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An Optimization Perspective of the Superiority of NOMA
Compared to Conventional OMA
DOI:
10.1109/TSP.2017.2725223
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Chen, Z., Ding, Z., Dai, X., & Zhang, R. (2017). An Optimization Perspective of the Superiority of NOMA Compared
to Conventional OMA. IEEE Transactions on Signal Processing. https://doi.org/10.1109/TSP.2017.2725223
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. X, OCT. 2016 1
An Optimization Perspective of the Superiority of
NOMA Compared to Conventional OMA
Zhiyong Chen, Zhiguo Ding, Member, IEEE, Xuchu Dai, Rui Zhang, Fellow, IEEE
Abstract—Existing work regarding the performance com-
parison between non-orthogonal multiple access (NOMA) and
orthogonal multiple access (OMA) can be generally divided
into two categories. The work in the first category aims to
develop analytical results for the comparison, often with fixed
system parameters. The work in the second category aims to
propose efficient algorithms for optimizing these parameters,
and compares NOMA with OMA by computer simulations.
However, when these parameters are optimized, the theoretical
superiority of NOMA over OMA is still not clear. Therefore,
in this paper, the theoretical performance comparison between
NOMA and conventional OMA systems is investigated, from
an optimization point of view. Firstly, sum rate maximizing
problems considering user fairness in both NOMA and various
OMA systems are formulated. Then, by using the method of
power splitting, a closed-form expression for the optimum sum
rate of NOMA systems is derived. Moreover, the fact that NOMA
can always outperform any conventional OMA systems, when
both are equipped with the optimum resource allocation policies,
is validated with rigorous mathematical proofs. Finally, computer
simulations are conducted to validate the correctness of the
analytical results.
Index Terms—Non-orthogonal multiple access (NOMA), or-
thogonal multiple access (OMA), power allocation, optimization.
I. INTRODUCTION
R
ECENTLY, non-orthogonal multiple access (NOMA)
has received extensive research interests due to its supe-
rior spectral efficiency compared to conventional orthogonal
multiple access (OMA) [1]–[3]. For example, NOMA has
been proposed to downlink scenarios in 3rd generation part-
nership project long-term evolution (3GPP-LTE) systems [4].
Moreover, NOMA has also been anticipated as a promising
multiple access technique for the next generation cellular
communication networks [5], [6].
Conventional multiple access techniques for cellular com-
munications, such as frequency-division multiple access
Manuscript received XXX XX, 2016. The associate editor coordinating the
review of this paper and approving it for publication was XXX.
This work was supported in part by the National Natural Science Foundation
of China (No. 61471334). The work of Z. Ding was supported by the Royal
Society International Exchange Scheme and the UK EPSRC under grant
number EP/N005597/1.
Z. Chen and X. Dai are with the Key Laboratory of Wireless-Optical
Communications, Chinese Academy of Sciences, School of Information
Science and Technology, University of Science and Technology of China.
Address: No. 96 Jinzhai Road, Hefei, Anhui Province, 230026, P. R. China
(email: zhiyong@mail.ustc.edu.cn, daixc@ustc.edu.cn).
Z. Ding is with the School of Computing and Communications, Lancaster
University, LA1 4YW, U.K. (email: z.ding@lancaster.ac.uk).
R. Zhang is with the Department of Electrical and Computer Engineering,
National University of Singapore, Singapore 117583, Singapore (e-mail:
elezhang@nus.edu.sg).
Digital Object Identifier XXXX/XXXX
(FDMA) for the first generation (1G), time-division multiple
access (TDMA) for the second generation (2G), code-division
multiple access (CDMA) used by both 2G and the third gener-
ation (3G), and orthogonal frequency division multiple access
(OFDMA) for 4G, can all be categorized as OMA techniques,
where different users are allocated to orthogonal resources,
e.g., time, frequency, or code domain to avoid multiple access
interference. However, these OMA techniques are far from the
optimality, since the spectrum resource allocated to the user
with poor channel conditions cannot be efficiently used.
To tackle this issue and further improve spectrum efficiency,
the concept of NOMA is proposed. The implementation of
NOMA is based on the combination of superposition coding
(SC) at the base station (BS) and successive interference
cancellation (SIC) at users [1], which can achieve the opti-
mum performance for degraded broadcast channels [7], [8].
Specifically, take a two-user single-input single-output (SISO)
NOMA system as an example. The BS serves the users at
the same time/code/frequency channel, where the signals are
superposed with different power allocation coefficients. At the
user side, the far user (i.e., the user with poor channel condi-
tions) decodes its message by treating the other’s message as
noise, while the near user (i.e., the user with strong channel
conditions) first decodes the message of its partner and then
decodes its own message by removing partner’s message from
its observation. In this way, both users can have full access to
all the resource blocks (RBs), moreover, the near user can
decode its own information without any interference from
the far user. Therefore, the overall performance is enhanced,
compared to conventional OMA techniques.
A. Related Literature
As a promising multiple access technique, NOMA and its
variants have attracted considerable research interests recently.
The authors in [1] firstly presented the concept of NOMA
for cellular future radio access, and pointed out that NOMA
can achieve higher spectral efficiency and better user fairness
than conventional OMA. In [2], the performance of NOMA
in a cellular downlink scenario with randomly deployed users
was investigated, which reveals that NOMA can achieve
superior performance in terms of ergodic sum rates. In [9], a
cooperative NOMA scheme was proposed by fully exploiting
prior information at the users with strong channels about
the messages of the users with weak channels. The impact
of user pairing on the performance of NOMA systems was
characterized in [10]. In [11], a new evaluation criterion was
developed to investigate the performance of NOMA, which
2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. X, OCT. 2016
shows that NOMA can outperform OMA in terms of the sum
rate, from an information-theoretic point of view.
Recently, various NOMA schemes have been proposed for
practical wireless communication systems, including sparse
code multiple access (SCMA) [12]–[14], multi-user shared
access (MUSA) [15] and pattern-division multiple access
(PDMA) [16]. These schemes are generally called code-
domain multiplexing in [6], since the users in these schemes
are multiplexed over the same time-frequency resources, but
are assigned different codes. Note that multiplexing in PDMA
can be carried out in both the code domain and spatial domain.
In addition to these code-domain NOMA schemes mentioned
above, some other NOMA schemes have also been investi-
gated. For example, in [17], bit-division multiplexing (BDM),
which allocates a certain amount of bits out of multiple
symbols as a sub-channel at bit level by exploiting the inherent
characteristic of multiple unequal error protection levels in
high-order modulation, was proposed and implemented for
scalable video broadcasting (SVB). In [18], interleave division
multiple access (IDMA), which performs interleaving of chips
after symbols are multiplied by spreading sequences, was
investigated and compared to direct sequence code division
multiple access (DS-CDMA), in terms of performance and
complexity.
To further improve spectral efficiency, the combination
of NOMA and multiple-input multiple-output (MIMO) tech-
niques, namely MIMO-NOMA, has also been extensively
investigated. In [19], a new design of precoding and detection
matrices for MIMO-NOMA was proposed. A novel MIMO-
NOMA framework for downlink and uplink transmission was
proposed by applying the concept of signal alignment in
[20]. To characterize the performance gap between MISO-
NOMA and optimal dirty paper coding (DPC), a novel
concept termed quasi-degradation for multiple-input single-
output (MISO) NOMA downlink was introduced in [21].
Then, the theoretical framework of quasi-degradation was fully
established in [22], including the mathematical proof of the
properties, necessary and sufficient condition, and occurrence
probability. Consequently, practical algorithms for multi-user
downlink MISO-NOMA systems were proposed in [23], by
taking advantage of the concept of quasi-degradation. Lately,
to optimize the overall bit error ratio (BER) performance of
MIMO-NOMA downlink, an interesting transmission scheme
based on minimum Euclidean distance (MED) was proposed
in [24].
B. Contributions
Recently, extensive efforts have been spent to identify the
superiority of NOMA over OMA, and these existing work
can be divided into two categories. The work in the first
category, e.g., [9], [10], aims to develop analytical results for
the comparison between NOMA and OMA, and often relies on
the use of fixed system parameters, such as power allocation
coefficients and other bandwidth resources. The work in the
second category, e.g., [25], [26], aims to propose efficient
algorithms to optimize these system parameters. However, the
obtained complicated solutions are not in closed-form expres-
sions and hence cannot be used for the analytical comparison
directly. As a result, computer simulations are often used
for the performance comparison. Therefore, a theoretic study
of the superiority of NOMA over OMA, when the system
parameters are optimized in both cases, is still missing. To
bridge the gap between the two categories, in this paper,
the theoretical performance comparison between NOMA and
OMA is evaluated, from an optimization point of view, where
optimal resource allocation is carried out to both multiple
access schemes.
Especially, two kinds of OMA systems are considered, i.e.,
OMA-TYPE-I and OMA-TYPE-II, which represent, respec-
tively, OMA systems with optimum power allocation and fixed
time/frequency allocation, and OMA systems with both opti-
mum power and time/frequency allocation. The contributions
of this paper can be summarized as follows.
1) The sum rate maximizing problems for both NOMA
and OMA systems are formulated, with consideration of
user fairness. Particularly, instead of using simple OMA
with fixed system parameters, more sophisticated OMA
schemes with joint power and time/frequency optimiza-
tion are considered.
2) The closed-form expression for the optimum sum rate
for NOMA systems is given, by taking advantage of the
power splitting method.
3) By deriving and analysing the minimum required power
of different systems, it is pointed out that the minimum
required power of NOMA is always smaller than or at
least equal to that of both OMA-TYPE-I and OMA-
TYPE-II systems.
4) The fact that the optimum sum rate of NOMA systems
is always larger than or equal to that of both OMA-
TYPE-I and OMA-TYPE-II systems with various user
fairness considerations is validated by rigorous mathe-
matical proofs.
C. Organization
The remainder of this paper is organized as follows. Sec-
tion II briefly describes the system model and the problem
formulation. Section III provides the optimal power allocation
policies as well as their performance comparison. Simulation
results are given in Section IV, and Section V summarizes this
paper.
II. PROBLEM FORMULATION
Consider a downlink communication system with one BS
and K users, where the BS and all the users are equipped with
a single antenna. By using NOMA transmission, the received
signal at user i is
y = h
i
x + n
i
, i = 1, 2, ..., K, (1)
where h
i
denotes the channel coefficient, and n
i
∼ CN(0, N
0
)
is the additive white Gaussian noise (AWGN) at user i. x =
K
i=1
√
P
i
s
i
is the superposition of s
i
’s with power allocation
policy P = {(P
1
, P
2
, ..., P
K
)|
K
i=1
P
i
= P }, s
i
represents
the data intended to convey to user i, P
i
denotes the power
allocated to user i, and P denotes the total power constraint.
CHEN et al.: AN OPTIMIZATION PERSPECTIVE OF THE SUPERIORITY OF NOMA COMPARED TO CONVENTIONAL OMA 3
For ease of analysis, we assume that |h
1
| ≥ |h
2
| ≥ ... ≥ |h
K
|
and the total bandwidth is normalized to unity in this paper.
In consideration of user fairness, herein, we introduce
the minimum rate constraint r
∗
. Mathematically, the power
allocation policy should guarantee the following constraint:
min
i
r
i
≥ r
∗
,
where r
i
is the achievable rate of user i in nats/second/Hz,
which is given by
r
i
= ln
1 +
P
i
|h
i
|
2
N
0
+ |h
i
|
2
i−1
j=1
P
j
. (2)
For the special case of i = 1, the summation in the denomi-
nator becomes 0, and the corresponding rate becomes
r
1
= ln
1 +
P
i
|h
i
|
2
N
0
.
Note that r
i
is achievable since the channels are ordered and
the user with strong channels can decode those messages sent
to the users with weaker channels.
Therefore, the optimization problem of maximizing the total
sum rate with the user fairness constraint for NOMA systems
can be formulated as follows:
R
N
, max
P
i
K
i=1
r
i
s.t. r
i
= ln
1 +
P
i
|h
i
|
2
N
0
+ |h
i
|
2
i−1
j=1
P
j
,
K
i=1
P
i
≤ P,
min
i
r
i
≥ r
∗
.
(3)
In traditional OMA systems, e.g., frequency division multi-
ple access (FDMA) or time division multiple access (TDMA),
time/frequency resource allocation is non-adaptively fixed, i.e.,
each user is allocated with a fixed sub-channel. For notational
simplicity, we refer to this type of OMA as OMA-TYPE-I in
this paper. Consequently, to optimize the power allocations,
the optimization problem of OMA-TYPE-I assuming equal
resource (time or frequency) allocation to all users can be
formulated as follows:
R
O1
, max
P
i
K
i=1
r
i
s.t. r
i
=
1
K
ln
1 +
P
i
|h
i
|
2
N
0
/K
,
K
i=1
P
i
≤ P,
min
i
r
i
≥ r
∗
.
(4)
Since the sub-channel allocations among users are not
optimized, some users may suffer from poor channel condi-
tions due to large path loss and random fading. Thus, the
optimization problem for jointly designing power and sub-
channel allocations is considered next. Specifically, the total
time/frequency is divided into N sub-channels to be orthog-
onally shared by K users (N ≥ K), and this optimization
problem can be formulated as follows:
R
OX
, max
P
i,n
,S
i
K
i=1
n∈S
i
r
i,n
s.t. r
i,n
=
1
N
ln
1 +
P
i,n
|h
i,n
|
2
N
0
/N
,
N
n=1
K
i=1
P
i,n
≤ P,
P
i,n
≥ 0, ∀i, n
n∈S
i
r
i,n
≥ r
∗
,
S
1
, S
2
, ..., S
K
are disjoint,
S
1
∪ S
2
∪ ... ∪ S
K
= {1, 2, ..., N},
(5)
where P
i,n
and h
i,n
are the power allocated to and the channel
coefficient of user i’s sub-channel n, respectively. S
i
is the set
of indices of sub-channels assigned to user i.
Note that the optimization problem in (5) is not a con-
vex problem. Fortunately, if we assume that h
i,n
= h
i
,
the optimization problem in (5) can be upper-bounded by
the following optimization problem by replacing the discrete
time/frequency allocation with a continuous one as follows:
R
O2
, max
P
i
,α
i
K
i=1
r
i
s.t. r
i
= α
i
ln
1 +
P
i
|h
i
|
2
α
i
N
0
,
K
i=1
P
i
≤ P,
min
i
r
i
≥ r
∗
,
K
i=1
α
i
= 1.
(6)
For notational simplicity, in this paper , we refer to the OMA
system with the optimization given in (6) as OMA-TYPE-II.
It is also important to point out that the optimization problem
in (5) has been well studied during the last two decades,
while the optimization problem in (6) is not often considered
in the literature. In particular, problem (6) is different from
the classical optimization problem for power and channel
allocation in OMA systems, which is studied in problem (5).
Particularly, problem (6) is the optimization problem for joint
power and time/frequency allocation in OMA systems. The
difference between problems (5) and (6) is that, taking FDMA
for example, the bandwidth of each subchannel in problem (5)
is fixed, while it needs to be optimized in problem (6), i.e., in
problem (6), the width of each frequency channel is allowed
to be dynamically changed.
Note that the optimization problems in (4) and (6) are
applicable to both TDMA and FDMA, due to the fact that
over all user orthogonal time slots the energy conservation
K
i=1
α
i
P
i
α
i
= P is established in TDMA and the effective
noise power becomes αN
0
in FDMA.
4 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. X, OCT. 2016
By observing the definitions of the three kinds of OMA
systems, it is implied that
R
O1
≤ R
OX
≤ R
O2
.
Therefore, to show the superiority of NOMA compared to
OMA, we only need to prove that
R
N
≥ R
O2
.
However, to dig out more sophisticated properties of these
OMA systems, OMA-TYPE-I and OMA-TYPE-II are both
considered in this paper. Moreover, different mathematical
skills need to be employed to prove the superiority of NOMA
compared to OMA-TYPE-I and OMA-TYPE-II, respectively.
III. OPTIMAL PERFORMANCE ANALYSIS
A. Closed-form Solution of NOMA
The optimum closed-form solution of NOMA is given in
Theorem 1.
Theorem 1. Given P and r
∗
, if
P
∗
N
, (e
r
∗
− 1)N
0
K−1
i=0
e
ir
∗
|h
K−i
|
2
≤ P, (7)
then, the optimization problem in (3) is feasible, and the
optimal solution can be written as
R
N
= Kr
∗
+ ∆r
N
, (8)
where
∆r
N
= ln
1 +
(P − P
∗
N
)|h
1
|
2
N
0
e
Kr
∗
. (9)
Proof: Following the idea introduced in [27], we split
the total power into two parts, 1) the minimum power for
supporting the minimum rate transmission, denoted by P
∗
N
,
2) the excess power, denoted by ∆P
N
. Denote the minimum
power for maintaining minimum rate transmission and the
excess power of user i by P
∗
i
and ∆P
i
, respectively. The
minimum power P
∗
i
is defined as follows. If all users are
allocated their minimum powers, then all users will achieve
the minimum rate. Mathematically, P
∗
i
is defined as
r
∗
= ln
1 +
P
∗
i
N
0
|h
i
|
2
+
j<i
P
∗
j
. (10)
Then, we have the following equalities.
P
i
= P
∗
i
+ ∆P
i
, P
∗
N
=
K
i=1
P
∗
i
,
∆P
N
=
K
i=1
∆P
∗
i
, P = P
∗
N
+ ∆P
N
.
(11)
It follows from the definition that the minimum power of
each user can be given by
P
∗
i
= (e
r
∗
− 1)
N
0
|h
i
|
2
+
j<i
P
∗
j
. (12)
Therefore, we can obtain the following expression for the sum
power of the minimum power P
∗
i
P
∗
N
=
K
i=1
P
∗
i
= (e
r
∗
− 1)N
0
K−1
i=0
e
ir
∗
|h
K−i
|
2
. (13)
From equation (10), we can have
P
∗
i
N
0
|h
i
|
2
+
j<i
P
∗
j
= e
r∗
− 1
=
(e
r
∗
− 1)
j<i
∆P
j
j<i
∆P
j
=
P
∗
i
+ (e
r
∗
− 1)
j<i
∆P
j
N
0
|h
i
|
2
+
j<i
P
∗
j
+
j<i
∆P
j
=
P
∗
i
+ (e
r
∗
− 1)
j<i
∆P
j
N
0
|h
i
|
2
+
j<i
(P
∗
j
+ ∆P
j
)
.
Therefore, the minimum rate r
∗
can also be written as
r
∗
= ln
1 +
P
∗
i
+ (e
r
∗
− 1)
j<i
∆P
j
N
0
|h
i
|
2
+
j<i
(P
∗
j
+ ∆P
j
)
. (14)
Then, the rate increment for user i can be calculated as
∆r
i
= ln
1 +
P
∗
i
+ ∆P
i
N
0
|h
i
|
2
+
j<i
(P
∗
j
+ ∆P
j
)
− r
∗
= ln
1 +
∆P
i
− (e
r
∗
− 1)
j<i
∆P
j
N
0
|h
i
|
2
+
j≤i
P
∗
j
+ e
r
∗
j<i
∆P
j
.
(15)
By defining
P
e
i
=
∆P
i
− (e
r
∗
− 1)
j<i
∆P
j
e
(K−i)r
∗
,
n
e
i
=
N
0
|h
i
|
2
+
j≤i
P
∗
j
e
(K−i)r
∗
,
we have
∆r
i
= ln
1 +
P
e
i
n
e
i
+
j<i
P
e
j
. (16)