NOMA in Distributed Antenna System for
Max-Min Fairness and Max-Sum-Rate

arXiv:1706.05314v1 [] 16 Jun 2017

Dong-Jun Han, Student Member, IEEE, Minseok Choi, Student Member, IEEE, and Jaekyun Moon Fellow, IEEE
School of Electrical Engineering
Korea Advanced Institute of Science and Technology
Daejeon, 34141, Republic of Korea
Email: {djhan93, ejaqmf}@kaist.ac.kr, jmoon@kaist.edu

Abstract—Distributed antenna system (DAS) has been deployed for over a decade. DAS has advantages in capacity especially for the cell edge users, in both single-cell and multi-cell environments. In this paper, non-orthogonal multiple access (NOMA)
is suggested in single-cell DAS to maximize user fairness and sumrate. Two transmission strategies are considered: NOMA with
single selection transmission and NOMA with blanket transmission. In a two-user scenario, the center base station (BS) uses
NOMA to serve both users and the remote radio units (RRUs)
serve only the weak user based on the transmission scheme. The
signals sent from the RRUs are not considered as interference
at both user sides. At one side, the signals from the RRUs are
used to detect the required data and at the other side, the signals
are used to perform successive interference cancellation (SIC).
The max-min performance is studied when instantaneous channel
gain information (CGI) or channel distribution information (CDI)
is known at the transmitter. A closed-form expression of the
upper bound is derived for the data rate of each user with only
CDI. In addition, the sum-rate is studied with a minimum rate
constraint when instantaneous CGI known at the transmitter. The
results show that NOMA with blanket transmission gives the best
performance compared to other schemes; conventional-NOMA,
conventional single selection scheme and joint-transmission (JT)
NOMA. It is also shown that with less transmit power, NOMA
with single selection transmission gives better performance than
conventional-NOMA.

I. I NTRODUCTION
UPPORTING a large number of users with high data rates
is one of the most rewarding challenges for future 5G
communications. To comply with this demand, small cells
have to be densely deployed in a cellular network [1]. The
distributed antenna system (DAS) is one way to increase cell
densification, and is viewed as a promising candidate for future
wireless communications [2]. In DAS, the remote radio units
(RRU) are geographically distributed and each coordinates
with a base station (BS). The BS is a central unit where
the main signal processing is performed and the transmission
scheme is determined. Because of the increased cell coverage,
reduced inter-cell interference and transmit power, DAS gives
advantages on capacity [3] and power consumption [4]. In
[5], the potential of DAS for massive multiple-input multipleoutput (MIMO) is shown relative to the conventional colocated system.
In terms of multiple access, non-orthogonal multiple access
(NOMA) is regarded as a potential candidate for 5G [6]–

S

[9]. NOMA gives higher spectral efficiency and capacity than
orthogonal multiple access (OMA), since multiple users use
the same frequency/time/code with different power levels relying on successive interference cancellation (SIC) performed
at the receivers [6]. In a two-user scenario, NOMA generally
allocates more power to the user with a lower channel gain
and a less power to the user with a higher channel gain. SIC
is performed at the user with the better channel condition.
In [7]–[10], sum-rate and fairness performances of NOMA
are studied with performance improvements shown compared
to OMA. Ergodic capacity of NOMA in MIMO systems is
studied in [7]. Sum-rate performance of NOMA is analyzed
in [8] with randomly deployed users, and fairness issues
are discussed in [9]. In [10], power allocation based on
proportional fairness scheduling was studied for both maxsum-rate and max-min-rate.
Recently, there have been some studies about NOMA in
coordinated multipoint (CoMP) systems [11], [12]. In [11],
NOMA is used in a coordinated two-point system to support
a cell-edge user. In [12], opportunistic selection of BSs is
studied with NOMA in a multi-cell CoMP system. In this
paper, NOMA is suggested in DAS. While the previous works
[11], [12] focused on using NOMA in the system where the
macro BSs cooperate, here NOMA is used in the system where
the RRUs within the macrocell cooperate with the center BS.
In particular, in our work the center BS always serves both near
and far users in the cell with enough power using NOMA. The
RRUs send data depending on the transmission scheme.
The main contrinutions of this paper are summarized as follows. NOMA is suggested in DAS for user fairness and sumrate maximization with two transmission schemes: NOMA
with a blanket transmission scheme, where all the RRUs
transmit data, and NOMA with a single selection transmission
scheme, where only one RRU is selected to transmit data.
The center BS always serves both users by NOMA. Max-min
fairness is studied for two cases. The first case is when the
instantaneous channel gain information (CGI) is known at the
transmitter. The second case is when the channel distribution
information (CDI), which is also called statistical channel state
information (CSI), is known at the transmitter. A closed-form
expression of the upper bound is derived for the data rate of
each user with only CDI. Sum-rates under a minimum rate
constraint are also studied with instantaneous CGI. Compared
to conventional NOMA, conventional single selection scheme

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Fig. 1. Two-user downlink scenario in single-cell distributed antenna system.

and joint-transmission (JT) NOMA, the proposed NOMA with
blanket transmission gives the best performance for both user
fairness and sum-rate. With a less transmit power, NOMA with
single selection transmission also gives better performance
than conventional NOMA.
This paper is organized as follows. Section II describes the
system model indcluding cellular architecture of DAS, channel model and transmission schemes. The max-min fairness
problem is studied in Section III, while the sum-rate problem
is considered in Section IV. Numerical results are shown
in Section V. Finally, conclusions along with discussions of
future works are given in Section VI.
II. S YSTEM M ODEL
A. Single Cell Distributed Antenna System
This paper considers the two-user downlink scenario in a
single-cell environment. The architecture of the DAS is shown
in Fig. 1. A cell include central BS and 6 RRUs having a single
antenna each. Let Pcen and Prru denote the transmit power of
the center BS and each RRU, respectively, assuming the same
power is allocated to every RRU. The total transmit power is
assumed to be P = Pcen + 6 × Prru. For convenience, the cell
radius is normalized as 1.
As in other previous studies [13], [14], we assume that one
is close to the center BS (near user) and the other is close to
the cell edge (far user). In particular, the near user is assumed
to be located in the fractional cell of the center BS.
B. Channel Model
The Rayleigh
fading channel from RRU i to user j is defined
p
as hi,j = Li,j gi,j for i ∈ {0, 1, ..., 6}, j ∈ {1, 2}, where the
center BS is indexed as i = 0. Li,j denotes the slow fading,
Li,j = dα1 , where di,j denotes the distance between user j
i,j
and RRU i, α is the pathloss factor. gi,j denotes i.i.d fast
fading component having the complex Gaussian distribution,
gi,j ∼ CN (0, 1).
This paper considers two cases, instantaneous CGI or only
CDI known at the transmitter. When the transmitter knows
the instantaneous CGI, the center BS is possible to decide

which user is the strong or weak one. User j is the strong one,
when j = arg maxk∈{1,2} |h0,k |2 . The other user is denoted
as the weak user. Note that the strong user with instantaneous
CGI does not always imply the near user. The reason is that
the randomly generated channel of the near user does not
guarantee a larger channel norm compared to the channel of
the other user which is located relatively far from the center
BS.
On the other hand, only CDI at the transmitter means that
the BS considers user j having a larger channel variance as
the strong user, j = arg maxk∈{1,2} L0,k . The other one is
denoted as the weak user. In this case, the strong user always
indicates the near user. For both CSI cases, the weak user
always decodes its information directly, and the strong user
decodes its own data after performing SIC.

C. Transmission Schemes
Single selection and blanket transmissions [3] are the basic transmission schemes in DAS. This paper is also based
on these schemes, but the difference is that the center BS
transmits the signal to both users by NOMA and the RRUs
employ these transmission schemes for only the weak user.
In other words, the RRUs only send data for the weak user
by single selection transmission or blanket transmission. The
signals sent from the RRUs are not considered as interference
at the both user terminals. The weak user receives the signals
from the RRUs as a part of its own data and, on the other
hand, the RRU signals help the strong user to perform SIC
before detecting its data.
1) NOMA with single selection transmission: The DAS
decides one of the RRUs to transmit the signals to the weak
user, whose channel gain between the weak user is the largest
among all RRUs. Here, the channel gain becomes either the
norm of the instantaneous channel or the channel variance,
depending on the knowledge of CSI at the transmitter. Let
users 1 and 2 are the weak and strong users, respectively.
Denote xi,j as the data symbol to user j from RRU i for
i ∈ {0, 1, ..., 6}, j ∈ {1, 2} where the center BS is again
indexed as i = 0. Then when RRU q is selected, the received
signals of users 1 and 2 are given by
y1 = h0,1 (x0,1 + x0,2 ) + hq,1 xq,1 + n1
y2 = h0,2 (x0,1 + x0,2 ) + hq,2 xq,1 + n2

(1)
(2)

where yj and nj are the received signal and noise at
user j ∈ {1, 2}, respectively. At this point, we let
E[|n1 |2 ] =E[|n2 |2 ] =σ 2 and Pj = E[|x0,j |2 ] for j ∈ {1, 2}.
We assume that each RRU or the center BS transmits signal
with its full power. Then, Pcen = P1 + P2 , Prru = E[|xp,j |2
for p ∈ {1, 2..., 6}, j ∈ {1}.
Since we consider the cases where the BS knows the
instantaneous CGI or the CDI, so the data rate of user 1 from
(1) becomes
Z1 = log2 (1 +

|h0,1 |2 P1 + |hq,1 |2 Prru
)
|h0,1 |2 P2 + σ 2

(3)

which is the rate without the exact channel value information
at the transmitter [15], [16]. Similarly, from (2), the rate for
user 1 (for SIC) becomes
|h0,2 |2 P1 + |hq,2 |2 Prru
).
Z2 = log2 (1 +
|h0,2 |2 P2 + σ 2

(4)

Since the information of user 1 should be decoded at both
user sides, the data rate of user 1 can be finally written as
R1 = min(Z1 , Z2 ) with instantaneous CGI at the transmitter.
If only CDI is known, the average rate should be discussed,
which beocomes R1 = E[min(Z1 , Z2 )]. After SIC, user 2 has
the rate of
|h0,2 |2 P2
R2 = log2 (1 +
).
(5)
σ2
for the instantaneous CGI case. With only CDI at the trans|h |2 P
mitter, R2 = E[log2 (1 + 0,2σ2 2 )].
2) NOMA with blanket transmission: All of RRUs transmit
the signals to the weak user simultaneously, while the BS
serves both users by NOMA. The received signal becomes
y1 = h0,1 (x0,1 + x0,2 ) +

6
X

hq,1 xq,1 + n1

(6)

hq,2 xq,1 + n2 .

(7)

q=1

y2 = h0,2 (x0,1 + x0,2 ) +

6
X
q=1

Similar to single selection transmission, we can obtain
P
|h0,1 |2 P1 + 6q=1 |hq,1 |2 Prru
)
Z1 = log2 (1 +
|h0,1 |2 P2 + σ 2
P6
|h0,2 |2 P1 + q=1 |hq,2 |2 Prru
Z2 = log2 (1 +
)
|h0,2 |2 P2 + σ 2

(8)
(9)

where Z1 , Z2 are the user rates from (6) and (7), respectively.
The only difference from single selection transmission is that
user 1 receives its data with much more power and SIC can
be performed better due to the increased power of user 1,
at user 2. After SIC, R2 is the same as in single selection
transmission.
III. M AX -M IN FAIRNESS P ROBLEM
A. Instantaneous CGI known at the Transmitter
Instantaneous CGI is the norm of the instantaneous channel
[17], which can be typically obtained by feedback from
the receivers. Without loss of generality, we assume that
|h0,1 |2 < |h0,2 |2 i.e., user 1 and user 2 are denoted as
weak and strong users, respectively. For NOMA with single selection transmission, the RRU q serves user 1 where
q = argmaxp∈{1,2...,6} |hp,1 |2 . For NOMA with blanket transmission, all RRUs are selected to send data for user 1.
Throughout this paper, we fix the transmit power of the
center as 0.5P , i.e., Pcen = 0.5P . Optimizing the portion for
the center BS is a topic for future research. The goal of this
paper is to optimize power allocation for users with fixed Pcen .
Let P1opt be the optimal power allocation for the weak user.
Then power of Pcen − P1opt is allocated to the strong user.
The max-min problem is formulated as
P1opt = argmax min(R1 , R2 ).
P1 ∈[0,Pcen ]

(10)

Recall that P1 denotes the power allocation for user 1 (weak
user). We fisrt state the following lemma.
Lemma 1. Suppose instantaneous CGI is known at the transmitter. R1 is an increasing function of P1 . R2 is a decreasing
function of P1 .
Proof: For computational convenience, let σ 2 = 1. For
both single selection and blanket transmissions,
−|h0,2 |2
|h0,1 |2
1
1
∂Z1
∂R2
=
=
<
0,
>0
∂P1
ln2 |h0,2 |2 P2 + 1
∂P1
ln2 |h0,1 |2 P2 + 1
(11)
∂Z2
|h0,2 |2
1
=
>0
(12)
∂P1
ln2 |h0,2 |2 P2 + 1
Since Z1 , Z2 are increasing functions of P1 , the minimum of
Z1 , Z2 is also an increasing function of P1 .
By Lemma 1, the following theorem can be obtained which
describes optimal power allocation.
Theorem 1. Suppose instantaneous CGI is known at the
transmitter and let the solutions for Z1 = R2 , Z2 = R2
with respect to P1 , be PZ1 , PZ2 respectively. Then, the optimal
power allocation with respect to P1 for the max-min problem
becomes
(
PR1 ∈ [0, Pcen ]
PR1 ,
opt
P1 =
(13)
0,
otherwise
where PR1 = max(PZ1 , PZ2 ).
Proof: By Lemma 1, min(R1 , R2 ) is maximized when
R1 = R2 . It can be shown that the solution of R1 = R2 with
respect to P1 , is PR1 . Note that the optimal solution should
be in the range of [0, Pcen ], i.e., P1opt ∈ [0, Pcen ]. If PR1 ∈
[0, Pcen ], the optimal solution becomes PR1 . If PR1 < 0, it
means that allocating all power Pcen to user 2 is the best
way to maximize the minimum rate, i.e., P1opt = 0. The case
PR1 > Pcen could not happen, since then, R2 (PR1 ) < 0 while
R1 (PR1 ) > 0. Thus, the optimal solution can be represented
as (13).
Closed-form expressions of PZ1 , PZ2 can be obtained by
solving the quadratic equations of Z1 = R2 , Z2 = R2 . Then,
the max-min(R1 , R2 ) becomes min(R1 (P1opt ), R2 (P1opt )).
B. Only CDI known at the Transmitter
With CDI, the order of distance from the center BS to
users are known. We denote the far and near user from
the center BS as user 1, user 2 respectively which means
L0,1 < L0,2 . Recall that the strong user implies the near
user with only CDI at the transmitter. For NOMA with single
selection, RRU q is selected to transmit data for user 1
where q = argmaxp∈{1,2...,6} Lp,1 . For NOMA with blanket
transmission, all RRUs are selected to serve user 1. For both
transmission schemes, SIC is performed near user 2.
For this scenario, we would like to maximize the minimum
capacity (average rate) of each user. Again, let P1opt be the
optimal power allocation for the weak user. The max-min
fairness problem is then formulated as
P1opt = argmax min(R1 , R2 )
P1 ∈[0,Pcen ]

(14)

where R1 , R2 are now average rates. After SIC, the average
rate for the near user can be written as
|h0,2 |2 P2
)]
(15)
R2 = E[log2 (1 +
σ2
for both NOMA with single selection transmission and blanket
transmission schemes. Define Ct (x) as the ergodic capacity of
an i.i.d MISO channel with t transmit antennas given by [18]
Ct (x) =
where En (x) =

R∞
1

t−1
1/x X

e
ln2

e−xt
tn dt.

k=0

1
Ek+1 ( )
x

(16)

From [18], it can be shown that

L0,2 P2
).
σ2
The rate for the far user can be written as
R2 = C1 (

R1 = E[min(Z1 , Z2 )]

(17)

(18)

where Z1 , Z2 are defined in (3), (4) for NOMA with single
selection transmission. Using the fact that
E[min(Z1 , Z2 )] ≤ min(E[Z1 ], E[Z2 ]),

(19)

the upper bound is derived for user 1 in closed-form. Define
the upper bound of R1 , as
R1UB , min(E[Z1 ], E[Z2 ]).

(20)

For detecting its own information at user 1,
|h0,1 |2 P1 + |hq,1 |2 Prru
)]
(21)
|h0,1 |2 P2 + σ 2
|h0,1 |2 Pcen + |hq,1 |2 Prru
= E[log2 (1 +
)]
(22)
σ2
2
|h0,1 | P2
)]
(23)
− E[log2 (1 +
σ2
L0,1 Pcen
L0,1 Pcen
=
C1 (
)
(24)
L0,1 Pcen − Lq,1 Prru
σ2
Lq,1 Prru
Lq,1 Prru
+
C1 (
)
(25)
Lq,1 Prru − L0,1 Pcen
σ2
L0,1 P2
).
(26)
− C1 (
σ2
Likewise, E[Z2 ], which is the average rate for detecting the
far user’s information at user 2, can be computed as
E[Z1 ] = E[log2 (1 +

|h0,2 |2 P1 + |hq,2 |2 Prru
)]
|h0,2 |2 P2 + σ 2
L0,2 Pcen
L0,2 Pcen
C1 (
)
=
L0,2 Pcen − Lq,2 Prru
σ2
Lq,2 Prru
Lq,2 Prru
+
C1 (
)
(2)
σ2
Lq,2 Prru − L Pcen

E[Z2 ] = E[log2 (1 +

(27)
(28)
(29)

0,0

L0,2 P2
).
(30)
σ2
Similarly, for the NOMA with blanket transmission scheme,
E[Z1 ] and E[Z2 ] become
− C1 (

E[Z1 ] =

6
X

πi,1 C1 (

Li,1 Qi
L0,1 P2
) − C1 (
)
σ2
σ2

(31)

πi,2 C1 (

Li,2 Qi
L0,2 P2
) − C1 (
)
σ2
σ2

(32)

i=0

E[Z2 ] =

6
X
i=0

respectively, where πi,j , Qi are defined as
πi,j =

6
Y

k=0,k6=i

Qi =

(

Pcen ,
Prru ,

Li,j Qi
Li,j Qi − Lk,j Qk

(33)

i=0
otherwise

(34)

for i ∈ {0, 1, ..., 6}, j ∈ {1, 2}.
First, we would like to obtain the upper bound of the
max-min value, which is max-min(R1UB , R2 ). Let the optimal
solution for the upper bound with respect to P1 , as P1UB . We
first state the following lemma.
Lemma 2. Suppose only CDI is known at the transmitter.
R1UB is an increasing functions of P1 . R2 is a decreasing
function of P1 . R1 is a non-decreasing function of P1 .
Proof: Only considering the expectation operation additionally in the proof of Lemma 1, it can be shown that E[Z1 ],
E[Z2 ] are increasing functions and R2 is a decreasing function
of P1 , which completes the proofs for R1UB , R2 .
Again by Lemma 1, min(Z1 , Z2 ) is an increasing function
of P1 . We want to show that E[min(Z1 , Z2 )] is a nondecreasing function of P1 . Since min(Z1 , Z2 ) is a function of
the channels and power P1 , we let Z(h̄, P1 )=min(Z1 , Z2 ). We
show that if y≥x then E[Z(h̄, y)]≥E[Z(h̄, x)]. Assume that
y≥x and let the probability density function of h̄ be fH̄ (h̄).
Then, since fH̄ (h̄)≥0 and Z(
R h̄, y)−Z(h̄, x)≥0, we can obtain
E[Z(h̄, y)]− E[Z(h̄, x)]= fH̄ (h̄)[Z(h̄, y)−Z(h̄, x)]dh̄≥0.
This completes the proof for R1 .
By Lemma 2, the optimal solution for the upper bound is
obtained when R2 = min(E[Z1 ], E[Z2 ]), if the solution exists
in the range of [0, Pcen ]. However, a closed form solution
of R2 = min(E[Z1 ], E[Z2 ]) is difficult to obtain due to the
integration operations in R2 , E[Z1 ], E[Z2 ]. Since R1UB is an
increasing function and R2 is a decreasing function of P1 , the
optimal solution P1UB can be directly obtained by a bisectionbased power allocation which was also used in [7]. Algorithm
1 provides the detail. Note that the algorithm includes the
cases where R2 cannot reach R1 for P1 ∈ [0, Pcen ], where the
optimal solution becomes P1UB = 0. Then, the upper bound
of the max-min becomes min(R1UB (P1UB ), R2 (P1UB )).
Algorithm 1 Bisection method for power allocation
1: Initialize u = 0, v = Pcen
2: while v − u ≥ ǫ
3: P1 = v + u/2
4: if R1 (P1 ) < R2 (P1 ) : u = P1
5: else v = P1
6: end while
7: Output: P1UB = P1
Although the above solution P1UB is not optimal for the
exact max-min(R1 , R2 ), we choose this suboptimal power
allocation for simulation since the closed-form expression for
R1 is diffucult to obtain, which means optimal power solution
is also difficult to obtain.

IV. M AX -S UM - RATE PROBLEM
The above discussions focused on maximizing the minimum
data rate for user fairness. In this section, sum-rate performance is studied under a minimum rate constraint as in [7].
Instantaneous CGI is assumed to be known at the transmitter.
The problem can be formulated as
P1opt = argmax (R1 + R2 )

(35)

subject to min(R1 , R2 ) ≥ Rt

(36)

P1 ∈[0,Pcen ]

where Rt is the minimum data rate constraint to guarantee
quality of service (QoS). The following Lemma is used and
Z1 , Z2 , R1 , R2 are all the same as in Section III-A.
Lemma 3. Suppose instantaneous CGI is known at the transmitter. Assuming |h0,1 |2 < |h0,2 |2 , R1 +R2 is a non-increasing
function of P1 .
Proof: For computational convenience, let σ 2 = 1.
Since |h0,1 |2 < |h0,2 |2 , for both single selection and blanket
transmissions,
∂Z1
(|h0,1 |2 − |h0,2 |2 )
∂R2
1
+
=
<0
∂P1
∂P1
ln2 (|h0,2 |2 P2 + 1)(|h0,1 |2 P2 + 1)
(37)
∂R2
∂Z2
+
=0
(38)
∂P1
∂P1
Since Z1 + R2 and Z2 + R2 are both non-increasing functions
of P1 , min(Z1 , Z2 ) + R2 is also a non-increasing function of
P1 , which completes the proof.
The overall outage event of the system is defined as the
event that any user in the system cannot achieve the required
minimum data rate. By Lemma 2, the following theorem can
be obtained, which describes the optimal power allocation.
Theorem 2. Suppose instantaneous CGI is known at the
transmitter and let the solution of Z1 = Rt , Z2 = Rt with
respect to P1 be PZ1 , PZ2 respectively. Then, the optimal
power allocation of the max-sum-rate problem becomes


PR1 ∈ [0, Pcen ] and R2 (PR1 ) ≥ Rt
PR1 ,
opt
P1 = 0,
PR1 < 0 and R2 (0) ≥ Rt


outage, otherwise
(39)
where PR1 = max(PZ1 , PZ2 ).
Proof: By Lemma 1 and 3, if the constraint
min(R1 , R2 ) ≥ Rt exists, R1 + R2 is maximized when
min(R1 , R2 ) = Rt . It can be shown that the solution of
R1 = Rt is PR1 .
First consider the case PR1 ∈[0, Pcen ]. If R2 (PR1 )≥ Rt ,
min(R1 (PR1 ), R2 (PR1 )) = Rt so PR1 becomes the optimal power solution. If R2 (PR1 )< Rt , then min(R1 (PR1 ),
R2 (PR1 )) < Rt so PR1 cannot be the solution (outage case).
Second, consider the case PR1 < 0. Since R1 is an
increasing function of P1 , the user 1’s rate is greater than or
equal to Rt even with zero power allocation by the center BS,
i.e., R1 (0)≥ Rt . Due to Lemma 3, if R2 (0)≥ Rt , allocating
all power to user 2 is the best way to maximize sum-rate so

the optimal power allocation with respect to P1 becomese 0. If
R2 (0)< 0, then min(R1 (PR1 ), R2 (PR1 )) < Rt so it becomes
an outage case.
At last, PR1 > Pcen implies outage since R1 < Rt for
the power allocation in the range of [0, Pcen ]. Therefore, the
optimal solution can be represented as (39).
Closed-form expressions of PZ1 , PZ2 can be obtained by
solving the linear equations: Z1 = Rt , Z2 = Rt . Then, the
max(R1 + R2 ) becomes R1 (P1opt ) + R2 (P1opt ).
V. N UMERICAL R ESULTS
In this section, numerical results are presented with the
power allocation methods studied above. The pathloss factor is
assumed to be 4 and the noise variance σ 2 is normalized to be
1. We also assume Pcen =0.5P . We compare our results with
conventional NOMA with total power of P at the center BS,
and conventional single selection scheme in DAS. In addition,
we make comparison with some kinds of blanket transmission
in DAS. Blanket transmission by frequency split can be a
candidiate, but joint-transmission (JT) NOMA [12] seems to
be a stronger candidate for comparison.
For conventional single selection scheme in DAS, we assume that user 2 is always served by the center BS and
user 1 is served by the RRU which has the best instantaneous channel gain or channel variance. Especially for this
scheme, we also consider the case Pcen =Prru = 71 P , since
user 1 receives relatively small signal power if we assume
Pcen = 0.5P , which would result in a low performance. For
JT-NOMA, all RRUs including the center BS allocate
P6 power
of ratio β to user 1. Let |h1 |2 =Pcen |h0,1 |2 +Prru q=1 |hq,1 |2
P6
and |h2 |2 =Pcen |h0,2 |2 + Prru q=1 |hq,2 |2 with instantaneous CGI at the transmitter.. Without loss of generalβ|h1 |2
ity assume |h1 |2 <|h2 |2 . Then, Z1 =log2 (1 + (1−β)|h
2
2 ),
1 | +σ
2

2

β|h2 |
(1−β)|h2 |
Z2 =log2 (1 + (1−β)|h
) and
2
2 ), R2 = log2 (1 +
σ2
2 | +σ
R1 = min(Z1 , Z2 ). If only CDI known at the transmitP6
P6
L Qi
(1−β)Li,1 Qi
ter, E[Z1 ]= i=0 πi,1 C1 ( i,1
),
σ2 )− Pi=0 πi,1 C1 (
σ2
P6
6
(1−β)Li,2 Qi
Li,2 Qi
)
−
π
C
(
)
E[Z2 ] =
π
C
(
2
i,2
1
2
i,2
1
i=0
i=0
σ
σ
P6
(1−β)Li,2 Qi
)
where
π
,
Q
are
and R2 =
π
C
(
i,j
i
i=0 i,2 1
σ2
defined in (33) and (34), respectively. The optimal power
allocation factor β is found numerically for JT-NOMA.

A. Max-Min Fairness
1) Instantaneous CGI known at the transmitter: Fig. 2
shows the average max-min value versus distance between the
cell center and the far user. Recall that the far user in this case
does not always indicate the weak user. The moving direction
of the far user is from the cell edge to the cell center, following
the line in Fig. 1. The near user is fixed at normalized distance
of 0.2 in the same line as the far user (see Fig. 1). The
transmit signal-to-noise ratio (SNR) is assumed to be 10 dB.
Here, the transmit SNR is defined as σP2 . For each location,
the max-min values are averaged by optimal power allocation
with 106 randomly generated channels. The difference among
the schemes is that the performance of conventional NOMA
increases linearly while the others does not. This is because for
conventional NOMA, the data rate of the far user is dominant

3.4

5.5

3.2

5
4.5

3

4

2.8

Average max-min

Average max-min

Proposed: NOMA with blanket (lower bound)
Proposed: NOMA with single selection (lower bound)
Conventional NOMA (upper bound)
Conventional NOMA (lower bound)
JT-NOMA (upper bound)
JT-NOMA (lower bound)
DAS single selection, Pcen=Prru
DAS single selection, Pcen=0.5P

2.6
2.4
2.2
2
1.8
1.6
0.6

0.7

0.75

0.8

3
2.5
2

Proposed: NOMA with blanket
Proposed: NOMA with single selection
Conventional NOMA
JT-NOMA
DAS single selection, Pcen=Prru
DAS single selection, Pcen=0.5P

0.65

3.5

1.5
1
0.5
0.85

0.9

0.95

1

0

2

Normalized distance from center BS to the far user

Proposed: NOMA with blanket
Proposed: NOMA with single selection
Conventional NOMA
JT-NOMA
DAS single selection, Pcen=Prru
DAS single selection, Pcen=0.5P

4.5

Average max-min

4
3.5
3
2.5
2
1.5
1
0.5
0

2

4

6

8

10

12

14

16

6

8

10

12

14

16

18

20

Transmit SNR [dB]

Fig. 2. Average max-min value with instantaneous CGI known at the
transmitter. Near user is fixed at normalized distance 0.2. Transmit SNR=10
dB.

5

4

18

20

Transmit SNR [dB]

Fig. 3. Average max-min versus transmit SNR with instantaneous CGI known
at the transmitter.

for the max-min performance since all the power is centralized.
However, for the proposed two schemes and JT-NOMA, the
data rate of the near user becomes dominant for the max-min
value when the far user is close to the RRU which is located
around the distance of 0.67. Since the near user is fixed, the
performance plot becomes almost flat near the distance of
0.67. For conventional single selection scheme in DAS, the
max-min value is maximized around the distance of 0.67 and
decreases as the far user approaches to the cell center, since
the signal power from the RRU decreases and interference
from the center BS increases. With only transmit power of
Pcen +Prru , the proposed NOMA with single selection scheme
gives better performance compared to conventional NOMA
with transmit power of P . NOMA with blanket transmission
scheme gives the best performance compared to others. This
indicates the advantage of introducing NOMA in DAS which
can give the best user fairness.
Fig. 3 shows the average max-min value versus transmit

Fig. 4. Average max-min versus transmit SNR with only CDI known at the
transmitter.

SNR. To consider the general case, the near user is assumed
to be uniformly distributed within the circle with radius 0.3,
and the far user (cell-edge user) is uniformly distirbuted within
the ring with radius 0.8 and 1. By generating random pair of
points, and allocating optimal power, average max-min value
is obtained. The results are consistent with Fig. 2, confirming
the advantage of the suggested schemes.
2) Only CDI known at the transmitter: The average maxmin value versus transmit SNR is shown in Fig. 4. All the
settings are the same as in Fig. 3. The upper bound of JTNOMA is obtained numerically by simulations while the upper
bound of other criteria are computed using the closed-form
expressions of R1UB , R2 with optimal power allocation of
P1UB . The lines labeled as lower bound, are obtained by
monte carlo simulations of R1 , R2 with power allocation of
P1UB . Since P1UB is not the optimal solution for the exact
max-min value, it can be viewed as a lower bound of maxmin(R1 , R2 ). Again, NOMA with blanket transmission gives
the best result, with better performance than the upper bound
of JT-NOMA. With less transmit power, NOMA with single
selection transmission performs better than the upper bound
of conventional NOMA.
B. Max-Sum-Rate
With Instantaneous CGI at the transmitter, Fig. 5 shows the
average sum-rate versus Rt . The settings are the same as in
Fig. 3. For the system outage case, i.e., min(R1 , R2 ) < Rt , the
sum-rate is assumed 0. The average sum-rate decreases as Rt
increases. This is because the increase of Rt requires center
BS to allocate more power to the weak user to achieve the
minimum data rate, which reduces the sum-rate according to
Lemma 3. One can also see that for a relatively small data rate
constraint, conventional NOMA gives the best performance.
This is because for conventional NOMA, total power P is
centralized. After allocating some power to ensure QoS, it
uses all the remaining power to serve the strong user (most
likely to be the near user but not always) which gives a large
sum-rate advantage. However, as the required minimum data

less transmit power, NOMA with single selection can give
better performance compared to conventional NOMA.
While we studied a two-user scenario, extending the result
to more than two users is an important issue. More extensive
choices of transmission schemes and optimization methods
should be considered to serve general k users. Solving joint
optimization of (Pcen , P1 ) is also an interesting topic for
future research. Multi-cell environment should be also studied
to consider the effect of inter-cell interference. Combining
cooperative NOMA with the proposed schemes is another idea
to improve user fairness and sum-rate.

7
6.5

Average sum-rate

6
5.5
5
4.5
Proposed: NOMA with blanket
Proposed: NOMA with single selection
Conventional NOMA
JT-NOMA
DAS single selection, Pcen=Prru
DAS single selection, Pcen=0.5P

4
3.5
3
0.5

1

1.5

2

R EFERENCES
2.5

Rt

Fig. 5. Average sum-rate versus Rt with instantaneous CGI known at the
transmitter. Transmit SNR=10 dB.

11
Proposed: NOMA with blanket
Proposed: NOMA with single selection
Conventional NOMA
JT-NOMA
DAS single selection, Pcen=Prru
DAS single selection, Pcen=0.5P

10
9

Average sum-rate

8
7
6
5
4
3
2
1
0
0

2

4

6

8

10

12

14

16

18

20

Transmit SNR [dB]

Fig. 6. Average sum-rate versus transmit SNR with instantaneous CGI known
at the transmitter. Rt = 2.

rate becomes bigger, it has relatively large performance losses
since ensuring Rt requires more power than other schemes.
Fig. 6 shows the average sum-rate versus transmit SNR with
the same setting as in Fig. 3. The minimum rate constraint is
assumed to be Rt = 2. With high transmit SNRs, conventional
NOMA gives the best sum-rate because Rt = 2 is a relatively
small data rate constraint around the transmit SNR of 20 dB.
For other SNR regions, the overall results are consistent with
the max-min analysis, confirming the advantage of NOMA in
DAS.
VI. C ONCLUSION

AND FUTURE WORK

In this paper, NOMA has been suggested in DAS for maxmin fairness and max-sum-rate. Power alloction was studied
when instantnaeous CGI or CDI known at the transmitter.
It was shown that the proposed NOMA with blanket transmission scheme gives the best max-min and max-sum-rate
performances in a single-cell environment. In addition, with

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