I Introduction
In multicarrier multiple access systems, the total frequency bandwidth is divided into subcarriers and assigned to users to optimize the spectrum utilization. Orthogonal multiple access (OMA), such as orthogonal frequencydivision multiple access (OFDMA) adopted in 3GPPLong Term Evolution 4G standards [1], only serves one user per subcarrier in order to avoid intracell interference and have lowcomplexity signal decoding at the receiver side. OMA is known to be suboptimal in terms of spectral efficiency [2].
The principle of multicarrier nonorthogonal multiple access (MCNOMA) is to multiplex several users on the same subcarrier by performing signals superposition at the transmitter side. Successive interference cancellation (SIC) is applied at the receiver side to mitigate interference between superposed signals. MCNOMA is a promising multiple access technology for 5G mobile networks as it achieves better spectral efficiency and higher data rates than OMA schemes [3, 4, 5].
Careful optimization of the transmit powers is required to control the intracarrier interference of superposed signals and maximize the achievable data rates. Besides, due to error propagation and decoding complexity concerns in practice [6], subcarrier allocation for each transmission also needs to be optimized. As a consequence, joint subcarrier and power allocation problems in NOMA have received much attention. In this class of problems, weighted sumrate (WSR) maximization is especially important as it can achieve different tradeoffs between sumrate performance and user fairness [7]. One may also consider to use the weights to perform fair resource allocation in a stable scheduling [8, 9].
Two types of power constraints are considered in the literature. On the one hand, cellular power constraint is mostly used in downlink transmissions to represent the total transmit power budget available at the base station (BS). On the other hand, individual power constraint sets a power limit independently for each user. The latter is clearly more appropriate for uplink scenarios [10, 11], nevertheless it can also be applied to the downlink [12, 13].
It is known that WSR maximization in OFDMA systems is polynomial time solvable if we consider cellular power constraint [14] but strongly NPhard with individual power constraints [15]. It has also been proven that MCNOMA with individual power constraints is strongly NPhard [12, 16]. Several algorithms are developed to perform subcarrier and/or power allocation in this setting. Fractional transmit power control (FTPC) is a simple heuristic that allocates a fraction of the total power budget to each user based on their channel condition [17]. In [10] and [11], heuristic user pairing strategies and iterative resource allocation algorithms are studied for uplink transmissions. A time efficient iterative waterfilling heuristic is introduced in [18] to solve the problem with equal weights. Reference [12] derives an upper bound on the optimal WSR and proposes a nearoptimal scheme using dynamic programming and Lagrangian duality techniques. This scheme serves as benchmark due to its high computational complexity, which may not be suitable for practical systems with low latency requirements.
To the best of our knowledge, it is not known if MCNOMA with cellular power constraint is polynomial time solvable or NPhard. Only a few papers have developed optimization schemes for this problem, which are either heuristics with no theoretical performance guarantee or algorithms with impractical computational complexity. For example, a greedy user selection and heuristic power allocation scheme based on differenceofconvex programming is proposed in [19]. The authors of [20] employ monotonic optimization to develop an optimal resource allocation policy, which serves as benchmark due to its exponential complexity. The algorithm of [12] can also be applied to cellular power constraint scenarios, but it has very high complexity as well.
Motivated by this observation, we investigate the WSR maximization problem in a downlink multicarrier NOMA system with cellular power constraint. Our contributions are as follows:

[leftmargin=*]

We propose a novel framework in which the original problem is divided into two polynomial time solvable subproblems:

The multicarrier power control (given a fixed subcarrier allocation) subproblem is nonconvex. By taking advantage of its separability property, we design an optimal and low complexity algorithm (MCPC) based on projected gradient descent.

The singlecarrier user selection subproblem is a nonconvex mixedinteger problem that we solve using dynamic programming (SCUS).
This work also aims to give an understanding on how each subproblem’s particular structure can facilitate the algorithm design. In that respect, the above MCPC and SCUS are basic building blocks that can be applied in a wide range of resource allocation problems. Our analysis of the separability property in MCPC and the combinatorial constraints in SCUS provide mathematical tools which can be used to study other similar resource allocation optimization problems.


Papers [19, 20] restrict the number of multiplexed users per subcarrier, denoted by , to be equal to 2. However, our result can be applied for an arbitrary number of superposed signals on each subcarrier, i.e., . The value of can be set depending on error propagation and decoding complexity concerns in practice.

We propose an efficient heuristic to solve the general WSR maximization problem by combining MCPC and SCUS. Numerical results show that it achieves nearoptimal sumrate and user fairness, as well as significant improvement in performance over OMA.
The paper is organized as follows. In Section II, we present the system model and notations. Section III formulates the WSR problem. We introduce our algorithms in Section IV and analyze their optimality and complexity. We show in Section V some numerical results, highlighting our solution’s sumrate and fairness performance, as well as their computational complexity. Finally, we conclude in Section VI.
Ii System Model and Notations
We define in this section the system model and notations used throughout the paper. We consider a downlink multicarrier NOMA system composed of one base station (BS) serving users. We denote the index set of users by , and the set of subcarriers by . The total system bandwidth is divided into subcarriers of bandwidth , for each . We assume orthogonal frequency division, so that adjacent subcarriers do not interfere each other. Moreover, each subcarrier experiences frequencyflat block fading on its bandwidth .
Let denotes the transmit power from the BS to user on subcarrier . User is said to be active on subcarrier if , and inactive otherwise. In addition, let be the channel gain between the BS and user on subcarrier , and be the received noise power. For simplicity of notations, we define the normalized noise power as . We denote by
the vector of all transmit powers, and
the vector of transmit powers on subcarrier .In power domain NOMA, several users are multiplexed on the same subcarrier using superposition coding. A common approach adopted in the literature is to limit the number of superposed signals on each subcarrier to be no more than . The value of is meant to characterize practical limitations of SIC due to decoding complexity and error propagation [6]. We represent the set of active users on subcarrier by . The aforementioned constraint can then be formulated as , where denotes the cardinality of a finite set. Each subcarrier is modeled as a multiuser Gaussian broadcast channel [6] and SIC is applied at the receiver side to mitigate intraband interference.
The SIC decoding order on subcarrier is usually defined as a permutation function over the active users on , i.e., . However, for ease of reading, we choose to represent it by a permutation over all users , i.e., . These two definitions are equivalent in our model since the Shannon capacity (2) does not depend on the inactive users , for which . For , returns the th decoded user’s index. Conversely, user ’s decoding order is given by .
In this work, we consider the optimal decoding order studied in [6, Section 6.2]. It consists of decoding users’ signals from the highest to the lowest normalized noise power:
(1) 
User first decodes the signals of users to and subtracts them from the superposed signal before decoding its own signal. Interference from users for is treated as noise. The maximum achievable data rate of user on subcarrier is given by Shannon capacity:
(2) 
where equality is obtained after normalizing by . We assume perfect SIC, therefore interference from users for is completely removed in (2).
Iii Problem Formulation
Let be a sequence of positive weights. The main focus of this work is to solve the following joint subcarrier and power allocation optimization problem:
()  
subject to  
The objective of is to maximize the system’s WSR. As discussed in Section I, this objective function has received much attention since its weights can be chosen to achieve different tradeoffs between sumrate performance and fairness [7]. The weights can also be tuned to perform fair resource allocation in a stable scheduling [8, 9]. Note that represents the cellular power constraint, i.e., a total power budget at the BS. In , we set a power limit of for each subcarrier . This is a common assumption in multicarrier systems, e.g., [14, 15]. Constraint ensures that the allocated powers remain nonnegative. Due to decoding complexity and error propagation in SIC [6], we restrict the number of multiplexed users per subcarrier to in .
Let us consider the following change of variables:
(3) 
We define and .
Proof:
See Appendix A. ∎
The advantage of this formulation is that it exhibits a separable objective function in both dimensions and . In other words, it can be written as a sum of functions , each only depending on one variable . In the following section, we take advantage of this separability property to design efficient algorithms to solve . The solution of can then be obtained by solving .
Iv Algorithm Design
Decomposing the joint subcarrier and power allocation problem into smaller subproblems allows us to develop efficient algorithms by taking advantage of each subproblem’s particular structure. The technical details are given below.
Iva SingleCarrier User Selection
In this subsection, we restrict the optimization search space to a single subcarrier. Given and a power budget , the singlecarrier user selection subproblem consists in finding a power allocation on satisfying the multiplexing and SIC constraint .
()  
subject to 
Let us introduce auxiliary functions that will help us in our algorithm design. For , and , assume that the consecutive variables are all equal to a certain value . We define as:
This simplification of notation is relevant for the analysis of SCUS (Algorithm 2) and also the coming MCPC (Algorithm 4). Indeed, if users are not active (i.e., ), then , therefore can be replaced by and are redundant with . If constraint is satisfied, up to users are active on each subcarrier. Thus, evaluating and optimizing the objective function of only requires operations.
We study the properties of in Lemma 2. Note that , therefore Lemma 2 also holds for functions . Fig. 1 shows an example of .
Lemma 2 (Properties of ).
Let , , and , we have:

If or , then is strictly increasing and concave on .

Otherwise when and , is strongly unimodal. It increases on and decreases on , where
Besides, is concave on and convex on , where
Proof:
The idea consists in studying the first and second derivatives of . Details are omitted due to space limitation. ∎
We present in Algorithm 1 the pseudocode MaxF which computes the maximum of on following the result of Lemma 2. MaxF only requires a constant number of basic operations, therefore its complexity is . For ease of reading, we summarize some system parameters of a given instance of , for all , as follows:
We propose to solve using Algorithm 2 (SCUS) based on dynamic programming (DP). This approach consists in computing recursively the elements of three arrays , , . Let and , we define as the optimal value of with no more than users multiplexed on subcarrier in constraint , and with the additional constraints and as follows:
()  
subject to  
It is interesting to note that is the optimal value of , since constraints and become trivially true for . Let be the optimal solution achieving . We define , which is also equal to due to constraint .
The idea of SCUS is to recursively compute the elements of for and through the following recurrence relation:
(4) 
where , and , , correspond to allocations where user and are each either active or inactive. A detailed analysis is given in Appendix B.
During SCUS’s iterations, the array keeps track of which previous element of (i.e., or or ) have been used to compute the current value function . More precisely, contains one of the following indices , or . This allows us to retrieve the entire optimal vector at the end of the algorithm (at lines 3238) by backtracking from index to .
When , no user can be active on this subcarrier due to constraint . Therefore, , , can be initialized to zero at lines 1 to 6. For simplicity, we also extend , and on the index and and initialize them to zero in lines 7 to 11.
Theorem 3 (Optimality and complexity of algorithm Scus).
Proof:
See Appendix B. ∎
IvB MultiCarrier Power Control
The following multicarrier power control subproblem consists in maximizing the WSR taking as input a fixed subcarrier allocation , for all , so that constraint can be ignored. Since inactive users on each subcarrier have no contribution on the data rates, i.e., and , we remove them in the definition of for the study of this subproblem. Without loss of generality, we then transform to and index the remaining active users on each subcarrier by . We formulate as a twostage optimization problem. The firststage is:
()  
subject to 
where , for , are intermediate variables representing each subcarrier’s power budget. denotes the power budget vector. The feasible set
(5) 
is chosen to satisfy and . is the optimal value of the secondstage:
()  
subject to  
The firststage consists in optimizing the allocated power budget among subcarriers . While the secondstage performs power allocation on each subcarrier , with a given allocation and power budget . Fig. 2 gives an overview of the nested optimization that we design to solve .
We solve the secondstage using Algorithm 3 (SCPC). The idea is to iterate over variables for to , and compute their optimal value at line 3. If the current allocation satisfies constraint , then gets value . Otherwise, the algorithm backtracks at line 6 and finds the highest index such that . Then, variables are set equal to at line 10. The optimality and complexity of this algorithm are presented in Theorem 4.
Theorem 4 (Optimality and complexity of algorithm Scpc).
Given subcarrier and a power budget , algorithm SCPC computes the optimal singlecarrier power control. Its worst case computational complexity is .
Proof:
See Appendix C. ∎
Proof:
See Appendix D. ∎
According to Lemma 5, is concave on the convex feasible set . Hence, the optimal multicarrier power allocation can be computed efficiently using projected gradient descent [21]. This MCPC scheme is described in Algorithm 4, in which the secondstage optimal value , defined in , is determined using SCPC as described in the auxiliary function in lines 12 to 13. The search direction at line 4 can be found either using numerical methods or an exact gradient formula (in the Proof of Lemma 5). Note that the step size at line 5 can be tuned by backtracking line search or exact line search [21]. We adopt the latter to perform simulations. The projection of on the simplex at line 6 can be computed efficiently [22], the details of its implementation are omitted here.
Theorem 6 (Optimality and complexity of algorithm Mcpc).
Proof:
It follows from Lemma 5 and classical convex programming results [21] that projected gradient descent converges to the optimal multicarrier power control (MCPC) in iterations. Each iteration requires to compute SCPC(), for . Thus, we derive from Theorem 4 and that each iteration’s worst case complexity is . ∎
IvC MultiCarrier Joint Subcarrier and Power Allocation
We design an efficient heuristic joint subcarrier and power allocation scheme, namely JSPA, which combines MCPC and SCUS. We present its principle in Fig. 3. JSPA is similar to the twostage MCPC depicted in Fig. 2. With the difference that subcarrier allocation is no longer provided as input, and has to be optimized in the secondstage. To this end, we replace the secondstage SCPC by the optimal singlecarrier user selection algorithm SCUS studied in Section IVA. Although SCUS is optimal, the returned is no longer concave. As a consequence, JSPA is not guaranteed to converge to a global maximum. Each iteration of JSPA requires to compute SCUS(), for . Thus, we derive from Theorem 3 that each iteration’s worst case complexity is . Nevertheless, we show by numerical results in the next section that it achieves nearoptimal weighted sumrate performance with low complexity.
V Numerical Results
We evaluate the WSR, user fairness performance and computational complexity of JSPA through numerical simulations. We compare our proposed scheme with the nearoptimal high complexity benchmark scheme LDDP introduced in [12], as well as FTPC with greedy subcarrier allocation, which is also considered in [12] and [17, 18, 19]. We consider a cell of diameter meters, with one BS located at its center and users distributed uniformly at random in the cell. The number of users varies from to , and the number of subcarriers is . Each point in the following figures are average value obtained over random instances. The simulation parameters and channel model are summarized in Table I.
Fig. 4 and 5 illustrate the WSR and complexity of JSPA and LDDP for different systems, i.e., OMA with , NOMA with and . In these two figures, weights are generated uniformly at random in . Algorithm LDDP requires to discretize the total power budget into power levels. Here, we choose . Due to the high computational complexity of LDDP, we only simulate it for to .
In Fig. 4, the performance loss of JSPA compared to LDDP is less than for any number of users . This indicates that JSPA also achieves nearoptimal WSR, close to that of LDDP. In addition, we see that the performance gain of NOMA systems compared to OMA is for and for .
In Fig. 5, we count the number of basic operations (additions, multiplications, comparisons) performed by each scheme, which reflects their computational complexity. As expected from Theorem 4, the complexity of JSPA increases linearly with (there is a constant difference between the curves of JSPA with , and in the semilog plot). In addition, each LDDP’s iteration has complexity [12]. Reference [13] shows that the number of power levels should be to achieve nearoptimal WSR. Thus, JSPA has lower asymptotic complexity than LDDP. For example with , LDDP requires power levels, and JSPA is asymptotically faster by a factor . In our simulations, JSPA runs within seconds on a common computer^{1}^{1}1with the following specifications: Python 3, Windows 7, 64bits, AMD A105750M APU with Radeon HD Graphics, and 8 GB of RAM. for , whereas LDDP requires 1600 times more operations for and .
Parameters  Value 

Cell radius  250 m 
Minimum distance from user to BS  35 m 
Carrier frequency  GHz 
Path loss model  dB, is in km 
Shadowing  Lognormal, 
Fading  Rayleigh fading with variance 1 
Noise power spectral density  174 dBm/Hz 
System bandwidth  MHz 
Number of subcarriers  
Number of users  to 
Total power budget  1 W 
Error tolerance  
Parameter  1 (OMA), 2 and 3 (NOMA) 
In Fig. 6 and 7, we implement a proportional fair scheduler on one frame, which is composed of time slots. The objective of this setup is to evaluate the fairness performance of our scheme when the weights are chosen according to a proportional fair scheduling. The channel state information is collected at the beginning of the frame. Let be user ’s data rate at time slot , and be user ’s average data rate prior to . In the proportional fair scheduling [23], is updated as follows:
The weight of user at time is then set as , in order to achieve a good tradeoff between spectral efficiency and user fairness [24]. At the end of the frame, we average each user ’s data rate over the entire frame, i.e., . We show respectively in Fig. 6 and 7, the proportional fairness index defined as
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