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IEEE COMMUNICATIONS LETTERS, VOL. 24, NO. 11, NOVEMBER 2020 2435
BER Analysis for Uplink NOMA in Asymmetric Channels
Fanbo Wei , Ting Zhou ,Member, IEEE, Tianheng Xu ,Member, IEEE,
and Honglin Hu ,Senior Member, IEEE
Abstract— The bit error rate (BER) analysis has been recog-
nized as an effective approach to investigate the problems in non-
orthogonal multiple access (NOMA) systems, such as inter-user
interference (IUI) and error propagation. However, the impact of
asymmetric channels on adaptive modulation and coding (AMC)
will bring great challenges to the BER analysis of NOMA systems.
In this letter, by exploiting the scaling characteristics of Euclidean
distance, we disclose the boundary effect of asymmetric channels
and derive the boundary value for paired users. In addition,
we provide closed-form BER expressions for uplink NOMA users
on both sides of the boundary value. Numerical results verify the
existence of the boundary value and conform to our derivations.
Index Terms—Bit error rate, non-orthogon al multiple access,
asymmetric channels, boundary value.
I. INTRODUCTION
WITH the development of wireless networks, the demand
for spectrum resources is growing rapidly [1], [2].
In recent years, non-orthogonal multiple access (NOMA) has
been recognized as a promising candidate technique for next-
generation networks [3]. Particularly, NOMA improves the
capacity of wireless systems when serving a combination of
strong and weak users. Many studies have confirmed the supe-
rior ability of NOMA to improve spectral efficiency [4]–[6].
As a key performance indicator (KPI) of wireless networks,
the bit error rate (BER) has been fully analyzed for NOMA
systems [5]–[7]. Specifically, based on the pulse shaping
technique, the research in [6] presents a theoretical BER analy-
sis of QPSK modulation for Fast Fourier Transform-based
NOMA (FFT-NOMA). The closed-form BER expressions of
uplink NOMA applying QPSK constellation are provided
in [7]. However, in existing researches, BER analysis mainly
focuses on the users of the same modulation order. Most
existing studies do not take into consideration the practical
impact of asymmetric channels on NOMA scheme, where the
received signal-to-noise ratios (SNRs) of users are significantly
different. Hence, theoretical research that investigates the
BER performance of uplink NOMA schemes in asymmetric
channels is pressingly needed.
Manuscript received April 12, 2020; revised May 15, 2020; accepted June 8,
2020. Date of publication June 18, 2020; date of current version Novem-
ber 11, 2020. This work was supported by the National Key Research and
Development Program of China (No. 2018YFB1802300), the National Natural
Science Foundation of China (No. 61801461), and the Shanghai Munici-
pality of Science and Technology Commission Project (Nos. 19511103102,
19511104204). The associate editor coordinating the review of this letter and
approving it for publication was Z. Qin. (Corresponding author: Ting Zhou.)
Fanbo Wei is with the Shanghai Advanced Research Institute, Chinese
Academy of Sciences, Shanghai 201210, China, and also with the School of
Electronic, Electrical and Communication Engineering, University of Chinese
Academy of Sciences, Beijing 100049, China.
Ting Zhou, Tianheng Xu, and Honglin Hu are with the Shanghai Advanced
Research Institute, Chinese Academy of Sciences, Shanghai 201210, China
(e-mail: zhouting@sari.ac.cn).
Digital Object Identifier 10.1109/LCOMM.2020.3003274
In practical systems, uplink NOMA schemes tend to pair
users with significant differences in channel conditions and
allocate more power to strong users to minimize inter-user
interference (IUI) [8], [9]. Thus, the asymmetric channel
is considered to be a key feature of the uplink NOMA
scheme [10], [11]. In order to improve channel utilization in
asymmetric channels, adaptive modulation and coding (AMC)
should be considered for the NOMA users in asymmetric
channels [12]. Nevertheless, different modulations will greatly
increase the difficulty of successive interference cancellation
(SIC) decoding process, and eventually bring in serious error
propagation to NOMA systems. Accordingly, in order to
well control the error propagation effect, BER analysis for
uplink NOMA systems becomes an urgent need. In this letter,
we define the inter-user gap (namely IUG) which measures
the SNR difference between two users and investigate the
BER performance of an uplink NOMA scheme in asymmetric
channels. Our contributions can be summarized as follows:
•Based on the scaling effect brought by different modu-
lations, the scaling of the weak user’s signals is mapped
to constellation diagram, which reflects the scaling of the
Euclidean distance between superimposed signals. Then,
the boundary effect in asymmetric channel is disclosed
and the boundary value for paired users is derived.
•To better guide the implementation of AMC technique in
practical uplink NOMA systems, the exact closed-form
BER expressions are derived for two users with different
modulations.
•Simulation results conform to our derivations and show
that the BER performance of the weak user achieves
remarkable gain when the IUG exceeds the boundary
value.
II. SYSTEM MODEL AND PROBLEM FORMULATION
We consider a typical uplink NOMA scheme with one
base station (BS) and two users. Define hias the channel
gain from the user-ito the BS, where i∈{1,2}is the
user index. Particularly, hican be expressed as hi=gi·
PL
−1(Di),wherePL
−1(Di)accounts for the path loss, and
Diaccounts for the distance from the user-ito the BS. Define
|gi|as the additive white Gaussian noise (AWGN) channel
gain. Without loss of generality, we assume that the NOMA
scheme has perfect synchronization and the channel gains
are perfectly known [13], [14]. Channel gains are sorted as
|h2|2>|h1|2[14]. The specific synchronization and channel
estimation can be referred to [15]–[17]. Due to the channel
difference, different modulations are needed for different users
to improve channel utilization. Considering there are many
combinations of modulations in practical NOMA systems,
we select the most classical modulations (user-1: BPSK
[18], [19], user-2: QPSK [6], [7]). Notations s1and s2
are the BPSK signal of user-1 and the QPSK signal of
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2436 IEEE COMMUNICATIONS LETTERS, VOL. 24, NO. 11, NOVEMBER 2020
Fig. 1. Illustration of the signal constellation for user-2: (a) the first state:
d1/√β1<d
2; (b) the second state: d1/√β2≥d2.
user-2 respectively. The received signal at the BS can be
expressed as
y=P1h1s1+P2h2s2+w
=P1g1·PL
−1(D1)s1+P2g2·PL
−1(D2)s2+w(1)
where Piand hirefer to the transmission power and the chan-
nel coefficient of users respectively, with P2|h2|2>P
1|h1|2.
Notation wis the AWGN with zero mean and variance σ2
w.
The SNRs of the paired users are defined as γ1=P1|h1|2/σ2
w
and γ2=P2|h2|2/σ2
w.
Fig. 1 illustrates the constellation points of user-2 with
the IUI of user-1. The orange circles denote user-2’s original
QPSK constellation points that are defined as cp(a1a2),where
a1,a2denotes the bits of user-2, and 2d2denotes the minimum
distance between constellation points. Due to the IUI from
user-1’s signals, each original QPSK constellation point is
transferred to two possible superimposed constellation points.
The superimposed constellation points are represented by
dotted circles in Fig. 1 and are defined as cp(a1a2,a
3),where
a3is the bit of user-1. Particularly, the minimum distance
between the superimposed constellation points is 2d1/√β,
where 2d1is the minimum distance between the original
BPSK constellation points of user-1, and βis the IUG between
the paired users, i.e., β=γ2/γ1. Note that in this letter, d1
and d2are defined as: d1=1,d2=√2/2respectively.
Since the minimum distance between the superimposed
constellation points is 2d1/√β, it can be observed that there
are two states for the IUG β. Referring to Fig. 1 (a), the first
state can be expressed as:
d1/β1<d
2,i.e., β1>d
2
1/d2
2=2.(2)
For the original constellation point cp(10) in the first
state, both its superimposed constellation points cp(10,1) and
cp(10,0) are located in the correct decision region before the
AWGN is introduced. Other points also have similar situations.
On the other hand, since each received signal at the BS
experiences a distinct channel gain, the transmission power of
the users in uplink NOMA cannot be shared. Thus, the trans-
mission power may has a limited range from the perspective
of energy preserving [20]. Considering the possible effects of
this case and referring to Fig. 1 (b), the second state can be
expressed as:
d1/β2≥d2,i.e., 1<β
2≤d2
1/d2
2=2.(3)
In the second state, some superimposed constellation points
fall directly into the error decision region for the 1st bit a1
without the AWGN, such as cp(10,0) and cp(00,1) .Inthis
situation, the power of IUI from user-1’s signals exceeds that
of user-2’s signals in phase-dimension. Consequently, it will
bring an extremely strong interference when decoding a1of
user-2 during SIC process. It should be pointed out that this
case does not exist when users have the same modulation,
where the superimposed constellation points never exceed the
decision boundary before the noise is introduced [7]. However,
this case do exist in the combinations of different modulations.
According to the above analysis, d2
1/d2
2that is approximately
equal to 3 dB can be regarded as a boundary value for paired
users in such an asymmetric channel.
III. CLOSED-FORM BER EXPRESSIONS
In this section, we consider the above two states and derive
the closed-form BER expressions of user-2 and user-1.
A. The BER of User-2 Applying QPSK in Different States
Theorem 1: Using the QPSK modulation, the BER of user-2
(strong user) in state 1 has a closed-form formulation as
BERS1
2=1
21
2Qd2−d1/√β1
σw/√2+1
2Qd2+d1/√β1
σw/√2
+Qd2
σw/√2,(4)
where Q(x)=1/√2π∞
xexp(−u2/2)du denotes
Qfunction.
Proof: Let us start with the first bit a1in point cp(10).
As shown in Fig. 1 (a), both the superimposed constellation
points of cp(10) (cp(10,1) and cp(10,0)) are located in the
correct decision region for a1,andaxis-Qis the decision
boundary. Since the distances from cp(10,1) and cp(10,0)
to their decision boundaries are (d2+d1/√β1)and (d2−
d1/√β1)respectively, a bit error will occur when the AWGN
exceeds (d2+d1/√β1)or (d2−d1/√β1). Considering the
four possible cases (four dashed rectangles) in which a1is
decoded erroneously, the error probability of a1is given by
PS1
a1=1
4Pr d2−d1
√β1
<|w|+Prd2+d1
√β1
<|w|
=1
2Qd2−d1/√β1
σw/√2+Q(d2+d1/√β1
σw/√2).(5)
Since user-1’s symbols are 1-dimensional, the decoding of
a3does not interfere with the decoding of a2. The error
probability of a2is as follows
PS1
a2=1
2Pr (d2<|w|)=Qd2
σw/√2.(6)
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WEI et al.: BER ANALYSIS FOR UPLINK NOMA IN ASYMMETRIC CHANNELS 2437
Therefore, in state 1, the BER expression of user-2 can be
derived as [21]
BERS1
2=1
2
2
k=1
PS1
ak.(7)
Using (5), (6), and (7), we obtain the previous equation (4).
Theorem 2. Using the QPSK modulation, the BER of user-
2 in state 2 is expressed as
BERS2
2=1
21
2Qd2−d1/√β2
σw/√2+1
2Qd2+d1/√β2
σw/√2
+Qd2
σw/√2.(8)
Proof: As shown in Fig. 1 (b), the point cp(10,0) falls
directly into the error decision region for a1in state 2. There-
fore, a1in cp(10,0) can only achieve the correct decoding
when the AWGN is less than d2−d1/√β2(≤0);otherwise,
a decoding error occurs. Accordingly, a1in cp(00,1) can only
be decoded correctly when the AWGN exceeds d1/√β2−d2.
Considering four dashed rectangles in Fig. 1 (b), the error
probability of a1in state 2 is given by
PS2
a1=1
4Pr d2−d1
√β2
<|w|+Prd2+d1
√β2
<|w|.
(9)
Here, we have d2−d1/√β2≤0, which can be regarded
as a sign of the increased interference in state 2. Since a2in
state 2 has the same error probability as in state 1, i.e.,
PS2
a2=P
S1
a2=1
2Pr(d2<|w|),(10)
the BER of user-2 in state 2 is obtained as
BERS2
2=1
2
2
k=1
PS2
ak.(11)
With (9), (10), and (11), we complete the proof. To obtain
further insight, we compare the theoretical BER of user-2 in
state 1 and state 2.
Lemma 1: For a given SNR, the theoretical BER perfor-
mance of user-2 in two states satisfies
BERS2
2>BER
S1
2.(12)
Proof: Please refer to Appendix A.
B. The BER of User-1 Applying BPSK in Different States
For simplicity, when a3is decoded after SIC process,
we assume that a2has been decoded correctly. Then only the
decoding result of a1would affect the decoding of a3. Hence
two cases need to be discussed while wrong decoding occurs at
a3: (I) a1is decoded correctly; (II) a1is decoded erroneously.
Theorem 3: Using the BPSK modulation, the BER of user-1
(weak user) in state 1 has a closed-form expression as
BERS1
1=P
S1
a3,I+P
S1
a3,II
=1
22Qd1/√β1
σw/√2−Qd2+d1/√β1
σw/√2
+Q2d2+d1/√β1
σw/√2+Qd2−d1/√β1
σw/√2
−Q2d2−d1/√β1
σw/√2,(13)
where PS1
a3,Iand PS1
a3,II represent the error probabilities of a3
in two different cases respectively.
Proof: For case I, we first consider the point cp(00,1)
in Fig. 1 (a). It can be observed that a1will be decoded
correctly when the AWGN goes beyond d1/√β1−d2.Further-
more, a wrong decoding will occur for a3when the AWGN
exceeds d1/√β1. Hence for cp(00,1), the range of win case
Iisw>d
1/√β1. Analogously, for cp(10,0), the range of w
in case I can be obtained as w<−d1/√β1. It can also be
inferred from Fig. 1 (a) that for cp(00,0) ,win case I satisfies
−d2−d1/√β1<w<−d1/√β1.Forcp(10,1), the limitation
of win case I is represented as: d1/√β1<w<d
1/√β1+d2.
Considering the four dashed rectangles in Fig. 1 (a),
the error probability of a3in case I is given by
PS1
a3,I=1
4Pr d1
√β1
<|w|
+Pr d1
√β1
<|w|<d
2+d1
√β1.(14)
However, the same approach does not apply to case II in
which the error propagation from user-2 is 2d2. Specifically,
for cp(00,1) as shown in Fig. 1 (a), a1will be decoded
erroneously when the AWGN is less than d1/√β1−d2.Inthis
situation, the distance from constellation point a3to its deci-
sion boundary will be changed from d1/√β1to 2d2−d1/√β1.
Thus, there will exist wrong decoding for a3when the noise
goes beyond −2d2+d1/√β1. Consequently, the range of win
case II is −2d2+d1/√β1<w<−d2+d1/√β1. Meanwhile,
the ranges of win other superimposed constellation points is
obtained as w>2d2+d1/√β1,w<−2d2−d1/√β1,and
d2−d1/√β1<w<2d2−d1/√β1.
Therefore, the error probability of a3in case II is
represented by
PS1
a3,II =1
4Pr 2d2+d1
√β1
<|w|
+Pr d2−d1
√β1
<|w|<2d2−d1
√β1.(15)
From (14) and (15), we obtain the previous equation (13).
Theorem 4. Using the BPSK modulation, the BER of user-1
in state 2 has a closed-form expression as
BERS2
1=P
S2
a3,I+P
S2
a3,II
=1
22Qd1/√β2
σw/√2−Qd2+d1/√β2
σw/√2
+Q2d2+d1/√β2
σw/√2+Qd2−d1/√β2
σw/√2
−Q2d2−d1/√β2
σw/√2.(16)
Proof: Similar to the proof of Theorem 3, the error prob-
ability of a3in case I can be obtained as follows
PS2
a3,I=1
4Pr d1
√β2
<|w|
+Pr d1
√β2
<|w|<d
2+d1
√β2.(17)
For case II, the error probability of a3can also be
expressed as
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2438 IEEE COMMUNICATIONS LETTERS, VOL. 24, NO. 11, NOVEMBER 2020
Fig. 2. The BER performance of the users with different βin NOMAWOC C .
PS2
a3,II =1
4Pr 2d2+d1
√β2
<|w|
+Pr d2−d1
√β2
<|w|<2d2−d1
√β2.(18)
Based on (17) and (18), we obtain the previous equa-
tion (16).Note that the derivation process of Theorem 1 to
Theorem 4 has superior compatibility. In addition to the
combination of BPSK and QPSK, similar derivations can be
further extended to other modulations (e.g., the combination
of QPSK and 16QAM).
IV. NUMERICAL RESULTS
In this section, the theoretical BER of an uplink NOMA
scheme and the existence of the boundary value are ver-
ified through simulation results. Simulations are designed
without and with channel coding (namely NOMAWOCC and
NOMAWCC ), which are implemented on AWGN channels.
Due to different modulations (user-1: BPSK, user-2: QPSK),
the distances from the original constellation points of two
users to their decision axes are d1=1and d2=√2/2.
Thus, the boundary value that satisfies β=d2
1/d2
2=2
is approximated by 3 dB here. The state 1 corresponds to
β>3 dB in the simulations, while the state 2 corresponds to
β≤3dB.
Fig. 2 shows the BER performance of NOMAWOCC.For
different states, this figure shows a perfect match between
the numerical results and the theoretical analysis (4), (8),
(12), (13) and (16). For β=2.5 dB, when SNR exceeds
a certain value, here is an unexpected phenomenon that the
BER performance deteriorates with the increase of SNR. The
reason is that, in this state, some superimposed constellation
points fall into the wrong decision region directly without
AWGN. Consequently, the BER shows a growing trend when
SNR increases (the AWGN decreases). On the other hand, for
β=3.5 dB, since the IUI is greatly reduced, the decreasing
trend of BER with the increase of SNR is as expected.
For practical applications, the BER of NOMAWCC is illus-
trated in Fig. 3. This figure shows that for a given SNR,
the BER performance of user-2 (strong user) is improved with
theincreaseofβ. Such a phenomenon is reasonable, because
when βis increased, the distance from the superimposed
constellation points to the original constellation points will
be closer. Thus, it is easier to achieve correct decoding. For
Fig. 3. The BER performance of the users with different βin NOMAWCC .
Fig. 4. The BER performance of the users with different βin NOMAWCC .
user-1 (weak user), it can be observed that although the BER
performance of user-2 has been improved, the BER perfor-
mance of user-1 still deteriorates with the increase of SNR
when βequals 2.5 dB and 3 dB. This is because in the SIC
process, due to hard decision and hard reconstruction, user-2
can only provide very limited help for user-1 to handle error
propagation effect [22]. For β=3.5 or 4 dB, the BER of user-1
drops dramatically at high SNR environment. To accurately
describe this phenomenon, we further investigate the impact
of βon the BER performance of user-1, as shown in Fig. 4.
Note that the SNR of user-1 is set as 13 dB and βis set
in [2.5, 4.0] dB. It can be seen that there is no significant
improvement in the BER performance until βexceeds 3.0 dB.
Therefore, β=d2
1/d2
2=2that is approximately equal to 3 dB
can be considered as a boundary value.
V. C ONCLUSION
In this letter, we revealed the boundary value effect that
directly affects the BER performance of NOMA systems.
Moreover, we derived the closed-form BER expressions of
an uplink NOMA scheme in asymmetric channels. Numerical
results confirmed the existence of the boundary value and
verified our derivations. With the guidance of boundary value
effect, it could be promising to take into account a flexible
AMC mechanism for NOMA systems in future work.
APPENDIX
PROOF OF THE LEMMA 1
By Theorem 1 and Theorem 2, we obtain PS1
a2=P
S2
a2.
Therefore, we only use the error probability of a1in different
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WEI et al.: BER ANALYSIS FOR UPLINK NOMA IN ASYMMETRIC CHANNELS 2439
state PS1
a1and PS2
a1to compare the theoretical BER of user-2,
which can be formulated as
PS1
a1=1
2Qd2−d1/√β1
σw/√2+Qd2+d1/√β1
σw/√2
=M(β1),(19)
PS2
a1=1
2Qd2−d1/√β2
σw/√2+Qd2+d1/√β2
σw/√2
=N(β2),(20)
respectively. Furthermore, we define that x1(β)= d2−d1/√β
σw/√2
and x2(β)=d2+d1/√β
σw/√2.SinceQfunction has the property
d(Q(x))
d(x)=−1
√2πexp −1
2x2=−g(x),(21)
we have
d(M(β1))
d(β1)
=1
2−d(x1(β1))
d(β1)g(x1(β1)) −d(x2(β1))
d(β1)g(x2(β1))
=β−3/2
1
4σw/√2[g(x2(β1)) −g(x1(β1))].(22)
As a result, β1and β2satisfy
β1>d
2
1/d2
2=2≥β2>1.(23)
Accordingly, we can find that |x1(β1)|<|x2(β1)|,and
g(x2(β1)) <g(x1(β1)). Using (22), the function M(β1)is
proved as a monotonically decreasing function of β1, and its
maximum value satisfies
Mmax <M(2) = 1
2[Q(0) + Q(2/σw)].(24)
Similarly, we have
d(N(β2))
d(β2)
=1
2−d(x1(β2))
d(β2)g(x1(β2)) −d(x2(β2))
d(β2)g(x2(β2))
=β−3/2
2
4σw/√2[g(x2(β2)) −g(x1(β2))].(25)
Since |x1(β2)|<|x2(β2)|,wehaveg(x2(β2)) <g(x1(β2)).
Hence the function N(β2)is also a monotonically decreasing
function of β2, and it has a minimum value
Nmin =N(2) = 1
2[Q(0) + Q(2/σw)] >M
max.(26)
With the help of (26), we prove that
BERS2
2>BER
S1
2.(27)
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