Power and Channel Allocation for Non-
Orthogonal Multiple Access in 5G Systems:
Tractability and Computation
Lei Lei, Di Yuan, Chin Keong Ho and Sumei Sun
Journal Article
N.B.: When citing this work, cite the original article.
©2016 IEEE. Personal use of this material is permitted. However, permission to
reprint/republish this material for advertising or promotional purposes or for creating new
collective works for resale or redistribution to servers or lists, or to reuse any copyrighted
component of this work in other works must be obtained from the IEEE.
Lei Lei, Di Yuan, Chin Keong Ho and Sumei Sun, Power and Channel Allocation for Non-
Orthogonal Multiple Access in 5G Systems: Tractability and Computation, IEEE
TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2016. 15(12), pp.8580-8594.
http://dx.doi.org/10.1109/TWC.2016.2616310
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-134309
1
Power and Channel Allocation for Non-orthogonal
Multiple Access in 5G Systems: Tractability and
Computation
Lei Lei
1
, Di Yuan
1
, Chin Keong Ho
2
, and Sumei Sun
2
1
Department of Science and Technology, Link
¨
oping University, Sweden
2
Institute for Infocomm Research (I
2
R), A
∗
STAR, Singapore
Emails: {lei.lei; di.yuan@liu.se}, {hock; sunsm}@i2r.a-star.edu.sg
Abstract—A promising multi-user access scheme, non-
orthogonal multiple access (NOMA) with successive interference
cancellation (SIC), is currently under consideration for 5G
systems. NOMA allows more than one user to simultaneously
access the same frequency-time resource and separates multi-user
signals by SIC. These render resource optimization in NOMA
different from orthogonal multiple access. We provide theoretical
insights and algorithmic solutions to jointly optimize power
and channel allocation in NOMA. We mathematically formulate
NOMA resource allocation problems, and characterize and ana-
lyze the problems’ tractability under a range of constraints and
utility functions. For tractable cases, we provide polynomial-time
solutions for global optimality. For intractable cases, we prove the
NP-hardness and propose an algorithmic framework combining
Lagrangian duality and dynamic programming (LDDP) to deliver
near-optimal solutions. To gauge the performance of the solutions,
we also provide optimality bounds on the global optimum.
Numerical results demonstrate that the proposed algorithmic
solution can significantly improve the system performance in both
throughput and fairness over orthogonal multiple access as well
as over a previous NOMA resource allocation scheme.
Index Terms—Non-orthogonal multiple access, resource allo-
cation, successive interference cancellation, 5G.
I. INTRODUCTION
Orthogonal multi-user access (OMA) techniques are used
in 4G long term evolution (LTE) and LTE-Advanced (LTE-
A) networks, e.g., orthogonal frequency division multiple
access (OFDMA) for downlink and single-carrier frequency
division multiple access (SC-FDMA) for uplink [1], [2]. In
OMA, within a cell, each user has exclusive access to the
allocated resource blocks. Thus, each subchannel or subcarrier
can only be utilized by at most one user in every time slot.
OMA avoids intra-cell interference, and enables single-user
detection/decoding and simple receiver design. However, by
its nature, orthogonal channel access is becoming a limiting
factor of spectrum efficiency.
In the coming decade, mobile data traffic is expected to
grow thousand-fold [3], [4]. Accordingly, the network capacity
must dramatically increase for 5G systems. Capacity scaling
for 5G is enabled by a range of techniques and schemes, e.g.,
cell densification, utilization of unlicensed spectrum, and ad-
vanced radio access schemes. New multi-user access schemes
have been investigated as potential alternatives to OFDMA and
SC-FDMA [3]–[5]. A promising scheme is the so-called non-
orthogonal multiple access (NOMA) with successive inter-
ference cancellation (SIC) [6]. Unlike interference-avoidance
multiple access schemes, e.g., OFDMA, multiple users in
NOMA can be assigned to the same frequency-time resource
so as to improve spectrum efficiency [6]. On one hand, this
results in intra-cell interference among the multiplexed users.
On the other hand, some of the interfering signals in NOMA
can be eliminated by multi-user detection (MUD) with SIC
at the receiver side. To enable this process, more advanced
receiver design and interference management techniques are
considered for 5G networks [4].
A. Related Works
From an information theory perspective, under the assump-
tion of simultaneous multi-user transmission via superposition
coding with SIC, capacity region and duality analysis have
been studied in [7]. The authors of [8] have provided an
analysis of implementing interference cancellation in cellular
systems. Towards future 5G communication systems, some
candidate access schemes are under investigation in recent
research activities, e.g., sparse code multiple access (SCMA)
[3], and NOMA [6].
In [9], the authors studied the capacity region for NOMA. In
[10], by assuming predefined user groups for each subchannel,
a heuristic algorithm for NOMA power allocation in downlink
has been proposed, and system-level simulations have been
conducted. In [11], the authors address various implementation
issues of NOMA. The authors in [12] considered a sum-rate
utility maximization problem for dynamic NOMA resource
allocation. In [13], outage performance of NOMA has been
evaluated. The authors derived the ergodic sum rate and outage
probability to demonstrate the superior performance of NOMA
with fixed power allocation. In [14], fairness considerations
and a max-min fairness problem for NOMA have been ad-
dressed. The fairness in NOMA can be improved via using and
adapting the so-called power allocation coefficients. For uplink
NOMA, the authors in [15] provided a suboptimal algorithm
to solve an uplink scheduling problem with fixed transmission
power. In [16], a weighted multi-user scheduling scheme is
proposed to balance the total throughput and the cell-edge
2
user throughput. In [17], the authors proposed a greedy-based
algorithm to improve the throughput in uplink NOMA. In
[18], the authors studied and evaluated user grouping/pairing
strategies in NOMA. It has been shown that, from the outage
probability perspective, it is preferable to multiplex users of
large gain difference on the same subcarrier. In [19], the
authors address subcarrier allocation and power assignment in
downlink NOMA, with the objective of balancing the through-
out with the number of scheduled users. The solution approach
uses matching theory, by first assuming equal power split and
applying a user-subchannel matching algorithm that converges
to a stable matching, followed by a water-filling phase for
power allocation. In [20], a monotonic optimization method
is developed for NOMA subcarrier and power allocation. The
method potentially approaches the global optimum, at the cost
of high complexity in the number of users per subcarrier. We
remark that there are other setups of SIC than that considered
in NOMA. An example is the interference channel in which
common information is transmitted for partial interference
cancellation, for which Etkin et al. [21] provided an analysis of
the resulting capacity region and trade-off from an information
theory perspective.
We note that in some studies of NOMA (e.g., [10], [14],
[18]), the aspect of subcarrier or subchannel allocation was
simplified or not addressed. However, to fully reach the
potential of NOMA, user-subchannel allocation is of high
significance due to fading. Assuming uniform subchannels
or using fixed rules for subchannel allocation may result in
significant performance loss. The importance of subchannel
allocation is evidenced by the growing interest in explicitly
taking this aspect into account in some recent works [12],
[19], [20].
Apart from investigation of NOMA performance in cellular
networks, from an optimization perspective, the complexity
and tractability analysis of NOMA resource allocation is of
significance. Here, tractability for an optimization problem
refers to whether or not any polynomial-time algorithm can be
expected to find the global optimum [1]. Tractability results
for resource allocation in OMA and interference channels have
been investigated in a few existing works, e.g., [1], [22], [23]
for OFDMA, [2] for SC-FDMA, and [23], [24] for interference
channel. For NOMA, to the best of our knowledge, no such
study is available in the existing literature.
B. Contributions
In spite of the existing literature of performance evaluation
for NOMA, there is lack of a systematic approach for NOMA
resource allocation from a mathematical optimization point of
view. The existing resource allocation approaches for NOMA
are typically carried out with fixed power allocation [13],
[15], [17], predefined user set for subchannels [10], or pa-
rameter tuning to improve performance, e.g., updating power
allocation coefficients [14]. Moreover, compared with OMA,
NOMA allows multi-user sharing on the same subchannel,
thus provides an extra dimension to influence the performance
in throughput and fairness. However, how to balance these
two key performance aspects in power and channel allocation
is largely not yet studied in the literature. In addition, little
is known on the computational complexity and tractability of
NOMA resource allocation.
In this paper, the solutions of joint channel and power
allocation for NOMA are subject to systematic optimization,
rather than using heuristics or ad-hoc methods. To this end, we
formulate, analyze, and solve the power and channel optimiza-
tion problem for downlink NOMA systems, taking into ac-
count practical considerations of fairness and SIC. We present
the following contributions. First, for maximum weighted-
sum-rate (WSR) and sum-rate (SR) utilities, we formulate
the joint power and channel allocation problems (JPCAP)
mathematically. Second, we prove the NP-hardness of JPCAP
with WSR and SR utilities. Third, we identify tractable cases
for JPCAP and provide the tractability analysis. Fourth, we
propose an algorithmic framework based on Lagrangian du-
ality and dynamic programming to facilitate problem solving.
Unlike previous works, our approach contributes to delivering
near-optimal solutions, as well as performance bounds on
global optimum to demonstrate the quality of our near-optimal
solutions. We use numerical results to illustrate the significant
performance improvement of the proposed algorithm over
existing NOMA and OFDMA schemes.
Our work extends previous study of user grouping in
NOMA. In [18], the number of users to be multiplexed on
a subcarrier is fixed, and performance evaluation consists
of rule-based multiplexing policies. In our case, for each
subcarrier, the number of users and their composition are
both output from solving an optimization problem. Later in
Section VII, results of optimized subcarrier assignment and
user grouping will be presented for analysis. The current paper
extends our previous study [12] in several dimensions. The
extensions consist of the consideration of the WSR utility
metric, a significant amount of additional theoretical analysis
of problem tractability, the development of the performance
bound on global optimality, as well as the consideration of
user fairness in performance evaluation.
The rest of the paper is organized as follows. Section II gives
the system models for single-carrier and multi-carrier NOMA.
Section III formulates JPCAP for WSR utility and provides
complexity analysis. Section IV analyzes the tractability for
special cases of JPCAP. In Section V, we provide the tractabil-
ity analysis for relaxations of JPCAP. Section VI proposes an
algorithmic framework. Numerical results are given in Section
VII. Conclusions are given in Section VIII.
II. SYSTEM MODEL
A. Basic Notation
We consider a downlink cellular system with a base station
(BS) serving K users. The overall bandwidth B is divided into
N subchannels, each with bandwidth B/N. Throughout the
paper, we refer to subchannel interchangeably with subcarrier.
We use K and N to denote the sets of users and subchannels,
respectively, and g
kn
to denote the channel gain between the
BS and user k on subcarrier n. Let p
kn
be the power allocated
to user k on subcarrier n. A user k is said to be multiplexed on
a subchannel n, if and only if p
kn
> 0. The power values are
3
subject to optimization. At the receiver, each user equipment
has MUD capabilities to perform multi-user signal decoding.
With SIC, some of the co-channel interference will be treated
as decodable signals instead of as additive noise.
B. NOMA Systems
To ease the presentation of the system model, for the
moment let us consider the case that all the K users can
multiplex on each subcarrier n in a multi-carrier NOMA
system (MC-NOMA) at downlink. For each subcarrier n, we
sort the users in set K in the descending order of channel
gains, and use bijection b
n
(k): K 7→ {1, 2, . . . , K} to represent
this order, where b
n
(k) is the position of user k ∈ K in the
sorted sequence. For our downlink system scenario, in [25]
(Chapter 6.2.2, pp. 238) it is shown that, with superposition
coding, a user can decode the data of another user with worse
channel gain, and this is not constrained by the specific power
split. The reason is that the first user, due to its better receiving
condition than the second user, can decode any data that the
second user can successfully decode. Consider one subcarrier
and two users k and h with gain g
k
> g
h
. User h does not
perform SIC, and its rate equals log(1 +
p
h
g
h
p
k
g
h
+η
), where η
denotes the noise, and p
k
and p
h
are the power levels. For
user k with better gain g
k
, the SINR for the data for user
h is
p
h
g
k
p
k
g
k
+η
>
p
h
g
h
p
k
g
h
+η
. Hence user k can decode the data
(at the rate governed by the right-hand side of the inequality)
of user h, and this is not dependent on the power relation.
Thus user k is able to perform SIC, by subtracting the re-
encoded signal intended for receiver h from the composite
signal. That is, user k on subcarrier n, before decoding its
signal of interest, first decodes the received interfering signals
intended for the users h ∈ K\{k} that appear later in the
sequence than k, i.e., b
n
(h) > b
n
(k). The interfering signals
with order b
n
(h) < b
n
(k) will not be decoded and thus treated
as noise. Hence, the interference after SIC for user k on
subcarrier n is
P
h∈K\{k}:b
n
(h)<b
n
(k)
p
hn
g
kn
, ∀k ∈ K, ∀n ∈ N . If
there are users having the same channel gain, then SIC applies
following the principle in [25] (Chapter 6.2.2), provided that
an ordering of the users is given. From the discussion, the
SINR of user k on subcarrier n is given below.
SINR
kn
=
p
kn
g
kn
P
h∈K\{k}:b
n
(h)<b
n
(k)
p
hn
g
kn
+ η
(1)
The noise power here equals the product of the power spectral
density of white Gaussian noise and the subcarrier bandwidth.
The rate of each user in NOMA is determined by the user’s
SINR after SIC. Thus, the achievable rate of user k on
subcarrier n is R
kn
= log(1 + SINR
kn
) nat/s with normalized
bandwidth
B
N
= 1.
For single-carrier NOMA systems (SC-NOMA), we omit
the subcarrier index. For convenience, the users k ∈
{1, . . . , K} in SC-NOMA are defined in the descending order
of channel gains, where g
1
≥ g
2
≥, . . . , ≥ g
K
. Thus the
user index also represents its position in the sequence, and
user k is able to decode the signal of user h if k < h.
We define SINR
k
=
p
k
g
k
P
h∈K\{k}:h<k
p
h
g
k
+η
, ∀k ∈ K. The
achievable rate of user k is R
k
= log(1 + SINR
k
) nat/s
with normalized bandwidth B = 1. Following the discussion
earlier, for two users 1 and 2 with g
1
> g
2
, the achievable
rates are log(1 +
p
1
g
1
η
) and log(1 +
p
2
g
2
p
1
g
2
+η
), respectively.
We use U
n
as a generic notation for the set of users multi-
plexed on subchannel n for MC-NOMA. For SC-NOMA, the
corresponding entity is denoted by U. We use M, 1 ≤ M ≤
K, to denote the maximum number of multiplexed users on a
subcarrier. The reason of having this parameter is to address
complexity considerations of implementing MUD and SIC. In
NOMA, the system complexity increases by M, because a
user device needs to decode up to M signals. The setting
of M depends on receiver’s design complexity and signal
processing delay for SIC [4], [8]. For practical implementation,
M is typically smaller than K. However, our optimization
formulations and the solution algorithm are applicable to
any value of M between one and K. We also remark that
the benefits of NOMA come with signaling overhead that is
necessary to facilitate SIC. Although signaling is outside the
scope of the current paper, it is of significance in practical
implementation. On the other hand, it has been shown in [11]
that NOMA remains superior to OMA in throughout when the
signaling overhead is accounted for. In addition, parameter M,
which limits the number of users per subcarrier, provides an
effective way to control the signaling cost. By the results to
be presented in Section VII, most of the improvements due to
NOMA is achieved for small M.
Two utility functions, WSR and SR, are considered
in this paper. The WSR utility is denoted by f
W
=
P
k∈K
w
k
P
n∈N
R
kn
, where w
k
is the weight coefficient
of user k ∈ K. Clearly, the selection of the weights has
strong influence on the resource allocation among the users.
In general, the weights can be used to steer the resource
allocation towards various goals, such as to implement ser-
vice class priority of users, and fairness (e.g., a user with
averagely poor channel receives higher weight). In our work,
the algorithmic approach is applicable without any assumption
of the specific weight setting. For performance evaluation, we
set the weights following proportional fairness. That is, for
one time slot, a user’s weight is set to be the reciprocal of
the average user rate prior to the current time slot [15]. As
a result, the resource allocation will approach proportional
fairness over time. The SR utility, a special case of WSR,
is defined as f
R
=
P
k∈K
P
n∈N
R
kn
. The term of SR utility
is used interchangeably throughput in this paper. For both SR
and WSR, SIC with superposition coding [25] applies to the
users multiplexed on the same subcarrier. As was discussed
earlier, the decoding does not rely on assuming specific, a
priori constraint on the power allocation among the users.
III. JOINT POWER AND CHANNEL ALLOCATION
In this section, we formulate JPCAP using WSR utility for
MC-NOMA. We use W-JPCAP to denote the optimization
problem. In general, JPCAP amounts to determining which
users should be allocated to which subcarriers, as well as
the optimal power allocation such that the total utility is
maximized. In the following we define the variables and
4
formulate W-JPCAP as P 1
WSR
below, where all p-variables
and x-variables are collected in vectors p and x, respectively.
p
kn
= allocated power to user k on subcarrier n.
x
kn
=
1 if user k is multiplexed on subcarrier n,
i.e., p
kn
> 0,
0 otherwise.
P 1
WSR
: max
x,p
X
k∈K
w
k
X
n∈N
R
kn
x
kn
(2a)
s.t.
X
k∈K
X
n∈N
p
kn
≤ P
tot
(2b)
X
n∈N
p
kn
≤ P
k
, ∀k ∈ K (2c)
X
k∈K
x
kn
≤ M, ∀n ∈ N (2d)
In P 1, the objective (2a) is to maximize the WSR utility,
where R
kn
is defined below.
R
kn
= log(1 +
p
kn
g
kn
P
h∈K\{k}:
b
n
(h)<b
n
(k)
p
hn
g
kn
+ η
), ∀k ∈ K, ∀n ∈ N
(3)
Constraints (2b) and (2c) are respectively imposed to ensure
that the total power budget and the individual power limit
for each user are not exceeded. The per-user power limit P
k
is introduced for practical considerations, such as regulatory
requirement on power towards a user device. Such a limit
is also very common in OMA (e.g., [23], [24]). Constraints
(2d) restrict the maximum number of multiplexed users on
each subcarrier to M . We remark that the power allocation is
represented by the p-variables of which the values are subject
to optimization, whereas the power limits P
tot
and P
k
, k ∈ K
are given entities. Suppose M users, say users 1, . . . M, are
allocated with positive power on channel n. If P
tot
≥
M
P
k=1
P
k
happens to hold, then all the M users may be at their respective
power limits, i.e., setting p
kn
= P
k
, k = 1, . . . , M , is feasible.
Otherwise, the M users can still be allocated with positive
power, though not all of them can be at the power limits.
Indeed, if user k is allocated with power p
kn
> 0 on channel
n, then typically p
kn
< P
k
unless user k is not allocated
power on any other channel than n. For the total power limit
P
tot
to be meaningful, one can assume
P
k∈K
P
k
> P
tot
without loss of generality, because otherwise the total power
limit P
tot
is not violated even if all users are allocated with
their respective maximum power, that is, (2b) becomes void
and should be dropped. We do not consider any further specific
assumptions on the relation between P
tot
and P
k
, k ∈ K, to
keep the generality of the system model.
Remark. We do not explicitly impose the constraint that
p
kn
> 0 if and only if x
kn
= 1. This is because setting
p
kn
> 0 and x
kn
= 0 is clearly not optimal, by the facts that
p
kn
> 0 will lead to rate degradation of other users due to the
co-channel interference (if there are other users on channel n),
and that for user k, p
kn
> 0 means power is consumed, but
x
kn
= 0 means no benefit as the rate in (2a) becomes zero.
Formulation P1
WSR
is non-linear and non-convex. The con-
cavity of the objective function (2a) cannot be established in
general, because of the presence of the binary x-variables and
the product of x and p. However, in complexity analysis, nei-
ther non-convexity nor non-linearity of a formulation proves
the problem’s hardness, as a problem could be inappropriately
formulated. Therefore, we provide formal hardness analysis
for W-JPCAP below.
Theorem 1. W-JPCAP is NP-hard.
Proof: We establish the result in two steps. First, we
conclude that if M = 1 in (2d), W-JPCAP is NP-hard, as
it reduces to OFDMA subcarrier and power allocation, for
which NP-hardness is provided in [22] and [23]. For general
MC-NOMA with M > 1, we construct an instance of W-
JPCAP and establish the equivalence between the instance
and the OFDMA problem considered in [23]. We consider
an instance of W-JPCAP with K users, N subcarriers, and
M = 2. Let denote a small value with 0 < <
1
e
KN
. The
total power P
tot
is set to NKP
k
. The power limit P
k
= 1 is
uniform for ∀k ∈ K, and the noise parameter η = . Among
the K users, we select an arbitrary one, denoted by
¯
k ∈ K,
and assign a dominating weight w
¯
k
= e
KN
and channel gain
g
¯
kn
= 1 on all the subcarriers, whereas the other users’
weights and channel gains are w
k
= and g
kn
≤
1
e
KN
,
∀k ∈ K\{
¯
k} and ∀n ∈ N . From above, the ratios
w
¯
k
w
k
and
g
¯
kn
g
kn
are sufficiently large such that allocating any power
p ≤ P
¯
k
to user
¯
k on any subcarrier n, the utility w
¯
k
R
¯
kn
>
max(
P
k∈K\{
¯
k}
P
n∈N
w
kn
R
kn
) for using the same power
budget p, since
P
k∈K\{
¯
k}
P
n∈N
w
kn
R
kn
is bounded by
KNe
−KN
log(1 +
e
−KN
p
), and w
¯
k
R
¯
kn
= e
KN
log(1 +
p
)
is clearly greater than KN e
−KN
log(1 +
e
−KN
p
). Thus,
allocating power to user
¯
k rather than other users is preferable
for maximizing utility.
Due to the uniform gain g
¯
kn
and the dominating weight
w
¯
k
for user
¯
k on all channels, the optimal power allocation
for user
¯
k is to uniformly allocate an amount of
P
¯
k
N
to each
subcarrier. Then the remaining problem is to allocate power
P
tot
−P
¯
k
= (NK −1)P
k
to the remaining K −1 users. Every
user k ∈ K\{
¯
k} is still subject to constraint (2c). Note that for
M = 2, each subcarrier now can accommodate one extra user
at most. Compared to the OFDMA problem in [23], W-JPCAP
has one extra total power constraint, i.e., (2b), however, recall
that P
tot
is set to NKP
k
, and for this value (2b) is in fact
redundant. Therefore, a special case of W-JPCAP with M > 1
is equivalent to the OFDMA problem in [23], and the result
follows.
IV. TRACTABILITY ANALYSIS FOR UNIFORM WEIGHTS
The hardness of W-JPCAP could have stemmed from sev-
eral sources, e.g., the structure of the utility function, discrete
variables, non-concave objective, and the constraints. We start
from investigating how the weight in the utility function
influences the problem’s tractability. The utility function can
affect the computational complexity in problem solving [22],
[24]. One example is that the SR maximization problem
with total power constraint in OFDMA is polynomial-time
