Next-generation communication systems require self-sustainability wireless nodes to maintain a high data rates network and guarantee quality of service (QoS) [1]. Radio frequency (RF) signal-based simultaneous wireless information and power transfer (SWIPT) is a promising technique for prolonging the lifetime of continuous network operation [2,3]. SWIPT can jointly extract information and replenish energy from the same signal by performing two circuits to separate the information processing and power transfer alternatively [4,5]. Non-orthogonal multiple access (NOMA) has also 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 next-generation communication networks [6]. NOMA can provide the same resource (e.g., time/frequency/code) for multiple users by using different a power level in one subcarrier [7,8]. The combination of NOMA and device-to-device (D2D) communication is essential for alleviating the traffic burden on future networks. In contrast to the traditional concept of “D2D pair”, the concept of “D2D group” involves several D2D receivers that are capable of receiving information from a single D2D transmitter. To further improve the system spectrum efficiency (SE), the full-duplex (FD) transceiver is considered as it can be adopted in simultaneous downlink (DL) and uplink (UL) transmission in the same frequency band [9]. However, FD NOMA communication is more susceptible to eavesdropping compared to conventional half-duplex (HD) OMA. Further, this situation also causes extra energy consumption in SWIPT.

In practical systems, secrecy is a critical concern for the design of wireless communication protocols due to the broadcast nature of the wireless medium [10]. Physical security techniques can improve secure wireless information transmission by generating more interference to potential eavesdroppers [11]. By adding artificial noise (AN) and projecting it onto the null space of information user channels in information transmit beamforming, the potential eavesdroppers would experience a higher noise floor and thus obtain less information about the messages transmitted to the legitimate receivers [12]. In SWIPT systems, the secure communication problem is severer because of larger power consumption in the energy harvesting (EH) task [13]. Besides, taking into account that the base station (BS) may not perfectly know the channel state information (CSI) of the roaming users (RU) also creates a potential vulnerability of ensuring secure communications [10].

Most of the studies mentioned above rely on one case that the BS can get the perfect knowledge of CSI. However, in practice, the BS always has imperfect CSI. To deal with it, we assume 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 [14]. In this paper, we consider D2D-aided FD NOMA-enhanced secure SWIPT communications with imperfect CSI, in which D2D receivers can reuse the same subcarrier occupied by the information transmit user to improve the spectrum utilization in the power domain NOMA. To the best of our knowledge, the existing works cannot use a power efficiency algorithm in secure NOMA- and D2D-enhanced FD SWIPT systems with channel estimation and energy constraints. To study the problem, we propose an algorithm that involves constraints from EH and secure information transmission tasks to jointly extract information and replenish energy. Hence, the proposed algorithm needs to transform the probabilistic non-convex optimization problem into a bounded convex optimization problem. Besides, the existence of a trade-off between UL and DL co-channel interference (CCI) in the FD system needs to be solved by a multi-objective optimization method. Considering all the sub-problems above, a Pareto optimal policy is able to be defined with semidefinite programming (SDP) relaxation [15]. After that, we can iteratively minimize the bounded power allocation coefficients for the objective function of the optimization problem by CVX and guarantee security and QoS, simultaneously.

The rest of the article is organized as follows. The channel model and problem formulation are introduced inSection 2. Then, the proposed algorithm is elaborated inSection 3. InSection 4, we talk about the simulation results, whileSection 5 finally draws the conclusions of this work.

Notation:$A^{-1},A^H,\mathrm{Tr}\left(A\right),\mathrm{Rank}\left(A\right)$ and$\det \left(A\right)$ denote the inverse, Hermitian transpose, trace, rank and determinant of matrix$A$, respectively;$\mathrm{diag}\left(x_1,...,x_M\right)$ is a diagonal matrix with the diagonal elements given by$\left\{x_1,...,x_M\right\}$, and$\mathrm{diag}\left(A\right)$ returns a diagonal matrix having the main diagonal elements of$A$ on its main diagonal;${\left[x\right]}^{+}$ stands for$\max \left\{0,x\right\}$ [16]. In addition, the abbreviations in this work are summarized inTable 1.

In this section, the considered FD NOMA and D2D network in SWIPT along with the channel models is presented. Further, in the formulation of this problem, we first define the secure transmission problem in SWIPT employing a resource allocation scheme and give the imperfect CSI channel model. Then, a non-convex optimization problem is presented with the resource allocation design.

In this part, we first solve the two sub-problems with the Pareto optimality. Then, we analyze the network computational complexity.

Assume that for minimizing total energy consumption, the transmit power of the UL CU first decreases which impairs the objects in Problem 1. The transmit power objects of Problem 1,${\Vert w_{DT}\Vert }^2$ and$\mathrm{Tr}\left(Z\right)$, which result in significant SI should decrease to satisfy the SINR requirement${\mathrm{\Gamma }}_{req}^{UL}$. Besides, as for the links between DUs, the low power of AN causes a higher risk on the information leakage to the RU. Thus, the DT lowers the transmit power to DRs to meet the security requirement$C_{DR-E}$. It seems to converge to a reasonable circumstance that each of the allocated user powers approximately achieves the minimum SINR requirements. However, in practice, the user QoS is dynamically changing all the time. For example,${\mathrm{\Gamma }}_{req}^{DT}$ increases when DT needs a higher bandwidth for a live high-quality video. A higher${\mathrm{\Gamma }}_{req}^{DT}$ directs to a higher${\Vert w_{DT}\Vert }^2$, which puts higher SI on the UL channel. This means$P_{UL}$ also needs to increase to compensate this interference for satisfying the minimum required QoS${\mathrm{\Gamma }}_{req}^{UL}$. As for security, the increment in the BS emission power also causes a higher risk on the information leakage which leads to another power increment of AN. This further impacts the RUs as they need more power to ensure QoS. Therefore, the objectives in the two sub-problems conflict with each other. When we pursue an object to minimize the system consumed power, the result tends to become higher.

To overcome the shortcomings, the multi-objective optimization (MOO) problem is adopted with the concept of Pareto optimality. We first denote sub-problem i as$Q_i\left(Z,w_{DT},w_{DR1},w_{DR2},P_{UL}\right)$. A resource allocation policy that can be set as$\left\{Z,w_{DT},w_{DR1},w_{DR2},P_{UL}\right\}$ can achieve the Pareto optimal if and only if all the set members satisfy$Q_i\left(Z,w_{DT},w_{DR1},w_{DR2},P_{UL}\right)\ge Q_i^{*},\forall i\in \left\{1,2\right\},$ where$Q_i^{*}$ represents the optimal objective value of Problem i. To capture the complete Pareto optimal set, we resort to the weighted Tchebycheff scheme that can jointly solve the trade-off between the two sub-problems in MOO as shown in Problem 3 [10].

Problem 3: where and

$\tau$ denotes an auxiliary optimization variable that targets the total transmit power minimization of the trusted UL and DL users. Variable${\lambda }_i\ge 0,{\displaystyle {\sum }_i{\lambda }_i=1},$ specifies the priority of thei-th objective compared to the other objectives and reflects the preference of the system operator [15].

As$C4$–$C6$ are non-convex constraints and$C4$–$C7$ have uncertain CSI, we consider using a linear relaxation approach to transform$C4$–$C6$ into a set of linear matrix inequalities (LMI) and the generalized S-procedure approach to solve the infinite number of inequality constraints$C4$–$C7$ produced by CSI uncertainty.

First, to handle the non-convex constraints$C4$–$C6$, we note the implications in the following equivalent transformation: and where:$R_{tol}^{CU}={\log }_2\left(1+{\xi }_{tol}^{CU}\right)$,$R_{tol}^{DT}={\log }_2\left(1+{\xi }_{tol}^{DT}\right)$, and$R_{tol}^{DR}={\log }_2\left(1+{\xi }_{tol}^{DR}\right)$.

Next, note that$\tilde{C}4$–$\tilde{C}6$ and$C7$ still involve an infinite number of inequality constraints [17], and we consider introducing the generalized S-procedure to solve it.

Let$f\left(X\right)=X^{H}AX+X^{H}B+B^{H}X+C$, and D⪰0. For$t\ge 0$,$f\left(X\right)$⪰0,$\forall X\in \left\{X|\mathrm{Tr}\left(DX{X}^H\right)\le 1\right\}$ is equivalent to

Since$\tilde{C}4$ involves two coupled estimation error variables, a slack matrix variable$M_{CU}\in$$\mathscr{H}$^M first needs to be introduced to handle it. In particular,$\tilde{C}4$ is equivalently represented by

Then, we apply (35) to$\tilde{C}4a$ and$\tilde{C}4b$ to get$\overline{C}4a$ and$\overline{C}4b$. for${\alpha }_{CU}\ge 0$, and for${\beta }_{CU}\ge 0$.

By substituting$L_E={\widehat{L}}_E+\mathsf{\Delta }{L}_E$ into (33), we have where$\mathsf{\Delta }{L}_E\in \left\{\mathsf{\Delta }{L}_E|\mathrm{Tr}\left({\epsilon }_E^{-2}\mathsf{\Delta }{L}_E\mathsf{\Delta }{L}_E^H\right)\le 1\right\}$ and$W_{DT}=w_{DT}{w}_{DT}^H$. Then, constraint$\tilde{C}5$ is equivalently written as where$B_{L_E}=\left[\begin{array}{cc}{\widehat{L}}_E& I_M\end{array}\right]$.

Similar to$\tilde{C}4,$$\tilde{C}6$ is equivalently represented by where$M_{DR}\in$$\mathscr{H}$^N and$W_{DR}=w_{DR1}^{}{w}_{DR1}^H+w_{DR2}^{}{w}_{DR2}^H$.

Then, we apply (35) to$\tilde{C}6a$ and$\tilde{C}6b$ to get$\overline{C}6a$ and$\overline{C}6b$. for${\alpha }_{DR}\ge 0$, and for${\beta }_{DR}\ge 0$. Similarly, constraint$C7$ is equivalently written as for$t_{RU}\ge 0$.

Now, the MOO problem (29) becomes a convex semidefinite programming (SDP) and can be written as

The convex problem (47) is efficiently optimized by the standard convex programming software named CVX [18]. Note that once we find a rank-one matrix to be the solution of the relaxed SDP problem, the matrix can also be used as a solution of the original problem of (47). Next, we proof the existence of an optimal solution of (47).

In the proposed secure FD-SWIPT network, the CSI received by a device is assumed to be bounded. This circumstance directs the infinite number of CSI values for a user to compute the accurate CSI of another one. Besides, considering the constraints, we treat the optimization problem as a non-convex problem. To directly apply theS-procedure method to the power minimization problem, we introduce several leverage parameters into the non-convex constraints with bounded CSI values. Then, the constraints can be written as equivalent matrices with linear form elements. At last, the non-convex problem is transformed into a convex form, which can be classified as the relaxed SDP problem.

The performance of the proposed MOO resource allocation algorithm is investigated in this section.Table 2 specifies the most important simulation parameters. The RU is equipped withN = 2 antennas. The cell BS is on the center of the cellular withM = 3 antennas with a radius of 1000 m. The small-scale fading is modeled by Rayleith distribution and the large-scale fading can be calculated by (d/d₀)^−η, whered is the distance between two network devices andd₀ is the reference distance, which is set to be 100 m. Besides, the path loss exponentη is set to be 3. We set the communication radius of the D2D group as 300 m.