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IEEE COMMUNICATIONS LETTERS 1
Average Block Error Rate of Downlink NOMA
Short-Packet Communication Systems in
Nakagami-mFading Channels
Jianchao Zheng, Qi Zhang, Member,IEEE, and Jiayin Qin
Abstract—Short-packet communication systems provide low
latency communications in fifth-generation wireless cellular net-
works. In this letter, we study the average block error rate
(BLER) of downlink non-orthogonal multiple access (NOMA)
short-packet communication systems. By invoking stochastic
geometry, we consider that the locations of NOMA users are
uniformly distributed in a disc and theoretically derive the
analytical expressions for average BLER in Nakagami-mfad-
ing channels. Numerical results illustrate that our theoretically
derived analytical average BLERs are almost the same as the
simulation results.
Index Terms—Block error rate (BLER), Nakagami-mfading
channels, non-orthogonal multiple access (NOMA), short-packet
communications.
I. INT ROD UC TI ON
Short-packet communications were introduced by Polyan-
skiy et al. to reduce the physical-layer communication latency
[1]. In [1], Polyanskiy et al. also introduced a new performance
metric, block error rate (BLER), to properly measure the
performance of short-packet communications. Inspired by [1],
Makki et al. derived the closed-form expressions for the out-
age probabilities of incremental redundancy hybrid automatic
repeat request (HARQ) for short-packet communications in
[2], [3]. For HARQ, if the initial transmission rate is higher
than the achievable rate of channel, an HARQ will help by
indicating the transmitter to reduce the transmission rate. The
received signals in different retransmission rounds can be com-
bined for decoding [2], [3]. In [4], the maximal achievable rate
for a given blocklength and error probability over quasi-static
multiple-input-multiple-output fading channels were studied.
The aforementioned works consider the conventional or-
thogonal multiple access (OMA) systems. Non-orthogonal
multiple access (NOMA) systems, which allow communica-
tion resources, such as time and frequency, to be shared by
all users, achieve a higher transmission rate than the OMA
systems [5]–[12]. In [6], the performance of NOMA with
randomly deployed users was investigated. In [7], by invoking
This work was supported in part by the National Natural Science Foundation
of China under Grant 61672549, in part by the Guangdong Natural Science
Foundation under Grant 2018B0303110016, and in part by the Guangzhou
Science and Technology Program under Grant 201804010445.
J. Zheng and Q. Zhang are with the School of Electronics and Information
Technology, Sun Yat-sen University, Guangzhou 510006, Guangdong, China
(e-mail: zhengjch8@mail2.sysu.edu.cn, zhqi26@mail.sysu.edu.cn). J. Qin is
with the School of Electronics and Information Technology, Sun Yat-sen
University, Guangzhou 510006, Guangdong, China, and also with the Xinhua
College, Sun Yat-sen University, Guangzhou 510520, Guangdong, China (e-
mail: issqjy@mail.sysu.edu.cn).
stochastic geometry, the physical layer security of NOMA
systems in large-scale networks was investigated. In [8],
secure beamforming and power allocation design optimization
problem which maximizes the sum achievable secrecy rate was
solved. In [9], a unified framework was derived to analyze
the outage behaviors of NOMA networks, where stochastic
geometry is employed to model the locations of NOMA users.
In [10], Wei et al. analyzed the asymptotic ergodic sum-rate
gain of NOMA over OMA in uplink cellular communica-
tion systems. In [11], Kara et al. derived the exact closed-
form bit error rate expressions under successive interference
cancelation (SIC) error for downlink NOMA systems. In
[12], the pairwise error probability of NOMA systems over
Nakagami-mfading channels was investigated. Cooperative
NOMA schemes were proposed in [13]–[15]. Combining the
short-packet communications with the NOMA systems, Yu et
al. derived the closed-form expressions for BLER of users in
[16]. Furthermore, in [16], the near-optimal power allocation
coefficients and blocklength to ensure the NOMA system
reliability were provided.
To our best knowledge, the research on downlink NOMA
short-packet communication systems where the locations of
NOMA users are modeled by stochastic geometry is still
missing in the literature. In this letter, we consider that the
locations of NOMA users are uniformly distributed in a disc
by invoking stochastic geometry. For short-packet commu-
nications, we theoretically derive the closed-form analytical
expressions for average BLER of the downlink NOMA users
in Nakagami-mfading channels [17].
II. SYSTEM MO DE L
Consider a single-input-single-output (SISO) downlink NO-
MA short-packet communication system, which consists of a
BS and Kusers, in Nakagami-mfading channels [17]. All
the users are uniformly distributed in a disc with radius r. The
BS is located at the center of the disc. The BS needs to send
information of Lbits to the kth user, k∈ K ={1,2,·· · , K}.
The channel between the BS and the kth user is expressed as
gk=hk(1 + da
k)−1
2(1)
where hkrefers to the small-scale fading coefficient for
Nakagami-mfading channel such that |hk|2is a Gamma
distributed random variable, dkrefers to the distance from the
BS to the kth user, and a≥2refers to the path loss exponent.
IEEE COMMUNICATIONS LETTERS 2
Without loss of generality, we assume that all channels follow
the order of
|g1| ≤ |g2| ≤ ·· · ≤ |gK|.(2)
Using the NOMA protocol, the BS broadcasts signal
x=
K
k=1 αkP sk(3)
with blocklength nwhere Prefers to the transmit power of
the BS, skrefers to the signal intended to the kth user, and
αkrefers to the power allocation factor of the kth user. From
(2), the power allocation factors should satisfy the following
conditions [6]
α1≥α2≥ ·· · ≥ αK.(4)
All the users use SIC to decode the signals according to the
ascending order from the signals intended to the 1st user to
those intended to the Kth user. Thus, the kth user decodes the
signals intended to the jth user with the signal-to-noise-and-
interference ratio (SINR)
γ(j)
k=αjP|gk|2
σ2+P|gk|2K
i=j+1 αi
(5)
where σ2denotes the variance of the additive Gaussian noise
at the kth user. From [4], given gk, the conditional BLER can
be approximated as
ϵ(j)
k≈Ψγ(j)
k, L, n(6)
where Ψ(γ, L, n),Q((C(γ)−L/n)/V(γ)/n),Q(x) =
1
√2π∞
xe−t2
2dt,C(γ) = log2(1 + γ), and V(γ) = (log2e)2·
(1 −(1 + γ)−2).
Since SIC is used, the kth user should decode the signals
intended to the jth user for all j < k before decoding its own
signals. Thus, given gk, the conditional BLER at the kth user
is expressed as
ϵk=ϵ(1)
k+1−ϵ(1)
kϵ(2)
k+·· ·
+1−ϵ(1)
k·1−ϵ(2)
k·· ·1−ϵ(k−1)
kϵ(k)
k.(7)
By realizing that skcan be decoded at the kth user only when
SIC is done successfully and generally ϵ(j)
kis small (e.g.,
10−3to 10−5) [16], then the conditional BLER in (7) can
be approximated as
ϵk≈min
1,
k
j=1
ϵ(j)
k
.(8)
III. ANALYTICAL AVE RAG E BLER
In this section, the analytical expressions for average BLER
of the downlink NOMA users in Nakagami-mfading channels
are theoretically derived.
At the kth user, the analytical average BLER to decode sk
is obtained by
E[ϵk]≈min
1,
k
j=1
Eϵ(j)
k
(9)
where
Eϵ(j)
k≈∞
0
Ψ(γ(j)
k, L, n)fγ(j)
k
(t)dt. (10)
Similar to [2], [16], we approximate Ψ(γ, L, n)by a linear
function as follows
ZL,n(γ) =
1, γ ≤vL,n
1
2−δL,n√n(γ−βL,n ), vL,n < γ < uL,n
0, γ ≥uL,n
(11)
where βL,n = 2L
n−1,δL,n = (2π(22L
n−1))−1
2,vL,n =
βL,n −1
2δ−1
L,nn−1
2, and uL,n =βL,n +1
2δ−1
L,nn−1
2. Substituting
(11) into (10), we have [16]
Eϵ(j)
k≈δL,n√nuL,n
vL,n
Fγ(j)
k
(t)dt (12)
From (12), to derive the analytical average BLER, we should
know the CDF of γ(j)
k. To derive the CDF of γ(j)
k, from (5),
we have
Fγ(j)
k
(t) = Pr |gk|2≤ζjt
ραj(ζj−t)
=F|gk|2ζjt
ραj(ζj−t)(13)
where F|gk|2(t)is the CDF of the ordered channel gain for
the k-th user, ρ=P/σ2, and ζj=αj/(K
i=j+1 αi). Using
order statistics [18] and binary series expansion, the CDF of
the ordered channels has a relationship with the unordered
channels as follows
F|gk|2(t) = φk
K−k
i=0
ξi,k F|˜gk|2(t)k+i(14)
where F|˜gk|2(t)is the CDF of the unordered channel gain for
the k-th user and
φk=K!
(K−k)!(k−1)! and ξi,k =K−k
i(−1)i
i+k.(15)
Note that in (1), the CDF of |hk|2is [17]
F|hk|2(t) = 1 −exp −mt
ηm−1
q=0
1
q!mt
ηq
(16)
where η= 1 denotes the average power of hk, i.e.,
E[|hk|2]. Following the derivations in [7], using the Gaussian-
Chebyshev quadrature [9], [19], we have
F|˜gk|2(t)≈
U
p=1
bp1−exp −mcpt
ηm−1
q=0
1
q!mcpt
ηq
(17)
where Uis a complexity-vs-accuracy tradeoff parameter and
bp=π
2U1−θp2(1 + θp),(18)
cp=1 + r
2(1 + θp)a
, θp= cos 2p−1
2Uπ.(19)
IEEE COMMUNICATIONS LETTERS 3
Substituting (17) into (13) and (14), from multinomial theorem
and Binomial series, we obtain
Fγ(j)
k
(t) = Φ (t−ζj) + Φ (ζj−t) Θ(t)(20)
where Φ(t)denotes the unit step function,
Θ(t) = φk
K−k
i=0
ξi,k
τ1+···+τU=i+ki+k
τ1,·· · , τU
U
p=1
bτp
p
τp
l=0 τp
l(−1)l
κ0+···+κm−1=ll
κ0,·· · , κm−1
m−1
q=0 1
q!κqmcp
ηqκq
Ω(t) (21)
and
Ω(t) = ζjt
ραj(ζj−t)qκq
exp −lmcpζjt
ηραj(ζj−t).(22)
where τp≥0,p∈ {1,·· · , U }and κq≥0,q∈ {0,··· , m −
1}are the multinomial distribution expansion coefficients. In
the expression of uL,n
vL,n Fγ(j)
k
(t)dt in (12), only Ω(t)contains
integral variable t. Let
y=ζjt
ραj(ζj−t).(23)
We have
t=ραjζjy
ζj+ραjy.(24)
Since dt
dy =ζ2
j
ραj(ρ−1α−1
jζj+y)2, we have
uL,n
vL,n
Ω(t)dt =
ζ2
j
ραj
ζjuL,n
ραj(ζj−uL,n)
ζjvL,n
ραj(ζj−vL,n)
yqκq
ρ−1α−1
jζj+y2exp (−ψpy)dy (25)
where ψp=lmcp
η. By letting z=ρ−1α−1
jζj+y, we have
uL,n
vL,n
Ω(t)dt =ζ2
j
ραj
exp ψpζj
ραj(26)
·λL,n
ϕL,n
(z−ρ−1α−1
jζj)qκq
z2exp (−ψpz)dz.
where
ϕL,n =ζjvL,n
ραj(ζj−vL,n)+ζj
ραj
,(27)
λL,n =ζjuL,n
ραj(ζj−uL,n)+ζj
ραj
.(28)
From Binomial series, we obtain
uL,n
vL,n
Ω(t)dt =ζ2
j
ραj
exp ψpζj
ραj(29)
·
qκq
ω=0 qκq
ω−ζj
ραjω
Υ(qκq−ω−2)
where
Υ(ϖ) = λL,n
ϕL,n
zϖexp (−ψpz)dz. (30)
Since for Nakagami-mfading channels we have m > 0, when
ϖ=qκq−ω−2 = −2, from [3.351.4] in [20], we have
Υ(ϖ) = ψ−ϖ−1
p[Ei (−ψpϕL,n)−Ei (−ψpλL,n )] (31)
where Ei(z) = −∞
−z
e−t
tdt is the exponential integral func-
tion. When ϖ=qκq−ω−2 = −1, from [3.352.1] in [20],
we have
Υ(ϖ) = Ei (−ψpλL,n)−Ei (−ψpϕL,n).(32)
When ϖ=qκq−ω−2≥0, from [3.351.1] in [20], we have
Υ(ϖ) = ψ−ϖ−1
p[Γ (ϖ+ 1, ψpλL,n)−Γ (ϖ+ 1, ψpϕL,n )] .
(33)
where Γ(z, t)is the lower incomplete gamma function.
Substituting (12), (20), (21), (31), (32), and (33) into (9),
we obtain
E[ϵk]≈min
1,
k
j=1
δL,n√nφk
K−k
i=0
ξi,k (34)
τ1+···+τU=i+ki+k
τ1,···, τUU
p=1
bτp
p
τp
l=0 τp
l(−1)l
κ0+···+κm−1=ll
κ0,·· · , κm−1m−1
q=0 1
q!κqmcp
ηqκq
·ζ2
j
ραj
exp ψpζj
ραjqκq
ω=0 qκq
ω−ζj
ραjω
Υ(ϖ).
When ρ=P/σ2→ ∞, we have
χj,ζjt
ραj(ζj−t)→0.(35)
In (17), by omitting O(χj), we have
F|˜gk|2(χj)≈
U
p=1
bp
m!mcpχj
ηm
∝ρ−mk (36)
where the equality m−1
q=0 1
q!zq= exp (z)−∞
q=m
1
q!zqis
used. Thus, we have E[ϵk]≈min{1, ϱ}where
ϱ=
k
j=1
K!
(K−k)!k!U
p=1
bp
m!mcpζjβL,n
ρηαj(ζj−βL,n )mk
∝ρ−mk.(37)
IV. NUMERICAL RES ULT S
In this section, numerical results are presented to verify
our analysis. All the users are uniformly distributed in a
disc with radius r. The noise spectral density is assumed to
be -174 dBm/Hz. The signal bandwidth is assumed to be
1 MHz. Thus, the variance of the additive Gaussian noise
is σ2=−114 dBm. The complexity-vs-accuracy tradeoff
parameter in (17) is U= 10. For each simulated curve,
IEEE COMMUNICATIONS LETTERS 4
0 5 10 15 20 25 30 35 40
P (dBm)
10-5
10-4
10-3
10-2
10-1
100
Average BLER
3rd User, Simul.
3rd User, Anal.
3rd User, Asymp.
2nd User, Simul.
2nd User, Anal.
2nd User, Asymp.
1st User, Simul.
1st User, Anal.
1st User, Asymp.
r=2000 m, a=3.5
r=500 m, a=5
Fig. 1. Average BLER versus P; performance of three-user downlink NOMA
short-packet communication system in Nakagami-mfading channels where
m= 2,L= 80 bits, and n= 100.
we produce 107randomly generated channel realizations and
compute the average BLERs.
In Fig. 1, we consider a downlink NOMA short-packet
communication system between a BS and K= 3 users in
Nakagami-mfading channels where m= 2. The BS needs
to send information of L= 80 bits to each user. The signal
blocklength sent from the BS is n= 100. The power allocation
factors for three users are α1= 0.80,α2= 0.16, and
α3= 0.04. We present the average BLERs of three users
where the two cases (r= 2000 m, the path loss exponent
is a= 3.5) and (r= 500 m, a= 5) are considered. From
Fig. 1, it is observed that our theoretically derived analytical
average BLERs are almost the same as the simulation results.
From Fig. 1, it is also found that at low P, the average BLER
of the 3rd user is higher than the 1st and 2nd users. This is
because at low P,ϵ(j)
kis non-negligible for k∈ {1,2,3}and
j∈ {1,2,3}. From (8), ϵ3=ϵ(1)
3+ϵ(2)
3+ϵ(3)
3is higher than
ϵ1=ϵ(1)
1and ϵ2=ϵ(1)
2+ϵ(2)
2since the former includes three
terms and the latter include only one or two terms. At high
P, the average BLER of the 3rd user is lower than the 1st
and 2nd users. This is because from (37), the diversity order
for the 3rd user is 6 whereas those for the 1st and 2nd users
are only 2 and 4, respectively. Thus, the cross points appear
in Fig. 1.
Remark: Since the average BLER analysis here provided the
BLER lower bound for short-packet communications [1]–[4],
the actual coding and modulation schemes are not included.
In Fig. 2, we present the average BLERs of two users for
different values of mwhere r= 1000 m, a= 4, and P= 15
dBm. The power allocation factors for two users are α1= 0.80
and α2= 0.2. We consider two cases, i.e., (L= 80 bits,
n= 100) and (L= 8000 bits, n= 10000). From Fig. 2, it is
found that the average BLERs of two users decrease with the
increase of m.
12345678
m
10-5
10-4
10-3
10-2
10-1
Average BLER
1st User, n=100, Simul.
1st User, n=100, Anal.
1st User, n=10000, Simul.
1st User, n=10000, Anal.
2nd User, n=100, Simul.
2nd User, n=100, Anal.
2nd User, n=10000, Simul.
2nd User, n=10000, Anal.
Fig. 2. Average BLER versus m; performance of two-user downlink NOMA
short-packet communication system in Nakagami-mfading channels where
r= 1000 m, a= 4, and P= 15 dBm.
V. CONCLUSION
In this letter, we have theoretically derived the analytical ex-
pressions for average BLER of downlink NOMA short-packet
communication systems in Nakagami-mfading channels. It is
shown through numerical results that our theoretically derived
analytical average BLERs match the simulation results.
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