# Optimization of Rate Allocation and Power Control for Rate Splitting
Multiple Access (RSMA)

Zhaohui Yang, , Mingzhe Chen, , Walid Saad, , and Mohammad Shikh-Bahaei Part
of the material in this paper was submitted to IEEE Globecom 2019 [1].Z. Yang
and M. Shikh-Bahaei are with the Centre for Telecommunications Research,
Department of Informatics, King’s College London, WC2B 4BG, UK, Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Chen is with Beijing Key
Laboratory of Network System Architecture and Convergence, Beijing University
of Posts and Telecommunications, Beijing, China 100876, Email:
<EMAIL_ADDRESS>Saad is with Wireless@VT, Bradley Department of
Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA, USA,
Email<EMAIL_ADDRESS>
###### Abstract

In this paper, the sum-rate maximization problem is studied for wireless
networks that use downlink rate splitting multiple access (RSMA). In the
considered model, each base station (BS) divides the messages that must be
transmitted to its users into a “private” part and a “common” part. Here, the
common message is a message that all users want to receive and the private
message is a message that is dedicated to only a specific user. The RSMA
mechanism enables a BS to adjust the split of common and private messages so
as to control the interference by decoding and treating interference as noise
and, thus optimizing the data rate of users. To maximize the users’ sum-rate,
the network can determine the rate allocation for the common message to meet
the rate demand, and adjust the transmit power for the private message to
reduce the interference. This problem is formulated as an optimization problem
whose goal is to maximize the sum-rate of all users. To solve this nonconvex
maximization problem, the optimal power used for transmitting the private
message to the users is first obtained in closed form for a given rate
allocation and common message power. Based on the optimal private message
transmission power, the optimal rate allocation is then derived under a fixed
common message transmission power. Subsequently, a one-dimensional search
algorithm is proposed to obtain the optimal solution of common message
transmission power. Simulation results show that the RSMA can achieve up to
15.6% and 21.5% gains in terms of data rate compared to non-orthogonal
multiple access (NOMA) and orthogonal frequency-division multiple access
(OFDMA), respectively.

###### Index Terms:

Rate splitting multiple access, sum-rate maximization, rate allocation, power
control.

##  I Introduction

Driven by the rapid development of advanced multimedia applications such as
virtual reality [2], next-generation wireless networks [3] must support high
spectral efficiency and massive connectivity. By splitting users in the power
domain, non-orthogonal multiple access (NOMA) can simultaneously serve
multiple users at the same frequency or time resource [4, 5, 6, 7, 8, 9].
Consequently, NOMA-based access scheme can achieve higher spectral efficiency
than conventional orthogonal multiple access [10, 11, 12]. However, using
NOMA, the users must decode all of the interference as they receive the
messages [11], which significantly increases the computational complexity
needed for signal processing. To solve this problem, the idea of rate
splitting multiple access (RSMA) was proposed in [13, 14, 15]. In RSMA, the
message transmitted to the users is divided into a common message and a
private message. The common message is a message that all users must receive
and the private message is a message that only a specific intended user wishes
to receive. To receive the common message, the users must decode the
interference from other users. In contrast, to receive the private message,
the users must only consider the interference from other users’ private
messages which can be treated as noise. Therefore, adjusting the split of
common and private messages can control the computational complexity and the
data rate achieved by RSMA. However, implementing RSMA in wireless networks
also face several challenges [15] such as the split of common and private
message, resource management for effective private message transmission, and
synchronization of message transmission.

Recently, a number of existing works such as in [15, 16, 17, 18, 19, 20, 21,
22] studied important problems related to RSMA. The work in [15] introduced
the challenges and opportunities of using RSMA for multiple input multiple
output (MIMO) based wireless networks. In [16], the authors proposed a
distributed rate splitting method to maximize the data rates of the users. The
authors in [17] evaluated the performance of RSMA, NOMA, and space-division
multiple access (SDMA) and showed that RSMA achieves better performance than
NOMA and SDMA. The authors in [18] investigated the use of linearly-precoded
rate-splitting method for simultaneous wireless information and power transfer
networks. In [19], the authors used RSMA to maximize the rate of all users in
downlink multi-user multiple input single output (MISO) systems under
imperfect channel state information at the transmitter. The work in [20]
studied the energy efficiency of the RSMA and NOMA schemes in a millimeter
wave downlink transmission scenario. The use of RSMA is investigated in [21]
for a downlink multiuser MISO system with bounded errors in the channel state
information at the transmitter. The authors in [22] analyzed the data rate of
using RSMA for two-receiver MISO broadcast channel with finite rate feedback.
However, most of the existing works such as in [15, 16, 17, 18, 19, 20, 21,
22] that only find the suboptimal power control solutions for RSMA in
different wireless systems such as MIMO and MISO, do not find optimal power
control and rate allocation solutions for RSMA in single-input single-output
(SISO) systems. Meanwhile, none of these existing works [15, 16, 17, 18, 19,
20, 21, 22] considers a successive interference cancelation (SIC) constraint
for the private message transmission in RSMA, which is needed to guarantee the
successful decoding of the common message.

The main contribution of this paper is an optimized rate allocation and power
control scheme for RSMA in a downlink SISO system. To our best knowledge, this
is the first work that finds, jointly, the optimal rate allocation solution
for common message transmission and the optimal power allocation for both
common and private message transmission. Our key contributions include:

  * •

We propose a wireless network that uses RSMA and in which one base station
(BS) transmits message to multiple users using RSMA scheme. To maximize the
data rate of the users, we optimize the allocation of the data rate of common
message transmission for each user as well as the transmit power that is used
to transmit the common and private messages.

  * •

We formulate the considered rate allocation and power control problem as an
optimization problem whose goal is to maximize the network sum-rate under both
rate and SIC constraints. To solve this problem, we first derive a closed-form
expression for the optimal transmit power of the private message. Our
fundamental analysis shows that, with the exception of one of the users, all
users are allocated with the minimum power to maintain the minimum rate
demand. Then, we characterize the finite solution space for the optimal rate
allocation. In order to obtain the optimal rate allocation and power control,
we propose a one-dimensional search algorithm that is shown to have linear
complexity for equal rate demand.

  * •

Simulation results show that the optimized RSMA algorithm can achieve up to
15.6% and 21.5% gains in terms of data rate compared to NOMA and orthogonal
frequency-division multiple access (OFDMA).

The rest of this paper is organized as follows. The system model and problem
formulation are described in Section II. The optimal solution is presented in
Section III. Simulation results are analyzed in Section IV. Conclusions are
drawn in Section V.

##  II System Model and Problem Formulation

Consider the downlink a single-cell wireless network that consists of one BS
servicing a set $\mathcal{K}$ of $K$ users using RSMA [13]. In RSMA, the
common message is decoded by all users, while the individual private message
is only decoded by each user. At the receiver side, each user first decodes
the common message and then decodes its private message using the previously
decoded common message.

Let the common message of all users be $s_{0}$ and the private message of each
user $k$ be $s_{k}$. The transmitted signal $x$ of the BS is expressed as:

$x=\sqrt{p_{0}}s_{0}+\sum_{k=1}^{K}\sqrt{p_{k}}s_{k},$ (1)

where $p_{0}$ is the transmit power of the common message $s_{0}$ and $p_{k}$
is the transmit power of the private message $s_{k}$ transmitted to user $k$.

The total received message at user $k$ can be given by:

$y_{k}=\sqrt{h_{k}}x+n_{k}=\sqrt{h_{k}p_{0}}s_{0}+\sum_{j=1}^{K}\sqrt{h_{k}p_{j}}s_{j}+n_{k},$
(2)

where $h_{k}$ represents the channel gain between user $k$ and the BS and
$n_{k}$ is the additive white Gaussian noise with variance $\sigma^{2}$. The
achievable rate of user $k$ decoding common message $s_{0}$ can be expressed
as:

$c_{k}=B\log_{2}\left(1+\frac{h_{k}p_{0}}{h_{k}\sum_{j=1}^{K}p_{j}+\sigma^{2}}\right),$
(3)

where $B$ is the bandwidth of the BS. Without loss of generality, the channel
gains are sorted in ascending order, i.e., $h_{1}\leq h_{2}\leq\cdots\leq
h_{K}$. To ensure that all users can successfully decode common message
$s_{0}$, the rate of common message should be chosen as [17]:

$\displaystyle\min_{k\in\mathcal{K}}c_{k}$
$\displaystyle=\min_{k\in\mathcal{K}}B\log_{2}\left(1+\frac{p_{0}}{\sum_{j=1}^{K}p_{j}+\frac{\sigma^{2}}{h_{k}}}\right)$
$\displaystyle=B\log_{2}\left(1+\frac{p_{0}}{\sum_{j=1}^{K}p_{j}+\frac{\sigma^{2}}{\min_{k\in\mathcal{K}}h_{k}}}\right)$
$\displaystyle\overset{\text{(a)}}{=}B\log_{2}\left(1+\frac{p_{0}}{\sum_{j=1}^{K}p_{j}+\frac{\sigma^{2}}{h_{1}}}\right)$
$\displaystyle=c_{1},$ (4)

where equality (a) follows from the fact that $h_{1}\leq h_{2}\leq\cdots\leq
h_{K}$.

To successfully implement SIC operation at the receiver, the transmit power of
each user must satisfy the following constraint [23]:

${h_{k}}p_{0}-{h_{k}}\sum_{j=1}^{K}p_{j}-{\sigma^{2}}\geq\theta,\quad\forall
k\in\mathcal{K},$ (5)

where $\theta$ is the minimum difference between the decoding signal power and
the non-decoded inter-user interference signal power plus noise power [24].
This minimum difference is required to distinguish the common message to be
decoded and the remaining non-decoded private message of all users (plus
noise). Based on the channel condition $h_{1}\leq h_{2}\leq\cdots\leq h_{K}$,
constraint (5) can be simplified as:

$p_{0}-\sum_{j=1}^{K}p_{j}\geq\frac{\theta+\sigma^{2}}{h_{1}}.$ (6)

Given the common message rate $c_{1}$ and the rate $a_{k}$ allocated to user
$k$, the constraint of each user $k$’s data rate of receiving common message
is given by:

$\sum_{k=1}^{K}a_{k}\leq c_{1},$ (7)

where (7) indicates that the total data rates of all users receiving common
message must be less than the rate of common message $c_{1}$.

After having decoded the common message $s_{0}$, each user can decode its
private message, the achievable rate of user $k$ decoding its private message
$s_{k}$ is given by:

$\displaystyle r_{k}$
$\displaystyle=B\log_{2}\left(1+\frac{h_{k}p_{k}}{h_{k}\sum_{j=1,j\neq
k}^{K}p_{j}+\sigma^{2}}\right).$ (8)

Given the common message rate $a_{k}$ and achievable private message rate
$r_{k}$, the total transmission rate of user $k$ in RSMA is:

$r_{k}^{\text{tot}}=a_{k}+r_{k}=a_{k}+B\log_{2}\left(1+\frac{h_{k}p_{k}}{h_{k}\sum_{j=1,j\neq
k}^{K}p_{j}+\sigma^{2}}\right).$ (9)

###  II-A Problem Formulation

Given the considered system model, our objective is to optimize the rate
allocation and power control so as to maximize the sum-rate under a total
power constraint and individual minimum rate requirements. Mathematically, the
sum-rate maximization problem for RSMA can be formulated as:

$\displaystyle\mathop{\max}_{\bm{a},\bm{p}}\quad$
$\displaystyle\sum_{k=1}^{K}{\left(a_{k}+B\log_{2}\left(1+\frac{h_{k}p_{k}}{h_{k}\sum_{j=1,j\neq
k}^{K}p_{j}+\sigma^{2}}\right)\right)},$ (10) s.t.
$\displaystyle\sum_{k=1}^{K}a_{k}\leq
B\log_{2}\left(1+\frac{h_{1}p_{0}}{h_{1}\sum_{j=1}^{K}p_{j}+\sigma^{2}}\right),$
(10a)
$\displaystyle{a_{k}+B\log_{2}\left(1+\frac{h_{k}p_{k}}{h_{k}\sum_{j=1,j\neq
k}^{K}p_{j}+\sigma^{2}}\right)}\geq R_{k},\quad\forall k\in\mathcal{K},$ (10b)
$\displaystyle p_{0}-\sum_{j=1}^{K}p_{j}\geq\frac{\theta+\sigma^{2}}{h_{1}},$
(10c) $\displaystyle\sum_{k=0}^{K}p_{k}\leq P,$ (10d) $\displaystyle
a_{k},p_{0},p_{k}\geq 0,\quad\forall k\in\mathcal{K},$ (10e)

where $\bm{a}=[a_{1},a_{2},\cdots,a_{K}]$,
$\bm{p}=[p_{0},p_{1},p_{2},\cdots,p_{K}]$, $R_{k}$ is the minimum rate demand
of user $k$, and $P$ is the maximum transmit power of the BS. Constraint (10a)
ensures that each user can decode the common message. The minimum rate
constraints for all users are given in (10b). Constraint (10c) shows the
successful SIC power requirement, and (10d) presents the maximum power
constraint.

Since the objective function is not concave, the sum-rate maximization problem
(10) is nonconvex. Moreover, the rate and power vectors are coupled in the
objective function and constraints, and hence, it is generally hard to solve
problem (10). Despite the nonconvexity and coupling of variables in problem
(10), the globally optimal solution to problem (10) can be effectively
obtained in Section III.

RSMA can potentially improve the rate of the network as shown in [13]. It is
therefore imperative to quantify the performance gains that RSMA can obtain
compared to conventional multiple access schemes. However, only suboptimal
resource allocation is obtained for the sum-rate optimization with RSMA in the
existing literature [15, 16, 17, 18, 19, 20, 21, 22]. Although the sum-rate
maximization problem in (10) investigates SISO, the optimal solution can be
derived, as shown in the following section, and this solution can serve as a
benchmark for the optimization with multiple antenna.

##  III Optimal Rate Allocation and Power Control

In this section, we first provide the optimal conditions of problem (10).
Then, based on these optimal conditions, the optimal private message
transmission power is obtained in closed form under a given rate allocation
and common message transmission power. Substituting the optimal private
message transmission power in problem (10), the optimal closed-form rate
allocation is then derived under a fixed common message transmission power.
Finally, a one-dimensional search algorithm is proposed to obtain the optimal
solution of problem (10). The proposed process for solving problem (10) is
summarized in Fig. 1.

Figure 1: Proposed approach for solving problem (10).

###  III-A Optimal Conditions

Before solving problem (10), we provide some optimal conditions, which will be
used to simplify problem (10).

###### Lemma 1

At the optimal solution $(\bm{a}^{*},\bm{p}^{*})$ of problem (10), the common
message constraint (10b) holds with equality, i.e.,
$\sum_{k=1}^{K}a_{k}^{*}=\log_{2}\left(1+\frac{h_{1}p_{0}^{*}}{h_{1}\sum_{j=1}^{K}p_{j}^{*}+\sigma^{2}}\right)$.

Lemma 1 can be easily proved by contradiction.

###### Lemma 2

At the optimal solution $(\bm{a}^{*},\bm{p}^{*})$ of problem (10), the maximum
power constraint (10d) holds with equality, i.e., $\sum_{k=0}^{K}p_{k}^{*}=P$.

Proof: See Appendix A. $\Box$

Applying Lemma 1 and substituting $\sum_{j=1,j\neq k}^{K}p_{j}=P-p_{0}-p_{k}$
from Lemma 2 to (10), we can then observe that problem (10) is equivalent to
the following problem:

$\displaystyle\mathop{\max}_{\bm{a},\bm{p}}\quad$
$\displaystyle\sum_{k=1}^{K}a_{k}+\sum_{k=1}^{K}B\log_{2}\left(\frac{h_{k}(P-p_{0})+\sigma^{2}}{h_{k}(P-p_{0}-p_{k})+\sigma^{2}}\right),$
(11) s.t.
$\displaystyle\sum_{k=1}^{K}a_{k}=B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})+\sigma^{2}}\right),$
(11a) $\displaystyle
a_{k}+B\log_{2}\left(\frac{h_{k}(P-p_{0})+\sigma^{2}}{h_{k}(P-p_{0}-p_{k})+\sigma^{2}}\right)\geq
R_{k},\forall k\in\mathcal{K},$ (11b) $\displaystyle\sum_{k=0}^{K}p_{k}=P,$
(11c) $\displaystyle p_{0}\geq\frac{P}{2}+\frac{\theta+\sigma^{2}}{2h_{1}},$
(11d) $\displaystyle a_{k},p_{k}\geq 0,\quad\forall k\in\mathcal{K}.$ (11e)

To solve problem (11), we can show that it is further equivalent to another
optimization problem, which admits a closed-form solution for the optimal
private message transmission power.

###### Lemma 3

The optimal solution of problem (11) is equivalent to the following problem:

$\displaystyle\mathop{\max}_{\bm{a},{\bm{p}}}\quad$ $\displaystyle
B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})+\sigma^{2}}\right)+\sum_{k=1}^{K}B\log_{2}\left(\frac{h_{k}(P-p_{0})+\sigma^{2}}{h_{k}(P-p_{0}-p_{k})+\sigma^{2}}\right),$
(12) s.t. $\displaystyle\sum_{k=1}^{K}a_{k}\leq
B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})+\sigma^{2}}\right),$
(12a) $\displaystyle
a_{k}+B\log_{2}\left(\frac{h_{k}(P-p_{0})+\sigma^{2}}{h_{k}(P-p_{0}-p_{k})+\sigma^{2}}\right)\geq
R_{k},\forall k\in\mathcal{K},$ (12b) $\displaystyle\sum_{k=0}^{K}p_{k}=P,$
(12c) $\displaystyle p_{0}\geq\frac{P}{2}+\frac{\theta+\sigma^{2}}{2h_{1}},$
(12d) $\displaystyle p_{k}\geq 0,0\leq a_{k}\leq R_{k},\quad\forall
k\in\mathcal{K}.$ (12e)

Proof: See Appendix B. $\Box$

Note that the maximum rate limitation $0\leq a_{k}\leq R_{k}$ is added in
constraint (12e), which will prove to be helpful in obtaining the optimal
private message transmission power in closed form.

###  III-B Optimal Private Message Transmission Power

Given the simplified problem in (12), next, we find the optimal private
message transmission power. Given rate allocation $\bm{a}$ and common message
power control $p_{0}$, problem (12) becomes

$\displaystyle\mathop{\max}_{\bar{\bm{p}}}\quad$
$\displaystyle\sum_{k=1}^{K}B\log_{2}\left(\frac{h_{k}(P-p_{0})+\sigma^{2}}{h_{k}(P-p_{0}-p_{k})+\sigma^{2}}\right),$
(13) s.t. $\displaystyle\sum_{k=1}^{K}p_{k}=P-p_{0},$ (13a) $\displaystyle
p_{k}\geq p_{k}^{\min},\quad\forall k\in\mathcal{K},$ (13b)

where $\bar{\bm{p}}=[p_{1},p_{2},\cdots,p_{K}]$ is a vector of power that is
allocated to each user for receiving private message and

$p_{k}^{\min}=\left(1-2^{\frac{a_{k}-R_{k}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{k}}\right).$
(14)

Due to constraint (12e), $p_{k}^{\min}$ is always non-negative.

Note that the fist term in objective function (12) is a constant with given
common message power control $p_{0}$, thus the fist term in objective function
(12) is omitted in problem (13). In (13b), $p_{k}^{\min}$ is used to meet the
minimum rate constraint in (12b), and problem (13) is feasible if and only if
$\sum_{k=1}^{K}p_{k}^{\min}\leq P-p_{0}$, which can be given as:

$\sum_{k=1}^{K}\left(1-2^{\frac{a_{k}-R_{k}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{k}}\right)\leq
P-p_{0}.$ (15)

Since objective function is convex, we can infer that the maximization problem
(13) is nonconvex. To effectively solve problem (13), the following theorem is
presented.

###### Theorem 1

For the optimal solution $\bar{\bm{p}}^{*}$ of problem (13), there exists one
$k$ such that $p_{k}^{*}=P-p_{0}-\sum_{j=1,j\neq k}^{K}p_{j}^{\min}$ and
$p_{j}^{*}=p_{j}^{\min}$, $\forall j\in\mathcal{K},j\neq k$.

Proof: See Appendix C. $\blacksquare$

From Theorem 1, the structure of the optimal solution of problem (13) is
revealed. Although problem (13) is nonconvex, the optimal solution can be
obtained in closed form, which can be given by the following theorem.

###### Theorem 2

For nonconvex problem (13), the optimal power allocation $\bar{\bm{p}}^{*}$ is

$p_{k}^{*}=P-p_{0}-\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{a_{j}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right),$
(16)
$p_{j}^{*}=\left(1-2^{\frac{a_{j}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right),\quad\forall
j\in\mathcal{K},j\neq k,$ (17)

and the optimal sum-rate of private message is

$\displaystyle
B\log_{2}\left(\frac{P-p_{0}+\frac{\sigma^{2}}{h_{k}}}{\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{a_{j}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)+\frac{\sigma^{2}}{h_{k}}}\right)+\sum_{j=1,j\neq
k}^{K}(R_{j}-a_{j}),$ (18)

where

$k=\arg\min_{j\in\mathcal{K}}2^{\frac{a_{j}-R_{j}}{B}}\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right).$
(19)

Proof: See Appendix D. $\blacksquare$

Theorem 2 states that it is optimal for the BS to allocate more power to the
user that can maximize the sum-rate while allocating the minimum transmit
power that can meet the data rate requirement for all other users.

For the special case with $a_{j}=R_{j}$, $\forall j\in\mathcal{K}$, we can
obtain $p_{j}^{\min}=0$ and
$k=\arg\min_{j\in\mathcal{K}}\frac{\sigma^{2}}{h_{j}}=K$ according to (19),
i.e., all the power should be allocated to the user with the highest channel
gain. This observation is trivial since allocating the maximum power to the
user with the highest channel gain will always improve the rate.

For the special case in which $\bm{a}=\bm{0}$, i.e., the broadcast channel
without SIC, we have the following corollary that follows from Theorem 2.

###### Corollary 1

For the downlink nonconvex sum-rate maximization problem in broadcast channel,
which is given by:

$\displaystyle\mathop{\max}_{\bm{p}}\quad$
$\displaystyle\sum_{k=1}^{K}{B\log_{2}\left(1+\frac{h_{k}p_{k}}{h_{k}\sum_{j=1,j\neq
k}^{K}p_{j}+\sigma^{2}}\right)},$ (20) s.t.
$\displaystyle{B\log_{2}\left(1+\frac{h_{k}p_{k}}{h_{k}\sum_{j=1,j\neq
k}^{K}p_{j}+\sigma^{2}}\right)}\geq R_{k},\>\forall k\in\mathcal{K},$ (20a)
$\displaystyle\sum_{k=1}^{K}p_{k}\leq P,$ (20b) $\displaystyle p_{k}\geq
0,\quad\forall k\in\mathcal{K},$ (20c)

the maximization problem in (20) is feasible if and only if

$\sum_{j=1}^{K}\left(1-2^{\frac{-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)\leq
P-p_{0},$ (21)

and the optimal power allocation $\bm{p}^{*}$ is to meet the minimum rate
requirements of all users except one user, i.e.,

$p_{k}^{*}=P-\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{-R_{j}}{B}}\right)\left(P+\frac{\sigma^{2}}{h_{j}}\right),$
(22)
$p_{j}^{*}=\left(1-2^{\frac{-R_{j}}{B}}\right)\left(P+\frac{\sigma^{2}}{h_{j}}\right),\quad\forall
j\in\mathcal{K},j\neq k,$ (23)

and the optimal objective value is

$\displaystyle
B\log_{2}\left(\frac{P+\frac{\sigma^{2}}{h_{k}}}{\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{-R_{j}}{B}}\right)\left(P+\frac{\sigma^{2}}{h_{j}}\right)+\frac{\sigma^{2}}{h_{k}}}\right)+\sum_{j=1,j\neq
k}^{K}R_{j},$ (24)

where

$k=\arg\min_{j\in\mathcal{K}}2^{\frac{-R_{j}}{B}}\left(P+\frac{\sigma^{2}}{h_{j}}\right).$
(25)

Corollary 1 provides the optimal power allocation that maximizes the downlink
sum-rate of the broadcast channel. From this result, we can see that more
power should be allocated to one of the users compared to allocating the
minimum transmit power to all other users. The user that should be allocated
more power is jointly determined by the channel gain and the minimum rate
demand, as shown in (25).

###  III-C Optimal Rate Allocation

In the previous subsection, the optimal power allocation vector $\bar{\bm{p}}$
can be obtained as a function of the rate allocation vector $\bm{a}$ and
common message power $p_{0}$. Thus, substituting the optimal power allocation
vector $\bar{\bm{p}}$ given in (16) and (17) in Theorem 2, the original
problem in (12) can be simplified as:

$\displaystyle\mathop{\max}_{\bm{a},p_{0}}\quad$ $\displaystyle
B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})+\sigma^{2}}\right)+\sum_{j=1,j\neq
k}^{K}(R_{j}-a_{j})$
$\displaystyle+B\log_{2}\left(\frac{P-p_{0}+\frac{\sigma^{2}}{h_{k}}}{\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{a_{j}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)+\frac{\sigma^{2}}{h_{k}}}\right),$
(26) s.t. $\displaystyle\sum_{j=1}^{K}a_{j}\leq
B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})+\sigma^{2}}\right),$
(26a) $\displaystyle
k=\arg\min_{j\in\mathcal{K}}2^{\frac{a_{j}-R_{j}}{B}}\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right),$
(26b)
$\displaystyle\sum_{j=1}^{K}\left(1-2^{\frac{a_{j}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)\leq
P-p_{0},$ (26c) $\displaystyle
p_{0}\geq\frac{P}{2}+\frac{\theta+\sigma^{2}}{2h_{1}},$ (26d) $\displaystyle
0\leq a_{j}\leq R_{j},\quad\forall j\in\mathcal{K},$ (26e)

where constraint (26b) is added since more power should be allocated to user
$k$ to maximize the sum-rate while other users are allocated the minimum power
to ensure the minimum rate constraint. Constraint (26c) follows from (15),
which ensures that the private message power control problem is feasible.

Due to objective function and constraints (26a)-(26c), problem (26) is
nonconvex and, hence, it is generally hard to directly optimize rate
allocation $\bm{a}$ and private power $p_{0}$. To solve problem (26), we first
fix the private message transmission power and derive the optimal rate
allocation. Given common message transmission power $p_{0}$, problem (26)
becomes

$\displaystyle\mathop{\max}_{\bm{a}}\quad$ $\displaystyle\sum_{j=1,j\neq
k}^{K}(R_{j}-a_{j})+B\log_{2}\left(\frac{P-p_{0}+\frac{\sigma^{2}}{h_{k}}}{\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{a_{j}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)+\frac{\sigma^{2}}{h_{k}}}\right),$
(27) s.t. $\displaystyle\sum_{j=1}^{K}a_{j}\leq c_{1},$ (27a) $\displaystyle
k=\arg\min_{j\in\mathcal{K}}2^{\frac{a_{j}-R_{j}}{B}}\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right),$
(27b)
$\displaystyle\sum_{j=1}^{K}\left(1-2^{\frac{a_{j}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)\leq
P-p_{0},$ (27c) $\displaystyle 0\leq a_{j}\leq R_{j},\quad\forall
j\in\mathcal{K}.$ (27d)

To solve (27), we first need to prove that the objective function (27) is
convex, which can be given by the following lemma.

###### Lemma 4

The objective function (27) is convex.

Proof: See Appendix E. $\blacksquare$

According to Lemma 4, the objective function (27) is convex. The maximum point
of the convex function always lies in the corner points, i.e., the optimal
solution of problem (27) lies in a finite space of potential candidates. For
the optimal solution of (27), we can prove the following:

###### Theorem 3

The optimal solution $\bm{a}^{*}$ of problem (27) exists in the following
three cases:

Case 1:

$a_{j}^{*}\in\\{0,R_{j}\\},\quad j\in\mathcal{K}.$ (28)

Case 2: there exists $l\in\mathcal{K}$ such that

$a_{j}^{*}\in\\{0,R_{j}\\},a_{l}^{*}\in\\{b_{1}|_{0}^{R_{k}},b_{2}|_{0}^{R_{k}}\\}\quad
j\in\mathcal{K},j\neq l,$ (29)

where

$\displaystyle b_{1}=$ $\displaystyle c_{1}-\sum_{j=1,j\neq k}^{K}a_{j},$ (30)

and

$\displaystyle b_{2}=$ $\displaystyle
R_{k}-B\log_{2}\left(P-p_{0}+\frac{\sigma^{2}}{h_{k}}\right)+B\log_{2}\left({\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{a_{j}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)+\frac{\sigma^{2}}{h_{k}}}\right).$
(31)

Case 3: there exists $m,n\in\mathcal{K}$,

$a_{j}^{*}\in\\{0,R_{j}\\},a_{m}^{*}=c_{1}-\sum_{j=1,j\neq
k}^{K}a_{j}^{*},\quad j\in\mathcal{K},j\neq m,n,$ (32)

and $a_{n}^{*}$ satisfies

$\displaystyle\left(1-2^{\frac{c_{1}-\sum_{j=1,j\neq
k,n}^{K}a_{j}^{*}-a_{n}^{*}-R_{m}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{m}}\right)+\left(1-2^{\frac{a_{n}^{*}-R_{n}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{n}}\right)$
$\displaystyle+\sum_{j=1,j\neq
m,n}^{K}\left(1-2^{\frac{a_{j}^{*}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)=P-p_{0}.$
(33)

In Theorem 3, Case 1 indicates that the corner points satisfy the constraint
in (27d), Case 2 indicates that the corner points satisfy the constraint in
(27d) as well as one constraint in (27a) or (27c), and Case 3 indicates that
the corner points satisfy the constraints in (27a), (27c), and (27d). Theorem
3 follows directly from the fact that maximizing a convex function lies in its
corner points. The optimal solution of problem (27) will be one of the corner
points in all three cases.

Since the left term of (3) is a convex function of $a_{n}^{*}$, at most two
solutions for $a_{n}^{*}$ satisfy (3). We then observe that there are $2^{K}$,
$2^{K}$, and $2^{K-1}$ points for Cases 1, 2 and 3, respectively. As a result,
there are $5\times 2^{K-1}$ potential candidates for the optimal solution of
problem (27) according to Theorem 3.

For the special case that all users have the same data rate requirement, the
optimal solution of problem (27) can be precisely formulated in the following
theorem.

###### Theorem 4

Given the data rate requirement of each user, i.e.,
$R_{1}=R_{2}=\cdots=R_{K}=R$, the optimal solution $\bm{a}^{*}$ of problem
(27) is:

i) if $\lfloor\frac{c_{1}}{R}\rfloor<K$,

$r_{j}^{*}=\left\\{\begin{array}[]{ll}\\!\\!R,&\text{if}\;j<l,\\\
\\!\\!c_{1}-(l-1)R,&\text{if}\;j=l,\\\
\\!\\!0,&\text{otherwise,}\end{array}\right.$ (34)

where $\lfloor\cdot\rfloor$ means round down and
$l=\lfloor\frac{c_{1}}{R}\rfloor$.

ii) if $\lfloor\frac{c_{1}}{R}\rfloor\geq K$,

$r_{j}^{*}=R,\quad\forall j\in\mathcal{K}.$ (35)

Proof: See Appendix F. $\blacksquare$

Theorem 4 provides the optimal rate allocation of problem (27) with equal rate
demand in closed form. Based on Theorem 4, we can determine the user that the
BS will allocate additional power using the following lemma.

###### Lemma 5

Given the data rate requirement of each user, i.e.,
$R_{1}=R_{2}=\cdots=R_{K}=R$, it is optimal to allocate the additional power
to the user with the highest channel gain, which is given by:

$k=\arg\min_{j\in\mathcal{K}}2^{\frac{a_{j}-R}{B}}\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)=K.$
(36)

Proof: Based on Theorem 5, we have $a_{1}^{*}\geq a_{2}^{*}\geq\cdots\geq
a_{K}^{*}$. Since
$\frac{\sigma^{2}}{h_{1}}\geq\frac{\sigma^{2}}{h_{2}}\geq\cdots\geq\frac{\sigma^{2}}{h_{K}}$,
we have

$\displaystyle
k=\arg\min_{j\in\mathcal{K}}2^{\frac{a_{j}^{*}-R}{B}}\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)=K.$
(37)

$\blacksquare$

###### Lemma 6

For a network with two users, $K=2$ and their data rate requirements are
$R_{1}=R_{2}=R$, the optimal solution $(\bm{a}^{*},p_{0}^{*})$ of problem (26)
is $a_{1}^{*}=R$, $a_{2}^{*}=0$ and

$p_{0}^{*}=\max\left\\{\left(1-2^{\frac{-R}{B}}\right)\left(P+\frac{\sigma^{2}}{h_{1}}\right),\frac{P}{2}+\frac{\theta+\sigma^{2}}{2h_{1}}\right\\}.$
(38)

Proof: See Appendix G. $\blacksquare$

According to Lemma 6, for a network that only has two users, it is optimal to
decode the messages of user 1 first in the common message, and the remaining
power is then all allocated to user 2. For $K=2$ users, NOMA can be viewed as
a special case of RSMA if $a_{2}=p_{1}=0$, i.e., the common message is
allocated to only user 1, while only user 2 has its unique private message.

###  III-D Optimal Rate Allocation and Power Control

To obtain the optimal rate allocation and power control of problem (12), we
propose a novel solution, shown in Algorithm 1 where $\xi$ is the minimum step
size for searching $p_{0}$. In Algorithm 1, the optimal common message power
$p_{0}$ is obtained by a one-dimensional search method, while the optimal rate
allocation $\bm{a}$ and private message power $\bar{\bm{p}}$ are accordingly
obtained in closed form given $p_{0}$.

Algorithm 1 Optimal Rate Allocation and Power Control

1: for $p_{0}=\frac{P}{2}+\frac{\theta}{2}+\frac{\sigma^{2}}{2h_{1}}:\xi:P$ do

2: Obtain the optimal rate allocation $\bm{a}$ of problem (27) according to
Theorem 3.

3: Calculate the optimal common message power allocation $\bar{\bm{p}}$ of
(13) according to Theorem 2.

4: end for

5: Obtain the optimal $p_{0}$ with the maximum objective value (12a).

In each step of Algorithm 1, the main complexity lies in solving (27) given
$p_{0}$. Since there are $5\times 2^{K-1}$ potential candidates for the
optimal solution of problem (27) according to Theorem 3, the complexity of
solving the problem in (27) is $\mathcal{O}(5\times 2^{K-1})$. As a result,
the complexity of Algorithm 1 is $\mathcal{O}(5L2^{K-1})$, where
$L=\mathcal{O}\left(\frac{Ph_{1}-\theta h_{1}-{\sigma^{2}}}{2h_{1}\xi}\right)$
denotes the number of iterations for searching $p_{0}$. In practice, we
consider a small number of users, i.e., $K$ is small due to the complexity of
decoding the common message, the computation complexity of Algorithm 1 can be
practical. To deal with a large number of users, the users can be classified
into different groups with small number of users in each group. The users in
different groups occupy different subchannels and users in the same group are
allocated to the same subchannel using RSMA. Moreover, for equal rate demand,
the complexity of solving the problem in (27) is $\mathcal{O}(K)$ according to
Theorem 4 and the complexity of Algorithm 1 is $\mathcal{O}(LK)$.

##  IV Simulation Results

In this section, we evaluate the performance of the proposed optimal rate
allocation and power control algorithm. $K$ users there are uniformly
distributed in a square area of size $300$ m $\times$ $300$ m. The path loss
model is $128.1+37.6\log_{10}d$ ($d$ is in km) [24] and the standard deviation
of shadow fading is $4$ dB.In addition, the bandwidth of the BS is $B=1$ MHz
and the noise power is $\sigma^{2}=-104$ dBm. Unless specified otherwise, the
system parameters are set as maximum transmit power $P=30$ dBm, equal rate
demand $R_{1}=R_{2}=\cdots=R_{K}=R=1$ Mbits/s and SIC detection threshold is
set as $\theta=-94$ dBm. The main system parameters are listed in Table I.

TABLE I: System Parameters Parameter | Value
---|---
Bandwidth of the BS $B$ | 1 MHz
noise power $\sigma^{2}$ | -104 dBm
maximum transmit power $P$ | 30 dBm
Minimal rate demand $R$ | 1 Mbits/s
SIC detection threshold $\theta$ | -94 dBm

The proposed optimal rate allocation and power control algorithm for rate
maximization of RSMA is labeled as ‘RSMA’. We compare with the proposed
algorithm with the optimal power control of NOMA for rate maximization [24],
which is labeled as ‘NOMA’. To compare conventional orthogonal multiple access
(OMA), we use a OFDMA system [25] as a baseline, which is labeled as ‘OFDMA’.

Figure 2: Sum-rate versus maximum transmit power of the BS ($K=2$ users)
Figure 3: Rate region for two users.

Fig. 2 shows how the sum-rate changes as the maximum transmit power of the BS
varies for a network having two users. From Fig. 2, we can see that the sum-
rate linearly increases with the logarithmic maximum transmit power of the BS.
This figure shows that RSMA achieves the same performance as NOMA in terms of
sum-rate, which verifies the theoretical finding in Lemma 6. However, RSMA
will dynamically allocate the rate of the common message to multiple users to
meet the rate demand, while the rate decoded in NOMA is allocated to only one
specific user. From Fig. 2, we can see that both RSMA and NOMA significantly
outperform conventional OFDMA. This is because the BS can simultaneously
transmit signals to all users by RSMA or NOMA at the same frequency, while the
BS transmits signal to different users in different resource blocks by OFDMA.

Figure 4: Sum-rate versus maximum transmit power of the BS ($K=3$ users and
$R_{1}=R_{2}=R_{3}=1$ Mbits/s). Figure 5: Sum-rate versus maximum transmit
power of the BS ($K=3$ users, $R_{1}=1.5$ Mbits/s, $R_{2}=0.5$ Mbits/s and
$R_{3}=1$ Mbits/s).

The rate region for two users is given in Fig. 3. It is shown that the rate of
user 1 is always greater than a fixed value when the rate of users 2 is
positive in NOMA. This is due to the fact that the allocated power of user 1
should be greater than a fixed value according to the successful SIC
constraint. The maximum rate of user 2 in RSMA is larger than that in NOMA due
to the benefit of rate splitting in RSMA.

Figs. 4 and 5 show the sum-rate sum-rate versus maximum transmit power of the
BS with equal rate demand and unequal rate demand for three users,
respectively. From these figures, we can see that RSMA always achieves the
best performance among all schemes. This is due to the fact that the number of
SIC in RSMA is once, while the number of SIC in NOMA can be twice, which
results in high power allocation to the users with low channel gain according
to the successful SIC power requirement [24, Eq. (3)] and leads to a low sum-
rate. These figures also show that RSMA outperforms NOMA particularly for low
maximum BS transmit power. Compared to OFDMA, RSMA is better due to the fact
that all users can be served with the whole bandwidth of the BS.

Figure 6: Sum-rate versus minimum rate demand ($K=3$ users, $R_{2}=R_{3}=1$ M
bits/s). Figure 7: Sum-rate versus SIC detection threshold ($K=3$ users).
Figure 8: CDF of the sum-rate resulting from RSMA, NOMA, and OFDMA for a
network with $K=3$ users. Figure 9: Sum-rate versus number of users.

Fig. 6 shows the sum-rate versus minimum rate demand. From this figure, RSMA
always achieves a better performance than NOMA and OFDMA. From Fig. 6, we can
observe that the sum-rate decreases slightly when minimum rate demand is low.
However, for a high minimum rate demand, the sum-rate decreases rapidly. This
is because a high minimum rate demand requires the BS to allocate more power
to the users with worse channel gains, which consequently degrades the sum-
rate performance. Fig. 6 also demonstrates that, as the minimum rate demand
increases, the sum-rates of OFDMA and NOMA decrease faster than RSMA. In
particular, RSMA can achieve up to 15.6% and 21.5% gains in terms of data rate
compared to NOMA and OFDMA, respectively. This is due to the fact that RSMA
exhibits a better spectrum efficiency compared to OFDMA and NOMA, and OFDMA
and NOMA are more sensitive to high minimum rate demand than RSMA.

Fig. 7 shows the sum-rate versus SIC detection threshold $\theta$. For both
RSMA and NOMA, we find that the sum-rate decreases as the SIC detection
threshold increases. This is due to the fact that, as the SIC detection
threshold increases, the BS must allocate more power to the common message in
RSMA and the user with worse channel gain in NOMA. For OFDMA, naturally, the
sum-rate remains the same when the SIC detection threshold increases. The
proposed RSMA algorithm outperforms the NOMA in terms of sum-rate,
particularly for cases with a high SIC detection threshold. Moreover, the sum-
rate decreases faster for NOMA than RSMA as the SIC detection threshold
increases, which implies that RSMA is more suitable for high SIC detection
threshold.

Fig. 8 shows the cumulative distribution function (CDF) of the sum-rate
resulting from RSMA, NOMA, and OFDMA for a network with $K=3$ users. From Fig.
8, the CDFs for RSMA and NOMA all improve significantly vs. OFDMA especially
for high sum-rate region, which shows that both RSMA and NOMA are suitable for
high sum-rate transmission. Moreover, we can find that RSMA outperforms NOMA
at regions with moderate data rates, i.e., 5-15 Mbits/s. This is because RSMA
can adjust the split between the common and private messages so as to control
the interference decoding and thus optimize the sum-rate of users.

The sum-rate versus number of users is given in Fig. 9. Clearly, the proposed
RSMA is always better than NOMA and OFDMA especially when the number of users
is large. When the number of users is large, the multiuser gain is more
apparent by the proposed RSMA compared to conventional NOMA and OFDMA. This is
because RSMA can effectively determine the rate of each user receiving common
message to meet its specific rate demand, while the SIC time of each user is
high for NOMA and the allocated bandwidth of each user is low for OFDMA when
the number of users is large. RSMA achieves better performance than NOMA and
OFDMA at the cost of additional computational complexity according to Section
III-D.

##  V Conclusions

In this paper, we have investigated the allocation of data rate of common
message transmission and the transmit power used for common and private
message transmission in a SISO RSMA system. We have formulated the problem as
a minimization problem. To solve this problem, we have derived the optimal
transmit power of private message in closed form. Then, we have characterized
the finite solution space for the optimal rate allocation. Finally, we have
proposed a one-dimensional search algorithm to find the optimal rate
allocation and power control solutions. Simulation results show that RSMA
achieves higher sum-rate than NOMA and OFDMA especially for low maximum
transmit power of the BS, high minimum rate demand of users, high SIC
detection threshold and large number of users.

##  Appendix A Proof of Lemma 2

We assume that the optimal solution $(\bm{a}^{*},\bm{p}^{*})$ of problem (10)
satisfies $\sum_{k=0}^{K}p_{k}^{*}<P$. We construct new power allocation by
scaling power $\bm{p}^{*}$, i.e.,

$p_{k}^{\prime}=\frac{P}{\sum_{j=0}^{K}p_{j}^{*}}p_{k}^{*}>p_{k}^{*},\quad\forall
k=0,1,\cdots,K.$ (A.1)

Given new power allocation vector
$\bm{p}^{\prime}=[p_{0}^{\prime},p_{1}^{\prime},\cdots,p_{K}^{\prime}]$, we
have

$\displaystyle
B\log_{2}\left(1+\frac{h_{1}p_{0}^{\prime}}{h_{1}\sum_{j=1}^{K}p_{j}^{\prime}+\sigma^{2}}\right)>B\log_{2}\left(1+\frac{h_{1}p_{0}^{*}}{h_{1}\sum_{j=1}^{K}p_{j}^{*}+\frac{\sigma^{2}{\sum_{j=0}^{K}p_{j}^{*}}}{{P}}}\right),$
(A.2) $\displaystyle
B\log_{2}\left(1+\frac{h_{k}p_{k}^{\prime}}{h_{k}\sum_{j=1,j\neq
k}^{K}p_{k}^{\prime}+\sigma^{2}}\right)>B\log_{2}\left(1+\frac{h_{k}p_{k}^{*}}{h_{1}\sum_{j=1,j\neq
k}^{K}p_{j}^{*}+\frac{\sigma^{2}{\sum_{j=0}^{K}p_{j}^{*}}}{{P}}}\right),$
(A.3)

and

$\displaystyle p_{0}^{\prime}-\sum_{j=1}^{K}p_{j}^{\prime}=$
$\displaystyle\frac{P}{\sum_{j=0}^{K}p_{j}^{*}}\left(p_{0}^{*}-\sum_{j=1}^{K}p_{j}^{*}\right)>p_{0}^{*}-\sum_{j=1}^{K}p_{j}^{*}\geq\theta+\frac{\sigma^{2}}{h_{1}},$
(A.4)

where the first inequalities in (A.2)-(A.4) follow from the fact that
$\frac{P}{\sum_{j=0}^{K}p_{j}^{*}}>1$.

According to (A.2)-(A.4), we can see that new solution
$(\bm{a}^{*},\bm{p}^{\prime})$ is feasible and the objective value (10) of new
solution is better than that of solution $(\bm{a}^{*},\bm{p}^{*})$, which
contradicts the fact that $(\bm{a}^{*},\bm{p}^{*})$ is the optimal solution.
Lemma 2 is proved.

##  Appendix B Proof of lemma 3

If the pair $(\bm{a},{\bm{p}})$ is feasible in problem (12), then the pair
$(\bm{a},{\bm{p}})$ is also feasible in problem (11) with the same objective
value. It follows from the fact that the optimal value of (12) is less than or
equal to the optimal value of (11).

Conversely, if the pair $(\bm{a},{\bm{p}})$ is feasible in (11), we can
construct a new pair $(\bm{a}^{\prime},{\bm{p}})$, where

$a_{k}^{\prime}=\min\\{a_{k},R_{k}\\},\quad\forall k\in\mathcal{K}.$ (B.1)

It can be shown that solution $(\bm{a}^{\prime},{\bm{p}})$ is feasible in
problem (12). Moreover, the objective value of problem (11) is the same as
problem (12). Thus, we conclude that the optimal value of (12) is greater than
or equal to the optimal value of (11). Hence, problem (12) is equivalent to
problem (11).

##  Appendix C Proof of Theorem 1

Assume that there exist $m$ and $n$ such that $p_{m}^{*}>p_{m}^{\min}$ and
$p_{n}^{*}>p_{n}^{\min}$ for the optimal solution $\bar{\bm{p}}^{*}$. Next, we
can show that there always exist feasible power $p_{m}^{\prime}$ and
$p_{n}^{\prime}$ with better objective value (13a).

To construct such $p_{m}^{\prime}$ and $p_{n}^{\prime}$, we substitute
$p_{j}=p_{j}^{*}$, $j\in\mathcal{K}$, $j\neq m,n$, into problem (13), which
yields

$\displaystyle\mathop{\max}_{p_{m},p_{n}}\quad$ $\displaystyle
B\log_{2}\left(\frac{h_{m}(P-p_{0})+\sigma^{2}}{h_{m}(P-p_{0}-p_{m})+\sigma^{2}}\right)+B\log_{2}\left(\frac{h_{n}(P-p_{0})+\sigma^{2}}{h_{n}(P-p_{0}-p_{n})+\sigma^{2}}\right),$
(C.1) s.t. $\displaystyle p_{m}+p_{n}=P-p_{0}-\sum_{j=1,j\neq
m,n}^{K}p_{j}^{*},$ (C.1a) $\displaystyle p_{m}\geq p_{m}^{\min},p_{n}\geq
p_{n}^{\min}.$ (C.1b)

According to (C.1a), we have

$p_{m}=P-p_{0}-\sum_{j=1,j\neq m,n}^{K}p_{j}^{*}-p_{n}.$ (C.2)

Combining (C.1b) and (C.2), we have

$p_{n}^{\min}\leq p_{n}\leq P-p_{0}-\sum_{j=1,j\neq
m,n}^{K}p_{j}^{*}-p_{m}^{\min}.$ (C.3)

From (C.2) and (C.3), we can see that problem (C.1) can be simplified as

$\displaystyle\mathop{\max}_{p_{n}}\quad$
$\displaystyle-B\log_{2}\left({h_{m}\left(\sum_{j=1,j\neq
m,n}^{K}p_{j}^{*}+p_{n}\right)+\sigma^{2}}\right)-B\log_{2}({h_{n}(P-p_{0}-p_{n})+\sigma^{2}}),$
(C.4) s.t. $\displaystyle p_{n}^{\min}\leq p_{n}\leq P-p_{0}-\sum_{j=1,j\neq
m,n}^{K}p_{j}^{*}-p_{m}^{\min}.$ (C.4a)

Due to the convexity of function $-\log(x)$, the objective function (C.4) is
convex. Since the maximization of a convex function always lies in the
boundary of the feasible solution, i.e., the optimal solution $p_{n}^{\prime}$
of problem (C.4) satisfies

$p_{n}^{\prime}\in\left\\{p_{n}^{\min},P-p_{0}-\sum_{j=1,j\neq
m,n}^{K}p_{j}^{*}-p_{m}^{\min}\right\\}.$ (C.5)

Further considering (C.2), we can construct

$p_{m}^{\prime}=P-p_{0}-\sum_{j=1,j\neq m,n}^{K}p_{j}^{*}-p_{n}^{\prime}.$
(C.6)

Based on the equivalence of problem (C.1) and problem (C.4),
$(p_{m}^{\prime},p_{n}^{\prime})$ is the optimal solution of problem (C.1).
According to (C.5) and (C.6), $p_{m}^{\prime}=p_{m}^{\min}$ or
$p_{n}^{\prime}=p_{n}^{\min}$ is always satisfied.

Since $(p_{m}^{\prime},p_{n}^{\prime})$ is the optimal solution of problem
(C.1) and $(p_{m}^{\prime},p_{n}^{\prime})\neq(p_{m}^{*},p_{n}^{*})$, we can
claim that solution

$(p_{1}^{*},\cdots,p_{m-1}^{*},p_{m}^{\prime},p_{m+1}^{*},\cdots,p_{n-1}^{*},p_{n}^{\prime},p_{n+1}^{*},\cdots,p_{K}^{*})$
(C.7)

is feasible with better objective value than solution $\bar{\bm{p}}^{*}$,
which contradicts the fact that $\bar{\bm{p}}^{*}$ is the optimal solution of
problem (13).

As a result, the proof of Theorem 1 is complete.

##  Appendix D Proof of Theorem 2

Substituting the optimal power allocation $p_{k}^{*}=P-p_{0}-\sum_{j=1,j\neq
k}^{K}p_{j}^{\min}$ and $p_{j}^{*}=p_{j}^{\min}$, $\forall
j\in\mathcal{K},j\neq k$ to problem (13) according to Theorem 1, we can obtain
the objective value (13a) as

$\displaystyle\sum_{j=1,j\neq
k}^{K}B\log_{2}\left(\frac{h_{j}(P-p_{0})+\sigma^{2}}{h_{j}(P-p_{0}-p_{j}^{\min})+\sigma^{2}}\right)+B\log_{2}\left(\frac{h_{k}(P-p_{0})+\sigma^{2}}{h_{k}\sum_{j=1,j\neq
k}^{K}p_{j}^{\min}+\sigma^{2}}\right).$ (D.1)

To maximize sum-rate (D.1), the optimal $k$ should be chosen as

$\displaystyle k=\arg\max_{m\in\mathcal{K}}$
$\displaystyle\sum_{j=1}^{K}B\log_{2}\left(\frac{h_{j}(P-p_{0})+\sigma^{2}}{h_{j}(P-p_{0}-p_{j}^{\min})+\sigma^{2}}\right)-B\log_{2}\left(\frac{h_{m}(P-p_{0})+\sigma^{2}}{h_{m}(P-p_{0}-p_{m}^{\min})+\sigma^{2}}\right)$
$\displaystyle+B\log_{2}\left(\frac{h_{m}(P-p_{0})+\sigma^{2}}{h_{m}\sum_{j=1,j\neq
m}^{K}p_{j}^{\min}+\sigma^{2}}\right)$
$\displaystyle=\arg\max_{m\in\mathcal{K}}\>$ $\displaystyle
B\log_{2}({h_{m}(P-p_{0}-p_{m}^{\min})+\sigma^{2}})-B\log_{2}\left({h_{m}\sum_{j=1,j\neq
m}^{K}p_{j}^{\min}+\sigma^{2}}\right)$
$\displaystyle=\arg\max_{m\in\mathcal{K}}$
$\displaystyle\frac{h_{m}(P-p_{0}-p_{m}^{\min})+\sigma^{2}}{h_{m}\sum_{j=1,j\neq
m}^{K}p_{j}^{\min}+\sigma^{2}}-1$ $\displaystyle=\arg\max_{m\in\mathcal{K}}$
$\displaystyle\frac{P-p_{0}-\sum_{j=1}^{K}p_{j}^{\min}}{\sum_{j=1}^{K}p_{j}^{\min}+\frac{\sigma^{2}}{h_{m}}-p_{m}^{\min}}$
$\displaystyle=\arg\min_{m\in\mathcal{K}}$
$\displaystyle\frac{\sigma^{2}}{h_{m}}-p_{m}^{\min}.$ (D.2)

Substituting (14) to (D), we have

$\displaystyle k$
$\displaystyle=\arg\min_{j\in\mathcal{K}}\frac{\sigma^{2}}{h_{j}}-\left(1-2^{\frac{a_{j}-R_{j}}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)=\arg\min_{j\in\mathcal{K}}2^{\frac{a_{j}-R_{j}}{B}}\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right).$
(D.3)

This completes the proof of Theorem 2.

##  Appendix E Proof of Lemma 4

Denote the objective function (27) as $f(\bm{a})$ and we have

$\displaystyle\frac{\partial^{2}f(\bm{a})}{\partial a_{m}a_{k}}=0,\quad\forall
m\in\mathcal{K},$ (E.1) $\displaystyle\frac{\partial^{2}f(\bm{a})}{\partial
a_{m}^{2}}=\frac{(\ln 2)u_{m}\left(d-\sum_{j=1,j\neq
m}^{K}u_{j}\right)}{B\left(d-\sum_{j=1}^{K}u_{j}\right)^{2}},\quad\forall
m\in\mathcal{K},$ (E.2)

and

$\displaystyle\frac{\partial^{2}f(\bm{a})}{\partial a_{m}a_{n}}=\frac{(\ln
2)u_{m}u_{n}}{B\left(\sum_{j=1}^{K}u_{j}+\frac{\sigma^{2}}{h_{k}}\right)^{2}},\quad\forall
m,n\in\mathcal{K},m\neq n,$ (E.3)

where

$u_{j}=2^{\frac{a_{j}-R}{B}}\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right),u_{k}=0,\quad\forall
j\neq k,$ (E.4)

and

$d={\sum_{j=1,j\neq
k}^{K}\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)+\frac{\sigma^{2}}{h_{k}}}.$
(E.5)

Then, we can obtain Hessian matrix of (27) as

$\nabla^{2}f(\bm{a})=\frac{(\ln
2)\left(\left(d-\sum_{j=1}^{K}u_{j}\right)\text{diag}(\bm{u})+\bm{u}^{T}\bm{u}\right)}{B\left(\sum_{j=1}^{K}u_{j}+\frac{\sigma^{2}}{h_{k}}\right)^{2}}$
(E.6)

where $\bm{u}=[u_{1},u_{2},\cdots,u_{K}]$. Since $u_{j}>0$ for all $j\neq k$
and $u_{k}=0$ according to (E.4), both $\text{diag}(\bm{u})$ and
$\bm{u}^{T}\bm{u}$ are positive semi-definite. Consequently, the Hessian
matrix of (27) is positive semi-definite. In consequence, (27) is a convex
function. This completes the proof of Lemma 4.

##  Appendix F Proof of Theorem 4

We first show that $a_{m}^{*}\geq a_{n}^{*}$ for all $m<n$ for the optimal
solution $\bm{a}^{*}$ of problem (27) with equal rate demand. This can be
proved by the contradiction method. If there exists $m<n$ such that
$a_{m}^{*}<a_{n}^{*}$, we can construct a new solution $\bm{a}^{\prime}$ with
$a_{m}^{\prime}=a_{n}^{*}$, $a_{n}^{\prime}=a_{m}^{*}$,
$a_{j}^{\prime}=a_{j}^{*}$ for $j\neq m,n$. Then, we can obtain

$\displaystyle\sum_{j=1}^{K}\left(1-2^{\frac{a_{j}^{\prime}-R}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{K}\left(1-2^{\frac{a_{j}^{*}-R}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)+\left(2^{\frac{a_{m}^{*}-R}{B}}-2^{\frac{a_{n}^{*}-R}{B}}\right)\left(\frac{\sigma^{2}}{h_{m}}-\frac{\sigma^{2}}{h_{n}}\right)$
$\displaystyle<$
$\displaystyle\sum_{j=1}^{K}\left(1-2^{\frac{a_{j}^{*}-R}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right),$
(F.1)

where the inequality follows from the fact that $h_{m}\leq h_{n}$ and
$a_{m}^{*}<a_{n}^{*}$. Based on (F), we can claim that the new solution
$\bm{a}^{\prime}$ is feasible with better objective value then solution
$\bm{a}^{*}$, which contradicts the fact that $\bm{a}^{*}$ is the optimal
solution.

Then, we show that the objective function (27) monotonically increases with
$a_{j}$ for all $j\neq k$. To show this, the first derivative of the objective
function (27) with respect to $a_{j}$ can be presented as:

$\displaystyle\frac{\partial f(\bm{a})}{\partial a_{j}}$
$\displaystyle=-1+\frac{2^{\frac{a_{j}-R}{B}}\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)}{\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{a_{j}-R}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)+\frac{\sigma^{2}}{h_{k}}}$
$\displaystyle\geq-1+\frac{2^{\frac{a_{k}-R}{B}}\left(P-p_{0}+\frac{\sigma^{2}}{h_{k}}\right)}{\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{a_{j}-R}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)+\frac{\sigma^{2}}{h_{k}}}$
$\displaystyle=\frac{P-p_{0}-\sum_{j=1}^{K}\left(1-2^{\frac{a_{j}-R}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)-\frac{\sigma^{2}}{h_{k}}}{\sum_{j=1,j\neq
k}^{K}\left(1-2^{\frac{a_{j}-R}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{j}}\right)+\frac{\sigma^{2}}{h_{k}}}$
$\displaystyle\geq 0,$ (F.2)

where $f(\bm{a})$ denotes the objective function (27). The first inequality in
(F) follows from constraint (27b), and the second inequality in (F) follows
from constraint (27c).

Based on (F), the objective function (27) increases with each rate $a_{j}$,
while (F) shows that it is optimal to allocate rate to the user that has the
lowest channel gain. As a result, the optimal rate allocation can be given in
(34) and (35). This completes the proof of Theorem 4.

##  Appendix G Proof of Lemma 6

According to Theorem 4, we consider the following three situations of the
optimal rate allocation.

1) If $\lfloor\frac{c_{1}}{R}\rfloor<1$, we can obtain that the optimal rate
allocation is $a_{1}^{*}=c_{1}$ and $a_{2}^{*}=0$. Substituting
$a_{1}^{*}=c_{1}$ in (II) and $a_{2}^{*}=0$ to problem (26) with $K=2$ and
$R_{1}=R_{2}=R$ yields

$\displaystyle\mathop{\max}_{p_{0}}\quad$ $\displaystyle
R+B\log_{2}\left(\\!1\\!+\\!\frac{\left(P+\frac{\sigma^{2}}{h_{1}}\right)2^{\frac{-R}{B}}-\frac{\sigma^{2}}{h_{1}}}{P-p_{0}+\frac{\sigma^{2}}{h_{1}}-\left(P+\frac{\sigma^{2}}{h_{1}}\right)2^{\frac{-R}{B}}+\frac{\sigma^{2}}{h_{2}}}\right),$
(G.1) s.t. $\displaystyle
P-p_{0}+\frac{\sigma^{2}}{h_{1}}-\left(P+\frac{\sigma^{2}}{h_{1}}\right)2^{\frac{-R}{B}}+\left(1-2^{\frac{-R}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{2}}\right)\leq
P-p_{0},$ (G.1a) $\displaystyle
B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})+\sigma^{2}}\right)\leq
R,$ (G.1b) $\displaystyle
p_{0}\geq\frac{P}{2}+\frac{\theta+\sigma^{2}}{2h_{1}}.$ (G.1c)

According to (G.1a), we have

$\displaystyle\frac{\sigma^{2}}{h_{1}}-\left(P+\frac{\sigma^{2}}{h_{1}}\right)2^{\frac{-R}{B}}\leq-\left(1-2^{\frac{-R}{B}}\right)\left(P-p_{0}+\frac{\sigma^{2}}{h_{2}}\right)\leq
0,$ (G.2)

which indicates that the objective function (G.1) monotonically increases with
$p_{0}$. As a result, the optimal $p_{0}$ of problem (G.1) is the maximum
feasible $p_{0}^{*}$, i.e., constraint (G.1b) holds with equality, which can
be given by

$p_{0}^{*}=\left(1-2^{\frac{-R}{B}}\right)\left(P+\frac{\sigma^{2}}{h_{1}}\right).$
(G.3)

In order to make sure that problem (G.1) is feasible, we must have

$p_{0}^{*}\geq\max\left\\{\frac{\sigma^{2}}{h_{2}}-\frac{\sigma^{2}}{h_{1}}+\frac{P}{1-2^{\frac{-R}{B}}},\frac{P}{2}+\frac{\theta+\sigma^{2}}{2h_{1}}\right\\},$
(G.4)

according to constraints (G.1a) and (G.1c).

2) If $1\leq\lfloor\frac{c_{1}}{R}\rfloor<2$, we can obtain that the optimal
rate allocation is $a_{1}^{*}=R$ and $a_{2}^{*}=c_{1}-R$, which simplifies the
problem in (26) as

$\displaystyle\mathop{\max}_{p_{0}}\>$ $\displaystyle
B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})\\!+\\!\sigma^{2}}\right)+B\log_{2}\left(\frac{{h_{2}}(P-p_{0})+{\sigma^{2}}}{{\sigma^{2}}}\right),$
(G.5) s.t. $\displaystyle
P-p_{0}+\frac{\sigma^{2}}{h_{2}}-\frac{P-p_{0}+\frac{\sigma^{2}}{h_{2}}}{P-p_{0}+\frac{\sigma^{2}}{h_{1}}}\frac{P+\frac{\sigma^{2}}{h_{1}}}{2^{\frac{R}{B}}}\leq
P-p_{0},$ (G.5a) $\displaystyle R\leq
B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})+\sigma^{2}}\right)<2R,$
(G.5b) $\displaystyle p_{0}\geq\frac{P}{2}+\frac{\theta+\sigma^{2}}{2h_{1}}.$
(G.5c)

Since $h_{1}\leq h_{2}$ and the first derivative of (G.5) with respect to
$p_{0}$ is

$\frac{1}{P-p_{0}+\frac{\sigma^{2}}{h_{1}}}-\frac{1}{P-p_{0}+\frac{\sigma^{2}}{h_{2}}}\leq
0,$ (G.6)

the objective function (G.5) decreases with power $p_{0}$. The optimal
$p_{0}^{*}$ can be obtained from constraints (G.5b) and (G.5c), which can be
given by:

$p_{0}^{*}=\max\left\\{\left(1-2^{\frac{-R}{B}}\right)\left(P+\frac{\sigma^{2}}{h_{1}}\right),\frac{P}{2}+\frac{\theta+\sigma^{2}}{2h_{1}}\right\\}.$
(G.7)

3) If $\lfloor\frac{c_{1}}{R}\rfloor\geq 2$, we can obtain that the optimal
rate allocation is $a_{1}^{*}=R$ and $a_{2}^{*}=R$. In this case, problem (26)
becomes

$\displaystyle\mathop{\max}_{p_{0}}\>$ $\displaystyle
B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})+\sigma^{2}}\right)+B\log_{2}\left(\frac{{h_{2}}(P-p_{0})+{\sigma^{2}}}{{\sigma^{2}}}\right),$
(G.8) s.t. $\displaystyle
B\log_{2}\left(\frac{h_{1}P+\sigma^{2}}{h_{1}(P-p_{0})+\sigma^{2}}\right)\geq
2R,$ (G.8a) $\displaystyle
p_{0}\geq\frac{P}{2}+\frac{\theta+\sigma^{2}}{2h_{1}},$ (G.8b)

where the objective value also decreases with $p_{0}$. Compared (G.8) and
(G.8), we can claim that the optimal $p_{0}^{*}$ should be achieved as the
minimum value, i.e., (G.7).

According to the above three cases, it is observed that $a_{1}^{*}=R$. Since
the objective values (G.1), (G.5) and (G.8) do not change with $a_{2}$, we can
show that $a_{2}^{*}=0$ is also optimal. This completes the proof of Lemma 6.

## References

  * [1] Z. Yang, M. Chen, W. Saad, and M. Shikh-Bahaei, “Downlink sum-rate maximization for rate splitting multiple access (RSMA),” _submitted to Proc. IEEE Global Commun. Conf._ , 2019.
  * [2] M. Chen, W. Saad, and C. Yin, “Virtual reality over wireless networks: Quality-of-service model and learning-based resource management,” _IEEE Trans. Commun._ , vol. 66, no. 11, pp. 5621–5635, Nov. 2018.
  * [3] W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless systems: Applications, trends, technologies, and open research problems,” _arXiv preprint arXiv:1902.10265_ , 2019.
  * [4] Y. Liu, Z. Qin, M. Elkashlan, Z. Ding, A. Nallanathan, and L. Hanzo, “Nonorthogonal multiple access for 5G and beyond,” _IEEE Proceedings_ , vol. 105, no. 12, pp. 2347–2381, Dec. 2017.
  * [5] M. Vaezi, Z. Ding, and H. V. Poor, _Multiple Access Techniques for 5G Wireless Networks and Beyond_. Springer, 2018.
  * [6] L. Dai, B. Wang, Y. Yuan, S. Han, C. l. I, and Z. Wang, “Non-orthogonal multiple access for 5G: Solutions, challenges, opportunities, and future research trends,” _IEEE Commun. Mag._ , vol. 53, no. 9, pp. 74–81, Sep. 2015\.
  * [7] Z. Ding, X. Lei, G. K. Karagiannidis, R. Schober, J. Yuan, and V. K. Bhargava, “A survey on non-orthogonal multiple access for 5G networks: Research challenges and future trends,” _IEEE J. Sel. Areas Commun._ , vol. 35, no. 10, pp. 2181–2195, Oct. 2017.
  * [8] Z. Yang, C. Pan, W. Xu, Y. Pan, M. Chen, and M. Elkashlan, “Power control for multi-cell networks with non-orthogonal multiple access,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 2, pp. 927–942, Feb. 2018.
  * [9] L. Yang, Q. Ni, L. Lv, J. Chen, X. Xue, H. Zhang, and H. Jiang, “Cooperative non-orthogonal layered multicast multiple access for heterogeneous networks,” _IEEE Trans. Commun._ , vol. 67, no. 2, pp. 1148–1165, Feb. 2019.
  * [10] K. Wang, Y. Liu, Z. Ding, and A. Nallanathan, “User association in non-orthogonal multiple access networks,” in _Proc. IEEE Int. Conf. Commun._ , Kansas City, MO, USA, May 2018, pp. 1–6.
  * [11] Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, and K. Higuchi, “Non-orthogonal multiple access (NOMA) for cellular future radio access,” in _Proc. IEEE Veh. Technol. Conf._ Dresden, German, June 2013, pp. 1–5.
  * [12] Z. Yang, W. Xu, C. Pan, Y. Pan, and M. Chen, “On the optimality of power allocation for NOMA downlinks with individual QoS constraints,” _IEEE Commun. Lett._ , vol. 21, no. 7, pp. 1649–1652, July 2017.
  * [13] B. Rimoldi and R. Urbanke, “A rate-splitting approach to the gaussian multiple-access channel,” _IEEE Trans. Inf. Theory_ , vol. 42, no. 2, pp. 364–375, Mar. 1996.
  * [14] T. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” _IEEE Trans. Inf. Theory_ , vol. 27, no. 1, pp. 49–60, Jan. 1981\.
  * [15] B. Clerckx, H. Joudeh, C. Hao, M. Dai, and B. Rassouli, “Rate splitting for MIMO wireless networks: A promising PHY-layer strategy for LTE evolution,” _IEEE Commun. Mag._ , vol. 54, no. 5, pp. 98–105, May 2016.
  * [16] J. Cao and E. M. Yeh, “Asymptotically optimal multiple-access communication via distributed rate splitting,” _IEEE Trans. Inf. Theory_ , vol. 53, no. 1, pp. 304–319, Jan. 2007.
  * [17] Y. Mao, B. Clerckx, and V. O. K. Li, “Rate-splitting multiple access for downlink communication systems: Bridging, generalizing, and outperforming SDMA and NOMA,” _EURASIP J. Wireless Commun. Network._ , vol. 2018, no. 1, pp. 1–54, May 2018.
  * [18] ——, “Rate-splitting for multi-user multi-antenna wireless information and power transfer,” _arXiv preprint arXiv:1902.07851_ , 2019.
  * [19] H. Joudeh and B. Clerckx, “Sum-rate maximization for linearly precoded downlink multiuser MISO systems with partial CSIT: A rate-splitting approach,” _IEEE Trans. Commun._ , vol. 64, no. 11, pp. 4847–4861, Nov. 2016\.
  * [20] A. Rahmati, Y. Yapıcı, N. Rupasinghe, I. Guvenc, H. Dai, and A. Bhuyany, “Energy efficiency of RSMA and NOMA in cellular-connected mmwave UAV networks,” _arXiv preprint arXiv:1902.04721_ , Feb. 2019.
  * [21] H. Joudeh and B. Clerckx, “Robust transmission in downlink multiuser MISO systems: A rate-splitting approach,” _IEEE Trans. Signal Process._ , vol. 64, no. 23, pp. 6227–6242, Dec. 2016.
  * [22] C. Hao, Y. Wu, and B. Clerckx, “Rate analysis of two-receiver MISO broadcast channel with finite rate feedback: A rate-splitting approach,” _IEEE Trans. Commun._ , vol. 63, no. 9, pp. 3232–3246, Sep. 2015.
  * [23] M. S. Ali, H. Tabassum, and E. Hossain, “Dynamic user clustering and power allocation for uplink and downlink non-orthogonal multiple access (NOMA) systems,” _IEEE access_ , vol. 4, pp. 6325–6343, Aug. 2016.
  * [24] M. S. Ali, E. Hossain, A. Al-Dweik, and D. I. Kim, “Downlink power allocation for CoMP-NOMA in multi-cell networks,” _IEEE Trans. Commun._ , vol. 66, no. 9, pp. 3982–3998, Sep. 2018.
  * [25] K. Seong, M. Mohseni, and J. M. Cioffi, “Optimal resource allocation for OFDMA downlink systems,” in _Proc. IEEE Int. Symp. Inf. Theory_ , Seattle, WA, USA, 2006, pp. 1394–1398.