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1
NOMA-UAV Networks via Updating Decoding
Order
Joint Location and Transmit Power Optimization for
Abstract—Unmanned aerial vehicle (UAV) can be combined
with non-orthogonal multiple access (NOMA) to achieve better
performance. However, jointly optimizing the location, transmit
power and decoding order for NOMA-UAV networks remains
difficult, due to the change of decoding order as a result of UAV
mobility. In this letter, a low-complexity scheme is proposed to
maximize the sum rate of NOMA-UAV networks via updating
decoding order, which can be decomposed into two steps. First,
the joint location and power optimization can be divided into
two non-convex sub-problems, which are further approximated
via successive convex optimization. Then, the decoding order is
updated according to the optimized UAV location. An iterative
algorithm is proposed to execute the two steps alternately. In
addition, the asymptotic performance is analyzed. Simulation
results demonstrate the effectiveness of the proposed scheme.
Index Terms—NOMA, power and location optimization, suc-
cessive interference cancellation, UAV.
I. INTRODUCTION
Recently, unmanned aerial vehicles (UAVs) have been wide-
ly used as carrying platforms of base stations in wireless
communications [1], [2]. Many recent studies have been
dedicated to UAV communications [3]–[7]. In [3], the joint
optimization of trajectory and transmit power was studied by
Wu et al. to maximize the sum rate in multi-UAV networks.
Yang et al. studied UAV energy tradeoff for the data collection
in UAV networks via trajectory optimization in [4]. Zhao et
al. [5] proposed a novel channel tracking scheme for UAV
mmWave multi-antenna systems. In [6], Gong et al. considered
Manuscript received August 10, 2020; revised September 4, 2020; accepted
September 7, 2020. The work was supported by the National Natural Science
Foundation of China (NSFC) under Grant 61871065 and 61971194, and also
supported by the project “The Verification Platform of Multi-tier Coverage
Communication Network for Oceans (LZC0020)”. The associate editor coor-
dinating the review of this paper and approving it for publication was M. R.
A. Khandaker. (Corresponding author: Nan Zhao.)
a UAV-assisted cellular network, applying the superimposed
training sequence with imperfect channel statistics. UAV-aided
jamming for secure communication with unknown location of
the eavesdropper was investigated in [7] by Nnamani et al.
On the other hand, to improve the spectrum efficiency,
non-orthogonal multiple access (NOMA) is emerging as a
crucial technique for future wireless networks [8]–[10]. In [8],
Chen et al. have proved that NOMA has a better performance
than orthogonal multiple access (OMA). A resource allocation
algorithm for NOMA networks was proposed by Chang et al.
to improve the secrecy energy efficiency [9]. In [10], Lei et
al. proposed a max-min transmit antenna selection scheme for
NOMA systems with secrecy outage performance analyzed.
Due to their advantages, it becomes natural to integrate
UAV and NOMA for enhancing the performance, and some
fundamental works have been done in [11], [12]. Liu et al. set
up a general framework for NOMA-UAV networks in [11].
Mei and Zhang proposed a cooperative NOMA scheme for
cellular-connected UAV networks in [12]. Recently, plenty of
research on resource allocation of NOMA-UAV network has
been conducted [13]–[17]. In [13], Tang et al. proposed a
UAV placement scheme to maximize the number of users
in a NOMA-UAV network. A joint placement and power
optimization scheme was proposed by Liu et al. for NOMA-
UAV networks in [14]. The joint optimization of altitude and
beamwidth was considered in a NOMA-UAV network by Nasir
et al. in [15]. Liu et al. proposed a distributed NOMA-UAV
scheme to assist emergency communications [16]. Wang et
al. [17] proposed a UAV-aided NOMA scheme with secure si-
multaneous wireless information and power transfer. Resource
allocation for optimizing the energy efficiency in NOMA-UAV
network was introduced in [18], [19].
Motivated by above works, we focus on the system design
for NOMA-UAV networks. The location and transmit power of
UAV are jointly optimized to maximize the sum rate. Different
from above works with fixed decoding order, we propose an
iterative algorithm to update the current decoding order in
the ascending order of channel gains after each iteration. Nu-
merical results show that the proposed scheme can effectively
improve the performance of NOMA-UAV networks.
II. SYSTEM MODEL AND PROBLEM FORMULATION
A. System Model
Consider a NOMA-UAV network with one UAV and K
ground users. All are equipped with a single antenna. Define
2
U
i
as the ith user, i ∈ K , {1, 2, . . . , K}. The superimposed
information is transmitted from the UAV to the users via
NOMA. The received signal at U
i
is given by
s
i
= h
i
K
j=1
z
j
+ n
i
, i ∈ K, (1)
where h
i
represents the channel coefficient from the UAV to
U
i
, and n
i
denotes the additive white Gaussian noise (AWGN)
at U
i
. z
j
is the message for U
j
with |z
j
|
2
= a
j
P
sum
= P
j
,
where P
sum
is the sum transmit power of UAV, a
j
is the power
coefficient of U
j
, and P
j
is the transmit power for U
j
.
Assume that the horizontal coordinates of U
i
is denoted as
q
i
= [x
i
, y
i
]
T
∈ R
2×1
. The UAV hovers at a fixed altitude
H and its horizontal coordinate is L = [X, Y ]
T
∈ R
2×1
. The
distance between the UAV and U
i
can be expressed as
d
i
=
H
2
+ ∥q
i
− L∥
2
. (2)
The probability of UAV-to-ground links dominated by line-
of-sight (LoS) can be expressed as
P
LoS
i
=
1
1 + a
0
exp(−b
0
(θ
i
− a
0
))
, (3)
where a
0
and b
0
denote the environment constants. θ
i
=
arcsin
H
d
i
represents the elevation angle between the UAV
and U
i
. According to an extensive survey for UAV channel
modeling [20], when the UAV is located high enough (e.g.,
120m), the LoS probability is approximate to 1. Thus, the
channel from the UAV to U
i
can be denoted as
h
i
=
ρ
0
d
−2
i
=
ρ
0
H
2
+ ∥q
i
− L∥
2
, (4)
where ρ
0
denotes the reference channel gain at the unit
distance.
According to NOMA, the successive interference cancella-
tion (SIC) is applied for users to receive their own messages,
and users with low channel gains are compensated by high
power allocation ratios. Therefore, according to the distance
from the UAV to users, we assume |h
1
|
2
≥ · · · ≥ |h
i
|
2
≥
· · · ≥ |h
K
|
2
> 0 with 0 < a
1
≤ · · · ≤ a
i
≤ · · · ≤ a
K
.
U
i
needs to decode the messages from U
i+1
to U
K
and
removes them from the superposed signal. Then, we can
express the signal-to-interference-plus-noise-ratio (SINR) for
U
i
(2 ≤ i ≤ K) as
SINR
i
=
|h
i
|
2
P
i
|h
i
|
2
i−1
k=1
P
k
+ σ
2
=
P
i
i−1
k=1
P
k
+
σ
2
|h
i
|
2
, (5)
where σ
2
is the power of AWGN at user receivers. When
i = 1, the received SINR can be calculated by
SINR
1
=
P
1
|h
1
|
2
σ
2
. (6)
Thus, the transmission rate for U
i
can be denoted as
R
i
= log
2
(1 + SINR
i
) , i ∈ K. (7)
We assume γ
i
denotes the SINR threshold of U
i
, which can
be expressed as
SINR
i
≥ γ
i
, i ∈ K. (8)
Accordingly, the rate threshold of U
i
is obtained from (8) as
η
i
= log
2
(1 + γ
i
), i ∈ K. (9)
B. Problem Formulation
Define P = {P
i
, i ∈ K}. To maximize the sum rate of
ground users via jointly optimizing L and P based on (7) and
(8), the optimization problem can be formulated as
max
L,P
i∈K
log
2
(1 + SINR
i
) (10a)
s.t. SINR
i
≥ γ
i
, (10b)
0 < P
1
≤ · · · ≤ P
i
≤ · · · ≤ P
K
, (10c)
K
i=1
P
i
≤ P
sum
. (10d)
Combining (5), (6) and (7), the problem (10) can be
rewritten as
max
L,P
i∈K
log
2
(1 + SINR
i
) (11a)
s.t.
P
i
i−1
k=1
P
k
+
σ
2
|h
i
|
2
≥ γ
i
, ∀i ∈ K\{1}, (11b)
P
1
|h
1
|
2
/σ
2
≥ γ
1
, (11c)
0 < P
1
≤ · · · ≤ P
i
≤ · · · ≤ P
K
, (11d)
K
i=1
P
i
≤ P
sum
. (11e)
In (11), the constraints (11b) and (11c) are non-convex with
respect to L and P. Thus, approximation will be used in the
next section.
Different from the fixed SIC order [14], the decoding order
is updated after each iteration according the ranking of channel
gains in this letter, with the stronger user decoded later.
III. ITERATIVE ALGORITHM FOR THE OPTIMIZATION
The problem (11) is difficult to solve due to its non-
convexity. Thus, we propose a scheme to optimize the location
and power alternately via successive convex optimization.
A. Transmit Power Optimization
First, we fix the UAV location and (11) becomes
max
P
i∈K
log
2
(1 + SINR
i
) (12a)
s.t.
P
i
i−1
k=1
P
k
+
σ
2
|h
i
|
2
≥ γ
i
, i ∈ K\{1}, (12b)
(11c), (11d), (11e). (12c)
(12c) is convex. (12b) is non-convex and its left-hand-side
can be replaced by R
i
. Thus, it can be changed into two
concave functions with respect to P as
R
i
= log
2
(1 + SINR
i
)
= log
2
1 +
|h
i
|
2
P
i
|h
i
|
2
i−1
k=1
P
k
+ σ
2
= log
2
|h
i
|
2
i
k=1
P
k
+σ
2
−log
2
|h
i
|
2
i−1
k=1
P
k
+σ
2
.
(13)
3
We need to approximate the second concave function via
the first-order Taylor expansion at a specific point to obtain
its global upper-bound. Define the transmit power in the rth
iteration and the second concave function as P
r
and
¯
R
i
,
respectively, and we have
¯
R
i
= log
2
|h
i
|
2
i−1
k=1
P
k
+ σ
2
≤
i−1
k=1
A
r
i
(P
k
− P
r
k
) + B
r
i
,
¯
R
[ub]
i
,
(14)
where A
r
i
and B
r
i
can be calculated as
A
r
i
=
|h
i
|
2
log
2
(e)
|h
i
|
2
i−1
l=1
P
r
l
+ σ
2
, (15)
B
r
i
= log
2
|h
i
|
2
i−1
k=1
P
r
k
+ σ
2
. (16)
Therefore, (12) becomes convex as
max
P
i∈K
log
2
(1 + SINR
i
) (17a)
s.t. log
2
|h
i
|
2
i
k=1
P
k
+σ
2
−
¯
R
[ub]
i
≥η
i
, ∀i ∈ K\{1}, (17b)
(11c), (11d), (11e), (17c)
which is convex and can be solved by CVX.
B. Location Optimization
Then, we fix the transmit power to transform (11) into
max
L
i∈K
log
2
(1 + SINR
i
) (18a)
s.t.
ρ
0
H
2
+ ∥q
i
− L∥
2
P
i
ρ
0
H
2
+ ∥q
i
− L∥
2
i−1
k=1
P
k
+ σ
2
≥ γ
i
,
∀i ∈ K\{1},
(18b)
P
1
ρ
0
σ
2
H
2
+ ∥q
1
− L∥
2
≥ γ
1
. (18c)
The constraints (18b) and (18c) are non-convex with respect
to L. For (18b), it can be split into two convex functions with
respect to ∥q
i
− L∥
2
as
R
i
=log
2
(1 + SINR
i
)
=log
2
1+
ρ
0
H
2
+ ∥q
i
− L∥
2
P
i
ρ
0
H
2
+ ∥q
i
− L∥
2
i−1
k=1
P
k
+ σ
2
=
˜
R
i
−
ˆ
R
i
,
(19)
where
˜
R
i
= log
2
ρ
0
H
2
+ ∥q
i
− L∥
2
i
k=1
P
k
+ σ
2
, (20)
ˆ
R
i
= log
2
ρ
0
H
2
+ ∥q
i
− L∥
2
i−1
k=1
P
k
+ σ
2
. (21)
Notice that
˜
R
i
is neither concave nor convex with respect to
L. Thus, we define the local point L
r
in the rth iteration and
derive the lower-bounded expression of
˜
R
i
via the first-order
Taylor expansion as
˜
R
i
= log
2
ρ
0
H
2
+ ∥q
i
− L∥
2
i
k=1
P
k
+ σ
2
≥
i
k=1
−C
r
i
∥q
i
−L∥
2
−∥q
i
−L
r
∥
2
+D
r
i
,
˜
R
[lb]
i
,
(22)
where C
r
i
and D
r
i
can be calculated as
C
r
i
=
P
k
ρ
0
H
2
+ ∥q
i
− L
r
∥
2
2
log 2(e)
ρ
0
H
2
+ ∥q
i
− L
r
∥
2
i
l=1
P
l
+ σ
2
, (23)
D
r
i
= log
2
ρ
0
H
2
+ ∥q
i
− L
r
∥
2
i
l=1
P
l
+ σ
2
. (24)
With (19) and (22), (18b) can be transformed into
˜
R
[lb]
i
−
ˆ
R
i
≥ η
i
.
(25)
(25) is still a non-convex constraint due to
ˆ
R
i
. Introducing
V = {V
i
= ∥q
i
− L∥
2
, ∀i},
ˆ
R
i
can be reformulated as
ˆ
R
i
= log
2
ρ
0
H
2
+ V
i
i−1
k=1
P
k
+ σ
2
, (26)
where the slack variable needs to satisfy
V
i
≤ ∥q
i
− L∥
2
, ∀i. (27)
In (27), ∥q
i
− L∥
2
is convex with respect to L, which can be
approximated via Taylor expansion at the given point L
r
as
∥q
i
− L∥
2
≥ ∥q
i
− L
r
∥
2
+ 2 (q
i
− L
r
)
T
(L − L
r
) . (28)
Now, (26) can replace
ˆ
R
i
in (25), which can be turned into
˜
R
[lb]
i
− log
2
ρ
0
H
2
+ V
i
i−1
k=1
P
k
+ σ
2
≥ η
i
.
(29)
(29) is a convex constraint because it is jointly concave with
respect to L
r
and V
i
.
For (18c), it can be regarded as
˜
R
i
with i = 1, which can
be solved in a similar way. Thus, (18) can be made convex as
max
L,V
i∈K
log
2
(1 + SINR
i
) (30a)
s.t.
˜
R
[lb]
i
− log
2
ρ
0
H
2
+ V
i
i−1
k=1
P
k
+ σ
2
≥η
i
,
∀i ∈ K\{1},
(30b)
V
i
≤ ∥q
i
− L
r
∥
2
+ 2 (q
i
− L
r
)
T
(L − L
r
) , (30c)
˜
R
[lb]
1
≥ η
1
, (30d)
which is convex and can be solved by CVX.
4
C. Iterative Algorithm
Based on Section III-A and Section III-B, (11) can be solved
iteratively using Algorithm 1. In Step 3, the decoding order is
updated according to the optimized UAV location.
Algorithm 1 Iterative Algorithm for (11)
Initialization: Set the geometric center of users as the starting
location L
0
=
K
i=1
q
i
/K. The initial decoding order and the
power P
0
are set according to L
0
and the minimum transmit
power. Set the initial index of iterations as r = 0.
while (R(P
r+1
, L
r+1
) − R(P
r
, L
r
) ≤ ϵ
1
) do
1. Solve (17) via L
r
, and obtain P
r+1
.
2. Solve (30) via P
r+1
, and obtain L
r+1
.
3. Update the decoding order according to L
r+1
.
4. Update: r = r + 1.
end
Output: R
∗
sum
= R(P
r
, L
r
).
The convergence of Algorithm 1 is proved in Proposition 1.
Proposition 1: Algorithm 1 is convergent.
Proof: Define the objective value of rth iteration as
R(P
r
, L
r
). In the (r + 1)th iteration, we obtain the objective
value R(P
r+1
, L
r
) by Step 1 of Algorithm 1, and it is the
lower bound of the original problem (11). Thus we have
R(P
r
, L
r
) ≤ R(P
r+1
, L
r
), (31)
for Step 2 of Algorithm 1, and we can obtain the objective
value R(P
r+1
, L
r+1
). Similarly, we have
R(P
r+1
, L
r
) ≤ R(P
r+1
, L
r+1
). (32)
Step 3 in Algorithm 1 can always adjust the current decoding
order in each iteration, and the sum rate will not decrease.
Thus, combining (31) with (32), we prove the objective value
of (11) is non-decreasing after each iteration, and is upper
bounded by a finite value. Algorithm 1 is convergent.
D. Analysis of the Last Decoding User
The last decoding user is the closest one to the UAV and last
decoded via SIC. The last decoding user is determined when
the UAV location is initialized, and will not change during
iterations, which is proved in Proposition 2. To simplify the
derivation, we introduce an auxiliary variable α
i
as
α
i
=
σ
2
|h
i
|
2
. (33)
Proposition 2: The last decoding user is not changed during
iterations and the optimal UAV location is getting closer to this
user with increasing transmit power.
Proof: Define U
1
as the initial last decoding user, and we
can always find suitable power allocation at L
0
to satisfy
R
i
(α
∗
i
) = R
1
(α
∗
1
), ∀i ∈ K\{1}. (34)
The derivative of R
i
(α
i
) and R
1
(α
1
) can be expressed as
R
′
i
(α
i
) = (log
2
(1 + SINR
i
(α
i
)))
′
=
−P
i
log
2
e
α
i
+
i−1
k=1
P
k
α
i
+
i
k=1
P
k
, 2 ≤ i ≤ K,
(35)
0 50 100 150 200 250 300 350 400
x (m)
0
50
100
150
200
250
300
350
400
y (m)
Ground User
Initial location of UAV
Optimal location of UAV
P
sum
= 4 mW
P
sum
= 5 mW
P
sum
= 20 mW
P
sum
= 100 mW
P
sum
= 10 mW
P
sum
= 50 mW
Fig. 1. The optimal UAV location with different values of P
sum
. η =
(1, 1, 1) bit/s/Hz.
R
′
1
(α
1
) =
−P
1
log
2
e
α
1
(α
1
+ P
1
)
. (36)
From (35) and (36), R
i
(α
i
) and R
1
(α
1
) decrease with α
i
and α
1
, respectively. To compare the first derivative values,
we change the two functions into absolute values as
|R
′
i
(α
i
)|
|R
′
1
(α
1
)|
=
P
i
α
1
(α
1
+ P
1
)
P
1
α
i
+
i−1
k=1
P
k
α
i
+
i
k=1
P
k
=
SINR
i
(α
i
)(α
1
+ P
1
)
SINR
1
(α
1
)
α
i
+
i
k=1
P
k
.
(37)
According to the decoding order at L
0
, we have h
∗
1
≥
h
∗
i
, i ∈ K \{1} and α
∗
1
≤ α
∗
i
. Thus, |R
′
i
(α
∗
i
)| < |R
′
1
(α
∗
1
)|
is met under the assumption in (34). In order to increase the
sum rate, the UAV location will approach U
1
. The user U
1
always has the best channel condition and the last decoding
user is not changed. We can observe that the increase of power
has a much greater influence on R
1
from (37). Therefore, with
the rate thresholds are satisfied, the optimal UAV location will
approach U
1
with larger P
sum
.
From Proposition 2, we can conclude that the UAV location
should move to the last decoding user, if we need to improve
the sum rate with higher transmit power.
IV. SIMULATION RESULTS AND DISCUSSION
In the simulation, assume that the UAV hovers at H = 150
m. We set σ
2
= −110 dBm and ρ
0
= −60 dB.
First, we set K = 3 and users are marked by red triangles, as
shown in Fig. 1. The optimal location of UAV is presented for
different transmit power, when η = (1, 1, 1) bit/s/Hz. From the
result, we can observe that the optimal UAV location becomes
closer to the last decoding order when P
sum
increases, which
is consistent with the conclusion from Proposition 2.
The sum rate of the proposed scheme is compared in Fig.
2 with different η according to the topology in Fig. 1 with the
same threshold for each user. The result shows that the sum
rate increases when the transmit power of the UAV is higher.
