I. INTRODUCTION
THE rapid advancement of social information has enabled the Internet of things (IoT) to gradually demonstrate its increasingly significant role in a wide range of industries [1]–[3]. Wireless-powered backscatter communication (WP-BackCom) as an important and low-power consumption IoT technology has been concerned by academia and industry [4]–[6]. In WP-BackCom networks, sensor nodes are capable of achieving continuous energy self-supply by using radio frequency (RF) energy harvesting (EH) technologies, and efficiently transmitting data by backscattering the surrounding RF signals to the receiver via an impedance-matching way [7]–[9].
A. Related Work
Resource allocation (RA) is a crucial technique in WP-BackCom networks, which can effectively enhance system performance by jointly optimizing system related parameters, such as transmit power, reflection coefficients, time factors, and so on [10]–[30]. Therefore, RA is a very popular research direction in the field of WP-BackCom networks. For instance, the authors in [10] investigated the problem of maximizing the sum rate in multi-carrier WP-BackCom networks. But only one user was considered. In [11], researchers designed a RA algorithm for WP-BackCom networks by jointly considering harvestthen-transmit (HTT) modes and backscatter communication modes. The proposed algorithm achieved the maximum system throughput by jointly optimizing transmit power, mode selection, and time factors. Furthermore, a fairness-based RA problem was studied in [13] to achieve the max-min throughput in an HTT-based WPBackCom network. Then, they extended the work [12] into the max-min energy efficiency (EE) problem [13]. To enhance spectrum utilization and allow more user accessing, a RA algorithm for non-orthogonal multiple access (NOMA)-enhanced WP-BackCom networks had been designed in [14], where the overall throughput of all backscatter devices was achieved. In [15], the total energy consumption was minimized via the joint optimization of traffic offloading and RA strategies in RF-powered backscatter mobile wireless networks. In addition, the issues of relay selection and power control were studied in a relay-aided BackCom network [16]. And the effective EE was maximized in [17]. Due to the impact of eavesdroppers, information security in WP-BackCom networks may not be satisfied. To this end, the authors in [18] investigated the problem of maximizing the achievable secrecy rate-energy region in WP-BackCom networks with a non-linear energy harvester and an eavesdropper by optimizing the backscatter time, power allocation ratios, and artificial noise (AN). Moreover, RA problems in WP-BackCom networks have been extended into the scenarios of full-duplex communication [19], [20], cognitive communication [21], [22], mobile edge computing (MEC)-based communication [23], [24], relay communication [25].
Despite the above works have made significant contributions to the optimal RA problems in WP-BackCom networks with ideal channel information, perfect channel state information (CSI) is difficult to obtain due to feedback delays, estimation errors, etc. To this end, the authors in [26] studied the robust RA problem for WP-BackCom with NOMA and channel uncertainties. Furthermore, based on Gaussian channel error models, a robust EE problem was studied in Backscatter-assisted cooperative vehicle-to-everything (V2X) communications with NOMA [27]. Moreover, a throughputmaximization RA algorithm with imperfect CSI was proposed for NOMA-based cognitive BackCom systems [28]. In [29], a robust RA problem for multi-tag ambient backscatter systems was studied to maximize the minimum user rate by jointly optimizing power allocation coefficients and backscatter time allocation factors, where the channel uncertainties were modeled by the non-convex chance constraints. Moreover, a robust EE-based RA optimization problem with non-linear EH models was studied in [6], where the max-min fairness and security transmission were also considered. In [30], the computation speed and the signal power were jointly optimized to achieve end-to-end latency minimization under imperfect CSI, where the proposed distributionally robust chance-constrained optimization problem was solved by using the Bernstein-typeinequality method and conditional value-at-risk method in the Gaussian distribution and arbitrary distribution of channel estimation errors, respectively.
B. Motivation and Contribution
Based on the above-mentioned works, most of them (i.e., [7]–[25]) have assumed the ideal channel conditions (e.g., perfect CSI) and perfect transmission environments (e.g., without information leakage), it is impossible to exactly obtain the CSI at the base station due to link delays and quantization errors. The CSI mismatch may distort the received signals at the terminals. Thus, robust RA problems with imperfect CSI have been concerned in [27]–[30], they have demonstrated that the robust design can improve the transmission robustness of signals (e.g., lower outage probabilities). However, the free transmission nature of electromagnetic waves also makes the wireless information easy to be wiretapped. Information security is also very important for WP-BackCom networks. In this case, there are a small amount of works focusing on secure transmission [18]. But the linear EH model is considered under perfect CSI. To this end, there is no open work to simultaneously study robustness transmission and secure communication in WP-BackCom networks under practical conditions.
In order to support massive user access and reduce power consumption, this paper studied a robust energy-efficiency optimization problem for NOMA-based multiple-input singleoutput (MISO) WP-BackCom networks under imperfect CSI and multiple eavesdroppers (Eves). The main contributions are listed as follows.
Different from the non-robust algorithms [7]–[25] and the non-secure algorithms [27]–[30], a robust secure RA problem under practical scenarios is formulated in NOMA-based WP-BackCom networks under both channel uncertainties and information security. The total EE of IoT devices is maximized by jointly optimizing the active beamforming and AN at the base station (BS), the transmission power of IoT devices and time factors.
Due to the channel uncertainties in the objective function and the constraints, the energy-efficient optimization problem belongs to a non-convex problem with infinitedimensional constraints. To this end, the robust optimization problem with uncertain parameters is transformed into a deterministic one by applying the worst-case approach, variable substitution, and successive convex approximation (SCA). The fractional objective function is converted into a subtraction form via Dinkelbach’s method. Finally, a block coordinate descent (BCD) based robust RA algorithm is proposed to tackle the above problem.
Simulation results demonstrate that the proposed algorithm has better security and stronger robustness compared to the existing algorithms.
The rest of the paper is organized as follows. Section II gives a NOMA-based WP-BackCom network with multiple Eves and channel uncertainties. In Section III, efficient techniques are used to obtain the solutions of robust RA. Section IV gives the simulation results. Section V concludes the paper.
Notations: [TeX:] $$\mathbf{A}^{\mathrm{H}}, \operatorname{rank}(\mathbf{A}), \text { and } \operatorname{Tr}\{\mathbf{A}\}$$ denote the Hermitian conjugate transpose, rank, and trace, respectively. [TeX:] $$\mathbf{A} \succeq \mathbf{0}$$ denotes a positive semi-definite matrix. [TeX:] $$\mathbb{E}[\cdot]$$ is the statistical expectation. [TeX:] $$\mathcal{C} \mathcal{N}\left(\mu, \delta^2\right)$$ denotes the Gaussian distribution with mean and variance [TeX:] $$\delta^2$$. [TeX:] $$\mathbb{C}^{M \times N}$$ denotes an [TeX:] $$M \times N$$ complex matrix. [TeX:] $$|\cdot|$$ denote the absolute operation. The abbreviations used in this paper are summarized in Table I.
II. SYSTEM MODEL AND PROBLEM FORMULATION
We consider a NOMA-based WP-BackCom network with multiple Eves as depicted in Fig. 1, where there is a N-antenna BS serving an information receiver (IR) through K singleantenna IoT devices. At the same time, there are M singleantenna Eves nearby the IR to wiretap wireless signals. In order to avoid co-channel interference, it is assumed that K IoT devices backscatter their information to the IR by using time division multiple access (TDMA), where T and [TeX:] $$\tau_0$$ denote the time frame length and EH time, respectively. [TeX:] $$\tau_1$$ is the active transmission time for IoT devices with NOMA. If [TeX:] $$\alpha_k$$ is the backscatter time of IoT device k, the corresponding EH time is [TeX:] $$\tau_0-\alpha_k,$$ and satisfies [TeX:] $$\sum_{k=1}^K \alpha_k+\tau_1 \leq T.$$ The sets of IoT devices and Eves are defined as [TeX:] $$\mathcal{K}=\{1, \cdots, K\}(\forall k \in \mathcal{K})$$ [TeX:] $$\mathcal{M}=\{1, \cdots, M\}(\forall k \in \mathcal{M})$$. According to the transmission model, the transmitted signal of the BS is
where [TeX:] $$\boldsymbol{w}_k \in \mathbb{C}^{N \times 1} \text { and } \boldsymbol{z}$$ denote the beamforming vector from the BS to IoT device k and AN vector, respectively. Besides, [TeX:] $$\boldsymbol{z}$$ is a Gaussian signal with mean zero and variance [TeX:] $$\boldsymbol{Z}$$; [TeX:] $$s_k$$ denotes the signal from the BS to IoT device k, and [TeX:] $$\mathbb{E}\left[\left|s_k\right|^2\right]=1.$$
A NOMA-based WP-BackCom network with multiple Eves.
Owing to the impact of obstacles and long-distance transmission, the direct link between the BS and the IR is invalid [31], [32]. Therefore, the received signals of the IR and the mth Eve during time slot k are
where [TeX:] $$\boldsymbol{g}_k \in \mathbb{C}^{N \times 1}$$ denotes the channel vector from the BS to IoT device k. [TeX:] $$f_{m, k} \text { and } h_k$$ denote the channel gains from IoT device k to the mth Eve and the IR, respectively. [TeX:] $$\beta_k$$ denotes the power reflection coefficient at IoT device k. [TeX:] $$n_k^I \sim \mathcal{C} \mathcal{N}\left(0, \sigma_k^2\right)$$ denotes the additive Gaussian white noise (AWGN) with mean zero and variance [TeX:] $$\sigma_k^2$$ at the IR during the kth time slot. [TeX:] $$n_{m, k}^I \sim \mathcal{C N}\left(0, \sigma_{m, k}^2\right)$$ denotes the AWGN with mean zero and variance [TeX:] $$\sigma_{m, k}^2$$ at the mth Eve during the kth time slot; [TeX:] $$c_k$$ denotes the information symbol of IoT device k, and [TeX:] $$\mathbb{E}\left[\left|c_k\right|^2\right]=1.$$
According to the decoding strategy of the successive interference cancellation (SIC) technique [33], the IR first decodes [TeX:] $$s_k,$$ and then detects [TeX:] $$c_k,$$ after removing [TeX:] $$s_k$$ from the received signal. As a result, the achievable throughput of the IR and Eve m during the kth time slot is
where [TeX:] $$r_k^{I R} \text { and } r_{m, k}^{I E}$$ denote the rate of the IR and Eve m during time slot k. Thus, the secrecy rate is
where [TeX:] $$(x)^{+}=\max (0, x).$$
Considering the non-linear EH model [34], [35], the actual harvested power of IoT device k is
where [TeX:] $$P^{s a t}$$ denotes the saturated power. [TeX:] $$\varepsilon_k \in[0,1]$$ denotes the energy conversion efficiency. [TeX:] $$P_k^{I N}$$ represents the power received by the kth IoT device, specifically,
where [TeX:] $$\boldsymbol{W}_k=\boldsymbol{w}_k \boldsymbol{w}_k^H$$ is the covariance matrix, and rank [TeX:] $$\left(\boldsymbol{W}_k\right)=1.$$
Based on the channel training techniques, the IR can obtain the estimated CSI [9]–[15]. Therefore, combining SIC and NOMA, the IR needs to perform a sequential virtual sorting algorithm to sort the signals in a descending order, such as [TeX:] $$h_1 \geq \cdots \geq h_K.$$ During [TeX:] $$\tau_1,$$ the IR initially decodes the signal, subsequently removes it from the received signals, and proceeds with decoding until all signals are successfully decoded [36]. Then, the SINR of the IR and the mth Eve are
where [TeX:] $$\sigma^2$$ denotes the noise power at the IR. [TeX:] $$\sigma_m^2$$ is the noise power at the mth Eve during the active transmission phase, and [TeX:] $$P_k$$ is the active transmit power of IoT device k. Thus, the achievable throughput of the IR and Eve m are
where [TeX:] $$r^{\mathrm{II}, \mathrm{R}} \text { and } r_m^{\mathrm{II,E}}$$ denote the corresponding rates of the IR and Eve m. Therefore, the secrecy rate is
As a result, the sum throughput is
Without loss of generality, defining [TeX:] $$E_k^{\mathrm{C}}$$ as the circuit energy consumption of IoT device k, the harvested energy satisfies
Therefore, the total system energy consumption is
To maximize system EE and guarantee the transmission requirement of each device, the RA optimization problem can be formulated as
where [TeX:] $$P^{\max }$$ is the maximum transmit power threshold at the BS. [TeX:] $$r_k^{I, \min }$$ denotes the minimum secrecy rate threshold during the EH phase. [TeX:] $$r^{\mathrm{II}, \mathrm{min}}$$ denotes the minimum secrecy rate threshold during the active transmission phase. The maximum transmit power at the base station is limited by [TeX:] $$C_1.$$ [TeX:] $$C_2 \text{ and } C_3$$ are the time constraints. [TeX:] $$C_4$$ denotes the EH constraint. [TeX:] $$C_5$$ is the minimum secrecy rate constraint. [TeX:] $$C_7$$ is the rank-one constraint. [TeX:] $$\mathbf{P 1}$$ is a non-convex problem that is difficult to solve.
Since perfect CSI cannot be obtained in practical systems, it is necessary to consider channel uncertainties in [TeX:] $$\mathbf{P 1}$$. Based on the bounded CSI error models1 in [26], we have
where [TeX:] $$\bar{f}_{m, k}$$. denotes the estimated channel. [TeX:] $$\Delta f_{m, k}$$ denotes the corresponding estimation error, [TeX:] $$\varsigma_{m, k} \geq 0$$ is the upper bound of the channel estimation error.
1 Due to the limited information processing capabilities, it is difficult to obtain the statistical channel estimation errors, so that we use the bounded channel uncertainties to model the estimation errors of CSI
Based on (17) and the worst-case approach, the robust form of [TeX:] $$\mathbf{P 1}$$ becomes.
[TeX:] $$\mathbf{P 2}$$ is a non-convex problem with the infinite-dimensional constraints.
III. ROBUST RA ALGORITHM
In this section, we first convert [TeX:] $$\mathbf{P 2}$$ into a deterministic form, then transform it into a convex one that can be tackled by effective convex optimization tools.
A. Transformation Process of Deterministic Form
In order to deal with the uncertainty in [TeX:] $$r_k^{I R}-\max _m\left(\max _{\Delta f_{m, k}} r_{m, k}^{I E}\right) \geq r_k^{I, \min },$$ based on the variable relaxation method, we have the following constraints
where [TeX:] $$v_k, \theta_k, \gamma_k, \omega_{m, k}, \chi_{m, k}, \text { and } \gamma_{m, k}$$ are the slack variables. Therefore, the above secrecy rate constraint can be relaxed as [TeX:] $$\gamma_k-\max _m \gamma_{m, k} \geq r_k^{I, \min }.$$
Because (20) is still non-convex, based on the SCA method and first-order Taylor expansion [26], (20) can be approximated as
where [TeX:] $$x_k^1, x_k^2 \text {, and } x_k^3$$ are the slack variables; [TeX:] $$\bar{x}_k^2 \text { and } \bar{x}_k^3$$ are the last iteration values of [TeX:] $${x}_k^2 \text { and } {x}_k^3,$$ respectively. Similarly, (21) can be approximated as
where [TeX:] $$y_{m, k}^1, y_{m, k}^2, \text { and } y_{m, k}^3$$ are the slack variables; [TeX:] $$\bar{y}_{m, k}^1 \text { and } \bar{\gamma}_{m, k}$$ are the last iteration values of [TeX:] $$y_{m, k}^1 \text { and } \gamma_{m, k},$$ respectively.
Based on the uncertainty set and the worst-case method, we have
To deal with [TeX:] $$r^{\mathrm{II}, \mathrm{R}}-\max _m\left(\max _{\Delta f_{m, k}} r_m^{\mathrm{II}, \mathrm{E}}\right) \geq r^{\mathrm{II}, \min } \text { in } \bar{C}_5,$$ the legitimate rate can be transformed into
Similar to (29), the secrecy rate under the active transmission phase can be expressed as
To deal with the non-convexity of (30), we have
where [TeX:] $$\vartheta, \psi, \mu_m, \varphi_m, \xi \text { and } \varpi_m$$ are the slack variables. Then, [TeX:] $$r^{\mathrm{II}, \mathrm{R}}-\max _m\left(\max _{\Delta f_{m, k}} r_m^{\mathrm{II}, \mathrm{E}}\right) \geq r^{\mathrm{II}, \min }$$ can be relaxed to [TeX:] $$\xi-\max _m \varpi_m \geq r^{\mathrm{II}, \min }.$$
Obviously, (31) and (32) are non-convex, and based on the SCA method and the first-order Taylor expansion, (31) can be relaxed as
where [TeX:] $$z^1, z^2, \text { and } z^3$$ are the slack variables; [TeX:] $$\bar{z}^2 \text { and } \bar{z}^3$$ are the values of the last iteration of [TeX:] $${z}^2 \text { and } {z}^3$$, respectively.
Similarly, (32) can be approximated as
where [TeX:] $$h_m^1, h_m^2, \text { and } h_m^3$$ are the slack variables. [TeX:] $$\bar{h}_m^1 \text { and } \bar{\varpi}_m$$ are the values of the last iteration of [TeX:] $$h_m^1 \text { and } \varpi_m.$$
Based on the Cauchy-Schwartz inequality, (35) can be converted into
Thus, (35) can be rewritten as
Therefore, the sum throughput becomes [TeX:] $$\bar{R}^{\text {SUM }}=\sum_{k=1}^K \alpha_k \gamma_k+\tau_1 \xi.$$ Defining [TeX:] $$\Lambda \triangleq \left\{x_k^1, x_k^2, x_k^3, y_{m, k}^1, y_{m, k}^2, y_{m, k}^3, z^1, z^2, z^3, h_m^1, h_m^2, h_m^3\right\}$$ and [TeX:] $$\Omega \triangleq\left\{v_k, \theta_k, \gamma_k, \omega_{m, k}, \chi_{m, k}, \gamma_{m, k}, \vartheta, \psi, \mu_m, \varphi_m, \xi, \varpi_m\right\},$$ [TeX:] $$\mathbf{P 2}$$ becomes the following deterministic one
Because of the coupled variables in the objective function and [TeX:] $$C_4$$, [TeX:] $$\mathbf{P3}$$ is still a non-convex optimization problem.
B. Robust RA Algorithm
[TeX:] $$\mathbf{P3}$$ is a non-linear fractional programming problem. Based on Dinkelbach’s method [37], the objective function can be reformulated into
where [TeX:] $$\eta \geq 0$$ is an auxiliary variable. It is obvious that [TeX:] $$f(\eta)=\bar{R}^{\mathrm{SUM}}-\eta E^{\mathrm{SUM}}\lt 0$$ holds if tends to positiveinfinity; otherwise [TeX:] $$f(\eta)=\bar{R}^{\mathrm{SUM}}-\eta E^{\mathrm{SUM}} \geq 0$$ holds. Thus, [TeX:] $$f(\eta)=\bar{R}^{\mathrm{SUM}}-\eta E^{\mathrm{SUM}}$$ is a strictly decreasing convex function with the variable . If [TeX:] $$\boldsymbol{W}_k^*, \boldsymbol{Z}^*, P_k^*, \alpha_k^*, \tau_1^*, \Lambda^* \text {, and } \Omega^*$$ are optimal solutions, the maximum EE can be expressed as
Moreover, the auxiliary variable [TeX:] $$\eta^*$$ is
Introducing the auxiliary variable t, [TeX:] $$\mathbf{P3}$$ becomes
Due to the coupled variables in the constraints, [TeX:] $$\mathbf{P4}$$ remains a non-convex problem. By using the alternating optimization technique, [TeX:] $$\mathbf{P4}$$ can be decomposed into two subproblems: 1) a subproblem of beamforming vectors, AN vectors, and active transmit power; 2) a time allocation subproblem. Under the fixed [TeX:] $$\alpha_k$$, [TeX:] $$\mathbf{P4}$$ can be relaxed as
[TeX:] $$\mathbf { P4-A }$$ is still a non-convex problem due to the rank-one constraint [TeX:] $$C_7.$$ To this end, based on the semi-definite relaxation (SDR) approach [38], [39], [TeX:] $$C_7$$ can be relaxed as a convex optimization problem that can be solved via the CVX toolbox [40]. If the matrix satisfies the rank-one condition, a feasible solution can be obtained via the eigenvalue decomposition method. Otherwise, the solution can be recovered via the Gaussian randomization method [41].
Under the fixed [TeX:] $$\boldsymbol{W}_k, \boldsymbol{Z}, P_k \text {, and } t,$$ the time allocation subproblem is
Since the constraints of [TeX:] $$\mathbf { P4-B }$$ are convex, [TeX:] $$\mathbf { P4-B }$$ is a convex optimization problem. Thus, a BCD-based robust RA algorithm is designed in Algorithm 1.
A BCD-based robust RA algorithm
C. Convergence Analysis
In this subsection, the convergence of the proposed algorithm is analyzed. Since both [TeX:] $$\mathbf { P4-A }$$ and [TeX:] $$\mathbf { P4-B }$$ are the standard convex optimization problems, the feasible solutions of [TeX:] $$\boldsymbol{W}_k, \mathbf{Z}, P_k, \alpha_k \text {, and } \tau_1$$ are optimal in each iteration. Therefore, the optimal value of [TeX:] $$\mathbf { P4 }$$ is non-decreasing or remains a constant after each iteration. Moreover, since [TeX:] $$\boldsymbol{W}_k, \boldsymbol{Z}, \text { and } P_k$$ are limited by the maximum transmit power constraint, and [TeX:] $$\alpha_k, \tau_1$$ are constrained by transmission time. As a result, the optimal objective of [TeX:] $$\mathbf { P4 }$$ is bounded by an upper bound. Therefore, the proposed algorithm is converged.
D. Complexity Analysis
The complexity of the proposed algorithm is analyzed in this subsection. Based on the computational complexity of the traditional interior-point method [11], we have
where [TeX:] $$\beta(\kappa)=\sum_{t=1}^p c_t+2(d-p)$$ is the barrier parameter. [TeX:] $$C=e \sum_{t=1}^p c_t^3+e^2 \sum_{t=1}^p c_t^2+e \sum_{t=p+1}^d q_t^2+e^3$$ denotes the iteration expenditure. [TeX:] $$\varpi$$ is the precision. p and (d − p) denote the number of semi-positive constraints and second-order cone constraints, respectively. [TeX:] $$c_t$$ is the dimension of the tth semipositive constraint. [TeX:] $$q_t$$ is the dimension of the tth second-order cone constraint. e denotes the number of optimal variables.
Defining [TeX:] $$\beta_A(\kappa), \beta_B(\kappa), C_A, C_B, \varpi_A, \varpi_B$$ as the barrier parameters, iteration expenditure, the precision of [TeX:] $$\mathbf{P4-A} \text{ and } \mathbf{P4-B}$$, [TeX:] $$\mathcal{O}\left(D_{\mathrm{A}}\right) \text { and } \mathcal{O}\left(D_{\mathrm{B}}\right)$$ as the complexity of [TeX:] $$\mathbf{P4-A} \text{ and } \mathbf{P4-B}$$, the computational complexity can be obtained as
where [TeX:] $$X_{\max }=\frac{1}{\sigma^2} \log \left(L_{\max }\right)$$ is the computational complexity of the BCD method. [TeX:] $$D_{\mathrm{A}}=\sqrt{\beta_A(\kappa)} C_{\mathrm{A}} \ln \left(\frac{1}{\varpi_{\mathrm{A}}}\right)$$ and [TeX:] $$D_{\mathrm{B}}=\sqrt{\beta_A(\kappa)} C_{\mathrm{B}} \ln \left(\frac{1}{\varpi_{\mathrm{B}}}\right)$$. Therefore, the complexity of the proposed algorithm can be obtained as
where [TeX:] $$\beta_A(\kappa)=(6 K+6) M+9 K+(K+1) N+8, C_{\mathrm{A}}=d_{\mathrm{A}} m_1+d_{\mathrm{A}}^2 m_2+d_{\mathrm{A}}^3, \beta_B(\kappa)=2 K+4, C_{\mathrm{B}}=d_{\mathrm{B}} b_1+d_{\mathrm{B}}^2 b_2+d_{\mathrm{B}}^3, d_{\mathrm{A}}=N^2(K+1)+6 M K+7 K+6 M+7, d_{\mathrm{B}}=K+1, m_1=N^3(K+1)+(6 K+6) M+9 K+8, m_2=N^2(K+1)+(6 K+6) M+9 K+8, \text { and } b_1=b_2=2 K+4.$$.
IV. SIMULATION RESULTS
In this section, we validate the effectiveness of the proposed algorithm. The Eves and IR are randomly distributed within a circle with a radius of 4 m centered at a distance of 10 m on the right of the BS, and IoT devices are placed at coordinates (5, 4) and (5, 8) respectively. Without loss of generality, the large-scale fading is characterized by the model [TeX:] $$P L(d)=\rho\left(d / d_0\right)^{-\bar{\alpha}} \text {, }$$ where [TeX:] $$\rho=-30 \mathrm{~dB}$$ represents the path loss, [TeX:] $$d_0=1 \mathrm{~m}$$ denotes the reference distance, d is the distance among transceivers, and [TeX:] $$\bar{\alpha}=3$$ is the path-loss exponent [42]. The spherical channel uncertainty has an upper bound of [0, 0.34] [43]. Other parameters are: [TeX:] $$\sigma_k^2=\sigma_{m, k}^2=\sigma^2=\sigma_m^2=-100 \mathrm{dBm},$$ N = K = M = 2, [TeX:] $$\beta_k=0.2, \varepsilon_k=0.2, \quad P^{\text {sat }}=10 \mathrm{~dBm}$$ [12], [TeX:] $$P^{\text {max }}=39 \mathrm{~dBm}$$ [9], [TeX:] $$r_k^{I, \min }=r^{I I, \min }=0.1 \mathrm{bit} / \mathrm{Hz}$$ [24], T = 1 s, [TeX:] $$\varpi=10^{-3}, L_{\max }=10^5, E_k^C=5 \times 10^{-2} \mathrm{~J}$$ [14]. Moreover, to demonstrate the superiority of the proposed algorithm, the comparison algorithms are explained as follows.
The non-robust algorithm with a non-linear EH model [14]: This algorithm does not consider the influence of channel uncertainties, while the nonlinear EH model is applied at the IoT devices.
The non-robust algorithm with a linear EH model [45]: The algorithm considers perfect CSI, while the EH model at each IoT device is linear.
Fig. 2 shows the convergence of the proposed algorithm. From the figure, the proposed algorithm has fast convergence. Moreover, the system EE decreases with the increasing [TeX:] $$E_k^C,$$ since the system EE is a decreasing function with [TeX:] $$E_k^C$$ according to the utility function in (17).
The convergence of the proposed algorithm.
Fig. 3 presents the system EE versus the upper bound of the CSI estimation error [TeX:] $$\varsigma_{m, k}$$ under different [TeX:] $$\beta_k.$$ It is obvious that the system EE decreases with the increase of [TeX:] $$\varsigma_{m, k}$$. A larger [TeX:] $$\varsigma_{m, k}$$ means the estimated channel is far from the true value, so that the received signal of the IR is heavily distorted. Additionally, the system EE increases under a bigger [TeX:] $$\beta_k,$$ since it can allocate more information rate under a larger reflection coefficient.
The system EE versus [TeX:] $$\varsigma_{m, k}$$ under different [TeX:] $$\beta_k.$$
Fig. 4 compares the system EE with [TeX:] $$\beta_k$$ under different algorithms. From the figure, the system EE improves as the increase of [TeX:] $$\beta_k$$ under different algorithms. The reason is that the increase of [TeX:] $$\beta_k$$ will cause a higher backscatter rate at the EH phase from (4), thereby improving the system EE. On the other hand, under the same [TeX:] $$\beta_k$$, the proposed algorithm has the highest EE, since the proposed algorithm considers the effect of the nonlinear EH model and channel uncertainties in practical systems, so as to improve the system EE while ensuring transmission robustness. Moreover, the total EE of the traditional algorithms is lower than that of our algorithm.
The system EE versus [TeX:] $$\beta_k$$ under different algorithms.
Fig. 5 presents the system EE versus [TeX:] $$\varsigma_{m, k}$$ under different algorithms. Obviously, the system EE decreases with the increasing [TeX:] $$\varsigma_{m, k}$$, since a bigger [TeX:] $$\varsigma_{m, k}$$ brings more channel uncertainties so that it needs more power to overcome the impact caused by channel estimation errors. Additionally, the proposed algorithm has the best EE. Because the channel uncertainty is considered in our algorithm design ahead of time, which reduces the impact of parameter perturbations.
The system EE versus [TeX:] $$\varsigma_{m, k}$$ under different algorithms.
Fig. 6 gives the outage probability versus [TeX:] $$\varsigma_{m, k}(\forall m, k)$$ under different algorithms. With the increase of [TeX:] $$\varsigma_{m, k},$$ the outage probabilities of different algorithms improve accordingly. Furthermore, the proposed algorithm has the lowest outage probability. The reason is that the CSI errors has been considered in the algorithm design ahead of time, it can overcome the impact of CSI estimation errors. Additionally, the non-robust algorithm with the linear EH model has the highest outage probability, since it considers an ideal EH model, which mismatches the practical system.
The outage probability versus [TeX:] $$\varsigma_{m, k}$$ under different algorithms.
V. CONCLUSION
In this paper, a robust secure RA algorithm with nonlinear EH models was proposed to maximize the system EE in a NOMA-based WP-BackCom network under imperfect CSI and multiple Eves. The joint design of active/passive beamforming vectors, AN, transmit power, and time slots were achieved under the secure rate constraint and the minimum EH constraint of each user, the maximum transmit power constraint at the BS, and the transmission time constraint. The non-convex problem was transformed into a convex one by applying a series of convex techniques, then a BCD-based robust algorithm was designed accordingly. The computation complexity and convergence of the proposed algorithm were also analyzed. Simulation results demonstrated the proposed algorithm had a lower outage probability and a higher system EE compared to the non-robust algorithm and the non-secure algorithm.