# Active RIS-Assisted Transmission Design for Wireless Secrecy Network with Energy Harvesting

Authors: Ping Li; Jinhong Bian
Journal: Mathematical Problems in Engineering
 (2023)

---

## Abstract

We investigated the secrecy transmission design in a simultaneous wireless information and power transfer (SWIPT) system, where an active reconfigurable intelligent surface (RIS) is utilized to enhance the secure information transmission as well as the wireless energy transmission. Specifically, we studied the fairness secrecy rate maximization by jointly optimizing the transmit beamformer and the reflecting coefficients, subject to the transmit power constraint and the nonlinear energy harvesting constraint. To solve the formulated nonconvex problem, we utilize the successive convex approximation to reformulate the problem and then propose an alternating optimization to address the approximated problem. The simulation result showed the performance of the proposed design as well as the superiority of active RIS when compared with other benchmarks.


---

## Body

## 1. Introduction

Future wireless networks will meet an increasing requirement for wireless applications such as high spectral efficiency and low power consumption [1]. Since battery capacity is limited, the dual use of radio frequency (RF) signals for enabling simultaneous wireless information and power transfer (SWIPT) has attracted intense interest [2].On the other hand, due to the broadcast character of the wireless channel, the signal sent to the legitimate user maybe eavesdropped by the eavesdropper (Eve) [3]. Normally, to achieve SWIPT efficiently, the energy receiver (ER) is deployed closer to the transmitter (Tx) than the information receiver (IR) to help energy harvesting (EH). However, when the ER tries to decode the confidential information sent to the IR, it will pose a great threat to communication security [4]. To address this issue, physical layer security (PLS) techniques have been proposed to improve the secrecy performance in wireless networks, which utilizes the randomness of the wireless channel to achieve secure communication at the physical layer [5].Nowadays, reconfigurable intelligent surface (RIS) has emerged as a promising technique for the wireless network. The RIS is a planar array with several low-cost reflecting elements, which can alter the incident signal passively [6]. Since it only reflects the received signal, the hardware and power cost for RIS are much lower than the traditional active transmitter or relay [7]. With these advantages, the RIS has sparked great research interests, such as the nonorthogonal multiple access (NOMA) networks [8], the SWIPT networks [9], the hybrid satellite-terrestrial networks [10], the multicell network [11], and the mobile edge computing (MEC) network in [12], where a passive RIS and relay-assisted MEC scheme was proposed.Meanwhile, since RIS can enhance or weak the signal power in different directions, thus it is beneficial to secure transmission design and optimization. To be specific, the authors of [13] investigated the secrecy rate maximization in a downlink multiple-input single-output (MISO) network assisted by a RIS. Then, the authors of [14] studied the secure transmission with multilayer RIS architecture. The authors of [15] studied the energy-efficient beamformer and cooperative jamming design for RIS-assisted networks. Also, the authors of [16] studied the RIS-enhanced secure scheme against both eavesdropping and jamming.However, the reflecting signals suffer from large-scale fading twice. To mitigate this “double fading” effect, a new concept called active RIS has been proposed in [17], where each unit is equipped with active amplification [18]. Then, the authors of [19] studied the optimization technique for active RIS. The authors of [20] compared whether active RIS is superior to passive RIS. Recently, the authors of [21] studied the active RIS-aided secure transmission, where a semidefinite relaxation (SDR) method was proposed to maximize the secrecy rate. Then, the authors of [22, 23] investigated the active RIS-assisted secure transmission for satellite terrestrial networks, and the results suggested the superiority of active RIS when compared with passive RIS in improving the secrecy rate and secrecy energy efficiency performances, respectively.Motivated by this, this work investigates the effect of an active RIS in the SWIPT network. Specifically, by considering individual IRs and ERs, as well as the nonlinear EH model, we handle the fairness secrecy rate maximization objective by jointly designing the transmit beamformer (BF) and the reflecting coefficient (RC), subject to the EH constraints. We utilize the successive convex approximation (SCA) method to recast the problem, and then, an alternating optimization (AO) algorithm is proposed. Besides, the computational complexity is analyzed. The simulation result showed the performance of the proposed design. Our main contribution is as follows:(1)
We propose to use active RIS to improve the secure performance of the SWIPT network, as well as to alleviate the “double fading.” Specifically, we aim to maximize the minimal secrecy rate of the IRs by jointly optimizing the transmit BF and the RC, subject to the nonlinear EH constraints and the transmit power constraints.(2)
The formulated problem is nonconvex due to the nonsmooth max-min secrecy rate objective and the nonconvex EH constraint. To handle this obstacle, we recast the original problem into a quasi-convex problem. Then, we propose an AO algorithm, which optimizes the variables alternatingly. Besides, the proposed algorithm enjoys polynomial-time computational complexity, which is beneficial to implement.(3)
Simulation results show the performance gains of the proposed designs as well as the superiority of active RIS, which not only effectively relieve the “double fading” effect but also enhance the secure performance when compared with other baselines.Although some works such as [2–4, 21] have studied the PLS issue or the energy harvesting issue, however, the differences between our work with these works can be summarized as follows: (1) the system model of our work is quite different with these works. Only the authors of [21] studied the active RIS-assisted secure transmission while the others are related to relay. In addition, the authors of [3, 21] are not related to SWIPT while the authors of [2, 4] are not focus on security communication; (2) the formulated problem and optimization method of our work are quite different with these works. We propose an AO with an SCA-based algorithm to maximize the minimal secrecy rate among these users while guaranteeing the harvested power at each ER exceeds the given threshold. Such a problem has not been studied in these works.The rest part is organized as follows. A system model description and problem formulation are given in Section2. Section 3 develops an AO-based optimization approach. Simulation results are illustrated in Section 4. Finally, Section 5 finishes the paper.Notations: The transpose, conjugate transpose, and trace of a matrix A are denoted as AT, AH, and TrA, respectively. AF denotes the Frobenius norm of matrix A. Diaga is a diagonal matrix with a on the main diagonal. Ra denotes the real part. CN0,A denotes a circularly symmetric complex Gaussian random vector with mean 0 and covariance A.


## 2. System Model and Problem Formulation

### 2.1. System Model

As shown in Figure1, the network consists of one Tx, one RIS, K IRs, and L ERs. The Tx has N antennas, and the RIS has M elements. In addition, each IR/ER is the single antenna. The channel between Tx and the k-th IR/l-th ER is denoted as hT,k∈CN×1 and gT,l∈CN×1. And the channel between Tx and the RIS is denoted as F∈CM×N, and the channel between the RIS and the k-th IR/l-th ER is denoted as hR,k∈CM×1 and gR,l∈CM×1.Figure 1
System model.Then, we introduce the mathematical model of active RIS. To be specific, the RC matrix of the active RIS is given byΦ=Diagϕ1,…,ϕM∈CM×M, where the RC of the m-th element is denoted as ϕm=αmejθm, with αm and θm being the amplitude and the phase within the intervals αm∈0,αm,max and θm∈0,2π. For active RIS, αm,max can be larger than 1.Letsk∈C denote the symbol for the k-th IR. The transmit signal x is(1)x=∑k=1Kwksk+w0,where wk∈CN×1 is the BF for the k-th IR, and w0∈CN×1 is the AN to deteriorate the reception of the information at the ER.Thus, the received signal at thek-th IR/ER is(2a)yi,k=hT,kH+hR,kHΦFx+hR,kHΦnr+ni,k,(2b)ye,l=gT,lH+gR,lHΦFx+gR,lHΦnr+ne,l,where ni,k and ne,l denote the noise at the k-th IR and the l-th ER with ni,k∼CN0,σi,k2 and ne,l∼CN0,σe,l2. In addition, nr is the noise at the RIS with nr∼CN0,σr2.Thus, the signal-to-interference-noise ratio (SINR) for thek-th IR is(3)Γk=hT,kH+hR,kHΦFwk2∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2.When thel-th ER try to eavesdrop the confidential message send to the k-th IR, the SINR is given by(4)Γk,l=gT,lH+gR,lHΦFwk2∑j=0,j≠kKgT,lH+gR,lHΦFwj2+σr2gR,lHΦ2+σe,l2.Thus, the secrecy rate for thek-th IR is(5)Rk=log21+Γk−maxllog21+Γk,l.And the harvested power at the antenna for thel-th ER is(6)Elin=∑k=0KgT,lH+gR,lHΦFwk2+σr2gR,lHΦ2.Here, we employ a nonlinear EH model, which is given as(7)Elprac=ΨpracElin≜Rl/1+e−alElin−bl−Rl/1+ealbl1−1/1+ealbl,where Elprac denotes the practical output power for the EH circuit, and Rl is a constant meaning the maximum harvested power when the EH circuit saturates. al and bl are constants determined by the circuit. For more details about the nonlinear EH model, readers can refer to [24].


### 2.2. Problem Formulation

Our problem is to designwk and Φ, such that the minimal secrecy rate among these IRs can be maximized while guaranteeing the harvested power at each ER is exceed the given threshold. Thus, the problem is(8a)maxwk,ΦGwk,Φ≜minkRk,(8b)s.t.Elprac≥Eth,(8c)∑k=0KΦFwk2+σr2Φ2≤Pr,(8d)∑k=0Kwk2≤Ps,(8e)αm≤αm,max,where Pr and Ps stand for the total power constraint for the RIS and Tx.


## 2.1. System Model

As shown in Figure1, the network consists of one Tx, one RIS, K IRs, and L ERs. The Tx has N antennas, and the RIS has M elements. In addition, each IR/ER is the single antenna. The channel between Tx and the k-th IR/l-th ER is denoted as hT,k∈CN×1 and gT,l∈CN×1. And the channel between Tx and the RIS is denoted as F∈CM×N, and the channel between the RIS and the k-th IR/l-th ER is denoted as hR,k∈CM×1 and gR,l∈CM×1.Figure 1
System model.Then, we introduce the mathematical model of active RIS. To be specific, the RC matrix of the active RIS is given byΦ=Diagϕ1,…,ϕM∈CM×M, where the RC of the m-th element is denoted as ϕm=αmejθm, with αm and θm being the amplitude and the phase within the intervals αm∈0,αm,max and θm∈0,2π. For active RIS, αm,max can be larger than 1.Letsk∈C denote the symbol for the k-th IR. The transmit signal x is(1)x=∑k=1Kwksk+w0,where wk∈CN×1 is the BF for the k-th IR, and w0∈CN×1 is the AN to deteriorate the reception of the information at the ER.Thus, the received signal at thek-th IR/ER is(2a)yi,k=hT,kH+hR,kHΦFx+hR,kHΦnr+ni,k,(2b)ye,l=gT,lH+gR,lHΦFx+gR,lHΦnr+ne,l,where ni,k and ne,l denote the noise at the k-th IR and the l-th ER with ni,k∼CN0,σi,k2 and ne,l∼CN0,σe,l2. In addition, nr is the noise at the RIS with nr∼CN0,σr2.Thus, the signal-to-interference-noise ratio (SINR) for thek-th IR is(3)Γk=hT,kH+hR,kHΦFwk2∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2.When thel-th ER try to eavesdrop the confidential message send to the k-th IR, the SINR is given by(4)Γk,l=gT,lH+gR,lHΦFwk2∑j=0,j≠kKgT,lH+gR,lHΦFwj2+σr2gR,lHΦ2+σe,l2.Thus, the secrecy rate for thek-th IR is(5)Rk=log21+Γk−maxllog21+Γk,l.And the harvested power at the antenna for thel-th ER is(6)Elin=∑k=0KgT,lH+gR,lHΦFwk2+σr2gR,lHΦ2.Here, we employ a nonlinear EH model, which is given as(7)Elprac=ΨpracElin≜Rl/1+e−alElin−bl−Rl/1+ealbl1−1/1+ealbl,where Elprac denotes the practical output power for the EH circuit, and Rl is a constant meaning the maximum harvested power when the EH circuit saturates. al and bl are constants determined by the circuit. For more details about the nonlinear EH model, readers can refer to [24].


## 2.2. Problem Formulation

Our problem is to designwk and Φ, such that the minimal secrecy rate among these IRs can be maximized while guaranteeing the harvested power at each ER is exceed the given threshold. Thus, the problem is(8a)maxwk,ΦGwk,Φ≜minkRk,(8b)s.t.Elprac≥Eth,(8c)∑k=0KΦFwk2+σr2Φ2≤Pr,(8d)∑k=0Kwk2≤Ps,(8e)αm≤αm,max,where Pr and Ps stand for the total power constraint for the RIS and Tx.


## 3. The Proposed Method

Firstly, we turn (8b) to the following constraint Elin≥Ωlin, where(9)Ωlin=bl−InMl/Eth1−1/1+ealbl+Ml/1+ealbl−1al.The detailed procedure is given in the Appendix. Thus, (8b) is recast as(10)∑k=0KgT,lH+gR,lHΦFwk2+σr2gR,lHΦ2≥Ωiin.


### 3.1. The AO Procedure

Firstly, we handle the max–min information rate objective. In fact, by introducing the slack variableτk, rk, and λk,l,∀k,∀l, (8a)–(8d) can be reformulated as(11a)maxwk,Φ,rk,τk,λk,lminτk,(11b)s.t.τk≤log2rkλk,l,∀k,∀l,(11c)1+hT,kH+hR,kHΦFwk2∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2≥rk,(11d)1+gT,kH+gR,kHΦFwk2∑j=0,j≠kKgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2≤1λk,l,(11e)8c,8d,8e(11b) is equivalent to 2τk+2+rk−λk,l2≤rk+λk,l2, which can be further recast as a second-order cone constraint as 2τk+2,rk−λk,l≤rk+λk,l.Then, (11c) can be reformulated as(12)hT,kH+hR,kHΦFwk2rk−1≥∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2.Besides, (11d) can be equivalently rewritten as(13)∑j=0,j≠kKgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2λk,l≥∑k=0KgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2.However, (12) and (13) are all nonconvex, since the difference between the two quadratic forms is neither convex nor concave. To handle the above obstacles, we follow the constrained convex procedure by replacing these functions with first-order expansions. Specifically, we denote fY,yv,v=vHYv/v−y for Y≽0 and v≥y, at a certain point v˜,v˜, and the first-order Taylor expansion of the function is FY,yv,v,v˜,v˜=2Rv˜HYv/v˜−y−v˜HYv˜/v˜−y2v−y [25].By adopting the above Taylor expansion, when fixingΦ, at the given points w˜k,r˜k,λ˜l,k, (12) can be transformed as(14)2Rw˜kHhT,kH+hR,kHΦFHhT,kH+hR,kHΦFwkr˜k−1−hT,kH+hR,kHΦFw˜k2r˜k−12rk−1≥∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2.Similarly, (13) can be approximated as(15)∑j=0,j≠kK2Rw˜jHgT,kH+gR,kHΦFHgT,kH+gR,kHΦFwjλ˜k,l−∑j=0,j≠kKgT,kH+gR,kHΦFw˜j2λ˜k,l2λk,l+σr2gR,kHΦ2+σe,l22λ˜k,l−λk,lλ˜k,l2≥∑k=0KgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2.Then, (13) can be approximated as(16)∑k=0K2Rw˜kHgT,lH+gR,lHΦFHgT,lH+gR,lHΦFwk−∑k=0KgT,lH+gR,lHΦFwk2+σr2gR,lHΦ2+σe,l2≥Ωiin.Following the above step, with givenΦ, we approximate (8a)–(8d) into the following convex problem with respect to (w.r.t.) wkk=0K.(17a)P1:maxwk,rk,λk,l,τkminτk,(17b)s.t.2τk+2,rk−λk,l≤rk+λk,l,∀k,∀l,(17c)14,15,16,8c,8d.Then, we will handle the problem w.r.t.Φ with given wkk=0K.Firstly, around the given pointΦ˜,r˜k,λ˜l,k, we have(18)2RwkHhT,kH+hR,kHΦF˜HhT,kH+hR,kHΦFwkr˜k−1−hT,kH+hR,kHΦF˜wk2r˜k−12rk−1≥∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2.Then, we obtain(19)∑j=0,j≠kK2RwjHgT,kH+gR,kHΦF˜HgT,kH+gR,kHΦFwjλ˜k,l−∑j=0,j≠kKgT,kH+gR,kHΦF˜wj2λ˜k,l2λk,l+2σr2RTrΦ˜HgR,kgR,kHΦλ˜k,l−σr2gR,kHΦ˜2λ˜k,l2λk,l+σe,l22λ˜k,l−λk,lλ˜k,l2≥∑k=0KgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2.Lastly, (10) can be recast as(20)∑k=0K2RwkHgT,lH+gR,lHΦF˜HgT,lH+gR,lHΦFwk−∑k=0KgT,lH+gR,lHΦFwk2+2σr2RTrΦ˜HgR,lgR,lHΦ−σr2gR,lHΦ2+σe,l2≥Ωiin.Thus, we reformulated (8a)–(8d) as the following convex problem w.r.t. Φ.(21a)P2:maxΦ,rk,λk,l,τkminτk,(21b)s.t.2τk+2,rk−λk,l≤rk+λk,l,∀k,∀l,(21c)18,19,20,8d,8e.By combining the above step, we obtain the entail algorithm, which is summarized as Algorithm 1, where each step can be solved by CVX [26].Algorithm 1: The AO algorithm
(1)
Initialize a feasible pointwk0k=0K,Φ0, choose η=0;(2)
repeat(a) Obtainwkηk=0K by solving (17);(b) ObtainΦη by solving (21);(c) Updatew˜kk=0K,Φ˜←wkηk=0K,Φη;(d)η←η+1;(3)
until Converge.(4)
Outputwk⋆k=0K,Φ⋆.It should be pointed out that, during the above procedure, we assume that the phase shift is continuous. In fact, we can extend Algorithm 1 to obtain the discrete phase shiftθmd conveniently. Specifically, we assume that θmd equally spaced takes τ values in the circle F≜0,2π/τ,…,2πτ−1/τ [27]. Then, we project θm into F and select the one which leads to the minimum distance to θm as θmd, i.e., θmd=2πq⋆/τ, where q⋆=argminθm−2πq/τ0≤q≤τ−1. Here, during each step of the iteration in Algorithm 1, we project the obtained θm to F, and set the obtained θmd as a fixed point to update wkk=0K [13].Now, we analyze the computational complexity of Algorithm 1. Specifically, to optimizewkk=0K, according to [28], the complexity is O3N2+2N. Then, to optimize Φ, the complexity is OM2+M. Hence, the complexity of Algorithm 1 is given by OTmax3N2+2N,M2+M, where T denotes the number of iterations.Then, we analyze the convergence of Algorithm 1. In fact, we have(22)Gwk,Φ≥Gηwk,Φ,Gwkη,Φη=Gηwkη,Φη,∀wk,∀Φ,where Gwkη,Φη and Gηwkη,Φη denote the corresponding objective values of (8a)–(8d) and (11a)–(11e), when wk,Φ←wkη,Φη. Therefore, we have(23)Gwkη+1,Φη+1≥aGηwkη+1,Φη+1>bGηwkη,Φη=Gwkη,Φη,where b holds because both wkη+1,Φη+1 and wkη,Φη are the feasible points and optimal solutions of (8a)–(8d), respectively. Thus, wkη+1,Φη+1 is better to (8a)–(8d) than wkη,Φη. Furthermore, due to (8c)–(8e), the sequence wkη,Φη is bounded. Then, according to [29], there exists a sequence wkη,Φη with a limit point wk⋆,Φ⋆, i.e.,(24)limγ⟶+∞Gwkη,Φη−Gwk⋆,Φ⋆=0,thus completes the proof.


## 3.1. The AO Procedure

Firstly, we handle the max–min information rate objective. In fact, by introducing the slack variableτk, rk, and λk,l,∀k,∀l, (8a)–(8d) can be reformulated as(11a)maxwk,Φ,rk,τk,λk,lminτk,(11b)s.t.τk≤log2rkλk,l,∀k,∀l,(11c)1+hT,kH+hR,kHΦFwk2∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2≥rk,(11d)1+gT,kH+gR,kHΦFwk2∑j=0,j≠kKgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2≤1λk,l,(11e)8c,8d,8e(11b) is equivalent to 2τk+2+rk−λk,l2≤rk+λk,l2, which can be further recast as a second-order cone constraint as 2τk+2,rk−λk,l≤rk+λk,l.Then, (11c) can be reformulated as(12)hT,kH+hR,kHΦFwk2rk−1≥∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2.Besides, (11d) can be equivalently rewritten as(13)∑j=0,j≠kKgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2λk,l≥∑k=0KgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2.However, (12) and (13) are all nonconvex, since the difference between the two quadratic forms is neither convex nor concave. To handle the above obstacles, we follow the constrained convex procedure by replacing these functions with first-order expansions. Specifically, we denote fY,yv,v=vHYv/v−y for Y≽0 and v≥y, at a certain point v˜,v˜, and the first-order Taylor expansion of the function is FY,yv,v,v˜,v˜=2Rv˜HYv/v˜−y−v˜HYv˜/v˜−y2v−y [25].By adopting the above Taylor expansion, when fixingΦ, at the given points w˜k,r˜k,λ˜l,k, (12) can be transformed as(14)2Rw˜kHhT,kH+hR,kHΦFHhT,kH+hR,kHΦFwkr˜k−1−hT,kH+hR,kHΦFw˜k2r˜k−12rk−1≥∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2.Similarly, (13) can be approximated as(15)∑j=0,j≠kK2Rw˜jHgT,kH+gR,kHΦFHgT,kH+gR,kHΦFwjλ˜k,l−∑j=0,j≠kKgT,kH+gR,kHΦFw˜j2λ˜k,l2λk,l+σr2gR,kHΦ2+σe,l22λ˜k,l−λk,lλ˜k,l2≥∑k=0KgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2.Then, (13) can be approximated as(16)∑k=0K2Rw˜kHgT,lH+gR,lHΦFHgT,lH+gR,lHΦFwk−∑k=0KgT,lH+gR,lHΦFwk2+σr2gR,lHΦ2+σe,l2≥Ωiin.Following the above step, with givenΦ, we approximate (8a)–(8d) into the following convex problem with respect to (w.r.t.) wkk=0K.(17a)P1:maxwk,rk,λk,l,τkminτk,(17b)s.t.2τk+2,rk−λk,l≤rk+λk,l,∀k,∀l,(17c)14,15,16,8c,8d.Then, we will handle the problem w.r.t.Φ with given wkk=0K.Firstly, around the given pointΦ˜,r˜k,λ˜l,k, we have(18)2RwkHhT,kH+hR,kHΦF˜HhT,kH+hR,kHΦFwkr˜k−1−hT,kH+hR,kHΦF˜wk2r˜k−12rk−1≥∑j=0,j≠kKhT,kH+hR,kHΦFwj2+σr2hR,kHΦ2+σi,k2.Then, we obtain(19)∑j=0,j≠kK2RwjHgT,kH+gR,kHΦF˜HgT,kH+gR,kHΦFwjλ˜k,l−∑j=0,j≠kKgT,kH+gR,kHΦF˜wj2λ˜k,l2λk,l+2σr2RTrΦ˜HgR,kgR,kHΦλ˜k,l−σr2gR,kHΦ˜2λ˜k,l2λk,l+σe,l22λ˜k,l−λk,lλ˜k,l2≥∑k=0KgT,kH+gR,kHΦFwj2+σr2gR,kHΦ2+σe,l2.Lastly, (10) can be recast as(20)∑k=0K2RwkHgT,lH+gR,lHΦF˜HgT,lH+gR,lHΦFwk−∑k=0KgT,lH+gR,lHΦFwk2+2σr2RTrΦ˜HgR,lgR,lHΦ−σr2gR,lHΦ2+σe,l2≥Ωiin.Thus, we reformulated (8a)–(8d) as the following convex problem w.r.t. Φ.(21a)P2:maxΦ,rk,λk,l,τkminτk,(21b)s.t.2τk+2,rk−λk,l≤rk+λk,l,∀k,∀l,(21c)18,19,20,8d,8e.By combining the above step, we obtain the entail algorithm, which is summarized as Algorithm 1, where each step can be solved by CVX [26].Algorithm 1: The AO algorithm
(1)
Initialize a feasible pointwk0k=0K,Φ0, choose η=0;(2)
repeat(a) Obtainwkηk=0K by solving (17);(b) ObtainΦη by solving (21);(c) Updatew˜kk=0K,Φ˜←wkηk=0K,Φη;(d)η←η+1;(3)
until Converge.(4)
Outputwk⋆k=0K,Φ⋆.It should be pointed out that, during the above procedure, we assume that the phase shift is continuous. In fact, we can extend Algorithm 1 to obtain the discrete phase shiftθmd conveniently. Specifically, we assume that θmd equally spaced takes τ values in the circle F≜0,2π/τ,…,2πτ−1/τ [27]. Then, we project θm into F and select the one which leads to the minimum distance to θm as θmd, i.e., θmd=2πq⋆/τ, where q⋆=argminθm−2πq/τ0≤q≤τ−1. Here, during each step of the iteration in Algorithm 1, we project the obtained θm to F, and set the obtained θmd as a fixed point to update wkk=0K [13].Now, we analyze the computational complexity of Algorithm 1. Specifically, to optimizewkk=0K, according to [28], the complexity is O3N2+2N. Then, to optimize Φ, the complexity is OM2+M. Hence, the complexity of Algorithm 1 is given by OTmax3N2+2N,M2+M, where T denotes the number of iterations.Then, we analyze the convergence of Algorithm 1. In fact, we have(22)Gwk,Φ≥Gηwk,Φ,Gwkη,Φη=Gηwkη,Φη,∀wk,∀Φ,where Gwkη,Φη and Gηwkη,Φη denote the corresponding objective values of (8a)–(8d) and (11a)–(11e), when wk,Φ←wkη,Φη. Therefore, we have(23)Gwkη+1,Φη+1≥aGηwkη+1,Φη+1>bGηwkη,Φη=Gwkη,Φη,where b holds because both wkη+1,Φη+1 and wkη,Φη are the feasible points and optimal solutions of (8a)–(8d), respectively. Thus, wkη+1,Φη+1 is better to (8a)–(8d) than wkη,Φη. Furthermore, due to (8c)–(8e), the sequence wkη,Φη is bounded. Then, according to [29], there exists a sequence wkη,Φη with a limit point wk⋆,Φ⋆, i.e.,(24)limγ⟶+∞Gwkη,Φη−Gwk⋆,Φ⋆=0,thus completes the proof.


## 4. Simulation Results

The deployment is given in Figure2, where there exist one Tx, one RIS, 2 IRs, and 2 ERs. The coordinates of Tx and RIS are 0m,0m,10m and 50m,10m,10m, while the IRs and ERs are randomly located in a circle with radius 5m and centered at 60m,0m,1.5m, respectively. The following settings are adopted: N=4, M=40, τ=4, [6], αn,max=10,∀n, [17]. Besides, σi,k2=σe,l2=−80dBm,∀k,∀l, and σr2=−80dBm [19], and al=1500, bl=0.0022, and Rl=3.9mW [24]. The other parameters are the same as those in [13].Figure 2
Simulation deployment.We compare the proposed scheme with the following baselines: (1) the discrete phase shift scheme; (2) the passive RIS with relay scheme [12]; (3) the passive RIS-assisted scheme; (4) the no RIS case, which are labeled as “Active RIS,” “Discrete,” “Relay” “Passive RIS,” “No RIS,” respectively.Figure3 shows the obtained secrecy rate versus the number of iterations with different N and M. From Figure 3, we can see that the secrecy rate always increases with the iteration numbers and converges within 20 iterations, which verifies the convergence of the AO method.Figure 3
Convergence.First, we show the secrecy rate versusPs in Figure 4. As we can see, the secrecy rate increases with Ps, and all RIS-aided schemes outperform the no-RIS-aided designs. Besides, the active RIS scheme significantly outperforms the passive RIS design, since the power amplification can alleviate the impact of “double fading.” Thus, the received signal power at the IRs is enhanced. The discrete phase shifts suffer certain performance losses when compared with the continuous case. However, the loss is very slight, which suggested the effectiveness of the discrete operation. In addition, we can see that the proposed design outperforms the passive RIS with a relay scheme, mainly due to the reason that the passive RIS with a relay scheme needs two continuous time slots to achieve information transmission [30].Figure 4
Secrecy rate versus the transmit power.Then, we show the secrecy rate versusM in Figure 5, where we can see that for all these methods, the secrecy rate tends to increase with M. This is because more signals can reach the RIS with larger M. Besides, active RIS obtains better performance than passive RIS.Figure 5
Secrecy rate versus the number of RIS elements.Next, we show the secrecy rate versus the EH constraint in Figure6, where we can see that the secrecy rate decreases with the EH threshold. Since with a higher threshold, more signals need to transmit in the ER’s channel, and thus, the secrecy rate tends to decrease.Figure 6
Secrecy rate versus the EH constraint.Lastly, Figure7 plots the secrecy rate versus the Tx-RIS distances, where the RIS moves along the x-axis from the Tx to the IR’s area. From this figure, we can see that the active RIS scheme outperforms the passive RIS scheme in the considered region. Moreover, for active RIS, the secrecy rate increases when RIS moves from the Tx to the IR’s area, while for passive RIS, the secrecy rate first decreases to a low point and then increases. Besides, whether for active RIS or passive RIS, when RIS moves away from the user area, the secrecy rate decreases. Thus, it is better to deploy the active RIS near the IRs.Figure 7
Secrecy rate versus the RIS location.


## 5. Conclusion

We investigated the active RIS-assisted secure SWIPT networks, where an AO algorithm was proposed to design the BF and RC to handle the fairness secrecy rate objective subject to the nonlinear EH constraints and the transmit power constraints. The simulation result verified the performance of the proposed scheme.



---

*Source: 8897781-2023-02-01.xml*