Joint user pairing and power allocation for downlink non-orthogonal multiple access systems

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Joint User Pairing and Power Allocation for
Downlink Non-Orthogonal Multiple Access Systems
Jie Mei, Lei Yao, Hang Long, and Kan Zheng
Wireless Signal Processing and Network Lab,
Key Laboratory of Universal Wireless Communication, Ministry of Education,
Beijing University of Posts and Telecommunications,
Beijing, China, 100876,
E-mail: meijie.wspn@bupt.edu.cn.
Abstract—Downlink non-orthogonal multiple access (NOMA),
where users are paired as user set and multiplexed in the power
domain, is a promising technology for fifth generation (5G)
communication system. This paper studies joint optimization
of user pairing and power allocation to maximize generalized
proportional fair metric subject to transmit power constraints.
This problem can be divided into two parts: user pairing and
power allocation. In order to reduce the computation burden,
we present a pre-defined multi-user pairing criterion to exclude
user sets, which are unsuitable for multiplexing. Furthermore, for
a given user set, a low complexity multi-user power allocation
scheme is proposed by exploiting the convexity of the optimization
problem. Simulation results show that the proposed user pairing
and power allocation scheme can significantly enhance the
downlink system performance and reduce complexity compared
to the existing schemes in NOMA systems.
Index Terms—Non-orthogonal multiple access (NOMA), multi-
user power allocation, user pairing, resource allocation (RA).
I. INT ROD UC TI ON
The booming data traffic growth exceeds the capacity in-
crease of wireless communication networks [1], [2]. Non-
orthogonal multiple access (NOMA) with users multiplexed in
the power domain is proposed as a powerful countermeasure.
In conventional OMA systems, each user is allocated with
radio resources exclusively. In NOMA systems, two or more
users can share the same radio resource, the transmitter super-
poses signals of the multiplexed users and the receiver decodes
its signal by successive interference cancellation (SIC). Theo-
retically, it has been proved that NOMA outperforms OMA in
both uplink and downlink [3]. Some researches show that both
the average and cell-edge user throughput can be improved by
NOMA compared to OMA in the downlink [3]-[6]. Besides,
recent research show that uplink NOMA can achieve system
capacity limit compared to OMA [9], [10]. Furthermore, in
some situations, like near far situation, the power domain
NOMA system has a significant performance gain over OMA
system.
In downlink NOMA with users multiplexed in power do-
main, there are several fundamental differences compared to
conventional OMA systems. Firstly, in NOMA, the scheduler
allocates more than one user on the same radio resource, which
requires the multi-user scheduling. Then, the scheduling metric
has to reflect both the spectral efficiency (e.g., user throughput)
and the system fairness (e.g., cell edge user throughput ). To
achieve these goals, the proportional fair (PF) based scheduler
for NOMA is proposed in [6], [10], [11]. Due to the multi-
user multiplexing in the power domain, the performances, such
as the average cell throughput, cell-center throughput, and
cell-edge throughput, etc., are closely related to the multi-
user power allocation scheme adopted. Tree-search based
transmission power allocation (TTPA) [4], fractional transmit
power control (FTPC) and fixed (channel-independent) power
allocation [6] are proposed for practical usage with the reduced
complexity and downlink signaling overhead.
However, the existing scheduling and transmission power
allocation schemes have several main drawbacks. Firstly, in
order to find the optimal user set that maximizes the PF
scheduling metric, all user sets are needed to be exhaustively
searched, which increases the complexity of user pairing,
especially when user sets have 3 or 4 users. For a given user
set, the existing power allocation schemes are sub-optimal and
only based on qualitative analysis. Therefore, the objective of
this paper is to find an optimal downlink multi-user power
allocation scheme for a given user set in NOMA systems while
reducing the complexity of user pairing. To emphasize clearly,
the authors have addressed in the following items.
•Usually the scheduler needs to inefficiently search all user
sets [6], [11]. In order to alleviate the computation bur-
den, we propose a pre-defined multi-user pairing criterion
to exclude a large part of unsuitable user sets, which are
not appropriate for power domain multiplexing. Thus, the
scheduler does not need to search all user sets, which can
significantly reduce the complexity of user pairing.
•To find the optimal power allocation solution for a given
user set, we formulate a power allocation problem based
on PF scheduling metric for NOMA system. However,
the multi-user power allocation problem is non-convex
and intractable to solve. Thus, uplink-downlink duality is
employed to transform the original problem to a normal
convex problem, which can be solved by using Karush-
Kuhn-Tucker (KKT) conditions with low complexity.
The remainder of the paper is organized as follows. Sec-
tion II describes the system model employing NOMA. Then
the multi-user pairing and the power allocation scheme are
proposed in Section III. Section V presents simulation results

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Channel gain at UE side
High Low
Decode UE3's signal
Decode UE2's signal
SIC of UE3's signal
Decode UE1's signal
Power
Frequency
Subband b
SIC of UE3's signal
SIC of UE2's Signal
W
User1 User2 User3
UE3
UE2
UE1
User set={UE1,UE2,UE3}
Fig. 1. Illustration of a 3-user NOMA with SIC at UE side.
on the performances of proposed user pairing and power
allocation scheme and compares it with existing schemes in
NOMA systems. Finally, Section VI concludes the paper.
II. SY ST EM MO DE L
As shown in Fig. 1, one single-antenna evolved NodeB (eN-
B) serves K2-antennas user terminals in a cell. We consider
a NOMA system with NCsubbands, where the bandwidth
of each subband is Wand the total bandwidth is NCW. In
each time interval, musers, user set Um
b={π1
b, π2
b, ..., πm
b},
are scheduled from Kusers to be served in subband b
(16b6NC), where πk
bindicates the k-th (16k6m)
user of user set Um
bin subband b, and mdenotes the number
of users multiplexed in subband b. The maximum size of the
user set in each subband is NM, namely 16m6NM.
A. Transmitter
Denote db(πk
b)as the transmitted modulation symbol of user
πk
bwith E[|db(πk
b)|2] = 1, and assume the transmitted symbols
of different users are independent in the same subband. Then,
the transmit symbol in subband bis given by a superposition
of the modulation symbol, i.e.
xb=
m

k=1 pb(πk
b)db(πk
b),(1)
where pb(πk
b)is the transmission power allocated to user
πk
bin subband b. Then, the power allocated to user set Um
b
can be denoted as vector p=pb(π1
b), pb(π2
b), ..., pb(πm
b)
and assume the total transmission power per subband is P,
therefore, user set Um
bshould satisfy
m

k=1
pb(πk
b) = P , (2)
pb(πb(k)) >0,1≤k≤m. (3)
B. Receiver
The received signal vector of user πk
bin subband b, can be
expressed as
yb(πk
b) = hb(πk
b)xb+wb(πk
b),(4)
where hb(πk
b)is the channel coefficient vector from the eNB
to user πk
bin subband band the channel coefficient includes
the large-scale path loss and small-scale fading coefficients,
term wb(πk
b)denotes the additive white gaussian noise plus
inter-cell interference vector of user πk
bin subband b. At
the receiver, applying maximal ratio combination (MRC) to
yb(πk
b)gives rise to
rb(πk
b) = hH
b(πk
b)yb(πk
b)/
hb(πk
b)

=
hb(πk
b)

m

k=1 pb(πk
b)db(πk
b) + nb(πk
b),(5)
where the average power of nb(πk
b)is Nb(πk
b) = E[|nb(πk
b)|2].
Furthermore, the channel gain of user πk
bin subband bis
defined as gb(πk
b) = ∥hb(πk
b)∥2/Nb(πk
b), the optimal decoding
order is in the order of the increasing channel gain. This
decoding order is reasonable, because users with lower channel
gains are allocated higher levels of transmit power than those
with higher channel gains, i.e.,
pbπk
b> pbπk−1
b,2≤k≤m. (6)
For simplicity, users in user set Um
bare sorted in the order
of decreasing channel gain, namely gb(πm
b)< ... < gb(π2
b)<
gb(π1
b). Thus, the signal-to-interference-plus-noise-power-ratio
(SINR) of user πk
bcan be given by
γbπk
b=gb(πk
b)pb(πk
b)
k−1

j=1
gb(πk
b)pb(πj
b)+1
.(7)
Therefore, the throughput of user πk
bin subband bis at time
interval twritten as
Rbπk
b|Um
b;t=Wlog21 + γbπk
b.(8)
C. Problem Formulation
From (7) and (8), it can be seen the multi-user scheduling
policy and multi-user power allocation significantly affect the
system performance, which are investigated in this paper.
In order to guarantee the fairness among the users in a cell,
the scheduler is based on PF metric for NOMA system [6],
[11]. Denote tas the time index with the unit of subframe
length and tcas the average time window, the average through-
put of user kper subband at time interval t+ 1 is defined as
T(k;t+ 1) = 1−1
tcT(k;t) + 1
tc1
NC
NC

b=1
Rb(k;t).(9)
Based on PF policy for NOMA system, we define the
following functions
fb(p|Um) =
m

k=1
Rb(πk|Um;t)
T(πk;t)=
m

k=1
w(πk)Rb(πk|Um;t),(10)
where function fb(p|Um)is given by summation of the PF
metric of all users in user set Um={π1, π2, ..., πm}occu-
pying subband band vector prepresents the power allocation
of user set Um. For user set Um, its PF metric Qb(Um)is
the maximum value of fb(p|Um), which depends on power
allocation vector p. Therefore, PF metric Qb(Um)for user

3
set Umin subband bcan be obtained by solving problem
P0:Qb(Um) = max
p
fb(p|Um)
s.t.(2),(3),(6),
At the eNB, the multi-user scheduler enumerates candidate us-
er sets generated by user pairing. Scheduler allocates subband
bto user set Um
bwith maximum PF scheduling metric, thus
this optimization problem can be formulated as
P:Um
b= argmax
Um
Qb(Um)
s.t.(2),(3),(6).
For subband b, the candidate user set maximizing the
scheduling metric is selected. However, there are two difficul-
ties in solving scheduling problem P: user pairing and power
allocation for a given user set.
III. MULTI -US ER SCHEDULING AND PO WE R ALLOCATIO N
This section first introduce a pre-defined user pairing crite-
rion and then propose the optimal multi-user transmit power
allocation scheme for a given user set.
A. Pre-defined User Pairing Criterion
Because problem Pis a combinational optimization prob-
lem, it can be solved by enumerating all candidate user sets
to find its optimal solution [6]. And the candidate user sets is
generated by user pairing. In existing user pairing scheme, all
possible user set is paired together and the number of candidate
user sets is
V=K
1+K
2+... +K
NM.
The computing burden of this approach increases heavily
with the parameter NMand K. However, for some user sets,
there is no optimal solution to problem P0. The idea of this
paper is to exclude a part of unsuitable user sets, which do
not have optimal power allocation solution. To this end, we
analyze the existence condition of optimal solution to problem
P0.
Lemma 1. In subband b, for arbitrary user set Um=
{π1, π2, ..., πm}, where the permutation of user set is sorted
in decreasing order of the channel gain
gb(π1)> gb(π2)> ... > gb(πm),
The necessary condition for the existence of optimal solution to
problem P0is the weighting factor of user with lower channel
gain is larger than user with higher channel gain
w(π1)< w(π2)< ... < w(πm).
Proof: First we show by contradiction that a user set with
two users, U2={π1, π2}, in subband b, with gb(π2)< gb(π1)
and w(π2)≤w(π1). It is easily seen that it is optimal to
transmit with full power to only the user with the stronger
channel gain, however, this solution does not satisfy the
constraint (3), (6). Similarly, when the user set with more than
two users, the above conclusion is also established. Thus, for
user sets do not satisfy conditions in Lemma 1, it does not
exist power allocation solution satisfying constraints in (3),
(6).
Therefore, when scheduler enumerating user sets to obtain
optimal user set for subband b, users can be paired as user set
only if they satisfy conditions in Lemma 1. It can significantly
decrease the computation burden, because it does not need to
enumerate all possible user sets. In the followings, we assume
user set satisfy the necessary condition in Lemma 1.
B. Optimal Transmission Power Allocation
To solve problem P, scheduler needs to enumerate candi-
date user sets based on the proposed user pairing criterion.
Then scheduler calculate PF scheduling metric for every
candidate user set. Specifically, for a given user set Um, we
obtain optimal power allocation solution that maximizes the
weighted sum of instantaneous user throughput in problem
P0. However, problem P0is a non-convex problem. To make
the problem tractable, we first remove constraint (6) then we
obtain a relaxed problem of P0as
P1: max
p
fb(p|Um)
s.t.(2),(3).
Although problem P1is still difficult to solve, we can
use uplink-downlink duality to transform this problem into a
well-structured convex problem which can be solved by KKT
condition. Furthermore, if the optimal solution of problem
meets P1constraint (6), it is also the optimal solution of
original problem P0; if otherwise, problem P0does not have
optimal solution. Define function ˜
fb(q|Um)as
˜
fb(q|Um) =
m

k=1
w(πk)RUL
b(πk|Um;t),
where RUL
b(πk|Um;k) = Wlog21 + Bk−1g(πk)q(πk)is
uplink user throughput, term Bk= 1 + m
j=k+1 g(πj)q(πj).
The uplink power allocated can be denoted as vector
q= [q(π1), q(π2), ..., q(πm)]. Using uplink-downlink duality
in [12], [13], we can obtain the problem:
P2: max
q
˜
fb(q|Um)
s.t.q>0,
m

k=1
q(πk) = P,
In uplink, the decoding order should be in the order of
decreasing channel gain of users links. And, the transform
relationship between vector qand downlink power allocation
vector p, which is introduced in [14], [15] is:
p(πk) = Bk−1Akq(πk), k = 1,2, ..., m, (11)
where term Ak= 1 + g(πk)k−1
j=1 p(πj).
Theorem 1. Problem P1and problem P2are equivalent.
Proof: The equality between uplink rate and downlink
rate. First, the downlink rate of user can be represented as
RDL
b(πk|Um;t) = Wlog21 + Ak−1g(πk)p(πk).

4
Then, using transformation formula (11), we can deduce
RDL
b(πk|Um;t) = Wlog21 + Ak−1gb(πk)Bk−1Akq(πk)
=RUL
b(πk|Um;t).
The downlink-uplink transformation satisfy sum power con-
straint. For user πmin user set Um, formula (11) can be
rewritten as
p(πm) = B−1
mAmq(πm)
=q(πm)
1 + gb(πm)
m−1

j=1
p(πj)
.
By adding m−1
j=1 p(πj)to both sides, we get
m

j=1
p(πj) = q(πm)+ [1+gb(πm)q(πm)] ·
m−2

j=1
p(πj)
+p(πm−1) [1 + gb(πm)q(πm)] .
We can further show that
m

j=1
p(πj) =
m

j=m−1
q(πj)+Bm−2·
m−2

j=1
p(πj).
Thus we can obtain m
j=1 p(πj) = m
j=1 q(πj)by induction.
Theorem 2. The optimal solution q⋆of problem P2can be
obtained by solving equation set
q⋆
µ−1=A−1b,(12)
where A∈Rm×mis a row full rank matrix, vector b∈Rm×1
and scalar µis lagrange multiplier.
Proof: For simplicity, we rewrite the objective function
˜
fbof problem P2in more convenient form
˜
fb(q|Um) =
m

k=1
∆klog2
1 +
m

j=k
g(πj)q(πj)
,
∆k=w(πk), k = 1,
w(πk)−w(πk−1),others,
where ∆k>0due to user set satisfy the pre-defined user
pairing criterion. Because problem P2is a convex problem,
we may consider the KKT conditions, which are necessary
and sufficient for optimality. And, the Lagrangian function of
problem P2is given by
L(q,v, µ)= ˜
fb(q|Um)+
m

k=1
vkq(πk)−µm

k=1
q(πk)−P,
where the parameter vand µare dual variables, By differen-
tiating, we obtain the KKT conditions:
P−
m

k=1
q(πk)=0 ,q > 0,
vk≥0,1≤k≤m,
vkq(πk) = 0 ,1≤k≤m,
k

i=1
∆i
1 +
m

j=i
gb(πj)q(πj)
=µ
gb(πk),1≤k≤m.
(13)
For equation set (13), we can rewrite it in the standard form
m

j=1
q(πj) = P ,
m

j=1
gb(πj)q(πj)−∆1
µ/g(π1)=−1,
m

j=k
gb(πj)q(πj)−∆k
µ(1/g(πk)−1/g(πk−1)) =−1,2≤k≤m.























⇐⇒ Aq
µ−1=b.
As we can see, matrix Ais row full rank which means
the equation has unique solution. Thus, we can directly solve
equations set to obtain the optimal solutions q⋆
q⋆
µ−1=A−1b.
Furthermore, the optimal solution p⋆of P1can be obtained
by using uplink-to-downlink transformation (11). We obtain
optimal power allocation solution for Umin subband b, if p⋆
satisfy constraint (6).
C. Proposed Scheduling and Power Allocation Scheme
In conclusion, the scheduling for NOMA can be summa-
rized as: After channel and interference estimation at the
UE receiver side, the channel gain is calculated and fed
back to the eNB. At the eNB side, scheduler calculate the
average throughput per subband of each user in the cell.
Then, scheduler begins to allocate subbands to user sets. For
subband b, the scheduler enumerate candidate user sets satisfy
the predefined user-pairing criterion. For every candidate user
set, scheduler obtains its optimal power allocation solution and
PF scheduling metric by solving problem Pand then allocate
subband bto the user set with maximum PF scheduling metric.
Detailed procedure is in Algorithm 1.
IV. SIM UL ATION AND ANA LYSI S
A. Simulation Setup
Here we evaluate the performance of proposed user-pairing
criterion and optimal power allocation scheme in downlink
multi-cell scenario. Besides, we compare to NOMA using
conventional multi-user pairing and power allocation scheme.
We assume universal frequency reuse is used. Detailed sim-
ulation parameters including channel model and system as-
sumptions are summarized in Table I. Besides, we assume the

5
Algorithm 1 Multi-user scheduling and power allocation
Initial setting: For subband b, from all Kusers, after
channel and interference estimation at the UE side, the
channel gain is calculated and fed back to the eNB.
Step 1: Based on predefined multi-user pairing criterion
in Lemma 1, scheduler enumerate candidate user sets
{Um: 1 ≤m≤NM}.
Step 2: For every candidate user set, scheduler obtains its
optimal power allocation solution. For arbitrary candidate
user set Um:
Step 2.1: Solving equation set (12), if its solution
q⋆satisfy constraint q>0; if otherwise, go to Step 2.3.
Step 2.2: Using transformation (11), we obtain p⋆to
P2. If p⋆satisfy constraint (6), the optimal power allocation
solution for user set Umin subband bis obtained.
Step 2.3: If the optimal power allocation exists, then
return p⋆and Qb(Um); if otherwise, Qb(Um) = 0.
Step 3: Scheduler allocate subband bto user set Um
bwith
maximum PF scheduling metric Qb(Um).
TABLE I
SYS TEM PA RAM ET ERS
Parameter Assumption
Carrier frequency 2 GHz
Overall band width 10 MHz
Subband width 2 MHz
Number of subbands 5
Cellular layout Hexagonal grid/ 19 sites/ 3 sectors per site
Inter-site distance 500 m
User number 10 users per cell
User distribution Uniform in cell
Total eNB TX power 43 dBm
eNB/User antenna gain 8 dBi/ 0 dBi
Number of eNB/User antennas 1/2
Channel model 3GPP urban macro spatial channel model
Thermal noise density -174 dBm/Hz
Shadowing standard deviation 8 dB
Scheduling policy/Traffic model PF/ full buffer
Distance-dependent Path loss (dB)
P LLOS(R) = 103.4 + 24.2log10 (R)
P LNLOS(R) = 131.1 + 42.8log10 (R)
Rin km
ideal channel and unquantized estimation without a feedback
delay. The resource allocation is updated every subframe.
In the simulation,the parameter, NMis settled from 1 to 4.
In addition, the NMof one corresponds to the OMA. The
subband allocation is updated every 1 ms and the averaging
time window tcis set to 100 ms.
B. Results and Analysis
Fig. 2 shows the user SINR of NOMA is less than OMA.
There are two main reasons for this. Firstly, while SIC applied
at user side, users multiplexed in the power domain may bring
more interference than OMA. In addition, the pairing users
actually allocated less power compared to OMA.
In Fig. 3 and Table II show the user throughput of NOMA
achieves better throughput than OMA for the entire region
of the cumulative distribution. Furthermore, the maximum
achievable user throughput of NOMA is much larger than
OMA. Because users in NOMA can share the same subband,
the throughput of user is increasing mainly due to more
Fig. 2. CDF of downlink user SINR.
Fig. 3. CDF of downlink user throughput.
bandwidth resource allocation. Although the pairing users are
allocated less power than OMA users, the gain obtained by
more allocated bandwidth compensates the degradation due to
power allocation. Overall, NOMA achieves performance gain
compare to OMA. However, when NMis four, the achievable
gain is relatively small compared to NMis three. This implies
that the parameter NMshould be determined based on both
performance gain and implement cost depending on NM.
Fig. 4 shows the mean and cell-edge user throughput as a
function of NM. We can see that NOMA using the proposed
TABLE II
PERFORMANCE GAIN FOR NOMA US IN G PROP OS ED PO WER A LL OCATI ON
SC HEM E
OMA
(NM= 1)
NOMA
(NM= 2)
NOMA
(NM= 3)
NOMA
(NM= 4)
Average user
TP (bps/Hz) 1.268 1.745(↑37%) 2.211(↑74%) 2.319(↑82%)
Cell-edge user
TP(bps/Hz) 0.160 0.210(↑21%) 0.490(↑206%) 0.565(↑253%)

6
Fig. 4. Average and cell-edge user TP of proposed power allocation scheme
and FTPC in [6].
TABLE III
USER PAIRING COMPLEXITY OF PROPOSED POWER ALLOCATION SCHEME
AN D FTPC IN [6 ]
power
allocation
schemes
NM= 2 NM= 3 NM= 4
Average canadidate
user sets E[V]
proposed 240(↓10%) 650(↓75%) 1322(↓62%)
FTPC 264 1140 3508
power allocation scheme increases both the mean and cell-edge
user throughput simultaneously compared to OMA (NM= 1).
The results show that the average user throughput of proposed
power allocation scheme outperforms FTPC in [6]. In FTPC,
scheduler allocates more power to cell-edge users, the cell-
edge throughput of proposed scheme is a little lower than
FTPC when NMis 3 or 4.
Furthermore, in Table III, the user pairing complexity,
indicated by E[V], of proposed scheme is lower than FTPC.
And, with increased NM, the complexity of proposed scheme
is significantly lower than FTPC, especially, when NMis four,
the E[V]of proposed scheme is only 38 per cent of the FTPC.
V. CONCLUSIONS
This paper investigates joint user pairing and multi-user
power allocation of downlink NOMA. We utilize PF-based
radio resource (subbands and transmission power) allocation.
A pre-defined user pairing criterion is introduced to alleviate
computation burden. Furthermore, we propose the optimal
power allocation achieves the high throughput with low com-
plexity. Simulation results indicate that NOMA significantly
enhances the system-level throughput performance compared
to OMA. Besides, it also indicates that NOMA with proposed
multi-user power allocation scheme in this paper outperforms
existing sub-optimal scheme. Unlike conventional scheme,
proposed power allocation scheme can not obtain power
allocation solution for arbitrary user set. Because, for some
user sets, the corresponding power allocation problem does
not have solution. On the other hand, although user pairing
complexity is reduced, the whole complexity of the multi-user
scheduling is still high due to enumeration. These issues need
to be further studied.
VI. ACK NOWLEDGMENT
The work was supported by the China Natural Science
Funding (61271183), the National Key Technology R&D
Program of China under Grant 2014ZX03003011-004 and
the Fundamental Research Funds for the Central Universities
under Grant 2014ZD03-02.
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