Subchannel and Power Allocation for Nonorthogonal Multiple Access Relay Networks with AmplifyandForward Protocol
Abstract
In this paper, we study the resource allocation problem for a singlecell nonorthogonal multiple access (NOMA) relay network where an OFDM amplifyandforward (AF) relay allocates the spectrum and power resources to the sourcedestination (SD) pairs. We aim to optimize the resource allocation to maximize the average sumrate. The optimal approach requires an exhaustive search, leading to an NPhard problem. To solve this problem, we propose two efficient manytomany twosided SD pairsubchannel matching algorithms in which the SD pairs and subchannels are considered as two sets of players chasing their own interests. The proposed algorithms can provide a suboptimal solution to this resource allocation problem in affordable time. Both the static matching algorithm and dynamic matching algorithm converge to a pairwise stable matching after a limited number of iterations. Simulation results show that the capacity of both proposed algorithms in the NOMA scheme significantly outperforms the conventional orthogonal multiple access scheme. The proposed matching algorithms in NOMA scheme also achieve a better userfairness performance than the conventional orthogonal multiple access.
I Introduction
With rapidly increasing demands in mobile services, wireless networks require an ever higher spectral efficiency and massive connectivity [2]. However, the capacity of conventional orthogonal frequency division multiple access (OFDMA) is not likely to afford the explosive growth of data traffic. As a result, finding new multiple access techniques to achieve high spectrum efficiency and massive connectivity have become a critical and urgent challenge to be resolved in the current wireless communication networks [3]. The nonorthogonal multiple access (NOMA) technique was discussed to be used in LTE network in [4], and is regarded as a promising technology for 5G network [5]. Unlike the OFDMA scheme, NOMA can accommodate multiple users in the same time and frequency domains by differentiating the users through power domain or code domain multiplexing. NOMA has the advantages of a low complexity receiver and high spectrum efficiency due to its multiplexing nature.
In the conventional OFDMA scheme, each subchannel can only be assigned to one user. In contrast, NOMA system allows multiple users to share the same subchannel to achieve multiplexing gains. However, this also leads to unavoidable cochannel interference. To tackle this problem, various multiuser detection (MUD) techniques have been proposed, such as the successive interference cancellation (SIC) techniques [6], which can be applied at the enduser receivers to decode the received signals and reduce the interuser interference effectively. In [7], NOMA transmitter and low complexity receiver were proposed and its performance was compared with the theoretical performance of SIC. In [8], the resource allocation and user scheduling problem was studied in a downlink NOMA network with a joint algorithm.
The relaying technology has been regarded as an effective method to extend the coverage and improve the system performance of wireless networks. By accommodating more users, we utilise NOMA in a relay network, where different users share the spectrum resource of the network. NOMA relay network has the advantage of providing massive connectivity and higher spectrum efficiency for the users. It also provides a larger power and coverage with the relay to improve the network performance. However, the design of NOMA relay is very challenging due to the complicating characteristics of relay networks.
Recently, there are some initial works on NOMA relay networks. In [9], the outage probability of an amplifyandforward (AF) relay network was derived and a lower bound of the outage probability was provided. In [10], joint network channel coding and decoding for halfduplex multiple access multiple relay channels in NOMA scheme was studied. The application of simultaneous wireless information and power transfer in NOMA network was studied in [11] and a cooperative protocol was developed. Cooperative NOMA is a technique that improves the quality of service of the network, where different users in a NOMA network cooperate with each other to enhance the performance. In [12], a fullduplex devicetodevice aided cooperative NOMA scheme was proposed to improve the outage performance of the weak users. A theoretical study on the selection of the cooperative NOMA was studied in [13]. Although there are some works in NOMA relay networks, few of them have considered the resource allocation problem in such a network. Most of the existing works of NOMA relay network [9, 10, 11] focused on the performance analysis, such as outage probability [9], or focus on code design [10].
In this paper, we consider the NOMA relay networks to enhance the access spectral efficiency and at the same time provide wide area coverage in a large scale network. An OFDM amplifyandforward (AF) relay assigns the subchannels and allocate different level of power to a set of source destination (SD) pairs, each of which consists of a source node and a destination node, the source node transmits through the relay to its paired destination node. Each SD pair can occupy multiple subchannels and each subchannel can be shared by multiple SD pairs. For the SD pairs sharing the same subchannel, the SIC technique is adopted to remove the interuser interference. Joint subchannel and power allocation is then formulated as a nonconvex optimization problem to maximize the total sumrate. The optimal solution for this problem is NPhard and requires exhaustive search. Therefore, an efficient low complexity resource allocation algorithm is required. The process of solving resource allocation problem can be associated with matching theory due to the structure of the system. In a NOMA relay network, the relay allocates a set of spectrum resources to the set of SD pairs. The allocation process can be solved with matching theory under NOMA protocol, and the design of matching algorithm plays an important part in the resource allocation problem.
We propose to utilize matching theory in the resource allocation problem in NOMA relay network. This problem is separated into two subproblems, a subchannel allocation problem and a power allocation problem. In the subchannel allocation problem, the set of SD pairs and the set of subchannels are both seeking to match with the opposite set to maximize their own profit. Therefore, we consider the SD pairs and subchannels as two sets of selfish and rational players aiming at maximizing their own profits. Motivated by this, we formulate the subchannel allocation problem as a manytomany two sided matching game with externalities in which interdependencies exist between the players’ preferences due to the cochannel interference. Two novel userspectrum matching algorithms extended from the GaleShapley algorithm [14] are proposed for the matching game formulation to reach a stable matching. For the power allocation problem, the water filling algorithm is utilized to enhance the power efficiency. We take proportional fairness into consideration and aim at maximizing a function related to the average throughput of all the users, so as to guarantee the throughput of the users at the edge of the cell with a poor channel condition.
The main contributions of this paper can be summarized as follows.

We formulate a joint subchannel and power allocation problem for a downlink NOMA network to maximize the average sumrate over each subchannel.

We represent the subchannel allocation problem equivalent to a manytomany matching game, and propose two matching algorithms considering externalities [15], that is, the change in the matching structure caused by interuser interference is fully embodied. We then utilize an iterative water filling algorithm [16] to allocate the power. The properties of our matching algorithm are then analyzed in terms of stability, convergence and complexity.

Simulation results show that the proposed matching algorithms in NOMA scheme outperform OFDMA scheme significantly in both capacity and fairness.
The rest of this paper is organized as follows. In Section II we describe the system model of the NOMA relay networks. In Section III, we formulate the optimization resource allocation problem as a manytomany twosided matching problem, and propose a matching algorithm, followed by the corresponding analysis. Simulation results are presented in Section IV, and finally we conclude the paper in Section V.
Ii System Model
We consider a singlecell oneway NOMA network as depicted in Fig.1, consisting of one OFDM AF relay and SD pairs. Each SD pair consists of one source node and one destination node, where the source node communicates with the destination node assisted by relay . Let denote the set of source nodes and denote the set of destination nodes. We assume that relay has full knowledge of the instantaneous channel side information (CSI), which varies in every time slot. In each time slot, based on the CSI of each channel, relay assigns a subset of nonorthogonal subchannels, denoted as , to the SD pairs and allocates different power over the subchannels. According to the NOMA protocol[17], one subchannel can be allocated to multiple SD pairs, one SD pair has access to multiple subchannels in the network, and each SD pair shares the same group of subchannels. We assume that at most SD pairs can have access to each subchannel. To guarantee fairness among the SD pairs, we set a maximum number of subchannels allocated to each SD pair [18]. Communication between the source nodes and destination nodes in each time slot consists of two phases, described specifically as follows.
In the first phase, the source nodes transmit signals to relay . We denote the th source node as and the th subchannel as . The transmitting power of over is denoted as p, satisfying for each , where P is the maximum transmit power of . We consider a block fading channel, for which the channel remains constant within each timeslot, but varies independently from one to another. The complex coefficient of between and relay consists of two parts, the smallscale fading and the largescale fading [19][20]. It is denoted by , where describes the Rayleigh fading channel gain of from to relay , describes the distance between relay and , with being the constant path loss coefficient. The Rayleigh fading channel gain is a smallscale fading, whose real part and imaginary part of the channel gain both obey the Gaussian distribution, and it varies in different time slots. The largescale fading is denoted as , which only depends on the distance between the source node and relay and remain constant in different time slots. Let be the transmitting information symbol of unit energy from over . The signal that relay receives from over is given by
(1) 
where is the additive white Gaussian noise (AWGN), and is the noise variance.
In the second phase, relay amplifies the signals received from each source node and broadcasts the superposed signals to the destination nodes [9]. Let denote the amplification factor of relay over and is the transmit power that relay allocates to over , where is the th destination node. The relation between and is given by
(2) 
Correspondingly, we denote as the complex coefficient of between relay and , and , where denotes the Rayleigh fading channel gain of from relay to , and is the distance between and relay . The smallscale fading channel gain differs in every time slot, while the largescale fading channel gain remain constant. Let be the set of SD pairs that have access to . The signal that receives from relay over is given by
(3) 
where is AWGN. It is assumed that . By substituting into , the equation can be rewritten as
(4)  
(5) 
From , we see that the equivalent channel gain of the a SD pair is determined by the channel gains of both the first and the second phases. We then normalize and extract the equivalent channel gain as
(6) 
The numerator part shows the channel fading and power of the target signal, and the denominator part describes the AWGN of both sourcerelay transmission and relaydestination transmission. After receiving the signals, the destination nodes perform SIC to reduce the interference from the source nodes of other SD pairs with a smaller equivalent channel gain over [21]. For example, for , if , first treats as the interference to and cancel in decoding ^{1}^{1}1For the downlink channel that we consider, [22] shows that a user can decode the signal of another user with worse channel gain, with any split of the total power.. The order for decoding is based on the increasing channel gains described above, which guarantees that the upper bound on the capacity region can be reached[23]. The interference that receives over is shown as below,
(7) 
Note that the noise and interference for over consists of three parts: the noise at , the amplified noise forwarded by relay , and interference from other source nodes. Therefore, the data rate of over is given by
(8) 
When considering a multitime slot scenario, we care more about the average throughput than the instantaneous throughput of the network. The average throughput that received per subchannel in the th time slot is defined [24] as
(9) 
Parameter denotes the time duration during which we calculate the average throughput for throughput averaging, and is the data rate of over in the th time slot, which can be calculated by .
We utilize proportional fairness in this model to guarantee the quality of service for the cell edge users. The scheduling metric of the SD pairs over subchannel is shown as follows [25],
(10) 
The scheduling metric for the set of SD pair is , and the SDsubchannel pairing that maximizes equation (10) for the destination nodes in over each subchannel will be adopted.
Our objective is to maximize over each subchannel by jointly allocating in each time slot. To better describe the resource allocation, we define a binary SD pairsubchannel pairing matrix , in which denotes that S and D are paired with . We also assume that relay has a maximum transmitted power of , and satisfy the following inequation . The objective and restrictions of problem can be formulated as follows:
(11a)  
(11b)  
(11c)  
(11d)  
(11e)  
(11f)  
(11g)  
(11h) 
where and shows that each subchannel can be allocated to at most SD pairs and each SD pair can have access to at most subchannels. and are the power restrictions of the source nodes and relay respectively. Constraints and show that the transmitting power is no less than 0.
It can be observed that is a nonconvex problem due to the binary constraint in and the existence of the interference term in the objective function [26]. However, in the complexity theory, nonconvexity can not prove the problem¡¯s hardness, as a problem could be inappropriately formulated. Therefore, we prove that the nonconvex optimization problem is also an NPhard one in the following theorem.
Theorem 1: The sumrate maximization problem in is NPhard.
Proof.
See Appendix A. ∎
Iii Many to Many Matching for NOMA
As shown in , the average sumrate of the network is determined by both power allocation and subchannel allocation. We decouple the resource allocation problem into two subproblems, subchannel allocation and power allocation and develop a suboptimal solution.
There are some previous works on resource allocation problems that have utilised matching theory. A distributed spectrum access algorithm for cognitive radio relay networks that results in a stable matching was proposed in [27] and [28]. In [29], a marketing noncooperation game was applied in proposing a dynamic spectrum sharing algorithm to reach the Nash equilibrium. In [30], the authors focused on relayaided D2D communication, and proposed a distributed solution approach using stable matching to allocate radio resources. In the subchannel allocation, we assume that each source node allocates its power equally over the subchannels, and the amplification coefficient of relay over each subchannel are the same. We recognize subchannel allocation as equivalent to a manytomany twosided matching problem between the set of SD pairs and the set of subchannels, which we will explain in details in Section III.A. After the subchannel allocation is performed, each source node allocates its own transmitting power over its matched subchannels by utilizing water filling algorithm [31], then relay determines its amplification coefficient over each subchannel.
Iiia Two Sided ManytoMany Matching Problem Formulation
In the subchannel allocation process, the SD pairs prefers to access the subchannels with good quality to achieve the best service, while relay aims at maximizing the throughput of the network by arranging which SD pairs can be assigned to each subchannel. The subchannel allocation problem can be considered by relay as a matching process in which the set of subchannels and the set of SD pairs match with each other. To better describe the matching process between the SD pairs and the subchannels^{2}^{2}2The outcome of the matching is the solution for relay to allocate the subchannels as the interests of the subchannels and relay are identical., we consider the set of SD pairs and the set of subchannels as two disjoint sets of selfish and rational players aiming to maximize their own interests. Since the source and destination nodes are already paired, we can transform the matching problem between the set of SD pairs and the set of subchannels into an equivalent matching process between source nodes and subchannels for convenience.
Let denote a matching pair if is assigned to in the th time slot. To depict the influence of the resource allocation for each player, we assume that each player has preferences over the subsets of the opposite set. The preference of each source node is based on its achievable data rates, while the preference of the subchannel is determined by . Note that the subchannel allocation is performed in each time slot according to the corresponding CSI, implying that the matching process can be considered separately for each time slot. Without loss of generality, we consider the matching process in time slot . We define the utility of subchannel as the product of the source nodes¡¯ average throughput over . Given and as two subsets of source node , ’s preference over different subsets of source nodes can be written as
(12) 
which implies that prefers to because the former subset of source nodes provides a larger utility than the latter one. Given and as two subsets of subchannel , the preference of over these subsets of subchannels can be represented as
(13) 
where and are the amplification coefficient of relay and the transmitting power of each source node. Inequality implies that prefers to due to a higher channel gain.
Definition 1: A preference list is an ordered set containing all the possible subsets of the opposite set for player (). Given are the subsets of the opposite set of player , player ’s preference list representing that are player ’s potential matching pairs and .
We denote as the set of preference lists of the source nodes and subchannels, where and are the preference lists of and , respectively. We also assume that the preferences of the source nodes and subchannels are transitive. The definition of transitive is shown as follows.
Definition 2: We say the preference of is transitive if for and , we have , where is a player of the matching game. , and are the subsets of player ’s opposite set.
With the definition of transitive and preference list, we can then formulate the optimization problem as a manytomany twosided matching game.
Definition 3: Given two disjoint sets, of the source nodes, and of the subchannels, a manytomany matching is a mapping from the set into the subsets of in the th time slot such that for every , and :
Conditions and state that each source node is matched with a subset of subchannels, and each subchannel can be allocated to a subset of source nodes. Condition and implies that each subchannel can be allocated to no more than SD pairs, and each SD pair can be allocated to no more than subchannels. Condition shows that the subchannels and SD pairs are matched mutually in each time slot. We define the strategy of player as the set of the players that are matched with , i.e., . A player’s strategy is obtained from the manytomany matching process, based on the preference of each player and the matching algorithm.
The matching model above is more complicated than the conventional twosided matching models for two main reasons. Firstly, for source node , its strategy is not only determined by itself, but also affected by the strategies of other source nodes, which is called externalities. Because of the externalities, the preference list of the source node varies with the matching structure^{3}^{3}3Matching structure describes the current matching condition between the set of source nodes and the set of subchannels.. For subchannel , the utility it can obtain from the player of the opposite set is affected by because of the cochannel interference. As a result, in our model the players should be matched with any subsets of the opposite set instead of a single player. Therefore, the number of potential matching combinations can be extremely huge with the increment of the players in each set. This makes the problem quite intractable even when the power allocation is not considered. Secondly, under the conventional definition of stable matching such as that in [14], there is no guarantee that a stable matching exists even in manytoone matchings. In that case, the matching is possible to be unstable in this matching model. For these reasons, the matching process of this model is sophisticated and there is no existing matching algorithm that can solve this problem efficiently. Therefore, to solve this matching problem, we develop two extended versions of the GaleShapley algorithm [14] and propose two new matching algorithms in Section III.B and Section III.C.
IiiB Static SD pairSubchannel Matching Algorithm
In this subsection, we propose a lowcomplexity static SD pairsubchannel matching algorithm (SSDSMA). To achieve the low complexity and operability of this algorithm, we assume that all the source nodes only construct their preference lists at the beginning of the matching. The preference lists of the source nodes are static and will not be changed throughout the matching process, which is the same as the conventional GaleShapley algorithm. In this matching model, every source node makes their decisions first according to their own preference lists. In each round of proposals from source nodes, each source node proposes itself to at most one subchannel and then wait the response from the subchannels. After all the incompletely matched^{4}^{4}4A source node is incompletely matched when it is matched with less than subchannels. source nodes have proposed themselves to the subchannels, the subchannels decide whether to accept the proposing source nodes. We define it as a static matching iteration when the source nodes propose themselves to the set of subchannels and at least one proposal is accepted.
It is also assumed that each source node has no idea of other source nodes’ preference when constructing their preference lists. That is, for each source node , the potential externalities brought by other source nodes are unpredictable at the beginning of the matching. As a result, the source nodes do not consider the impact of potential cochannel interference brought by other source nodes when constructing preference lists. We can then simplify the preference list of the source nodes in SSDSMA in the following method. The subsets of can be replaced by the subchannels in in the preference lists of the source nodes. Given and as two different subchannels, can be simplified as
(14) 
We then define the static preference list of the source nodes.
Definition 4: A static preference list of a source node is an ordered set containing all the possible subchannels for source node (). Given as the subchannels to which is possible to access with, ’s static preference list represents that are ’s potential matching pairs and .
Let’s denote by the set of static preference lists of the source nodes, where is the preference lists of . However, the preference lists of the subchannels are unreducible in static matching iterations because there still exist interdependencies between the source nodes who share the same subchannel in SSDSMA. The key idea of SSDSMA is that each source node proposes itself to the most preferred subchannel which has not refused it. The subchannels then decide whether to accept these proposals by judging if it can bring benefits to the itself. When all the proposed subchannels have responded to the proposing source nodes, this static matching iteration is performed and the source nodes will check if it is necessary to perform the next static matching iteration. We now describe how the subchannels choose the proposing source nodes by introducing the concept of blocking pair.
Definition 5: Given a matching and a pair with ) and . , is a blocking pair if . . .
The existence of blocking pair has the following necessary conditions. Firstly, and have never been matched with each other. That is to say, have not proposed itself to before, and is still in ’s preference list. Secondly, the matching of and can increase both of their utility. A matching is blocked by when both side of the players prefers to be matched with each other.
With the definition above, we can describe the strategy of each subchannel as below. To reduce the complexity, the subchannels do not construct the whole preference lists in advance, as will be analyzed in Section III.E. When receives the proposal from source node , and form a blocking pair if can provide a higher sumrate than over . Under this condition, will accept the proposal from . However, there is a special case when the subchannel has already been paired with source nodes before accepting the proposal from . In that case, has to give up one of the matched source nodes in the subset . The subchannel then calculates every possible where and . Afterwards, chooses to match with . Suppose that gives up the matching with , will not form a blocking pair any longer because will not propose itself to again. The process of the proposed static SD pairsubchannel matching algorithm is to find and eliminate the potential blocking pairs.
We now describe the whole process of SSDSMA. The specific details of the proposed SSDSMA are described in Table I, consisting of an initialization phase and a matching phase.
In the initialization phase, each source node calculates the rate that every subchannel can provide and then constructs its static preference list in the order of the corresponding rates. In the matching phase, each source node that has been paired with less than subchannels proposes itself to the most preferred subchannel in its static preference list if there is any, and remove the subchannel from its static preference list. After all the proposing source nodes have proposed themselves to the set of subchannel, the subchannels that have received proposals will decide whether to accept the proposals of the source nodes or not. The subchannel will accept the proposal from a source node if it can increase its throughput over itself by matching with this source node. If the subchannel has already matched with source node before accepting the current proposal, it will unmatch with one of the matched source nodes that causes the minimum loss of throughput over it. After all the proposing source nodes have received the responses from the corresponding subchannels, they will check if they are still willing to make any proposals. A source node will make a proposal when it is matched with less than subchannels and still has a nonempty static preference list. If any source node wants to make a new proposal, another static matching iteration will be performed. The SSDSMA terminates when no source node would like to make new proposals.

However, in the matching process, the changes of the source nodes’ strategies lead to a dynamic cochannel interference for each source node. As a result, the preference of the source nodes is likely to be changed because of externalities. To adjust the SD pairsubchannel matching with externalities, we will introduce the novel dynamic SD pairsubchannel matching algorithm in the following subsection.
IiiC Dynamic SD pairSubchannel Matching Algorithm
In this subsection, to develop a subchannel allocation scheme that fully depicts the interaction between the SD pairs caused by cochannel interference, we present the dynamic SD pairsubchannel matching algorithm (DSDSMA). Different from the SSDSMA, the preference lists of the source nodes are adjusted dynamically according to the current matching structure in DSDSMA. The DSDSMA contains a sequence of SSDSMA iterations, and we do not use a fixed static preference list for each source node throughout the DSDSMA process. In each round of the SSDSMA iteration, it is necessary for each source node to adjust its static preference list. Because when SSDSMA is performed, the strategies of some source nodes will be changed, and the cochannel interference over each subchannel is also changed respectively.
The preference lists of the source nodes are adjusted dynamically after a SSDSMA iteration is performed. In DSDSMA, the set of static preference list is extended to different sets of static preference lists, denoted as , where is the index of SSDSMA iteration in the DSDSMA. The static preference lists of the source nodes in the th SSDSMA iteration is given by . For example, means that in the th SSDSMA iteration, the preference relation satisfies . After a SSDSMA iteration is performed, the matching structure of the network may have been changed and ’s preference over the set of subchannels may be different. Player then constructs a new static preference list according to the current matching structure before the th SSDSMA iteration. The difference between and is caused by the signaltosignal interference of the source nodes in the th SSDSMA iteration.
To reduce the complexity of the DSDSMA, we try to avoid the repeated matching proposals in different SSDSMA iterations. For a source node , there may exist some subchannels that never accept source node ’s proposal in the current matching structure, which have been proved in the past matching iterations. Hence we define the concept of forbidden pair for the source nodes as follows.
Definition 6: For and , if and , is a forbidden pair for over . It is denoted as .
In this matching model, the strategies of source nodes affect the decision of the subchannels. For different matching structures, a subchannel may make different decisions on the same proposal. For example, will accept ’s proposal when , while may refuse ’s proposal when , if the cochannel interference brought by reduces ’s utility. However, the decision of a subchannel over the same proposal is always the same when the strategies of the source nodes are fixed. The definition of forbidden pair shows that the proposal of a source node will not be accepted by a subchannel if it has been refused with the same strategies of the source nodes before. Each time a proposal of the source node is refused, the current matching structure will be recorded in its forbidden pair, and it will not propose itself to the same subchannel under the same matching structure.
We then describe the whole process of DSDSMA to solve the subchannel matching problem with externalities. In each iteration of SSDSMA, the source nodes will amend their static preference lists with the current matching structure and refresh their forbidden pairs each time they are refused by a subchannel. After one SSDSMA matching process is performed, it will figure out if the DSDSMA is over, if not, it will return to the beginning of the SSDSMA and start the next iteration.
Table IV shows the detailed steps of the DSDSMA. The DSDSMA starts with the process of SSDSMA, where the source nodes construct their static preference lists of the first SSDSMA matching iteration. Then the source nodes that have been matched with less than subchannels propose themselves to the most preferred subchannels in their current nonempty static preference lists, and remove the corresponding subchannels from their current static preference lists. The subchannels then decide whether to accept the proposals or not according to the criteria of SSDSMA. If a proposal from source node is refused by a subchannel , will add the current matching of into its forbidden pair. The process of SSDSMA completes when no source node is willing to make any proposal with their current static preference lists.
In the next SSDSMA matching iteration, source nodes first figure out if there exist any forbidden pairs with the current matching structure. If there is any, the corresponding subchannel in the forbidden pairs will not be listed in the static preference list of the source node in the following SSDSMA matching iteration. The set of source nodes then construct their static preference lists of the next SSDSMA matching iteration according to the current strategies of the source nodes, and the next SSDSMA matching process is performed. The DSDSMA process completes when the SSDSMA matching process performs only one static matching iteration. That is, no source node proposes itself to any subchannel in the first static matching iteration of the SSDSMA matching process.

IiiD Water Filling Power Allocation
Power allocation can be implemented after the SD pairsubchannel matching. We divide the power allocation into two phases. In the first phase, the transmitting power of source nodes is allocated through the water filling algorithm, which can be presented as
(15) 
where
(16) 
is the water filling level of over , and is the set of subchannels allocated to .
In the second phase, relay allocates its amplification coefficient over different subchannels. We assume that the maximum power that relay allocates to each subchannel is identical, i.e., . To maximize the sum data rate, relay provides the maximum power level over every subchannel, so that the amplification coefficient can be given by
(17) 
IiiE Stability, Convergence and Complexity
IiiE1 Stability and Convergence
With the definition of blocking pair and the transitive preference list explained above, we then introduce the conception of pairwisestability as below and prove that the proposed SSDSMA and DSDSMA both converge to a pairwise stable matching.
Definition 6: A matching is defined as pairwise stable if it is not blocked by any pair which does not exist in .
Lemma 1: If the proposed SSDSMA converges to a matching , then is a pairwise stable matching.
Proof.
If is not a pairwise stable matching, it means that there exists a pair , such that , , , and , . According to the the proposed static SD pairsubchannel matching algorithm, must have proposed itself to before since it can provide a higher utility than . We assume that eliminate in the th static matching iteration, denoted as While only accepts the proposals that provide a larger benefit, we have . Finally, we have , , and , which is contradictory to the transitive property of the preference list. Hence, lemma 1 is proved. ∎
Lemma 2: If the proposed DSDSMA converges to a matching , then is a pairwise stable matching.
Proof.
If is not a pairwise stable matching, it means that there exists a pair , such that , , , and , . According to the the proposed DSDSMA, will propose itself to in the following SSDSMA matching process since it can provide a higher utility than . The only possibility that is a pairwise stable matching is that is a forbidden pair of over , which means that has been refused by under the same matching structure earlier in this DSDSMA. The reason refused under this condition is that has a larger product of average throughput than . It can be denoted as , , which is contradictory to the assumption. Hence, lemma 2 is proved. ∎
Theorem 2: The proposed SSDSMA converges to a pairwise stable matching after a limited number of static matching iterations in each time slot.
Proof.
As shown in Table I, in each iteration, every source node will propose itself to the mostpreferred subchannel in its static preference list. No matter the proposal is accepted or not, the source node will remove this subchannel from its static preference list and will not propose itself to this subchannel again. As the matching goes on, the potential choices for each source node keeps decreasing. So the number of iterations is no more than , where is the number of subchannels, and the proposed SSDSMA will converge within iterations. According to Lemma 1, the proposed SSDSMA converges to a pairwise stable matching. ∎
Theorem 3: The proposed DSDSMA converges to a pairwise stable matching after a limited number of iterations in each time slot.
Proof.
In the process of DSDSMA, each source node is possible to propose itself to the same subchannel in different SSDSMA matching process. However, with the definition of forbidden pair, when each time a source node proposes itself to a subchannel, there are only two possibilities. One possibility is that the subchannel accepts the proposal and does not disconnect any matched pairs, so that the matched pair of SD pairsubchannel increases. The other possibility is that the subchannel rejects the matching with a source node, and the disconnected source node will add the current matching into its forbidden pair. As the matching goes on, the total number of the matched pair and forbidden pair of each source node over the subchannels keeps increasing. For each source node, there is no more than forbidden pairs over each subchannel, where is the number of source nodes, is the maximum number of SD pairs that can have access to the same subchannel, so the total number of forbidden pair is limited. Because each subchannel can be matched with no more than source nodes, the total number of matched pair is also limited. As a result, the total number of matched pair and forbidden pair is limited. The process of DSDSMA completes with a finite times’ proposals. According to Lemma 2, the proposed DSDSMA converges to a pairwise stable matching. ∎
IiiE2 Complexity
In this part we calculate and compare the complexity of the proposed SSDSMA, DSDSMA, and the optimal exhaustive search, to analyse the feasibility of each matching algorithm.
Theorem 4: The complexity of the optimal exhaustive search is . The iteration number of SSDSMA is , and the complexity of SSDSMA is .
Proof.
For the optimal exhaustive search, relay exhaustively searches the best subset of SD pairs over every subchannel. Since every source node and subchannel can be paired with each other, there exists possible combinations and the complexity of the optimal exhaustive search is .
For the SSDSMA, it contains two phases: the initialization phase and the matching phase. In the initialization phase, every source node constructs its own static preference list. The initialization of each preference list is considered as a sorting problem with the complexity of , and the total complexity of the initialization phase is . ^{5}^{5}5If we also construct the preference list of the subchannels, it can be proved that there is an extra complexity of . That is why we do not construct the whole preference list of the subchannels at once. In the matching phase, the number of iterations is no more than , and in each iteration, at most source nodes make proposals, so the complexity is . The total complexity of the proposed SSDSMA is . ∎
Theorem 5: The complexity of DSDSMA in each iteration is . The upper bound of proposal number in DSDSMA is .
DSDSMA contains a sequence of SSDSMA iterations. In each iteration, the SD pairs adjust their preference lists and propose the matching phase of SSDSMA. The complexity of DSDSMA in each iteration equals to the complexity of SSDSMA, that is, . However, the strict upper bound for the number of iteration in DSDSMA is hard to obtain, and the reason is given as following. The number of iteration is determined by the relation ship between the adjusted preference list and the matched subchannels for each SD pair. However, both of the two factors for a SD pair are affected by any other SD pairs. The solution for the strict complexity of DSDSMA is impeded by the complicated externalities in the network.
To evaluate the complexity of DSDSMA, we give the upper bound of total proposal number in DSDSMA. As shown in Theorem 3, each time a SD pair propose itself to a subchannel, either the total connection number increases or the number of forbidden pair increases. The maximum number of total connection is , and the maximum number of total forbidden pair is . The upper bound of proposal number in DSDSMA is .
The whole process of the twosided resource allocation problem for the NOMA relay network is shown in Table III.

Iv Simulation Results
In this section, we evaluate the performance of the proposed SSDSMA and DSDSMA in NOMA scheme considering proportional fair. We compare the proposed algorithms with the OFDMA scheme and the optimal exhaustive search. In the OFDMA scheme, each subchannel can only be allocated to one SD pair at one time, while one SD pair may have access to multiple subchannels. For the optimal exhaustive search, each source node and each subchannel can be paired with any number of players from the opposite side as long as they want.
In the simulation, most of the parameters are set based on the existing LTE/LTEAdvanced specifications [32] [33]. The radio resource allocation is updated every 1 ms, and the user throughput averaged over 10 ms is measured. We set and as 8 and 3 respectively and all curves are generated by averaging over 1000 instances of the algorithms. Table IV is the parameters of the simulation.
Parameter  Value 

cell range  200m square 
number of subchannels (K)  10 
number of SD pairs (N)  5 to 50 
peak power of source node  46 dBm 
peak power of relay  86 dBm 
noise  additive white Gaussian noise 
noise variance ()  174 dBm 
path loss coefficient  3.76 
center frequency  2GHz 
bandwidth  4.5MHz 
scheduling interval  1 ms 
averaging interval of user throughput  10 ms 
8  
3  
scheduler  proportional fairness 
fading  Rayleigh fading 
Fig.2 (a) shows the number of proposals vs. CDF of the number of proposals with 5, 15 and 25 SD pairs in different matching algorithms. It is illustrated that the number of proposal increases with the increment of the SD pair number. The CDF curves for SSDSMA is steeper than DSDSMA, which means that the number of static iterations in SSDSMA is more stable than DSDSMA. It is also shown that the number of iteration for SSDSMA is smaller than DSDSMA with the same number of SD pairs. The difference between two algorithms becomes larger with more SD pairs, which is determined by the nature of the algorithms. We also give the theoretical curve for the largest proposal number of SSDSMA in the matching phase, which agree with our simulation curve.
In Fig.2 (b), we simulate the number of SD pairs vs. the running time with different matching algorithms. The running time of each matching algorithm reflects its complexity and is shown in exponential form. The running time of SSDSMA and DSDSMA grow gradually with the increment of SD pair, and the running time of DSDSMA is always a little bit larger than that of the SSDSMA. The running time of the exhaustive search increase rapidly with the number of SD pairs, and is much larger than the proposed algorithms, which further proved its infeasibility.
Fig.3 illustrates the relation between average sumrate and the number of SD pairs and shows the comparison of average sumrate of the proposed NOMA schemes, OFDMA scheme, and the optimal exhaustive search. As proved in Section III.E, the complexity of the optimal exhaustive search increases exponentially with the number of SD pairs and subchannels. To get the simulation result of the optimal exhaustive search in a regular time, we have to decrease the number of SD pairs and subchannels in the simulation. Here we set the number of subchannels as 3, and the number of SD pairs is reduced to 312, while other parameters are still the same as in Table IV. The curve of optimal exhaustive search is generated based on averaging over 100 instances of the algorithms.
The performance of the proposed DSDSMA and SSDSMA in NOMA scheme outperform the OFDMA scheme significantly because it achieves a more efficient utilization of spectrum resource. The average sumrate increases as the number of SD pair grows, and the growth becomes slower as turns larger because of the saturation of channel capacity. But when the number of SD pairs is much larger than the number of subchannels, the average sumrate continues to increase at a low speed due to the multiuser gain. When comparing the SSDSMA with DSDSMA, we can observe that with the increment of SD pairs, the advantage of DSDSMA over SSDSMA in the average sumrate of the network becomes larger. That is because when the network gets more crowded, the impact of the externalities will be more significant. DSDSMA can adjust its matching based on the externalities and provide a larger average sumrate.
It can be noted the DSDSMA can reach almost of the average sumrate of the optimal exhaustive search when there are only 3 SD pairs in the network. The average sumrate of the SSDSMA is close to that of the DSDSMA and the optimal exhaustive search with 3 SD pairs in the network. However, the performance gap of the proposed algorithms to the optimal exhaustive search increases with the increment of SD pairs. It makes sense since the complexity of the optimal exhaustive search is significantly more than that of the proposed algorithms, as the number of SD pair increases. It can also be noted that the proposed algorithm can always achieves a significantly higher sum rate compared to the OFDMA scheme.
Fig.4 depicts the number of scheduled SD pairs vs. the number of SD pairs in different matching structures. The scheduled SD pairs in OFDMA scheme is no more than the number of subchannels since each subchannel can be matched with no more than one SD pair. In NOMA scheme, the number of scheduled SD pair increases almost linearly when there are less than SD pairs, because there are plenty of spare spectrum resources. When increases, the number of scheduled SD pair becomes saturated and increases slowly due to the multiuser gain. SSDSMA slightly outperforms DSDSMA in terms of the number of scheduled SD pairs when the network gets crowded.
Fig.5 shows the number of SD pairs vs. the average rate of each celledge SD pair in the network in different matching structures. We define the total distance of a SD pair as the distance between the source node and relay plus the distance between relay and the destination node. The celledge SD pairs are those with a total distance of more than 160 m. In the matching with the proposed algorithms, the average rate of celledge SD pairs outperform the OFDMA scheme obviously. That is because in each time slot, there are more access SD pairs in NOMA scheme than in OFDMA scheme, and the cell edge users have a larger chance to access the channels in NOMA scheme. The cell edge users have a relatively high average rate when there are only a few SD pairs in the network. The average rate decreases rapidly as the number of SD pairs becomes larger because of the limited spectrum resource. The average rate of the celledge SD pairs turn to be stable in a crowded network. The only difference between SSDSMA and DSDSMA lies in the middle of the curve, where the average rate of the cell edge SD pair in DSDSMA is slightly lower than that in the SSDSMA. In OFDMA scheme, the average rate of celledge SD pair tend to be very low when there are over 10 SD pairs in the network due to its small number of accessed SD pairs.
Fig. 6 shows the average sumrate vs. the number of SD pairs for different and in the proposed DSDSMA. When is fixed, it can be seen that the average sumrate is higher with a larger . With fixed, the average sumrate also increases with . When increases, the marginal increment of the average sumrate becomes smaller. Due to the increased intersignal interference, it is more difficult for a new coming SD pair to further enhance the average sumrate over the subchannels. As increases, the marginal increment of the average sumrate also becomes smaller as the possibility that a matching is denied by the restriction of decreases.
V Conclusion
In this paper, we studied the resource allocation problem in a NOMA wireless network with a oneway OFDM AF relay by optimizing the subchannel assignment and the power allocation. By formulating the problem as a manytomany twosided matching problem, we proposed two near optimal SD pairsubchannel matching algorithms in which the SD pairs and subchannels can be matched and converge to a stable matching. The average sumrate of the proposed SSDSMA and DSDSMA in NOMA scheme are higher than that of the conventional OFDMA network and close to that of optimal exhaustive search. The proposed matching algorithms can serve more users and provide better service to the celledge users when compared with the conventional OFDMA scheme. The proposed SSDSMA has a lower complexity while DSDSMA performs better in average sumrate of the whole network.
Appendix A
Proof of Theorem 1. The sumrate maximization problem in (11a) is NPhard.
Proof.
The proof of this theorem can be divided into two parts, and .
When , becomes an joint power and subchannel allocation problem in the traditional OFDMA system, which has been proved to be NPhard in [34].
When , we proof that the problem is NPhard even when we omit the power allocation problem and allocate the power equally over each subchannel. Since the 3dimensional matching problem (3DM problem) has been proven to be NPcomplete in [35], we try to construct a instance which can be proved to be equal with a 3DM problem. When the decision problem of this specific instance is proved to be NPcomplete, the instance of with equal power allocation is an NPhard problem [36].
We construct a instance where and . Suppose that relay R allocates the power equally over each subchannel for the accessed SD pairs, and the SD pairs are separated into two disjoint sets and such that , is the whole set of SD pairs , and and . For each subchannel, we assume that it is allocated to one SD pair from and the other from . Let be a collection of ordered triples , where = According to , the sumrate of any triple can be set as . We need to determine whether there exists a set so that = , , where any and do not contain the same components. For , it is a 3DM if the followings hold: = ; For any two distinct triples, and , we have . If we set to an infinite negative, the problem we formed will be reduced to a 3DM decision problem. Therefore, the decision problem of this instance is NPcomplete, and the corresponding instance is NPhard. Since a special case of is proved to be NPhard, the sumrate maximization problem in is NPhard. ∎
References
 [1] S. Zhang, B. Di, L. Song, and Y. Li, “Radio Resource Allocation for Nonorthogonal Multiple Access (NOMA) Relay Network Using Matching Game,” in Proc. IEEE Int. Commun. Conf., pp. 16, Kuala Lumpur, May 2016.
 [2] L. Dai, B. Wang, Y. Yuan, S. Han, C. I, and Z. Wang, “Nonorthogonal Multiple Access for 5G: Solutions, Challenges, Opportunities, and Future Research Trends,” IEEE Commun. Mag., vol. 53, no. 9, pp. 7481, Sep. 2015.
 [3] J. Thompson, X. Ge, H. Wu, R. Irmer, H. Jiang, G. Fettweis, and S. Alamouti, “5G Wireless Commnucation Systems: Prospects and Challenges,” IEEE Commun. Mag., vol. 52, no. 2, pp. 6264, Feb. 2014.
 [4] Y. Yuan, Z. Yuan, G. Yu, CH. Hwang, PK. Liao, A. Li, and K. Takeda, “Nonorthogonal transmission technology in LTE evolution,” IEEE Commun. Mag., vol. 54, no. 7, pp. 6874, Jul. 2016.
 [5] K. Higuchi and A. Benjebbour, “Nonorthogonal Multiple Access(NOMA) with Successive Interference Cancellation for Future Radio Access,” IEICE Trans. Commun., vol. E98B, no. 3, Mar. 2015.
 [6] J. Liberti, S. Moshavi, and P. Zablocky, “Successive interference cancellation,” U.S. Patent 8670418 B2, Mar. 2014.
 [7] C. Yan, A. Harada, A. Benjebbour, Y. Lan, A. Li, and H. Jiang, “Receiver Design for Downlink NonOrthogonal Multiple Access (NOMA),” in Proc. IEEE Veh. Tech. Conf., pp. 16, Glasgow, May 2015.
 [8] B. Di, S. Bayat, L. Song, and Y. Li, “Radio Resource Allocation for Downlink NonOrthogonal Multiple Access (NOMA) Networks using Matching Theory,” in Proc. IEEE Global Commun. Conf., pp. 16, San Diego, Dec. 2015.
 [9] J. Men and J. Ge, “NonOrthogonal Multiple Access for Multipleantenna Relaying Networks,” IEEE Commun. Lett., vol. 19, no. 10, pp. 16861689, Oct. 2015.
 [10] A. Mohamad, R. Visoz, and A. O. Berthet, “Code Design for MultipleAccess MultipleRelay Wireless Channels with NonOrthogonal Transmission,” in Proc. IEEE Int. Conf. Commun., pp. 23182324, London, Jun. 2015.
 [11] Y. Liu, Z. Ding, M. Elkashlan, and H. V. Poor, “Cooperative NonOrthogonal Multiple Access with Simultaneous Wireless Information and Power Transfer,” IEEE J. Sel. Areas Commun., vol. 34, no. 4, pp. 938953, Mar. 2016.
 [12] Z. Zhang, Z. Ma, M. Xiao, Z. Ding, and P. Fan, “Full Duplex DevicetoDevice Aided Cooperative NonOrthogonal Multiple Access,” IEEE Trans. Veh. Tech. vol. pp, no. 99. pp. 11. Aug. 2016.
 [13] Z. Ding, H. Dai, and H. V. Poor, “Relay Selection for Cooperative NOMA,” IEEE Wireless Commun. Lett., vol. 5, no. 4, pp. 416419, Jun. 2016.
 [14] A. Roth and M. Sotomayor, “TwoSided Matching: A Study in Game Theoretic Modeling and Analysis,” Cambridge, UK: Cambridge Univ. Press, 1992.
 [15] M. Pycia and M. B. Yenmez, “Matching with Externalities,” Socail Science Electronic Publishing, May, 2015.
 [16] R. H. Gohary, Y. Huang, Z. Q. Luo, and J. S. Pang, “A Generalized Iterative WaterFilling Algorithm for Distributed Power Control in the Presence of a Jammer,” IEEE Trans. Signal Process., vol. 57, no. 7, pp. 26602674, Feb. 2009.
 [17] Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, and K. Higuchi, “Nonorthogonal Multiple Access (NOMA) for Cellular Future Radio Access,” in Proc. IEEE Veh. Tech. Conf., pp. 15, Dresden, Jun. 2013.
 [18] S. Bayat, R. Louie, Z. Han, B. Vucetic and Y. Li, “Distributed User Association and Femtocell Allocation in Heterogeneous Wireless Networks,” IEEE Trans. Commun., pp. 30273043, vol. 62, no. 8, Aug. 2014.
 [19] H. Q. Ngo, A. Ashikhmin, H. Yang, E. G. Larsson, and T. L. Marzetta, “Cellfree massive MIMO versus small cells,” IEEE Trans. Wireless Commun., 2016, submitted.
 [20] P. Liu, S. Jin, T. Jiang, Q. Zhang, and M. Matthaiou, “Pilot power allocation through user grouping in multicell massive MIMO systems,” IEEE Trans. Commun., 2016, submitted.
 [21] J. Umehara, Y. Kishiyama, and K. Higuchi, “Enhancing User Fairness in Nonorthogonal Access with Successive Interference Cancellation for Cellular Downlink,” in Proc. Inter. Conf. Commun. System, pp. 15, Omaha, Nov. 2012.
 [22] D. Tse, “Downlink AWGN Channel” in Fundamentals of Wireless Communication, Cambridge, UK: Cambridge Univ. Press, 2005, ch. 6, sec. 2, pp. 238242.
 [23] S. Wang, L. Chen, Y. Yang, and G. Wei, “Quantization in Uplink MultiCell Processing with Fixedorder Successive Interference Cancellation Scheme under Backhaul Constraint,” in Proc. IEEE Veh. Tech. Conf., pp. 15, Seoul, May 2014.
 [24] N. Otao, Y. Kishiyama, and K. Higuchi, “Performance of nonorthogonal access with SIC in cellular downlink using proportional fairbased resource allocation,” in Proc. IEEE Int. Symp. Wireless Commun. System, pp. 476480, Paris, Aug. 2012.
 [25] M. Kountouris and D. Gesbert, “Memorybased opportunistic multiuser beamforming,” in Proc. IEEE Int. Symp. Inform. Theory, pp. 14261430, Adelaide, Sept. 2005.
 [26] L. Wolsey and G. Nemhauser, “Integer and Combinatorial Optimization,” John Wiley Sons, USA, 2014.
 [27] S. Bayat, R. Louie, B. Vucetic, and Y. Li, “Dynamic decentralised algorithms for cognitive radio relay networks with multiple primary and secondary users utilising matching theory,” Trans. Emerging Telecommun. Technol., vol. 24, no. 5, pp. 486502, May, 2013.
 [28] S. Bayat, R. Louie, Y. Li and B. Vucetic, “Cognitive radio relay networks with multiple primary and secondary users: distributed stable matching algorithms for spectrum access,” in Proc. IEEE Int. Conf. Commun., pp. 16, Kyoto, Jun. 2011.
 [29] P. Lin, J. Jia, Q. Zhang, and M. Hamdi, “Dynamic Spectrum Sharing With Multiple Primary and Secondary Users,” IEEE Trans. Veh. Tech., vol. 60, no. 4, pp. 17561765, Mar. 2011.
 [30] M. Hasan, and E. Hossain, “Distributed Resource Allocation for RelayAided DevicetoDevice Communication Under Channel Uncertainties: A Stable Matching Approach,” IEEE Trans. Commun., vol. 63, no. 10, pp. 38823897, Aug. 2015.
 [31] W. Yu, W. Yhee, S. Boyd, and J. Cioffi, “Iterative WaterFilling for Gaussian Vector MultipleAccess Channel,” IEEE Trans. Inf. Theory, pp. 322, vol. 50, no. 1, Jan. 2004.
 [32] 3GPP TR 25.996, “Spatial channel model for Multiple Input Multiple Output (MIMO) simulations,” Release 12, Sept. 2014.
 [33] 3GPP TS 36.213, “Evolved Universal Terrestrial Radio Access (EUTRA) Physical Layer Procedures,” Release 12, Sept. 2014.
 [34] Y. Liu and Y. Dai, “On the Complexity of Joint Subcarrier and Power Allocation for Multiuser OFDMA Systems,” IEEE Tran. Signal Process., vol. 62, no. 3, pp. 583596, Feb. 2014.
 [35] V. Kann, “Maximum Bounded 3dimensional Matching is MAX SNPcomplete,” IEEE Info. Process. Lett., vol. 37, no. 1, pp. 2735, Jan. 1991.
 [36] M. Sipser, “Introduction to the Theory of Computation,” Cengage Learning, USA, 2012.