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A True Power Allocation Constraint for
Non-Orthogonal Multiple Access with M-QAM
Signalling
Ferdi KARA, Hakan KAYA
Wireless Communication Technologies Laboratory (WCTLab)
Department of Electrical and Electronics Engineering
Zonguldak Bulent Ecevit University
Zonguldak, TURKEY 67100
Email: {f.kara,hakan.kaya}@beun.edu.tr
Abstract—Non-orthogonal multiple access (NOMA) scheme is
seen as a strong candidate for future wireless networks thanks to
its spectral efficiency. Thus, NOMA has attracted a great recent
attention and the power allocation (PA) in NOMA has been
analyzed in terms of various perspectives. However, all these
studies analyze only the Shannon limit and the detection at the
users have not been considered in detail. In this paper, we propose
a true PA constraint for a detectable NOMA signal design in
multi-user NOMA schemes with arbitrary M-QAM signalling.
With the proposed PA constraint, the successful detection at
the receivers are guaranteed. We evaluate the proposed PA with
computer simulations. We also compare with other PA strategies
in the literature and present that if the provided PA constraint
is not satisfied, none of the users can detect it symbols and all
users have the worst bit error rate (BER) performance (i.e., 0.5).
Finally, we demonstrated that for a reliable communication, the
number of users and/or the modulation orders of the users should
be limited.
Index Terms—NOMA, power allocation, error performance
I. INTRODUCTION
Non-orthogonal multiple access (NOMA) allows multi-
users sharing the same resource blocks, hence the massive
machine type communication becomes feasible by surpassing
the available resource bottleneck (time, frequency, code etc.)
[1]. To this end, in the recent years, NOMA has been envi-
sioned as one of the strongest candidates for future wireless
networks and has been investigated for various aspects of
wireless communications such as multiple access, coordinated
multi-point, mobile edge computing.
In NOMA, multi users’ symbols are superimposed with
different power allocation (PA) coefficients at the transmitter
and conveyed to the users on the same resource block. Thus,
an inter-user-interference (IUI) occurs and this IUI is mitigated
at the receivers by the successive interference canceler (SIC).
As expected, the PA has dominant effect on the performance
of NOMA, hence a great deal number of studies has been
devoted to optimizing PA in terms of various constraints such
as sum rate, min-rate, outage probability, bit error rate, energy
efficiency etc. In [2], user pairing is implemented and then
in a two-user single-input-single-output (SISO) system, the
optimum PA is derived with a proportional fairness objective.
Then, the PA is optimized in a multi-user SISO network
under outage probability constraints of the users [3]. The
PA constraint is defined in [4] to achieve higher spectral
efficiency than OMA scheme for all NOMA users. The PA
is optimized in a SISO network to maximize sum rate and
minimum rate of users by considering proportional fairness
[5]. In [6], physical layer security is considered in a SISO
NOMA scheme and the PA is optimized to maximize the
secrecy sum rate. In two user SISO and MIMO schemes,
the PA is optimized to maximize sum rate by guaranteeing
minimum rate requirements of both users [7]. Optimal PA is
derived in closed form for user fairness and weighted sum
rate maximization under the impact of SIC [8]. Moreover,
joint optimization of PA with other resource problems such
as user pairing/scheduling/clustering [9]–[12], channel and
sub-carrier allocations [13], [14] have been also addressed
in various networks in the literature. In more recent papers,
the fundamentals of power allocation problem are defined to
achieve better outage performance than conventional OMA
networks [15] and the analytical framework is conducted for a
dynamic power and channel allocation problem [16] in multi-
user systems.
However, all aforementioned studies approach the PA opti-
mization problem in terms of theoretical Shannon limit by con-
sidering SINR definitions. Nevertheless,when an actual modu-
lator and demodulator are implemented. these PA optimization
algorithms do not guarantee successful decoding at the re-
ceivers and the users may have severe bit/symbol/block error
rate (BER/SER/BLER) performances compared to the OMA
schemes [17]–[19]. To address this, the PA optimization is
defined in [20] to minimize the maximum BER performance of
users in a two-user downlink NOMA. Then, the PA constraints
are defined in NOMA-based cooperative relaying [21] and
cooperative NOMA schemes [22] to achieve minimum BER
and maximum capacity with target BER constraints, respec-
tively. According to [17]–[22], although the PA optimization
solutions in the literature [2]–[16] seem optimum in terms of
information theoretical perspective, the users can not detect
their symbols when an actual modulator and demodulator are
implemented. To this end, in this paper, we define a true PA
constraint which guarantees successful decoding at the users
in multi-user downlink schemes with arbitrary modulations.
The rest of this paper is organized as follows. The Section II
introduces the multi-user downlink NOMA scheme with the
transmitter and receiver structures. Then, in Section III, the
novel PA constraint is defined for a detectable signal design
at the receivers and theoretical proof is conducted. In section
IV, simulation results are presented to evaluate the proposed
PA constraint. Moreover, comparisons with the PA schemes in
the literature are also presented in Section IV. Finally, Section
V discusses the results and concludes the paper.
II. SY ST EM MO DE L
In this paper, we consider a downlink scenario where a
transmitter -BS- sends data to Lend users (UEi, i =
1,2,...L). All nodes are assumed to have a single antenna. We
assume a non-orthogonal multiple access (NOMA) scheme,
thus the BS transmits all users’ symbols at the same resource
block (frequency, time, code).
A. Transmitter Side
The BS implements a superposition coding, hence the total
superposition coded symbol at the transmitter is given by
xsc =
L
X
i=1
√αixi(1)
where αiis the power allocation (PA) coefficient for the ith
user. In this paper, αi< αi+1 is assumed and PL
i=1 αi= 1.
xiis the baseband modulated symbol of ith user with Mi-ary
modulation. E|xi|2= 1,∀i. This superimposed symbol is
conveyed to the users. Therefore, The received signal at the
each user is given by
yi=√P xschi+ni, i = 1,2, . . . , L (2)
where Pis the transmit power of the BS. hiand niare the flat
fading channel coefficient between the BS and ith user and the
additive Gaussian noise at the receiver i, respectively. Whereas
ni∼CN (0, N0)is defined, hican be any wireless channel
model according to applications (indoor, outdoor) such as
Rayleigh fading, Nakagami-mfading, Rician fading etc.
B. Receiver Side
Since the PA coefficient of the Lth user has the highest
value, the Lth user detects its own symbols by pretending the
other users’ symbol as noise. Thus, the maximum likelihood
(ML) detection at the Lth user is given as
ˆxL= argmin
kyL−pP αLhLxL,k
2
, k = 1,2, . . . , ML,
(3)
where xL,k denotes the kth point in the ML-ary constellation.
On the other hand, the ith user should implement iterative
successive interference cancelers (SIC) to detect its own
symbols where it firstly detects the jth users symbols (i.e.,
j=i+ 1, i + 2, . . . , L) and subtracts these symbols from the
received signal. Hence, the detection process at the ith user is
given by
ˆxi= argmin
ky(L−i+1)
i−pP αihixi,k
2
, k = 1,2, . . . , Mi,
(4)
where
y(L−i+1)
i=y(L−j+1)
i−√P hi√αjˆxj, j =i+ 1, i + 2, . . . , L
(5)
where y(1)
i,yiand
ˆxj= argmin
ky(L−j+1)
i−pP αjhixj,k
2
, k = 1,2, . . . , Mj,
(6)
III. POWE R ALL OC ATION CONSTRAINT
The PA coefficients have dominant effect on the perfor-
mance of users. In case of wrongly chosen of PA coefficients,
none of users can detect its symbols correctly. Therefore, the
PA coefficients should be chosen carefully. In order to design
a detectable signal constellation for each user, by considering
the ML detection and the iterative SIC operations, the PA
constraint is given by
Theorem 1:
αi>(Mi−1) Pi−1
j=1 qαj
Mj−1(pMj−1)2
,(7a)
s.t a1>0,(7b)
s.t PL
i=1 αi= 1 (7c)
where (7.b)guarantees the minimum power for the first user
(last user in SIC order) and (7.c)limits the total power
consumption.
Proof:
As seen in (1), the total signal constellation consists of the
weighted sum of the each users constellations. Thus, it scatters
and differs from any well-known signal constellations (e.g., M-
QAM). One can easily see that the PA coefficients have domi-
nant effects on this newly-obtained signal constellation. If the
PA coefficients are chosen in such way that the symbols across
the decision boundaries of ML decoding, none of the users can
detect their symbols. Therefore, the PA coefficients should be
selected not to cause too much scattering. Without loss of the
generality, in Fig. 1.a , we present the signal constellation of
3 user case with Mi= 4 ∀iwhen the binary information
is b3,1b3,2= 00. Since the α3has the highest value within
αi, the UE3implements only an ML decoding. In this case,
if the received signal exceeds the decision boundary of the
ML decoding, an erroneous detection occurs. Therefore, none
of the symbols, within the newly-obtained constellation after
superposition coding, should not across the decision boundary.
As given in Fig. 1.a, the constellation points (A,B,C,D,E,F,G)
(a) Constellation for the 3rd user (b) Constellation for the 2nd user after correct SIC
Fig. 1. Total Constellation of Conventional NOMA for L= 3 users and Mi= 4-QAM
are mostly likely points to across the decision boundaries of 4-
QAM ML detector (i.e., in-phase or quadrature axes). Hence,
in order to detect UE3symbols, it should be guaranteed that
rα3
2>rα2
2+rα1
2.(8)
Otherwise, regardless of the channel effects and the additive
noise, the symbols in those points (A,B,C,D,E,F,G) across the
ML decision region and the symbols are always detected erro-
neously. Then, we assume that a correct SIC is implemented at
UE2, in this case, the remaining signal constellation at U E2is
given in Fig. 1.b. Based on the ML decision rule, as discussed
above, to detect UE2symbols correctly, the PA coefficient
should meet rα2
2>rα1
2.(9)
Otherwise, the UE2symbols (in H,J,K points) cannot be
detected.
Now, we extend this rule for general M-QAM constellations
with Lusers. Mi-QAM modulation can be considered as √Mi
times √Mi-PAM modulations in in-phase and/or quadrature
axes with total E[|xi|2]
√Mienergies for each. The distance between
two neighbour points in √Mi-PAM constellation is given by
2di= 2s3
2 (Mi−1).(10)
According to (3) after the superposition coding (e.g. (1)), if
the scattered points exceeds the the decision boundary of two
neighbours (i.e., half of the distance between two constellation
points), an erroneous detection occurs. As discussed above, it
is mostly likely that the points which are the furthest from
the original constellation point (most scattered) across the
decision boundary. In the √Mi-PAM constellation, for the
furthest point, the distance from the origin is given by
di,max = (pMi−1)di.(11)
Therefore, by considering the SIC order, the PA coefficient of
ith user should be greater than the weighted total of the PA
coefficients and dmax of the users which are later in the SIC
order. Hence, the PA constraint is given by
√αidi>
i−1
X
j+1
√αjdj,max.(12)
By substituting (10) and (11) into (12), the (7a) is obtained.
So the proof is completed.
IV. NUMERICAL RES ULTS
In this section, we present computer simulations to evaluate
the proposed PA constraint. In the simulations, we assume that
all users have the same modulation order (e.g., Mi=M∀i).
Firstly, we present BER performances of 2 and 3 users cases
over AWGN channel (the simplest wireless channel model) in
Fig. 2. In Fig. 2.a, in 2-users case, when M= 16, we present
BER performances for [α1, α2] = [0.05,0.95] and [α1, α2] =
[0.1,0.9]. As seen that since the latter PA does not meet with
the proposed constraint, the users cannot detect their symbols
over even AWGN channels (even though channel fading effect
has not been considered) although this PA is mostly chosen
for capacity an outage performances of NOMA systems in
literature. In Fig. 2.b, in 3-users case, M= 4 and the PAs are
[α1, α2, α3] = [0.1,0.2,0.7] and [α1, α2, α3] = [0.1,0.3,0.6].
The same discussions are valid for also 3-users case.
Then, we present simulations over Rayleigh fading channels
in Fig. 3-5 for L= 2,3,4, respectively for M= 4,16,64.
TABLE I
PA COE FFICI EN TS IN T HE SIMULATIONS
Commonly Used PA (αi)
L M Proposed PA (αi) Values Ref
2 4 αi= [0.1181,0.8819]
αi= [0.2,0.8] [23]2 16 αi= [0.0207,0.9793]
2 64 αi= [0.0045,0.9955]
3 4 αi= [0.0261,0.1948,0.7791] αi= [0.1667,0.3333,0.5] [23]
3 16 αi= [0.0012,0.0588,0.9400] αi= [0.05,0.25,0.7]] [24]
3 64 αi= [0.0001,0.0154,0.9845]
4 4 αi= [0.0063,0.0473,0.1893,0.7571] αi= [0.1,0.2,0.3,0.4] [23]
4 16 αi= [0.0001,0.0037,0.0586,0.9377] αi= [0.02,0.05,0.18,0.75] [24]
4 64 αi= [0.0001,0.0037,0.0586,0.9377]
0 5 10 15 20 25 30
SNR(dB)
10-5
10-4
10-3
10-2
10-1
100
BER
User 1, i=[0.05, 0.95]
User 2, i=[0.05, 0.95]
User 1, i=[0.1, 0.9]
User 2, i=[0.1, 0.9]
(a) 2-users case M= 16
0 5 10 15 20 25 30
SNR(dB)
10-3
10-2
10-1
100
BER
User 1, i=[0.1, 0.2, 0.7]
User 2, i=[0.1, 0.2, 0.7]
User 3, i=[0.1, 0.2, 0.7]
User 1, i=[0.1, 0.3, 0.6]
User 2, i=[0.1, 0.3, 0.6]
User 3, i=[0.1, 0.3, 0.6]
(b) 3-users case M= 4
Fig. 2. BER performance of NOMA over AWGN channel
For the reliability of the simulations, we also provide the
theoretical BER curves when L= 2,M= 4 [25] and
L= 3,M= 4 [26]. Moreover, we present comparisons for the
different PA coefficients which are the proposed fixed-PA (e.g.,
ai=L−i+1
µwhere µensures PL
i=1 ai= 1) for multi-user
scenarios in the pioneer paper [23] and the used PA in the BER
performance paper of multi-user NOMA with only BPSK [24].
The PA coefficients in the simulations are given in Table I. In
the Rayleigh fading simulations, since it is more practicable,
we do not use instantaneous channel ordering. We assume that
0 5 10 15 20 25 30
SNR(dB)
10-3
10-2
10-1
100
BER
User 1, M=4
User 2, M=4
User 1, M=16
User 2, M=16
User1, M=64
User 2, M=64
Theoretical, M=4
(a) The proposed PA
0 5 10 15 20 25 30
SNR(dB)
10-3
10-2
10-1
100
BER
User 1, M=4
User 2, M=4
User 1, M=16
User 2, M=16
User 1, M=64
User 2, M=64
Theoretical, M=4
(b) The fixed-PA in [23]
Fig. 3. BER performance of two-user NOMA over Rayleigh fading channel
when σ2= 0dB ∀i
all users have the same variances, hence hi∼CN (0, σ2)∀i
where σ2= 0dB is assumed. As seen from the all figures,
the proposed PA outperforms significantly the PA schemes in
the literature. Indeed, it is proved that if the proposed PA
constraint is not satisfied (see Fig.3.b, Fig 4.b, Fig. 4.c, Fig.
5.b and Fig.5.c), none of the users can detect its symbols.
Thus, it is very crucial to select the true PA coefficients for a
reliable communication. The fixed-PA strategy in the pioneer
paper [23] works only if the number of user is equal to
0 5 10 15 20 25 30
SNR(dB)
10-3
10-2
10-1
100
BER
User 1, M=4
User 2, M=4
User 3, M=4
User 1, M=16
User 2, M=16
User 3, M=16
User1, M=64
User 2, M=64
User 3, M=64
Theoretical, M=4
(a) The proposed PA
5 10 15 20 25 30
SNR(dB)
10-2
10-1
100
BER
User 1, M=4
User 2, M=4
User 3, M=4
User 1, M=16
User 2, M=16
User 3, M=16
User1, M=64
User 2, M=64
User 3, M=64
Theoretical, M=4
(b) The fixed-PA in [23]
0 5 10 15 20 25 30
SNR(dB)
10-2
10-1
100
BER
User 1, M=4
User 2, M=4
User 3, M=4
User 1, M=16
User 2, M=16
User 3, M=16
User 1, M=64
User 2, M=64
User 3, M=64
Theoretical, M=4
(c) The PA in [24]
Fig. 4. BER performance of three-user NOMA over Rayleigh fading channel
when σ2= 0dB ∀i
L= 2 and the modulation order is M≤4. In all other
schemes, the NOMA signal is not detectable. The proposed
PA of BPSK in [24] works only for M= 4, however, with
the increase of modulation order this PA strategy also does
not provide a detectable NOMA signal. On the other hand,
with the proposed PA constraint, regardless of the number
of users and the modulation order, all user can detect their
0 5 10 15 20 25 30
SNR(dB)
10-2
10-1
100
BER
User 1, M=4
User 2, M=4
User 3, M=4
User 4, M=4
User 1, M=16
User 2, M=16
User 3, M=16
User 4, M=16
User1, M=64
User 2, M=64
User 3, M=64
User 4, M=64
(a) The proposed PA
0 5 10 15 20 25 30
SNR(dB)
0.2
0.25
0.3
0.35
0.4
0.45
BER
User 1, M=4
User 2, M=4
User 3, M=4
User 4, M=4
User 1, M=16
User 2, M=16
User 3, M=16
User 4, M=16
User1, M=64
User 2, M=64
User 3, M=64
User 4, M=64
(b) The fixed-PA in [23]
0 5 10 15 20 25 30
SNR(dB)
10-2
10-1
100
BER
User 1, M=4
User 2, M=4
User 3, M=4
User 4, M=4
User 1, M=16
User 2, M=16
User 3, M=16
User 4, M=16
User 1, M=64
User 2, M=64
User 3, M=64
User 4, M=64
(c) The PA in [24]
Fig. 5. BER performance of four-user NOMA over Rayleigh fading channel
when σ2= 0dB ∀i
symbols. Nevertheless, with the increase of Land/or M, for a
detectable signal design, the PA constraint becomes too strict,
thus the the users with lower PA coefficients have poor BER
performances. For instance, even in a two-user NOMA, when
M= 64, according to (7), a2>0.98 should be satisfied.
Therefore, the UE1is allocated with too low transmit power
so that its performance is decreased. We can also see that with
the increase of L. For example, in Fig. 5.a, UE1has very poor
performance although the proposed PA meets (7). Thus, the
number of users in a resource block and/or the modulation
order of the users should be limited in NOMA schemes.
V. CONCLUSION
In this paper, we provide a novel PA constraint for a
reliable communication in multi-user NOMA schemes with
arbitrary modulation order. We prove that if the proposed
PA constraint is not meet, none of the users can detect its
symbols and NOMA does not work when an actual modulator
and demodulator are implemented. The PA optimization in
NOMA schemes should not be handled only in terms of
information theoretical perspective, we should also consider
the signal detection at the users. Furthermore, we demonstrate
that with the increase of numbers of users in a resource block
and/or modulation order of the users, the PA constraint is too
strict and NOMA users have poor performances. Thus, the
user scheduling/pairing and PA optimization should be further
investigated along with the modulation order constraint. These
are seen as future research topics.
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