# Secrecy Energy Efficiency Optimization for Reconfigurable Intelligent Surface-Aided Multiuser MISO Systems

Authors: Jinhong Bian; YuanYuan Wang; Feng Zhou
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

Currently, the reconfigurable intelligent surface (RIS) has been applied to improve the physical layer security in wireless networks. In this paper, we focus on the secure transmission in RIS-aided multiple-input single-output (MISO) systems. Specifically, by assuming that only imperfect channel state information (CSI) of the eavesdropper can be obtained, we investigated the robust secrecy energy efficiency (SEE) optimization via jointly designing the active beamforming (BF), artificial noise (AN) at Alice, and the passive phase shifter at the RIS. The formulated problem is hard to handle due to the complicated secrecy rate expression as well as the infinite constraints introduced by the CSI uncertainties. By utilizing the Taylor expansion, we transformed the fractional programming into a convex problem, while all the constraints are approximated via the successive convex approximation and constrained concave-convex procedure. Then, by using the extended S-Lemma, we transform the infinite constraints into linear matrix inequality, which is convex. Finally, an alternate optimization (AO) algorithm was proposed to solve the reformulated problem. Simulation results demonstrated the performance of the proposed design.


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## Body

## 1. Introduction

Recently, the energy efficiency (EE) has been regarded as an important aspect in assessing the performance of communication systems, which is defined as the ratio of the information rate to the total power consumption [1]. Meanwhile, the reconfigurable intelligent surface (RIS) has been emerged as a promising technique to improve the performance such as coverage, throughput, spectrum efficiency (SE), and EE of future wireless networks [2, 3].To be specific, Fang et al. [4] studied the EE optimization of RIS-enabled nonorthogonal multiple access (NOMA) networks, where a semidefinite relaxation- (SDR-) based algorithm was proposed to optimize the transmit and passive BF. Then, [5] investigated the EE-oriented resource allocation for RIS-assisted uplink systems by using a block coordinate descent- (BCD-) based method. Then, Le et al. [6, 7] studied the EE optimization in RIS-assisted cell-free multiple-input single-output (MISO) and multiple-input multiple-output (MIMO) network, respectively. Moreover, Yang et al. [8–10] investigated the EE optimization of distributed wireless network, device-to-device network, and wireless energy harvesting network, respectively. Recently, Yuo et al. [11] studied the EE and SE tradeoff in RIS-aided MIMO network, where an iterative mean-square error minimization approach was developed to optimize the phase shifter. On the other hand, in [12], Wang et al. studied the cooperative hybrid NOMA-based mobile-edge computing networks. In [13], Wang et al. studied the delay-sensitive secure NOMA transmission for Internet-of-Things (IoT) networks. Then, Wang et al. [14] investigated the outage-driven link selection for secure buffer-aided networks. Moreover, He et al. [15] investigated a NOMA-enabled framework for relay deployment and network optimization in airborne access vehicular ad hoc networks.Besides the benefits of improving the SE or EE, the application of RIS is also appealing in secure communication application. For example, Dong and Wang [16] studied the secure MIMO transmission enhanced by a RIS, where a BCD-based method was proposed to design the precoding and phase shifter. Then, a minorization-maximization- (MM-) based method was proposed to optimize the precoding, artificial noise (AN) covariance and the phase shifter for secure MIMO network in [17]. Actually, if the phase shifts are given, the optimization problem reduces to a conventional beamforming (BF) design problem in MISO network, which has been extensively studied in the literature. Inspired by this consideration, alternating optimization (AO) algorithms are usually applied to decouple the optimization variables. Then, the BF vectors at the Tx are typically obtained by using existing BF design methods such as the weighted minimum mean square error (WMMSE) algorithm [18]. On the other hand, the optimization methods of the phase shifts in RIS-assisted network can be summarized as follows: (a) optimization techniques for continuous phase shifts: (1) relaxation and projection; (2) SDR; (3) majorization-minimization (MM) algorithm; (4) manifold approach; (5) element-wise BCD; (6) alternating direction method of multipliers (ADMM) based algorithm; (7) penalty convex-concave procedure (PCCP); (8) barrier function penalty; (9) accelerated projected gradient; (10) deep reinforcement learning. (b) Optimization techniques for discrete phase shifts: (1) rounding method; (2) binary mode selection method; (3) negative square penalty [19].The performance index of these works were commonly measured by the criterion called secrecy rate, which is the capacity of conveying information to the legitimated users (Bobs), while keeping it confidential from the eavesdroppers (Eves) [20]. However, for network design with both security and EE requirements, it is beneficial to combine these two metrics into an individual target, which is termed as the secrecy EE (SEE) and defined as the ratio between the achieved secrecy rate to the total power consumption. Specifically, Wu et al. [21] studied the SEE maximization in RIS-assisted cognitive radio system, where a Dinkelbach and second order cone programming- (SOCP-) based method was proposed. Also, Wang et al. [22] studied the robust BF and cooperative jamming design in a RIS-assisted MISO network to maximize the SEE, where an AO algorithm was developed. Recently, Liu et al. [23] studied the SEE in RIS-aided secure simultaneous wireless information and power transfer (SWIPT) network, where an AO-based method was proposed.The acquirement of the channel state information (CSI) is an important issue in communication network. Currently, the methods of the channel estimation for the RIS-aided network can be divided into two folds. The first one is to estimate the transmitter (Tx)-RIS channel and the RIS-user channel separately by installing some active elements at the RIS, which requires more hardware and power consumption [24]. The second method is to estimate the cascaded Tx-RIS-user channel, e.g., the product of the Tx-RIS channel and the RIS-user channel, where the main advantage is that no extra hardware and power cost are needed [25], and actually the cascaded channel is sufficient for the joint BF design [26]. However, in secure communication scenario, since the Eves are commonly passive nodes, it is very hard to obtain the perfect CSI of the Eves.In existing works, the CSI of the Eves are commonly model as the bounded error model and the statistical error model. For the bounded CSI errors, in [27], Zhou et al. studied the worst-case robust BF design for RIS-aided MISO communication systems to maximize the worst case information rate. Then, Yu et al. [28] studied the worst-case robust secrecy design in RIS-assisted network with bounded CSI errors. Besides, for the statistical error model, in [29], Zhao et al. studied the outage-constrained robust BF for RIS-aided wireless communication. Also, in [30, 31], the authors studied the outage-constrained BF in RIS-assisted secure network. Then, in [20, 32], the authors studied both the two kinds of robust design in RIS-aided multiuser system and secure SWIPT system, respectively. Recently, in [33], Hong et al. proposed a novel On/Off scheme for the RIS to improve the robustness. However, these works mainly focused on the secrecy rate optimization; the SEE in RIS-aided multiuser downlink network has not been studied yet.Motivated by these observations, in this paper, we investigate the robust SEE design in a downlink multiuser MISO network. The main contributions are summarized as follows:(i)
by assuming that only imperfect CSI of the Eve can be obtained, we aim to maximize the worst case SEE by jointly designing the active BF and AN at Alice, and the passive phase shifter at the RIS. The formulated problem is hard to handle due to the complicated secrecy rate expression as well as the infinite constraints introduced by the CSI uncertainty(ii)
differently with the commonly used Dinkelbach algorithm, which turn the fractional objective into a subtractive form, we equivalently transform it into a more tractable reformulation with linear objective via successive convex approximation (SCA) and constrained concave-convex procedure (CCCP). Then, by using the extended S-Lemma, we transform the infinite constraints into linear matrix inequality, which is convex(iii)
finally, an AO algorithm was proposed to solve the reformulated problem. Simulation results demonstrated the performance of the proposed design and provide some meaningful insights. (1) RIS plays more important role than AN in enhancing the SEE; (2) less number of quantization bits can lead to higher SEE; (3) the acquirement of the CSI is important in secrecy transmission designThe rest of this paper is organized as follows. A system model and problem formulation is given in Section2. Section 3 investigates the joint BF, AN, and phase shifter design, wherein a SCA and CCCP based iterative approach is proposed. Simulation results are illustrated in Section 4. Section 5 concludes this paper.Notations: Throughout the paper, we use the upper case boldface letters for matrices and lower case boldface letters for vectors. The conjugate, transpose, conjugate transpose, and trace of matrix A are denoted as A∗, AT, AH, and TrA, respectively. a=vecA means to stack the columns of A into a. A±0 means that A is positive semidefinite. Besides, Diagx1,⋯,xN represents a diagonal matrix with x1,⋯,xN on the main diagonal. ·2 and ·F denote the Euclidean norm and Frobenius norm, respectively. I indicates an identity matrix. Moreover, Ra, a, and ∠a denote the real part, the modulus, and the angle of a complex number a, respectively. The distribution of a circularly symmetric complex Gaussian (CSCG) random vector with mean x and covariance Σ is denoted by CNx,Σ. In addition, ∘ means the Hadamard production; ⊗ denotes the Kronecker product, and E· means the mathematical expectation, respectively.


## 2. System Model and Problem Formulation

### 2.1. System Model

We consider a downlink multiuser MISO system as shown in Figure1, which consists of one transmitter (Alice), one RIS, K Bobs, and K Eves. The Alice and the RIS are equipped with N antennas and M reflecting elements, respectively, while all Bobs and Eves are single antenna nodes. We denote F∈ℂN×M, gk∈ℂN×1, hk∈ℂM×1, ge,k∈ℂN×1, and he,k∈ℂM×1, as the channels between Alice and RIS, between Alice and the k-th Bob, between RIS and the k-th Bob, between Alice and the k-th Eve, and between RIS and the k-th Eve, respectively. In addition, we assume that a RIS controller is utilized to exchange the CSI between Alice and the RIS. Besides, the controller adjusts the phase shift and amplitude of each passive element at the RIS to achieve passive BF so that the signal power is improved at the Bobs or reduced at the Eves.Figure 1
The system model for the RIS-assisted secure network.Alice sendsK independent data streams for each Bob in the same frequency band, under the thread of the Eves. Let us denote sk as the confidential message intended to the k-th Bob, with Esk2=1. Since AN is injected to degrade the Eve’s channel, the transmitted signal is given as x=∑k=1Kwksk+v, where wk∈ℂN×1 denotes the BF vector intent to the k-th Bob, and v∈ℂN×1 denotes the AN vector.In this work, we assume that each Eve only eavesdrop the signal send to the nearest Bob. Thus, the received signals at thek-th Bob and the k-th Eve are, respectively, given by
(1a)yk=gkH+hkHΘHF∑k=1Kwksk+v+nk,(1b)ye,k=ge,kH+he,kHΘHF∑k=1Kwksk+v+ne,k,where nk and ne,k are the zero-mean additive Gaussian noise at the k-th Bob and Eve, with variance σk2 and σe,k2, respectively.Here, we assume that the Bobs have the priori knowledge about the AN vectorv. Thus, Bobs can cancel the interference and the actually received signal at the k-th Bob is given by yk=gkH+hkHΘHF∑k=1Kwksk+nk.Here,Θ denotes an M×M diagonal reflection coefficient matrix (also known as the passive BF matrix), which can be written as Θ=Diagθ1,⋯,θM, θm=ejφm,φm∈0,2π,∀m∈MΔ=1,⋯,M. Let Q denote the number of quantization bits for phase-shift control per RIS element, respectively. We thus have
(2)θm∈XdΔ=θmθm=ejφm,φm∈S,where SΔ=0,2π/L,⋯,2πL−1/L with L=2Q, i.e., the discrete phase-shift values are assumed to be equally spaced in a circle. Furthermore, by letting Q⟶∞, the model in (2) becomes the case with continuous phase shifts, i.e., θm∈XcΔ=θmθm=1. In practice, due to the hardware limitation, it is costly to achieve continuous phase shift on the reflecting elements. However, it is meaningful to investigate the continuous phase shift design, since the optimization method for the continuous phase shift is useful to the discrete phase shift design.In fact, by denotingθ=θ1,⋯,θMT, we have
(3)gkH+hkHΘHFwk=θHDiaghkHF⏟Hkwk+gkHwk=θH1⏟θ∧HHk,gkT⏟H¯kwk=θ∧HH¯kwk,and
(4)ge,kH+he,kHΘHFwk=θHDiaghe,kH⏟He,kFwk+ge,kHwk=θ∧HHe,k,ge,kT⏟H¯e,kwk=θ∧HH¯e,kwk.Thus, the signal-to-interference-plus-noise ratio (SINR) at thek-th BoB and the k-th Eve can be given as
(5a)Γk=θ∧HH¯kwk2∑i=1,i≠kKθ∧HH¯kwi2+σk2,(5b)Γe,k=θ∧HH¯e,kwk2∑i=1,i≠kKθ∧HH¯e,kwi2+θ∧HH¯e,kv2+σe2,respectively.Hence, the information rates at thek-th Bob and the Eve are, respectively, given by
(6a)Ckwk,v,θ^=log21+Γk,(6b)Ce,kwk,v,θ^=log21+Γe,k.Thus, the secrecy rate for the RIS-assisted network is given by(7)Rswk,v,θ^=min∀k∈1,⋯,KCkwk,v,θ^−Ce,kwk,v,θ^.Similar to [11], the total power consumption of the network is model as
(8)Pwk,v,θ^=Psηs+Pc+MPRISQ,where Ps=∑k=1Kwk2+v2 denotes the transmit power; 0≤ηs≤1 is the power amplifier efficiency for Alice. In addition, the constant Pc denotes the total circuit power consumption for Alice and the receivers. Without loss of generality, we assume ηs=1 in the following part, [10, 21, 23]. Besides, PRISQ denotes the per unit hardware dissipated power at the RIS with a Q-bit resolution phase shifter. From (11), we can observe that the RIS operates without consuming any transmit power, since RIS is a passive device which do not change the magnitude of the reflected signals. (According to [3], the typical power consumption values of each phase shifter are 1.5, 4.5, 6.0, and 7.8mW for 3-, 4-, 5-, and 6-bit resolution phase shifting.)According to several related works such as [3–5], the linear power consumption model is rational when the following assumptions holds. First, the static circuit power Pc is independent of the data rate. Second, the power amplifier at Alice operates in the linear region, thus a constant power offset can approximate the hardware power consumption. Typical wireless communication transceivers satisfy these two assumptions generally.Therefore, the SEE for the RIS-aided network is defined as(9)SEEwk,v,θ^=Rswk,v,θ^Pwk,v,θ^.


### 2.2. CSI Error Models

In this work, both imperfect direct link and cascaded link are considered, i.e., we modelHe,k and ge,k as
(10)He,k=H^e,k+ΔHe,k,ge,k=g^e,k+Δge,k,where He,k and ge,k denote the true CSI; H^e,k and g^e,k denote the estimated CSI; ΔHe,k and Δge,k denote the associated CSI error, respectively. Specifically, the following bounded CSI models are considered, where the CSI error is assumed to lie in a region with a given bound, i.e.,
(11)He,k=ΔHe,k2≤εe,k,Ge,k=Δge,k2≤χe,k,where εe,k and χe,k denote the respective sizes of the bounded channel error region. As for the statistical error model, since it can be transformed into the bounded model by the proposed method in [20, 32], we focus on the bounded error model in this work due to the generality.


### 2.3. Problem Formulation

Our objective is to maximize the SEE by jointly designingwk, v, and θ^, subject to the maximum transmit power constraint and the unit modulus constraint (UMC). Here, we first relax the discrete phase shift design to the continuous phase shift design, then project the obtained solution to the discrete set. Mathematically, our problem is formulated as
(12(a))P1:maxwk,v,θ^SEEwk,v,θ^(12(b))s.t.∑k=1Kwk22+v22≤Pmax,(12(c))θ^m=1,∀m∈M,θ^M+1=1.Notably, we find thatP1 is nonconvex [34], which is hard to solve. In the next section, we will develop a tractable solution to P1 through convex approximation.


## 2.1. System Model

We consider a downlink multiuser MISO system as shown in Figure1, which consists of one transmitter (Alice), one RIS, K Bobs, and K Eves. The Alice and the RIS are equipped with N antennas and M reflecting elements, respectively, while all Bobs and Eves are single antenna nodes. We denote F∈ℂN×M, gk∈ℂN×1, hk∈ℂM×1, ge,k∈ℂN×1, and he,k∈ℂM×1, as the channels between Alice and RIS, between Alice and the k-th Bob, between RIS and the k-th Bob, between Alice and the k-th Eve, and between RIS and the k-th Eve, respectively. In addition, we assume that a RIS controller is utilized to exchange the CSI between Alice and the RIS. Besides, the controller adjusts the phase shift and amplitude of each passive element at the RIS to achieve passive BF so that the signal power is improved at the Bobs or reduced at the Eves.Figure 1
The system model for the RIS-assisted secure network.Alice sendsK independent data streams for each Bob in the same frequency band, under the thread of the Eves. Let us denote sk as the confidential message intended to the k-th Bob, with Esk2=1. Since AN is injected to degrade the Eve’s channel, the transmitted signal is given as x=∑k=1Kwksk+v, where wk∈ℂN×1 denotes the BF vector intent to the k-th Bob, and v∈ℂN×1 denotes the AN vector.In this work, we assume that each Eve only eavesdrop the signal send to the nearest Bob. Thus, the received signals at thek-th Bob and the k-th Eve are, respectively, given by
(1a)yk=gkH+hkHΘHF∑k=1Kwksk+v+nk,(1b)ye,k=ge,kH+he,kHΘHF∑k=1Kwksk+v+ne,k,where nk and ne,k are the zero-mean additive Gaussian noise at the k-th Bob and Eve, with variance σk2 and σe,k2, respectively.Here, we assume that the Bobs have the priori knowledge about the AN vectorv. Thus, Bobs can cancel the interference and the actually received signal at the k-th Bob is given by yk=gkH+hkHΘHF∑k=1Kwksk+nk.Here,Θ denotes an M×M diagonal reflection coefficient matrix (also known as the passive BF matrix), which can be written as Θ=Diagθ1,⋯,θM, θm=ejφm,φm∈0,2π,∀m∈MΔ=1,⋯,M. Let Q denote the number of quantization bits for phase-shift control per RIS element, respectively. We thus have
(2)θm∈XdΔ=θmθm=ejφm,φm∈S,where SΔ=0,2π/L,⋯,2πL−1/L with L=2Q, i.e., the discrete phase-shift values are assumed to be equally spaced in a circle. Furthermore, by letting Q⟶∞, the model in (2) becomes the case with continuous phase shifts, i.e., θm∈XcΔ=θmθm=1. In practice, due to the hardware limitation, it is costly to achieve continuous phase shift on the reflecting elements. However, it is meaningful to investigate the continuous phase shift design, since the optimization method for the continuous phase shift is useful to the discrete phase shift design.In fact, by denotingθ=θ1,⋯,θMT, we have
(3)gkH+hkHΘHFwk=θHDiaghkHF⏟Hkwk+gkHwk=θH1⏟θ∧HHk,gkT⏟H¯kwk=θ∧HH¯kwk,and
(4)ge,kH+he,kHΘHFwk=θHDiaghe,kH⏟He,kFwk+ge,kHwk=θ∧HHe,k,ge,kT⏟H¯e,kwk=θ∧HH¯e,kwk.Thus, the signal-to-interference-plus-noise ratio (SINR) at thek-th BoB and the k-th Eve can be given as
(5a)Γk=θ∧HH¯kwk2∑i=1,i≠kKθ∧HH¯kwi2+σk2,(5b)Γe,k=θ∧HH¯e,kwk2∑i=1,i≠kKθ∧HH¯e,kwi2+θ∧HH¯e,kv2+σe2,respectively.Hence, the information rates at thek-th Bob and the Eve are, respectively, given by
(6a)Ckwk,v,θ^=log21+Γk,(6b)Ce,kwk,v,θ^=log21+Γe,k.Thus, the secrecy rate for the RIS-assisted network is given by(7)Rswk,v,θ^=min∀k∈1,⋯,KCkwk,v,θ^−Ce,kwk,v,θ^.Similar to [11], the total power consumption of the network is model as
(8)Pwk,v,θ^=Psηs+Pc+MPRISQ,where Ps=∑k=1Kwk2+v2 denotes the transmit power; 0≤ηs≤1 is the power amplifier efficiency for Alice. In addition, the constant Pc denotes the total circuit power consumption for Alice and the receivers. Without loss of generality, we assume ηs=1 in the following part, [10, 21, 23]. Besides, PRISQ denotes the per unit hardware dissipated power at the RIS with a Q-bit resolution phase shifter. From (11), we can observe that the RIS operates without consuming any transmit power, since RIS is a passive device which do not change the magnitude of the reflected signals. (According to [3], the typical power consumption values of each phase shifter are 1.5, 4.5, 6.0, and 7.8mW for 3-, 4-, 5-, and 6-bit resolution phase shifting.)According to several related works such as [3–5], the linear power consumption model is rational when the following assumptions holds. First, the static circuit power Pc is independent of the data rate. Second, the power amplifier at Alice operates in the linear region, thus a constant power offset can approximate the hardware power consumption. Typical wireless communication transceivers satisfy these two assumptions generally.Therefore, the SEE for the RIS-aided network is defined as(9)SEEwk,v,θ^=Rswk,v,θ^Pwk,v,θ^.


## 2.2. CSI Error Models

In this work, both imperfect direct link and cascaded link are considered, i.e., we modelHe,k and ge,k as
(10)He,k=H^e,k+ΔHe,k,ge,k=g^e,k+Δge,k,where He,k and ge,k denote the true CSI; H^e,k and g^e,k denote the estimated CSI; ΔHe,k and Δge,k denote the associated CSI error, respectively. Specifically, the following bounded CSI models are considered, where the CSI error is assumed to lie in a region with a given bound, i.e.,
(11)He,k=ΔHe,k2≤εe,k,Ge,k=Δge,k2≤χe,k,where εe,k and χe,k denote the respective sizes of the bounded channel error region. As for the statistical error model, since it can be transformed into the bounded model by the proposed method in [20, 32], we focus on the bounded error model in this work due to the generality.


## 2.3. Problem Formulation

Our objective is to maximize the SEE by jointly designingwk, v, and θ^, subject to the maximum transmit power constraint and the unit modulus constraint (UMC). Here, we first relax the discrete phase shift design to the continuous phase shift design, then project the obtained solution to the discrete set. Mathematically, our problem is formulated as
(12(a))P1:maxwk,v,θ^SEEwk,v,θ^(12(b))s.t.∑k=1Kwk22+v22≤Pmax,(12(c))θ^m=1,∀m∈M,θ^M+1=1.Notably, we find thatP1 is nonconvex [34], which is hard to solve. In the next section, we will develop a tractable solution to P1 through convex approximation.


## 3. Proposed SEE Optimization Method

In this section, we will propose a SCA and CCCP based method to convertP1 into a solvable reformulation.First, by introducing slack variablesrb and re, we recast P1 as
(13(a))P2:maxwk,v,θ^,rb,rerb−re∑k=1Kwk22+v22+Pc+MPRISQ,(13(b))s.t.θ∧HH¯kwk2∑i=1,i≠kKθ∧HH¯kwi2+σk2≥2rb−1,(13(c))θ∧HH¯e,kwk2∑i=1,i≠kKθ∧HH¯e,kwi2+θ∧HH¯e,kv2+σe,k2≤2re−1.(13(d))12b,12c.Then, via introducing auxiliary variablesp=p1,1,⋯,p1,K,p2,1,⋯,p2,K,p3T and q=q1,1,⋯,q1,K,q2,1,⋯,q2,K,q3T; P2 can be transformed into the following problem
(14(a))P3:maxwk,v,θ^,rb,re,p,qrb−re∑k=1Kwk22+v22+Pc+MPRISb(14(b))s.t.wkHH¯kHθ^θ∧HH¯kwk≥p1,k,(14(c))∑i=1,i≠kKwiHH¯kHθ^θ∧HH¯kwi+σk2≤1p2,k,(14(d))p1,k≥p32/p2,k,p32≥2rb−1,(14(e))wkHH¯e,kHθ^θ∧HH¯e,kwk≤q1,k2,(14(f))∑i=1,i≠kKwiHH¯e,kHθ^θ∧HH¯e,kwi+vHH¯e,kHθ^θ∧HH¯e,kv+σe,k2≥q2,k,(14(g))q1,k2/q2,k≤q3,q3≤2re−1,(14(h))12b,12c.Furthermore, to convert the fractional objective into a linear reformulation, we introduce auxiliary variablesa1, a2, and a3, P3 can be recast as the following problem
(15(a))P4:maxwk,v,θ^,rb,re,p,q,a1,a2,a3≥0a3(15(b))s.t.rb−re≥a1,(15(c))∑k=1Kwk22+v22+Pc+MPRISb≤1a2,(15(d))14b−14h,(15(e))a32≤a1a2.where we use the trick that maximize a32 is equivalent to maximize a3 to linear the objective.P4 is still hard to solve due to the nonconvex constraints (15(c)) and (15(d)). In the following, we will utilize the SCA and CCCP to approximate these constraints. Firstly, we handle the first part of (14(d)) and (14(g)). In fact, p1,k≥p32/p2,k is equivalent to p1,kp2,k≥p32. Moreover, p1,kp2,k≥p32 is equivalent to p1,k+p2,k2−p1,k−p2,k2≥4p32, which can be further rewritten as
(16)2p3,p1,k−p2,k2≤p1,k+p2,k,which is joint convex with respect to (w.r.t.) p1,k,p2,k,p3. Similarly, a32≤a1a2 and q1,k2/q2,k≤q3 can be equivalent rewritten as
(17)2a3,a1−a22≤a1+a2,2q1,k,q2,k−q32≤q2,k+q3.Then, we turn to the right hand side of (14(c)), (14(e)), (14(g)) and (15(c)) to handle the nonconvex functions 1/p2,k, q1,k2, 2re, and 1/a2. In fact, in the i+1-th iteration, these constraints can be replaced by the following linear constraints
(18(a))∑i=1,i≠kKwiHH¯kHθ^θ∧HH¯kwi+σk2≤2p¯2,k−p2,kp¯2,k2,(18(b))wkHH¯e,kHθ^θ∧HH¯e,kwk≤2q1,kq¯1,k−q¯1,k2,(18(c))u3≤2r¯ere−r¯eln2+1−1,(18(d))∑k=1Kwk22+v22+Pc+MPRISb≤2a¯2−a2a¯22,where p¯2,k, q¯1,k, r¯e, and a¯2 are the optimal values of p2,k, q1,k, re, and a2 in the l-th iteration, respectively.Then, we focus on the nonconvex (14(b) and 14(f)). In fact, these constraints can be turn to convex constraints w.r.t. a given variable when fixing the others. Actually, the left hand side of (14(b)) is convex w.r.t. wk or θ^ when fixing the others. Since a convex function can be lower-bounded or under-estimated by the first order Taylor expansion around a given point, thus, in the i+1-th iteration, (14(b)) can be approximated as
(19)2RwkHH¯kHθ^θ∧HH¯kw¯k−w¯kHH¯kHθ^θ∧HH¯kw¯k≥p1,k,where w¯k is the optimal values of wk in the l-th iteration.Then, (14(f)) can be approximated as
(20)∑i=1,i≠kK2RwiHH¯e,kHθ^θ∧HH¯e,kw¯i−w¯iHH¯e,kHθ^θ∧HH¯e,kw¯i+2RvHH¯e,kHθ^θ∧HH¯e,kv¯−v¯HH¯e,kHθ^θ∧HH¯e,kv¯+σe,k2≥q2,k,where v¯ is the optimal values of v in the l-th iteration.The remain task is to handle the CSI uncertainties in (18(b) and (20)). First, we denote H~e,k=H∧e,k,g∧e,kT, and H¨e,k=ΔHe,k,Δge,kT; it is known that H¯e,k=H~e,k+H¨e,k. Besides, the following equation holds
(21)ΔHe,kF≤εe,k⇒vecΔHe,k2≤εe,k.Then, we denote(22)I1Δ=Diag1...1⏟MN0...0⏟N,I2Δ=Diag0...0⏟MN1...1⏟N.Thus, the following relationship holds(23)h¨e,kHI1h¨e,k≤εe,k2,h¨e,kHI2h¨e,k≤χe,k2,where h¨e,k=vecH¨e,k. We find the following extended S-Lemma is useful to handle the CSI uncertainty.Lemma 2 (see [20]).
Define the function with variablex∈ℂn×1,
(24)fjx=xHAjx+2RebjHx+cj,j=0,⋯,J,where Aj=AjH∈ℂn×n, bj∈ℂn×1, and cj∈ℝ. The condition fjx≥0j=1J⇒f0x≥0 holds if and only if there exists λj≥0j=1J such that
(25)A0b0b0Hc0−∑j=1JλjAjbjbjHcj≽0.To utilize Lemma2, we first denote h¯e,k=vecH¯e,k, h~e,k=vecH~e,k. By invoking the identity TrAHBCD=vecAHDT⊗BvecC, (18(b)) can be rewritten as
(26)h~e,kHwkwkHT⊗θ^θ∧Hh~e,k+h¨e,kHwkwkHT⊗θ^θ∧Hh¨e,k+2Rh~e,kHwkwkHT⊗θ^θ∧Hh¨e,k≤2q1,kq¯1,k−q¯1,k2.Following Lemma2, (26) holds if the following linear matrix inequality (LMI) holds
(27)−wkwkHT⊗θ∧θ∧H+α1,kI1+α2,kI2−wkwkHT⊗θ∧θ∧Hh~e,k−h~e,kHwkwkHT⊗θ∧θ∧Hρk≽0,where α1,k≥0,α2,k≥0k=1K are the slack variables, and ρk=−h~e,kHwkwkHT⊗θ^θ∧Hh~e,k+2q1,kq¯1,k−q¯1,k2−α1,kεe,k2−α2,kχe,k2.Similarly, by denotingΦk=∑i=1,i≠kKwiHw¯i+w¯iHwi−w¯iHw¯i+vHv¯+v¯Hv−v¯Hv¯, (20) is equivalent to
(28)h~e,kHΦkT⊗θ^θ∧Hh~e,k+h¨e,kHΦkT⊗θ^θ∧Hh¨e,k+2Rh~e,kHΦkT⊗θ^θ∧Hh¨e,k+σe,k2≥q2,k.Then, by Lemma2, we find that (28) holds if the following LMI holds
(29)ΦkT⊗θ~θ∧H+β1,kI1+β2,kI2ΦkT⊗θ∧θ∧Hh~e,kh~e,kHΦkT⊗θ∧θ∧Hπk≽0where β1,k≥0,β2,k≥0k=1K are the slack variables, and πk=h~e,kHΦkT⊗θ^θ∧Hh~e,k−σe,k2−β1,kεe,k2−β2,kχe,k2.Combining these steps, around given pointw¯k,v¯,θ¯,p¯2,k,p¯3,q¯1,k,q¯2,k,r¯e,a¯2, we obtained the following approximated subproblem w.r.t. wk,v(30a)P5:maxwk,v,rb,re,p,q,a1,a2a3(30b)s.t.p3≥2rb−1,12b,14e,15b,16,17,18a,18c18d,19,25,27,which is convex and can be efficiently solved by the optimization toolbox, such as CVX [35].Nextly, we handle these constraints w.r.t.θ^ with fixed wk,v. Here, we mainly focus on the difference between the counterpart of the optimization w.r.t. wk,v. Firstly, (23) can be approximated as
(31)2Rθ¯HHkwkwkHHkHθ^−θ¯HHkwkwkHHkHθ¯≥p1,k,where θ¯ is the optimal values of θ^ in the l-th iteration.Then, for convenient, we denoteΩk=vvH+∑i=1,i≠kKwiwiH, thus, (14(c)) can be approximated as
(32)θ∧HHk∑i=1,i≠kKwiwiHHkHθ^+σk2≤2p¯2,k−p2,kp¯2,k2,and (14(f)) can be approximated as
(33)2Rθ¯HHe,kΩkHe,kHθ^−θ¯HHe,kΩkHe,kHθ¯+σe,k2≥q2,k.Then, by Lemma2, we find that (33) holds if the following LMI holds
(34)ΩkT⊗θ∧θ¯H+β1,kI1+β2,kI2ΩkT⊗θ∧θ¯Hh~e,kh~e,kHΩkT⊗θ∧θ¯Hπk≽0,where β1,k≥0,β2,k≥0k=1K are the slack variables, and πk=h~e,kHΩkT⊗θ^θ¯Hh~e,k−σe,k2−β1,kεe,k2−β2,kχe,k2.To this end, the only nonconvexity inP4 is the UMC (12(c)). In the following, we will utilize the penalty CCP method to handle (12(c)). According to the penalty CCP principle, (12(c)) can be first equivalently transformed into 1≤θ∧m2≤1. Then, we introduce the following Lemma to handle the nonconvex part 1≤θ∧m2.Lemma 3 (see [32]).
Letb be a complex scalar variable, then the following inequality holds
(35)b2≥b¯†b+b†b¯−b¯†b¯,for any fixed point a~.Via Lemma3, the subproblem w.r.t. θ^ can be formulated as
(36(a))P6:maxθ^,rb,re,p,q,a1,a2,a3,ba3−λl∑m=1Mbm,(36(b))s.t.θ∧m2≤1+bm,∀m,(36(c))−θ¯m∗θ^m−θ^m∗θ¯m+θ¯m∗θ¯m≤bm−1,∀m,(36d)p3≥2rb−1,12b,14e,15b,16,17,18c,18d,29,30,32,where θ¯m is the obtained θ^m in the previous iteration; b=b1,⋯,bMT∈ℂM×1 is the slack variable for the UMC, and λ≥0 is the penalty multiplier to scale the penalty item ∑m=1Mbm, which can control the feasibility of θ^. Here, λl is updated by λl=minγλl−1,λmax, where the upper bound λmax is used to avoid numerical problems. More details can refer to [27, 32].To this end, we have solvedP1 efficiently. The whole procedure is summarized in Algorithm 3, where WSRl is the obtained optimal value in the l-th iteration for P1. In addition, ς denotes the accuracy.Algorithm 1: The AO Algorithm for P1.
1.Initialization: l=1, set Ps, F, hk, he, gk, ge and the initial value wk1,v1,θ∧1.2:    Repeat(a) Obtainwkl,vl with fixed θ^ by solving P5.(b) Obtainθ∧l with fixed wk,v by solving P6.(c)l=l+1.3:UntilWSRl+1−WSRl≤ς.4:Outputwkå,vå,θ∧å.Thus, we have finished the algorithm in the continuous phase shift case. While for the discrete phase shift case, at the end of Algorithm1, we project the obtained θm into the discrete set. In particular, we denote the solution of the two cases as θmc and θmd, respectively. Then, we map θmc to obtain θmd, i.e., θmd=ejϕuå, where uå=argmin1≤u≤τθmc−ejϕq.


## 4. Simulation Results

In this section, we provide simulation results to assess the performance of the proposed algorithm. The simulation scenario is shown in Figure2, where there are one Alice, one RIS, 4 Bobs, and 4 Eves. A three dimensional coordinate system is utilized to represent the positions of these nodes, where Alice and RIS are deployed at (0 m, 0 m, and 10 m) and (50 m, 10 m, and 10 m), while all Bobs/Eves are randomly located in a circle centered at (50 m, 0 m, and 2 m) and with radius 5 m. In fact, the electromagnetic feature of each reflecting element can be adapted. Thus, by proper designing the phase of each reflecting element, the incident signal can be reflected to different direction to serve different users. Besides, the distance between these users is commonly smaller than the distance between the Alice to RIS, and the RIS is deployed more closer to the users than the Alice. Thus, the RIS does not need to form a very thin reflecting beam.Figure 2
The deployment of the simulation scenario.Unless otherwise specified, the simulation settings are assumed as follows:N=4, M=40, Q=3, Pmax=−20dBW, Pc=−40dBm, and σk2=σe,k2=−80dBm,∀k. The large-scale path loss is given by PL=PL0−10αlog10d/d0, where PL0 is the path loss at the reference distance d0; d is the link distance; α is the path loss exponent. More details about the simulation parameters can refer to [36]. As for the CSI uncertainty, we use a parameter to model the uncertainty, which is given as εe,k2=εe,k2H^e,kF2, and χe,k2=ρe,k2g^e,k22, respectively, where εe,k and ρe,k measure the relative number of the CSI errors and are set as εe,k2=10−4 and ρe,k2=10−4 [32], respectively. In addition, for the parameters of the CCCP method, we set λ1=20, λmax=10, and γ=0.8 [27].Here, we compare the proposed method with the following benchmarks: (1) the no AN design, e.g., settingz=0 while only optimizing wk and θ^; (2) the random RIS design, e.g., choosing θ^ randomly; (3) the no RIS-assisted case; (4) Alice can obtain ideal CSI of the Eves, which can be seen as the upper bound of the proposed design; (5) the penalty dual decomposition (PDD) method in [20]; (6) the SDR with Gaussian randomization (GR) methods [4]. These designs are labeled as “Proposed method”, “No AN design”, “Random RIS”, “No RIS case”, “Ideal CSI case”, “PDD method”, and “SDR with GR”, respectively.Firstly, we investigate the convergence behaviour of the proposed method by comparing the obtained SEE with the number of iterations. Figure3 shows the convergence behaviour with different number of N and M via fixing all the other parameters. From this figure, we can see that the proposed method can always converge to the optimal solution in almost 20 iterations, which suggests the convergence of the proposed method. Besides, it should be pointed that in the simulation, we utilize the obtained objective value of P5, e.g., a3 in (50) to judge the secrecy energy efficiency. This is mainly due to the fact that there exists a penalty term −λl∑m=1Mbm in the objective of P6, thus using the obtained objective value of P5 to judge the secrecy energy efficiency is more straightforward.Figure 3
Convergence behaviour.Then, we compare the SEE of these schemes versus the transmit power budgetPmax, the result is shown in Figure 4. From this figure, we can see that the proposed method achieves the very closed performance with the PDD method, and outperforms the SDR with GR method. In addition, for all these methods, the SEE tends to increase with Pmax only when Pmax is smaller than a threshold and then the SEE approaches a constant when Pmax is larger than the threshold value. This is due to the fact that there exists a unique maximizer of the transmit power for SEE maximization and the SEE saturates when the power budget exceeds the value of this maximizer. Consequently, the actual transmit power remains constant when the maximum SEE is attained, leading to saturated SEE in high Pmax regimes. To provide more insights for this results, the corresponding secrecy rate is reported versus Pmax in Figure 5, where we can see that the secrecy rate increases rapidly with Pmax when Pmax is small and then increases slowly when Pmax is large. Thus, using all the available power budget to maximize the secrecy rate is not a sound strategy from the perspective of the energy efficiency. Besides, it is also observed that the no AN method achieves nearly performance with the proposed method, while the no RIS and random RIS methods suffer certain performance loss, especially for the no RIS design, which suggests that RIS play a more important role than AN in improving the performance.Figure 4
The SEE versusPmax.Figure 5
The secrecy rate versusPmax.Next, we show the obtained SEE of the proposed method versusN and M; the result is shown in Figure 6. From this figure, we can see that the SEE increases monotonically with N and M. This is due to that larger N or M leads to higher secrecy rate, thus to improve the SEE. In addition, we can see that M plays a more important role on the SEE performance than N, since the RIS utilizes passive elements, no radio frequency chains will be required, thus lower power consumption can be obtained.Figure 6
The SEE versusN and M.Moreover, we compare the SEE of these schemes versus the number of quantization bitsQ; the result is shown in Figure 7. From this figure, we can see that for all these RIS-aided methods, the SEE decreases monotonically with Q. This is mainly due to the fact that the increase of Q can only increase the secrecy rate slightly but lead a significant increase of the quantization energy consumption, thus decrease the SEE.Figure 7
The SEE versusQ.Finally, we show the SEE of these schemes versus the CSI uncertainty levelεe,k2 and ρe,k2 in Figures 8 and 9, respectively. From these two figures, we find that the SEE decreases when the uncertainty level increases. This phenomenon indicated that the CSI uncertainty level has nonnegligible impact on the secure performance. Since the larger CSI uncertainty level, the higher probability that the confidential information leakage to the Eve’s channel. In addition, by comparing Figure 8 with Figure 9, we can see that the CSI uncertainty of the cascaded channel plays a more dominating role than that of the direct channels in the SEE performance.Figure 8
The SEE versusεe,k2.Figure 9
The SEE versusρe,k2.


## 5. Conclusion

In this paper, we have investigated the robust SEE optimization in a RIS-assisted wiretap channels. Specifically, we aim to maximize the worst case SEE via jointly designing the BF, the AN covariance, and the phase shifter. Different with the commonly used Dinkelbach algorithm, we transformed the fractional programming into a solvable problem with linear objective, while all the constraints are approximated via SCA and CCCP. Then, an iterative method was proposed to solve the approximated convex problem. Simulation results demonstrated the performance of the proposed design.



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*Source: 5498172-2022-10-07.xml*