Optimal power allocation in NOMAbased twopath successive AF relay systems
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Abstract
Due to the characteristic of transmitting multiplexed signals in superposed mode over the same spectrum, nonorthogonal multiple access (NOMA) technology is deemed as a promising way to improve spectral efficiency in fifth generation (5G) networks. In this paper, we develop a NOMA cooperative system based on the twopath successive relaying concept, in which the data at the source node is divided into two parallel parts and is transmitted to the destination in superposed mode via the assistance of two amplifyandforward (AF) relays. On the condition that the transmit power of the individual nodes and the entire system are all constrained, the maximization of achievable rate is formulated as an optimization problem. Following the guidelines of KarushKuhnTucher (KKT) conditions, the dual decomposition method is adopted to obtain the closedform expressions of the optimal power allocation. Moreover, to balance the achievable rate between two superposed signals, which is equivalent to minimizing the required spectrum bandwidth, a power allocation scheme between the superposed signals is proposed. In order to verify the effectiveness and efficiency of the proposed power allocation scheme, we conduct extensive numerical simulation on some realistic system setup. The results demonstrate that our analytical insights about the optimal power allocation are aligned with the simulation outcome.
Keywords
Nonorthogonal multiple access (NOMA) Successive AF relay KarushKuhnTucker (KKT) Optimal power allocationAbbreviations
 5G
Fifth generation
 AF
Amplifyandforward
 DF
Decodeandforward
 FIC
Full interference cancellation
 KKT
KarushKuhnTucher
 NOMA
Nonorthogonal multiple access
 OFDM
Orthogonal frequency division multiplexing
 QoS
Qualityofservice
1 Introduction
In wireless networks, spectrum efficiency and energy efficiency are two most important metrics to consider, as their improvement is one of the promising means to realize broadbandgreen communication concept. Compared to the conventional orthogonal multiple access (OMA) scheme, nonorthogonal multiple access (NOMA) technique can further improve spectrum efficiency and system achievable rate, as the interference cancelation between difference users can be successfully implemented as long as one user’s received signaltointerferenceplusnoise ratio (SINR) for the other user’s signal is larger than or equal to the received SINR of the other user for its own signal [1]. Therefore, it has been deemed as a promising technique to mitigate spectrum congestion in the fifth generation (5G) networks [2, 3, 4]. On the other hand, with the assistance of relay(s), greater system capacity can be obtained with the same power consumption or lower power consumption is required while keeping the system capacity intact [5, 6]. However, the spectral efficiency of cooperation system will be reduced greatly if the relay operates in halfduplex mode, as such relay(s) cannot transmit and receive signals simultaneously. Hence, twopath relay protocol is commonly used in cooperation systems. Consequently, when NOMA and twopath relay cooperation techniques are combined effectively, the performance of spectral efficiency and power consumption can be improved greatly. Currently, NOMAbased relaying systems have been an emerging research topic in the recent days.
For NOMAbased relaying systems, when relays operate in halfduplex decodeandforward (DF) mode, many system models and resource allocation schemes already have been proposed. In [7], for a system in which NOMA technique is applied in both direct and relay transmissions, analytical expressions for outage probability and ergodic sum capacity are derived. In [8], a NOMAbased relaying system is proposed to improve spectral efficiency as well as its achievable capacity is investigated. A bufferaided NOMA relaying system is proposed in [9], its performance is investigated, and an adaptive transmission scheme for such system is proposed in [10]. For a system with slowly faded NOMAequipped multiplerelay channels, the benefit of joint network channel coding and decoding is studied in [11]. In [12], an analytical framework for a NOMAbased relaying system is developed, and then, its performance over Rician fading channels is studied. In [13], the impact of relay selection on the performance of cooperative NOMA is studied, and then, a twostage relay selection strategy is proposed. In [14], a novel signal detection scheme for a simple NOMAbased relaying system is proposed, and then, the ergodic sum rate and outage performance of the system are investigated. In [15], based on Alamouti spacetime blockcoded NOMA technique, a twophase cooperative DF relaying scheme is proposed. In [16], a dualhop cooperative relaying scheme using NOMA is proposed, where two sources communicate with their corresponding destinations in parallel over the same channel via a common relay. To maximize the throughput of a NOMAequipped wireless network with multiple relays, in [17], a novel approach to dynamically select an optimal relay mode and optimal transmit power is proposed.
On the other hand, when relays operate in halfduplex amplifyandforward (AF) mode, many schemes have been proposed to improve the performance of NOMAbased relaying systems. In [18], a NOMAbased multiantennaequipped relaying network is designed, and then, its outage performance is analyzed. When a base station communicates with multiple mobile users simultaneously through the help of a relay over Nakagamim fading channels, the overall performance is analyzed in [19]. For a NOMAequipped singlecell relay network, where an OFDMbased AF relay allocates its spectrum and power resources to sourcedestination (SD) pairs, a manytomany twosided SD pairsubchannel matching algorithm is proposed in [20]. In [21], a joint power allocation and relay beamforming design problem is investigated, and then, an alternating optimizationbased algorithm is proposed to maximize the achievable rate. In [22], the outage performance of a cooperative NOMAequipped relay system is studied, and then, an accurate closedform approximation of the outage probability is derived. In [23], when multiple users transmit messages to two destinations under the help of multiple AF relays, an optimal relay selection criterion is proposed to improve outage performance, and closedform analytical expressions for the outage probability are derived. In [24], a relayaided NOMA technique is proposed for uplink cellular networks, where the cooperative relay transmission is used to accommodate more than one user per orthogonal resource block in the context of interferencelimited scenarios.
To enhance system flexibility, in [25], for NOMAequipped cooperative networks with both the DF and AF relaying protocols, where one base station communicates with two mobile users with the aid of multiple relays, a twostage relay selection strategy is proposed while considering different qualityofservice (QoS) requirements of the users. In [26], the analytical performance of a NOMAequipped cellular system, with multiple AF and DF relays, is analyzed and then compared with that of four traditional schemes. To enable continuous signal transmission, a fullduplex cooperative NOMA system between dual users is proposed in [27] and then the outage probability and achievable ergodic sum capacity of the corresponding system are investigated. In [28], a NOMAbased multipair twoway relay network is developed, and then, a rate splitting scheme and a successive group decoding strategy are proposed to optimize its performance.

We have shown the idea of a NOMAbased novel twopath relaying system, where the target information of the source node is divided into two equal streams and they are transmitted to the destination via two halfduplex AF relays in superposed mode. Because of adopting the idea of superposition model in the NOMA technique, intuitively, lower level of system frequency is required compared to a system with OMA scheme. In such system, while targeting on the maximization of the system capacity, the power allocation among the source and relay nodes is an optimization problem given that there are power constraints at the individual nodes as well as the entire system. The dual decomposition method is adopted to obtain the closed form expressions of the optimal power allocation.

Once the optimal power allocation at each node is known, in order to separate the superposed received messages at the destination, the optimal power allocation between the two data streams is formulated as an optimization problem while targeting on the minimization of the required frequency band. Similar to the solution scheme of the first problem, the dual decomposition method is adopted to obtain the closed form expression of the optimal power allocation.

We conduct extensive numerical simulation in order to verify the effectiveness and accuracy of the proposed power allocation scheme from different angles. Under different scenarios, we verify the analytical insights of the proposed power allocation scheme with the simulation results.
The rest of the paper is organized as follows. In Section 2, a novel NOMAbased twopath AF relay system model is presented, and then, the corresponding optimization problem is formulated. The optimal power allocation strategy is proposed in Section 3. We evaluate the proposed scheme in Section 4. Finally, we conclude the paper with some future research direction in Section 5.
2 System description
In Fig. 1, we demonstrate a novel NOMAbased twopath successive relay system. The system consists of a source node, a destination node, and two relays that operate in AF mode. For notational simplicity, the channel gain between two corresponding terminals is denoted as h_{i}, i∈{1,2,⋯,6}. It is supposed that the relays work in halfduplex mode. Therefore, when the source node transmits signals continuously at every time slot, relay 1 and relay 2 receive the transmitted signals and forward them to the destination successively in an alternative manner^{1}.
For such tworelay cooperative systems, the location of the relays affects system performance greatly. Hence, to maximize the system achievable rate, the relays should be put on the optimal positions. Generally speaking, when the interference between two relays can be canceled completely, the relays are usually put in the same place due to the sake of management. Hence, the distance from the relays to the source and the destination are equal.
The superposed signal, transmitted from the source node, can be written as \(\sqrt {a_{1} p_{s}}s_{1} + \sqrt {a_{2} p_{s}}s_{2}\), where p_{s} denotes transmit power, s_{1} and s_{2} are transmitted data symbols with E[s_{1}^{2}]=E[s_{2}^{2}]=1, a_{1} and a_{2} are power allocation coefficients between the transmitted signals s_{1} and s_{2} while satisfying a_{1}+a_{2}=1 and a_{1}>a_{2} constraints. In each slot, the destination node tries to recover the superposed signals and then decode s_{1} and s_{2} individually through the successive interference cancelation (SIC) technology. Similar to the system model in [6, 30], it is assumed that the channel between the source node and the destination node is in deep fade, and hence, the direct link between them can be ignored. The channel gain between two nodes is constant in a symbol duration, and it is inversely proportional to the fourth power of the distance between the transmitter and the receiver. Noise signal of each link follows Additive White Gaussian Noise (AWGN) distribution with zero mean and σ^{2} variance.
where η_{d}[n] is the additive Gaussian noise with zero mean and σ^{2} variance.
In (3), α_{1}h_{1}h_{6}x_{s}[ n−1] is the desired signal to be recovered, α_{1}h_{6}η_{1}[ n−1]+η_{d}[n] is the additive noise at the destination, and the term α_{1}h_{3}h_{6}x_{2}[ n−1] is the interference from relay 2 which should be canceled or suppressed.
It is assumed that the transmit power of the relays are equal in the system. Hence, the corresponding amplifying factor at the relays can be represented as \(\alpha _{1} = \sqrt {p_{r}}/\sqrt {\left h_{1}\right ^{2} p_{s} + \left h_{3}\right ^{2} p_{r} + \sigma _{1}^{2}}\) and \(\alpha _{2} = \sqrt {p_{r}}/\sqrt {\left h_{2}\right ^{2} p_{s} + \left h_{4}\right ^{2} p_{r} + \sigma _{2}^{2}}\), where the transmit power at both relay 1 and relay 2 are considered as the same and it is p_{r}. Without loss of generality, we assume that all the additive Gaussian noise be independent and identically distributed (i.i.d) with zeromean and σ^{2} variation.
3 Achievable rate analysis and optimal power allocation
3.1 Achievable rate analysis and problem formulation
As all the additive noise is assumed to have zero mean and unit variance, after substituting the expression of α_{1}, the achievable rate in the odd symbol duration can be shown as C_{odd}=log_{2}[1+h_{1}^{2}h_{5}^{2}h_{6}^{2}p_{s}p_{r}/{h_{1}^{2}h_{5}^{2}p_{s}+(h_{5}^{2}h_{6}^{2}+h_{3}^{2}h_{6}^{2}+h_{3}^{2}h_{5}^{2})p_{r}+h_{5}^{2}}].
In the same way, the achievable rate in the even symbol duration can be obtained, which is shown as C_{even}=log_{2}[1+h_{2}^{2}h_{5}^{2}h_{6}^{2}p_{s}p_{r}/{h_{2}^{2}h_{6}^{2}p_{s}+(h_{5}^{2}h_{6}^{2}+h_{4}^{2}h_{5}^{2}+h_{4}^{2}h_{6}^{2})p_{r}+h_{6}^{2}}]. Maximizing the system achievable rate is equivalent to maximizing the achievable rate at both the odd and even symbol duration. However, under given constraints on transmit power, it is impossible to obtain maximal achievable rate in both the even and odd symbol duration simultaneously due to the variation channel fading. Therefore, we investigate the system achievable rate at the odd and even symbol duration separately while realizing the optimal power allocation. As both the processes and schemes are similar, we only present the scheme to maximize the achievable rate at the odd symbol duration for brevity, and the maximal achievable rate in the even symbol duration can be obtained in the similar manner. For notational simplicity, we use f_{i} to denote h_{i}^{2}, i∈{1,2,⋯,6} hereafter.
where \(p_{max}^{s}\) and \(p_{max}^{r}\) are the maximal transmit power that the source node and the corresponding relay node can provide, respectively, and p_{max} is the total transmit power due to the consideration of interference threshold to other users when a cognitive scenario is investigated.
It is easy to see the formulation in (8) is a convex problem, so there is a optimal solution. However, though p_{r} only appears in the numerator of expression, p_{s} appears both in the numerator and denominator. Furthermore, three constrains on them should be considered jointly. Hence, it is difficult to obtain the solution straightforwardly.
3.2 The optimization of the achievable rate
where λ_{i}, i∈{1,2,3} are Lagrange multipliers with constraint λ_{i}≥0.
According to (10), the scheme to achieve the optimally allocated power is presented as follows.
On the condition that \(p_{max}^{s} + p_{max}^{r} \leq p_{max}\) holds, it is straightforward to see that the maximal achievable rate can be obtained by setting \(p_{s} = p_{max}^{s}\) and \(p_{r} = p_{max}^{r}\). Therefore, we focus on the \(p_{max}^{s} + p_{max}^{r} > p_{max}\) case. To maximize the system achievable rate, p_{s}+p_{r}=p_{max} should be set.
On the condition that \(p_{max}^{s} + p_{max}^{r} > p_{max}\) holds, it is unlikely that both λ_{2}>0 and λ_{3}>0 occur, which implies that \(p_{r} = p_{max}^{r}\) and \(p_{s} = p_{max}^{s}\) should be set. Hence, there is no available solutions for this case. Consequently, to maximize the system achievable rate in the odd symbol duration, three cases need to be considered: (1) λ_{2}=0 and λ_{3}>0, (2) λ_{3}=0 and λ_{2}>0, and (3) λ_{2}=0 and λ_{3}=0.
Case 1: λ_{2}=0 and λ_{3}>0
For this case, based on (10e), \(p_{r} = p_{max}^{r}\) can be concluded. Combining with the p_{s}+p_{r}=p_{max} condition, \(p_{s}= p_{max}  p_{max}^{r}\) can be derived.
From (11), when a > 0 holds, it can be concluded that \(p_{max}^{r} > \left (b + \sqrt {b\left (b a p_{max}\right)}\right)/a\) or \(p_{max}^{r} < \left (b \!\!{\vphantom {\sqrt {b(b a p_{max})}}}\right. \left.\sqrt {b(b a p_{max})}\right)/a\) should hold. Moreover, it can be testified that \(\left (b + \sqrt {b(b a p_{max})}\right)/a > p_{max}\) and \(0 < \left (b \!\!{\vphantom {\sqrt {b(b a p_{max})}}}\right. \left.\sqrt {b(b a p_{max})}\right)/a < p_{max}\) hold. Hence, the available scenario is \(p_{max}^{r} < \left (b  \sqrt {b(b a p_{max})}\right)/a\) under the a>0 condition^{2}. On the other hand, if a<0 holds, the range of \(p_{max}^{r}\) can be shown as \(\left (b + \sqrt {b(b a p_{max})}\right)/a < p_{max}^{r} < \left (b  \sqrt {b(b a p_{max})}\right)/a\). As \(b  \sqrt {b(b a p_{max})} <0\) holds, \(p_{max}^{r}\) locates in a positive range which is practical. Furthermore, as \(p_{s} = p_{max}  p_{max}^{r}\) and \(p_{max}^{r} < \left (b  \sqrt {b(b a p_{max})}\right)/a\) hold, \(p_{s} > \left [b + a p_{max} +\!\!{\vphantom {\sqrt {b(b a p_{max})}}}\right. \left.\sqrt {b(b  {ap}_{max}}\right ]/a\) can be concluded, which implies that \(p_{max}^{s} > \left [b + a p_{max} + \sqrt {b(b  {ap}_{max}}\right ]\) should hold.
To summarize, to maximize the system capacity, \(p_{r} = p_{max}^{r}\) and \(p_{s}= p_{max}  p_{max}^{r}\) should be set for the scenario that \(p_{max}^{r} < \left (b  \sqrt {b(b {ap}_{max})}\right)/a\) and \(p_{max}^{s} > \left [b + a p_{max} + \sqrt {b(b  {ap}_{max}}\right ]/a\) hold.
Case 2: λ_{3}=0 and λ_{2}>0
It is easy to see that Δ=4(−ap_{max}+b)^{2}−4a(ap_{max}−b)p_{max}=4b(b−ap_{max})>0 always holds. Hence, when a>0 and \(p_{max}^{s} > 0\) are considered, \(0 < p_{max}^{s} < \left (b + a p_{max} + \sqrt {b(b{ap}_{max})}\right)/a\) can be derived, which locates in a positive range. Hence, it is consistent with practical scenarios. When a<0, \(p_{max}^{s} > \left (b + a p_{max}  \sqrt {b(b{ap}_{max})}\right)/(2a)\) can be derived or \(p_{max}^{s} < \left (b + {ap}_{max} + \sqrt {b(b{ap}_{max})}\right)/a\) should hold. Moreover, it is easy to prove that \(0<\left (b + {ap}_{max} + \sqrt {b(b{ap}_{max})}\right)/a < p_{max}\) and \(\left (b + a p_{max}  \sqrt {b(b{ap}_{max})}\right)/a> p_{max}\) hold. Hence, the feasible solution is \(p_{max}^{s} < \left (b + {ap}_{max} +\!\!{\vphantom {\sqrt {b(b a p_{max})}}}\right. \left. \sqrt {b(b{ap}_{max})}\right)/a\) when a<0.
To summarize, when \(p_{max}^{s} < \left (b + a p_{max} + {\vphantom {\sqrt {b(b a p_{max})}}}\right. \left.\sqrt {b(b{ap}_{max})}\right)/a\) and \(p_{max}^{r}\! > \left (b  \sqrt {b(b{ap}_{max})}\right)/a\), the optimally allocated power can be set as \(p_{s} = p_{max}^{s}\) and \(p_{r} = p_{max}  p_{max}^{s}\).
Case 3: λ_{2}=0 and λ_{3}=0.
As allocated power should satisfy the \(p_{r} \leq p_{max}^{r}\) and \(p_{s} \leq p_{max}^{s}\) constraints, the applicable scenario is \(p_{max}^{r} \geq \left (b  \sqrt {b(bap_{max})}\right)/{a}\) and \(p_{max}^{s} \geq \left (b +\!\!{\vphantom {\sqrt {b(b a p_{max})}}}\right. \left. ap_{max} + \sqrt {b(bap_{max})}\right)/a\).

When \(p_{max}^{r} < \left (b  \sqrt {b(b ap_{max})}\right)/a\) and \(p_{max}^{s} > \left (b + a p_{max} + \sqrt {b(b  ap_{max})}\right)/a\) hold, \(p_{r} = p_{max}^{r}\) and \(p_{s} = p_{max}  p_{max}^{r}\) can be set.

When \(p_{max}^{r} > \left (b  \sqrt {b(bap_{max})}\right)/a\) and \(p_{max}^{s} < \left (b + a p_{max} + \sqrt {b(bap_{max})}\right)/a\) hold, \(p_{s} = p_{max}^{s}\) and \(p_{r} = p_{max}  p_{max}^{s}\) should be set.

When \(p_{max}^{r} \geq \left (b  \sqrt {b(bap_{max})}\right)/{a}\) and \(p_{max}^{s} \geq \left (b + ap_{max} + \sqrt {b(bap_{max})}\right)/a\), \(p_{r} = \left (b  \sqrt {b(bap_{max})}\right)/a\) and \(p_{s} = \left (b + ap_{max} + \sqrt {b(bap_{max})}\right)/a\) should be set.

When \(p_{max}^{r} < \left (b  \sqrt {b(bap_{max})}\right)/{a}\) and \(p_{max}^{s} < \left (b + ap_{max} + \sqrt {b(bap_{max})}\right)/a\) hold, \(p_{r} =p_{max}^{r}\) and \(p_{s} = p_{max}^{s}\) should be set.

On the condition that \(p_{max}^{r} \!< \!\left (d \,\, \sqrt {d(d cp_{max})}\right)\!/c\) and \(p_{max}^{s} > \left (d + c p_{max} + \sqrt {d(d  cp_{max})}\right)\!/c\) hold, \(p_{r} = p_{max}^{r}\) and \(p_{s} = p_{max}  p_{max}^{r}\) can be set.

On the condition that \(p_{max}^{r} \!\!>\! \!\left (d  \sqrt {d(dcp_{max})}\right)\!/c\) and \(p_{max}^{s} \!<\! \left (d\! + c p_{max} + \sqrt {d(dcp_{max})}\right)/c\) hold, \(p_{s} = p_{max}^{s}\) and \(p_{r} = p_{max}  p_{max}^{s}\) should be set.

On the condition that \(p_{max}^{r} \!\geq \! \left (d \,\, \sqrt {d(dcp_{max})}\right)\!/{c}\) and \(p_{max}^{s} \geq \left (d + cp_{max} + \sqrt {d(dcp_{max})}\right)/c\), \(p_{r} = \left (d  \sqrt {d(dcp_{max})}\right)/c\) and \(p_{s} = \left (d + cp_{max} + \sqrt {d(dcp_{max})}\right)/c\) should be set.

On the condition that \(p_{max}^{r} \!\!<\! \left (d\! \! \sqrt {d(dcp_{max})}\right)/{c}\) and \(p_{max}^{s} < \left (d + cp_{max} + \sqrt {d(dcp_{max})}\right)/c\), \(p_{r} =p_{max}^{r}\) and \(p_{s} = p_{max}^{s}\) should be set.
3.3 The optimization of the required frequency band
From (7), \(\left h_{tot}\right ^{2} = \alpha _{1}^{2} \left h_{1}\right ^{2} \left h_{6}\right ^{2}\) and \(\sigma _{tot}^{2} = \left (\alpha _{1}^{2} \left h_{6}\right ^{2} + 1 + \alpha _{1}^{2} \left h_{3}\right ^{2} \left h_{6}\right ^{2}/\left h_{5}\right ^{2}\right) \sigma ^{2}\) can be obtained. It can be proved that \(a_{2} = \sigma _{tot}^{2}/\left (\sigma _{tot}^{2} + \sqrt {\sigma _{tot}^{2} \left (\sigma _{tot}^{2} + \left h_{tot}\right ^{2} P_{s}\right)}\right) < 1/2\), which is consistent with the a_{1}+a_{2}=1 and a_{2}<a_{1} conditions.
4 Performance evaluation
In this section, a linear network is simulated to verify the effectiveness of our proposed schemes. Generally speaking, wireless channel is affected by Rayleigh and shadow fadings. For the resultant channels (in between the source, the relays and the destination) in the considered system, Rayleigh fading is statistically the same, and hence, we only consider the shadow fading in the simulation. To simplify the simulation process, the parameters of the shadow fading, represented by f_{i}=h_{i}^{2}, i∈{1,2,⋯,6}, are generated randomly in the (0,1) range. Furthermore, as two relays are put together, their distance towards the source node and the destination node are equal, and so the allocated power in the odd and even symbol duration are the same. Consequently, for the sake of brevity, we only evaluate the performance of the odd symbol duration. Without loss of generality, it is furthermore assumed that all the Gaussian white additive noises are independent and identically distributed (i.i.d.) with zero mean and unit variance. For the sake of notational simplicity, all the transmit power and power consumption are normalized by the power of the additive noise in the following simulations.
From Figs. 5 and 6, it can be seen that the achievable rate in case 3 is the largest. Moreover, the power consumption of case 2, case 1, and case 3 are equal. It implies that when \(p_{max}^{r} \geq (b  \sqrt {b(bap_{max})})/{a}\) and \(p_{max}^{s} \geq (b + ap_{max} + \sqrt {b(bap_{max})})/a\) hold, the largest achievable rate can be obtained under the same power consumption. The \(p_{max}^{r} \geq (b  \sqrt {b(bap_{max})})/{a}\) and \(p_{max}^{s} \geq (b + ap_{max} + \sqrt {b(bap_{max})})/a\) conditions imply that the optimally allocated power between the relay node and the source node always satisfy their individual constrains. Hence, the resultant power allocation is globally optimal in which the achievable rate is the largest.
5 Conclusion
For a NOMAequipped cooperative relay system with the SIC mode, where the cognitive relays operate in the AF mode, a power allocation scheme was proposed to maximize the achievable rate under some predetermined transmit power constraints. Based on the KKT conditions, network scenarios were divided into two different types. The power allocation to maximize the achievable rate of each type was presented in a closedform expression. Then, based on the optimally allocated source power, a power allocation scheme was proposed for the transmit data streams while targeting on the minimization of the required frequency band. Extensive simulations were conducted to evaluate the performance of the proposed power allocation schemes. As of future research, we would like to extend the singleuser scenario to the multiuser one while keeping the same NOMAbased cooperative relay idea. Incorporating massive millimeter multipleinput multipleoutput (MIMO) technology in such a system could be another possible research direction.
Footnotes
 1.In our considered system, though it is a simple point to point system, when NOMA technique and superposition transmission are taken, two data streams can be transmitted in parallel mode with the same frequency spectrum. In this way, the spectral efficiency can be improved greatly.
 2.
As a>0 and b>0 hold, it can be concluded that \((b + \sqrt {b(b  a p_{max})})/a > b/a = (f_{1} f_{5} p_{max} + f_{3} f_{5} p_{r} + f_{5})/(f_{1} f_{5}  f_{5} f_{6}  f_{3} f_{6}  f_{3} f_{5}) > p_{max}\), which conflicts with the p_{r}+p_{s}=p_{max} constraint, and so it should be excluded. Moreover, it can also be concluded that \(0< (b\sqrt {b(b  {ap}_{max})})/a = b p_{max}/(b + \sqrt {b(b a p_{max})}) < (b p_{max})/b = p_{max}\), which is practical under the given constraint. Hence, the feasible scenario is \(p_{max}^{r} < (b  \sqrt {b(b {ap}_{max})})/a\) when a>0.
 3.
The solution of (16) can be expressed as \(p_{r} = [b \pm \sqrt {b(b  a p_{max})}]/a\). As b−ap_{max}>0 always holds, when a>0, the solution \(p_{r} = [b  \sqrt {b(bap_{max})}]/a < p_{max}\) is feasible. On the other hand, when a<0, the solution \(p_{r} = [b  \sqrt {b(bap_{max})}]/a < p_{max}\) is feasible. Consequently, the available solution is (18) no matter for a>0 or a<0.
Notes
Acknowledgements
Not applicable.
Funding
This work was supported by the Open Project Program of the Key Laboratory of Universal Wireless Communications (2016KFKT2016104), Ministry of Education, the Beijing University of Posts and Telecommunications, the National natural Science Foundation of China (61771414) and the Natural Science Foundation of Hunan Province of China (2017JJ2249).
Availability of data and materials
Not applicable.
Authors’ contributions
SW was responsible for the mathematical derivation and paper writing. SC was responsible for the numerical simulation. RR was responsible for the derivation checking and language smoothing. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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