I Introduction
Driven by the rapid development of advanced multimedia applications, nextgeneration wireless networks [2] must support high spectral efficiency and massive connectivity. In consequence, rate splitting multiple access (RSMA) has been recently proposed as an effective approach to provide more general and robust transmission framework compared to nonorthogonal multiple access (NOMA) [3, 4, 5, 6, 7, 8] and spacedivision multiple access (SDMA). However, implementing RSMA in wireless networks also faces several challenges [9] such as decoding order design and resource management for message transmission.
Recently, a number of existing works such as in [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] have studied a number of problems related to the implementation of RSMA in wireless networks. In [9], the authors outlined the opportunities and challenges of using RSMA for multiple input multiple output (MIMO) based wireless networks. The authors in [10] developed a rate splitting algorithm for the maximization of users’ data rates. The authors in [12] developed an algorithm to optimize the users’ sumrate in downlink RSMA under imperfect channel state information (CSI). The authors in [12] optimized users’ sumrate in downlink multiuser multiple input single output (MISO) systems under imperfect CSI. The work in [13] showed that RSMA can achieve better performance than NOMA and SDMA. In [14], the application of linearlyprecoded rate splitting is studied for multiple input single output (MISO) simultaneous wireless information and power transfer (SWIPT) broadcast channel systems. The authors in [15] investigated the rate splittingbased robust transceiver design problem in a multiantenna interference channel with SWIPT under the normbounded errors of CSI. The work in [16] developed a transmission scheme that combines rate splitting, common message decoding, clustering and coordinated beamforming so as to maximize the weighted sumrate of users. In [17], the energy efficiency of the RSMA and NOMA schemes is studied in a downlink millimeter wave transmission scenario. The authors in [18] used RSMA for a downlink multiuser MISO system with bounded errors of CIST. The data rate of using RSMA for tworeceiver MISO broadcast channel with finite rate feedback is studied in [19]. Our prior work in [20] investigated the power management and rate splitting scheme to maximize the sumrate of the users. However, most of the existing works such as in [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] studied the use of RSMA for the downlink rather than in the uplink. In fact, using RSMA for uplink data transmission can theoretically achieve the optimal rate region [21]. Moreover, none of the existing works in [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] jointly considered the optimization of power management and message decoding order for uplink RSMA. In practical RSMA deployments, the message decoding order will affect the transmission rate of the uplink users and, thus, it must be optimized.
The main contribution of this paper is a novel framework for optimizing power allocation and message decoding for uplink RSMA transmissions. Our key contributions include:

We consider the uplink of a wireless network that uses RSMA, in which each user transmits a superposition of two messages with different power levels and the base station (BS) uses a successive interference cancellation (SIC) technique to decode the received messages. The power allocation and decoding order problem is formulated as an optimization problem whose goal is to maximize the sumrate of all users under proportional rate constraints.

The nonconvex sumrate maximization problem with discrete decoding variable and transmit power variable is first transformed into an equivalent problem with only the rate splitting variable. Then, the optimal solution of the rate splitting is obtained in closed form. Based on the optimal rate splitting of each user, the optimal transmit power can be derived under a given decoding order. Finally, the optimal decoding order is determined by exhaustive search. To reduce the computational complexity, a lowcomplexity RSMA scheme based on user pairing is proposed to show near sumrate performance of RSMA without user pairing.

We provide closedform optimal solutions for sumrate maximization problems in uplink NOMA, frequency division multiple access (FDMA), and time division multiple access (TDMA). Simulation results show that RSMA achieves better performance than NOMA, FDMA, and TDMA in terms of sumrate.
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. Section IV presents a lowcomplexity sumrate maximization scheme. The optimal solutions of sumrate maximization for NOMA, FDMA and TDMA are provided in Section V. Simulation results are analyzed in Section VI. Conclusions are drawn in Section VII.
Ii System Model and Problem Formulation
Consider a single cell uplink network with one BS serving a set of users using RSMA. In uplink RSMA, each user first transmits a superposition code of two messages to the BS. Then, the BS uses a SIC technique to decode the messages of all users [21].
The transmitted message of user is given by:
(1) 
where is the transmit power of message from user .
The total received message at the BS can be given by:
(2) 
where is the channel gain between user and the BS and is the additive white Gaussian noise. Each user has a maximum transmission power limit , i.e., .
To decode all messages from the received message , the BS will use SIC. The decoding order at the BS is denoted by a permutation . The permutation belongs to set defined as the set of all possible decoding orders of all messages from users. The decoding order of message from user is . The achievable rate of decoding message is:
(3) 
where is the bandwidth of the BS, is the power spectral density of the Gaussian noise. The set in (3) represents the messages that are decoded after message .
Since the transmitted message of user includes messages and , the achievable rate of user is given by:
(4) 
Our objective is to maximize the sumrate of all users with proportional rate constraints. Mathematically, the sumrate maximization problem can be formally posed as follows:
(5)  
s.t.  (5a)  
(5b)  
(5c) 
where , is defined in (4), and . is a set of predetermined nonnegative values that are used to ensure proportional fairness among users. The fairness index is defined as
(6) 
with the maximum value of 1 to be the greatest fairness case in which all users would achieve the same data rate [22]. With proper unitization, we set
(7) 
Although it was stated in [21] that RSMA can reach the optimal rate region, no practical algorithm was proposed to compute the decoding order and power allocation. It is therefore necessary to quantify the uplink performance gains that RSMA can obtain compared to conventional multiple access schemes.
Due to the nonlinear equality constraint (5a) and discrete variable , problem (5) is a nonconvex mixed integer problem. Hence, it is generally hard to solve problem (5). Despite the nonconvexity and discrete variable, we will next develop a novel algorithm to obtain the globally optimal solution to problem (5).
Iii Optimal Power Allocation and Decoding Order
In this section, an effective algorithm is proposed to obtain the optimal power allocation and decoding order of sumrate maximization problem (5).
Iiia Optimal SumRate Maximization
Let be the sumrate of all users. Given this new variable , problem (5) can be rewritten as:
(8)  
s.t.  (8a)  
(8b)  
(8c) 
where is the sumrate of all users since according to (7) and (8a).
Problem (8) is challenging to solve due to the decoding order variable with discrete value space. To handle this difficulty, we provide the following lemma, which can be used for transforming problem (8) into an equivalent problem without decoding order variable .
Lemma 1
In RSMA, under a proper decoding power order and splitting power allocation , the optimal rate region can be fully achieved, i.e.,
(9) 
where is an empty set and means that is a nonempty subset of .
Lemma 1 follows directly from [21, Theorem 1]. Based on Lemma 1, we can use the rate variable to replace the power and decoding variables. In consequence, problem (8) can be equivalently transformed to
(10)  
s.t.  (10a)  
(10b) 
where . In problem (10), the dimension of the variable is smaller than that in problem (8). Moreover, the discrete decoding order variable is replaced by rate variable in problem (10). Regarding the optimal solution of problem (10), we have the following lemma.
Lemma 2
For the optimal solution of problem (10), there exists at least one such that .
Proof: See Appendix A.
Theorem 1
Proof: According to Lemma 2, there exists at least one such that
(13) 
To ensure the feasibility of (10b), the optimal is given by (11). Then, according to (10a), the optimal is determined as in (12).
From (11), one can directly obtain the optimal sumrate of problem (10) in closed form, which can be helpful in characterizing the rate performance of RSMA.
Having obtained the optimal solution of problem (10), we still need to calculate the optimal of the original problem (8). Next, we introduce a new algorithm to obtain the optimal of problem (8).
Substituting the optimal solution of problem (10) into problem (8), we can obtain the following feasibility problem:
find  (14)  
s.t.  (14a)  
(14b)  
(14c) 
Due to the decoding order constraint (14c), it is challenging to find the optimal solution of problem (14). To solve this problem, we first fix the decoding order to obtain the power allocation and then exhaustively search . Given decoding order , problem (14) can be simplified as:
find  (15)  
s.t.  (15a)  
(15b)  
(15c) 
Note that the equality in (14a) is replaced by the inequality in (15a). The reason is that any feasible solution to problem (14) is also feasible to problem (15). Meanwhile, for a feasible solution to problem (15), we can always construct a feasible solution to problem (14).
To verify the feasibility of problem (15), we can construct the following problem by introducing a new variable :
(16)  
s.t.  (16a)  
(16b)  
(16c) 
Proposition 1
Proof: On one side, if is a feasible solution of problem (15), we can show that is a feasible solution of problem (16), which indicates that the optimal objective value of problem (16) should be equal to or larger than 1.
On the other side, if the optimal solution of problem (16) satisfies , we can show that is a feasible solution of problem (15).
Problem (16) is nonconvex due to constraints (16a). To handle the nonconvexity of (16), we adopt the difference of two convex function (DC) method, using which a nonconvex problem can be solved suboptimally by converting a nonconvex problem into convex subproblems. In order to obtain a near globally optimal solution of problem (16), we can try multiple initial points , which can lead to multiple locally optimal solutions. Thus, a near globally optimal solution can be obtained by choosing the locally optimal solution with the highest objective value among all locally optimal solutions. To construct an initial feasible point, we first arbitrarily generate that satisfies linear constraints (16b)(16c), and then we set:
(17) 
By using the DC method, the left hand side of (16a) satisfies:
where represents the value of at iteration , and the inequality follows from the fact that is a concave function and a concave function is always no greater than its firstorder approximation. By substituting the left term of constraints (16a) with the concave function , problem (16) becomes convex, and can be effectively solved by the interior point method [23].
The optimal sumrate maximization algorithm for RSMA is provided in Algorithm 1, where is the number of initial points to obtain a near globally optimal solution of nonconvex problem (16).
IiiB Complexity Analysis
In Algorithm 1, the major complexity lies in solving problem (10) and problem (14). To solve (10), from Theorem 1, the complexity is since the set has nonempty subsets. According to steps 212, a near globally optimal solution of problem (14) is obtained via solving a series of convex problems with different initial points and decoding order strategies. Considering that the dimension of variables in problem (16) is , the complexity of solving convex problem in step 6 by using the standard interior point method is [23, Pages 487, 569]. Since the network consists of users and each user transmits a superposition two messages (there are messages in total), the decoding order set consists of elements. Given initial points, the total complexity of solving problem (14) is . As a result, the total complexity of Algorithm 1 is .
In practice, we consider small
to reduce the SIC complexity, the computational 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 frequency bands and users in the same group are allocated to the same frequency band using RSMA
[24, 25]. For the special case with , we can show that the optimal optimal decoding order and power allocation of problem (14) can be obtained in closed form.IiiC RSMA with Two Users
Based on Lemma 1, the rate region of RSMA with two users can be expressed by:
(18) 
where
(19) 
According to Algorithm 1, the computational complexity needed to obtain the boundary point (as shown in Lemma 2 the optimal point to minimize always lies in the boundary point) of the rate region for RSMA is high. In the following, we introduce a lowcomplexity method to obtain all boundary points of the rate region in twouser RSMA.
In twouser RSMA, only one user needs to transmit a superposition code of two messages and the other user transmits one message. Without loss of generality, user 1 only transmits one message , i.e., the transmit power for message is always 0.
Lemma 3
For twouser RSMA, the optimal decoding order is , and . For the boundary rate , we consider the following three cases.
Case (1) , : the optimal power allocation is
(20) 
Case (2) , : the optimal power allocation is
(21) 
Case (3) , , : the optimal power allocation is
(22) 
Proof: See Appendix B.
Iv LowComplexity SumRate Maximization
According to Section IIIB, the computational complexity of sumrate maximization for RSMA is extremely high. In this section, we propose a lowcomplexity scheme for RSMA, where users are classified into different pairs^{1}^{1}1In this paper, we assume that the user pairing is given, which can be obtained according to matching theory [24] or the order of channel gains [25]. and each pair consists of two users. RSMA is used in each pair and different pairs are allocated with different frequency bands. Assume that users are classified into pairs, i.e., . The set of all pairs is denoted by .
For pair , the allocated fraction of bandwidth is denoted by . Let denote the data rate of user in pair . According to Lemma 1, we have:
(23) 
(24) 
where denotes the channel gain between user in pair and the BS, and is the maximal transmission power of user in pair .
Similar to (5), the sumrate maximization problem for RSMA with user pairing can be formulated as:
(25)  
s.t.  (25a)  
(25b)  
(25c)  
(25d)  
(25e) 
where , , and is a set of predetermined nonnegative values that are used to ensure proportional fairness among users with .
Introducing a new variable , problem (25) can be rewritten as:
(26)  
s.t.  (26a)  
(26b)  
(26c)  
(26d)  
(26e) 
To solve problem (26), we can use the bisection method to obtain the optimal solution. Denote the optimal objective value of problem (26) by .
We can conclude that problem (26) is always feasible with and infeasible with . This motivates us to use the bisection method to find the optimal , as shown in Fig. 1, where is the value of in the th iteration and is the initial value interval of . To show the feasibility of problem (26) for each given , we solve a feasibility problem with constraints (26a)(26e). With given , the feasibility problem of (26) becomes
find  (27)  
s.t.  (27a)  
(27b)  
(27c)  
(27d)  
(27e) 
Substituting (27a) into (27c) and (27d), we have:
(28) 
(29) 
It can be proved that is a monotonically increasing function. Thus, to satisfy (28) and (29), bandwidth fraction should satisfy:
(30) 
where
(31) 
(32) 
(33) 
and is the LambertW function.
V SumRate Maximization for Uplink NOMA/FDMA/TDMA
To evaluate the performance gain of the RSMA scheme proposed in Sections IIIV, and for comparison purposes, we will solve the sumrate maximization problems for uplink NOMA, FDMA and TDMA schemes.
Va Noma
Without loss of generality, the channel gains are sorted in descending order, i.e., . In NOMA, the BS first decodes the messages of users with high channel gains and then decodes the messages of users with low channel gains by subtracting the interference from decoded strong user. The achievable rate of user with NOMA is calculated as [26]:
(35) 
where is the transmit power of user . The transmission power has a maximum transmit power limit , i.e., we have , .
Similar to (8), the sumrate maximization problem for uplink NOMA can be given by:
(36)  
s.t.  (36a)  
(36b) 
where . To obtain the optimal solution of problem (36), we provide the following theorem.
Theorem 2
Proof: See Appendix C.
VB Fdma
In FDMA, each user will be allocated a fraction of the BS bandwidth. Let denote the fraction of bandwidth allocated to user . Then the data rate of user is:
(40) 
Note that user transmits with maximum power in (40) since there is no interuser interference and large power leads to high data rate. Due to limited bandwidth, we have .
Similar to (8), the sumrate maximization problem for uplink FDMA can be given by:
(41)  
s.t.  (41a)  
(41b)  
(41c) 
where . Regarding the optimal solution of problem (41), we provide the following theorem.
Theorem 3
Proof: See Appendix D.
VC Tdma
In TDMA, each user will be assigned a fraction of time to use the whole BS bandwidth. Let be the fraction of time allocated to user . The data rate of user is:
(44) 
with .
Similar to (8), the sumrate maximization problem for uplink TDMA can be given by:
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