Robust Secure Energy Efficiency Optimization in the MIMO Wiretap Channel with Energy Harvesting

This work considers a secure multiple-input multiple-output (MIMO) channel where the energy receiver (ER) is a potential eavesdropper (Eve). Specifically, we aim to maximize the secrecy energy efficiency (SEE) via jointly designing the transmit precoding matrix and the artificial noise (AN) covariance at the base station (BS), as well as the power splitting (PS) ratio at the desired receiver (DR), subject to the constraints of the transmit power budget and harvested energy threshold. To handle the formulated highly nonconvex fractional problem, we apply the successive convex approximation (SCA) method to transform the objective and constraint into a tractable form. Then, a penalty-based iterative algorithm is proposed. Finally, simulation results validate the performance of the proposed design.


Introduction
Simultaneous wireless information and power transfer (SWIPT) and energy efficiency (EE) transmission are considered two prominent approaches to the ever increasing demand for energy in the context of next-generation communication [1]. More specifically, in SWIPT-based communications, the receivers are able to salvage electromagnetic energy from radiofrequency (RF) signals. On the other hand, energy-efficient communication is aimed at achieving a certain rate-energy balance by maximizing the EE, defined as the number of delivered bits per unit energy [2].
Traditionally, SWIPT and EE are often independently investigated. Recently, there is growing interest in combining these two concepts together to maximize the EE in the SWIPT system in [3]. Several authors characterized the capacity and energy tradeoffs for SWIPT under different system setups, such as the multiple-input multiple-output (MIMO) channel in [4] and the amplify-and-forward (AF) relay networks [5].
It is a fundamental requirement for the next-generation wireless communication systems to provide secure communication. As a complement to traditional cryptographic encryption, a large amount of literature has been devoted to the physical-layer security (PLS), especially in multiantenna systems [6]. Taking full advantage of the spatial degrees of freedom (DoF) offered by multiple transmit antennas, the transmitter could enhance the overall security by the means of transmit beamforming (BF) and artificial noise (AN) [7]. The effectiveness of transmit BF and AN has also been verified in the secure SWIPT system [8].
Since the goal of secrecy throughput maximization may conflict with the aim of harvested energy maximization, secure EE (SEE) can be considered a performance metric for studying the tradeoff between these two goals. Specifically, for the SEE design, in [9], the authors investigated an artificial noiseaided energy efficiency optimization in the MIMO channel with SWIPT, while in [10], the authors investigated the secure and energy-efficient BF design for multiuser downlink multiple-input single-output (MISO) channel with SWIPT.
From a more practical view, the channel state information (CSI) of the eavesdropper (Eve) may not be perfect obtained by the transmitter due to the existence of channel estimation, quantization, or feedback errors; thus, robust secrecy design has been widely investigated. Specifically, in [11,12], the authors studied the robust secrecy design in the MISO channel, based on the probabilistic constrained robust model and the worst case robust model, respectively, while in [13], the authors investigated the robust harvested energy efficiency optimization for secure MIMO SWIPT systems. In [14], the authors investigated the robust BF designs for secrecy MIMO SWIPT systems, while in [15,16], the authors investigated the outage constrained robust secrecy rate maximization and secrecy energy efficiency maximization design with SWIPT, respectively. The above works were focused on the linear RF energy harvesting (EH) model. Furthermore, by considering a practical nonlinear EH model, in [17], the authors studied the robust BF design in a nonorthogonal multiple access (NOMA) network relying on SWIPT. In [18], the authors studied the computation efficiency maximization in wireless-powered mobile edge computing networks. Moreover, in [19], the authors investigated the AN-aided secure cognitive BF for cooperative MISO-NOMA using SWIPT, while in [20], the authors studied the EE optimization in secure MISO SWIPT systems with a nonlinear EH model. However, these works mainly focus on the MISO channel, and the SEE in the MIMO SWIPT channel has not been investigated yet.
Motivated by these observations, in this work, we focus on the robust SEE design for a SWIPT-based MIMO wiretap channel, where the desired receiver (DR) employs the power splitting (PS) scheme to facilitate the SWIPT and AN is emitted by the transmitter to improve the security. Specifically, we aim to maximize the ratio of the secrecy rate with the total power consumption, by jointly designing the precoding matrix, AN covariance, and the PS ratios. Different from the commonly used Dinkelbach method, we apply the successive convex approximation (SCA) method to transform the constraints into tractable forms, where the fractional objective is turned into exponential functions. Then, to reduce the performance loss introduced by the rank relaxation, a penalty-based iterative algorithm is proposed. Finally, numerical results indicate the performance of the proposed design and provide some meaningful insights.
Our contributions are summarized as follows: (1) To the best of our knowledge, this is the first work to study the SEE optimization in a MIMO SWIPT network. The formulated problem is nonconvex due to the nonconvex objective, the nonconvex constraints, the coupled variables, and the infinite constraints arising from the CSI uncertainty (2) Different from the Dinkelbach algorithm, we do not transform the fractional programming into a subtractive form. However, we directly handle the fractional objective, which is no need of the one-dimensional search procedure, thus reducing the computational complexity. Specifically, a SCA method is utilized to decompose and reformulate the original nonconvex problem, while the CSI error is tackled by the Sprocedure. Then, a penalty-based iterative algorithm is proposed to optimize the BF, the AN covariance, and the PS ratio The rest of this work is organized as follows. The system model and problem statement are given in Section 2. Section 3 investigates the joint precoding, AN, and PS design, where a SCA-based algorithm is proposed. Simulation results are provided in Section 4. Section 5 concludes this work.
1.1. Notations. Throughout this work, boldface lowercase and uppercase letters denote vectors and matrices, respectively. The conjugate, transpose, conjugate transpose, and trace of matrix A are denoted as A † , A T , A H , and TrðAÞ, respectively. a = vecðAÞ denotes stacking the columns of matrix A into a vector a. A ≽ 0 indicates that A is a positive semidefinite matrix. kak denotes the Euclidean norm of vector a. e denotes the element-wise product. I is an identity matrix with proper dimension. CN ð0, IÞ denotes a circularly symmetric complex Gaussian random vector with mean 0 and covariance I. ½x + indicates max ð0, xÞ.

System Model and Problem Statement
2.1. System Model. Let us consider a MIMO downlink system as shown in Figure 1, which consists of one transmitter, one DR, and M energy receivers (ERs). In the network, the DR employs a PS scheme to extract information and power simultaneously while the ER is potential Eve. It is assumed that the transmitter, the DR, and each ER are equipped with N T , N D , and N E antennas, respectively. The channel coefficients between the transmitter and the DR, as well as the In this work, we assume that only imperfect ERs' CSI can be attained. Similar to [13], the imperfect CSI can be modeled as where G m denotes the estimate of the true G m , ΔG m denotes their respective error, and ε m represents the respective size of the bounded error region.

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The information bearing the signal vector s ∈ ℂ d×1 (d ≤ N T ) is precoded by the precoding matrix W ∈ ℂ N T ×d , and AN is used to help energy transfer and improve security. Thus, the transmit signal vector x ∈ ℂ N T ×1 can be expressed as where v ∈ ℂ N T ×1 is the AN vector with v~CN ð0, ZÞ. Without loss of generality (W.l.o.g.), we assume that E½ss H = I. The total transmit power is where μ is the transmitter power amplifier efficiencies, respectively. In addition, P c is the total circuit power consumption for the network. Then, the received signals at the DR and the m-th ER can be expressed as where n D ∈ ℂ N D ×1 and n E,m ∈ ℂ N E ×1 are the noise vectors at the DR and the m-th ER with n D~C N ð0, σ 2 D IÞ and n E,m~C N ð0, σ 2 E,m IÞ, respectively. The received signals at the DR are divided into two streams. One is used to information decoding (ID), and the other is for EH. Specifically, by denoting ρ as the PS ratio at the DR, the signal received for ID and EH at the DR can be expressed as where n P ∈ ℂ N D ×1 is the additional processing noise at the DR with n P~C N ð0, σ 2 P IÞ. It should be noted that, when the harvested power is relatively small, the nonlinear EH model can be approximated as a linear EH model [20]. Thus, in this work, we adapt the linear EH model due to the simplicity. Furthermore, in the simulation part, we will compare the SEE performance in linear and nonlinear EH models, respectively.
Accordingly, the achievable secrecy rate can be expressed as where C D and C E,m denote the mutual information at the DR and the m-th ER, respectively, and are given by On the other hand, the received RF power at the DR and the m-th ER can be expressed as where α and η m denote the energy transform efficiency at the DR and the m-th ER, respectively. Therefore, the SEE for the MIMO network is defined as 2.2. Problem Statement. In this work, we aim to maximize the SEE via jointly designing the transmit precoding matrix and the AN covariance at the transmitter, as well as the PS ratio at the DR, subject to the constraints of the transmit power budget and harvested energy threshold. Specifically, the SEE maximization problem subject to the worst case secrecy rate and minimum EH threshold constraint can be formulated as  Figure 1: The secure MIMO SWIPT system.

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where E R,th and E m,th are the minimum harvested energy threshold at the DR and the m-th ER, respectively. In addition, R th is the minimum secrecy rate and P s is the maximum transmit power.
In fact, there exists another commonly used robust design, e.g., the probabilistic constrained robust design [21], where the authors studied the robust trajectory and transmit power optimization for secure unmanned aerial vehicle-(UAV-) enabled networks and proposed a fair comparison between the performance of the worst case design and the outage case design by setting the radii of the bounded uncertainty regions to a certain value, which is denotes the inverse cumulative distribution function of a chi-square random variable with 2N T N E + 2 degrees of freedom, φ denotes the outage probability, and ϖ m is the region for the probabilistic CSI uncertainty; e.g., we assume that vecðΔG m Þ~CN ð0, ϖ 2 m IÞ. Thus, in this work, we mainly focus on the bounded CSI uncertainty, since the proposed method can be extended to the probabilistic constrained case.

A SCA-Based Method for the SEE Design
P1 is a typical fractional programming, different from the commonly used Dinkelbach algorithm in [22]; we will propose an effective method to handle this obstacle.
Firstly, we need to turn R s ðρ, W, ZÞ into a solvable reformation. Via denoting Q = WW H and t = 1/ρ, we obtain the following problem: where γ R and γ k are given as and γ 1 and γ 2 are the auxiliary variables.
The main difficulty of (11) is that (11b) and (11c) are the difference of convex functions; thus, P2 is still nonconvex. To overcome this difficulty, we will approximate these terms via the first-order Taylor expansion at a fixed point.
Specifically, around given point fQ,Z,tg, we havẽ where A = H HZ H + σ 2 D I +tσ 2 P I and B m = G m H ðQ +ZÞ G m + σ 2 E,m I, respectively. Thus, we have the following approximated problem: Nextly, we will focus on the CSI uncertainties in G m . By introducing slack variables fζ m , θ m g, the constraint γ m ≤ γ 2 can be rewritten as To transform (15a) and (15b) into deterministic reformations, we introduce the following lemma. Lemma 1 [13]. For any positive semidefinite matrix A, the inequality holds jI + Aj ≥ 1 + TrðAÞ and the equality holds if and only if rank ðAÞ ≤ 1.
According to Lemma 1, we obtain the following relationship: Then, we vectorized the equation G m = G m + ΔG m ; e.g., we denote g m = vecðG m Þ, g m = vecð G m Þ, Δg m = vecðΔG m Þ; it is easy to know that kΔG m k F ≤ ε m ⟹ kΔg m k 2 ≤ ε m . 4 Wireless Communications and Mobile Computing By using the identity TrðAB H CDÞ = vecðBÞ H ðA T ⊗ CÞ vecðDÞ, we transform (16) to Nextly, we introduce the following lemma to handle the channel uncertainty.
Lemma 2 [23]. Define the function where provided that there exists a point x 0 such that f 1 ðx 0 Þ < 0.
By using Lemma 2 with respect to Δg m , (17) can be transformed to the following linear matrix inequality (LMI): where ς m ≥ 0 is the auxiliary variable. Following a similar way, (15b) can be transformed as where Besides, the ER's harvested energy constraint (11e) can be turned into where λ m ≥ 0 is the auxiliary variable. By combining these steps, we obtain the following problem: The remaining task is to handle the fractional objective (23a). Different from the commonly used Dinkelbach method, which turns the fractional objective into a subtractive form, we will change (23a) into exponential functions. Then, the monotonicity of exponential functions will be utilized to simplify the objective [24]. Specifically, via introducing slack variables x 1 and x 2 , (23) can be turned into where Ω ≜ fQ, Z, t, γ 1 , γ 2 , ζ m , θ m , ς m , υ m , λ m , x 1 , x 2 g denotes the set of all optimization variables. The only nonconvex part in (24) is (24d). Via the firstorder Taylor expansion, around given pointx 2 , (24d) can be approximated as To this end, we turn (10) into the following problem: s:t: 22b ð Þ, 22c ð Þ, 23 ð Þ: Then, the remaining task is to handle the rank relaxation introduced by Lemma 1. In the following, we will propose a penalty-based method. Firstly, by invoking Lemma 1, we have ln jI + G H m ZG m j is a concave function w.r.t Z. To handle the concavity, the penalty-based method is utilized by putting the constraint into the objective function. Then, (26) can be recast as s:t: 22b ð Þ, 22c ð Þ, 23 ð Þ, ð28bÞ where κ is a penalty factor penalizing the violation of constraint rank ðG H m ZG m Þ = 1. In addition, the upper bound of ln jI + G H m ZG m j can be obtained by using the first-order

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Taylor approximation as Based on (29), (28) can be approximated as s:t: 22b ð Þ, 22c ð Þ, 23 ð Þ, 26c ð Þ: ð30bÞ (30) is convex w.r.t Z, which can be efficiently solved by the convex programming toolbox CVX [25]. The entail iterative algorithm for (30) is summarized in Algorithm 1, in which fQ ðiÞ , Z ðiÞ , t ðiÞ g is the obtained optimal solution in the i-th iteration, respectively. In addition, SEE ðiÞ is the optimal value of (30) in the i-th iteration; κ denotes the stopping criterion, namely, the tolerance ½h.
Here, we analyze the computational complexity of the proposed method. Since (26) involves the LMI constraints and the second-order cone (SOC) constraints, we can estimate the complexity based on the framework in [26]. Specifically, the complexity for a κ-optimal solution to (26) is on the order of ln ð1/κÞ ffiffiffi ffi ϖ p ϑ, where ϖ = 6N T N E + 2N T + 15, ϑ = 6nðN T N E + 1Þ 3 + 36n 2 ðN T N E + 1Þ 2 + 2nN 3 T + 4n 2 N 2 T , and n = Oð2N 2 T Þ, respectively. On the other hand, when the Dinkelbach algorithm is used, the computational complexity is τ ln ð1/κÞ ffiffiffi ffi ϖ p ϑ, where τ denoted the numbers of onedimensional search, ϖ = 6N T N E + 2N T + 12, ϑ = 6n ðN T N E + 1Þ 3 + 36n 2 ðN T N E + 1Þ 2 + 2nN 3 T + 4n 2 N 2 T , and n = Oð2N 2 T Þ, respectively. From this comparison, we can conclude that the proposed method can achieve lower complexity than the Dinkelbach algorithm, mainly due to the fact that the proposed method does not conduct the onedimensional search procedure.

Simulation Results
In this section, we provide some numerical results to testify the availability of our proposed scheme. Unless specified, the simulation setting is assumed as follows: N T = d = 4, N D = 2, N E = 2, M = 2, P s = 10dBW, P c = 0dBW, μ = 1, R th = 2bit/s/ Hz, E R,th = E m,th = −50dBW, ∀m, σ 2 D = σ 2 P = σ 2 E,m = −80dBW, and α = η m = 1, ∀m. In addition, each entry of H and G m is randomly generated by CN ð0, 10 −3 Þ, and the channel uncertainties are ε m = 10 −6 , ∀m. In addition, we also compare our design with several other methods: (1) the proposed method in the case of perfect CSI, which can be seen as the benchmark of the robust design; (2) the no AN design, e.g., setting Z = 0 while only optimizing W and ρ; (3) the fixed PS ratio design, e.g., setting PS ratio ρ = 0:5 while only optimizing W and Z; (4) the nonlinear EH model, e.g., the relationship between the input power E in and output power E out of the EH circuit is modeled as E out = ððM/ð1 + e ð−aðE in −bÞÞ ÞÞ − MΩÞ/ð1 − ΩÞ, where M is a constant denoting the maximum harvested power at the receiver when the EH circuit is saturated, a reflects the nonlinear charging rate with [1] Initialization:i = 1, set P s , E R,th , E m,th , R th . [2] repeat a) Obtain fQ ðiÞ , Z ðiÞ , t ðiÞ g and SEE ðiÞ with fixed fQ,Z,tg via solving (30). b) Update fQ,Z,tg = fQ ðiÞ , Z ðiÞ , t ðiÞ g. c).i = i + 1.
Algorithm 1: Iterative algorithm for (30).  Wireless Communications and Mobile Computing respect to the input power, b is related to the minimum turn-on voltage of an EH circuit, and Ω = 1/ð1 + e ab Þ, where we set M = 20mW, a = 6400, and b = 0:003 [13]; and (5) the Dinkelbach method. These methods are labeled as "proposed design," "perfect CSI," "no AN," "fixed PS," "nonlinear EH," and "Dinkelbach," respectively. Firstly, we investigate the convergency of the proposed design. Figure 2 shows several examples of the convergence behavior with random channel realizations. From this figure, we can see that the proposed method can always converge to the optimal solution within limited iterative numbers.
Nextly, Figure 3 plots the SEE versus the transmit power P s . From Figure 3, we can see that the efficiency firstly increases and then decreases with P s for all these methods while the proposed design outperforms the other designs; this is because when P s is relatively small compared to P c ,     Wireless Communications and Mobile Computing the SEE metric is mainly determined by the information rate in the numerator. By contrast, when P s becomes large, the SEE metric is also determined by the transmit power in the denominator. As a result, the SEE reaches a saturated point. In addition, the proposed method can obtain better performance than several other benchmarks, and the performance gap between the proposed method and the Dinkelbach method is negligible. Besides, the linear EH model can achieve higher SEE than the nonlinear EH model, which is mainly due to that more output power can be obtained for the linear EH circuit when given the same input power.
Then, Figure 4 shows the SEE versus the harvested power threshold E th . From Figure 4, we find that with the increase in E th , the SEE decreases for all the methods. However, the proposed design always achieves a higher efficiency than the other designs. While the no AN method suffers obvious performance loss. This is mainly due to AN's dual role: AN is able to degrade Eve's reception and act as ER's energy source at the same time, which compensates for the loss of secrecy incurred by the SWIPT scheme. Besides, dynamic PS is beneficial to coordinate ID and EH; thus, higher SEE can be obtained than the fixed PS scheme.
Lastly, Figure 5 shows the SEE versus the CSI uncertainty level ε m . From Figure 5, we find that with the increase in ε m , the SEE decreases for all the methods. Since with the increase in ε m , the robust secrecy rate tends to decrease in the condition of the same transmit power, the SEE decreases. This result suggests that the CSI uncertainty level has a huge impact on the SEE performance.

Conclusion
In this work, we have investigated the secrecy design in the MIMO SWIPT channel. Specifically, we formulated the SEE problem by jointly optimizing the precoding matrix, the AN covariance, and the PS ratio with multiple constraints. We utilized the SCA and penalty method to solve the formulated highly nonconvex problem. Simulation results validated that significant SEE improvement can be obtained with the proposed method.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.   Wireless Communications and Mobile Computing