MUs, the merge-and-split rule is applied, which is defined as follows.
Definition 1: Consider two sets of coalitions GA = MUAi ∪MUAj1 ... ∪MUAjn and GB = MUBi ∪MUBk1 ... ∪MUBkl,which are two potential coalition groups for MUi. Here, MUAimeans that MU i is in the coalition group A. For MU i, if andonly if its utility in group A (denoted by U(GA)) is larger thanits utility in group B (U(GA) > U(GB)), the coalition GA is
preferred over GB by Pareto order, denoted by GMUiA BGMUi
B .• Merge: For any individual MU from i to jn, if GA BMUi,MUj1 , ...,MUjn and GA = MUi∪MUj1 , ...∪MUjn, then merge MUi,MUj1 , ...,MUjn to GA,denoted by MUi,MUj1 , ...,MUjn → GA.
• Split: For any coalitions GA and GB , if GMUiA B GMUi
B ,then split GA into MUi,MUj1 , ...,MUjn and mergeit into a new coalition GB , denoted by GA ∪MUk1 ...∪MUBkl → GB ∪MUj1 ... ∪MUjn.
By the merge-and-split rule, a stable coalition formationresult can be found as a Pareto optimal solution [29]. We noticethat if a cluster can achieve the maximal power reduction, itwill be a choice with the maximum utility to each clustermember, and thus it has a higher chance of being formed. Theupdate rate κ(t) will be changed after each iteration. There-fore, within a single iteration, if there is no cluster with thesame utility, the Pareto optimal solution is unique. Moreover,we define the Dc stable as in [30], where the existence andconvergence proof are also available in our cases. For thedifferent iterations, the adjustment of κ(t) may lead to theconvergence on different Pareto optimal solutions. Then, wecan find one solution with the minimal power consumption asthe optimal solution by the following approach.
1) Initialization: Let the best result of power consumptionand the update rate be initialized as P ∗ = 0 and k(t) = 0,respectively, when the current update time is t = 1.
2) Iteration:• Step 1: Randomly group all MUs into different clusters
and employ the split-and-merge rule on each MU to formcoalitions. Then, find the potential cluster and repeatcoalition formation process until none of MU changesits strategy to improve the utility.
• Step 2: If the current total power consumption∑Nj=1 Pj
is lower than P ∗, record the solution and update P ∗ =∑Nj=1 P
∗j . Then, update κ(t+1) by (40) and let t = t+1
until t > T .Here T is a predefined maximum iteration depending on the
number of MUs, so that t = T means the end of the iteration.
VI. NUMERICAL RESULTSA. Single-cluster Performance Analysis
In this section, we evaluate the performance of a givenMIMO-NOMA cluster and the proposed clustering approach.In the simulation, we first demonstrate the impact of threemain factors on the NOMA clustering and ultimately thesystem performance in terms of power reduction. After that,we will focus on illustrating the superiority of our proposedclustering approach by comparing with two existing ones inthe literature. Since MU are randomly distributed under oursettings, the probability of forming a competitively large sizecluster is very low (the reason will be given in Fig. 6).
Consider a single cell network with a radius of 400m anda centrally located BS. The number of antennas at the BS isM = 20 or M = 40. The variance of Gaussian noise is σ2
u=−135dBm. The SINR requirement is δ=4dB. Similar to theexisting works [31], [32], the channel model settings include:the 3GPP long term evolution (LTE) pathloss parameters (α=
9
Distance(m)100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.2
0.4
0.6
0.8
1
1.2
(c-1)Radius of MU1 is 100m.
Distance(m)100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.2
0.4
0.6
0.8
1
1.2
(c-2)Radius of MU1 is 250m.
Radius of MU1 (m)
100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.005
0.01
0.015
0.02
0.025
0.03
(c-3)Radius of MU2 is 250m.
(c) Four slices of 3-D map
Real100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.2
0.4
0.6
0.8
1
1.2
(c-4)Radius of MU2 is 400m.
Distance(m)100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.2
0.4
0.6
0.8
1
(d-1)Radius of MU1 is 100m.
Distance(m)100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.2
0.4
0.6
0.8
1
(d-2)Radius of MU1 is 250m.
Radius of MU1 (m)
100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.002
0.004
0.006
0.008
0.01
0.012
(d-3)Radius of MU2 is 250m.
(d) Four slices of 3-D map
Radius of MU1 (m)
100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.2
0.4
0.6
0.8
1
(d-4)Radius of MU2 is 400m.
Fig. 2: 3-D map for power reduction in MIMO-NOMA1 and MIMO-NOMA2.
3.76 and β= 10−14.81), the Rayleigh fading with zero meanand unit variance (Γ
(n)i ∼ CN(0, 1)), a log-normal shadowing
γi ∼ N(0, 8)dB, and the transmit antenna power gain G =9dB. The channel coefficient between MU i and the BS’s mthantenna is modeled as
h(m)i = Γ
(m)i
√Gβd−αi γi, (45)
where di is the radius of MU i (i.e., the distance between MUi and the BS).
For explanation purpose, we focus on a 2-MU cluster. Theimpacts of three factors on the performance of power reduc-tion are analyzed: the radius of MUs, the radius differencebetween two MUs, and the channel correlation coefficient.We randomly generate the locations and channel gain vectorsof 10 MUs, with 5 cell-center MUs locating within radius[100, 150]m, and 5 cell-edge MUs locating within radius[346, 400]m. Then, we select one cell-center MU, namelyMU1, and one cell-edge MU, namely MU2, to form a MIMO-NOMA cluster. The correlation between these two MUs andthe shadowing coefficient are fixed by −0.2402−0.1579i and1.4dB, respectively. The only thing to be changed is the radiusof MU1 or MU2 from 110m to 350m. The radius of themshould satisfy a condition that MU1 is always smaller thanMU2 to keep MU1 always to be a cell-center MU comparedwith the location of MU2. Simulation results for MIMO-NOMA1 and MIMO-NOMA2 are shown in Fig. 2.
Fig. 2(a) shows the variance of power reduction resultedfrom clustering with respect to the radiusR1(R2) of the clustermemberMU1(MU2), under the MIMO-NOMA1 scenario. Toobserve the variance, we select four cross section views asshown in Fig. 2(c) by fixing one MU’s radius while changingthe other. As shown in Figs. 2(c-1) and 2(c-2), power reductionincreases with R2 if the radius of MU1 is given. However,given the radius of MU2 as shown in Figs. 2(c-3) and 2(c-4), the variance of power reduction with respect to R1 isnot that obvious unless MU2 locates at the cell edge. Bycomparing Figs. 2(c-2) with 2(c-4), we can see that a largerpower reduction is obtained when MUs are both close to thecell edge. Therefore, for MIMO-NOMA1, we can concludethat the power reduction is mainly determined by the radiusof the cell-edge MU, so that a cell-edge MU is the necessarycondition to form a beneficial cluster.
These observations imply the existence of radius thresholdsin separating the area of the cell-center and the cell-edge,which can be used to narrow down the searching space ofbeneficial NOMA clusters. To evaluate the minimal radius fora cell-edge MU, we generate a pair of MUs which includeMU1 (radius is fixed to 100m) and MU2 (radius changesfrom 100m to 400m). The Rayleigh fading coefficients aregenerated randomly with a sample quantity of 2000, while theshadowing coefficient is fixed by 1.4dB. Simulation showsthat when the radius of MU2 is smaller than 233m, the
10
Radius of MU2 (m)
100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.2
0.4
0.6
0.8
1
(a)Radius of MU1 is 100m.
Approximate valueAccurate value
Radius of MU2 (m)
100 200 300 400
%
0
2
4
6
8(b) Differents.
Radius of MU1 (m)
100 200 300 400
Po
we
r re
du
ctio
n(w
)
0
0.2
0.4
0.6
0.8
1
(c)Radius of MU2 is 400m.
Approximate valueAccurate value
Radius of MU1 (m)
100 200 300 400
%
0
2
4
6
8(d) Differents.
Fig. 3: Comparison, MU1 radius is fixed in (a) and (b),MU2 radius is fixed in (c) and (d).
power reduction is less than 0.01w for most of channel gaincorrelation coefficients. Therefore, the radius threshold for acell-edge MU can be selected by 233m.
The simulation results in MIMO-NOMA2 is shown in Fig. 2(b) and (d). As shown in Fig. 2(d-1), power reduction increaseswith R2 if MU1 is located at the cell center. However, differentfrom MIMO-NOMA1, if a MIMO-NOMA2 cluster only hascell-center MUs (as the case in Fig. 2(d-3)) or cell-edge MUs(as shown in Fig. 2(d-2)), the power reduction is close to zero.In Fig. 2(d-4), power reduction decreases with R1 if MU2 islocated at the cell edge. Thus, a beneficial cluster needs a cell-center MU and a cell-edge MU. In addition, by comparingFigs. 2(a) and (b), for a same pair of MUs, it is shown thatMIMO-NOMA1 can achieve a better power efficiency thanMIMO-NOMA2.
Since the result of MIMO-NOMA2 is an approximate solu-tion, we have to discuss its accuracy, and show the comparisonbetween the accurate results and the approximate ones in Fig.3. The percentage of error is equal to the accurate result minusthe approximate result and divide by the approximate result.From Figs. 3(b) and 3(d), we can observe that the approximateresults are nearly the same as the accurate ones when thedistance between MU1 and MU2 is sufficiently large. Thepercentage of error is less than 7% in this case. Besides, wenotice that the area of the percentage of error larger than 0.1%in 3(b) (or 3(d)) is [237, 240]m (or [169, 202]m), which isa small area when compared with that lower than 0.1% [241,400]m (or [100, 168]m). Therefore, to improve the accuracyof results, we can suitably set up radius thresholds for bothcell-center MUs and cell-edge MUs and the minimum distancebetween them.
For obtaining the radius threshold of a cell-edge MU, thesimulation process is the same as that of MIMO-NOMA1and the radius threshold is 236m for power reduction largerthan 0.01w. For deriving the radius threshold of a cell-centerMU, we fix the radius of MU2 as 400m and change theradius of MU1 from 100m to 400m. The results show thatthe radius threshold is 285m for power reduction larger than0.01w. To ensure the percentage of error less than 0.1%, the
minimum distance between them is 185m. Note that since thelog-normal shadowing is fixed as 1.4 under our settings, weneed to consider the real log-normal shadowing value beforewe employing these radius threshold.
Fig. 4 shows the effects of channel gain correlation coeffi-cient on power reduction in the vertical view, where the colorin color bar from dark to light means the amount of powerreduction from low to high. The channel gain correlationcoefficient between MU1 and MU2 is a randomly generatedcomplex value and other settings are fixed. The radius of MU1
and MU2 are fixed as 100m and 400m as shown in Figs. 4(a)and 4(b), and 100m and 270m as shown in Figs. 4(c) and 4(d),respectively. From these figures, we can observe that the powerreduction is positively associated with the absolute value ofthe real part of correlation coefficient while weakly associatedwith the imaginary part. There is a gap around zero of x-axis which indicates that if two MUs’ correlation coefficientis within this gap, they can not form a beneficial NOMAcluster. We compare MIMO-NOMA1 with MIMO-NOMA2 inthe same radius condition by noticing Figs. 4(a) and 4(b) orFigs. 4(c) and 4(d), and find that the gap in MIMO-NOMA1 issmaller than that in MIMO-NOMA2. It further illustrates thatthe solution space of MIMO-NOMA1 is larger than that ofMIMO-NOMA2. Besides, by comparing Figs. 4(a) with 4(c)or Figs. 4(b) with 4(d), the gap becomes broadened, and thevalue of color bar is reduced, when MU2 is getting close tothe cell center. This is because R2 has the larger influence onpower reduction. The detailed explanation is in appendix C.
B. MU clustering Result
To evaluate the performance of our proposed MU cluster-ing approach (power-reduction based approach), two existingapproaches in literature are also simulated as benchmarks:the channel gain-correlation based approach [20] and thechannel gain-difference based approach [21]. Both of themare the two-stage optimization, where MU clustering andsystem optimization apply independently handled. The mainprocedures of these two approaches are listed as follows.
1) The channel gain-correlation based approachStep 1: Generate a metric vector vi,j for all MUs as
vi,j =
0∣∣∣hi · hj∣∣∣, if
∣∣∣10 log |hi|2−10 log |hj |2∣∣∣≤3dB,
Otherwise,(46)
where hi and hi are channel gains of MU i with and withoutpath-loss coefficient, respectively.
Step 2: Group MUs into MIMO-NOMA clusters accordingto the descending order of vi,j .
Step 3: Calculate the beamforming matrix and the powerreduction. Note that the stronger user in a cluster is the MUwith a larger |hi|.
2) The channel gain-difference based approachStep 1: Generate a metric vector πi,j representing the
channel gain-difference as
πi,j =
||hi| − |hj ||,
0,if|hi·hj ||hi||hj | > 0.4,
Otherwise.(47)
Step 2: Group MUs into MIMO-NOMA clusters accordingto the descending order of πi,j .
11
Real-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Imag
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
(a) MIMO-NOMA1(R1=100m,R
2=400m)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Real-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Ima
g
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
(b) MIMO-NOMA2(R1=100m,R
2=400m)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Real-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Imag
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
(c) MIMO-NOMA1(R1=100m,R
2=250m)
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Real-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Imag
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
(d) MIMO-NOMA2(R1=100m,R
2=250m)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Fig. 4: Vertical view of correlation coefficient results
Step 3: In this approach, the channel gain matrix is com-posed by the channel gains of MIMO-OMA MUs and thestronger MUs (i.e., with a larger |hi|) in clusters. Then, cal-culate the beamforming matrix and the total power reduction.
The channel gain-correlation based approach is used forMIMO-NOMA1 (denoted by MIMO-NOMA1-CO), and thechannel gain-difference based approach is used for MIMO-NOMA2 (denoted by MIMO-NOMA2-GD).We further denotethe proposed power-reduction based approach for MIMO-NOMA1 as MIMO-NOMA1-PO and for MIMO-NOMA2 asMIMO-NOMA2-PO, and both of them are managed by thetraditional coalition game approach. In the following simula-tions, the BS is equipped withM=40antennas, and all MUsare distributed within a radius range of [100,500]m. Since thedifferent MUs’ distribution and channel gains may result ina large difference on total power consumption, we compareand evaluate the system performance by a normalized averagepower consumption (denoted by NPC), i.e., for the result of50 sets of randomly generated data, normalize them by theresults of MIMO-OMA, and then calculate the average value.
From Fig. 5, we notice that the normalized power consump-tion is decreased with the number of MUs for all approaches. Itmeans that NOMA-MIMO can reduce more power consump-tion in the large-scale system. Besides, MIMO-NOMA1 isbetter than MIMO-NOMA2 in improving the energy efficiencyas it is more flexible on power coefficient settings. In addition,compared with the approach in literature, MIMO-NOMA1-PO (MIMO-NOMA2-PO) is better than MIMO-NOMA1-CO(MIMO-NOMA2-GD) in the scenario of MIMO-NOMA1(MIMO-NOMA2). Thus, we can conclude that the power-reduction based approach obviously outperforms both thechannel gain-correlation based and the channel gain-difference
Fig. 5: Power reduction results comparison for differentclustering approaches.
based approaches. Moreover, the performance improvementbecomes more obvious with the number of MUs even in thecase of the same number of MIMO-NOMA clusters as shownin Fig. 5(b). It results from the fact that the power-reductionbased approach is a joint optimization approach, so that itcan be more efficient in finding an optimum MIMO-NOMAcluster set than the counterparts.
The results of fixed (denoted as PO2) and flexible (denotedas PO3) cluster size conditions are compared in Fig. 6(a) and(c). For the case of fixed (or flexible) cluster size, a MIMO-NOMA cluster can only include 2 MUs (2 or 3 MUs). Inthis figure, MIMO-NOMA1-PO2-M1 (MIMO-NOMA2-PO3-M1) denotes the result of power-reduction based approach withfixed (flexible) cluster size condition by the traditional coali-
(b) Tranditional Coalition Game vs Improved Coalition Game
MIMO-NOMA1-PO3-M1
MIMO-NOMA1-PO3-M2
Number of MU
5 10 15 20 25 30 35
Diffe
ren
ce
0
0.005
0.01
0.015
0.02(d) Comparisons Between M1 and M2
MIMO-NOMA1-PO2-M1 vs MIMO-NOMA1-PO2-M2
MIMO-NOMA2-PO2-M1 vs MIMO-NOMA2-PO2-M2
MIMO-NOMA1-PO3-M1 vs MIMO-NOMA1-PO3-M2
MIMO-NOMA2-PO3-M1 vs MIMO-NOMA2-PO3-M2
Fig. 6: Results with different cluster size limitations areshown in (a) and (c). Results for MIMO-NOMA 1 with
different game approaches are shown in (b) and (d).
tion game approach in MIMO-NOMA1 (MIMO-NOMA2)scenario. Simulation results show that the result with theflexible cluster size condition is better than without that in bothMIMO-NOMA1 and MIMI-NOMA2. Besides, the differencebetween fixed and flexible cluster sizes are gradually increasedwith the number of MUs as shown in Fig. 6(c), and thedifference is more obvious in MIMO-NOMA1 than MIMO-NOMA2. However, we notice that in Fig. 6(c), the differencein MIMO-NOMA1 decreases when the number of MUs islarger than 30. It means that a larger size cluster may notalways be better than a smaller one in terms of averagepower reduction, because i) the power reduction hinges on thechannel gain correlation coefficient and ii) the MU densityincreases with the number of MUs. In summary, MIMO-NOMA1 with the flexible cluster size condition performsbetter than the other cases, and the cluster size is affectedby the MU density and distribution.
The results based on the improved coalition game (denotedas M2) and the traditional coalition game (denoted as M1)are compared in 6(b) and (d). In Fig. 6(b), we only comparetwo results between MIMO-NOMA1-PO3-M1 and MIMO-NOMA1-PO3-M2. The result of MIMO-NOMA1-PO3-M1(MIMO-NOMA1-PO3-M2) is obtained by employing MIMO-NOMA1 with a flexible cluster size condition and operating by
the traditional (improved) coalition game. We notice that theimproved coalition game achieves a lower power consumptionthan the traditional coalition game, and such improvementhas an increasing trend as the number of MUs increases.Moreover, for the case of MIMO-NOMA2 or in the fixedcluster size condition, results are in the same tendency asshown in Fig. 6(d). Furthermore, Fig. 6(d) compares M1 andM2 for 4 different cases, and the curves show the differencebetween them, e.g., the result of MIMO-NOMA1-PO2-M1minus that of MIMO-NOMA1-PO2-M2. Thus, the differencemeans that M1 has much more power consumption than M2. Inthis figure, the difference value of MIMO-NOMA1 is largerthan that of MIMO-NOMA2. Moreover, we observe that asmall difference appears as the solution space is small (i.g.,with a small quantity of MUs, or in MIMO-NOMA2 cases).Besides, we notice a significant increase in Fig. 6(d) on bothMIMO-NOMA1-PO3 and MIMO-NOMA2-PO3 curves whenthe number of MUs is larger than 30 and both of them have theflexible cluster size condition. Therefore, we can conclude thatthe improved coalition game can find a better result than thetraditional coalition game approach, especially when a largequantity of MUs and the flexible cluster size condition areconsidered.
VII. CONCLUSION
In this paper, we formulate a joint optimization problemfor MU clustering in the MIMO communication system tominimize the total power consumption for the BS and proposea cluster beamforming strategy to calculate the beamformingvector and power allocation coefficients for MIMO-NOMAclusters. Based on the proposed approach and the deriveddecoding order, a closed-form expression for cluster beam-forming vector is obtained. Furthermore, we show that ourproposed approach can avoid the peer effect when MUs aregrouped in different clusters. Then, we carefully design animproved coalition game for MU clustering, by which the sizeof a NOMA cluster can be flexible and the performance ofthe optimal results is close to that of the global optimal solu-tion in terms of power consumption minimization. Moreover,we compare two different MIMO-NOMA scenarios for MUclustering and find that MIMO-NOMA1 (allocated a powercoefficient set for each MU) is superior than MIMO-NOMA2(allocated a single power coefficient for each MU) in powerreduction. In the simulation part, we analyze the three factorsthat may affect the performance of MIMO-NOMA clusteringto theoretically explain that our proposed clustering approachis a high-efficient algorithm compared with other existingapproaches. Furthermore, we conduct two comparisons todemonstrate that the proposed cluster beamforming strategyand the improved coalition game approach are effective in sig-nificantly improving energy efficiency for the MIMO systemwith a large quantity of MUs.
APPENDIX
A. MUs’ decoding orderThe decoding order is based on the principle that the power
consumption should be minimized.
13
In MIMO-NOMA1, if signals are first decoded on MUj , theoptimal function is shown in (13). If signals are first decodedon MUi, the optimal function can be rewritten as
f′(w∗i ,w
∗j ) =
∥∥w∗j∥∥2 + ‖w∗i ‖2(∣∣hHi w∗j ∣∣2δ + 1). (48)
By the same way, the optimal solution of (48) is
λ′
=−‖w′j‖2w′iw
′j
‖w′i‖2‖w′j‖2(1+δ)−δ|w′iw′j|2and β
′= − w′jw
′i
‖w′j‖2. If
f(wi,wj) < f′(w∗i ,w
∗j ), we will adopt the decoding order
in (13). To compare (13) and (48), we substitute the optimalsolutions into each functions and let
f(wi,wj)−f′(w∗i ,w
∗j ) =
δ∣∣∣w′iw′j∣∣∣6
(‖w′i‖2‖w′j‖2(1+δ)−δ|w′iw′j|2)2
(∥∥∥w′i∥∥∥2−∥∥∥w′j∥∥∥2)(δρ2−(1 + δ))(ρ2−1),
(49)
where ρ= |w′iw′j|‖w′i‖‖w′j‖
. Since coefficient ρ<1, ‖w′i‖2<∥∥w′j∥∥2 is
the necessary condition for f(wi,wj)−f′(w∗i ,w
∗j )<0.
For MIMO-NOMA2, we can employ the same way to provethat the decoding order of MUi and MUj cannot be changed.If signals are first decoded on MUj , we take the optimalsolution χ = − w′iw
′j
‖w′i‖2into (35) and get
g(χ)=(∥∥∥w′i∥∥∥2χ2+2χw
′
iw′
j+∥∥∥w′j∥∥∥2)=
∥∥∥w′j∥∥∥2− ∣∣∣w′iw′j∣∣∣2‖w′i‖2.(50)
If signals are first decoded onMUi, the optimal functionwill bemin g
′(β∗, λ∗)=(
∥∥∥w′i∥∥∥2χ∗2+2χ∗w′
iw′
j+∥∥∥w′j∥∥∥2)((δ+1)+ 1
χ∗2).
As the same way, we obtain χ∗ = − w′iw′j
‖w′i‖2. The minimal
power consumption is
g′(χ∗) = (1 + δ)(
∥∥∥w′j∥∥∥2 − ∣∣∣w′iw′j∣∣∣2‖w′i‖2
). (51)
Thus, NOMA decoding should be always started from the MU
with a maximum∥∥∥w′j∥∥∥2. For a cluster with the size larger than
2, the decoding order still need to satisfy this condition.
B. Proof for the nonexistence of δ+1χ2 ≥ 1
If δ+1χ2 ≥ 1, the approximated problem of (34) becomes
min g(χ) = (∥∥∥w′i∥∥∥2χ2 + 2χw
′
iw′
j +∥∥∥w′j∥∥∥2) δ+1
χ2
= (δ + 1)(∥∥∥w′j∥∥∥2y2 + 2w
′
iw′
jy +∥∥∥w′i∥∥∥2).
(52)
Since ∂2g(y)∂y2 = 2
∥∥∥w′j∥∥∥2 > 0, the result can be obtained from
∂g(y)∂y = 0. Thus, χ∗ = 1/y = −
∥∥∥w′j∥∥∥2w′iw′j
is the approximate
solution. Comparing χ∗2 with χ2 in (34), we get
χ2 =(w′iw′j)
2
‖w′i‖4=
∥∥∥w′j∥∥∥2‖w′i‖2
(w′iw′j)
2
‖w′i‖2‖w′j‖2,
χ∗2 =
∥∥∥w′j∥∥∥4(w′iw′j)
2=
∥∥∥w′j∥∥∥2‖w′i‖2
∥∥∥w′i∥∥∥2∥∥∥w′j∥∥∥2(w′iw′j)
2.
(53)
Since χ2 is based on the assumption of δ+1χ2 1, χ2 must
be larger than χ∗2. However, this conclusion is a contrast to
the fact that(w′iw′j)
2
‖w′i‖2‖w′j‖2≤ 1 . Therefore, the solution for the
case of δ+1χ2 ≥ 1 does not exist.
C. Explanation for the correlation coefficient results
Fig. 5 shows that the power reduction is mainly positiveassociated with the absolute value of the real part of correlationcoefficient while weakly associated with the imaginary part.This is caused by the ZF-beamforming and the proposedcluster beamforming strategy. Let
ρR =w′iw′j
‖w′i‖‖w′j‖=
(<(w′i)<(w
′j)+=(w
′i)=(w
′j))
‖w′i‖‖w′j‖. (54)
From (58) and (59), we notice that ∆Pk is positive relatedwith ρR. Then, we need to prove ρR is mainly related withthe real part of correlation coefficient R as shown in (57).According to (1), we have
W′∗W
′=
∥∥∥w′1∥∥∥2 w
′∗1 w
′
2 ... w′∗1 w
′
n......
...w′∗n w
′
1 w′∗n w
′
2 ...∥∥∥w′n∥∥∥2
=QT (HH∗)Q
=
∥∥∥h′1∥∥∥2 h
′∗1 h
′
2 ... h′∗1 h
′
n......
...h′∗n h
′
1 h′∗n h
′
2 ...∥∥∥h′n∥∥∥2
QTQ,Q = (HH∗)−1,
where, HH∗ and Q are both symmetric matrixes, in whichdiagonal elements is real. Thus, QTQ is a real symmetricmatrix. We notice that only off-diagonal elements in Q arerelated with the imaginary part of correlation coefficient, andwhich is smaller than the value of diagonal elements. Due tow′
iw′
j is the real part of w′∗i w
′
j , which is corresponding tothe real part of [h
′∗i h
′
1, ...,h′∗i h
′
n], ρR is mainly related withthe real part of correlation coefficient.
Here, we give a two-antenna and two-MU example asa special case for discussion. We assume that two MUs (iand j) form a MIMO-NOMA1 cluster with channel gains[a+Bi,D+ei] and [a+bi, d+ei], respectively. Based on (1),the beamforming vectors w
′
i and w′
j in the OM state can beobtained by
w′
i1 = (d−D)(ad+ be)/[F 2 +G2],
w′
i2 = (eF − dG)/[F 2 +G2],
w′
i3 = (aF + bG)/[F 2 +G2],
w′
i4 = (b−B)(ad+ be)/[F 2 +G2],w′
j1 = (D − d)(aD +Be)/[F 2 +G2],
w′
j2 = (−eF +DG)/[F 2 +G2],
w′
j3 = (−aF − bG)/[F 2 +G2],
w′
j4 = (B − b)(aD +Be)/[F 2 +G2],
(55)
where F = a(d − D) + e(b − B) and G = dB − bD. Thechannel gain correlation coefficient between MU1 and MU2
and γ =(a2(d−D)2+e2(b−B)2+(Bd−Db)2)((a(d−D)+e(b−B))2+(Bd−Db)2) < 1. Note that R is the real
part of correlation coefficient, γ is a variable which is relatedto e(b−B) and a(d−D). If they are equal to zero, ‖w′i‖
2
and∥∥w′j∥∥2 are both infinite. However, in the simulation of
Fig. 4, radius of two MUs are unchanged, and channel gainsin 40 antennas are generated by random valuables followingCN(0, 1). It indicates that ‖hi‖2 and ‖hj‖2 can only bechanged in a small range. Since MU j is first decoded, wehave (D2 +B2) > (d2 + b2) by condition ‖wi′‖2< ‖wj ′‖2.Then, we take results of λ and β into (12) as
∆Pk = ρ2R
∥∥∥w′j∥∥∥2 +
∥∥∥w′i∥∥∥21
ρ2R
(1+δ)−δ
=R2(‖hi‖2+‖hj‖2θ) 1(a2(d−D)2+e2(b−B)2+(Bd−Db)2) ,
x =
∣∣∣w′iw′j∣∣∣2‖w′i‖2‖w′j‖2
, θ = x(δ−2)+(δ+1)(1+δ−x)2 < 1.
(57)
From (58), we notice that ∆Pk is directly related tothe real part of correlation coefficient R2 while weakly re-lated to the imaginary part of correlation coefficient through(a2(d−D)2+e2(b−B)2+(Bd−Db)2). Based on this struc-ture, we can explain that, first of all, MIMO-NOMA1 is alwaysbetter than MIMO-OMA as∆Pk>0.However, not all beneficialMIMO-NOMA1 cluster has ∆Pk>0.01w. Therefore, the gapin Fig. 4(c) is larger than that in Fig. 4(a) if we reduce R2 andmake ‖hj‖2 small. Second, ∆Pk increases if we increase R1
or R2, which can be observed from comparing the value ofcolor bar between Fig. 4(a) and Fig. 4(c). Last, the influenceof the imaginary part increases with R2.
We conduct the same example for MIMO-NOMA2. In thiscase, the channel gain correlation coefficient between MU1
and MU2 (denoted as ρ), ‖w′i‖2and
∥∥w′j∥∥2are the same as
in (56). Then, if we assume λ = 1 and β = − w′iw′j
‖w′i‖2, the
cluster beamforming vector ‖vk‖2and power reduction can becalculated as
‖vk‖2 =∥∥∥w′j∥∥∥2 − ∣∣∣w′iw′j∣∣∣2
‖w′i‖2,
∆Pk =∥∥∥w′j∥∥∥2ρ2R +
∥∥∥w′i∥∥∥2(δ + 2− δ+1ρ2R
)
= σ2δ(a2(d−D)2+e2(b−B)2+(Bd−Db)2) (
R2
γ ‖hi‖2
+((2 + δ)γ − (1 + δ)γ γ2
R2 )‖hj‖2).
(58)
Note that if ((2 + δ)γ− (1 + δ)γ γ2
R2 ) > 0, we have γ2
R2 < 2
which makes ρ > 1/√
2. In Fig. 4(b), since ρ < 0.6, we haveγ2
R2 > 2 and ((2 + δ)γ− (1 + δ)γ γ2
R2 ) < −δγ. Moreover, sinceR2
γ < γ < 1 and δ > 1, ‖hi‖2 should be larger than ‖hj‖2to make ∆Pk > 0, which indicates that a beneficial MIMO-NOMA2 cluster should include both a cell-center MU and acell-edge MU. Furthermore, if we increase |R|, ∆Pk will beincreased more efficiently than that when we decrease γ. Forthe case of a cluster with the size larger than 2, according to(26), we will have∥∥∥w′l∥∥∥2−‖wl‖2=∥∥∥w′l∥∥∥21+ρ2R
1−ρ2R(ρ2R2+ρ2R3−2ρRρR2ρR3). (59)
where, ρR2 (ρR3) is calculated as the same formula in (54)
between i and l (j and l).
D. SIC decoding conditions
The achievable rate (or SINR) should be larger than apredefined threshold ζδ in order to ensure successful SICdecoding [33]. We transform (8d) into a constraint which canbe judged by power coefficients.
For MIMO-NOMA1, the condition to make xj successfullydecode on MU i is that γji > ζδ = ζγi. Through taking thebeamforming results into this inequality, we will have λ2 >(1 + δ)ζ in a 2-MU cluster. Similarly, for an n-MU cluster, ifi, j ∈ 1, 2, ..., n and i < j, the signal from MU j − 1 (andj) will be decoded at MU i, and the SINR after SIC operationcan be denoted as
γj−1i =|hHi wj−1
√pj−1|2
(∑j−2ε=1 |hHi wε
√pε|2+σ2)
,
γji =|hHi wj
√pj|2
(∑j−1ε=1 |hHi wε
√pε|2+σ2)
,
=|hHi wj
√pj|2
( 1
γj−1i
+1)|hHi wj−1√pj−1|2 >
|hHi wj|2|hHi wj−1|2
γj−1i
1+γj−1i
.
Therefore, ifγji>γj−1i , we have
∣∣hHi wj∣∣2>(1+γj−1i )∣∣hHi wj−1∣∣2
for any i+2<j <n. If γji >ζδ, we have∣∣hHi wj∣∣2>ζδ(1 +
1
γj−1i
)∣∣hHi wj−1∣∣2. Then, we observe that
γi = pi
(∑i−1ε=1 |hHj wε
√pε|2+σ2)
= δ,
γi+1i =
|hHi wi+1√pi+1|2
(pi+∑i−1ε=1 |hHj wε
√pε|2+σ2)
,
=|hHi wi+1
√pi+1|2
pi(1+1δ )
>|hHi wi+1|2
(1+ 1δ )
> ζδ.
Therefore, we have∣∣hHi wi+1
∣∣2 > (1 + δ)ζ. Taking a 3-MUcluster for example, the condition for successful decoding isthat (1 + δ)ζ < λ24, λ
21 and λ21(1 + 1
γji)ζδ < λ22.
For MIMO-NOMA2, the condition for successful decodingis that µε+1
µε> minζδ(1 + 1
γε1), ..., ζδ(1 + 1
γεε−1), ζ(1 + δ).
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Jiefei Ding is currently pursuing a Ph.D. degree inUniversity of Manitoba, Canada. She received B.Sc.degree from Xiangtan University, China, in 2012,the M.Sc. degree from the Guangdong Universityof Technology, China in 2015. She visited HongKong University of Science and Technology as apostgraduate visiting internship student in 2015. Shereceived the best paper award from INISCOM in2015. Her research interests include game theory,resource management of cloud computing, C-RAN,MIMO and NOMA technologies in 5G networks.
Jun Cai (M’04-SM’14) received the B.Sc. andthe M.Sc. degrees from Xi’an Jiaotong University,Xi’an, China, in 1996 and 1999, respectively, and thePh.D. degree from the University of Waterloo, ON,Canada, in 2004, all in electrical engineering. FromJune 2004 to April 2006, he was with McMasterUniversity, Hamilton, ON, as a Natural Sciences andEngineering Research Council of Canada Postdoc-toral Fellow. Since July 2006, he has been with theDepartment of Electrical and Computer Engineering,University of Manitoba, Winnipeg, MB, Canada,
where he is currently a Professor. His current research interests includeenergy-efficient and green communications, dynamic spectrum managementand cognitive radio, radio resource management in wireless communicationsnetworks, and performance analysis. Dr. Cai served as the TPC Co-Chairfor IEEE VTC-Fall 2012 Wireless Applications and Services Track, IEEEGlobecom 2010 Wireless Communications Symposium, and IWCMC 2008General Symposium; the Publicity Co-Chair for IWCMC in 2010, 2011, 2013,and 2014; and the Registration Chair for QShine in 2005. He also servedon the editorial board of the Journal of Computer Systems, Networks, andCommunications and as a Guest Editor of the special issue of the Associationfor Computing Machinery Mobile Networks and Applications. He receivedthe Best Paper Award from Chinacom in 2013, the Rh Award for outstandingcontributions to research in applied sciences in 2012 from the University ofManitoba, and the Outstanding Service Award from IEEE Globecom in 2010.
Changyan Yi (S’16-M’18) received the B.Sc. de-gree from Guilin University of Electronic Technol-ogy, China, in 2012, the M.Sc. and Ph.D. degreesfrom University of Manitoba, MB, Canada, in 2014and 2018, respectively. He is currently working asa Research Associate in Electrical and ComputerEngineering, University of Manitoba, Canada. Hewas awarded Chinese Government Award for Out-standing Students Abroad in 2017, A. Keith DixonGraduate Scholarship in Engineering for 2017-2018,Edward R. Toporeck Graduate Fellowship in En-
gineering for 2014-2017 (four times), University of Manitoba GraduateFellowship (UMGF) for 2015-2018, and IEEE ComSoc Student Travel Grantfor IEEE Globecom 2016. His research interests include algorithmic gametheory, queueing theory and their applications in radio resource management,wireless transmission scheduling and network economics.
