Impact of User Pairing on 5G Non-Orthogonal Multiple Access
Abstract
Non-orthogonal multiple access (NOMA) represents a paradigm shift from conventional orthogonal multiple access (MA) concepts, and has been recognized as one of the key enabling technologies for 5G systems. In this paper, the impact of user pairing on the performance of two NOMA systems, NOMA with fixed power allocation (F-NOMA) and cognitive radio inspired NOMA (CR-NOMA), is characterized. For F-NOMA, both analytical and numerical results are provided to demonstrate that F-NOMA can offer a larger sum rate than orthogonal MA, and the performance gain of F-NOMA over conventional MA can be further enlarged by selecting users whose channel conditions are more distinctive. For CR-NOMA, the quality of service (QoS) for users with the poorer channel condition can be guaranteed since the transmit power allocated to other users is constrained following the concept of cognitive radio networks. Because of this constraint, CR-NOMA has different behavior compared to F-NOMA. For example, for the user with the best channel condition, CR-NOMA prefers to pair it with the user with the second best channel condition, whereas the user with the worst channel condition is preferred by F-NOMA.
I Introduction
Multiple access in 5G mobile networks is an emerging research topic, since it is key for the next generation network to keep pace with the exponential growth of mobile data and multimedia traffic [1] and [2]. Non-orthogonal multiple access (NOMA) has recently received considerable attention as a promising candidate for 5G multiple access [3, 4, 5, 6]. Particularly, NOMA uses the power domain for multiple access, where different users are served at different power levels. The users with better channel conditions employ successive interference cancellation (SIC) to remove the messages intended for other users before decoding their own [7]. The benefit of using NOMA can be illustrated by the following example. Consider that there is a user close to the edge of its cell, denoted by , whose channel condition is very poor. For conventional MA, an orthogonal bandwidth channel, e.g., a time slot, will be allocated to this user, and the other users cannot use this time slot. The key idea of NOMA is to squeeze another user with better channel condition, denoted by , into this time slot. Since ’s channel condition is very poor, the interference from will not cause much performance degradation to , but the overall system throughput can be significantly improved since additional information can be delivered between the base station (BS) and . The design of NOMA for uplink transmissions has been proposed in [4], and the performance of NOMA with randomly deployed mobile stations has been characterized in [5]. The combination of cooperative diversity with NOMA has been considered in [8].
Since multiple users are admitted at the same time, frequency and spreading code, co-channel interference will be strong in NOMA systems, i.e., a NOMA system is interference limited. As a result, it may not be realistic to ask all the users in the system to perform NOMA jointly. A promising alternative is to build a hybrid MA system, in which NOMA is combined with conventional MA. In particular, the users in the system can be divided into multiple groups, where NOMA is implemented within each group and different groups are allocated with orthogonal bandwidth resources. Obviously the performance of this hybrid MA scheme is very dependent on which users are grouped together, and the aim of this paper is to investigate the effect of this grouping. Particularly, tn this paper, we focus on a downlink communication scenario with one BS and multiple users, where the users are ordered according to their connections to the BS, i.e., the -th user has the -th worst connection to the BS. Consider that two users, the -th user and the -th user, are selected for performing NOMA jointly, where . The impact of user pairing on the performance of NOMA will be characterized in this paper, where two types of NOMA will be considered. One is based on fixed power allocation, termed F-NOMA, and the other is cognitive radio inspired NOMA, termed CR-NOMA.
For the F-NOMA scheme, the probability that F-NOMA can achieve a larger sum rate than conventional MA is first studied, where an exact expression for this probability as well as its high signal-to-noise ratio (SNR) approximation are obtained. These developed analytical results demonstrate that it is almost certain for F-NOMA to outperform conventional MA, and the channel quality of the -th user is critical to this probability. In addition, the gap between the sum rates achieved by F-NOMA and conventional MA is also studied, and it is shown that this gap is determined by how different the two users’ channel conditions are, as initially reported in [8]. For example, if , it is preferable to choose , i.e., pairing the user with the best channel condition with the user with the worst channel condition. The reason for this phenomenon can be explained as follows. When is small, the -th user’s channel condition is poor, and the data rate supported by this user’s channel is also small. Therefore the spectral efficiency of conventional MA is low, since the bandwidth allocated to this user cannot be accessed by other users. The use of F-NOMA ensures that the -th user will have access to the resource allocated to the -th user. If is small, the -th user’s channel quality is similar to the -th user’s, and the benefit to use NOMA is limited. But if , the -th user can use the bandwidth resource much more efficiently than the -th user, i.e., a larger will result in a larger performance gap between F-NOMA and conventional MA.
The key idea of CR-NOMA is to opportunistically serve the -th user on the condition that the -th user’s quality of service (QoS) is guaranteed. Particularly the transmit power allocated to the -th user is constrained by the -th user’s signal-to-interference-noise ratio (SINR), whereas F-NOMA uses a fixed set of power allocation coefficients. Since the -th user’s QoS can be guaranteed, we mainly focus on the performance of the -th user offered by CR-NOMA. An exact expression for the outage probability achieved by CR-NOMA is obtained first, and then used for the study of the diversity order. In particular, we show that the diversity order experienced by the -th user is , which means that the -th user’s channel quality is critical to the performance of CR-NOMA. This is mainly because of the imposed SINR constraint, where the -th user can be admitted into the bandwidth channel occupied by the -th user, only if the -th user’s SINR is guaranteed. As a result, with a fixed , increasing does not bring much improvement to the -th user’s outage probability, which is different from F-NOMA. If the ergodic rate is used as the criterion, a similar difference between F-NOMA and CR-NOMA can be observed. Again take the scenario described in the last paragraph as an example. If , in order to yield a large gain over conventional MA, F-NOMA prefers the choice of , but CR-NOMA prefers the choice of , i.e., pairing the user with the best channel condition with the user with the second best channel condition.
Ii NOMA With Fixed Power Allocation
Consider a downlink communication scenario with one BS and mobile users. Without loss of generality, assume that the users’ channels have been ordered as , where denotes the Rayleigh fading channel gain between the BS and the ordered -th user. Consider that the -th user and the -th user, , are paired to perform NOMA.
In this section, we focus on F-NOMA, where the BS allocates a fixed amount of transmit power to each user. In particular, denote and as the power allocation coefficients for the two users, where these coefficients are fixed and . According to the principle of NOMA, since . The rates achievable to the two users are given by
(1) |
and
(2) |
respectively, where denotes the transmit SNR. Note that the -th user can decode the message intended for the -th user successfully and is always achievable at the -th user, since
On the other hand, an orthogonal MA scheme, such as time-division multiple-access (TDMA), can support the following data rate:
(3) |
where . In the following subsections, the impact of user pairing on the sum rate and the individual user rates achieved by F-NOMA is investigated.
Ii-a Impact of user pairing on the sum rate
In this subsection, we focus on how user pairing affects the probability that NOMA achieves a lower sum rate than conventional MA schemes, which is given by
(4) |
The following theorem provides an exact expression for the above probability as well as its high SNR approximation.
Theorem 1.
Suppose that the -th and -th ordered users are paired to perform NOMA. The probability that F-NOMA achieves a lower sum rate than conventional MA is given by
(5) | ||||
where , , , and . At high SNR, this probability can be approximated as follows: ,
(6) |
where , i.e., is a constant and not a function of .
Proof.
See the appendix. ∎
Theorem 1 demonstrates that it is almost certain for F-NOMA to outperform conventional MA, particularly at high SNR. Furthermore, the decay rate of the probability is approximately , i.e., the quality of the -th user’s channel determines the decay rate of this probability.
Ii-B Asymptotic studies of the sum rate achieved by NOMA
In addition to the probability , it is also of interest to study how large of a performance gain F-NOMA offers over conventional MA, i.e.,
where is a targeted performance gain. The probability studied in the previous subsection can be viewed as a special case by setting . An interesting observation for the cases with is that there will be an error floor for , regardless of how large the SNR is. This can be shown by studying the following asymptotic expression of the sum rate gap:
(7) | ||||
which is not a function of SNR. Hence the probability can be expressed asymptotically as follows:
(8) | ||||
When , , which is consistent with Theorem 1, since
When , (8) implies that the probability can be expressed asymptotically as follows:
(9) |
Directly applying the joint probability density function (pdf) of the users’ channels shown in (31), the probability can be rewritten as follows:
(10) | ||||
which is quite complicated to evaluate. In [9], a simpler pdf for the ratio of two order statistics has been provided as follows:
where and . By using this pdf, the addressed probability can be calculated as follows:
(11) | ||||
Ii-C Impact of user pairing on individual user rates
Careful user pairing not only improves the sum rate, but also has the potential to improve the individual user rates, as shown in this section. We first focus on the probability that F-NOMA can achieve a larger rate than orthogonal MA for the -th user which is given by
(12) | ||||
After some algebraic manipulations, the above probability can be further rewritten as follows:
(13) | ||||
where .
By applying a series expansion, the above probability can be rewritten as follows:
(14) | ||||
Again applying the results in (41) and (42), the above equation can be approximated as follows:
(15) |
which means that decays at a rate of .
On the other hand, the probability that the -th user can experience better performance in a NOMA system than in orthogonal MA systems is given by
Following similar steps as previously, we obtain the following:
(16) |
Interestingly in (16) is very much similar to in (13), which yields the following:
(17) | ||||
and its high SNR approximation is given by
(18) |
As can be seen from (15) and (18), the two users will have totally different experience in NOMA systems. Particularly, a user with a better channel condition is more willing to perform NOMA since , which is not true for a user with a poor channel condition. Furthermore, it is preferable to pair two users whose channel conditions are significantly distinct, since (15) and (18) implies that should be as small as possible and should be as large as possible.
(19) | ||||
Iii Cognitive Radio Inspired NOMA
NOMA can be also viewed as a special case of cognitive radio systems [10] and [11], in which a user with a strong channel condition, viewed as a secondary user, is squeezed into the spectrum occupied by a user with a poor channel condition, viewed as a primary user. Following the concept of cognitive radio networks, a variation of NOMA, termed as CR-NOMA, can be designed as follows. Suppose that the BS needs to serve the -th user, i.e., a user a with poor channel condition, due to either the high priority of this user’s messages or user fairness, e.g., this user has not been served for a long time. This user can be viewed as a primary user in a cognitive radio system. The -th user can be admitted into this channel on the condition that the -th user will not cause too much performance degradation to the -th user.
Consider that the targeted SINR at the -th user is , which means that the choices of the power allocation coefficients, and , need to satisfy the following constraint:
(20) |
This means that the maximal transmit power that can be allocated to the -th user is given by
(21) |
which means that if . Note that the choice of in (21) is a function of the channel coefficient , unlike the constant choice of used by F-NOMA in the previous section.
Since the -th user’s QoS can be guaranteed due to (20), we only need to study the performance experienced by the -th user. Particularly the outage performance of the -th user is defined as follows:
(22) |
and the following theorem provides an exact expression for the above outage probability as well as its approximation.
Theorem 2.
Proof.
See the appendix. ∎
Theorem 2 demonstrates an interesting phenomenon that, in CR-NOMA, the diversity order experienced by the -th user is determined by how good the -th user’s channel quality is. This is because the -th user can be admitted to the channel occupied by the -th user only if the -th user’s QoS is met. For example, if the -th user’s channel is poor and its targeted SINR is high, it is very likely that the BS allocates all the power to the -th user, and the -th user might not even get served.
Recall from the previous section that F-NOMA can achieve a diversity gain of for the -th user, and therefore the diversity order achieved by CR-NOMA could be much smaller than that achieved by F-NOMA, particularly if . This performance difference is again due to the imposed power constraint shown in (21).
It is important to point out that CR-NOMA can strictly guarantee the -th user’s QoS, and therefore achieve better fairness compared to F-NOMA. In particular, the use of CR-NOMA can ensure that a diversity order of is achievable to the -th user, and admitting the -th user into the same channel as the -th user will not cause too much performance degradation to the -th user. Particularly the SINR experienced by the -th user is strictly maintained at the predetermined level .
Sum rate achieved by CR-NOMA
Without sharing the spectrum with the -th user, i.e, all the bandwidth resource is allocated to the -th user, the following rate is achievable:
(23) |
It is easy to show that the use of CR-NOMA always achieves a larger sum rate since
(24) | ||||
This superior performance gain is not surprising, since the key idea of CR-NOMA is to serve a user with a strong channel condition, without causing too much performance degradation to the user with a poor channel condition.
In addition, it is of interest to study how much the averaged rate gain CR-NOMA can yield, i.e., . This averaged rate gain can be calculated as follows:
(25) | ||||
In general, the evaluation of the above equation is difficult, and in the following we provide a case study when . Particularly, the joint pdf of the channels for this special case can be simplified and the averaged rate gain can calculated as follows:
(26) | ||||
After some algebraic manipulations, the above equation can be rewritten as follows:
Now applying Eq. (3.352.2) in [12], the average rate gain can be expressed as follows:
(27) | ||||
where denotes the exponential integral.
Iv Numerical Studies
In this section, computer simulations are used to evaluate the performance of two NOMA schemes as well as the accuracy of the developed analytical results.
Iv-a NOMA with fixed power allocation
In Fig. 1, the probability that F-NOMA realizes a lower sum rate than conventional MA, i.e., , is shown as a function of SNR. and . As can be seen from both figures, F-NOMA almost always outperforms conventional MA, particularly at high SNR. The simulation results in Fig. 1 also demonstrate the accuracy of the analytical results provided in Theorem 1. For example, the exact expression of shown in Theorem 1 matches perfectly with the simulation results, whereas the developed approximation results become accurate at high SNR.
Another important observation from Fig. 1 is that increasing , i.e., scheduling a user with a better channel condition, will make the probability decrease at a faster rate. This observation is consistent to the high SNR approximation results provided in Theorem 1 which show that the slope of the curve for the probability is a function of . In Fig. 2, the probability is shown with different choices of . Comparing Fig. 1 to Fig. 2, one can observe that never approaches zero, regardless of how large the SNR is. This observation confirms the analytical results developed in (11) which show that the probability is no longer a function of SNR, when . It is interesting to observe that the choice of a smaller is preferable to reduce , a phenomenon previously reported in [8].
In Fig. 3, two different but related probabilities are shown together. One is , i.e., the probability that it is beneficial for the user with a poor channel condition to perform F-NOMA, and the other is , i.e., the probability that the user with a strong channel condition prefers conventional MA. In Section II.C, analytical results have been developed to show that both and are decreasing with increasing SNR, which is confirmed by the simulation results in Fig. 3. The reason that is reduced at a higher SNR is that the -th user’s rate in an F-NOMA system becomes a constant, i.e.,
Iv-B Cognitive radio inspired NOMA
In Fig. 4 the -th user’s outage probability achieved by CR-NOMA is shown as a function of SNR. As can be seen from the figure, the exact expression for the outage probability developed in Theorem 2 matches the simulation results perfectly. Recall from Theorem 2 that the diversity order achievable for the -th user is . Or in other words, the slope of the outage probability is determined by the channel quality of the -th user, which is also confirmed by Fig. 4. For example, when increasing from to , the outage probability is significantly reduced, and its slope is also increased. To clearly demonstrate the diversity order, we have provided an auxiliary curve in the figure which is proportional to . As can be observed in the figure, this auxiliary curve is parallel to the one for , which confirms that the diversity order achieved by CR-NOMA is .
Since Theorem 2 states that the diversity order of is not a function of , an interesting question is whether a different choice of matters. Fig. 5 is provided to answer this question. While the use of a larger does bring some reduction of , the performance gain of increasing is negligible, particularly at high SNR. This is because the channel quality of the -th user becomes a bottleneck for admitting the -th user into the same channel.
In Fig. 6 the performance of CR-NOMA is evaluated by using the ergodic data rate as the criterion. Due to the use of (21), the -th user’s QoS can be satisfied, and therefore we only focus on the -th user’s data rate, which is the performance gain of CR-NOMA over conventional MA. Fig. 6 demonstrates that, by fixing , it is beneficial to select two users with better channel conditions. While Fig. 5 shows that changing with a fixed does not affect the outage probability, Fig. 6 demonstrates that user pairing has a significant impact on the ergodic rate. Specifically, when fixing the choice of , pairing it with a user with a better channel condition can yield a gain of more than bit per channel use (BPCU) at dB. Another interesting observation from Fig. 6 is that with a fixed , increasing will improve the performance of CR-NOMA, which is different from F-NOMA. For example, when , Fig. 2 shows that the user with the worst channel condition, , is the best partner, whereas Fig. 6 shows that the user with the second best channel condition, i.e., , is the best choice.
V Conclusions
In this paper the impact of user pairing on the performance of two NOMA systems, NOMA with fixed power allocation (F-NOMA) and cognitive radio inspired NOMA (CR-NOMA), has been studied. For F-NOMA, both analytical and numerical results have been provided to demonstrate that F-NOMA can offer a larger sum rate than orthogonal MA, and the performance gain of F-NOMA over conventional MA can be further enlarged by selecting users whose channel conditions are more distinctive. For CR-NOMA, the channel quality of the user with a poor channel condition is critical, since the transmit power allocated to the other user is constrained following the concept of cognitive radio networks. One promising future direction of this paper is that the analytical results can be used as criteria designing distributed approaches for dynamic user pairing/grouping.
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Proof for Theorem 1: Observe that the sum rate achieved by NOMA can be expressed as follows:
On the other hand the sum rate achieved by conventional MA is given by
(28) |
Now the addressed probability can be written as follows:
(29) | ||||
After some algebraic manipulations, this probability can be rewritten as follows:
(30) | ||||
The right-hand side of the above inequality is non-negative since the -th user will get less power than the -th user, i.e., . Note that the joint pdf of and is given by [13]
(31) |
In addition, the marginal pdf of is given by
(32) |
By applying the above density functions, the addressed probability can be expressed as follows:
(33) | ||||
Note that the integral range for in is , and this range implies that , which causes an additional constraint on , i.e., . By applying the binomial expansion, the joint pdf can be further written as follows:
Therefore the probability can now be evaluated as follows:
(34) |
On the other hand, can be calculated as follows:
(35) | ||||
By applying the binomial expansion, can be written as follows:
(36) | ||||
To find high SNR approximations for and , first observe that the integral in (34) is calculated for the range of . At high SNR, the two functions and can be approximated as follows:
Define . It is straightforward to show
for , since . Therefore at high SNR, we can have the following approximation: