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Zero-forcing Oriented Power Minimization for
Multi-cell MISO-NOMA Systems: A Joint User
Grouping, Beamforming and Power Control
Perspective
Yaru Fu, Mingshan Zhang, Lou Sala¨
un, Chi Wan Sung, and Chung Shue Chen
Abstract—Future wireless communication systems have been
imposed high requirement on power efficiency for operator’s
profitability as well as to alleviate information and commu-
nication technology (ICT) global carbon emission. To meet
these challenges, the power consumption minimization problem
for a generic multi-cell multiple input and single output non-
orthogonal multiple access (MISO-NOMA) system is studied
in this work. The associated joint user grouping, beamforming
(BF) and power control problem is a mixed integer non-convex
programming problem, which is tackled by an iterative dis-
tributed methodology. Towards this end, the near-optimal zero-
forcing (ZF) BF is leveraged, wherein the semiorthogonal user
selection (SUS) strategy is applied to select BF users. Based
on these, the BF vectors and BF users are determined for
each cell using only local information. Then, two distributed
user grouping strategies are proposed. The first one, called
channel condition based user clustering (CCUC), performs user
grouping in each cell based on the channel conditions. This
is conducted independently of the power control part and has
low computational complexity. Another algorithm, called power
consumption based user clustering (PCUC), uses both the channel
conditions and inter-cell interference information to minimize
each cell’s power consumption. In contrary to CCUC, PCUC
is optimized jointly with the power control. Finally, with the
obtained user grouping and BF vectors, the resultant power
allocation problem is optimally solved via an iterative algorithm,
whose convergence is mathematically proven given that the
problem is feasible. We perform Monte-Carlo simulation and
numerical results show that the proposed resource management
methods outperform various conventional MISO schemes and the
non-clustered MISO-NOMA strategy in several aspects, including
power consumption, outage probability, energy efficiency, and
connectivity efficiency.
Index Terms—Beamforming, connectivity efficiency, energy
efficiency, inter-cell interference, multiple input single output
(MISO), non-orthogonal multiple access (NOMA), optimal power
control, user clustering.
This work was partially supported by a grant from the Research Grants
Council of the Hong Kong Special Administrative Region, China (Project
No. CityU 11216416). A part of the work was carried out at LINCS
(www.lincs.fr). Yaru Fu and Mingshan Zhang contributed equally to this work.
(Corresponding author: Chung Shue Chen)
C. S. Chen, L. Sala¨
un, and M. Zhang are with Bell Labs, Nokia Paris-
Saclay, 91620 Nozay, France (e-mail: chung shue.chen@nokia-bell-labs.com,
lou.salaun@nokia-bell-labs.com, mshzhang@bjtu.edu.cn). M. Zhang is now
with Beijing Jiaotong University.
Y. Fu and C. W. Sung are with the Dept. of Electrical Engineering, City
University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong SAR
(e-mail: albert.sung@cityu.edu.hk, yaru fu@sutd.edu.sg). Y. Fu is now with
Singapore University of Technology and Design.
I. INTRODUCTION
The dramatically increased demand of data traffic as well as
the explosive growth of innovative applications, e.g., intelli-
gent transportation system, intelligent computing, and internet-
of-things (IoT), have imposed great challenges for future
wireless cellular networks to provide massive connectivity
and satisfied user quality-of-service (QoS). Non-orthogonal
multiple access (NOMA) has been included by the third
generation partnership project long-term evolution advanced
(3GPP-LTE-A) and been recognized as a promising radio
access solution for addressing the massive access challenge
[1] and for its superior spectrum efficiency compared to
conventional orthogonal multiple access (OMA) strategies [2].
Specifically, in NOMA scheme, multiple users with diverse
power levels can be accommodated within the same resource
block simultaneously with the aid of advanced transmission
scheme, i.e., superposition coding, and progressive reception
technique, that is successive interference cancellation (SIC).
Therefore, transmit power of the multiplexed users should
be assigned properly as it is crucial for successful interfer-
ence elimination and signal decoding. The power allocation
for uplink and downlink single-carrier NOMA systems are
investigated in [3] and [4], respectively. The joint subcarrier
and power allocation problem for multi-carrier NOMA is
analyzed in [5]–[7], and its NP-hardness is shown in [8].
Additionally, there are some other works that study power
allocation problems for NOMA enabled multi-access edge
computing (MEC) networks [9], NOMA assisted unmanned
aerial vehicle (UAV) systems [10], NOMA in IoT systems
[11] and NOMA based wireless caching networks [12]–[14].
In the meantime, multiple-input and multiple-output (MI-
MO) is another promising technique to address 5G and beyond
(B5G) challenges. In regard to the resource management for
multi-antenna systems, we explore two works that are worth
mentioning. In [15], the energy-efficient beamforming (BF) for
a single-cell multiple-input and single-output (MISO) network
is investigated. Thereof, the authors propose a branch-and-
reduce-and-bound approach to attain the optimal solution.
In order to decrease the computational complexity of the
optimal algorithm, two time-efficient approximate methods
are designed. One adopts zero-forcing BF (ZF-BF), with
which the maximization problem recasts to be a concave-
convex fractional programming. The other one is bases on
2
the sequential convex approximation algorithm to find a sta-
tionary point. In [16], the BF and power control problem is
studied for MISO broadcast channel with the consideration
of various linear constraints. Numerically efficient algorithms
are provided to solve the general weighted sum rate problem
under different BF policies, i.e., dirty paper coding (DPC) and
ZF. Nevertheless, the designed algorithms cannot be applied to
MIMO-NOMA networks directly due to the distinct principle
of NOMA when compared to that of the conventional OMA
based strategies. As far as MIMO-NOMA, it was shown by
[17] that the application of MIMO in NOMA is able to
achieve an enhanced system performance compared to pure
NOMA and pure MIMO. The notion of MIMO-NOMA was
first introduced in [18], which has attracted much attention
[19]–[29] since then. Specifically, in [19], the ergodic capacity
maximization problem for a single-cell MIMO-NOMA system
is studied, wherein only statistical channel state information is
available at the transmitter side. In [20], the authors investigate
a NOMA beamforming system, in which each BF vector is
capable of accommodating multiple users. The optimal precod-
ing scheme for a two-user MISO-NOMA system is obtained
in [21] via Newton’s iterative algorithm to minimize total
power. A joint user scheduling and power allocation algorithm
is designed to optimally solve the sum rate maximization
problem for millimeter wave multiuser MISO-NOMA system
in [22]. To reduce the computational complexity, a sub-optimal
strategy is also developed. In addition, [23] investigates the
user clustering, BF and power allocation problem for sum
rate maximization of a single-cell MIMO-NOMA system, in
which the number of antennas per user is more than that of the
base station. The power minimization oriented precoding for
a single-cell uplink MIMO-NOMA is studied in [24] with the
consideration of jamming attacks, wherein the authors design
the precoders at the transmitters and equalizers at the receiver
jointly.
For multi-cell scenarios, [25] studies the sum throughput
maximization problem for multi-cell MIMO-NOMA systems
subject to the users’ QoS requirements, wherein only two
users are superimposed using NOMA scheme. In [26], several
interference alignment based coordinated BF strategies are
developed for a two-cell MIMO-NOMA network to enhance
the cell-edge users’ throughput and system fairness. Besides,
[27] maximizes the weighted sum rate of the strongest users
in a coordinated two-cell MIMO-NOMA system, where an
efficient majorization-minimization oriented iterative algorith-
m is designed. A joint design of BF and power allocation
for multi-cell MIMO-NOMA networks is proposed in [28],
in which the base stations (BSs) adopt coordinated multipoint
transmission in the downlink. A recent work [29] investigates
the interference alignment based rate maximization problem
for a three-cell MIMO-NOMA network, where each cell only
has two subscribers. Therein, the non-convex optimization
problem is solved by a successive convex approximation
method. The above discussed multi-cell works mainly focus
on rate maximization [25]–[29]. To the best of our knowledge,
power minimization problem for generic multi-cell MISO-
NOMA systems has not been addressed yet, which is also
an important research direction for NOMA since future green
communication systems target at achieving a 10-fold enhance-
ment in power efficiency.
The aforementioned discussion motivates us to investigate
the power minimization problem for the downlink of a general
multi-cell MISO-NOMA system with arbitrary number of
cells and users, in which each cluster can multiplex arbitrary
number of subscribers. We summarize the main contributions
of our work as follows:
1) We first explain the impact of user grouping, BF and
power control on the power cost of a generic multi-
cell MISO-NOMA network. Then, the total power mini-
mization problem is formulated taking into account each
user’s data rate requirement and a maximum number of
multiplexed users per cluster constraint. The formulated
problem is a mixed integer non-convex programming
problem, which is in general intractable to be solved
directly.
2) To handle this challenging power minimization prob-
lem, an efficient distributed iterative resource allocation
methodology is proposed. First, the BF vectors are com-
puted by ZF-BF, and semiorthogonal user selection (SUS)
algorithm is applied to select the BF users in each cell.
At this point, each beam only serves one BF user. Each
remaining user is then grouped with a BF user, such
that the BS serves them on the same beam through
signal superposition. Users in the same group (i.e., clus-
ter) performs SIC to decode the superposed signal. We
propose two user grouping strategies, namely CCUC and
PCUC. Finally, with the obtained user grouping and BF
vectors, the power control problem is solved iteratively.
The proposed schemes (BF, user grouping and power
control) are all distributed in the sense that they only
require local information available at each cell, i.e., the
link gains between each user and their serving BS, the
inter-cell interference estimated at each user’s equipment.
3) The proposed user grouping strategies are metric-
oriented, namely channel conditions based user clustering
(CCUC) and power consumption based user clustering
(PCUC). CCUC performs user grouping in each cel-
l based on the channel conditions. This is conducted
independently of the power control part and has low
computational complexity. In contrast, PCUC uses both
the channel conditions and inter-cell interference infor-
mation to minimize each cell’s power consumption in
a distributed manner. PCUC is based on the linear-sum
assignment problem and is optimized jointly with the
power control.
4) Given the user clustering and BF results, the original
optimization problem can be transformed into a non-
convex power control problem, denoted by P. The fea-
sibility of problem Pis characterized by an associated
problem defined as Q. Through rigorous analysis, we
show that the optimal values of the two problems are
equivalent and there exists a one-to-one mapping between
their optimal solutions. Fortunately, problem Qfalls into
the paradigm of classical power control theory and can be
optimally solved by an iterative algorithm with provable
3
convergence property. Such an optimal iterative power
control scheme as been proposed in our previous work [4]
for single-input single output (SISO) NOMA systems. In
this work, we will go further by generalizing it to MISO-
NOMA systems, as well as developing user grouping and
beamforming on top of the iterative power control.
5) Monte-Carlo simulation results show that our proposed
joint user grouping, BF and power allocation algorithms
outperform various conventional MISO schemes and the
non-clustered MISO-NOMA strategy in several aspects,
including outage rate, power consumption, energy ef-
ficiency and connectivity efficiency, suitable for future
systems.
Note that a preliminary result of this work has been pub-
lished in [30], in which power minimization problem for
a specific two-cell MISO-NOMA system under distributed
power control is studied. Here, we consider the more general
problem with arbitrary number of cells and users, wherein
each cluster can multiplex arbitrary number of subscribers.
Note that cluster based multi-antenna NOMA is considered
for supporting massive access in B5G systems [1], [17].
The rest of this paper is organized as follows. We describe
the multi-cell MISO-NOMA system model and formulate
the power minimization problem in Section II. Section III
presents the proposed user grouping and BF methods. With
the obtained user grouping and BF vectors, the original op-
timization problem is transformed into a power minimization
problem, which is denoted as P. In Section IV, we show the
feasibility as well as the optimality of Panalytically. Based
on which, the optimal power allocation algorithm is designed,
whose convergence is proved as well. The performance of
the proposed resource management schemes are evaluated by
extensive computer simulations and are compared to various
benchmarks in Section V. Finally, we conclude the paper in
Section VI.
II. SYSTEM DESCRIPTION AND PROB LEM FO RM ULATI O N
In this section, we first describe the multi-cell MISO-
NOMA system model and then the problem formulation.
A. System Description
The downlink of a MISO-NOMA system that consists of
Mcells is considered. Denote by M,{1,2, . . . , M }the
index set of all cells. In each cell m∈ M, there is one
base station (BS) serving a number of Kmusers. Each BS
is equipped with an antenna array of Nelements for the
downlink transmission and each user is equipped with a single
antenna. Define Km,{1,2, . . . , Km}as the set of the indices
of all subscribers who are associated with BS m. Without
loss of generality, we assume that Kmis independent from
(i.e., can be larger or smaller than) Nfor all m∈ M. For
m∈ M,m′∈ M and i∈ Km, define gi,m,m′∈C1×Nas
the channel gain vector between BS m′and user iwho is
attached to BS m. We focus on the scenario where each BS
m∈ M transmits |Jm|beams to provide multiuser downlink
transmission. Therein, each beam j∈ Jmserves one cluster
with Lj,m subscribers performing NOMA pattern, in which
Jmindicates the index set of all the clusters in cell m. Let
wj,m ∈CN×1be the corresponding beamforming (BF) vector
of cluster jin cell m, where m∈ M and j∈ Jm. Besides,
denote by Uj,m the index set of users who are assigned
to the j-th cluster by BS m. In addition, let pi,m be the
allocated power to user iby BS m. Furthermore, we define
pj,m ,(pi,m)i∈Uj,m as the power vector of users in the j-th
cluster of cell mand let pm,(p1,m,p2,m , . . . , p|Jm|,m)be
the power allocation strategy of all the users in cell m.
We use ||z||1to represent the l1-norm of any given vector
z, which equals to the sum of all the components in z. For
m∈ M and j∈ Jm, let qj,m ,||pj,m||1be the transmit
power summation of all users in j-th cluster of cell m. In
addition, denote by q,(q1,1, q2,1, . . . , q|JM|,M )the power
vector of all the clusters. Furthermore, we define
q−j,m ,(q1,1, q2,1, . . . , qj−1,m, qj+1,m , . . . , q|JM|,M ),(1)
where m∈ M and j∈ Jm.
In our MISO-NOMA system, the messages of users that
belong to the same cluster are transmitted via superposition
coding. For expression simplicity, we assume user iis assigned
to the j-th cluster of cell m. In addition, we assume frequency-
flat block fading channel. With aforementioned definitions, the
received signal at user iin cell m,yi,m, can be expressed as
follows:
yi,m =gi,m,mwj,m xj,m +gi,m,m
j′∈Jm,j′=j
wj′,mxj′,m
+
M
m′=1,m′=m
l∈Jm′
gi,m,m′wl,m′xl,m′+ni,m,
(2)
where ni,m ∼ CN(0, σ2
i,m)indicates the received noise
signal at user iin cell m,xj,m represents the superposed
signal of users in cluster jof cell m. Note that xj,m ,
i∈Uj,m √pi,msi,m , where si,m is the desired message of user
i. Thereof, E[|si,m|2] = 1 is satisfied for i∈ Kmand m∈ M.
Denote by ˆ
Ii,m the inter-cluster interference plus noise value
of user iwho is assigned to cluster jin cell m, and thus it
can be obtained due to (2) as follows:
ˆ
Ii,m ,
j′∈Jm,j′=j
qj′,m|gi,m,m wj′,m|2
+
M
m′=1,m′=m
l∈Jm′
ql,m′|gi,m,m′wl,m′|2+σ2
i,m,
(3)
where the first and second items on the right side of (3) refer
to the intra-cell interference and inter-cell interference at user i
in cell m, respectively. For notation simplicity, we use Ii,m to
represent the normalized inter-cluster interference plus noise
value of user iwho is attached to BS m, i.e.,
Ii,m ,ˆ
Ii,m/|gi,m,m wj,m|2,(4)
where ˆ
Ii,m is given by (3).
Since SIC is adopted among users within each cluster, the
decoding order of the multiplexed users should be considered.
For m∈ M and j∈ Jm, we adopt Πj,m to indicate the
4
set of all possible permutations of the elements in Uj,m.
For instance, if Uj,m ={1,2,3}, then we have Πj,m =
(1,2,3),(2,1,3),(1,3,2),(3,1,2),(2,3,1),(3,2,1).
Furthermore, we define an auxiliary vector ˆ
πj,m =
(ˆπj,m(1),ˆπj,m(2),...,ˆπj,m(Lj,m)) ∈Πj,m as the decoding
order of users in the j-th cluster of cell m. Let ˆπj,m(l), where
l∈ {1,2, . . . , Lj,m}, be its l-th element, which indicates that
subscriber ˆπj,m(l)first decodes the messages of ˆπj,m(1) to
ˆπj,m(l−1), subtracting these signals and finally decodes its
own message by treating the signals of the remaining users
in this cluster as noise. For multi-cell NOMA scenario, the
communication quality of each user depends not only on its
link gain but also on its suffered interference. Therefore, the
decoding order of subscribers in cluster jof cell mshould be
determined by their normalized interference plus noise values
[4], i.e., Ii,m,i∈ Uj,m . Since for i∈ Uj,m the value of Ii,m
is determined by q−j,m,πj,m is characterized as a function
of q−j,m and it is expressed as:
πj,m(q−j,m ) = (πj,m(1), πj,m (2), . . . , πj,m (Lj,m)) ∈Πj,m ,
(5)
so that the following conditions are met:
1) The normalized interference plus noise values, i.e.,
Iπj,m(l),m ,l∈ {1,2, . . . , Lj,m}, are permuted in de-
scending order as Iπj,m(1),m ≥Iπj,m (2),m ≥ ··· ≥
Iπj,m(Lj,m ),m.
2) When there is a tie, i.e., Iπj,m(l),m =Iπj,m (l′),m for l <
l′, then πj,m(l)< πj,m (l′).
For m∈ M, denote by wm,(w1,m,w2,m , . . . , w|Jm|,m)
and w,(w1,w2, . . . , wM)the BF strategies of cell m
as well as the system, respectively. In addition, let p,
(p1,p2, . . . , pM). For m∈ M and i∈ Km, we define
Ri,m(p,w)as the transmission data rate of subscriber ifrom
cell m. In addition, let R′i,m(ˆ
πj,m,p,w)be an auxiliary
function given below:
R′i,m(ˆ
πj,m,p,w),
Wlog2
1 + pi,m
Lj,m
l′=ˆπ−1
j,m(i)+1 pˆπj,m(l′),m +Ii,m
,(6)
in which Wis the system bandwidth, ˆ
πj,m ∈Πj,m, and
ˆπ−1
j,m(i)represents the decoding order of user iin ˆ
πj,m.
Specifically, ˆπ−1
j,m(i) = lif ˆπj,m(l) = i. With aforementioned
definitions, we have
Ri,m(p,w) = R′i,m (πj,m(q−j,m ),p,w),(7)
where m∈ M, i ∈ Km. Note that q−j,m is given by (1),
which is a function of p.
B. Problem Formulation
In this article, we aim to minimize the total transmit power
of the system while taking into account each user’s data rate
requirement and a maximum number of multiplexed users per
cluster. For m∈ M and i∈ Km, denote by ri,m the required
minimum transmission rate of user iwho is attached to BS
m. The power minimization problem is defined below:
min
m∈M
i∈Km
pi,m (8)
subject to
C1 : Ri,m(p,w)≥ri,m , m ∈ M, i ∈ Km,
C2 : Lj,m ≤L, m ∈ M, j ∈ Jm,
C3 : Uj,m ∩ Uj′,m =∅, m ∈ M, j ∈ Jm, j ′∈ Jm, j =j′,
C4 : pi,m ≥0, m ∈ M, i ∈ Km.
(9)
Note that C1indicates the minimum data rate requirement of
each user, C2restricts the number of users served in cluster
jof cell mto be no greater than L, where Lis a parameter
which jointly considers the practical limitations of receiver
design complexity and the error propagation due to SIC, C3
is used to make sure that each user is scheduled by a single
beam at most, and C4refers to the non-negativity of allocated
power.
The minimization problem (8) is a mixed integer non-linear
and non-convex programming problem, which is in general
intractable to solve directly [22]. Therefore, in the following
sections, a two-fold methodology is designated. Specifically,
in step 1, the user clustering and BF are determined. Based
on the obtained results, an optimal power allocation algorithm
is developed in step 2 to solve the resultant power allocation
problem.
III. B E AM FOR MIN G AN D USER CLUSTERING
ALGORITHMS
In this section, we first introduce the proposed BF strategy.
Based on which, two metric-oriented user clustering schemes
are designed. One is based on the channel condition and the
other rests with the metric of power cost.
A. Beamforming Design
It is known that dirty paper coding (DPC) is the optimal
method to achieve the capacity region of multi-user MIMO
broadcast channel systems. However, DPC is difficult to use
in practical system as it employs brute-force searching [20].
Instead, zero-forcing BF (ZF-BF) is a near-optimal and easy-
to-implement substitute [31]. Therefore, in this article, we
apply ZF as the BF policy. In our NOMA system, the BF
vectors of each cell are generated by some of its associated
users’ channel vectors. We call these users BF users and their
multiplexed users in each beam are named as matched users.
It is assumed that the channel state information of all users
are perfectly known by the BSs.
For m∈ M, denote by Tz,m the index sets of the
selected BF users in cell m. In addition, let H(Tz,m ),
{([gT
i,m,m])i∈Tz,m }T, where (·)Tcorresponds to the transpose
of a matrix. Since the beam vectors of the ZF-oriented BF
strategy are determined by the channel information of BF
users, the corresponding ZF-BF vectors that generated by the
link gains of users in Tz,m can be expressed as follows:
wm=H†(Tz,m) = H∗(Tz,m)(H(Tz,m)H∗(Tz,m))−1,(10)
5
where m∈ M, the operators (·)†and (·)∗represent the Moore
Penrose pseudoinverse and the conjugate transpose of a matrix,
respectively.
In this work, semiorthogonal user selection (SUS) algorithm
is adopted to select the BF users, which has been proven to
achieve the asymptotic optimal performance of DPC [31]. SUS
is conducted in an iterative manner. Its main idea is to select
a BF user from a constructed user group denoted by TSU S
during each iteration, and the set TSU S is renewed at the
end of each iteration such that the users in this group are
semiorthogonal to the BF user set. The procedure of BF user
selection continues until Nusers are selected or the user group
TSU S is empty. It is worth noting that the number of clusters
is less than the number of antennas if the algorithm terminates
according to the second condition. The orthogonality among
BF users is controlled by the SUS factor α, which is a small
positive constant between 0 and 1. In this work, the values
of αfor different system settings are chosen empirically by
simulations, which will be discussed later in Section V-A.
The implementation of the above SUS scheme is detailed in
Algorithm 1. With the output of Algorithm 1, the BF vectors
for each cell can be computed by (10).
Algorithm 1 SUS algorithm for BF users selection for cell m
Input: gi,m,m for i∈ Km.
Output: Tz,m.
1: Initialization: Tz,m =∅,TSU S ={1, ..., Km}and j= 1.
2: while j≤Nand TSU S =∅do
3: Search for a new BF user for cluster jof cell m.
4: for i∈ TSU S do
5: if j== 1 then
6: hi:= gi,m,m.
7: else
8: hi:= gi,m,m −
j−1
j′=1
gi,m,mh∗
(j′)
∥h(j′)∥2h(j′), in which x∗
depicts the conjugate transpose of a vector x.
9: end if
10: end for
11: Denote i∗:= arg max
i∈TSU S ∥hi∥.
12: Update h(j):= gi∗,m,m.
13: Adjust Tz,m ← Tz,m {i∗}.
14: Update TSU S ← {i|i∈ TSU S , i =i∗,|gi,m,m h∗
(j)|
∥gi,m,m∥∥h(j)∥<
α}, where αis the SUS factor, which is a small positive
constant.
15: j←j+ 1.
16: end while
B. User Clustering Methods
In this subsection, two user clustering approaches are de-
signed with the prior information of BF vectors obtained in
Section III-A.
1) Channel Condition based User Clustering (CCUC): In
this scheme, the clustering is determined by the link quality
of each user. The interference information is not required, i.e.,
the user grouping decision as well as the power control opti-
mization are mutually independent. CCUC algorithm is imple-
mented in an iterative way. At each iteration a matched user is
assigned to a given BF user according to the channel quality
oriented metric until all the users are clustered. Without loss of
generality, we take cell mas an example. Before detailing the
algorithm, we first give some definitions. Let Tz,j,m and T˜z,j,m
be the BF user and the matched users in the j-th cluster of cell
m, respectively. Thereby, the index set of users in cluster jof
cell mcan be obtained as Uj,m =Tz,j,m T˜z,j,m. Besides, we
define T˜z ,m =T˜z ,1,m ... T˜z ,|Jm|,m as the index set of all
the selected matched users. Moreover, let Tm=Tz,m T˜z,m
be the index set of the already selected users. In the beginning
of the algorithm, we assume the user index sets are empty, i.e.,
Tz,m =∅,T˜z,m =∅. In each iteration, the cluster-matched
user pair (j∗, i∗)with the highest correlation is chosen as
stated, i.e., the matched user i∗is then grouped in cluster j∗.
We repeat the aforementioned step until each of the subscribers
is allocated into a cluster. The pseudo-code of the CCUC
strategy in cell mis summarized in Algorithm 2.
Algorithm 2 The proposed CCUC algorithm for cell m
1: Initialization: Tz,m =∅,T˜z,m =∅,Tm=Tz,m T˜z,m,
and T˜z ,m =T˜z ,1,m ... T˜z ,|Jm|,m.
2: Determine Tz,m via selecting BF users according to Al-
gorithm 1.
3: while Tm=Kmdo
4: Match a new user to a cluster such that they have the
highest correlation value, i.e.,
5:
(j∗, i∗),arg max
(j,i)j∈Tz,m\{j′||Tz ,j′,m |=L−1, j′∈Tz,m },i∈Km\Tm
(
|gi,m,m ·g∗
Tz,j,m,m,m |
∥gi,m,m∥·∥gTz,j,m ,m,m∥).
6: T˜z ,j∗,m ← T˜z ,j∗,m {i∗}.
7: end while
8: Form the user indices set for each cluster in cell m, i.e.,
9: for j:= 1,2, ..., |Tz,m|do
10: Uj,m ← Tz,j,m T˜z,j,m.
11: end for
2) Power Consumption based User Clustering (PCUC):
The main idea of PCUC algorithm is to assign users to
different clusters such that the power consumption of the BS
is minimized. Before expounding the details about PCUC,
we first demonstrate the needed transmit power per cluster
to meet the multiplexed users’ data rate requirements under a
predetermined decoding order. We take an arbitrary cluster j
in cell mas an example. Suppose the decoding order ˆ
πj,m is
given, and we aim to meet the minimum data rate constraint
per subscriber in this cluster, which is expressed as follows:
R′i,m(ˆ
πj,m,p)≥ri,m , i ∈ Uj,m.(11)
Let ˜
pj,m(ˆ
πj,m,q−j,m )be the minimum power expenditure
of cluster jin BS m, where the least data rate requirement
per subscriber within this cluster is met. Therein, the prior
information of the BF strategy w, the decoding order as well
as the transmit power of other clusters, q−j,m, are given. For
6
notation simplicity, we define
γi,m ,2
ri,m
W−1(12)
as the required SINR of subscriber iin cell m, where i∈ Km
and m∈ M. Specifically, we have the following Lemma 1.
Lemma 1. For m∈ M,j∈ Jm, the minimum power
consumption of cluster jin cell munder decoding order
ˆ
πj,m ∈Πj,m is given as follows:
˜
pj,m(ˆ
πj,m,q−j,m ) =
Lj,m
l=1
χl,m(ˆ
πj,m)Iˆπj,m(l),m ,(13)
where
χl,m(ˆ
πj,m),
l′<l
(γˆπj,m (l′),m + 1)γˆπj,m (l),m .(14)
Proof. Obviously, the minimum sum power of all users in
cluster jis obtained given that the data rate constraints in
(11) achieves with equality. Firstly, we consider the power of
users pˆπj,m (Lj,m),m and pˆπj,m(Lj,m −1),m. According to (6), we
have
pˆπj,m (Lj,m),m =γˆπj,m(Lj,m ),mIˆπj,m (Lj,m),m ,(15)
pˆπj,m (Lj,m−1),m =γˆπj,m(Lj,m −1),m
×(pˆπj,m (Lj,m),m +Iˆπj,m(Lj,m −1),m).(16)
Based on (15) and (16), we obtain that
Lj,m
l=Lj,m−1
pˆπj,m (l),m =γˆπj,m (Lj,m −1),mIˆπj,m(Lj,m −1),m+
(γˆπj,m (Lj,m−1),m + 1)γˆπj,m(Lj,m ),mIˆπj,m (Lj,m),m .
(17)
Then, we calculate the power of user pˆπj,m (Lj,m−2),m , which
is as follows:
pˆπj,m (Lj,m−2),m =γˆπj,m(Lj,m −2),m
×(pˆπj,m (Lj,m−1),m +pˆπj,m(Lj,m ),m +Iˆπj,m(Lj,m−2),m ).
(18)
Based on (17) and (18), the required power of the last three
users is obtained. We repeat the aforementioned steps, the
summation of all the Lj,m users’ transmit power is achievable,
i.e.,
Lj,m
l=1
pˆπj,m (l),m =
Lj,m
l=1
l′<l
(γˆπj,m (l′),m + 1)
×γˆπj,m (l),mIˆπj,m(l),m ,
(19)
which completes the proof.
Now we go back to the PCUC algorithm, whose inputs
are the link gains of each user in the cell, as well as the
inter-cell interference estimated at each user equipment side.
Since the inter-cell interference may vary in time due to the
distributed iterative power control, PCUC is recomputed at the
beginning of each power control iteration using the most up-to-
date inter-cell interference information. To be more specific,
at iteration t, the power consumptions of each user in cell
munder all the possible clustering policies are calculated.
We represent the performance of each policy in a cost matrix
Cm∈RKm×L|Jm|, whose (i, L(j−1) + l)-th component,
denoted by Cm
i,L(j−1)+l, indicates the power cost of user iin
terms of satisfying its data rate requirement, if it is assigned
to the j-th cluster of cell mand is in the l-th decoding order.
For simplicity, we assume that the data rate requirement of
each user in the system is the same, and the corresponding
SINR requirement is denoted by γ. In accordance with (19),
the value of Cm
i,Lj+lcan be calculated as follows:
Cm
i,Lj+l=γ(γ+ 1)(l−1) I(t−1)
i,m ,(20)
where i∈ Km,l= 1,2, ..., L, and j∈ Jm.I(t−1)
i,m represents
the normalized interference plus noise value of user iwho is
attached to BS m, and is assumed to be assigned to cluster
jduring the t-th iteration. It is worth noting that I(t−1)
i,m is
computed based on the power control at the (t−1)-th iteration.
Based on the aforementioned analysis, the user cluster-
ing problem can be transformed to the linear sum as-
signment (LSA) problem, wherein we aim to obtain a
Boolean matrix Xmto minimize the BS’s power cost, i.e.,
ij,l Cm
i,L(j−1)+lXm
i,L(j−1)+l, where Xmindicates the clus-
tering strategy and has the same dimensions as Cm. In
addition, Cm
i,L(j−1)+lrepresents the (i, L(j−1) + l)-th entry
of Xm, whose value takes 1 if and only if user iis decoded
in the l-th order among all the users in cluster j.Xmis op-
timized such that the resultant clustering strategy induces the
minimum power consumption. Hence, the clustering decision
optimization can be formulated as the following LSA problem:
min
i
j,l
Cm
i,L(j−1)+lXm
i,L(j−1)+l,(21)
subjects to j,l Xm
i,L(j−1)+l= 1 and iXm
i,L(j−1)+l≤1.
These two constraints guarantee that the solution of Boolean
matrix Xmhas the features that each row has exactly one
‘1’, and each column has at most one ‘1’. In other words,
each user is assigned to exactly one cluster, and each cluster
contains at most Lusers, which correspond to the conditions
C2and C3in (9). According to [32], problem (21) can be
solved in polynomial time by Hungarian algorithm, which is
also known as Kuhn-Munkres algorithm. For brevity, details
of this algorithm are omitted here. The pseudo-code of the
t-th iteration of our PCUC strategy in cell mis summarized
in Algorithm 3.
Before ending this section, we claim that the PCUC user
clustering method needs to be executed jointly with the power
control algorithm during each iteration, which will be pre-
sented in the following section. Indeed, inter-cell interference
may change from one power control iteration to the other,
therefore PCUC is recomputed at the beginning of each
power control iteration using the most up-to-date inter-cell
interference information. These information can be measured
or estimated locally and be conducted through some periodical
or broadcast messages similar to that suggested in [33].
The signaling transmission associated energy consumption is
neglected compared with that of data packet transmission [34]
and thus is not considered in this paper. The same assumption
7
Algorithm 3 The t-th iteration of PCUC algorithm for cell m
Input: I(t−1)
i,m for i∈ Km, and Tz,m .
Output: Uj,m for j∈ |Jm|.
1: Initialization: U1,m, ..., U|Jm|,m =∅.
2: Formulate the LSA cost matrix C, i.e.,
3: for i:= 1,2, ..., Kmdo
4: for j:= 1,2, ..., |Jm|do
5: for l:= 1,2, ..., L do
6: Calculate the required power of user iassuming
that user iis assigned to a cluster jand is the l-th
decoded user, i.e.,
Cm
i,L(j−1)+l=γ(γ+ 1)(l−1)I(t−1)
i,m ,
7: where I(t−1)
i,m is calculated based on (4).
8: end for
9: end for
10: end for
11: Decide the boolean matrix Xmof user clustering by
Hungarian algorithm, such that the BS has the minimum
transmit power, i.e.,
min
i
j,l
Cm
i,L(j−1)+lXm
i,L(j−1)+l.
12: for j:= 1,2, ..., |Jm|do
13: Determine the user set of each cluster jbased on Xm.
Uj,m ← Uj,m {i|
l
Xm
i,L(j−1)+l= 1}.
14: end for
15: Iterative power control of t-th iteration according to Al-
gorithm 4.
was made in existing work [25]–[29] as well.
IV. POW ER AL LOC ATION ALGORITHM WITH FI XED US ER
GROU PIN G AND BF
In this section, we investigate the optimal power allocation
algorithm for the transmit power minimization problem with
given user grouping and the BF strategy w. This power control
problem is then expressed as follows:
min
m∈M
i∈Km
pi,m (22)
subject to
C1′:Ri,m(p)≥ri,m , m ∈ M, i ∈ Km,(23)
C4 : pi,m ≥0, m ∈ M, i ∈ Km.(24)
Note that wis ignored in the expression of Ri,m since it
is given. We denote the above problem by P, which is still
non-convex due to the non-convexity of Ri,m(p). In addition,
define P ⊆ Rn†as the feasible power region of P, wherein
n†,M
m=1 |Jm|
j=1 Lj,m.
A. The Optimal Users Decoding
In this subsection, we prove that the decoding order defined
in (5) is optimal, which requires the least power cost when
compared to that of the other decoding orders. Denote by
pj,m(q−j,m )the minimum power expenditure of cluster jto
meet the data rate requirements of its managed subscribers.
According to the foregoing analysis, we have
pj,m(q−j,m ) = ˜
pj,m(πj,m (q−j,m),q−j,m ),(25)
where ˜
pj,m is well defined in Lemma 1.
Lemma 2. πj,m (q−j,m )is the optimal decoding order for
the multiplexed users in cluster jof cell msince it has the
minimum power cost among all the possible decoding orders,
i.e.,
pj,m(q−j,m ),min
ˆ
π†
j,m∈Πj,m
˜
pj,m(ˆ
π†
j,m,q−j,m ), m ∈ M, j ∈ Jm.
(26)
Proof. Let ˆ
π∗
j,m ,πj,m(q−j,m ). We need to prove that
˜
pj,m(ˆ
π∗
j,m,q−j,m )≤˜
pj,m(ˆ
π†
j,m,q−j,m )is satisfied for
ˆ
π†
j,m ∈Πj,m. Without loss of generality, it is assumed that
ˆ
π†
j,m = (ˆπ†
j,m(1),...,ˆπ†
j,m(Lj,m )) ∈Πj,m \ {π∗
j,m}
achieves the minimum power cost in (26), i.e.,
˜
pj,m(ˆ
π†
j,m,q−j,m )<˜
pj,m(ˆ
π∗
j,m,q−j,m ).(27)
Based on the definition of decoding order in Section II-A,
there must exist a l < Lj,m such that
Iˆπ†
j,m(l),m < Iˆπ†
j,m(l+1),m ,(28)
since ˆ
π†
j,m and ˆ
π∗
j,m are two distinct decoding orders.
By exchanging the orders of users ˆπ†
j,m(l)and ˆπ†
j,m(l+
1), a new decoding order is obtained, which is de-
noted as ˆ
π◦
j,m = (ˆπ†
j,m(1), . . . , ˆπ†
j,m(l−1),ˆπ†
j,m(l+
1),ˆπ†
j,m(l),ˆπ†
j,m(l+ 2), . . . , ˆπ†
j,m(Lj,m )). In the following, we
reveal that
˜
pj,m(ˆ
π◦
j,m,q−j,m )<˜
pj,m(ˆ
π†
j,m,q−j,m ),(29)
which arises a contradiction to that ˆ
π†
j,m achieves the mini-
mum power in (26).
It is easy to observe that, the needed transmit power of
users ˆπ†
j,m(˜
l)for ˜
l > l + 1 where ˆπ†
j,m(˜
l)∈ Uj,m, will not
be affected after changing the orders of these two adjacent
users, i.e., users ˆπ†
j,m(l)and ˆπ†
j,m(l+ 1). Define p†and p◦as
the total required power of users ˆπ†
j,m(l)and ˆπ†
j,m(l+ 1) in
π†
j,m and π◦
j,m, respectively. Therefore, we only need to prove
that p◦< p†, since the power cost of user ˆπ†
j,m(˜
l)under the
order π◦
j,m is strictly less than that under the decoding order
π†
j,m for ˜
l < l with the assumption that p◦< p†. By some
manipulations, it can be obtained from (6) that
p†=γˆπ†
j,m(l),m (
Lj,m
l′=l+2
pˆπ†
j,m(l′),m +Iˆπ†
j,m(l),m )
+ (γˆπ†
j,m(l),m + 1)(
Lj,m
l′=l+2
pˆπ†
j,m(l′),m +Iˆπ†
j,m(l+1),m )γˆπ†
j,m(l+1),m ,
8
whereas
p◦=γˆπ†
j,m(l+1),m (
Lj,m
l′=l+2
pˆπ†
j,m(l′),m +Iˆπ†
j,m(l+1),m )
+ (γˆπ†
j,m(l+1),m + 1)(
Lj,m
l′=l+2
pˆπ†
j,m(l′),m +Iˆπ†
j,m(l),m )γˆπ†
j,m(l),m .
Consequently, we have
p◦−p†=γˆπ†
j,m(l+1),m γˆπ†
j,m(l),m (Iˆπ†
j,m(l),m −Iˆπ†
j,m(l+1),m ),
which is negative according to (28). This completes the proof.
B. Cluster Power Control Problem, Q
In this subsection, we first give the formulation of Qand
show its property of standard. Some technical details are
required. Then, we analyze the feasibility of Q. Finally, we
prove that the optimal solution to Qis unique.
1) The Formulation of Q:The related cluster
power control problem Qhas the objective function
min m∈M j∈Jmqj,m and subjects to the following
constraints:
qj,m ≥pj,m(q−j,m ), m ∈ M, j ∈ Jm,(30)
qj,m ≥0, m ∈ M, j ∈ Jm,(31)
whose feasible power region is defined as Q ⊆ Rn‡, wherein
n‡,m∈M |Jm|.
Define I(q),(Ij,m(q))m∈M,j ∈Jmas the interference
function of Q, each item of which can be taken as the
total interference that the corresponding cluster needs to over-
come. According to (30) and Lemma 2, we have Ij,m(q),
pj,m(q−j,m ) = minˆ
πj,m∈Πj,m ˜
pj,m(ˆ
πj,m,q−j,m ), where m∈
Mand j∈ Jm. The standard property of I(q)is discussed
below:
Lemma 3. The interference function I(q)is standard.
Proof. In accordance with Lemma 1, ˜
pj,m(ˆ
πj,m,q−j,m )is
expressed below via substituting (4) into (13), i.e.,
˜
pj,m(ˆ
πj,m,q−j,m ) =
Lj,m
l=1
βl,m(ˆ
πj,m)
j′∈Jm,j′=j
qj′,m|gˆπj,m(l),m,m wj′,m|2+
M
m=1,m′=m
l′∈Jm′
ql′,m′|gˆπj,m (l),m,m′wl′,m′|2+σ2
ˆπj,m (l),m
,
(32)
where βl,m(ˆ
πj,m),χl,m (ˆ
πj,m)
|gˆπj,m(l),m,m wj,m|2and χl,m(ˆ
πj,m)is
given by (14). Intuitively, ˜
pj,m(ˆ
πj,m,q−j,m )is an affine
function to q−j,m, satisfying the three properties of standard
[35]. Since the minimization operation preserves the property
of standard [35], Ij,m(q)is standard for m∈ M and j∈ Jm.
This completes the proof.
2) The Feasibility of Q:In this paragraph,
we show the feasibility of Q. Define ˆ
π,
(ˆ
π1,1,ˆ
π2,1, . . . , ˆ
π1,M , . . . , ˆ
π|JM|,M )and Π,
Π1,1×Π2,1× ··· × Π1,M × ··· × Π|JM|,M . The power
optimization subproblem with ˆ
π∈Πis considered:
min
m∈M
j∈Jm
qj,m (33)
subject to
qj,m ≥˜
pj,m(ˆ
πj,m,q−j,m ), m ∈ M, j ∈ Jm,(34)
qj,m ≥0,∀m∈ M, j ∈ Jm.
Since ˆ
πj,m ∈Πj,m and |Πj,m|=Lj,m !, we have
M
m=1 |Jm|
j=1 (Lj,m!) such power control subproblems in total.
Based on (32), we see that
qj,m ≥
Lj,m
l=1
βl,m(ˆ
πj,m)×
j′∈Jm,j′=j
qj′,m|gˆπj,m(l),m,m wj′,m|2+
m′∈M,m′=m
l′∈Jm′
ql′,m′|gˆπj,m (l),m,m′wl′,m′|2+σ2
ˆπj,m (l),m
.
(35)
The optimization problem (33) can be rewritten into the
following problem with matrix form:
min
m∈M
j∈Jm
qj,m (36)
subject to
(I−D(ˆ
π))q≥d(ˆ
π),(37)
q≥0,(38)
where Irepresents the identity matrix, d(ˆ
π),
(Lj,m
l=1 βl,m(ˆ
πj,m)σ2
ˆπj,m (l),m)m∈M,j ∈JmT
, in which
(x)Tindicates the transpose of vector x, and D(ˆ
π)is an
off-diagonal matrix whose (u, v)-th element, where u=v, is
expressed as follows:
La,m
l=1
βl,m(ˆ
πa,m)|gˆπa,m(l),m,m′wb,m′|2,
so that the listed two criteria are met:
1) m−1
l=1 |Jl|< u ≤m
l=1 |Jl|,
2) m′−1
l=1 |Jl|< v ≤m′
l=1 |Jl|,
where a,u−m−1
l=1 |Jl|and b,v−m′−1
l=1 |Jl|.
The above problem (36) is characterized by the decoding
orders of all clusters, i.e., ˆ
π. For ˆ
π∈Π, define F(ˆ
π)as
the feasible power region of the corresponding power control
subproblem. It is clear that F(ˆ
π) = m∈M,j∈JmFj,m (ˆ
πj,m),
where Fj,m(ˆ
πj,m),q≥0:qj,m ≥˜
pj,m(ˆ
πj,m,q−j,m ).
Therefore, we have the following Lemma 4.
Lemma 4. The feasible region of problem Qis given as
follows:
Q=
ˆ
π∈ΠF(ˆ
π).(39)
9
Proof. See Appendix B.
3) The Optimality of Q:In this part, we depict that the
optimal solution to Qis unique. We observe that the optimal
power strategy to Qachieves when the inequalities in (30) sat-
isfy with equalities. Thereby, the following result is obtained.
Theorem 5. Given that Qis non-empty, Qhas a unique
optimal solution.
Proof. Since Qis feasible, by Lemma 4, there must exist some
of the power control subproblems which are feasible. In addi-
tion, the optimal solution to each subproblem is unique [36].
Therein, the one who needs the least power consumption is
the optimal solution to Q.
In the following, we show the uniqueness. Denote by q∗,
(q∗
1,1, q∗
2,1, . . . , q∗
|JM|,M )and q†,(q†
1,1, q†
2,1, . . . , q†
|JM|,M )
two diverse optimal strategies. Intuitively, ||q∗||1=||q†||1.
Due to the fact that q†and q∗are different, an α > 1must
exist so that αq∗≥q†as well as αq∗
j,m =q†
j,m are satisfied
for some m∈ M and j∈ Jm. As a consequence, we have
αq∗
j,m =αpj,m(q∗
−j,m)>pj,m (αq∗
−j,m)≥pj,m (q†
−j,m) =
q†
j,m, which arises a contradiction.
C. Characterization of P
In this subsection, we first analyze the feasibility of P. Based
on which, we show that problems Pand Qhave the same
optimal values and one-to-one mapping relationship.
1) The Feasibility of P:A mapping Φfrom domain Q
to domain Pis defined. For any q∈ Q, we must have
qj,m ≥pj,m(q−j,m )for m∈ M,j∈ Jm. By letting the data
rate requirements of users in Uj,m hold with strict equalities,
we can find p†
j,m such that ||p†
j,m||1=pj,m (q−j,m)≤qj,m .
Denote by ε,qj,m/||p†
j,m||1≥1and pj,m =εp†
j,m.
It is easy to observe that pj,m ∈ P, and then we have
||pj,m||1=qj,m . The mapping Φ : Q→Pis then
defined as Φ(q),(p1,1,p2,1, . . . , p|JM|,M ), where pj,m for
m∈ M and j∈ Jmis characterized by the aforementioned
procedure. We note that Φ(q)∈ P for q∈ Q. Similarly,
another mapping Θfrom domain Pto domain Qis defined
as Θ(p1,1,p2,1, . . . , p|JM|,M ),(q1,1, q2,1, . . . , q|JM|,M ),
where qj,m =||pj,m||1for m∈ M and j∈ Jm.
It is easy to see that Θ(p1,1,p2,1, . . . , p|JM|,M )∈ Q if
(p1,1,p2,1, . . . , p|JM|,M )∈ P. With the above definitions, we
have the following result.
Theorem 6. Pis non-empty if and only if there exists some
ˆ
π∈Πsuch that the Perron-Frobenius eigenvalue of D(ˆ
π)is
less than 1.
Proof. Based on the definitions of Φand Θ, it is easy to check
that Pis feasible when and only when Qis feasible. According
to [36], the sufficient and necessary condition for a power
control problem having feasible solutions is that the Perron-
Frobenius eigenvalue of D(ˆ
π)is less than 1. Given the above
two facts as well as Lemma 4, we have Theorem 6.
2) The Optimality of P:The optimal power allocation to
problem Pobtains when the inequalities in (23) hold with
equalities. We will show that the optimal solution to Pis
Algorithm 4 The t-th power iteration of cluster jin cell m
Input: I(t−1)
i,m for i∈ Uj,m.
Output: q(t)
j,m and p(t)
j,m.
1: Calculate the optimal decoding order, πj,m, based on (5),
with inputs I(t−1)
i,m for i∈ Uj,m.
2: for l:= 1,2, . . . , Lj,m,do
3: Calculate χl,m by (14), i.e., χl,m :=
l′<l(γπj,m (l′),m + 1)γπj,m(l),m .
4: end for
5: Determine the least required transmit power of cluster j
at BS mas follows:
q(t)
j,m :=
Lj,m
l=1
χl,mI(t−1)
πj,m(l),m .
6: Determine the power allocation for user πj,m(Lj,m),
which is quoted below:
p(t)
πj,m(Lj,m ),m := γπj,m(Lj,m ),mI(t)
πj,m(Lj,m ),m.
7: for l:= Lj,m −1, Lj,m −2, ..., 1,do
8: Calculate the allocated power to user πj,m(l)in cell m
as follows:
p(t)
πj,m(l),m := γπj,m (l),m
l′>l
p(t)
πj,m(l′),m +I(t−1)
πj,m(l),m .
9: end for
10: return q(t)
j,m and (p(t)
i,m)i∈Uj,m .
unique and there is a one-to-one mapping property between
the optimal solutions to Pand Q.
Lemma 7. Problems Qand Phave the same optimal values.
Proof. Let p∗and q∗be the optimal power strategies to P
and Q, respectively. Thereby, we have ||q∗||1=||Φ(q∗)||1≥
||p∗||1and ||q∗||1≤ ||Θ(p∗)||1=||p∗||1. This completes the
proof.
Theorem 8. Given that Pis feasible, then its optimal solution,
denoted by p∗, is unique, which satisfies p∗= Φ(q∗), wherein
q∗is the optimal power allocation policy of Q.
Proof. Since Pis feasible, Qis also feasible. By Theorem 5,
the optimal strategy of Q,q∗, is unique. Based on Lemma 7,
p∗= Φ(q∗)is an optimal solution. Let p‡be another optimal
strategy to P. According to Lemma 7, Θ(p‡)is also optimal
to Qand equals to q∗, which means ||p‡
j,m||1=||p∗
j,m||1=
q∗
j,m is satisfied for m∈ M and j∈ Jm. Since the optimal
power allocation for Pobtains when each of the users achieves
exactly its minimum data rate requirement, we have p‡
i,m =
p∗
i,m under any fixed q∗
−j,m, which represents p‡=p∗.
The above obtained results uncover that we can get the op-
timal allocation policy to Pvia solving Q. In next subsection,
the optimal power control algorithm for Qis designed.
D. Optimal Power Allocation Algorithm
10
In this subsection, we describe the proposed optimal power
allocation to solve P, which has guaranteed convergence
performance and optimality given that Pis feasible, i.e., P
is non-empty.
Let q(t)
j,m be the total transmit power of all users in the
j-th cluster of cell mat iteration t, where m∈ M and
j∈ Jm. In addition, define q(t)= (q(t)
1,1, q(t)
2,1, . . . , q(t)
|JM|,M )
and q(t)
−j,m = (q(t)
1,1, q(t)
2,1, . . . , q(t)
j−1,m, q(t)
j+1,m,·· · , q (t)
|JM|,M ). In
the t-th iteration, the transmit power allocation for subscribers
within each cluster is conducted by their associated BS. With
aforementioned definitions, the iterative algorithm is given
below:
q(t)
j,m =pj,m(q(t−1)
−j,m )for m∈ M, j ∈ Jm.(40)
We re-write (40) in vector form, i.e.,
q(t)=I(q(t−1)),(41)
whose convergence is presented in the following theorem.
Theorem 9. Given that Pis feasible, i.e., Pis non-empty, the
iterative power control algorithm stated in (40) converges to
the unique optimal power strategy q∗with any fixed initial
point.
Proof. Since Pis non-empty, Pis feasible and thus Qis
also feasible according to the definitions of previously defined
mappings. By Theorem 5, the optimal strategy of Q,q∗,
is unique. In accordance with Lemma 3, (41) is standard.
Therefore, the iterative power allocation converges to a unique
fixed point with any predetermined initial power values [35].
The proof is completed since the optimal solution to Qis a
fixed point.
With the optimal strategy q∗, the optimal solution of the o-
riginal minimization problem can be obtained via the mapping
Φ. For expression simplicity, we summarize the pseudo code
of the t-th iteration of our designed optimal power allocation
in Algorithm 4.
So far, the design for the joint user clustering, BF and power
control algorithm is completed. Before ending this section, we
declare that our proposed resource management algorithms are
applicable to the usage scenarios with or without BS’s transmit
power budget taken into consideration. To be more specific, for
each BS, if the total required power in terms of satisfying all
its associated users’ minimum data rate requirements is larger
than the budget, outage occurs, as demonstrated in Section V,
wherein a maximum value of total transmit power per BS is set
to analyze the outage performance of our proposed versatile
methodologies.
V. SIMULATION RESULTS
In this section, Monte-Carlo simulation is used to evaluate
the performance of our proposed joint user grouping, BF
and power allocation strategies for multi-cell MISO-NOMA
system. In the simulation, we consider a 7-cell (M= 7)
network adopting wrap-around technique with 12 transmit
antennas (N= 12) for each BS. The inter-site distance is set
to 500 meters, and Kmusers are uniformly distributed within
TABLE I
SIMULATION PAR AM ET ER S
Parameters Values
Inter-site distance 500 m
Minimum distance from user to BS 35 m
Distance-dependent path loss 128.1 + 37.6 log10(d)dB, din km
Shadowing Log-normal, standard deviation 8 dB
Small scale fading Rayleigh fading with variance 1
Noise power spectral density -174 dBm/Hz
Throughput calculation Shannon’s capacity formula
System bandwidth, W10 MHz
Carrier frequency 2 GHz
Maximum number of iterations 100
Transmit power budget of a BS 0.1 W
Number of simulated BSs, M7
Number of antennas for each BS, N12
Maximum number of users per cluster, L K
Number of simulation instances for each case 10000
Number of users per cell, K8, 9, . . . , 13
User data rate requirement, R(0.2, 0.4, . . . , 1.4) Mbits/s
the hexagonal cell m, where m∈ M. The carrier frequency
is set to be 2 GHz, and the system bandwidth Wis set to
be 10 MHz. The noise power spectral density at each receiver
is assumed equal to −174 dBm/Hz. The radio propagation
model follows [37], which includes three components, i.e.,
the distance-dependent path loss, shadow fading as well as
the small-scale fading. In detail, the distance-dependent path
loss is given by 128.1+37.6 log10(d), where dis the Euclidean
distance between the BS (transmitter) and the user (receiver) in
km. The shadow fading follows a log-normal shadowing model
with a standard deviation of 8dB. Each user also experiences
independent Rayleigh fading with variance 1as small scale
fading. In addition, we assume all the subscribers have the
same data rate requirement, i.e., ri,m =Rfor m∈ M and
i∈ Km. Besides, each cell is assumed to have the same
number of subscribers, i.e., Km=Kfor m∈ M. Moreover,
the maximum number of users per cluster, i.e., L, is assumed
to be equal to K. Note that our proposed schemes are also
applicable to the scenarios with different values of Kmand
ri,m for m∈ M and i∈ Km. It is worth mentioning that the
plots are obtained by simulating 10000 independent instances
of randomly generated network realization. We summarize the
simulation parameters in Table I.
We use conventional ZF assisted OMA based MISO scheme
as a baseline, named as MISO-OMA-ZF, to compare its
performance with the proposed clustering enabled MISO-
NOMA schemes, which are denoted by MISO-NOMA-CCUC
and MISO-NOMA-PCUC, respectively. Specifically, in MISO-
OMA-ZF scheme, users in the same cell are grouped to
different clusters exclusively and ZF-BF is applied. In MISO-
NOMA-CCUC, the CCUC approach is adopted to perform
user clustering, which is optimized independently of the power
control. In contrast, MISO-NOMA-PCUC scheme requires the
power control information to perform user clustering at each
iteration, and thus is a joint user clustering and power alloca-
tion paradigm. Note that in the proposed two strategies, ZF-BF
and the optimal power control algorithm (i.e., Algorithm 4) are
used for the beamforming and power allocation, respectively.
Therein, SUS algorithm is applied to do the ZF user selection.
11
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
α
0.0
0.2
0.4
0.6
0.8
1.0
System outage rate
K = 8
K = 9
K = 10
K = 11
K = 12
K = 13
K = 14
K = 15
Fig. 1. System outage rate of MISO-NOMA-CCUC scheme under different
αand Kwith N= 12 and R= 1.0Mbits/s.
To show the effectiveness of our clustering based MISO-
NOMA strategies, the non-clustered MISO-NOMA approach
(i.e., |JM|= 1) [4] is also taken into consideration. We
declare that the proposed ZF-BF as well as the power control
algorithms are also applicable for the non-clustered scheme,
which is denoted by MISO-NOMA. To be more specific,
in MISO-NOMA strategy, the BS transmits the superposed
signals of all users within each cell. And therein the BF vector
is calculated based on the user with the highest channel gain.
Nevertheless, it should be noted that, this approach is less
practical for implementation due to the limitations of receiver
design complexity and the error propagation issue induced
by SIC, especially when the user number is large. Moreover,
for comprehensive comparison, the performance of other BF
techniques for MISO-OMA system such as matched filter
(MF) [30] and minimum mean square error (MMSE) [38] are
also evaluated. They are named respectively as MISO-OMA-
MF and MISO-OMA-MMSE.
A. Optimal Value of αin SUS
In this subsection, we explain how the SUS factor αis
chosen, as it significantly influences the performance of our
proposed MISO-NOMA strategies. Indeed, if αis too large,
the inter-cluster interference will be large due to the high
correlation between clusters, while if αis too small, more
users will be grouped in a cluster. In this work, the near-
optimal values of the SUS factor αfor different scenarios
are searched through empirical simulations for both MISO-
NOMA-CCUC and MISO-NOMA-PCUC schemes, wherein
αvaries from 0.1to 0.8with step size equal to 0.1. The best
value of αis chosen such that the minimum system outage rate
can be achieved, which is usually in our first concern. Note
that the system outage rate is defined by the proportion of the
number of instances where the total power consumption of
users in each cell exceeds the power budget of their associated
BS. An example of the system outage rate under various α
and Kwhen R= 1.0Mbits/s is shown in Fig. 1. The optimal
choice of αfor different Kcan be taken at the minimum point
of each curve. It can be seen that the optimal αdecreases
0 2 4 6 8
Number of iterations
0.05
0.10
0.15
0.20
0.25
0.30
Transmit power (mW)
BS0
BS1
BS2
BS3
BS4
BS5
BS6
(a) Without user outage.
0 10 20 30 40 50 60 70
Number of iterations
0.00
0.02
0.04
0.06
0.08
Transmit power (W)
BS0
BS1
BS2
BS3
BS4
BS5
BS6
(b) With at least one user in outage.
Fig. 2. Convergence behavior of MISO-NOMA-PCUC with or without user
outage.
with K, from 0.6to 0.3for Kincreasing from 8to 15. In
the following simulations, the found best empirical value of α
is adopted. It should be noted that the optimality of αcan be
further improved by more exhaustive search.
B. Convergence Performance
In most communication systems, the transmit power of each
BS is usually limited. Given that the power consumption of
BS mexceeds its power limit, we say that an outage occurs.
Generally, the outage is due to the fact that the power control
problem is infeasible, i.e., some of the users require very high
transmit power1for meeting their data rate requirements. In
this subsection, we investigate the convergence behavior of our
designed joint user clustering and power control algorithm,
as illustrated in Fig. 2(a) and Fig. 2(b). We plot the transmit
power of each BS during the iterations. These results are taken
from two typical instances in the simulation of MISO-NOMA-
PCUC scheme. Specifically, Fig. 2(a) depicts an instance
where there is no outage in the system. In this case, the power
1The required transmit power for some users may be larger than the power
budget of their BS. In such case, we say that these users are in outage and
cannot be served. In practice, their allocated power will be then simply set to
zero during the power control iterations.
12
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Data rate constraint (Mbits/s)
10−5
10−4
10−3
10−2
10−1
100
System outage rate
MISO-OMA-ZF
MISO-OMA-MMSE
MISO-OMA-MF
MISO-NOMA
MISO-NOMA-CCUC
MISO-NOMA-PCUC
(a) System outage rate vs. data rate requirement, K= 12.
8 9 10 11 12 13
Number of users per cell
10−5
10−4
10−3
10−2
10−1
100
System outage rate
MISO-OMA-ZF
MISO-OMA-MMSE
MISO-OMA-MF
MISO-NOMA
MISO-NOMA-CCUC
MISO-NOMA-PCUC
(b) System outage rate vs. number of users per cell, R= 1.4Mbits/s.
Fig. 3. System outage rate performance.
of all the 7 BSs can converge to their fine values rapidly. The
power allocation is feasible, i.e., the data rate requirement of
each user in the system can be satisfied. On the other hand,
Fig. 2(b) illustrates an instance where there is at least one
user in outage. It can be seen that the transmit power of the
7 BSs tends to diverge in the first 19 iterations. At the 19-
th iteration, the power of BS 1 exceeds the set maximum
power limit 0.1Watt. We abandon the outage users (i.e., set
the transmit power for them to zero) according to common
practice since their data rate requirements cannot be satisfied
given the power budget of the base stations. As a result, in the
example, after the 19-th iteration, the transmit power of each
BS in the system decreases naturally and finally converges to
a feasible solution which satisfies the data rate requirements
of the rest of the users.
C. Outage Performance
In this subsection, we evaluate the system outage rate and
the user outage rate performance of all the schemes. The user
outage rate is defined by the number of users in outage divided
by the total number of users in the system. On the other hand,
as aforementioned, the system outage rate is defined by the
percentage of the number of instances where there exists at
least one user in outage.
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Data rate constraint (Mbits/s)
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
User outage rate
MISO-OMA-ZF
MISO-OMA-MMSE
MISO-OMA-MF
MISO-NOMA
MISO-NOMA-CCUC
MISO-NOMA-PCUC
(a) User outage rate vs. data rate requirement, K= 12.
8 9 10 11 12 13
Number of users per cell
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
User outage rate
MISO-OMA-ZF
MISO-OMA-MMSE
MISO-OMA-MF
MISO-NOMA
MISO-NOMA-CCUC
MISO-NOMA-PCUC
(b) User outage rate vs. number of users per cell, R= 1.4Mbits/s.
Fig. 4. User outage rate performance.
The system outage rate and user outage rate under vari-
ous user data rate requirements are shown in Fig. 3(a) and
Fig. 4(a), respectively. Thereof, it is worth mentioning that, the
outage probabilities of MISO-NOMA-PCUC under R= 0.2
Mbits/s are zero in both figures. Since the logarithmic y-axis
is adopted, the values of MISO-NOMA-PCUC at R= 0.2
Mbits/s in both Fig. 3(a) and Fig. 4(a) are neglected. From both
figures, we can see that the three conventional OMA based
MISO schemes have higher system outage rate and user outage
rate when compared to that of our designed MISO-NOMA
strategies. Since in the three MISO-OMA baselines, the chosen
BF vectors could lead to great intra-cell and also inter-cell
interference given that users have highly correlated channels,
resulting in high outage probability. Besides, the non-clustered
MISO-NOMA method has the highest system outage rate
than all the other schemes. This is due to the higher power
requirement with respect to decode the superposed signals by
SIC. In contrast, the clustered MISO-NOMA enabled schemes
would group the highly correlated users in a same cluster and
use SIC at the receivers to separate their signals. This would
reduce the interference and achieve lower outage rate. From
Fig. 3(a) and Fig. 4(a), we can also see that MISO-NOMA-
PCUC outperforms MISO-NOMA-CCUC in the outage rate.
This is because MISO-NOMA-PCUC considers the interfer-
13
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Data rate constraint (Mbits/s)
10−6
10−5
10−4
10−3
10−2
10−1
100
Transmit power (W)
MISO-OMA-ZF
MISO-OMA-MMSE
MISO-OMA-MF
MISO-NOMA
MISO-NOMA-CCUC
MISO-NOMA-PCUC
(a) Power consumption vs. data rate requirement, K= 12.
8 9 10 11 12
Number of users per cell
10−3
10−2
10−1
Transmit power (W)
MISO-OMA-ZF
MISO-OMA-MMSE
MISO-OMA-MF
MISO-NOMA-CCUC
MISO-NOMA-PCUC
(b) Power consumption vs. number of users per cell, R= 1.4Mbits/s.
Fig. 5. Power consumption.
ence of each user and also their corresponding transmit power
in order to decide the user clustering, while MISO-NOMA-
CCUC is only based on the channel link gains.
The system outage rate and user outage rate under different
number of users are plotted in Fig. 3(b) and Fig. 4(b),
respectively. In the conventional OMA based MISO schemes,
at most Nusers can be supported at the same time, where Nis
the number of antennas. Note that when more users are active
in a cell, time scheduling would be required. Nevertheless,
MISO-NOMA schemes can support more users per cell, i.e.,
K > N is manageable under MISO-NOMA. In addition,
we can see that MISO-NOMA-PCUC outperforms all other
schemes in the outage rate performance, while the user outage
rate of MISO-NOMA and MISO-OMA-MF tends to one when
Ris large. Besides, the performance gain of MISO-NOMA-
PCUC compared to the other schemes is small when there are
fewer users per cell. This indicates that when Kis small, the
effect of user grouping would be less important.
D. Power Consumption
In this subsection, the power consumption of the system is
calculated for different schemes. It is worth noting that, to be
fair in comparison, only the instances in the simulation which
have feasible solution in all the schemes are considered in the
calculation. Each point in Fig. 5(a) and Fig. 5(b) is obtained
by averaging over all the feasible instances.
The power consumption under different user data rate
requirement Ris shown in Fig. 5(a). It can be seen that the
power cost of all schemes increase with the increasing R. In
addition, for any given R, the conventional MISO-OMA-MF
scheme requires the highest power consumption, while MISO-
NOMA-PCUC has the lowest power consumption. MISO-
OMA-ZF consumes slightly higher power than MISO-OMA-
MMSE, because MMSE takes into account the effect of noise
[30], [38]. From Fig. 5(a), on average MISO-NOMA-PCUC
saves about 47% and 94% of the power when compared to
MISO-NOMA-CCUC and MISO-OMA-ZF, respectively.
The power consumption under different number of users
is shown in Fig. 5(b).2Therein, the data rate requirement of
each user is set to be 1.4Mbits/s. It is shown that MISO-
NOMA-PCUC outperforms the other schemes in the power
consumption, especially when the number of users increases.
For example, when K= 12, MISO-NOMA-PCUC helps
to save power by 51% and 91% when compared to MISO-
NOMA-CCUC and MISO-OMA-ZF, respectively. It should be
noted that the performance gain is more remarkable for larger
K, because the performance of OMA based MISO strategies
are degraded dramatically due to the inter-cluster interference
when Kis large.
E. Energy Efficiency and Connectivity Efficiency
Energy efficiency (EE) is an important metric for future
wireless communication systems [4]. Here, we follow the
definition in [39] and consider the following notion:
EE =Total amount of transmitted data
Total radiation energy consumption (bits/joule).(42)
The energy efficiency of the schemes under different data rate
requirement is shown in Fig. 6(a). We can see that the energy
efficiency of all the schemes decreases with the increasing
R. In addition, all the NOMA based schemes outperform the
OMA aided strategies. Moreover, MISO-NOMA-PCUC still
has the highest energy efficiency among all the schemes.
Besides, we define the following metric, namely connec-
tivity efficiency (CE), to indicate the number of users that
can be supported per unit power consumption. The number
of supported users in the system is normalized by the total
transmit power consumption as follows:
CE =Total number of supported users
Total transmit power consumption (W−1).(43)
Fig. 6(b) shows the connectivity efficiency of the different
schemes under different data rate requirement. From Fig. 6(b),
we can see that the connectivity efficiency for each scheme de-
creases with the increasing R. Among which, MISO-NOMA-
PCUC outperforms the other five schemes as expected. This is
also due to the fact that MISO-NOMA-PCUC has both lower
user outage rate and lower total power consumption. Further-
more, NOMA enabled strategies achieve higher connectivity
2The power consumption of MISO-NOMA is not evaluated because its
system outage rate tends to one when R= 1.4Mbits/s.
14
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Data rate constraint (Mbits/s)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Energy efficiency (bits/joule)
1e11
MISO-OMA-ZF
MISO-OMA-MMSE
MISO-OMA-MF
MISO-NOMA
MISO-NOMA-CCUC
MISO-NOMA-PCUC
(a) Energy efficiency vs. data rate requirement, K= 12.
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Data rate constraint (Mbits/s)
0
100
200
300
400
500
600
Connectivity efficiency (number of users/mW)
MISO-OMA-ZF
MISO-OMA-MMSE
MISO-OMA-MF
MISO-NOMA
MISO-NOMA-CCUC
MISO-NOMA-PCUC
(b) Connectivity efficiency vs. data rate requirement, K= 12.
Fig. 6. Energy efficiency and connectivity efficiency.
efficiency compared to all the OMA oriented schemes, show-
ing the effectiveness of NOMA.
VI. CONCLUSION
In this paper,we investigate the power minimization problem
for a generic multi-cell MISO-NOMA system through a joint
user grouping, BF and power control perspective. The mixed
integer and non-convex programming problem is solved by
a two-fold methodology. Firstly, the BF and user clustering
are determined by the SUS algorithm and the proposed
user clustering strategies (PCUC and CCUC), respectively.
We can see that PCUC outperforms CCUC, since it uses
the information of the transmit power and interference level
during iterations when conducting user grouping while CCUC
only considers user’s channel link gain. Subsequently, with
the obtained user clustering and BF, we solve the power
allocation problem optimally via an iterative algorithm with
provable convergence guarantee. Numerical results show that
our proposed schemes outperform conventional MISO-OMA
strategies and non-clustered MISO-NOMA policy in several
aspects, including power consumption, outage rate, energy
efficiency, and connectivity efficiency. One may also consider
in the future to use NOMA and multi-antenna techniques for
mobile edge computing, fog radio access, mission-critical IoT,
and massive machine type communications in B5G systems.
APPENDIX A
PROO F OF LEM MA 4
Based on (30) and (31), Qis as follows:
Q=
m∈M,j∈Jmq≥0:qj,m ≥pj,m (q−j,m).(44)
According to Lemma 2, the above equation can be transformed
into
Q=
m∈M,j∈Jm
ˆ
πj,m∈Πj,m q≥0:qj,m ≥˜
pj,m(ˆ
πj,m,q−j,m )
=
m∈M,j∈Jm
ˆ
πj,m∈Πj,m
Fj,m(ˆ
πj,m).
(45)
Since Fj,m(ˆ
πj,m)is purely determined by the decoding order
of users in cluster jof cell m, (45) can be further expressed
as follows:
Q=
m∈M,j∈Jm
ˆ
π∈ΠFj,m(ˆ
πj,m).(46)
Besides, intersection of unions is in general a superset of union
of intersections, i.e.,
ˆ
π∈ΠF(ˆ
π) =
ˆ
π∈Π
m∈M,j∈JmFj,m (ˆ
πj,m)
⊆
m∈M,j∈Jm
ˆ
π∈ΠFj,m(ˆ
πj,m) = Q.(47)
Given any feasible power vector of problem Q,q∈ Q, in
accordance with (44) and Lemma 2, there must exist ˆ
π∗,
(ˆ
π∗
1,1,ˆ
π∗
2,1, . . . , ˆ
π∗
|JM|,M )∈Πsuch that
qj,m ≥pj,m(q−j,m ) = ˜
pj,m(ˆ
π∗
j,m,q−j,m ), m ∈ M, j ∈ Jm.
(48)
In accordance with (44), we have
q∈
m∈M,j∈JmFj,m (ˆ
π∗
j,m)⊆
ˆ
π∈Π
m∈M,j∈JmFj,m (ˆ
πj,m),
which uncovers that
Q ⊆
ˆ
π∈ΠF(ˆ
π).(49)
Follows from (47) and (49), the proof is completed.
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