Optimal User Pairing for Achieving Rate Fairness in Downlink NOMA Networks

Abstract

In this paper, a downlink non-orthogonal multiple access (NOMA) network is studied. We investigate the problem of jointly optimizing user pairing and beamforming design to maximize the minimum rate among all users. The considered problem belongs to a difficult class of mixed-integer nonconvex optimization programming. We first relax the binary constraints and adopt sequential convex approximation method to solve the relaxed problem, which is guaranteed to converge at least to a locally optimal solution. Numerical results show that the proposed method attains higher rate fairness among users, compared with traditional beamforming solutions, i.e., random pairing NOMA and beamforming systems.

Full text

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Optimal User Pairing for Achieving Rate
Fairness in Downlink NOMA Networks
Van-Phuc Bui, Phu X. Nguyen, Hieu V. Nguyen, Van-Dinh Nguyen, and Oh-Soon Shin
School of Electronic Engineering &Department of ICMC Convergence Technology,
Soongsil University, Korea (E-mail: osshin@ssu.ac.kr)
Abstract—In this paper, a downlink non-orthogonal mul-
tiple access (NOMA) network is studied. We investigate
the problem of jointly optimizing user pairing and beam-
forming design to maximize the minimum rate among all
users. The considered problem belongs to a difficult class
of mixed-integer nonconvex optimization programming. We
first relax the binary constraints and adopt sequential convex
approximation method to solve the relaxed problem, which is
guaranteed to converge at least to a locally optimal solution.
Numerical results show that the proposed method attains
higher rate fairness among users, compared with traditional
beamforming solutions, i.e., random pairing NOMA and
beamforming systems.
Index Terms—Beamforming, convex optimization, non-
orthogonal multiple access (NOMA).
I. INTRODUCTION
Non-orthogonal multiple access (NOMA) technique is
being considered as a promising candidate of the fifth gen-
eration networks (5G) [1], [2]. By allowing multiple users
to share the same time-frequency resources, NOMA adopts
the successive interference cancellation (SIC) technique
at users with better channel conditions that significantly
improves the system performance in terms of the spectral
efficiency and edge throughput of users with poorer channel
conditions [3].
By following the principle of NOMA, two users with
more distinct channel conditions should be paired [4].
Thus, the optimal user pairing is crucial to further improve
the performance of NOMA systems. In particular, random
user pairing schemes were proposed in [3]–[6], where a
near user is randomly paired with one farther from the base
station (BS). Nonetheless, a general criteria for dynamic
user pairing was not investigated. Recently, the optimal
user pairing was studied in [7], where a minimum rate con-
straint for all NOMA users is additionally imposed. All of
the aforementioned NOMA works focused on either single-
antenna scenarios and/or random user pairing schemes.
In the power domain NOMA, the BS should allocate a
higher portion of transmission power to users with poor
This research was supported in part by Basic Science Research Program
through the National Research Foundation of Korea (NRF) funded by
the Ministry of Education (No. 2017R1D1A1B03030436) and by the
Ministry of Science and ICT (No. NRF-2017R1A5A1015596) and in part
by the BK21 plus Program through NRF grant funded by the Ministry of
Education (No. 31Z20150313339).
channel conditions from the viewpoint of fairness, which
will generate strong interference to near users.
Motivated by the above discussion and shortcoming of
the previous works, in this paper, we propose a dynamic
user pairing and beamforming optimization problem for
downlink (DL) NOMA systems from the perspective of
fairness among users, which is inspired from the fact that
the users with poor channel conditions may get a very
low (even zero) throughput. The dynamic user pairing
is done by introducing binary variables, which results in
a difficult class of mixed-integer nonconvex optimization
problem. The exhaustive search over all possible cases
of user pairing provides the optimal performance of the
considered problem but with prohibitive complexity. The
main contributions of this paper are two-fold: (i)We
formulate a general optimization problem of dynamic user
pairing and beamforming design, which has not been
reported in the literature; (ii)For efficient and practical
implementations, we relax the binary constraints and pro-
pose a low-complexity iterative algorithm based on the
inner approximation (IA) framework for its solution [8],
which only solves a sequence of simple convex programs.
Numerical results are provided to confirm that our proposed
scheme is efficient in terms of the rate fairness.
Notation: Vectors are denoted by bold lower-case letters.
wH,wTand w∗are the Hermitian transpose, normal
transpose and conjugate of a vector w, respectively. E{·}
denotes the expectation and <{a}returns the real part of
a complex number a.Rand Cdenote the set of all real
and complex numbers, respectively. k·k denotes a vector’s
Euclidean norm.
II. SY ST EM MO DE L AN D PROB LE M FOR MU LATI ON
A. System Model
We consider a downlink NOMA system consisting
of one BS, a set of Mnear users (UEs), denoted by
M,{1,2,· · · , M }, and a set of Nfar UEs, denoted
by N,{1,2,· · · , N }, as illustrated in Fig. 1. In addi-
tion, UE(1, m)and UE(2, n)indicate the m-th UE with
m∈ M in inner-zone and the n-th UE with n∈ N
in outer-zone, respectively. For notational convenience,
G,{(1,1),· · · ,(1, M ),(2,1),· · · ,(2, N)}is defined as
the set of all UEs. The BS is equipped with Lantennas
while each user has a single antenna. Then, the received
arXiv:1812.11490v1 [cs.IT] 30 Dec 2018

2
UE(2, N −1)
R
UE(2, N)
BS
d
UE(1, m)
UE(2,2)
UE(1,1) UE(1, M)
UE(2, n)
UE(1,1)
UE(2,1)
Fig. 1. An illustration of DL NOMA consisting of one BS and (M+N)
users.
signal at UE(i, j)can be expressed as
yi,j =X
(i0,j0)∈G
hH
i,j wi0,j0xi0,j 0+ni,j,(i, j )∈ G (1)
where hi,j ∈CL×1and wi,j ∈CL×1are the channel
vector from the BS to UE(i, j)and the beamforming
vector, respectively. xi,j with E{|xi,j|2}= 1 and ni,j ∼
CN (0, σ 2
i,j )are the transmitted symbol and the additive
white Gaussian noise (AWGN) at UE(i, j), respectively.
In this paper, we assume that NOMA beamforming is
exploited for pairs of two users, UE(1, m)and UE(2, n)
∀m∈ M,∀n∈ N . To do so, we introduce new binary
variables αm,n ∈ {0,1}as
αm,n =1,if UE(1, m)and UE(2, n)are paired,
0,otherwise.
(2)
In each pair, the SIC technique will be adopted at UE(1, m)
by following NOMA principle. In particular, UE(1, m)
first decodes the message intended to UE(2, n)and sub-
tracts it before decoding UE(1, m)’s message. On the other
hand, UE(2, n)will decode its own message directly by
treating UE(1, m)’s message as noise. By defining α,
[αm,n]m∈M,n∈N ,w1,[w1,m ]m∈M,w2,[w2,n ]n∈N
and w,[wH
1wH
2]H, the signal-to-interference-plus-noise
ratio (SINR) at UE(1, m)and UE(2, n)can be written as
γ1,m(w,α) = hH
1,mw1,m 2
Ξm(w,α),(3a)
γ2,n(w,α) = min
m∈MnhH
2,nw2,n 2
Φn(w),hH
1,mw2,n 2
αm,nΨm,n (w)o(3b)
where Ξm(w,α),Φn(w)and Ψm,n(w)are defined as
Ξm(w,α) = X
m0∈M\{m}hH
1,mw1,m02
+X
n0∈N
(1 −αm,n0)hH
1,mw2,n02+σ2
1,m,
Φn(w) = X
(i,j)∈G\{(2,n)}hH
2,nwi,j 2+σ2
2,n,
Ψm,n(w) = X
(i,j)∈G\{(2,n)}hH
1,mwi,j 2+σ2
1,m.
In (3b), |hH
2,nw2,n |2
Φn(w)and |hH
1,mw2,n |2
αm,nΨm,n (w)are the SINRs
of UE(2, n)decoded at UE(2, n)and UE(1, m), respec-
tively. Clearly, if αm,n = 0,∀m, n, then γ2,n(w,α) =
|hH
2,nw2,n |2
Φn(w).
B. Problem Formulation
From (3), the achievable throughput for UE(i, j)can be
derived as
Ri,j (w,α) = log2(1 + γi,j (w,α)),(i, j)∈ G .(4)
Herein, we aim to maximize the minimum rate among all
UEs, called max-min rate (MMR) for short. Accordingly,
an optimization problem can be mathematically formulated
as
max
w,α
min
(i,j)∈G Ri,j (w,α)(5a)
s.t.kwk2≤Pmax
BS ,(5b)
αm,n ∈ {0,1},∀m∈ M,∀n∈ N ,(5c)
X
n∈N
αm,n ≤1,∀m∈ M,(5d)
X
m∈M
αm,n ≤1,∀n∈ N (5e)
where (5b) represents the transmit power constraint at the
BS, Pmax
BS and constraints (5c), (5d), (5e) establish the
criteria for user pairing. Specifically, constraints (5d) and
(5e) ensure that each UE(i, j)can opportunistically pair to
one UE only. We can see that (5a) is nonsmooth and non-
concave and (5c) corresponds to binary constraints, leading
to a mixed-integer nonconvex optimization of problem (5).
Thus, problem (5) is intractable and it may not be possible
to convert the problem into an equivalent convex one.
Following some recent studies on wireless communication
system designs [9], [10], we aim at solving (5) based on
the application of IA method, which efficiently provides a
locally optimal solution with low complexity.
III. PROP OS ED IT ER ATIV E ALGORITHM
The main difficulty of solving (5) is to handle the binary
constraints (5c). To overcome this issue, we first relax
αm,n ∈ {0,1}to 0≤αm,n ≤1and rewrite (5c) as
max
w,α
min
(i,j)∈G γi,j (w,α)(6a)
s.t.0≤αm,n ≤1,∀m∈ M,∀n∈ N ,(6b)
(5b),(5d),(5e) (6c)
where the optimal solutions of MMR problem and max-
min SINR problem are identical. We can observe that the
feasible sets are convex (quadratic and linear constraints),
and thus only the objective function (6a) remains noncon-
cave. In order to apply the IA method, we introduce a new
variable βto re-express (6) equivalently as
max
w,α,β 1/β (7a)
s.t.1/β ≤γi,j ,(i, j)∈ G,(7b)
(5b),(5d),(5e),(6b).(7c)

3
We remark that the equivalence between (6) and (7) is
guaranteed due to the fact that constraints (7b) must hold
with equality at optimum [10].
Concavity of (7a): The function 1/β is convex and thus
its first order approximation at a feasible point β(κ)found
at iteration κis given by
1
β≥2
β(κ)−β
(β(κ))2
which is a linear function.
Convexity of (7b): Here, we consider two cases of i∈
{1,2}due to different structures of the SINR functions.
(i) With i= 1: constraint (7b) is 1/β ≤γ1,j, which can
be revised as
Ξm(w,α)≤hH
1,mw1,m 2β. (8)
Next, we introduce new variables τ,{τm,n}m∈M,n∈N
to decompose (8) into the following two constraints:
(8) ⇔(hH
1,mw2,n02≤τm,n0, m ∈ M, n0∈ N ,(9a)
Ξm(w,α,τ)≤hH
1,mw1,m 2β(9b)
where Ξm(w,α,τ) = X
m0∈M\{m}hH
1,mw1,m02+
X
n0∈N
(1 −αm,n0)τm,n0+σ2
1,m.Since (9a) is convex con-
straint, we only need to handle the non-convexity of (9b).
Constraint (9b) is innerly convexified as
b
Ξ(κ)
m(w,α,τ)≤f(κ)
1,m(w)β, m ∈ M.(10)
where b
Ξ(κ)
m(w,α,τ)is a convex upper bound of
Ξm(w,α,τ)by using [10, Eq. (B.1)]:
b
Ξ(κ)
m(w,α,τ),X
m0∈M\{m}hH
1,mw1,m02+σ2
1,m+
X
n0∈N 1
2
1−α(κ)
m,n0
τ(κ)
m,n0
τ2
m,n0+1
2
τ(κ)
m,n0
1−α(κ)
m,n0
(1 −αm,n0)2
and f(κ)
1,m(w)is a lower bound of the convex function
hH
1,mw1,m 2given as [10, Eq. (22)]:
hH
1,mw1,m 2≥2<{(hH
1,mw(κ)
1,m)H(hH
1,mw1,m )}
−|hH
1,mw(κ)
1,m|2,f(κ)
1,m(w).
(ii) With i= 2: constraint (7b) is replaced by









1/β ≤|hH
2,nw2,n |2
Φn(w),∀n∈ N ,(11a)
1/β ≤|hH
1,mw2,n |2
αm,nΨm,n (w),∀m∈ M,∀n∈ N .(11b)
Similarly to (9b), constraints in (11) are innerly covexified
as
(Φn(w)≤f(κ)
2,n (w)β, ∀n∈ N ,(12a)
Ψm,n(w)≤˜
f(κ)
m,n(w,α)β, ∀m∈ M,∀n∈ N (12b)
where f(κ)
2,n (w)is a lower bound of |hH
2,nw2,n |2given as:
f(κ)
2,n (w),2<{(hH
2,nw(κ)
2,n)H(hH
2,nw2,n )}−|hH
2,nw(κ)
2,n|2
and ˜
f(κ)
m,n(w,α)is a lower bound of hH
1,mw2,n 2/αm,n
derived by using [10, Eq. (38)]:
˜
f(κ)
m,n(w,α),2<{(hH
1,mw(κ)
2,n)H(hH
1,mw2,n )}
α(κ)
m,n
−|hH
1,mw(κ)
2,n|2
(α(κ)
m,n)2(αm,n ).
We should note that constraints (10) and (12) can be
expressed as second-order cone (SOC) constraints, which
are obviously convex ones.
Summing up, problem (6) can be approximated as the
following convex program at iteration (κ+ 1):
max
w,α,β,τ
η,2
β(κ)−β
(β(κ))2(13a)
s.t.(5b),(5d),(5e),(6b),(9a),(10),(12).(13b)
We have numerically observed that some values of αm,n
are very close to binary but not exactly binary values at the
optimum. This makes (5) infeasible. Therefore, we further
introduce the rounding function after obtaining the optimal
solution of problem (13) as
α?
m,n =jα(κ)
m,n +1
2k, m ∈ M, n ∈ N .(14)
The proposed algorithm is summarized in Algorithm
1. Specifically, Algorithm 1 consists of two phases:
In phase 1, we successively solve (13) to achieve
(w(?),α(?), β(?),τ(?)). In phase 2, we first use the round-
ing function (14) to force αinto the nearest Boolean
values, and then resolve problem (13) for a fixed value of α
to find the optimal solution w(?). Moreover, due to the fact
that the IA method is employed, Algorithm 1 converges to
a stationary point, which also satisfies the Karush-Kuhn-
Tucker (KKT) conditions of (6). The detailed proof can be
done by following the same steps in [8], [10].
Complexity analysis: Since problem (13) has x=
(2(M+N)+3M N + 1) quadratic and linear constraints
and y=L(M+N)+2MN +1 optimization variables , the
per-iteration complexity of solving (13) is Ox2.5(y2+x)
[11].
IV. NUMERICAL RES ULT S
We consider a small-cell network serving 8 UEs with M
= 3 UEs in inner-zone and N= 5 UEs in outer-zone. Other
important parameters are included in Table I. Algorithm
1 is terminated when the increase in the objective value
between two consecutive iterations is less than 10−3.
Fig. 2 depicts the averaged MMR performance of our
proposed algorithm (NOMA-Optimal) with two other re-
source allocation schemes as a function of Pmax
BS . The

4
Algorithm 1 Proposed Iterative Algorithm to Solve (6)
Initialization: Set κ= 0 and generate feasible initial
points (w(0),α(0) , β(0) ,τ(0)).
Phase 1:
1: repeat
2: Solve the convex program (13) to compute the
optimal solution (w?,α?, β?,τ?).
3: Update (w(κ+1),α(κ+1) , β(κ+1) ,τ(κ+1)) =
(w?,α?, β?,τ?).
4: Set κ=κ+ 1.
5: until Convergence
6: Output-1: (w?, β?,τ?)=(w(κ), β (κ),τ(κ)),
α?is updated by (14).
Phase 2:
7: Run steps 1- 5 again to find beamformers wwith fixed
α.
8: Output-2: (w?,α?).
TABLE I
SIMULATION PARAMETERS
Parameters Value
Bandwidth 20 [MHz]
Noise power density -174 [dBm/Hz]
Path loss from the BS to a UE, σPL 140 + 37.6log10(d)[dB]
Shadowing standard deviation 8 [dB]
Radius of the cell (R)100 [m]
Coverage of near UEs (d)50 [m]
Distance between BS and the nearest UE ≥5 [m]
solution of NOMA-Random [3] is found by using Algo-
rithm 1 with αm,n being randomly chosen. The results
are averaged over 100 random channel realizations. As
expected, the MMR of Algorithm 1 is higher than that of
NOMA-Random and beamforming schemes, of which the
gaps are about 0.5 bps/Hz and 1 bps/Hz, respectively. This
further demonstrates the effectiveness of jointly optimizing
user pairing and beaforming design.
In Fig. 3, we illustrate the convergence behavior of
Algorithm 1 for different number of antennas at the BS. We
can see that increasing the number of antennas Lrequires
more number of iterations to converge. Nevertheless, it
requires a few iterations to achieve the optimal solution,
i.e., about 17 iterations for L= 16.
V. CONCLUSIONS
This paper studied a DL NOMA network, where the
optimal user pairing is investigated. We formulated a max-
min rate optimization problem to jointly optimize user
pairing and beamforming design, and then designed an
efficient iterative algorithm based on the IA method to
solve it. The effectiveness of the proposed algorithm has
been demonstrated by the numerical results.
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