Power allocation for D2D aided cooperative NOMA system with imperfect CSI

Abstract

Non-orthogonal multiple access (NOMA) has become one of the promising technologies for 5G, which can improve the spectrum resource utilization and system throughput along with the support of effective resource allocation algorithms. Most previous works on NOMA-enhanced cooperative relay systems assume perfect channel state information (CSI), which degrades the performance of their schemes severely when used in imperfect CSI environment. In this article, a power allocation algorithm in D2D aided cooperative NOMA communications with imperfect CSI is proposed. In the proposed algorithm, the probabilistic non-convex optimization problem is transformed into a non-probabilistic convex optimization problem by evaluating the channel gains and using the successive convex programming (SCP) which approximately gives the lower bound of the maximum transmission rate. In the SCP enhanced power allocation algorithm (SCPPAA), we iteratively obtain the sub-optimal power allocation coefficients for the optimization problem by Lagrangian dual multiplier method and Karush–Kuhn–Tucker conditions. This program is divided into two layers for updating the power allocation coefficients and the multipliers respectively. Numerical results demonstrate that our algorithm has a fast convergence performance and the algorithm in D2D-based cooperative NOMA scheme has significant sum-data-rate advantages compared with it in traditional ways.

Introduction

Non-orthogonal multiple access (NOMA) has recently become a key issue of the novel energy and spectrum efficient technologies due to a higher network capacity compared with orthogonal multiple access (OMA) in the fifth generation (5G) environment [1]. NOMA can provide the same resource (e.g., time/frequency/code) for multiple users by using different power level in one subchannel [2, 3]. To facilitate a balance between system throughput and user fairness, NOMA gives less power to the users with better channel conditions and vice versa [4]. In [5] and [6], to improve QoS, an NOMA radio network with simultaneous wireless information and power transfer is studied under a non-linear energy harvesting model. In [7], a method for predefining a minimum transmission rate for each user to guarantee QoS is focused on. And in [8, 9], a new secrecy transmission paradigm and an advanced resource allocation algorithm for uplink and downlink NOMA systems are proposed respectively. The energy efficiency (EE) is studied in a NOMA enabled heterogeneous cloud radio access network (H-CRAN) in [10] in which key technologies in 5G network are discussed to be properly implemented that can be applied in NOMA H-CRANs to improve EE. Most of the algorithms mentioned above show that NOMA technology is capable of satisfying the requirements of the 5G wireless communication standard from different aspects, especially in promoting EE and spectrum efficiency (SE) and supporting more network links. To fully exploit prior information get from NOMA, a cooperative NOMA transmission scheme is proposed in [11].

Apart from NOMA, the device-to-device (D2D) communication has been an essential way to alleviate the upcoming traffic pressure on the near future networks. Owing to rapid development of radio resource management algorithms and new peer discovery methods, D2D communication has been made a significant contribution to increasing SE and EE of 3GPP Long Term Evolution system by sharing spectrum resources with cellular users (CUs) [12]. By using D2D technology, cellular network users can directly communicate with each other, and thus it offloads data traffic of the base station (BS) in the more and more dense cellular network. Recently, many works have been prompted about combing D2D with other technologies in different environments [13].

The promising applications of NOMA technology in D2D communications have been put forward to further improve the potential benefits of EE and SE from the algorithms mentioned above and many models with excellent technologies have been presented [14]–[20]. In [14, 15] models of NOMA-based D2D communications for cooperative relaying systems are proposed. The concept of cooperative relay system (CRS) and the schemes of CRS using NOMA (C-NOMA) and OMA (C-OMA) are also talked about in the simulation [14]. And in [16, 17] the systems are also combine with energy harvesting. Unlike the traditional concept of “D2D pair”, the concept of “D2D group” in which several D2D receivers are capable of receiving information from one D2D transmitter is presented in [18]. In [19], the resource allocation problem of an NOMA-based cellular network is modeled as a Lagrangian function with KKT conditions, in which there are only two D2D power parameters. Different from the matching method of channel assignment for D2D users in [18], the optimization problem is solved by the sub-gradient method [20]. Through the above analysis, we note that NOMA can provide a fair transmission condition with Pareto optimality in power allocation from the game theory and D2D communications are effective means to improve the network capacity through increasing the number of accessed user devices. In [14], a D2D aided cooperative relaying system using NOMA is proposed to enhance the spectral efficiency. Recall that although D2D can unprecedentedly increase the spectrum efficiency, it divides part of the energy from the cellular network [18, 21]. And as D2D links reuse the same spectrum allocated to the cellular users, they may impose more interference on the network [22, 23]. For solving this problem, in [24], a contract-based cooperative spectrum sharing mechanism is proposed which can maximize the sum data rate of D2D links without deteriorating the performance of cellular links. In [25], two modes are developed for cooperatively using full duplex D2D communications and cellular networks. And a scheme named D2D cooperative relay scheme is proposed for human-to-human (H2H) communication to relieve the congestion produced by machine-to-machine (M2M) [26]. Besides that, when the network user number is two, the power allocation scheme can give a global optimization solution in a cooperative relaying system using NOMA [27]. And in [28], a novel power allocation method for multisource multirelay cooperative transmission networks with imperfect CSI is proposed while minimizing outage probability.

Most of the studies mentioned above rely on one case that the BS can get the perfect knowledge of the channel state information (CSI). However, in practice, the BS always has imperfect CSI. To deal with it, we assume that a channel estimation error model where the BS only knows the estimated channel gain and a prior knowledge of the variance of the estimation error [29]–[31]. In this paper, we consider D2D aided downlink single-cell cooperative NOMA communications with imperfect CSI, in which a D2D device can reuse the same subchannel occupied by a cellular user to improve the spectrum utilization. To the best of our knowledge, the existing works cannot use power allocation algorithms in NOMA and D2D enhanced cooperative relaying systems with channel estimation. Considering all the problems mention above, the proposed algorithm first transforms the probabilistic non-convex optimization problem into a non-probabilistic convex optimization problem with successive convex programming (SCP). Then it iteratively computes the sub-optimal power allocation coefficients for the optimization problem by Lagrangian dual multiplier method. The main contributions of this work can be summarized as follows:

  1. (1)

     To the best of our knowledge, the proposed algorithm is firstly used in a cooperative relaying system for the D2D underlaying cellular network combining with the NOMA technology, which is a candidate technology for future networks. And the D2D communication is introduced to offload traffic from the BS and increase network capacity. Moreover, by reusing the SIC decoding results from the prior time phase in the cooperative relaying system, the user can improve communication reliability by comparing the signals from different sources.

  2. (2)

    Considering imperfect CSI in the power allocation optimization problem, we firstly transform the probabilistic problem to a non-probabilistic problem. By using statistical methods to more precisely estimate parameters in the objective function, the outage probability limit constrain in the optimization problem can be solved. Then, we transform the non-convex problem into a convex problem by using power allocation optimization with SCP and find a sub-optimal solution which has a lower computational complexity cost compared with the exhausted search algorithm. At last, by jointly using the Lagrangian dual multiplier method with KKT conditions and the gradient descent method, the power allocation coefficients can be obtained iteratively, so the proposed algorithm can achieve the optimal performance closely.

  3. (3)

    The proposed algorithm is evaluated by numerous times of Monte Carlo simulations. Numerical results show that the proposed algorithm has a fast converge speed. And the power allocation scheme used in an NOMA and D2D aided cooperative relay system with imperfect CSI can achieve a higher sum data rate compared with that used in networks without D2D or that in OMA systems.

The rest of the paper is organized as follows. The channel model and problem formulation are introduced in Section 2. The proposed joint user scheduling, tree-based search power allocation and link selecting algorithm is elaborated in Section 3. In Section 4, the simulation results are presented, while Section 5 finally draws conclusions of the paper.

Network model and problem formulation

System model

We focus on an NOMA-based single-cell downlink scenario in a cooperative communication system which requires a relatively fair way to allocate power to the devices to improve the system capacity and we also use D2D with cooperative relay to further improve the SE. Figure 1 shows the system model of the D2D aided cooperative NOMA (DC-NOMA) system, where a BS, a relay (CU2), a cell user (CU1) and a D2D user (DU) are considered. CU2 is chosen to be the relay because it has the best channel state in the user group and the reason is explained in the next part of this section. As illustrated in Fig. 1, the BS transmits signals to CU2 directly in the first transmission phase. In the next phase the BS and CU2 transmit signals to CU1 respectively and CU2 also sends messages to DU at the same time. The two phases of the DC-NOMA system is described in the following.

Fig. 1
figure1

System model of DC-NOMA

Direct transmission phase

In this phase, the BS sends messages to the CUs based on the NOMA principle, i.e., the BS sends \(\sum\nolimits_{k = 1}^{2} {\sqrt {a_{k} P_{B} } } x_{k}\), where \(x_{k}\) is the modulated symbol, which represents the transmitted message for the \(k{\text{ - th}}\) CU (\(x_{1}\) for CU1,\(x_{2}\) for CU2), and \(a_{k} P_{B}\) denotes the allocated power to the \(k{\text{ - th}}\) CU, where \(P_{B}\) is the total transmit power of the BS, and \(a_{k}\) is the power allocation coefficient for message \(x_{k}\). It is assume that, without loss of generality, the users are ordered according to their CSI at the BS, i.e., \(a_{1} > a_{2}\) represents CU2 has better channel quality than CU1, where \(a_{1} + a_{2} = 1\). The received messages at CU2 is given by

$$y_{2} = h_{2} \sum\nolimits_{k = 1}^{2} {\sqrt {a_{k} P_{B} } } x_{k} + n_{2} ,$$
(1)

where the noise term \(n_{\left( \right)} \sim N_{C} \left( {0,\sigma^{2} } \right)\) is a complex additive white Gaussian noise (AWGN) in different channel at each receiver, and \(h_{2}\) denotes the Rayleigh fading channel gain from BS to CU2.

By treating symbol \(x_{1}\) as noise and using SIC to detect \(x_{2}\) at the end of this phase for CU2, the received signal-to-interference-and-noise ratio (SINR) at CU2 to detect the other user’s message is given by

$$SINR_{2,1} = \frac{{\left| {h_{2} } \right|^{2} a_{1} P_{B} }}{{\left| {h_{2} } \right|^{2} a_{2} P_{B} + \sigma^{2} }},$$
(2)

After the first user’s messages is decoded, the CU2 can decode its own information, the received SINR at CU2 is given by

$$SINR_{2} = \frac{{\left| {h_{2} } \right|^{2} a_{2} P_{B} }}{{\sigma^{2} }},$$
(3)

The conditions under which the second user can decode its own signal is given by \(\log_{2} \left( {1 + SINR_{2,1} } \right) > R_{1}\), where \(R_{1}\) denotes the first user’s targeted data rate for transmitting its own signal.

Cooperative phase

In this phase, the users of DC-NOMA system communicate with each other via short range channels. Particularly the second phase consists of two kinds of users including two CUs and a DU. CU2 broadcasts the superposed messages of CU1 along with the DU and BS also sends messages to DU during this phase in the same time slot. The superposition-coded signal from CU2 during this phase is \(\sqrt {b_{1} P_{R} } x_{1} + \sqrt {b_{2} P_{R} } x_{3}\), where \(P_{R}\) is the total transmit power of CU2, and \(b_{1}\) is the power allocation coefficient for \(x_{1}\), and \(b_{2}\) denotes the power allocation coefficient for \(x_{3}\), which represents the signal for the DU. It is assume that the DU always has higher channel quality than CU1, i.e., \(b_{1} > b_{2}\) where \(b_{1} + b_{2} = 1\). According to the NOMA protocol, the received signals at CU1 from the BS and CU2 are given by, respectively

$$y_{1} = h_{1} \sum\nolimits_{k = 1}^{2} {\sqrt {a_{k} P_{B} } } x_{k} + n_{1} ,$$
(4)

Similary, the received signals at CU1 from CU2 is

$$y_{1,2} = h_{1,2} \left( {\sqrt {b_{1} P_{R} } x_{1} + \sqrt {b_{2} P_{R} } x_{3} } \right) + n_{1,2} ,$$
(5)

where \(h_{1}\) and \(h_{1,2}\) denote the Rayleigh fading channel gains respectively from BS and CU2 to CU1. And CU1 can decode its own information with the following SINR:

$$SINR_{1,1} = \frac{{\left| {h_{1} } \right|^{2} a_{1} P_{B} }}{{\left| {h_{1} } \right|^{2} a_{2} P_{B} + \sigma^{2} }},$$
(6)

where \(SINR_{1,1}\) means the received SINR at CU1 from the BS,

$$SINR_{1,2} = \frac{{\left| {h_{1,2} } \right|^{2} b_{1} P_{R} }}{{\left| {h_{1,2} } \right|^{2} b_{2} P_{R} + \sigma^{2} }},$$
(7)

where \(SINR_{1,2}\) means the received SINR at CU1 from CU2,

$$SINR_{1} = \min \left\{ {SINR_{1,1} , \, SINR_{1,2} } \right\},$$
(8)

where \(SINR_{1}\) means the received SINR at CU1 from the two phases. As the achievable rate of DF relaying is dominated by the weakest link.

Recall that, without D2D, the capacity of C-NOMA and C-OMA at CU1 is lower than that of DC-NOMA [14]. During this phase, the received signal at DU is

$$y_{3} = h_{3} \left( {\sqrt {b_{1} P_{R} } x_{1} + \sqrt {b_{2} P_{R} } x_{3} } \right) + n_{3} ,$$
(9)

where \(h_{3}\) denotes the Rayleigh fading channel gain from CU2 to DU. By treating symbol \(x_{1}\) as noise and using SIC to detect \(x_{3}\) at the end of this phase for the DU, the received SINR at DU to detect the message of CU1 is given by

$$SINR_{3,1} = \frac{{\left| {h_{3} } \right|^{2} b_{1} P_{R} }}{{\left| {h_{3} } \right|^{2} b_{2} P_{R} + \sigma^{2} }},$$
(10)

After the first user’s messages is decoded, the CU2 can decode its own information, the received SINR at CU2 is given by

$$SINR_{3} = \frac{{\left| {h_{3} } \right|^{2} b_{2} P_{R} }}{{\sigma^{2} }},$$
(11)

The conditions under which the DU can decode its own signal is given by \(\log_{2} \left( {1 + SINR_{3,1} } \right) > R_{1}\).

Channel model, instantaneous channel capacity and outage capacity

In this network, we assume that BS cannot get the perfect channel state information (CSI) for the perfect CSI is difficult to obtain in practice. Under this circumstance, we investigate the power optimization by assuming the small scale fading channel can be well estimated by BS. We define \(h_{m} , \, m \in \left\{ {1,2,3} \right\}\)

$$h_{m} = D_{m} g_{m} ,$$
(12)

where \(D_{m} = d_{m}^{{ - \frac{1}{2}}}\) and \(g_{m} \sim N_{C} \left( {0,1} \right)\) account for path loss coefficient and Rayleigh fading channel gain between the BS, the relay and other devices.\(d_{1}\) is the distance between CU1 and the BS.\(d_{2}\) is the distance between CU2 and the BS.\(d_{3}\) is the distance between CU2 and the DU.

Since the path loss and shadowing are large scale fading factors and are slowly varying, we assume that the path loss and shadowing coefficients \(D_{m}\) is perfectly estimated at BS. By using the minimum mean square error channel estimation error model, we can model the Rayleigh fading coefficient among the BS and the devices as

$$g_{m} = \hat{g}_{m} + e_{m} ,$$
(13)

where \(g_{m}\) is the realistic Rayleigh fading channel coefficient, and \(\hat{g}_{m} \sim N_{C} \left( {0, \, 1 - \sigma_{e}^{2} } \right)\) is the estimated channel gain, and \(e_{m} \sim N_{C} \left( {0, \, \sigma_{e}^{2} } \right)\) is the estimated error. It is assume that, in this paper,\(\hat{g}_{m}\) and \(e_{m}\) are uncorrelated.

According to the Shannon’s capacity formula and the DF relaying principle, the maximum achievable data rate of the devices in the network is given by

$$C_{m} = \log_{2} \left( {1 + {\text{SINR}}_{m} } \right), \, m \in \left\{ {1,2,3} \right\},$$
(14)

In practice, with the estimated fading channel coefficient \(\hat{g}_{m}\), the realistic data rate can exceed the maximum achievable data rate easily. Therefore, we adopt the outage probability as a metric to measure the performance of the case when the scheduled data rate exceeds the achievable data rate with imperfect CSI. Referring to the formulas (3), (6), (7), (8) and (11), the received SINR from BS to CU1 under this circumstance is given by

$$\Phi_{1,1} = \frac{{\left| {\hat{h}_{1} } \right|^{2} a_{1} P_{B} }}{{\left| {\hat{h}_{1} } \right|^{2} a_{2} P_{B} + \sigma^{2} }},$$
(15)

where \(\hat{h}_{1} = D_{1} \hat{g}_{1}\) and represents the estimated channel gain of CU1.

The received SINR from CU2 to CU1 is

$$\Phi_{1,2} = \frac{{\left| {\hat{h}_{1,2} } \right|^{2} b_{1} P_{R} }}{{\left| {\hat{h}_{1,2} } \right|^{2} b_{2} P_{R} + \sigma^{2} }},$$
(16)

where \(\hat{h}_{1,2} = D_{1,2} \hat{g}_{1,2}\) and represents the estimated channel gain between the CU1 and CU2 D2D link from the BS.\(D_{1,2} = d_{1,2}^{{ - \frac{1}{2}}}\) and \(d_{1,2}\) is the distance between CU1 and CU2.

According to the Eqs. (15) and (16), we can obtain the received SINR at CU1 for the two phases with imperfect CSI.

$$\Phi_{1} = \min \left\{ {\Phi_{1,1} , \, \Phi_{1,2} } \right\},$$
(17)

The received SINR at CU2 is

$$\Phi_{2} = \frac{{\left| {\hat{h}_{2} } \right|^{2} a_{2} P_{B} }}{{\sigma^{2} }},$$
(18)

And the received SINR at DU is

$$\Phi_{3} = \frac{{\left| {\hat{h}_{3} } \right|^{2} b_{2} P_{R} }}{{\sigma^{2} }},$$
(19)

where \(\hat{h}_{m} = D_{m} \hat{g}_{m} \, {.}\) So the scheduled data rate of the user \(m, \, m \in \left\{ {1,2,3} \right\}\) can be written as

$$r_{m} = \log_{2} \left( {1 + \Phi_{m} } \right),$$
(20)

where \(r_{1}\) denotes the transmission rate of CU1.\(r_{2}\) is the transmission rate of CU2 and \(r_{3}\) is the transmission rate of DU.

Thus the network’s average outage sum rate can be defined as [32].

$$R_{sum} \left( {a_{k} ,b_{k} } \right) = \sum\limits_{m = 1}^{3} {r_{m} } \Pr \left[ {r_{m} \le C_{m} |\hat{g}_{m} } \right], \, m \in \left\{ {1,2,3} \right\},$$
(21)

where \(C_{m}\) represents the upper bound of \(r_{m}\).

Optimization problem formulation

In order to maximize the sum data rate for the DC-NOMA system, the power allocation optimization problem for DC-NOMA system can be formulated as

$$\begin{gathered} \, \mathop {\max }\limits_{{a_{k} ,b_{k} }} R_{sum} \left( {a_{k} ,b_{k} } \right), \hfill \\ s.t. \, C1:Pr\left[ {C_{m} < r_{m} |\hat{g}_{m} } \right] \le \varepsilon_{out} , \, m \in \left\{ {1,2,3} \right\}, \hfill \\ \, C2:\Phi_{m} \ge \Phi_{m}^{thr} , \, m \in \left\{ {1,2,3} \right\}, \hfill \\ \, C3:a_{k} \ge 0, \, b_{k} \ge 0, \, \forall k \in \left\{ {1,2} \right\}, \hfill \\ \, C4:\sum\nolimits_{k = 1}^{2} {a_{k} \le 1, \, } \sum\nolimits_{k = 1}^{2} {b_{k} \le 1} , \hfill \\ \end{gathered}$$
(22)

where C1 is the requirement of the channel outage probability \(\varepsilon_{out}\); C2 is imposed to restrict the data rate of each user must be larger than the constraint of the minimum data rate, and \(\Phi_{m}^{thr}\) is the SINR threshold for the \(m{\text{ - th}}\) user; C3 ensures that the power allocation coefficient of each device is non-negative, and C4 limits the upper bound of the power allocated to each user.

Solution to the optimization problem

The optimization problem (22) here is a non-convex problem as the existence of the co-channel interference with respect to \(a_{k}\) and \(b_{k}\). So the objective function in (18) is a non-convex function with non-convex probabilistic constraints. And the global optimal power allocation is rather difficult to be obtained in practice with affordable systematic and computational efficient approaches and the optimal problem cannot be solved with acceptable complexity in polynomial time. To decrease the complexity cost, we transform the probabilistic mixed problem into a non-probabilistic problem and then we apply sequential convex programming to efficiently solve the power optimization problem.

Optimization problem transformation

According to [30], we first rewrite the data rate in (16) as

$$r_{m} = \log_{2} \left( {1 + \Phi_{m} } \right) = {\text{log}}_{2} \left( {1 + \frac{{\alpha_{m} }}{{\beta_{m} }}} \right),$$
(23)

where \(\Phi_{m} = \frac{{\alpha_{m} }}{{\beta_{m} }} = 2^{{r_{m} }} - 1\) and \(\alpha_{m} = \beta_{m} \left( {2^{{r_{m} }} - 1} \right)\), and the achievable data rate is given by

$$C_{m} = \log_{2} \left( {1 + SINR_{m} } \right) = {\text{log}}_{2} \left( {1 + \frac{{\mu_{m} }}{{\upsilon_{m} }}} \right), \, \forall m,$$
(24)

where \(\mu_{1} = D_{1}^{2} \left| {g_{1} } \right|^{2} a_{1} P_{B} ,\) \(\mu_{1,2} = D_{1,2}^{2} \left| {g_{1,2} } \right|^{2} b_{1} P_{R} , \,\) \(\mu_{2} = D_{2}^{2} \left| {g_{2} } \right|^{2} a_{2} P_{B} ,\) \(\mu_{3} = D_{3}^{2} \left| {g_{3} } \right|^{2} b_{2} P_{R} {,}\) \(\upsilon_{1} = D_{1}^{2} \left| {g_{1} } \right|^{2} a_{2} P_{B} { + }\sigma^{2} ,\) \(\upsilon_{1,2} = D_{1,2}^{2} \left| {g_{1,2} } \right|^{2} b_{2} P_{R} + \sigma^{2} ,\) \(\upsilon_{2} = \upsilon_{3} = \sigma^{2} .\) The bound of the outage probability can be written as

$$Pr\left[ {C_{m} < r_{m} |\hat{g}_{m} } \right] \le \varepsilon_{out} = Pr\left[ {SINR_{m} < \Phi_{m} |\hat{g}_{m} } \right] \le \varepsilon_{out} ,$$
(25)

According to the total probability theorem, the outage probability can be given by

$$\begin{aligned} {\text{ }}Pr\left[ {SINR_{m} < \Phi _{m} |\hat{g}_{m} } \right] = & Pr\left[ {\frac{{\mu _{m} }}{{\upsilon _{m} }} < 2^{{r_{m} }} - 1|\hat{g}_{m} } \right] \\ & {\text{ = }}\Pr \left[ {E1} \right] \cdot Pr\left[ {\mu _{m} \le \alpha _{m} |\hat{g}_{m} } \right] \\ & + \Pr \left[ {E2} \right] \cdot Pr\left[ {\mu _{m} > \alpha _{m} |\hat{g}_{m} } \right], \\ \end{aligned}$$
(26)

where \(Pr\left[ {E1} \right] = Pr\left[ {\frac{{\mu_{m} }}{{\upsilon_{m} }} < 2^{{r_{m} }} - 1|\mu_{m} \le \alpha_{m} , \, \hat{g}_{m} } \right]\) and \(Pr\left[ {E2} \right] = Pr\left[ {\frac{{\mu_{m} }}{{\upsilon_{m} }} < 2^{{r_{m} }} - 1|\mu_{m} > \alpha_{m} , \, \hat{g}_{m} } \right]\).

According to [33], we can transform the outage probability constraint C1 to the following formulas

$$Pr\left[ {\mu_{m} \le \alpha_{m} |\hat{g}_{m} } \right] = \frac{{\varepsilon_{out} }}{2},$$
(27)
$$Pr\left[ {\upsilon_{m} \ge \beta_{m} |\hat{g}_{m} } \right] \le \frac{{\varepsilon_{out} }}{2},$$
(28)

By integrating the probabilistic constraints in (27) and (28) into (22), we can transform the target problem into a revised optimization problem as follows. For CU1, with the Markov inequality, we can get

$$\begin{aligned} Pr\left[ {\upsilon _{1} \ge \beta _{1} |\hat{g}_{1} } \right] & = Pr\left[ {D_{1}^{2} \left| {g_{1} } \right|^{2} a_{2} P_{B} \ge \beta _{1} - \sigma ^{2} |\hat{g}_{1} } \right] \\ & \le \frac{{E\left[ {D_{1}^{2} \left| {g_{1} } \right|^{2} a_{2} P_{B} } \right]}}{{\beta _{1} - \sigma ^{2} }} \\ & = \frac{{D_{1}^{2} \left| {g_{1} } \right|^{2} a_{2} P_{B} }}{{\beta _{1} - \sigma ^{2} }} \\ \end{aligned}$$
(29)

According to (28), let the right side of (29) equal to \(\varepsilon_{out} /2\), we can get

$$\frac{{D_{1}^{2} \left| {g_{1} } \right|^{2} a_{2} P_{B} }}{{\beta_{1} - \sigma^{2} }} = \frac{{\varepsilon_{out} }}{2},$$
(30)

Because \(\left| {g_{1} } \right|^{2} \sim N_{C} \left( {\hat{g}_{1} , \, \sigma_{e}^{2} } \right)\) is a non-central chi-squared distributed random variable with two degrees of freedom, the left side of (27) can be rewritten as

$$\begin{aligned} Pr\left[ {\mu _{m} \le \alpha _{m} |\hat{g}_{m} } \right] = & Pr\left[ {D_{1}^{2} \left| {g_{1} } \right|^{2} a_{1} P_{B} \le \alpha _{1} |\hat{g}_{1} } \right] \\ & = \Pr \left[ {\left| {g_{1} } \right|^{2} \le \frac{{\alpha _{1} }}{{D_{1}^{2} a_{1} P_{B} }}} \right] \\ & = F_{{\left| {g_{1} } \right|^{2} }} \left( {\frac{{\alpha _{1} }}{{D_{1}^{2} a_{1} P_{B} }}} \right) \\ & = 1 - Q_{1} \left( {\sqrt {\frac{{2\left| {\hat{g}_{1} } \right|^{2} }}{{\sigma _{e}^{2} }}} ,{\text{ }}\sqrt {\frac{2}{{\sigma _{e}^{2} }}\frac{{\alpha _{1} }}{{D_{1}^{2} a_{1} P_{B} }}} } \right), \\ \end{aligned}$$
(31)

where \(Q_{1} (a,b) = exp\left( { - \frac{{a^{2} + b^{2} }}{2}} \right)\sum\nolimits_{k = 0}^{\infty } {\left( \frac{a}{b} \right)^{k} I_{k} \left( {ab} \right)}\) is the first-order Marcum Q-function and \(I_{k} \left( {ab} \right)\) is the \(k{\text{ - th}}\) order modified Bessel function of the first kind. Based on (27), let (31) equal to \(\varepsilon_{out} /2\), then we can get

$$\alpha_{1} = F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\frac{{\varepsilon_{out} }}{2}} \right) \cdot D_{1}^{2} a_{1} P_{B} ,$$
(32)

Based on \(\left| {g_{1} } \right|^{2} = \left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} , \, \beta_{1} = \alpha_{1} /\left( {2^{{r_{1} }} - 1} \right)\), (31) and (33), we can get

$$\frac{{D_{1}^{2} \left| {g_{1} } \right|^{2} a_{2} P_{B} }}{{\alpha_{1} /\left( {2^{{r_{1} }} - 1} \right) - \sigma^{2} }} = \frac{{D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} } \right)a_{2} P_{B} }}{{\frac{{F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} a_{1} P_{B} }}{{2^{{r_{1} }} - 1}} - \sigma^{2} }} = \frac{{\varepsilon_{out} }}{2}.$$
(33)

With the same process, we can get the data rate of the devices in the DC-NOMA network as following:

$$\tilde{r}_{m} = \log_{2} \left( {1 + \tilde{\Phi }_{m} } \right),$$
(34)

where \(\tilde{r}_{m}\) is similar to \(r_{m}\). And \(\tilde{\Phi }_{1,1}\) denotes the received SINR from BS to CU1

$$\tilde{\Phi }_{1,1} = \frac{{\varepsilon_{out} F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} a_{1} P_{B} }}{{\varepsilon_{out} \sigma^{2} + 2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} } \right)a_{2} P_{B} }},$$
(35)

\(\tilde{\Phi }_{1,2}\) is the received SINR from CU2 to CU1

$$\tilde{\Phi }_{1,2} = \frac{{\varepsilon_{out} F_{{\left| {g_{1,2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1,2}^{2} b_{1} P_{R} }}{{\varepsilon_{out} \sigma^{2} + 2D_{1,2}^{2} \left( {\left| {\hat{g}_{1,2} } \right|^{2} + \sigma_{e}^{2} } \right)b_{2} P_{R} }},$$
(36)

\(\tilde{\Phi }_{1}\) is the received SINR of CU1

$$\tilde{\Phi }_{1} = \min \left\{ {\tilde{\Phi }_{1,1} , \, \tilde{\Phi }_{1,2} } \right\},$$
(37)

\(\tilde{\Phi }_{2}\) is the received SINR at CU2

$$\tilde{\Phi }_{2} = \frac{{\varepsilon_{out} F_{{\left| {g_{2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{2}^{2} a_{2} P_{B} }}{{2\sigma^{2} }},$$
(38)

and \(\tilde{\Phi }_{3}\) is the received SINR at DU

$$\tilde{\Phi }_{3} = \frac{{\varepsilon_{out} F_{{\left| {g_{3} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{3}^{2} b_{2} P_{R} }}{{2\sigma^{2} }},$$
(39)

And the transformed average sum rate of the DC-NOMA network can be written by

$$\tilde{R}_{sum} = \sum\limits_{m = 1}^{3} {\left( {1 - \varepsilon_{out} } \right)\tilde{r}_{m} } ,$$
(40)

The power allocation optimization for maximizing the system sum rate can be reformulated as

$$\begin{gathered} \mathop { \, \max }\limits_{{a_{m} ,b_{m} }} \, \tilde{R}_{sum} , \hfill \\ s.t. \, C1:\tilde{\Phi }_{m} \ge \Phi_{m}^{thr} , \, \forall m, \hfill \\ \, C2:a_{k} \ge 0, \, b_{k} \ge 0, \, \forall k \in \left\{ {1,2} \right\}, \hfill \\ \, C3:\sum\nolimits_{k = 1}^{2} {a_{k} \le 1, \, } \sum\nolimits_{k = 1}^{2} {b_{k} \le 1} , \hfill \\ \end{gathered}$$
(41)

Since the non-probabilistic optimization problem has co-channel interference, the transformed problem is still non-convex. In the next part of this section, we propose an iterative algorithm to find optimal solution for this challenging problem.

The successive convex programming power allocation algorithm (SCPPAA)

In this section, we propose the power allocation scheme for each user in the system. Because the global optimal solution of the non-convex problem is difficult to obtain with respect to \(a_{m}\) and \(b_{m} \, {.}\) So we use sequential convex programming to derive local optimal solutions by solving a sequence of problems to decrease the computational complexity.

The objective function in (41) can be rewritten as

$$\mathop {\max }\limits_{{a_{k} ,b_{k} }} \, \tilde{R}_{sum} = \left( {1 - \varepsilon_{out} } \right)\sum\nolimits_{m = 1}^{3} {\log_{2} \left( {1 + \tilde{\Phi }_{m} } \right)} ,$$
(42)

As proved in [34], we have

$$\log_{2} \left( {1 + \tilde{\Phi }_{1} } \right) \ge c_{1} \log_{2} \tilde{\Phi }_{1} + d_{1} ,$$
(43)
$$\log_{2} \left( {1 + \tilde{\Phi }_{2} } \right) \ge c_{2} \log_{2} \tilde{\Phi }_{2} + d_{2} ,$$
(44)
$$\log_{2} \left( {1 + \tilde{\Phi }_{3} } \right) \ge c_{3} \log_{2} \tilde{\Phi }_{3} + d_{3} ,$$
(45)

where \(c_{1}\) is defined as

$$c_{1} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{1} }}{{1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{1} }},$$
(46)

\(c_{2}\) is defined as

$$c_{2} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{2} }}{{1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{2} }},$$
(47)

\(c_{3}\) is defined as

$$c_{3} = \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{3} }}{{1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{3} }}.$$
(48)

The \(d_{1}\),\(d_{2}\) and \(d_{3}\) are defined as

$$d_{1} = \log_{2} \left( {1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{1} } \right) - \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{1} }}{{1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{1} }}\log_{2} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{1} ,$$
(49)
$$d_{2} = \log_{2} \left( {1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{2} } \right) - \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{2} }}{{1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{2} }}\log_{2} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{2} ,$$
(50)
$$d_{3} = \log_{2} \left( {1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{3} } \right) - \frac{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{3} }}{{1 + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{3} }}\log_{2} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{3} ,$$
(51)

respectively. And the equalities in (43), (44) and (45) are satisfied when \(\tilde{\Phi }_{1} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{1}\), \(\tilde{\Phi }_{2} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{2}\) and \(\tilde{\Phi }_{3} = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\gamma }_{3}\), respectively.

Based on (43), (44) and (45), the lower bound of the target function in (44) can be expressed as

$$\tilde{R}_{sum} \ge \left( {1 - \varepsilon_{out} } \right)\sum\nolimits_{m = 1}^{3} {\Theta_{m} } ,$$
(52)

where \(\Theta_{m} , \, m \in \left\{ {1,2,3} \right\}\) can be defined as

$$\Theta_{m} = c_{m} \log_{2} \tilde{\Phi }_{m} + d_{m} , \, \forall m,$$
(53)

Set \(a_{1} = 2^{{s_{1} }}\),\(a_{2} = 2^{{s_{2} }}\),\(b_{1} = 2^{{t_{1} }}\) and \(b_{2} = 2^{{t_{2} }}\). We can get a new optimization problem from (42) and (54) as follows:

$$\begin{gathered} \mathop { \, \max }\limits_{x,t} \sum\nolimits_{m = 1}^{3} {\Theta_{m} } , \hfill \\ s.t. \, C1:\tilde{\Phi }_{m} \ge \Phi_{m}^{thr} , \, \forall m, \hfill \\ \, C2:\sum\nolimits_{k = 1}^{2} {2^{{s_{k} }} \le 1, \, } \sum\nolimits_{k = 1}^{2} {2^{{t_{k} }} \le 1} . \hfill \\ \end{gathered}$$
(54)

It can be proved that the new formulated problem is a concave problem [18]. By rearranging \(\Theta_{m}\), we can get that \(\Theta_{1,1}\) denotes the function of transmission from BS to CU1

$$\begin{aligned} \Theta_{1,1} \left( {2^{{s_{1} }} } \right) = & c_{1,1} \left[ {\varepsilon_{out} F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} s_{1} P_{B} } \right. \\ & \left. { - \;\log_{2} \left( {\varepsilon_{out} \sigma^{2} + 2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{s_{2} }} P_{B} } \right)} \right] + d_{1} \\ \end{aligned}$$
(55)

\(\Theta_{1,2}\) denotes the function of transmission from CU2 to CU1

$$\begin{aligned} \Theta_{1,2} \left( {2^{{t_{1} }} } \right) = & c_{1,2} \left[ {\varepsilon_{out} F_{{\left| {g_{1,2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1,2}^{2} t_{1} P_{R} } \right. \\ & \left. { - \log_{2} \varepsilon_{out} \sigma^{2} + 2D_{1,2}^{2} \left( {\left| {\hat{g}_{1,2} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{t_{2} }} P_{R} } \right] + d_{1} , \\ \end{aligned}$$
(56)

\(\Theta_{1}\) is the transmission function of CU1

$$\Theta_{1} = \min \left\{ {\Theta_{1,1} , \, \Theta_{1,2} } \right\},$$
(57)

\(\Theta_{2}\) represents the function of CU2

$$\Theta_{2} \left( {2^{{s_{2} }} } \right) = c_{2} \left[ {\varepsilon_{out} F_{{\left| {g_{2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{2}^{2} s_{2} P_{B} - \log_{2} \left( {2\sigma^{2} } \right)} \right] + d_{2} ,$$
(58)

and \(\Theta_{3}\) is the function of DU

$$\Theta_{3} \left( {2^{{t_{2} }} } \right) = c_{3} \left[ {\varepsilon_{out} F_{{\left| {g_{3} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{3}^{2} t_{2} P_{R} - \log_{2} \left( {2\sigma^{2} } \right)} \right] + d_{3} ,$$
(59)

where \(\Theta_{m} , \, m \in \left\{ {1,2,3} \right\}\) are concave functions of \(s_{k}\) and \(t_{k}\). Thus the problem can be solved by its dual problem. And because the log-sum-exp function is convex, we can conclude that the summation of the three parts of the objective function is a standard convex optimization problem. We first use Lagrangian function to solve the primal problem of (54) which is an associated dual problem. The functions of the first phase can be written by

$$\begin{aligned} L\left( {s_{k} , \, \lambda_{k}^{1} } \right) = & \Theta_{1,1} \left( {2^{{s_{1} }} } \right) + \Theta_{2} \left( {2^{{s_{2} }} } \right) \\ & + \;\lambda_{1}^{1} \left( {1 - \sum\nolimits_{k = 1}^{2} {2^{{s_{k} }} } } \right){ + }\lambda_{2,1}^{1} \left( {{\tilde{\Phi }}_{{1,1}} - {\Phi }_{{1,1}}^{{{\text{thr}}}} } \right){ + }\lambda_{2,2}^{1} \left( {\tilde{\Phi }_{2} - \Phi_{2}^{thr} } \right), \\ \end{aligned}$$
(60)

and the second phase can be written by.

$$\begin{aligned} L\left( {t_{k} , \, \lambda_{k}^{2} } \right) = & \Theta_{1,2} \left( {2^{{t_{1} }} } \right) + \Theta_{3} \left( {2^{{t_{2} }} } \right), \\ & + \;2\lambda_{1}^{2} \left( {1 - \sum\nolimits_{k = 1}^{2} {2^{{t_{k} }} } } \right){ + }\lambda_{2,1}^{2} \left( {{\tilde{\Phi }}_{1,2} - {\Phi }_{{1,2}}^{{{\text{thr}}}} } \right){ + }\lambda_{2,2}^{2} \left( {\tilde{\Phi }_{3} - \Phi_{3}^{thr} } \right), \\ \end{aligned}$$
(61)

where \(\lambda_{k}^{j} , \, j, \, k \in \left\{ {1,2} \right\}\) are the Lagrange multipliers according to the constraints C1 and C2 in (57) that are the KKT conditions of the power allocation optimization problem. Therefore, the dual problem of (54) is

$$\begin{gathered} \mathop { \, \min }\limits_{{\lambda_{k}^{j} }} \mathop {\max }\limits_{{s_{k} ,t_{k} }} \left( {L\left( {s_{k} , \, \lambda_{k}^{1} } \right) + L\left( {t_{k} , \, \lambda_{k}^{2} } \right)} \right) \hfill \\ s.t.\lambda_{k}^{j} \ge 0, \, \forall j,k, \hfill \\ \end{gathered}$$
(62)

To solve (54) and its dual problem, we decompose it into an inner layer and an outer layer. Firstly, in the inner layer, we update the power allocation coefficients \(s_{k}\) and \(t_{k}\) iteratively to tighten the lower bound of (52) until convergence. Then in the outer layer, we update the dual variables \(\lambda_{k}^{j}\) iteratively with the gradient descent method [19].

In the inner layer of the power allocation algorithm, we first fix the Lagrange multipliers, and the problem is transformed to a standard convex optimization problem with KKT conditions.

When decoding the signals for the users, we first let

$$\begin{aligned} \frac{{\partial L\left( {s_{k} ,{\mkern 1mu} \lambda _{k}^{1} } \right)}}{{\partial s_{1} }} = & c_{{1,1}} \varepsilon _{{out}} F_{{\left| {g_{1} } \right|^{2} }}^{{ - 1}} \left( {\varepsilon _{{out}} /2} \right) \cdot D_{1}^{2} P_{B} \\ & - \lambda _{1}^{1} 2^{{s_{1} }} \ln 2 + \lambda _{{2,1}}^{1} \frac{{\varepsilon _{{out}} F_{{\left| {g_{1} } \right|^{2} }}^{{ - 1}} \left( {\varepsilon _{{out}} /2} \right) \cdot D_{1}^{2} P_{B} 2^{{s_{1} }} \ln 2}}{{\varepsilon _{{out}} \sigma ^{2} + 2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma _{e}^{2} } \right)2^{{s_{2} }} P_{B} }} = 0, \\ \end{aligned}$$
(63)
$$\begin{aligned} \frac{{\partial L\left( {s_{k} , \, \lambda_{k}^{1} } \right)}}{{\partial s_{2} }} = & - c_{1,1} \frac{{2^{{s_{2} }} }}{{\varepsilon_{out} \sigma^{2} + 2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{s_{2} }} P_{B} }} \\ & + \;c_{2} \varepsilon_{out} F_{{\left| {g_{2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{2}^{2} P_{B} - \lambda_{1}^{1} 2^{{s_{2} }} \ln 2 \\ & - \;\lambda_{2,1}^{1} \frac{{\left( {2\ln 2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{s_{2} }} P_{B} } \right)\left( {\varepsilon_{out} F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} 2^{{s_{1} }} P_{B} } \right)}}{{\left( {\varepsilon_{out} \sigma^{2} + 2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{s_{2} }} P_{B} } \right)^{2} }} \\ & + \;\lambda_{2,2}^{1} \frac{{\varepsilon_{out} F_{{\left| {g_{2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{2}^{2} P_{B} 2^{{s_{2} }} \ln 2}}{{2\sigma^{2} }} = 0, \\ \end{aligned}$$
(64)
$$\begin{aligned} \frac{{\partial L\left( {t_{k} , \, \lambda_{k}^{2} } \right)}}{{\partial t_{1} }} = & c_{1,2} \varepsilon_{out} F_{{\left| {g_{1,2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1,2}^{2} P_{R} \\ & - \;\lambda_{1}^{2} 2^{{t_{1} }} \ln 2 + \lambda_{2,1}^{2} \frac{{\varepsilon_{out} F_{{\left| {g_{1,2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1,2}^{2} P_{R} 2^{{t_{1} }} \ln 2}}{{\varepsilon_{out} \sigma^{2} + 2D_{1,2}^{2} \left( {\left| {\hat{g}_{1,2} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{t_{2} }} P_{R} }} = 0, \\ \end{aligned}$$
(65)
$$\begin{aligned} \frac{{\partial L\left( {t_{k} , \, \lambda_{k}^{2} } \right)}}{{\partial t_{2} }} = & - c_{1,2} \frac{{2^{{t_{2} }} }}{{\varepsilon_{out} \sigma^{2} + 2D_{1,2}^{2} \left( {\left| {\hat{g}_{1,2} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{t_{2} }} P_{R} }} \\ & + \;c_{3} \varepsilon_{out} F_{{\left| {g_{3} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{3}^{2} P_{R} - \lambda_{1}^{2} 2^{{t_{2} }} \ln 2 \\ & - \;\lambda_{2,1}^{2} \frac{{\left( {2\ln 2D_{1,2}^{2} \left( {\left| {\hat{g}_{1,2} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{t_{2} }} P_{R} } \right)\left( {\varepsilon_{out} F_{{\left| {g_{1,2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1,2}^{2} 2^{{t_{1} }} P_{R} } \right)}}{{\left( {\varepsilon_{out} \sigma^{2} + 2D_{1,2}^{2} \left( {\left| {\hat{g}_{1,2} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{t_{2} }} P_{R} } \right)^{2} }} \\ & + \;\lambda_{2,2}^{2} \frac{{\varepsilon_{out} F_{{\left| {g_{3} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{3}^{2} P_{R} 2^{{t_{2} }} \ln 2}}{{2\sigma^{2} }} = 0, \\ \end{aligned}$$
(66)

respectively.

When \(\varepsilon_{out} \sigma^{2} \ll 1\), (63) can be rewritten as

$$\begin{aligned} \frac{{\partial L\left( {s_{k} , \, \lambda_{k}^{1} } \right)}}{{\partial s_{1} }} = & c_{1,1} \varepsilon_{out} F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} P_{B} \\ & - \;\lambda_{1}^{1} 2^{{s_{1} }} \ln 2 + \lambda_{2,1}^{1} \frac{{\frac{{\varepsilon_{out} F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} P_{B} 2^{{s_{1} }} \ln 2}}{{\varepsilon_{out} \sigma^{2} }}}}{{1 + \frac{{2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{s_{2} }} P_{B} }}{{\varepsilon_{out} \sigma^{2} }}}} \\ & \approx \;c_{1,1} \varepsilon_{out} F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} P_{B} \\ & - \;\lambda_{1}^{1} 2^{{s_{1} }} \ln 2 + \lambda_{2,1}^{1} \frac{{\varepsilon_{out} F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} P_{B} 2^{{s_{1} }} \ln 2}}{{2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} } \right)2^{{s_{2} }} P_{B} }} = 0, \\ \end{aligned}$$
(67)

and we apply the approximation to (62, 63, 64, 65 and 66). Therefore, by deduction and simplification, the optimal power allocation policy of CU2 in the \(i{\text{ - th}}\) iteration can be derived as

$$2^{{s_{2} (i)}} = \frac{{\lambda_{2,1}^{1} \varepsilon_{out} F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} P_{B} 2^{{s_{1} (i)}} \ln 2}}{{\left( {2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma_{e}^{2} } \right)P_{B} } \right)\left( {\lambda_{1}^{1} 2^{{s_{1} (i)}} \ln 2 - c_{1,1} \varepsilon_{out} F_{{\left| {g_{1} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1}^{2} P_{B} } \right)}},$$
(68)

the optimal power allocation policy from BS to CU1 in the \(i{\text{ - th}}\) iteration can be derived as

$$\begin{aligned} 2^{{s_{1} (i)}} = & \frac{{2\left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma _{e}^{2} } \right)2^{{s_{2} (i)}} }}{{\lambda _{{2,1}}^{1} \varepsilon _{{out}} F_{{\left| {g_{1} } \right|^{2} }}^{{ - 1}} \left( {\varepsilon _{{out}} /2} \right)\ln 2}}\left( {\frac{{ - c_{{1,1}} (i)}}{{2D_{1}^{2} \left( {\left| {\hat{g}_{1} } \right|^{2} + \sigma _{e}^{2} } \right)P_{B} }}} \right. \\ & + \;c_{2} (i)\varepsilon _{{out}} F_{{\left| {g_{2} } \right|^{2} }}^{{ - 1}} \left( {\varepsilon _{{out}} /2} \right) \cdot D_{2}^{2} P_{B} - \lambda _{1}^{1} (i)2^{{s_{2} \left( i \right)}} \ln 2 \\ & \left. { + \;\lambda _{{2,2}}^{1} (i)\frac{{\varepsilon _{{out}} F_{{\left| {g_{2} } \right|^{2} }}^{{ - 1}} \left( {\varepsilon _{{out}} /2} \right) \cdot D_{2}^{2} P_{B} 2^{{s_{2} \left( i \right)}} \ln 2}}{{2\sigma ^{2} }}} \right) \\ \end{aligned}$$
(69)

the optimal power allocation policy of the DU in the \(i{\text{ - th}}\) iteration can be derived as.

$$2^{{t_{2} (i)}} = \frac{{\lambda_{2,1}^{2} (i)\varepsilon_{out} F_{{\left| {g_{1,2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1,2}^{2} P_{R} 2^{{t_{1} (i)}} \ln 2}}{{\left( {2D_{1,2}^{2} \left( {\left| {\hat{g}_{1,2} } \right|^{2} + \sigma_{e}^{2} } \right)P_{R} } \right)\left( {\lambda_{1}^{2} (i)2^{{t_{1} (i)}} \ln 2 - c_{1,2} (i)\varepsilon_{out} F_{{\left| {g_{1,2} } \right|^{2} }}^{ - 1} \left( {\varepsilon_{out} /2} \right) \cdot D_{1,2}^{2} P_{R} } \right)}},$$
(70)

and the optimal power allocation policy from CU2 to CU1 in the \(i{\text{ - th}}\) iteration can be derived as

$$\begin{aligned} 2^{{t_{1} (i)}} = & \frac{{2\left( {\left| {\hat{g}_{{1,2}} } \right|^{2} + \sigma _{e}^{2} } \right)2^{{t_{2} (i)}} }}{{\lambda _{{2,1}}^{2} (i)\varepsilon _{{out}} F_{{\left| {g_{{1,2}} } \right|^{2} }}^{{ - 1}} \left( {\varepsilon _{{out}} /2} \right)\ln 2}}\left( {\frac{{ - c_{{1,2}} (i)}}{{2D_{{1,2}}^{2} \left( {\left| {\hat{g}_{{1,2}} } \right|^{2} + \sigma _{e}^{2} } \right)P_{R} }}} \right. \\ & + \;c_{3} (i)\varepsilon _{{out}} F_{{\left| {g_{3} } \right|^{2} }}^{{ - 1}} \left( {\varepsilon _{{out}} /2} \right) \cdot D_{3}^{2} P_{R} - \lambda _{1}^{2} (i)2^{{t_{2} \left( i \right)}} \ln 2 \\ & + \;\lambda _{{2,2}}^{2} (i)\frac{{\varepsilon _{{out}} F_{{\left| {g_{3} } \right|^{2} }}^{{ - 1}} \left( {\varepsilon _{{out}} /2} \right) \cdot D_{3}^{2} P_{R} 2^{{t_{2} \left( i \right)}} \ln 2}}{{2\sigma ^{2} }}, \\ \end{aligned}$$
(71)

where \(i \in \left\{ {0,I_{\max } } \right\}\) represents the number of iterations and \(I_{\max }\) is the maximum value of the iteration index.

Then as for the out layer, using the power allocation scheme, the dual variables can be solved with gradient descent and the Lagrange multipliers can be updated by

$$\lambda_{1}^{1} \left( {i + 1} \right) = \left[ {\lambda_{1}^{1} \left( i \right) - \xi_{1}^{1} \left( i \right)\left( {1 - \sum\nolimits_{k = 1}^{2} {2^{{s_{k} (i)}} } } \right)} \right]^{ + } ,$$
(72)
$$\lambda_{2,1}^{1} \left( {i + 1} \right) = \left[ {\lambda_{2,1}^{1} \left( i \right) - \xi_{2,1}^{1} \left( i \right)\left( {{\tilde{\Phi }}_{{1,1}} (i) - {\Phi }_{{1,1}}^{{{\text{thr}}}} } \right)} \right]^{ + } ,$$
(73)
$$\lambda_{2,2}^{1} \left( {i + 1} \right) = \left[ {\lambda_{2,2}^{1} \left( i \right) - \xi_{2,2}^{1} \left( i \right)\left( {{\tilde{\Phi }}_{2} (i) - {\Phi }_{2}^{{{\text{thr}}}} } \right)} \right]^{ + } ,$$
(74)

where \(\xi_{k}^{j} \left( i \right), \, j, \, k \in \left\{ {1,2} \right\}\) is the positive step size at the \(i{\text{ - th}}\) iteration. And we need to find appropriate step sizes for the convergence of the optimal problem.\(\lambda_{k}^{2}\) can be solved similarly. Since the primal problem of (54) is transformed to a standard convex optimization problem we can prove the convergence of the iteration in chapter IV. The proposed power allocation algorithm based on successive convex programming is shown in Algorithm 1.

figuree

Numerical results and discussions

In this section, we present the performance of the successive convex programming enhanced power allocation algorithm (SCPPAA) with imperfect CSI. We study the convergence performance of SCPPAA. And we investigate the data rate performance when applying this algorithm to three different circumstances including the SCPPAA with perfect CSI, the SCPPAA without D2D and the SCPPAA in OMA to study the potential benefits of the proposed D2D based NOMA scheme and explore how much the channel estimation of the imperfect CSI can influence the system performance. The specific parameter value settings are summarized in Table 1.

Table 1 Propose the simulation parameters of the study

Convergence of the proposed algorithm

The power allocation coefficient of each user versus the number of iterations of SCPPAA is shown in Fig. 2. We set the initial power allocation coefficients as 0.5. The results in Fig. 2 are averaged over 1000 independent adaptation processes which have different distributions of users in one cell. It can be observed that with \(\Delta = 0.01\) the proposed algorithm has a fast converge rate within 10 iterations in average.

Fig. 2
figure2

Power allocation rate of each user versus the number of iterations for Algorithm 1 with \(\Delta\) = 0.01

Figure 3 displays the sum data rate of the whole system versus the number of the iteration index of Algorithm 1 with imperfect CSI. In the figure, the network sum data rate becomes stable within around ten steps of iterations and the changing speed gradually slows down after five steps of iterations averagely. Summarizing the results obtained from Figs. 2 and 3, we can observe the convergence of the power allocation algorithm with the same parameters.

Fig. 3
figure3

The sum data rate versus the number of the iteration index for Algorithm 1 with \(\Delta\) = 0.01

Sum data rate versus locations of devices in the network

Figure 4 displays the sum data rate of the algorithm in four different circumstances mentioned before versus the distance between the CUs and the BS. In the four circumstances, the sum data rate of the network decreases with a changing speed that gradually slows down as the CUs moving away from the BS while the distance between CU2 and the D2D user (DU) is always 20 m. From Fig. 4, the sum data rate of the proposed algorithm is smaller than that in SCPPAA without D2D when the CUs are close to the BS. This is because when a device is located closely to the BS, it may get a larger data rate than accessing to a CU in the network.

Fig. 4
figure4

Plot of the sum data rate with respect to the average distance between CUs and the BS which is changed from 30 to 240 m

Figure 5 displays the sum data rate of SCPPAA in four different circumstances versus the distance between the CU2 and the DU. In the figure, the network sum data rate decreases with a changing speed that gradually slows down as the DU moving away from CU2 while the average distance between CUs and the BS is always 100 m. From Fig. 5, the sum data rate of SCPPAA with imperfect CSI is smaller than that in SCPPAA without D2D when the distance between the DUs is larger than 20 m. This is because the DU may locate closer to the BS than the CU2, and it may get a larger data rate than accessing to a CU which has lower transmission power than the BS.

Fig. 5
figure5

Plot of the sum data rate with respect to the average distance between CU2 and DU which is changed from 10 to 80 m

Summarizing the results obtained. From Figs. 4 and 5, we observe that the data rate of SCPPAA with perfect CSI is always larger than that of SCPPAA with imperfect CSI. This is because with perfect CSI, we do not consider the impact of outage probability on the sum data rate and we can cut down the boundary condition of it in the optimization problem. However, when the outage probability is introduced into the network, the SINR of each user decreases along with the data rate.

Sum data transmission rate versus transmitting power

Figure 6 compares the sum data rates of SCPPAA in different circumstances and with different transmitting power of BS. In the figure, when the transmit SINR is about 10 dB the increasing speed of the sum data rate changed from high to low which corresponds with the parameters of the objective function and after that the changing speed gradually slows down as the transmit SINR increases. When the transmit SINR is 40 dB, SCPPAA with perfect CSI achieves 10.7% rate gain over it with imperfect CSI. And SCPPAA with imperfect CSI achieves 12.2% and 21.5% rate gain over it without D2D and it in OMA. Thus in the four circumstances, the algorithm in NOMA with perfect CSI or imperfect CSI attains the remarkable performance gain over it without D2D and it in OMA.

Fig. 6
figure6

Sum data rate versus transmit SINR:\(\rho_{R} = P_{R} /\sigma^{2} {.}\)

Figure 7 illustrates how the increasing of the transmit SINR affects the sum data rate of the network with different transmit power of CU2 which is also the relay node. The figure shows that in the four circumstances, the algorithm in NOMA with perfect CSI or imperfect CSI attains the remarkable performance gain over it without D2D and it in OMA. When the transmit SINR is 30 dB, SCPPAA with perfect CSI achieves 9.3%, 21.7% and 30.8% rate gain over it with imperfect CSI, SCPPAA without D2D and SCPPAA in OMA, respectively.

Fig. 7
figure7

Sum data rate versus transmit SINR:\(\rho_{B} = P_{B} /\sigma^{2} {.}\)

From the two figures, we can note that the data rate performance is influenced by the transmitting power of each device in the network and the transmit SINR. And the two figures also give comparison of the algorithm under four circumstances to demonstrate the superiority of the NOMA based D2D scheme. Comparing the two figures, we can get that the increment of the transmitting power of BS bring a higher rate gain than the transmitting power of CU2 with the same increment of transmitting power.

Sum data rate versus transmit sin r with different outage probability

Let \(P_{S} = P_{B} + P_{R}\) and \(\rho_{S} = P_{S} /\sigma^{2} {.}\) In Fig. 8, the sum data rate of the proposed algorithms under four circumstances mentioned before versus the transmit SINR is shown with two different outage probabilities. From Fig. 8, the sum data rate of the network increases with the rising of the transmit SINR in a gradually changed speed from high to low. Because of the decreasing of the outage probability, the probability of the users receiving more information from the network increases, which leads to the increasing sum data rate of the whole system. In addition, it also depict that the NOMA based on D2D scheme achieves larger data rate than the conventional OMA way and the scheme without D2D.

Fig. 8
figure8

Sum data rate versus transmit SINR \(\rho_{S}\) with different outage probability

Figure 9 illustrates the performance in sum data rate of the proposed algorithm in different environment mentioned in Sects. 1 and 2 with imperfect CSI. It can be observed from the figure that SCPPAA in DC-NOMA with imperfect CSI is always more beneficial than SCPPAA in C-NOMA and C-OMA with the sane outage probability. When the transmit SINR is 30 dB, SCPPAA in DC-NOMA achieves 24.9% and 33.7% rate gain over it in C-NOMA and C-OMA, respectively.

Fig. 9
figure9

Sum data rate versus transmit SINR \(\rho_{S}\) with different outage probability

Conclusion

In this paper, we propose a power allocation algorithm based on successive convex programming for the D2D-based NOMA-enhanced cooperative communication system to solve the power allocation optimization problem. To deal with the outage probability constraint of the objective function with imperfect CSI, we use the Markov inequality and Marcum Q-function for evaluating the channel gains. After that, the non-probabilistic non-convex optimization problem can be transformed to a standard convex optimization problem by successive convex programming. In the successive convex programming enhanced power allocation algorithm, we iteratively obtain the sub-optimal power allocation coefficients for the optimization problem by Lagrangian dual multiplier method and Karush–Kuhn–Tucker (KKT) conditions. This program is divided into two layers for updating the power allocation coefficients and the multipliers respectively in low complexity which has a fast convergence speed. Moreover, we conduct simulations to evaluate the performance of the proposed algorithm and the sum data rate of the proposed algorithm under four different circumstances. Numerical results demonstrate that our algorithm has a well convergence performance and the algorithm in D2D-based cooperative NOMA scheme has significant sum-data-rate advantages compared with it in traditional ways.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 61601109); the Fundamental Research Funds for the Central Universities (Grant No. N152305001).

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Correspondence to Jingpu Wang.

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Wang, J., Song, X., Dong, L. et al. Power allocation for D2D aided cooperative NOMA system with imperfect CSI. Wireless Netw (2021). https://doi.org/10.1007/s11276-021-02561-x

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Keywords

  • Non-orthogonal multiple access (NOMA),
  • Device-to-device (D2D)
  • Cooperative relaying system
  • Imperfect channel state information (CSI)
  • Successive convex programming (SCP)
  • Power allocation