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On the Performance of Non-Orthogonal Multiple
Access in Short-Packet Communications
Yuehua Yu, He Chen, Yonghui Li, Zhiguo Ding, and Branka Vucetic
Abstract—This letter investigates the performance of non-
orthogonal multiple access (NOMA) in short-packet communica-
tions. Specifically, we aim to answer a fundamental question – for
given reliability requirements of users, how much physical-layer
transmission latency can NOMA reduce when compared to its
orthogonal multiple access counterparts in the finite blocklength
regime? To this end, we derive closed-form expressions for
the block error rates of users in NOMA. Based on this, we
further obtain the near-optimal power allocation coefficients
and blocklength to ensure certain reliability. Numerical results
validate our theoretical analysis and demonstrate the superior
performance of NOMA in reducing transmission latency.
Index Terms—Non-orthogonal multiple access (NOMA), short-
packet, finite blocklength (FBL), transmission latency
I. INTRODUCTION
Internet of Things (IoT) has become one key driver for
the development of 5G communications [1]. Different from
the traditional human-operated broadband services, many IoT
applications, particularly in industrial control and automation,
require ultra-reliable and low-latency communications (uRLL-
C) [2]. To reduce the physical-layer transmission latency,
uRLLC usually uses short-packets with finite blocklength
(FBL) codes, wherein a small number of symbols is adopted
in each data block to reduce the transmission time interval.
This calls for a fundamentally different system design and
performance analysis because the conventional Shannon’s for-
mula, which assumes the infinite blocklength [3], is no longer
applicable. In [4], Polyanskiy et al. developed a pioneering
framework for short-packet communications. Specifically, for
a given set of the Shannon capacity C, blocklength mand
the block error rate (BLER) ε, the maximal achievable rate
R∗(m, ε)can be tightly approximated by
R∗(m, ε)≈ C − V
mQ−1(ε) + Olog m
m,(1)
where Q−1(·)denotes the inverse of the Gaussian Q-function
Q(x) = ∞
x
1
√2πe−t2
2dt, and Vis the channel dispersion [4,
Def. 1]. Inspired by [4], the impact of FBL on the performance
of various communication setups has been extensively studied,
see e.g., [5]–[8] and references therein.
On the other hand, non-orthogonal multiple access (NO-
MA) has emerged as a promising solution to significantly
improve the spectral efficiency for 5G communications [9],
[10]. In particular, it offers great potential to achieve low-
latency by serving multiple users simultaneously with multiple
data streams superimposed in the power domain [11]. In this
sense, it is natural to integrate the short-packets and NOMA
techniques to achieve ultra-low latency communications. As
the first effort towards this, in this letter we aim to answer a
fundamental question: for given reliability constraints of users,
how much physical-layer transmission latency (measured in
Y. Yu, H. Chen, Y. Li and B. Vucetic are with the School of Electrical and
Information Engineering, University of Sydney, NSW 2006, Australia (email:
{yuehua.yu, he.chen, yonghui.li, branka. vucetic}@sydney.edu.au).
Z. Ding is with the School of Computing and Communications, Lancaster
University, Lancaster LA1 4YW, U.K. (email: z.ding@lancaster.ac.uk).
channel uses) can NOMA reduce when compared to its or-
thogonal multiple access (OMA) counterparts? The answer to
this question is non-trivial. This is because in NOMA, multiple
users are simultaneously severed, and thus the common block-
length needs to concurrently satisfy multi-users’ reliability
constraints. Furthermore, the blocklength design is affected by
the received signal-to-noise-plus-interference ratios (SINRs),
which are further affected by the power allocation among the
served users. In this case, the common blocklength minimiza-
tion is coupled with the power allocation design, which makes
the optimization problem more complex. In [12], the joint
power and coding-rate optimization problem of NOMA with
short-packets was studied, however, it was tackled implicitly
and no-closed form expressions were derived.
The contributions of this letter can be summarized as
follows. We characterize the transmission latency reduction
of NOMA over its OMA counterparts in short-packet com-
munications. In particular, the closed-form expressions of the
average reliability performance of users, measured by BLER,
are derived. Based on this, the asymptotically optimal power
allocation and blocklength are obtained for high SNR scenar-
ios. Finally, the achievable transmission latency reduction of
NOMA compared to OMA is characterized in closed-form.
The analysis is verified by simulations which demonstrate the
superior performance of NOMA in the FBL regime.
II. SY ST EM MO DE L AN D PROB LE M FOR MU LATI ON
In this paper, a NOMA downlink system with short-packets
is investigated. Considering the strong co-channel interference
and heavy system overhead for user coordination, we consider
that only two users1, denoted by ui,i={1,2}, are paired
to perform NOMA [13]. Let hidenote the channel coefficient
between the base station (BS) and ui. Without loss of gen-
erality, we assume |h1|2≥ |h2|2. According to the NOMA
principle, the BS will broadcast 2
i=1 √αiP si, where siis the
message to ui,Pis the total transmit power, and αiis the
power allocation coefficient satisfying α1≤α2and α1+α2=1.
The received signals at uiis given by
yi=hi√α1P s1+√α2P s2+ni,(2)
where niis the complex additive white Gaussian noise with
variance σ2
i. For simplicity, we assume σ2
i=σ2.
At u2, it decodes s2directly by treating s1as interference.
Then the received SINR of decoding s2at u2is given by
γ22 =α2|h2|2/α1|h2|2+ 1/ρ,(3)
where ρ=P/σ2is the transmit signal-to-noise ratio (SNR).
When given the blocklength m, the number of data bits
Nito ui, the Shannon capacity C(x) = log2(1 + x)and
the channel dispersion V(x) = 1−1
(1+x)2(log2e)2, the
1Note that the two-user form of NOMA is an elementary block of NOMA,
which has been included in the 3GPP-LTE Advanced recently [14]. Extending
our work to the general case with more than two users in a NOMA group has
been left as a future work.
2
instantaneous BLER of decoding s2at u2, denoted by ε2, can
be approximated as
ε2=ε22 ≈QC(γ22)−N2
m
V(γ22)/m ,Ψ(γ22 , N2, m),(4)
which can be obtained by omitting the item O(log m/m)in
(1) when m>100 channel uses2and re-arranging the resultant
equation [5]–[8]. In this case, the average BLER of decoding
s2at u2can be given by ¯ε2=E(ε2), where E(·)is the
expectation operator.
In contrast, u1will perform successive interference cancel-
lation (SIC). More specifically, it first decodes s2by treating
s1as interference. Similarly, the received SINR of decoding
s2at u1is given by
γ12 =α2|h1|2/α1|h1|2+ 1/ρ,(5)
and the instantaneous BLER of decoding s2at u1can be
approximated as ε12 ≈Ψ(γ12, N2, m).
If s2can be successfully decoded and removed, u1then can
decode s1in an interference-free manner with
γ11 =α1ρ|h1|2,(6)
and ε11 ≈Ψ(γ11, N1, m). By realizing that s1can be
recovered only when SIC is done successfully and generally
ε1iis small in uRLLC (e.g., 10−3to 10−5) [2], then the
instantaneous overall BLER at u1can be approximated as
ε1=ε12 + (1 −ε12)ε11 =ε12 +ε11 −ε12 ε11 ≈ε12 +ε11.(7)
Similarly, the average overall BLER of decoding s1at u1can
be approximated by ¯ε1≈¯ε12 + ¯ε11 =E(ε12) + E(ε11 ).
In this case, when given the average reliability constraints
¯εth
iand (3)-(7), the blocklength mcan be represented as a func-
tion of the power allocation coefficients given by M(α1, α2).
In order to reduce the common blocklength, we would like to
solve the following
P:m∗= min
α1,α2M(α1, α2),(8a)
s.t. ¯εi≤¯εth
i, i ∈ {1,2}(8b)
0< αi<1, α1+α2= 1.(8c)
By noting that α2= 1 −α1,¯εij is a decreasing function of m
since ∂Ψ(γij ,Nj,m)
∂√m>0, and Q(x)is a decreasing function of
x, we thus can simplify (8) as
P:m∗= min
α1M(α1),(9a)
s.t. ¯εi= ¯εth
i, i ∈ {1,2}(9b)
0< α1<1.(9c)
III. PROB LE M SOLVING
To solve the optimization problem (9), in this section, we
first derive closed-form expressions of the average BLER ¯εifor
both users in NOMA and then obtain the minimum blocklength
mifor each user. Based on these expressions, the optimal
power allocation coefficient α∗
1and the minimum common
blocklength m∗are then achieved. The blocklength reduction
of NOMA in short-packet communications compared to its
OMA counterparts is finally identified.
2The approximation expression (4) can be applied to any value of m.
However, as shown in [4], [8], the approximated BLER is very tight when
m > 100. But it becomes less accurate when m < 100. Fortunately, the range
of the blocklength in many practical applications is in the order of ∼100
channel uses [8].
A. Average BLER of Users ¯
εiin NOMA
We first have the average BLER ¯εij given by
¯εij ≈∞
0
QC(γij )−Nj
m
V(γij )/m fγij (x)dx, i, j ={1,2},(10)
where fX(x)denotes the probability density function of a
random variable X. Unfortunately, due to the complex form of
QC(γij )−Nj
m/V(γij )/m, it is difficult to find a closed-
form expression for ¯εij . Motivated by this, we thus follow [8]
to implement QC(γij )−Nj
m/V(γij )/m≃Zij with
Zij (γij )(a1)
=1, γij ≤νj,m,
1
2−δj,m√m(γij −βj,m ), νj,m <γij <µj,m,
0, γij ≥µj,m,
(11)
where (a1)is obtained by using the linearization technique
for the function QC(γij )−Nj
m
√V(γij )/m at γij =βj,m,δj,m =
1/2π22Nj/m −1,βj,m = 2
Nj
m−1,νj,m =βj,m −1
2δj,m√m,
and µj,m =βj,m +1
2δj,m√m. In this case, ¯εij then can be
evaluated as
¯εij ≈∞
0
Zij (γij )fγij (x)dx=δj,m√mµj,m
νj,m
Fγij (x)dx, (12)
where FX(x)denotes the cumulative distribution function
(CDF) of a random variable X.
In this letter, we assume the independent and identically
distributed Rayleigh-fading channel as in [5]–[8]. By applying
the order statistic, Fγij (x)can be written as
Fγ22 (x)=1−e−2x
λρ(α2−α1x),(13)
Fγ12 (x) = 1−e−x
λρ(α2−α1x)2,(14)
Fγ11 (x) = 1−e−x
λρα12,(15)
where λ=E(|hi|2), and the power allocation coefficients
should satisfy the condition α2−α1x > 0.
With Fγ22(x)in (13), the average BLER at u2can be
evaluated in a closed-from and given in Proposition 1.
Proposition 1: Given the number of transmit data bits N2
and the blocklength m, the average BLER ¯ε2at u2can be
tightly approximated as
¯ε2≈1−2δ2,mϖ e 2ϖ
α2√m
α1Ei −2ϖ
ϑν2,m −Ei −2ϖ
ϑµ2,m
−δ2,m√m
α1e−˜ν2,m ϑν2,m −e−˜µ2,m ϑµ2,m ,(16)
where ϖ=α2
λρα1,ϑν2,m =α2−α1ν2,m and ϑµ2,m =α2−
α1µ2,m,˜ν2,m =2ν2,m
λρϑν2,m ,˜µ2,m =2µ2,m
λρϑµ2,m , and Ei(x) =
−∫∞
−x
e−t
tdtis the exponential integral function.
Proof: See Appendix A.
Similarly, based on the CDF in (14)-(15), the average BLER
of u1can be obtained and given in Proposition 2.
Proposition 2: Given the number of transmit data bits N1
and the blocklength m, the average BLER ¯ε1at u1can be
tightly approximated as in (17).
Proof: See Appendix B.
The above two expressions (16) and (17) can be used
to evaluate the average BLER of users in NOMA systems.
However, due to the complex structure of (16) and (17), we
3
¯ε1≈1 + 2δ2,m ϖe ϖ
α2√m
α1eϖ
α2Ei −2ϖ
ϑν2,m −eϖ
α2Ei −2ϖ
ϑµ2,m −Ei −ϖ
ϑν2,m + Ei −ϖ
ϑµ2,m +δ2,m√m
α1e−˜ν2,m
2ϑν2,m
×e−˜ν2,m
2−2−e−˜µ2,m
2ϑµ2,m e−˜µ2,m
2−2+1 −2δ1,mα2√m
ϖ
e−µ1,mϖ
α2
e−µ1,mϖ
α2
4−1
−e−ν1,mϖ
α2
e−ν1,mϖ
α2
4−1
.(17)
cannot further optimize the power allocation coefficients α1
and find the minimum common blocklength m∗. We are thus
motivated to seek for a simpler expression for ¯εi.
B. Power and blocklength optimization at high SNR
Revisiting (12), we can find that ¯εij is an integral over
[νij , µij ]and µij −νij =2π(22Ni/m −1)/m is a small number
since Niis usually small in short-packet communications
[2]. In this case, by using the first order Riemann integral
approximation b
af(x)dx= (b−a)fa+b
2, the asymptotic
expression of ¯ε2for high SNR can be approximated as
¯ε2≈1−e−2β2,m
λρ(α2−α1β2,m)ρ→∞
≈2β2,m/(λρ(α2−α1β2,m )) .(18)
By substituting (18) into the optimization condition (9b), i.e.,
¯ε2= ¯εth
2, we can attain the minimum blocklength m2for u2
to meet the target average BLER ¯εth
2as below:
m2≈N2/log22 + λρ¯εth
2/2 + α1λρ¯εth
2.(19)
Similarly, by using the first order Riemann integral approxi-
mation, ¯
ε1can be approximated as
¯ε1≈1−e−β1,m
λρα12
+1−e−β2,m
λρ(α2−α1β2,m)2
ρ→∞
≈(β1,m/(λρα1))2+¯εth
2/22.(20)
By substituting (20) into the optimization condition (9b), i.e.,
¯ε1= ¯εth
1, we can obtain the minimum blocklength m1for u1
to meet the target average BLER ¯εth
1as below:
m1≈N1/log21 + α1λρ¯εth
1−¯εth
2/22.(21)
After deriving the approximated expression of m1and m2
at high SNR, we can obtain the approximated optimal power
allocation coefficient α∗
1given in Lemma 1.
Lemma 1: The optimal power allocation coefficient α∗
1can
be solved under the condition that m1=m2.
Proof: The optimal blocklength should meet the re-
liability constraints of users simultaneously, i.e, m∗=
min (max (m1, m2)). By taking the derivation of m1with
respect to α1, we have ∂ m1
∂α1=−N1ϱ1ln 2
(1+ϱ1α1) ln(1+ϱ1α1)2<0where
ϱ1=λρ˜εth
1and ˜εth
1=¯εth
1−¯εth
2/22. That is, m1is a decreasing
function of α1. In contrast, ∂m2
∂α1=N ϱ2ln 2
(2+ϱ2α1)(ln 2+ϱ2
2+ϱ2α1)2>0where
ϱ2=λρ¯εth
2, i.e., m2is an increasing function of α1. In this case,
the optimal α∗
1to minimize m∗can be attained by solving
m1=m2. The proof is completed.
According to Lemma. 1, at high SNR, the optimal power
allocation coefficient α∗
1can be approximated as
α∗
1≈2 + λρ¯εth
2/2N1
N2−1/λρ˜εth
1.(22)
We can see from (22) that α∗
1is an increasing function of N1
but a decreasing function of ¯εth
1. That is, when the number of
transmitted bits N1or the reliability constraint ¯εth
1are more
Transmit SNR ρ (dB)
2 6 10 14 18 22 26 30
Average BLER
10-5
10-4
10-3
10-2
10-1
100
NOMA: u1-simulation
NOMA: u1-analytical Eq. (17)
NOMA: u1-analytical Eq. (20)
NOMA: u2-simulation
NOMA: u2-analytical Eq. (16)
NOMA: u2-analytical Eq. (18)
Solida line: α1=0.1
Dashed line: α1=0.2
Fig. 1: ¯εivs. ρ,m= 100 channel
uses.
Power allocation coefficient α1
0.02 0.04 0.06 0.08 0.10 0.06 0.07 0.08 0.09 0.10
Min. blocklength mi (channel uses)
100
200
300
400
500
600
700
800
900 NOMA: minimum blocklength for u1 (m1)
NOMA: minimum blocklength for u2 (m2)
α1
* by using (22)
Fig. 2: mivs. α1,ρ=35dB.
demanding, more power should be allocated to u1. In contrast,
when the number of transmitted bits N2or the reliability
constraint ¯εth
2are more stringent, more power will be allocated
to u2and thus α∗
1decreases. By substituting (22) to (19),
we can approximate the minimum common blocklength to
guarantee the target average reliability of users at high SNR
as below:
m∗≈N2/log22 + λρ¯εth
2/2 + α∗
1λρ¯εth
2.(23)
C. Comparison to OMA
In conventional OMA (e.g., TDMA) systems, the transmis-
sion is interference-free and thus the minimum sum block-
length ˆm∗is the summation of ˆmifor i∈ {1,2}, where ˆmiis
the minimum blocklength for uiin OMA. Similarly, at high
SNR, ˆmican be approximated as
ˆm1≈N1/log21 + λρ¯εth
1,(24)
ˆm2≈N2/log22 + λρ ¯εth
2/2,(25)
In this case, when the reliability constraints ¯εth
1and ¯εth
2are
at the similar level, the blocklength reduction of NOMA
compared to OMA can be characterized as
∆m= ˆm∗−m∗
≈N1/log21 + λρ¯εth
1= ˆm1.(26)
Remark 1: From (26), we can see the blocklength reduction
of NOMA compared to OMA is approximated to ˆm1. That is,
given the reliability constraints, OMA needs ˆm1+ ˆm2channel
uses to serve u1and u2. However, in NOMA, by properly
optimizing the power allocation among the served users, only
ˆm2channel uses are needed to serve u1and u2simultaneously.
Moreover, we can see from (27) that ∆mis an increasing
function of N1.
IV. NUMERICAL STU DI ES
In this section, we present some numerical results to val-
idate our theoretical analysis in section III. Without loss of
generality, we assume imperfect SIC and E(|hi|2)= 1,
¯εth
1= 10−5,¯εth
2= 10−4,N1=N2= 80 bits, and the
minimum blocklength should be larger than 100 channel uses
to ensure the tightness of the Polyanskiy bound as in [8].
In Fig. 1, the analytical expressions of the average BLER
for NOMA are validated using Monte Carlo simulations. We
can see that the derived analytical expressions match well with
4
Transmit SNR ρ (dB)
30 31 32 33 34 35
Min. blocklength (channel uses)
0
200
400
600
800
1000
1200 NOMA: m∗by Eq.(23)
OMA: ˆm∗by Eq. (24)+ (25)
OMA: ˆm1by Eq. (24)
∆mby Eq. (26)
Fig. 3: Minimum Blocklength vs. transmit SNR ρ,α∗
1is
obtained with (22).
the simulation counterparts, which verifies the correctness of
our theoretical analysis in section III.
In Fig. 2, the minimum common blocklength for u1and u2
for various power allocation coefficient α1is presented. We
can see that m1is a decreasing function of α1while m2is
an increasing function of α1. In particular, the optimal power
allocation coefficient α∗
1derived from (22) coincides with the
Lemma 1 wherein m∗=m1=m2.
In Fig. 3, the minimum sum blocklength of NOMA is
compared to that of OMA. We can observe that the minimum
blocklength m∗in NOMA is much smaller than ˆm∗in OMA.
Moreover, we can see that the blocklength reduction ∆mcan
be tightly approximated by that of ˆm1, which validates the
approximation derived in (26).
V. CONCLUSION
In this letter, the physical-layer transmission latency reduc-
tion enabled by NOMA in short-packet communications has
been investigated. Specifically, the closed-form expressions for
the BLER of users were firstly derived in the finite blocklength
regime. Based on this, the near-optimal power allocation and
blocklength were further obtained to guarantee required user
reliabilities. The numerical results validated the theoretical
analysis and demonstrated the superior performance of NOMA
over OMA in short-packet communications.
APPENDIX A
PROO F OF PROPOSITION. 1
Given N2and m, the average BLER at u2, i.e., ¯ε2can be
approximated as below:
¯ε2≈δ2,m√mµ2,m
ν2,m
Fγ22 (x) dx
z=2x
λρ(α2−α1x)
= 1 −2δ2,m√mα2
λρα2
1˜µ2,m
˜ν2,m
e−z
(2
λρα1+z)2dz
t=z+2
λρα1
= 1 −2δ2,m√mα2e2
λρα1
λρα2
1˜µ2,m+2
λρα1
˜ν2,m+2
λρα1
e−t
t2dz
(a2)
= 1 −2δ2,mϖ e 2ϖ
α2√m
α1
×Ei −2ϖ
ϑν2,m −Ei −2ϖ
ϑµ2,m
−δ2,m√m
α1e−˜ν2,m ϑν2,m −e−˜µ2,m ϑµ2,m
where step (a2) is based on [15, Eq. (3.351.4)]. In particular,
ϖ=α2
λρα1,ϑν2,m =α2−α1ν2,m,ϑµ2,m =α2−α1µ2,m ,
˜ν2,m =2ν2,m
λρϑν2,m
and ˜µ2,m =2µ2,m
λρϑµ2,m
.
APPENDIX B
PROO F OF PROPOSITION. 2
As ¯ε1is a function of both ¯ε12 and ¯ε11, we first evaluate ¯ε1j,
j={1,2}, respectively. Similar to the proof of Proposition 1,
we have
¯ε12 ≈δ2,m√mµ2,m
ν2,m
Fγ12 (x) dx
= 1 + 2δ2,mϖe ϖ
α2√m
α1eϖ
α2Ei −2ϖ
ϑν2,m
−eϖ
α2Ei −2ϖ
ϑµ2,m −Ei −ϖ
ϑν2,m + Ei −ϖ
ϑµ2,m
+δ2,m√m
α1e−˜ν2,m
2ϑν2,m e−˜ν2,m
2−2
−e−˜µ2,m
2ϑµ2,m e−˜µ2,m
2−2 (27)
For ¯ε11, we have
¯ε11 ≈δ1,m√mµ1,m
ν1,m
Fγ11 (x) dx
= 1 −2δ1,mα2√m
ϖ
e−µ1,mϖ
α2
e−µ1,mϖ
α2
4−1
−e−ν1,mϖ
α2
e−ν1,mϖ
α2
4−1
.(28)
By substituting (27)-(28) to (7) and after some algebraic
manipulations, ¯ε1can be rephrased as in (17).
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