Fundamentals of Power Allocation Strategies for Downlink Multiuser NOMA with Target Rates
Abstract
For a downlink multiuser nonorthogonal multiple access (NOMA) system with successive interference cancellation receivers, the fundamental relationship between the user channel gains, target rates, and power allocation is investigated. It is shown that the power allocation strategy is completely derived as functions of the users’ target rates and the successive decoding order, and not of the channel gains, so long as the target rates have the same ordering as the user channel gains. It is proven that the total irremovable interference at each receiver has a fundamental upperlimit that is a function of the target rates, and the outage probability is one when this limit is exceeded. The concept of wellbehaved power allocation strategies is defined and its properties are derived. It is shown that any power allocation strategy that has outage performance equal to orthogonal multiple access (OMA) is always wellbehaved. Furthermore, it is proven that there always exists a wellbehaved power allocation strategy that has improved outage probability performance over OMA. A specific wellbehaved strategy is derived to demonstrate the improvement of NOMA over OMA for users. Simulations validate the theoretical results, and show that the best performance gains are obtained by the users with weaker channels.
I Introduction
Due to the rapidly increasing demand for higher datarates, more connected users and devices, and diversity of deployments, powerdomain nonorthogonal multiple access (NOMA) is being sought to help improve the capacity and user multiplexing of downlink and uplink cellular systems [1]. The 3rd Generation Partnership Project has already begun the study items for both downlink NOMA for LTEAdvance [2], and for uplink NOMA Powerdomain NOMA in New Radio (NR) [3]. With the attention that NOMA receives from the academic, private, and standard sectors, it is only a matter of time before NOMA is implemented in a future standard release.
Powerdomain NOMA takes advantage of the superposition coding (SC) and disparate transmit powers for signals at the transmitter, and employs a successive interference cancellation (SIC) enabled receiver in order to successively remove interference according to the ordered received powers of the signals. In the downlink, the basestation (BS) transmits the signals to multiple users over a common transmission period and bandwidth. Each user then obtains its own signal by employing SIC to decode and remove the interference of signals with greater power and lower datarates than its own. This is in contrast to orthogonal multiple access (OMA), which assigns nonoverlapping time slots (or frequency subbands) to each user in order to avoid interference. Since channel capacity increases linearly with time and frequency, and only logarithmically with transmit power, NOMA can outperform OMA in terms of achieving higher data rates [4].
Although it is proven in [11] that there always exists a power allocation approach for NOMA that can outperform OMA for the general multiuser NOMA case in terms of information capacity, this power allocation strategy relies on having perfect channel state information at the transmitter, which is not a realistic assumption in practical systems. It is more realistic that, based on channel state information (CSI) and target datarates provided to the BS by the users, the BS selects the remaining transmission parameters, which are then conveyed to the user immediately before the data is transmitted via a control channel. Therefore, it is important for a DL NOMA system to leverage this type of information when selecting the power allocation strategy for the multiple users, and it is beneficial to find a power allocation strategy that allows the system to always outperform the legacy OMA approach.
In this work, it is demonstrated that for users, the transmit power allocation strategy is completely derived with respect to the associated target data rates. It is subsequently proven that the power allocation strategy is such that the received interference coefficient for each signal must be below a certain threshold in order to avoid certain outages. A key property of finding a suitable power allocation strategy is then outlined and proven to satisfy the interference requirement, and such strategies possessing this property are referred to as wellbehaved. It is then proven that there always exists a power allocations strategy such that all users will have NOMA outage probability performance equal to that of OMA, and that such a strategy is always wellbehaved. Finally, the approach to derive a wellbehaved power allocation strategy such that a user can achieve better outage probability performance with NOMA over OMA is outlined.
Ii Previous Work and Current Contribution
The outage probability of NOMA was investigated in [6], where multiple users transmit simultaneously to multiple receivers, and it was shown the outage probability is improved when NOMA is combined with HARQ vs OMA with HARQ. However, this work uses a uniformly flat power allocation approach. The authors in [7] are the first work to show that the power allocation and interference coefficients of each user are fundamentally dependent on the particular user’s required rate, and thus the wrong selection of coefficients can lead to an outage with probability equal to 1. The usage of NOMA in a cogintiveradio inspired approach was mentioned in [8], where a user with weak channel condition is seen as the primary user and is provided as much power as needed in order to achieve its minimum rate, and the user with stronger channel is treated as the secondary user and receives any remaining power not allocated to the weaker user, and the outage probability of both is shown to clearly depend on pairing users with stronger channels.
A couple of works have focused on utilizing the rate achieved using OMA as the minimum rate required by a NOMA user, leading to a focus on finding the power allocation coefficients that achieve this condition, and demonstrate the outage performance for NOMA outperforms the outage performance of OMA. The region of power allocation coefficients that allow NOMA to outperform OMA in the downlink is first defined for the twouser case in [9]. The authors in [10] then use a power allocation approach in this region to analyze the outage performance and diversity orders of two paired users, according to their relative channel gains, and extend the work to the uplink case. In [11], the power allocation coefficients for a multiuser NOMA system that always outperforms OMA are proven to always have a sum less than or equal to 1, and hence show that a valid power allocation strategy for NOMA always exists that outperforms OMA in terms of capacity, while using less power than OMA.
More recent works have focused on optimizing the power allocation strategy. The authors in [14] propose a joint optimization of user pairing and power allocation by optimizing a cost function dependent on the instaneous achievable rates and a metric based on proportional fairness. The scheduling and power allocation algorithm that solves the optimization problem is compared to the fractional transmit power control algorithm and shown to improve performance for the user with stronger channel, while performance is not always improved for that of the weaker channel user. In [15], the authors seek to optimize the sumrate of a multiuser downlink NOMA system by using a constraint based on the total power allocated to the signals at each SIC stage, and its relation to the minimum required rate for that signal to be decoded. The authors in [16] study several algorithms that solve the NOMA power allocation problem, and point out that not many existing works had considered the strict constraint for the power allocations to follow the order of SIC decoding in their algorithms. They propose to incorporate the matching algorithm and optimum power allocation, and found that the constraint has a significant impact on the power allocation solution, which also yields superior performance over existing schemes. In [17], the authors propose a joint resource (bandwidth) and power allocation approach that optimizes a cost function that is an affine function of the power allocations and bandwidths. It is demonstrated that this algorithm can outperform the approach of simply optimizing the power allocations with fixed bandwidths, as well as the baseline OMA approach. In [18], the authors derive a weighted sumrate maximization algorithm to find the power allocation per subcarrier for a pair of users, and show that the performance of their approach improves as the diversity order of the system increases. In [19], the authors study the power allocation approach for multitiered cellular networks with cellcenter UEs and a celledge UE that is eligible for coordinated multipoint (CoMP) transmission. A joint power optimization algorithm is formulated, including target rates for each UE, and due to the prohibitive complexity, a distributed power optimization problem is formulated and shown to exhibit near optimum performance. The constraint on the power allocation coefficients relies on a linear function derived from the SINR for each SIC stage.
In the previous works, the power allocation constraints either do not consider the necessary requirements for successful SIC performance, or use constraints that do not give the fundamental relationship between power allocation and outage, such that an outage event is certainly avoided. In other words, the target rates and power allocation required to ensure whether successful SIC performance is even feasible at each stage of SIC decoding for each user is not directly considered in the works above, and this can cause unnecessary unavoidable outages to occur for multiple users. In fact, this phenomenon was described and demonsrated in [20] for twouser cacheaided NOMA systems, and in [21] for multiuser downlink NOMA systems with QoS constraint.
The main contribution of this work is to provide a comprehensive theoretical treatment of the power allocation strategy, and how it is related to the decoding order and the associated target rates of the users. In particular, this work provides the following:

The fundamental maximum irremovable interference that a particular signal can tolerate before its outage probability is equal to 1, regardless of how strong the channel SNR gains are.

The definition of a wellbehaved power allocation strategy, which causes the outage thresholds to be lesser for the signals of users that are earlier in the SIC decoding order. This condition is then shown to always constrain the irremovable interference to an acceptable value.

The baseline power allocation strategy that achieves outage performance equal to that of OMA is then derived in closedform, and is used to then derive the conditions for acquiring a power allocation strategy where all users have superior NOMA outage performance over OMA. With these conditions, the exact approach for increasing the power allocation beyond the baseline strategy is outlined in detail, and a quick example of a power allocation strategy that satisfies all of these conditions is presented along with its performance.
The necessity of such results in further studies regarding NOMA is clear, in the sense that when performing numerical studies of different algorithms for complex cellular deployments, the search for appropriate power allocation strategies can be restricted to avoid the strategies that in certain scenarios may have certain outages. In other words, the search space for the multiuser power allocation strategies can be greatly reduced with the knowledge that a large portion of the search space leads to certain outages for a subset of the users.
Iii System Model
Consider a wireless downlink system serving users. The basestation will transmit signals, each containing the information for one of the users. Let the signal for user be , , , and are transmitted with transmit SNR through a wireless channel with SNR gain . The channel gains of users are ordered as .
In the case of OMA, the received signal at user is given by
(1) 
where is the receiver thermal noise. Since user is given of the time (frequency) for its orthogonal slot (subband), the capacity of user using OMA is then given by
(2) 
For the NOMA system, user has power allocation coefficient , such that . The received signal at user is
(3) 
Using successive interference cancellation, the receiver at user will decode the messages of users in ascending order, starting with . Therefore, user will perform SIC on the signals of user , which have the form
(4) 
until it can obtain the intended signal at user given by
(5) 
where are the signals that are sent with messages encoded at rates beyond the capacity region of user , and thus are treated as interference.
Given the order of the channel gains, each user will decode the messages intended for each user before decoding its own message. Let the power allocation coefficients be , then each user has capacity
(6) 
and user has capacity
(7) 
Meanwhile, for each user to achieve its capacity, it must have the capacity to decode the messages sent to all users , and subtract their signals from the composite signal received. The capacity for user to decode user ’s message is given by
(8) 
Iv NOMA power allocation for minimum QoS rate systems
Suppose that each user has its information transmitted at a target rate , such that . A user , will experience an outage during the decoding process of its own information if any of the following occurs:
(9) 
for any . Call the sum of all the users’ target rates , and the average of the rates . Define the following events based on the specific signal user needs to detect and decode, where , and ,
(10)  
The outage event at user can then be described as
(11) 
For , , because user does not perform SIC in order to decode its own signal. From this definition of the outage, the following theorem is obtained.
Theorem 1.
For a user DL NOMA system with user target rates and power allocation coefficients , such that , call , which is the interference coefficient in the received signal that users will use to detect and decode user ’s information. If such that , then for user and ,
(12) 
and thus SIC will fail for all users .
Proof:
For any specific user , suppose that and . Since , it follows that
(13) 
The events and can be written in the form
(14) 
Since and ,
(15) 
Therefore solving equation (14) for leads to
(16)  
Therefore, since , this condition makes .
Now suppose that , then it must be true that . This will avoid the previous impossible event. However, if this is true, then events and gives rise to the inequality
(17) 
where the value inside the parentheses must be greater than zero in order to avoid the certain outage situation from equation (16). Therefore,
(18) 
It must be true that by definition of power allocation coefficients, however
(19) 
Therefore if , then having and requires , which is not possible.
Hence, for any user with , . ∎
Theorem 1 demonstrates that there is a fundamental relationship between the set of target rates and associated power allocation coefficients , . It also demonstrates that as these target rates increase, the values of decrease rapidly, indicating the well known fact that only users with large channel SNR gains should have large target rates.
Note that this does not indicate that the rate for user is guaranteed if , since the total power available for allocation to users may be less than to begin with. In fact, a bound that is more case specific to the actually selected power allocation coefficients, as outlined in [7], is
(20) 
However, although it is a more strict bound, it is dependent on the specific case of power allocation coefficients, whereas the bound provided in theorem 1 is a fundamental upperlimit on the received interference coefficient that cannot be exceeded by any power allocation scheme. So a set of power allocation coefficients that satisfy equation (20) also satisfies theorem 1.
With the assumption that the power allocation coefficients are selected such that , it should also be noted that in order for the SIC process to reach a decoding stage , it does so with a certain probability at each user . In other words, if users are going to avoid an outage, they must sequentially decode messages successfully in the process. Given the sequential nature of the decoding process, it is therefore desirable that the initial decoding stages have higher successful decoding success, i.e. lower outage probabilities.
In light of theorem 1, the events can be rewritten as
(21)  
This means that the outage events can be expressed as
(22)  
It is not desirable that a user ’s outage probability be primarily dictated by the success or failure of the earlier decoding stages. Since each event in equation (22) is determined by a finite length interval in , it is clear that , such that
(23)  
(24) 
This leads to the following proposition.
Proposition 1.
For a user DL NOMA system with target rates , if the associated power allocation coefficients are selected such that
(25) 
then
(26) 
and , satisfying the requirement from theorem 1.
Proof:
If , then it is true that
(27) 
If this is true, then it is true that
(28)  
(29) 
From the above, it is easy to show that
(30) 
To show that the condition above implies that satisfy theorem 1, it is sufficient to show that any power allocation coefficients satisfying equation (30) satisfy the inequality
(31) 
based on equation (15), according to the theorem. So if equation (30) holds , then for any and , it is easily shown that
(32)  
So
(34)  
(35) 
Hence, these power allocation coefficients satisfy the requirement in theorem 1. ∎
The above proposition provides the relationship between the power allocation coefficients in order to have a wellbehaved SIC outage event. In other words, the outage probability to decode user ’s information should not be bounded by having outage event during an SIC stage, but is bounded by successful decoding of its own information. Also, note that this condition also favors the decoding probability of users with lower channel SNR gains, as it places a lesser upperbound on the amount of irremovable interference received, i.e. the power allocation coefficients of users with larger channel SNR gains are restricted to a smaller region. From here on, any power allocation strategy which satisfies proposition 1, and by extension theorem 1, will be defined as being a wellbehaved strategy.
The final desired requirement for a NOMA power allocation strategy is that the outage probability performance is equal to or better than the outage probability performance of OMA. Define the event such that a user experiences an outage in a OMA system as
First, this is formally defined in the following.
Definition 1.
For a user :

the power allocation coefficient is defined as the exact power allocation required such that user achieves the same outage probability performance as it would achieve using OMA. In other words, ;

The power allocation is defined as the minimum power allocation such that , which can only be applied when all users also have power allocation ;

The interference coefficient .
Any power allocation strategy that improves the outage probability performance over OMA can be written as . If , then all users will achieve the same outage probability performance as OMA. It is clear that a user can only use in order to achieve the same outage probability performance as OMA if all users also use . It can now be shown that there always exists a power allocation strategy such that all users NOMA outage probabilities are equal to or lower than the respective OMA outage probabilities. The following theorem demonstrates this fundamental property.
Theorem 2.
For a user DL NOMA system with channel gains and associated rates with the condition , there always exists a power allocation strategy such that at least one user has NOMA outage event , and for ,
(36) 
meaning that the NOMA outage probability performance can always be at least as good or better than the OMA outage probability performance for every user.
Proof:
If , it must at least be true that s.t. , and then demonstrate that . To show that s.t. , begin with and equate
(37) 
Then for , equate
(38) 
This creates the recursive relationship which can be solved to find
(39) 
From here on, the power allocation strategy that satisfies equations (37, 39) will be called . In order for this to be a valid power allocation strategy, the sum of the coefficients must be proven to always be less than or equal to 1. Let the interference coefficient for user using this power allocation strategy be called , which can be found easily by noting from equation (38) that
(40)  
(41)  
(42) 
Since the function is a monotonically decreasing function in , then
So given that
then
So clearly the sum is less than 1. To complete the proof, only one strategy that satisfies the conditions stated in the theorem is needed, so let user have the power allocation coefficient
(43) 
which leads to . ∎
Based on the previous theorem, it is clear that the improvement of the outage probability performance of NOMA over OMA is based on the design of the additional power allocation to each coefficient , and the strategy are the starting point. In the case that user power allocation is not equal to , then clearly all users will have to allocate additional power in order for . Furthermore, any power allocation strategy should be demonstrated to be wellbehaved. The approach and implications of a user having wellbehaved NOMA power allocation strategy which demonstrates better outage probability performance over OMA is discussed in the next section.
V Wellbehaved power allocation strategies that improve NOMA outage probability performance over OMA
In order to determine how to construct a wellbehaved power allocation strategy which improves NOMA outage probability performance over OMA, the power allocation strategy that satisfies theorem 2 must be generalized. Since a power allocation coefficient for user signal can be described by , , and , then
(44)  
for . Note that by definition 1, iff , and iff . Furthermore, the portion of the interference coefficient resulting from the additional irremovable interference coefficient caused by the terms (the expression in the parenthesis above) can be expressed as
(45) 
So the general interference coefficient for user can be written as .
The total available power allocation coefficient for user is a function of . This is because in a DL NOMA system, the goal is to improve the overall outage performance, and the outage performance of users with weaker channels is usually the performance bottleneck. Furthermore, improving the outage performance of the weaker users first helps in the design of a power allocation strategy that is wellbehaved. The total available power allocation coefficient for user is then found by noting that
(46) 
The sum of the additional power allocation to users , is given by
(47) 
So the additional power allocation coefficient for user is a function of .
Using the generalized expression of the power allocation strategy that satisfies theorem 2, the properties of can be found such that the power allocation strategy is wellbehaved.
Theorem 3.
For users with channel gains , scheduled to receive signals with power allocation strategy , the power allocation strategy is wellbehaved if any combination of the following are true:

and , meaning and , for any ;

(48)
Proof:

Since , and and , then proposition 1 is used to show that the following is always true for :
(49) (50) which is true because .

For this case, the minimum allowable power allocation coefficient for users is . If , user power allocation coefficient leads to . The power allocation coefficient for user is then given by