Average Power Minimization for Downlink NOMA Transmission with Partial HARQ

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Average Power Minimization for Downlink NOMA
Transmission with Partial HARQ
Yanqing Xu∗, Donghong Cai†, Zhiguo Ding‡, Chao Shen∗, and Gang Zhu∗
∗State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China
†Institute of Mobile Communications, Southwest Jiaotong University, Chengdu, China
‡School of Electrical and Electronic Engineering, The University of Manchester, UK
Email: {xuyanqing,chaoshen,gzhu}@bjtu.edu.cn, cdhswjtu@163.com, zhiguo.ding@manchester.ac.uk
Abstract—In this paper, we consider the average transmit
power minimization problem in a non-orthogonal multiple access
(NOMA) system under the strict outage constraints of users.
In particular, we assume the base station (BS) only knows
the statistical channel state information (CSI) of the users and
the users have heterogeneous quality of service requirements.
Without loss of generality, we assume user 1 is a delay-aware user
with low target rate requirement and user 2 is a delay-tolerant
user with high target rate requirement. Due to the high target rate
requirement of user 2, it may fail to decode its own information.
Thus we proposed a partial hybrid automatic repeat request
(HARQ) scheme to guarantee the communication reliability of
user 2. By deriving the outage probability of users as closed-form
expressions, the average transmit power minimization problem
boils down to a nonconvex optimization problem and challenging
to solve. We then use a successive convex approximation (SCA)
based algorithm to handle the approximated problem iteratively,
which can guarantee to converge to at least a stationary point
of the problem. The simulation results show the efficacy of the
proposed NOMA transmission scheme and SCA-based algorithm.
Index Terms—Non-orthogonal multiple access, partial hybrid
retransmission repeat request, successive convex approximation
I. INTRODUCTION
As a promising candidate for 5G, the non-orthogonal mul-
tiple access (NOMA), has been intensively studied recently
[1, 2]. Different to the conventional orthogonal multiple access
(OMA) scheme, NOMA allows the transmitter severs multiple
users at the same resource block but with different power levels
simultaneously; while at the receiver side, sophisticated multi-
user detection, e.g., the successive interference cancellation
(SIC), technique can be exploited at the receivers who have
stronger channel conditions to remove the interference from
the other receivers with poorer channel conditions. Conse-
quently, the system spectral efficiency can be greatly increased
[3].
Traditionally, the system design of NOMA assumes that
the transmitter has the perfect channel state information (CSI)
feedback, e.g., [4, 5]. However, this assumption is not practical
The work of Y. Xu and C. Shen were supported by the Fundamen-
tal Research Funds for the Central Universities (No. 2018YJS206), and
the NSFC (61871027 and 61725101), Beijing NSF (L172020), and Ma-
jor projects of Beijing Municipal Science and Technology Commission
(Z181100003218010). The work of Z. Ding was supported by the UK
Engineering and Physical Sciences Research Council under grant number
EP/P009719/1 and by H2020-MSCARISE-2015 under grant number 690750.
as the channel condition changes rapidly in wireless communi-
cations. In view of this, by assuming the base station (BS) only
knows the statistical CSI, [6] investigated the power allocation
problem to respectively maximize the fairness data rate and
energy efficiency of the system under outage constraints. Due
to the imperfect CSI, the communication reliability of the
system can be badly influenced. To combat this, the hybrid
automatic repeat request (HARQ) can be incorporated to im-
prove the communication reliability of the system, e.g., [7, 8].
In particular, [7] considered the power assignment problem
with the assumption that the channel keeps invariant during
retransmissions for a packet. Then [8] extended the work of [7]
to the cases that the channels in different retransmissions are
independent and identically distribution (i.i.d.). However, both
[7] and [8] focused on the point-to-point system. To benefit
the advantage of NOMA, recently the HARQ transmission
scheme was applied to the system design in a two user
NOMA system in [9–12]. In particular, [9, 10] assumed the
communication is successful only when the both users can
decode their packets and [11, 12] mainly focused on the outage
performance analysis of the considered system.
In this paper, we consider the power allocation problem
in a partial HARQ aided NOMA system to minimize the
average transmit power under outage constraints for each user.
Different to [11, 12], we use the derived closed-form outage
probabilities to further optimize the system performance. Also
unlike [9, 10], where conventional HARQ scheme is used,
more flexible partial HARQ with chase combining (HARQ-
CC) strategy is applied in our system to minimize the average
transmit power. However, the derived outage probabilities
bring great challenges to our system design. By properly
utilizing the structure of the original problem, we propose
a successive convex approximation (SCA) based algorithm
to iteratively solve the relaxed problem. What’s more, the
proposed algorithm can be guaranteed to converge to at least a
stationary point of the problem. In the simulation section, the
efficiency of the proposed NOMA transmission scheme and
SCA-based algorithm is verified.
II. SYSTEM MODEL AND PROBL EM FORMULATION
Consider a downlink NOMA transmission scenario where
a single-antenna BS serves two single-antenna users. Here,
we assume that the BS only knows the statistical channel
978-1-5386-4920-6/18/$31.00 ©2018 IEEE

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Fig. 1. An illustration of a partial HARQ enabled NOMA system.
knowledge, i.e., the distances of users to the BS and the distri-
bution of small-scale fading, of the CSI. Such an assumption
is practical for real applications, as limited CSI feedback is
needed. Here we assume user 1 is located farther from the BS
than user 2 and the two users have heterogeneous quality of
service (QoS) requirements. In particular, user 1 is a delay-
aware user with low data rate requirement, e.g., this user is
carrying out phone calls or other types of real time tasks; user
2 is a delay-tolerant user with high data rate requirement, e.g.,
this user is downloading a movie or other delay-tolerant tasks.
Note that user 1 has a low target rate requirement, thus
it can be served with high reliability. However, due to the
high target rate requirement and channel fading, user 2 may
fails to decode its information. In view of this, a partial
HARQ-CC scheme is applied in our system to improve the
communication reliability performance of user 2. In the partial
HARQ scheme, the BS sends a new packet to user 1 during
each time slot; while for user 2, the same packet will be sent.
In each time slot, user 2 combines the received information
from the BS with maximum ratio combination (MRC) to do
joint decoding. If user 2 succeed to decode its information, a
one-bit acknowledgement (ACK) information will be returned
to the BS, otherwise a non-acknowledgement (NACK) will
be sent to the BS. If the BS receives a NACK, then it will
retransmit the same packet to use 2 until Tretransmissions is
reached, where Tis the maximum tolerable retransmissions
to guarantee the latency requirement of user 2.
The NOMA transmission scheme is used to improve the
spectral efficiency of the system. As per the NOMA ra-
tionale, the transmit signal of the BS in t-th time slot is
xt=√p1tx1t+√p2tx2t, where p1tand p2tare the transmit
power, x1tand x2tare the unit-power signal for user 1 and
user 2 respectively. The received signal at user kin the t-th
time slot is given by
yt=¯
hkt
!1+dα
k
(√p1tx1t+√p2tx2t)+nkt,k=1,2,(1)
where dkis the distance of user kto the BS, αis the path loss
component, ¯
hkt ∼CN(0,1) models the small-scale Rayleigh
fading and nkt ∼CN(0,σ
2
k)is the additive white Gaussian
noise (AWGN) at user kin the t-th time slot.
According to the NOMA principle, user 2 first carries out
SIC to remove the information of user 1 and then decodes
its own information. The received signal-to-interference-noise-
ratio (SINR) and signal-to-noise-ratio (SNR) to decode x1tand
x2tcan be described as
SINRx1t
2t=p1th2tλ2
p2th2tλ2+1,(2a)
SNRx2t
2t=
t
"
ℓ=1
p2ℓh2ℓλ2,(2b)
where h2t=|¯
h2t|2and λ2=1
(1+dα
2)σ2
2. Note that (2b) is
due to the HARQ-CC protocol is applied at user 2, thus it
can jointly decodes its information by combining the received
signals from the former time slots. On the other hand, user 1
decodes its information directly by treating x2tas noise, then
its received SINR to decode x1tis given by
SINRx1t
1t=p1th1tλ1
p2th1tλ1+1,(3)
where h1t=|¯
h1t|2and λ1=1
(1+dα
1)σ2
1. The outage event is
defined as the received SINRs and SNRs of users are smaller
than their target SNR requirements, i.e., γ1and γ2. Hence,
the outage probabilities of users in the t-th time slot can be
described as
P1t= Pr (SINRx1t
1t<γ
1),(4a)
P2t=1−Pr(SINRx1t
2ℓ≥γ1,for ℓ=1,..,t,SNRx2t
2t≥γ2).(4b)
To guarantee the communication reliability of users, the
following outage constraint should be satisfied with
P1t≤δ1,∀t, (5a)
P2T≤δ2,(5b)
where δk,k =1,2, is the tolerable outage probability of user
k. By letting P20 =1, we define the average transmit power
of user 2 in the t-th time slot as Pavg,t =p2tP2,t−1. Based
on the above definitions, we formulate the average transmit
power minimization problem as follows:
min
p1,p2
T
"
t=1
(p1t+p2tP2,t−1)(6a)
s.t.P1t≤δ1,∀t, (6b)
P2T≤δ2,(6c)
p1t>p
2t≥0,(6d)
p1t+p2,t ≤Pmax,(6e)
where (6b) and (6c) are the outage constraints, constraint
(6d) is due to the superposition coding requires a power gap
between two signals for SIC and meanwhile guarantee the
user-fairness. constraint (6e) is the maximum transmit power
constraint with Pmax denoting the maximum transmit power.
Problem (6) is challenging to solve mainly due to the outage
probabilities don’t have closed-form expressions. Thus in the
following we first present the outage probability analysis of
the partial HARQ enabled NOMA system.

III. OUTAGE PROBABILITY ANALYSIS
First, let’s focus on the outage probability of user 1 in each
time slot. Recall user 1’s outage probability as
P1t=Pr#p1th1tλ1
p2th1tλ1+1 <γ
1$(7a)
=Pr(p1th1tλ1<γ
1p2th1tλ1)(7b)
=Pr#h1t<γ1
(p1t−γ1p2t)λ1$.(7c)
Note the fact that the channel gain h1tfollows the exponential
distribution, thus the outage probability of user 1 in each time
slot can be written as
P1t=1−e−γ1
(p1t−γ1p2t)λ1,∀t. (8)
Due to the fact that SINRx1t
2t’s in (4b) are independent,
thus the outage probability of user 2 after Ttime slots can be
rewritten as
P2T=1−Pr(SNRx2t
2T≥γ2|SINRx1t
21 ≥γ1, ..., SINRx1t
2T≥γ1)
×%T
t=1 Pr(SINRx1t
2t≥γ1).(9)
Notice that to guarantee a high communication reliability,
the tolerable outage probabilities of users are generally small
numbers, e.g., δk= 10−2,k =1,2,or even smaller. Also note
that user 2 has a better channel condition than user 1 with a
high probability due to its smaller path loss (closer to the BS).
Thus the outage probability of SIC procedure to decode x1tat
user 2 will be even smaller than δ1, i.e., Pr(SINRx1t
2t≥γ1)>
1−δ1≈1. Hence, the outage probability of user 2 after T
time slot transmissions can be approximated as
P2T≈Pr (SNRx2t
2T<γ
2)(10a)
=Pr&T
"
t=1
p2th2tλ2<γ
2'.(10b)
The approximated outage probability of user 2 in (10b) is
given in the following theorem.
Theorem 1 The outage probability in (10b) can be described
as a closed-form expression as follow:
P2T≈
M
"
m=1
wm
T
%
t=1
1
1+mλ2ln 2
γ2p2t
,(11)
where wmis given by
wm=(−1) M
2+m
min{m, M
2}
"
n=⌊(m+1 )
2⌋
nM
2(2n)!
m(M
2−n)!n!(n−1)!(m−n)!(2n−m)! .
Proof: The proof of Theorem 1 is relegated to Appendix
A.
IV. SOLVING THE AVERAGE POWE R MINIMIZATION
PROBL EM
In this section, based on the derived outage probabilities in
(8) and (11), we solve the power allocation problem in (6).
By inserting the outage probabilities in (8) and (11), problem
(6) can be rewritten as
min
p1,p2
T
"
t=1 &p1t+p2t
M
"
m=1
wm
t−1
%
ℓ=1
1
1+gmp2ℓ'(12a)
s.t.1−e−γ1
(p1t−γ1p2t)λ1≤δ1,∀t, (12b)
M
"
m=1
wm
T
%
t=1
1
1+gmp2,t ≤δ2,(12c)
p1t>p
2t≥0,(12d)
p1t+p2,t ≤Pmax,(12e)
where gm=mλ2ln 2
γ2. Note that constraint (12b) can be written
as a convex one as follows:
p1t−γ1p2t≤γ1
−λ1ln(1 −δ1).(13)
So the main problem in solving problem (12) is due to
the outage probability of user 2, which is a complicated
multiplication and the parameter wms have negative values,
i.e., w2,w
4,w
6,...
. In the following subsection, we will show
how to use a convex approximation based algorithm to solve
this problem.
A. Successive Convex Approximation Method for Problem (12)
First, introduce auxiliary variables utand vto the objective
function of problem (12) and rewrite it as
min
T
"
t=1
p1t+p21 +v(14a)
s.t.
T
"
t=2
p2tut≤v, (14b)
M
"
m=1
wm
t−1
%
ℓ=1
1
1+gmp2ℓ≤ut.(14c)
then by defining
ezm,t !1
1+gmp2t
.(15)
problem (12) can be rewritten as
min
p1,p2
ut,v,{zm,t}
T
"
t=1
p1t+p21 +v(16a)
s.t.p
1t−γ1p2t≤γ1
−λ1ln(1 −δ1),(16b)
T
"
t=2
p2tut≤v, (16c)
M
"
m=1
wme(zm,1+...+zm,t−1)≤ut,(16d)
M
"
m=1
wme(zm,1+...+zm,T )≤δ2,(16e)
ezm,t =1
1+gmp2t
,(16f)
p1t>p
2t≥0,(16g)
p1t+p2,t ≤Pmax.(16h)

Algorithm 1 SCA-based algorithm for solving (6)
1: Initialization: Set r=0, given a set of feasible point, and the
desired accuracy ϵ.
2: repeat
3: Solve problem (19) by standard convex solver, and update
ar+1 by ar=!ur
t
pr
2t
4: Set r←r+1.
5: until desired accuracy is achieved.
6: Output: the obtained {p1t}and {p2t}.
Problem (16) is still nonconvex due the constraints (16c)
- (16f). However, comparing the original problem (12), the
nonconvex constraints can be approximated by convex ones,
and then it allows us to solve the reformulated problem iter-
atively by using a successive convex approximation method.
Specifically, in the r-th iteration, around a feasible point of
the original problem, one need to do convex approximation of
the nonconvex constraints. For instance, constraint (16c) can
be approximated by the arithmetic-geometric-mean inequality
as
T
"
t=2 &(arp2t)2+#ut
ar$2'≤v, (17)
where arcan be updated by ar=*ur−1
t
pr−1
2t
with ur−1
tand
pr−1
2tbeing the value of variable utand p2tin the (r−1)-th
iteration. On the other hand, by performing first-order Tylor
expansion, constraint (16d) can be approximated by
M/2
"
b=1
w2b−1e!t−1
ℓ=1 z2b−1,ℓ+
M/2
"
b=1
w2be!t−1
ℓ=1 z(r−1)
2b,ℓ
&1+
t−1
"
ℓ=1
z2b,ℓ−
t−1
"
ℓ=1
z(r−1)
2b,ℓ'≤ut.(18)
The other constraints can be treated similarly. As a result, the
problem (16) can be approximated by
min
p1,p2
ut,v,{zm,t}
T
"
t=1
p1t+p21 +v(19a)
s.t.(16b),(16g),(16h),(17),(18),(19b)
M/2
"
b=1
w2b−1e!T
t=1 z2b−1,t +
M/2
"
b=1
w2be!T
t=1 z(r−1)
2b,t
&1+
T
"
t=1
z2b,t −
T
"
t=1
z(r−1)
2b,t '≤ut,(19c)
ez(r−1)
m,t +1+zm,t −z(r−1)
m,t ,−1
1+gmp(r−1)
2t
+gm
+1+gmp(r−1)
2t,2+p2t−p(r−1)
2t,=0.(19d)
which is convex problem and thus can be efficiently solved by
the interior point method based solver, e.g., CVX. The proce-
dure of the SCA-based algorithm is outlined in Algorithm 1,
and in fact, we can draw the following proposition.
0 5 10 15 20 25 30 35 40
SNR(dB)
10-5
10-4
10-3
10-2
10-1
100
Outage of user 2
Analysis Results
Simulation Results
T = 1
T = 2
T = 3
Fig. 2. Outage probability of user 2 with γ1=0.2,γ2=2, and p1
p2
=3
2.
Proposition 1 The proposed algorithm can continuously de-
crease the power consumption gap between two successive
iterations and guarantee the generated power consumption
sequence converges to at least a stationary point.
Proof: The proof of Proposition 1 is similar as that in
[13, Theorem 1], thus we omit it here. "
V. S IMULATION RESULTS
In this section, simulation results are given to show the
accuracy of the derived outage probability and the performance
of the Algorithm 1. Without loss of generality, in the simula-
tions, we assume the users have the same outage requirements,
i.e., δ1=δ2=δ, the target SNRs of users are given by
γ1=0.2,γ
2=2. The distances of users to the BS are d1=8,
d2=4, and the path loss exponent is set to be α=2. The
noise power is set to be σ2
1=σ2
2=0.1. The FDMA and equal
power allocation (EPA, the transmit power keeps constant in
different time slots) schemes are used as a benchmarks to
evaluate the performance of the proposed NOMA transmission
scheme and the proposed power allocation scheme.
Fig. 2 shows the accuracy of the derived outage probability
of user 2, where the red curves are the analysis results plotted
based on (11) and the blue curves are the simulation results
and plotted based on (10b). One can observe that the derived
closed-form expression can approximate the outage probability
with high quality even with low-SNR and small number of
retransmissions.
Fig. 3 presents the relationship between the total average
power and the outage requirements of users. It can be observed
that the average transmit power decreases with the increase
of the value of outage requirement (larger value denotes
less strict outage requirement). One can also see that the
proposed NOMA scheme can achieve the best performance
and the performance gap between the proposed adaptive power

allocation scheme and the EPA scheme. Furthermore, the
performance of the proposed successive convex approximation
based algorithm is very close to that of the optimal perfor-
mance by the exhaustive search method.
VI. CONCLUSION
In this paper, we have considered the average transmit
power minimization problem under outage constraints in a
downlink NOMA transmission system with partial HARQ.
In particular, by deriving the closed-form expressions of the
outage probabilities, the average transmit power minimization
problem has been boiled down to a nonconvex optimization
problem. With the aid of the successive convex approximation
method, the problem has been solved iteratively and can
be guaranteed to converge to a least a stationary point of
the problem. Simulation results have shown the accuracy of
the derived outage probability and also the efficiency of the
proposed transmission scheme and SCA-based algorithm.
APPENDIX A
PROOF O F THEOREM 1
Firstly, by defining Xt=p2,th2,tλ2and X!
{X1,...,X
T}, the joint probability density function of
X1,...,X
Tcan be written as
fX(x1,..,x
T)=
T
%
t=1
fXt(xt)(20a)
=1
λT
2&T
%
t=1
1
p2,t 'e−
T
!
t=1
xt
p2,tλ2.(20b)
Then its moment-generating function can be written as
MX(s)=-+∞
0
fX(x)esxdx (21a)
=&T
%
t=1
1
p2,tλ2'T
%
t=1 -+∞
0
e−xt(1+sλ2p2,t)
p2,tλ2dxt(21b)
=&T
%
t=1
1
p2,tλ2'T
%
t=1
p2,tλ2
1−sλ2p2,t
(21c)
=
T
%
t=1
1
1−sλ2p2,t
.(21d)
Applying the inverse Laplace transform to (21d), we have
FX(x)=L−1#L#-+∞
0
fX(x)dx$$ (22a)
=L−1#1
sMX(−s)$(22b)
=L−1&1
s
T
%
t=1
1
1+sλ2p2,t '(22c)
≈
M
"
m=1
wm
T
%
t=1
1
1+mλ2ln 2
xp2,t
,(22d)
where (22d) is obtained based on the Gaver-Stehfest procedure
[14]. Thus the outage of user 2 can be given by P2,T =
FX(γ2). This completes the proof. "
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Outage Requirement δ
5
10
15
20
25
Average Transmit Power
NOMA:Optimalw/ExhaustiveSearch
NOMA:6XERptimalw/Alg.1
NOMA:EPA
OMA:FDMA
Fig. 3. Total average transmit power versus the outage requirements with
T=2and Pmax =40watt.
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