AbstractâIn this paper, we study the resource allocation and scheduling problem for a downlink non-orthogonal multiple access (NOMA) network where the ...
Radio Resource Allocation for Downlink Non-Orthogonal Multiple Access (NOMA) Networks using Matching Theory Boya Di∗ , Siavash Bayat† , Lingyang Song∗ , and Yonghui Li‡ , of Electrical Engineering and Computer Science, Peking University, Beijing, China. † Electronics Research Institute, Sharif University of Technology, Tehran, Iran. ‡ School of Electrical and Information Engineering, The University of Sydney, Sydney, Australia. ∗ School
Abstract—In this paper, we study the resource allocation and scheduling problem for a downlink non-orthogonal multiple access (NOMA) network where the base station (BS) allocates the spectrum resources and power to the set of users. We aim to optimize the sub-channel assignment and power allocation to achieve a balance between the number of scheduled users and total sum-rate maximization. To solve the above problem, we propose a many-to-many two-sided user-subchannel matching algorithm in which the set of users and sub-channels are considered as two sets of players pursuing their own interests. The algorithm converges to a pair-wise stable matching after a limited number of iterations. Simulation results show that the proposed algorithm can approach the performance of the upper bound and greatly outperforms the OFDMA scheme.
I. I NTRODUCTION Orthogonal Frequency Division Multiple Access (OFDMA) has been widely adopted in the 4th generation (4G) mobile communication systems such as LTE and LTE-Advanced to combat the narrow-band interference. To further increase the data rates of multiple access transmissions, various new multiple access techniques have been proposed such as Interleave Division Multiple Access (IDMA) [1], Low Density Spreading (LDS) [2], and Non-Orthogonal Multiple Access (NOMA). Among these techniques, NOMA has a lower receiver complexity and achieves significant improvement in spectral efficiency by allowing multiple users to share the same sub-channel in power domain, and thus, it has been considered as a promising candidate for future access technologies [3] due to its potential in improving frequency efficiency. Unlike the OFDMA scheme in which one sub-channel can only be assigned to one user, multiple users can share the same sub-channel in the NOMA scheme, creating the interuser interference. To tackle this problem, various multi-user detection (MUD) techniques such as the Successive Interference Cancellation (SIC) technique [4] can be applied at the end-user receivers to decode the received signals. Recently different aspects of the NOMA scheme have been discussed in several works [5]−[7]. In [5], the concept of basic NOMA with SIC was introduced and its performance was compared with the traditional OFDMA scheme through a system-level evaluation. A low-complexity power allocation method for NOMA with SIC receiver was discussed in [6] by exploiting a tree search algorithm. In [7], the authors discussed the user fairness in an uplink NOMA scheme for the wireless networks.
Unfortunately, so far few works have studied the subchannel allocation problems in the NOMA scheme. Some initial sub-channel allocation schemes have been adopted in the existing works [5]− [7], however, they are far from the optimal scheme. The sub-channel allocation for maximizing the total sum-rate of the system for NOMA is a non-trivial problem and remains as an open problem in the literature due to its combinatorial nature and co-channel interference. In this paper we consider a downlink NOMA wireless network in which the base station (BS) assigns the sub-channels to a set of users and allocates different levels of power to them. Each user can utilize multiple sub-channels and each sub-channel can be shared by multiple users. For the users sharing the same sub-channel, the SIC technique is adopted to remove the inter-user interference. The sub-channel and power allocation is then formulated as a non-convex optimization problem to maximize the total sum-rate. To tackle the above problem, we decouple the sub-channel and power allocation problems, and show that sub-channel allocation can be considered as a multivariate matching process in which the users and sub-channels are two sets of players to be matched with each other to achieve the maximum sumrate. We thus solve this problem by utilizing the matching theory [8], which provides an adaptive and low-complexity framework, making it suitable to solve the resource allocation problem with combinatorial nature [9] [10]. We then formulate the sub-channel allocation problem as a many-to-many twosided matching problem with externalities in which interdependencies exist between the players’ preferences due to the co-channel interference. A novel user-subchannel matching algorithm (USMA) is developed to solve this problem, and a stable matching can be formed via the proposed algorithm after a small number of iterations. The proposed matching algorithm is analyzed in terms of its stability, convergence, and complexity. Simulation results show that the USMA can achieve a performance close to the upper bound, and performs much better than the OFDMA scheme. The rest of this paper is organized as follows. In Section II, we describe the system model. In Section III, we formulate the optimization problem as a many-to-many two-sided matching problem, and propose a user-subchannel matching algorithm, followed by the corresponding analysis. Simulation results are presented in Section IV. Finally, we conclude the paper in
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technique [4]. The receiver of user Mj ∈ Sk can cancel the interference from any other user Mi in Sk with a lower 2 2 channel gain |hk,i | /σni 2 < |hk,j | /σnj 2 , i.e., user Mj can decode and cancel the signal from user Mi satisfying the above condition [5]. User Mj regards the signals of those users with channel gains than its own channel gain as noise, then it decodes xk,j from received signal yj 1 . Therefore, the sum-rate of user Mj is given by 2 pk,j |hk,j | , (2) log2 1 + 2 Rj = σn + Ik,j
SNS. S.
S.
where Ik,j is the interference that user Mj receives from other users in Sk over sub-channel SC k , 2 2 2 (3) Ik,j = |hk,i | > |hk,j | pk,i |hk,j | . i∈ M| σni 2
Fig. 1.
System model of the NOMA networks.
Section V.
σnj 2
The total sum-rate over SC k is then given by 2 pk,j |hk,j | RSCk = . log2 1 + 2 σn + Ik,j
(4)
j∈Sk
II. S YSTEM M ODEL Consider a downlink single-cell NOMA network as shown in Fig. 1, in which a single BS transmits the signals to a set of mobile users denoted by M = {1, · · · , M }. The BS divides the available bandwidth into a set of sub-channels, denoted by K = {1, · · · , K}. We assume that the BS has the full knowledge of channel state information (CSI). Based on the CSI of each channel, the BS assigns a subset of nonorthogonal subchannels to the users and allocates different levels of power to them. According to the NOMA protocol [5], one sub-channel can be allocated to multiple users, and one user can receive from the BS through multiple sub-channels. The power allocated to user Mj ∈M over sub-channel SC k is denoted by pk,j , satisfying j∈M pk,j ≤ Pk and Pk = Ps /K where Ps is the total transmitted power of the BS. We consider a block fading channel, for which the channel remains constant within a time-slot, but varies independently from one to another. The complex coefficient of SC k between user Mj and the BS is denoted by hk,j = gk,j /D (dj ), where gk,j denotes the Rayleigh fading channel gain, dj is the distance between user Mj and the BS, and D (·) is the path loss function. Let Sk be the set of active users over subchannel SC k , and xk,i be the transmitted symbol of user Mi over sub-channel SC k . The signal that user Mj receives over sub-channel SC k is then given by √ yj = hk,j pk,i xk,i + nj , (1) i∈Sk
where nj ∼ CN 0, σn 2 is the additive white Gaussian noise (AWGN) for user Mj , and σn 2 is the noise variance. Since sub-channel SC k can be utilized by a subset of users, Sk , the signal of any user Mj ∈ Sk will cause the interference to other users in Sk . To demodulate the target signal, each user Mj adopts the successive interference cancellation (SIC)
Note that the SIC technique performed at the user receiver may cause considerable complexity at the receiver since as the number of users over the same sub-channel grows, the implementation complexity of SIC increases. Therefore, considering the complexity caused by decoding, we assume that at most qu users can share one sub-channel at the same time, i.e., |Sk | ≤ qu . Given a proper value of qu , the decoding complexity at the receiver is much reduced to a tolerable level. Due to the scarce spectrum resources, we also assume that one sub-channel is assigned to at least ql users, which guarantees the number of scheduled users. To better describe the allocation of the sub-channels to the set of users, we introduce a K × M sub-channel matrix Γ in which the binary element γk,j denotes whether sub-channel SC k is allocated to user Mj . The system performance can be evaluated by the total sum-rate of all users, and our objective is to maximize the total sum-rate of the system by setting the variables {pk,j , γk,j } as well as improving the number of scheduled users. The optimization problem is then formulated as: 2 pk,j |hk,j | max (5a) log2 1 + 2 γk,j ,pk,j σn + Ik,j j∈M k∈K s.t.:ql ≤ γk,j ≤ qu , ∀k ∈ K, (5b) j∈M
γk,j ∈ {0, 1} , ∀k ∈ K, j ∈ M, pk,j ≤ Pk = Ps /K, ∀k ∈ K,
(5c) (5d)
j∈M
pk,j ≥ 0, ∀k ∈ K, j ∈ M, Rj ≥ Rmin , ∀j ∈ {M|γk,j = 1} , ∀k ∈ K.
(5e) (5f)
1 The order for decoding based on the increasing channel gains described above guarantees that the upper bound on the capacity region can be reached.
Constraints (5b) and (5c) ensure that each sub-channel can only be assigned to at most qu and at least ql users. Due to the limited transmitted power of the BS, the power variables must satisfy constraint (5d) and (5e). Constraint (5f ) guarantees the minimum data rates of those scheduled users. Note that this optimization problem is a non-convex optimization problem due to the binary constraint in (5c) as well as the existence of the interference term in the objective function [12]. Therefore, it may be too complex to solve this problem by utilizing the conventional centralized methods, especially in a dense network. We then separately solve the power and sub-channel allocation problems in Section III. Specifically, to facilitate the allocation of the sub-channels to the users as well as modeling the relationship between these interactive users, we utilize the matching theory and consider the set of users and the sub-channels as two disjoint sets of selfish and rational players that would like to match with each other such that their own benefits (the corresponding sum-rate) are maximized. III. M ANY- TO -M ANY M ATCHING FOR NOMA As observed in the objective function (5a), the multi-user power allocation and the sub-channel allocation are coupled with each other in terms of the total sum-rate. Considering the computational complexity, we decouple the power allocation and the sub-channel allocation, and obtain a suboptimal solution. The BS first assumes that the transmitted power is equally allocated to the users sharing the same sub-channel, then (5) can be formulated as a many-to-many two-sided matching problem, which can be solved by utilizing the matching theory. Afterwards, the BS allocates the transmitted power to the scheduled users over each sub-channel according to the waterfilling algorithm [11], as will be discussed in Section III.C. A. Many-to-many Two-sided Matching Problem Formulation To better show the dynamic interactions between the users and the sub-channels, we consider the set of users, M, and the set of sub-channels, K, as two disjoint sets of selfish and rational players aiming to maximize their own interests. If subchannel SC k is assigned to user Mj , then we say Mj and SC k are matched with each other and form a matching pair. To describe the competition behavior and decision process of each player, we assume that each player has preferences over the players of the other set. The preference of each player is based on the achievable data rates, and we then denote the set of preference lists of the users and sub-channels as P = {P (M1 ) , · · · , P (MM ) , P (SC1 ) , · · · , P (SCK )} , (6) where P (Mj ) and P (SCk ) are the preference lists of user Mj and sub-channel SC k , respectively. Each sub-channel is assigned to at least ql users, and its preference over different subset of the set of users can be represented as T SCk T , T ⊆ M, T ⊆ M ⇔ RSCk (T ) > RSCk (T ) , (7) which means SC k prefers the users in T to those in T because the former set provides higher profits than the latter one.
Note that each user can utilize one or more sub-channels. The relations of preferences over these sub-channels can be represented as 2
2
SCk Mj SCk ⇔ |hk,j | /σnj 2 > |hk ,j | /σnj 2 ,
(8)
SC k
because SC k which implies that Mj prefers SC k to provides higher channel gain than SC k . In the literature, various properties of preferences have been described and studied for various scenarios [8], [9], and different characteristics of preference lists may lead to different versions of matching algorithms and stability. In our scenario, the users’ and sub-channels’ preferences are depicted as transitive and substitutable. By calling a preference list as transitive, we mean that if T SCk T and T SCk T , then T SCk T . Given a player i in M ∪ K, for its preferred set S belonging to the opposite set of i that contains member k and k , we can say that if k ∈ S, then k ∈ S\ {k }. Therefore, the preference over the set of users and sub-channels is substitutable. With the notion of the preference list, we can then formulate the optimization problem as a many-to-many two-sided matching problem. Definition 1: Given two disjoint sets, M = {1, 2, · · · , M } of the users, and K = {1, 2, · · · , K} of the sub-channels, a many-to-many matching Ψ is a mapping from the set M ∪ K into the set of all subsets of M ∪ K such that for every Mj ∈ M, and SC k ∈ K: 1) Ψ (Mj ) ⊆ K; 2) Ψ (SCk ) ⊆ M; 3) ql ≤ |Ψ (SCk )| ≤ qu ; 4) SCk ∈ Ψ (Mj ) ⇔ Mj ∈ Ψ (SCk ). Condition 1) states that each user is matched with a subset of sub-channels, and condition 2) represents that each subchannel is matched with a subset of users. Taking into account the number of scheduled users and the tolerable complexity of the decoding technique at the user receiver, we set the size of SCk larger than ql and smaller than qu . Note that the matching model that was formulated here is more complicated than the conventional two-sided matching models. Firstly, in our model any member of each set can be matched with any subset of the opposite set. Since the number of potential subsets of each set can be quite large (especially when the set has many members), thus the number of possible matching combinations can be very large. Secondly, there exist interdependencies between the users who share the same subchannel. Thus each sub-channel needs to select a satisfying subset of users to match with, rendering the problem to be rather intractable even when the power allocation part is not considered. Therefore, to solve this matching problem, we develop an extended version of the Gale-Shapley algorithm [8] and propose a new matching algorithm in Section III.B. B. Many-to-Many Matching Algorithm for NOMA Inspired by the traditional Gale-Shapley algorithm, we propose the modified, and low-complexity USMA, which is performed by the BS without extra signaling cost compared
TABLE I U SER -S UBCHANNEL M ATCHING A LGORITHM (USMA) Step 1: Initialization. (a) Construct the list of all sub-channels {SCM atch } to record the number of users that sub-channel is currently matched with. (b) Each sub-channel and user sets its preference list {P (SCk )} and {P (Mj )}. Step 2: Matching Procedure. (a) Each user Mj ∈ M proposes itself to preferred sub its most hk,j 2 /nj , ∀j ∈ M. channel in P (Mj ): SCk = arg max k∈P(Mj ) (b) If |SCM atch (k)| < ql , then SC k adds Mj to SCM atch (k), and Mj removes SC k from P (Mj ). (c) Else if ql ≤ |SCM atch (k)| < qu , (i) SC k keeps Mj ’s offer and Mj removes SC k from P (Mj ) only if SCM atch (k) ∪ {Mj } SCk SCM atch (k). (ii) SC k updates SCM atch (k). (d) Else if |SCM atch (k)| ≥ qu , (i) SC k selects a set of qu users Squ satisfying Squ SCk S , S ⊂ SCM atch (k) ∪ {Mj }. (ii) SC k sets SCM atch (k) = Squ , and rejects other users. (iii) The rejected users remove SC k from their preference lists. (e) If no user is willing to make new offers to the sub-channels, go to Step 3, else go back to Step 2(a). Step 3: End of the algorithm.
to the traditional centralized methods. The key idea of the USMA is that if a user wants to match with a sub-channel, it proposes itself to that sub-channel, i.e., this user makes an offer to its preferred sub-channel. Each sub-channel has the right to accept or reject these offers. When all the users make an offer once, we say one round of proposals is performed. Before we describe the USMA in detail, we first introduce the concept of blocking pair to explain how the sub-channels choose among different offers from the proposing users. Definition 2: Given a matching Ψ and a pair (Mj , SCk ) with Mj ∈ / Ψ (SCk ) and SCk ∈ / Ψ (Mj ). (Mj , SCk ) is a blocking pair if 1) SSCk Ψ (SCk ), S ⊆ {Mj } ∪ Ψ (SCk ), and Mj ∈ S; 2) SCk Mj SCl , and SCl ∈ Ψ (Mj ). With the above definition, we can then describe the action of each sub-channel when it makes decisions among different offers. Since each sub-channel SC k aims at maximizing its own utility, every time it forms a blocking pair with Mj , it keeps the offer of Mj and rejects other users, i.e., the decision process of SC k is a process of eliminating the potential blocking pairs. We now describe the proposed USMA to solve the resource allocation problem in (5a)-(5f ). The specific details of our proposed algorithm are described in Table I, consisting of an initialization phase and a matching phase. In the initialization phase, each sub-channel and user sets its preference list according to the channel states. In the matching phase, at each round, every user proposes itself to its most preferred subchannel which can provide a higher sum-rate than Rmin and has not rejected it before, and the sub-channel that receives fewer than ql offers keeps these offers, while the sub-channel that receives more than ql offers only selects less than qu + 1 offers, and rejects the other users. The matching process stops when no user is willing to make new offers.
C. Water Filling Power Allocation Given the sub-channel allocation, the power allocation can be implemented. To reduce the complexity of power allocation caused by the SIC technique, we set Ik,j = 2 i∈Sk /{j} pk,i |hk,j | for simplicity. The water-filling solution can be expressed as follows: + 1 ∗ , (9) pk,j = λk − 2 |hk,j | /σn 2 where
1 λk = |Sk |
Pk +
1
i∈Sk
|hk,i | /σn 2
2
.
(10)
is the water-filling level of user Mj over SC k . D. Stability, Convergence and Complexity With the definition of blocking pair and the substitutable preference lists explained above, we then define the conception of pair-stability as below and prove that the USMA can converge to a pairwise-stable matching. Definition 3: A matching Ψ is defined as pairwise stable if it is not blocked by any pair which does not exist in Ψ. Lemma 1: If the USMA converges to a matching Ψ∗ , then ∗ Ψ is a pairwise-stable matching. Proof: Suppose that there exists a pair (Mj , SCk ) which doesn’t exist in Ψ∗ such that SSCk Ψ (SCk ), S ⊆ {Mj } ∪ Ψ (SCk ), Mj ∈ S, and SCk Mj SCl , SCl ∈ Ψ (Mj ). According to Step 2-2 and 2-3 in the USMA, Mj proposed to SC k and was subsequently rejected at iteration t, imt t plying that SCM atch (k) SCk {Mj } ∪ SCM atch (k). Given f inal t SCM atch (k) SCk SCM atch (k), we can infer that Mj ∈ / f inal (k) ∪ {M } according to the transitivity of the SCM j atch preference lists, which is inconsistent with the supposition. Since the pair (Mj , SCk ) is arbitrary, the matching Ψ isn’t blocked by any pair, which is pairwise stable. Theorem 1: The proposed USMA converges to a pairwisestable matching Φ∗ after a limited number of iterations. Proof: With the details of the USMA in Table I, we show that this algorithm will end within a limited number of iterations. Since in each iteration, each Mj ∈ M proposes to its most preferred sub-channel which hasn’t rejected it in the former iterations, so as the number of iterations increases, the set of choices for each Mj becomes smaller. There are K sub-channels in the network, so the number of proposals that each Mj makes to the sub-channels is no larger than K, and thus, the total number of iterations is no more than K. Therefore, the USMA converges to a final matching Φ∗ which is also pairwise-stable according to Lemma 1. Theorem 2: The complexity of the optimal exhaustive search is O K · 2M , while the complexity of the USMA is O M K2 . Proof: For the optimal exhaustive search, the BS searches all possible candidate user sets over each sub-channel, then selects the most profitable user set and performs the power allocation for each sub-channel. Since the number of candidate
IV. S IMULATION R ESULTS
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Total sum−rate
user sets is 2M − 1 and there are K sub-channels, the complexity can be expressed by O K · 2M . For the USMA, the complexity comes from two steps of the algorithm, i.e., the sorting phase in Step 1(b) and the matching procedure in Step 2. In the sorting phase, each user obtains its preference list of K sub-channels, in which the complexity is O M K 2 , while in the matching procedure, each user will propose as most M K times, resulting in the total complexity of O M K 2 .
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Total sum-rate vs. number of the users (K = 6).
Fig. 2.
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In this section, we evaluate the performance of the proposed USMA, and compare its performance with the OFDMA scheme and the upper bound. In the OFDMA scheme, we assume that each sub-channel can only be assigned to one user, and the total power of this sub-channel Pk is allocated to this user. To guarantee the number of scheduled users, we utilize the proportional fair scheduling algorithm [13] to assign the sub-channels to users. To depict the upper bound, we set qu = M , i.e., a sub-channel can be shared by all users. Then an optimal exhaustive search over all subsets of users as well as the power allocation utilizing the water filling algorithm [12] are performed over each sub-channel to maximize the sum-rate. The upper bound is unrealistic since the receiver demodulating complexity is particular high, and it is highly likely that the number of scheduled users is pretty small, as shown in Fig. 3. For the simulations, we set the BS’s peak power, Ps to 46dBm, noise variance, σ 2 to -90dBm, ql to 2, and we assume that all users are uniformly distributed in an square area with the size of length 250m. We obtain our simulation results as shown below, and all curves are generated based on averaging over 10000 instances of the algorithms. Figure 2 illustrates the total sum-rate vs. the number of users M with the number of sub-channels K = 6. We find out that the total sum-rate increases with the number of users, and the rate growth becomes slower as M increases. As the number of users grows, the BS tends to allocate the sub-channels to the users with higher channel gains, thereby improving the total sum-rates. When the number of users is much larger than the number of sub-channels, the total sum-rate continues to increase due to the multiuser diversity gain but grows at a slower speed. From Fig. 2, we can see that the performance of the proposed USMA algorithm is much better than that of the OFDMA scheme and reaches 96.7% of the upper bound when M = 30 and pu = 3. This is because in the OFDMA scheme, one sub-channel can only be assigned to one user, and thus, the BS does not take full advantage of the spectrum resources. When qu = 3, the USMA performs better than the case in which qu = 2. This is because the BS has more choices over the set of users to be assigned to each sub-channel. Figure 3 shows the number of scheduled users vs. the number of users with K = 5 within one time-slot. We find that when the number of users is smaller than the number of subchannels, all the users have access to the spectrum resources in both the NOMA and the OFDMA schemes. As the number of users exceeds the number of sub-channels, only up to K users can access the spectrum resources simultaneously in the
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Upper bound USMA(qu = 4)
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OFDMA 5 4 3 2 1
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Fig. 3.
Number of scheduled users vs. number of the users (K = 6).
OFDMA scheme, since in this scheme one sub-channel can only be assigned to one user. Thus in the OFDMA scheme, the percentage of the scheduled users drops rapidly. In the NOMA scheme, as the number of users grows, the number of scheduled users tends to a fixed value which is smaller than Kqu but is still much larger than that of the OFDMA scheme. Note that the number of scheduled users is higher when qu becomes larger, since more users have the opportunity to access the sub-channels. Fig. 3 also reflects that the upper bound is unrealistic since only few users have access to the sub-channels, which greatly harms the interests of other users. Figure 4 shows the average sum-rate of the scheduled users vs. the number of users with the number of sub-channels K = 6. When the number of users is small, the average sum-rate of the scheduled users decreases as the number of users increases. As the number of users becomes larger, the number of scheduled users tends to a fixed value, as shown in Fig. 3, while the total sum-rates keeps increasing due to multi-user diversity gain, and thus, the average sumrate of each scheduled user increases. Fig. 4 also shows that the average sum-rate of each scheduled user in the NOMA scheme is higher than that in the OFDMA scheme, even
Average data rate of the scheduled users
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OFDMA 30
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Fig. 4. Average sum-rate of the scheduled users vs. number of the users (K = 6).
Upper Bound USMA(qu = 4) USMA(q = 3) u
700
OFDMA(q = 1) u
Total sum−rate
VI. ACKNOWLEDGEMENT This work was supported in part by the National 973 Project under Grant 2013CB336700, by the National Natural Science Foundation of China under Grants 61222104, U1301255, and 61450110084, by National Science and Technology Major Project under Grant 2013ZX03003003, by the Ph.D. Programs Foundation of Ministry of Education of China under Grant 20110001110102, by the Australian Research Council (ARC) under Grants DP150104019, FT120100487 and LP150100994.
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a balance between the number of scheduled users and the maximization of the total sum-rate. By formulating the problem as a many-to-many two-sided matching problem, we proposed a near optimal user-subchannel matching algorithm in which the users and sub-channels can be matched and form a stable matching. According to the simulation results, we achieved the following conclusions. The minimum and maximum numbers of the users sharing one sub-channel have great influence on the number of scheduled users and the total sum-rates, and thus, they should not be set too high considering the processing ability of the receiver as well as the achievable data rates of each user. The total sum-rate, average sum-rate of the scheduled users, and the number of scheduled users in the NOMA scheme are higher than those in the OFDMA scheme regardless of the numbers of the sub-channels and users.
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R EFERENCES
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Fig. 5.
Total sum-rate vs. number of the sub-channels (M = 15).
though proportional fair user scheduling is utilized in the latter one, implying that we can guarantee the profits of more users utilizing the NOMA scheme. When qu = 3, the average sumrate is lower than the case with qu = 2, because more users with relatively low channel gains can share the sub-channels when the maximum number of the users over each sub-channel qu = 3. Figure 5 depicts the total sum-rate vs. the number of subchannels with the number of users M = 15. In the NOMA scheme, the total sum-rate increases with the number of subchannels since more users have access to the sub-channels and each user’s profit grows. In the OFDMA scheme, when the number of sub-channels is larger than the number of users, the total sum-rate remains fixed due to the exclusivity, which implies that the NOMA scheme performs better than the OFDMA scheme regardless of the numbers of the subchannels and the users. V. C ONCLUSION In this paper, we studied the resource allocation problem in a downlink NOMA wireless network by optimizing the subchannel assignment and the power allocation, while achieving
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