I Introduction
Ia Evolution of NOMA: From Fully SCSIC to HybridNOMA
The rapid growing demands for high data rate services along with energy constrained networks necessitates the characterization and analysis of wireless communication systems. It is proved that the capacity region of degraded singleinput singleoutput (SISO) Gaussian broadcast channels (BCs) can be achieved by performing linear superposition coding (SC) at the transmitter side combined with coherent multiuser detection algorithms, like successive interference cancellation (SIC) at the receivers side [1, 2, 3]. We call the latter technique as fully SCSIC (FSIC), where the signal of all the users are multiplexed. The achievable rate region of FSIC, frequency division multiple access (FDMA), and time division multiple access (TDMA) are analyzed in [4, 5].
The SIC complexity is cubic in the number of multiplexed users [6]. Another issue is error propagation, which increases with the number of users [6]. Hence, FSIC is still impractical for a large number of users, and can be considered as a benchmark for performance comparison. To tackle the above practical limitations, a lowcomplexity, yet suboptimal, technique called nonorthogonal multiple access (NOMA) is proposed in [7] which is based on SCSIC and orthogonal multiple access (OMA) such as FDMA/TDMA. In NOMA, users are grouped into multiple clusters, and SCSIC is performed among users within each cluster. The clusters operate in isolated resource blocks, based on FDMA/TDMA^{1}^{1}1The spatial domain multiple access (SDMA) can also be introduced on NOMA, where clusters are isolated by zeroforcing beamforming [6].. NOMA is considered as a promising candidate solution for the beyond5G (B5G)/6th generation (6G) wireless networks [6, 7, 8, 9, 10, 11]. The overall achievable rate of users can be increased by introducing the multicarrier technology to the fading channels. The concept of NOMA is also introduced on multicarrier systems, where SCSIC is combined with OFDMA, called OFDMANOMA, multicarrier NOMA or HybridNOMA. In HybridNOMA, each user can occupy more than one subcarrier, and SCSIC is applied to each isolated subcarrier. As a result, all the users can benefit from the multiplexing gain [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].
IB Related Works and Open Problems
The resource allocation in downlink singlecell NOMA under individual minimum rate demands of users can be classified into the following three categorizes:
IB1 Fsic
The FSIC technique is also referred to as singlecarrier/singlecluster NOMA [23, 24, 25]. In our previous work [25], we derived the closedform of optimal powers for user FSIC under the individual minimum rate demands of users to maximize users sumrate as well as minimize total power consumption. The work in [26] addresses the problem of simultaneously maximizing users sumrate and minimizing total power consumption defined as a utility function in FSIC. However, the analysis in [26] is affected by a detection constraint for successful SIC which is not necessary, since SISO Gaussian BCs are degraded. The closedform of optimal power coefficients for the wellknown fractional energy efficiency (EE) maximization problem is still an open problem.
IB2 Noma
The joint power allocation and user clustering in NOMA is proved to be strongly NPhard [13, 19, 20]. In this way, the latter two problems are decoupled in most of the prior works. For any given set of clusters, the optimal power allocation for sumrate/EE maximization in NOMA is more challenging compared to FSIC. In NOMA, there exists a competition among multiple clusters to get the cellular power. Actually, the optimal power allocation in NOMA includes two components: 1) Intercluster power allocation: optimal power allocation among clusters to get the cellular power budget; 2) Intracluster power allocation: optimal power allocation among multiplexed users to get cluster’s power budget. From the optimization perspective, the analysis in [25] is also valid for NOMA with predefined (suboptimal) power budget for each cluster, e.g., [27, 28]. In this case, the intracluster power allocation can be equivalently decoupled into multiple singlecarrier NOMA subproblems. To the best of our knowledge, the only work that address the globally optimal power allocation in NOMA is [29]. In [29], each cluster has only two users, and all the analysis is based on allocating more power to each weaker user to guarantee successful SIC, which is not necessary, due to the degradation of SISO Gaussian BCs [3, 10]. Therefore, the globally optimal power allocation for sumrate maximization in NOMA under the individual users minimum rate demands is still an open problem.
The fractional EE maximization problem in NOMA is addressed by [30, 31, 32, 33, 34, 29]. The EE maximization problem is solved by using the suboptimal differenceofconvex (DC) approximation method [30], Dinkelbach algorithm with Fmincon optimization software [31], and Dinkelbach algorithm with subgradient method [32, 33, 34]. Despite the potentials, the closedform of optimal powers for the EE maximization problem is not yet obtained by the prior works. The EE maximization problem was also addressed by [29] for only twouser case per cluster, however the formulated closedform of optimal power coefficients among users within each cluster is derived based on the condition allocating more power to the weaker user which is not necessary [10]. Besides, the Karush–Kuhn–Tucker (KKT) optimality conditions analysis for EE maximization problem is not yet addressed by the literature. It is still unknown what are the optimal power coefficients among multiplexed users, and which user deserves additional power rather than its minimum rate demand?
IB3 HybridNOMA
The works on HybridNOMA mainly focus on achieving the maximum multiplexing gain, where each user receives different symbols on the assigned subcarriers. The works in [12, 13, 14, 15, 16, 17, 18, 19, 20] address the problem of weighted sumrate/sumrate maximization without guaranteeing individual peruser minimum rate demands over all the assigned subcarriers. In HybridNOMA with minimum rate constraints, [21] proposes a suboptimal power allocation strategy for the EE maximization problem based on the combination of the DC approximation method and Dinkelbach algorithm. Also, a suboptimal penalty function method is used in [22]. Finding the optimal power allocation for sumrate/EE maximization problem in HybridNOMA is challenging, and we consider it as our future work^{2}^{2}2The main challenge is the nonconvexity of the minimum rate constraints in HybridNOMA. A suboptimal scheme is to define a specific minimum rate demand for each user on each subchannel, e.g., [32, 33, 34]. In this case, HybridNOMA can be equivalently transformed to a NOMA system, however the solution is still suboptimal..
IC Our Contributions and Paper Organization
In this work, we address the problem of finding optimal power allocation for the sumrate and EE maximization problems of the general downlink singlecell NOMA systems including multiple clusters each having an arbitrary number of users. Our main contributions are listed as follows:

We show that for the three main objective functions as total power minimization, sumrate maximization and EE maximization, only the clusterhead^{3}^{3}3The user with the highest decoding order which cancels the signal of all the other multiplexed users. users deserve additional power while all the other users get power to only maintain their minimal rate demands^{4}^{4}4For the total power minimization problem, the clusterhead users also get power to only maintain their minimal rate demands..

The feasible power allocation region in NOMA is defined as the intersection of closed boxes along with affine cellular power constraint. Then, the optimal value for the power minimization problem is obtained in closed form.

For the sumrate/EE maximization problem, we show that the NOMA system can be transformed to an equivalent virtual OMA system. Each cluster acts as a virtual OMA user whose effective channel gain is obtained in closed form. Moreover, each virtual OMA user requires a minimum power to satisfy its multiplexed users minimum rate demands, which is obtained in closed form.

A very fast waterfilling algorithm is proposed to solve the sumrate maximization problem in NOMA. The EE maximization problem is solved by using the Dinkelbach algorithm with inner Lagrange dual with subgradient method or barrier algorithm with inner Newton’s method. Different from [30, 31, 32, 33, 34], the closedform of optimal powers among multiplexed users is applied to further reduce the dimension of the problems, thus reducing the complexity of the iterative algorithms, as well as increase the accuracy of the solution, which is a winwin strategy.

We propose a necessary and sufficient condition for the equal intercluster power allocation strategy to be optimal. In contrast to FDMA, the equal intercluster power allocation strategy in NOMA can be suboptimal even if all the users in different clusters have the same channel conditions while different minimum rate demands.

We propose a sufficient condition such that the EE maximization problem in NOMA turns into the sumrate maximization problem where the BS operates at its maximum power budget. The latter condition is characterized according to the effective channel gain of the virtual OMA users as well as constant which is derived from the closedform of optimal powers among multiplexed users in each cluster .
Extensive numerical results are provided to evaluate the performance of FSIC, NOMA with different maximum number of multiplexed users, and FDMA in terms of outage probability, minimum total power consumption, maximum sumrate and EE. The performance comparison between NOMA and FSIC brings new insights on the suboptimalitylevel of NOMA due to user grouping based on FDMA/TDMA. In this work, we answer the question "How much performance gain can be achieved if we increase the order of NOMA clusters, and subsequently, decrease the number of user groups?" for a wide range of number of users and their minimum rate demands. The latter knowledge is highly necessary since multiplexing a large number of users would cause high complexity cost at the users’ hardware. The complete source code of the simulations including a user guide is available in [35].
Ii NOMA: OMABased SCSIC
Iia Network Model and Achievable Rates
Consider the downlink channel of a multiuser system, where a BS serves users with limited processing capabilities in a unit time slot of a quasistatic channel. The set of users is denoted by . The maximum number of multiplexed users is . FSIC is infeasible when . In this case, NOMA is applied, where users are grouped into clusters^{5}^{5}5In NOMA, the condition implies that .. We assume that each cluster operates in an isolated subchannel based on FDMA. To this end, the total bandwidth (Hz) is equally divided into isolated subchannels with the set , where the bandwidth of each subchannel is . The set of multiplexed users on subchannel is denoted by , in which is the binary channel allocation indicator, where if user occupies subchannel , we set , and otherwise, . In NOMA, each user belongs to only one cluster [27, 28, 29, 30, 31, 32, 33, 34]. As a result, we have , or equivalently, . According to the above, we have , where indicates the cardinality of a finite set. The exemplary models of FSIC, NOMA, and FDMA are illustrated in Figs. 1(a)1(c).
Each subchannel can be modeled as a SISO Gaussian BC. The transmitted signal by the BS on subchannel is formulated by where and are the modulated symbol from Gaussian codebooks, and transmit power of user on subchannel , respectively. Obviously, . The received signal at user on subchannel is
(1) 
where is the (generally complex) channel gain from the BS to user on subchannel , and is the additive white Gaussian noise (AWGN). We assume that the perfect channel state information (CSI) is available at the BS as well as users^{6}^{6}6The extension of the analysis in this paper to the imperfect CSI scenario with outage constraints [32] is straightforward..
In NOMA, SCSIC is applied to each multiuser subchannel according to the optimal CNRbased decoding order [1, 2, 3, 4, 5]. Let . Then, the CNRbased decoding order is indicated by , where represents that user fully decodes (and then cancels) the signal of user before decoding its desired signal on subchannel . Moreover, the signal of user is fully treated as noise at user . In summary, the SIC protocol in each isolated subchannel is the same as the SIC protocol of FSIC. We call the stronger user as the user with higher decoding order in the user pair . In each subchannel , the index of the clusterhead user is denoted by . When , the single user can be defined as the clusterhead user on subchannel since it does not experience any interference. Therefore, NOMA is a combination of SCSIC and OMA. In each subchannel, the clusterhead user does not experience any interference. The signaltointerferenceplusnoise ratio (SINR) of each user for decoding the desired signal of user on subchannel is [3]. User is able to fully decode the signal of user if and only if , where is the SINR of user for decoding its own signal . According to the Shannon’s capacity formula, the achievable rate (in bps) of user after successful SIC is given by [2, 3, 25]
where
is the vector of allocated powers to all the users on subchannel
. For the user pair with , the condition or equivalently holds independent of . Accordingly, at any , the achievable rate of each user on subchannel is equal to its channel capacity formulated by [3](2) 
IiB Optimization Problem Formulations
Assume that the set of clusters, i.e., is predefined. The general power allocation problem for maximizing userssumrate under the individual minimum rate demand of users in NOMA is formulated by
(3a)  
s.t.  (3b)  
(3c)  
(3d) 
where (3b) is the peruser minimum rate constraint, in which is the individual minimum rate demand of user . (3c) is the cellular power constraint, where denotes the maximum available power of the BS. (3d) is the maximum persubchannel power constraint, where denotes the maximum allowable power on subchannel^{7}^{7}7We do not impose any specific condition on . We only take into account in our analysis to keep the generality, such that , as special case. . For convenience, we denote the general power allocation matrix as .
Iii Solution Algorithms
In this section, we propose globally optimal power allocation algorithms for the sumrate and EE maximization problems. The closedform of optimal powers for the total power minimization problem is also derived to characterize the feasible set of our target problems.
Iiia SumRate Maximization Problem
Here, we propose a waterfilling algorithm to find the globally optimal solution of (3). The sumrate of users in each cluster, i.e., is strictly concave in , since its Hessian is negative definite [25]. Due to the isolation among subchannels, the overall sumrate in (3a) is strictly concave in . Besides, the power constraints in (3c) and (3d) are affine, so are convex. The minimum rate constraint in (3b) can be equivalently transformed to the following affine form as Accordingly, the feasible set of (3) is convex. Summing up, problem (3) is convex in . Let us define as the power consumption of cluster . Problem (3) can be equivalently transformed to the following joint intra and intercluster power allocation problem as
(5a)  
s.t.  (5b)  
(5c)  
(5d)  
(5e) 
In the following, we first convert the feasible set of (5) to the intersection of closedboxes along with the affine cellular power constraint.
Lemma 1.
The feasible set of (5) is the intersection of , and cellular power constraint , where the lowerbound constant is given by in which
Proof.
The feasibility of (5) can be determined by solving the power minimization problem as
(6) 
Assume that the feasible region of (6) is nonempty. The problem (6) is also convex with affine objective function. The strong duality holds since (5c) and (5e) hold with strict inequalities. In the KKT optimality conditions analysis in Appendix C of [25], we proved that in FSIC, the maximum power budget does not have any effect on the optimal powers obtained in the power minimization problem when the feasible region is nonempty. And, at the optimal point, all the multiplexed users get power to only maintain their minimal rate demands. In this system, if the strong duality holds, the problem can be equivalently decoupled into singlecarrier power minimization problems. In each subchannel , regardless of the clusters power budget vector , the optimal power of each user can be obtained in closed form given by where . Since the solution of (6) is unique, it can be easily shown that the optimal powers are indeed componentwise minimum. In other words, for any feasible , we have . Therefore, is indeed the lowerbound of . According to (5e), we guarantee that any satisfying (5c) is feasible, and the proof is completed. ∎
The feasibility of (5) can be directly determined by utilizing Lemma 1. In the following, we find the closedform of optimal intracluster power allocation as a linear function of .
Proposition 1.
For any given feasible , the optimal intracluster powers can be obtained by
(7) 
and
(8) 
where , .
Proof.
For any given feasible , (5c) and (5e) can be removed from (5). Then, problem (5) can be equivalently divided into intracluster convex power allocation subproblems, where for each subchannel , we find the optimal powers according to Appendix B in [25]. It is proved that at the optimal point of sumrate maximization problem in FSIC, all the multiplexed users with lower decoding order get power to only maintain their minimum rate demands. And the rest of the available power will be allocated to the clusterhead user. ∎
In contrast to FSIC, in NOMA, there is a competition among clusterhead users to get the rest of the cellular power. It is worth noting that at the optimal point, the cellular power constraint is active, meaning that all the remaining cellular power will be distributed among clusterhead users. According to (8), the optimal power of the clusterhead user can be obtained as a function of given by
(9) 
where , and are nonnegative constants. According to Proposition 1 and (9), the optimal value of (5) for the given can be formulated in closed form as
(10) 
Hence, (5) is equivalently transformed to the following intercluster power allocation problem
(11a)  
s.t.  (11b)  
(11c) 
In the objective function (11a), we substituted the closedform of optimal intracluster powers as a function of . Constraint (11b) is the cellular power constraint, and (11c) is to ensure that the individual minimum rate demands of all the users within each cluster is satisfied. The objective function (11a) is strictly concave, and the feasible set of (11) is affine. Accordingly, problem (11) is convex. For convenience, let , where . Then, (11) is equivalent to the following convex problem as
(12a)  
s.t.  (12b) 
where , , , and . The equivalent OMA problem (12) can be solved by using the waterfilling algorithm [38, 39]. After some mathematical manipulations, the optimal can be obtained as
(13) 
such that satisfies (12b). Also, is the dual optimal corresponding to constraint (12b). For more details, please see Appendix A. The pseudocode of the bisection method for finding is presented in Alg. 1.
After finding , we obtain by . Then, we find the optimal powers according to Proposition 1.
Remark 1.
In the transformation of (3) to (12), the NOMA system is equivalently transformed to a virtual OMA system including a single virtual BS with maximum power , and virtual OMA users operating in subchannels with maximum allowable power . Each cluster is indeed a virtual OMA user whose channel gain is , where is a function of the minimum rate demand of users with lower decoding order in cluster , and the channel gain of the clusterhead user, whose index is . Each virtual user has also a minimum power demand in order to guarantee the minimum rate demand of its multiplexed users in . For any given virtual power budget , the achievable rate of each virtual OMA user is the sumrate of its multiplexed users.
Based on the definition of virtual users for the sumrate maximization problem in NOMA and the KKT optimality conditions analysis, the exemplary systems in Figs. 1(a)1(c) are equivalently transformed to their corresponding virtual OMA systems shown in Figs. 1(d)1(f). Note that FDMA is a special case of NOMA, where each subchannel is assigned to a single user. Hence, each OMA user acts as a clusterhead user, and subsequently, the virtual users are identical to the real OMA users, i.e., , , and . As a result, each user in FDMA deserves additional power. It can be shown that the analysis for finding the optimal power allocation in NOMA and FDMA are quite similar, and the only differences are and . Finally, both of them can be solved by using the waterfilling algorithm for any number of users and clusters.
IiiB Equal InterCluster Power Allocation
The intercluster power allocation is necessary when , i.e., there is at least one cluster which is not allowed to operate at its maximum power . In this case, the distributed intracluster power allocation leads to violating the cellular power constraint (3c), since in the distributed power allocation among clusters, the constraint (3d) will be active. Alternatively, when , we guarantee that . There are a number of works, e.g., [27, 28], assuming , i.e., equal intercluster power allocation. In this case, the optimal intracluster power allocation can be obtained by Proposition 1.
It can be shown that in FDMA, the equal power allocation strategy is optimal if and only if 1) it is feasible; 2) all the users have exactly the same channel gains normalized by noise.
Remark 2.
In NOMA, the equal intercluster power allocation strategy is optimal if and only if 1) it is feasible; 2) all the virtual OMA users have the same channel gains, i.e., .
Proposition 2.
Consider two clusters with the same number of multiplexed users, where the corresponding users according to the decoding orders in different clusters have exactly the same channel gains. The equal intercluster power allocation strategy may not be optimal if the corresponding users with lower decoding orders have different minimum rate demands.
Proof.
The optimality condition in Remark 2 implies that
(14) 
The equivalent channel gain , and thus (14) only depends on the minimum rate demand of users with lower decoding order and the channel gain of the clusterhead users (see Remark 1). Based on (14), for the case that the clusterhead users have exactly the same channel gains, i.e., , the equal intercluster power allocation strategy is optimal if and only if . According to the definition of in (9), one simple case that for some is considering different minimum rate demands for the users with lower decoding order. ∎
Corollary 1.
The equal intercluster power allocation strategy can be optimal for completely diverse channel gains of users in different clusters and heterogeneous minimum rate demands of users. The unique condition for the optimality of equal intercluster power allocation strategy is given by (14).
In contrast to OMA, the individual minimum rate demand of users with lower decoding order plays an important role on the performance of equal intercluster power allocation strategy, while their channel conditions do not significantly affect the performance of this strategy.
IiiC Energy Efficiency Maximization Problem
In this subsection, we find a globally optimal solution for problem (4). According to Lemma 1, the feasible set of (4) is affine, so is convex. Moreover, the sumrate function in the numerator of the EE function in (4) is strictly concave in . The denominator of the EE function is an affine function, so is convex. Therefore, problem (4) is called a concaveconvex fractional program with a pseudoconcave objective function [36, 37]. The pseudoconcavity of the objective function in (4) implies that any stationary point is indeed globally optimal and the KKT optimality conditions are sufficient if a constraint qualification is fulfilled [36]. Hence, the problem (4) can be solved by using the iterative Dinkelbach algorithm [36, 37]. In this algorithm, we iteratively solve the following problem
(15) 
where is the fractional parameter, and is strictly concave in . This algorithm is described as follows: We first initialize parameter such that . At each iteration , we set , where is the optimal solution obtained from the prior iteration . After that, we find by solving (IIIC) in which . We repeat the iterations until , where is a tolerance tuning the accuracy of the optimal value. The pseudocode of the Dinkelbach algorithm for solving (4) is presented in Alg. 2.
It is verified that Alg. 2 converges to the globally optimal solution with polynomial time complexity depending on the number of variable and constraints in (IIIC). Similar to the transformation of (3) to (5), we define as the power consumption of cluster . The main problem (IIIC) can be equivalently transformed to the following joint intra and intercluster power allocation problem as
(16) 
The feasible set of problems (5) and (16) is equal, thus the feasibility of (16) can be characterized by Lemma 1.
Proposition 3.
Proof.
When is fixed, the second term in (16) is constant. Hence, the objective function of (16) can be equivalently rewritten as maximizing users sumrate given by , which is independent of . Hence, for any given feasible , problems (16) and (5) are identical. Accordingly, Proposition 1 also holds for any given and in (16). ∎
According to Proposition 3, the KKT optimality conditions analysis for the sumrate maximization problem (3) holds for the EE maximization problem (4). Subsequently, similar to the equivalent transformation of (5) to (11), problem (16) can be equivalently transformed to the following parametric form of the intercluster power allocation problem as
(17a)  
s.t.  (17b) 
where and are defined in (9). Note that since and are constants, the term can be removed from (16), so is removed in (17a) during the equivalent transformation. The differences between problems (11) and (17) are the additional term in , and also inequality constraint (17b). We prove that the activation of cellular power constraint (17b) depends on the value of , such that at the optimal point, the BS may not operate at its maximum power budget.
Proof.
The optimal solution of (17) is unique if and only if the objective function (17a) is strictly concave. For the case that the concave function in (17a) is increasing in , we can guarantee that at the optimal point, the cellular power constraint (17b) is active. In other words, for the case that , for any , the optimal satisfies . In this case, the cellular power constraint (17b) can be replaced with , thus the optimization problem (17) can be equivalently transformed to the sumrate maximization problem (11) whose globally optimal solution is obtained by Alg. 1. In the following, we find a sufficient condition, where it is guaranteed that , for any . The condition can be rewritten as After some mathematical manipulations, the latter inequality is rewritten as
(18) 
The righthand side of (18) is a constant providing an upperbound for the region of such that . The inequality in (18) holds for any , if and only if , and the proof is completed. ∎
For the case that Proposition 4 does not hold, Alg. 1 is suboptimal for (17) since the objective function in (17a) is decreasing in a subset of the feasible domain of . In this case, similar to the transformation of (11) to (12), we define , where . Then, the problem (17) is rewritten as
(19a)  
s.t.  (19b) 
where , , , and . The equivalent convex problem (19) can be solved by using the Lagrange dual method with subgradient algorithm or interior point methods (IPMs) [40]. The derivations of the subgradient algorithm for solving (19) is provided in Appendix B. Moreover, the derivations of the barrier algorithm with inner Newton’s method for solving (19) is provided in Appendix C. According to the above, depending on the value of at each Dinkelbach iteration, (IIIC) can be solved by using Alg. 1 or subgradient/barrier method. The pseudocodes of our proposed algorithms for solving (IIIC) in Step 3 of Alg. 2 based on the subgradient and barrier methods are presented in Algs. 3 and 4, respectively.
After finding via Algs. 3 or 4, we find the optimal intracluster power allocation by using (7) and (8). The duality gap in Alg. 4 after iterations is , where is the initial , and is the stepsize for updating in the barrier method. Therefore, after exactly barrier iterations, Alg. 4 achieves suboptimal solution [40].
Corollary 2.
In both the sumrate and EE maximization problems, in each cluster, only the clusterhead user deserves additional power, and all the other users get power to only maintain their minimal rate demands. The analysis proves that in the sumrate maximization problem, the BS operates at its maximum power budget. However, for the EE maximization problem, the BS may operate at lower power depending on the condition in Proposition 4.
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