# Transmission Design in Secure URLLC Network Assisted by Reconfigurable Intelligent Surfaces

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

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## Abstract

In this work, we focus on the secure transmission in reconfigurable intelligent surface- (RIS-) aided ultrareliable low-latency communications (URLLC) network. Specifically, we investigate the average effective secrecy rate (AESR) optimization via jointly designing the transmit beamforming, the artificial noise covariance at the transmitter (Tx), and the phase shifter at the RIS. Particulary, the AESR in URLLC network contains not only the commonly used information rate expression based on long codewords but also two penalty terms about the decoding error probability introduced by finite block-length coding. Thus, the AESR is a nonconcave objective. In addition, the unit modulus constraint of the reflecting coefficient at the RIS is nonconvex, which makes the formulated problem difficult to handle. To solve the formulated highly nonconvex problem, we first decouple the complicated objective into a linear function by introducing several slack variables. Then, we address the nonconvex constraints by the first-order expression and the penalty concave-convex procedure (PCCP) technique. Thus, an alternating optimization (AO) technique is proposed to solve the reformulated problem, where the convergence can be guaranteed by rigorous proof, and the computational complexity is a polynomial function of the main system parameters. Simulation results demonstrated the performance of the proposed design as well as the superiority of RIS in improving the secrecy performance when comparing with other baselines.


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## Body

## 1. Introduction

Recently, ultrareliable and low-latency communications (URLLC) has emerged as an important aspect for the Internet-of-Things (IoT) networks to enable critical applications, which require ultralow transmission latency (less than1ms) and extremely high reliability (packet error probability (PEP) less than 10−9) with relatively low data rate [1]. Different from conventional communications that based on long codewords, URLLC generally relies on the ultrashort-packet transmission, which is introducing the decoding error probability [2]. However, the main challenge to ensure high reliability in URLLC transmission is the random nature of the propagation channel due to multipath fading [3].Recently, the newly proposed reconfigurable intelligent surfaces (RIS) has drawn great attention in industry and academia. Fundamentally, an RIS is a kind of metasurface consisting of an array of passive elements, each of which can adjust the phase-shift and amplitude of the incident signals [4]. Thus, by controlling the parameters of the electromagnetic (EM) waves that impinge on the surface, RIS can generate passive reflection beamforming (BF) and create a favorable propagation environment to expand the coverage, improve transmission quality, and enhance security. Moreover, since the processing is only reflecting the received signal with no decoding and encoding operation, RIS consumes much less power than the active transmitter (Tx) or relay [5].With these above advantages, RIS has attract great research attention. To be specific, the authors in [6] investigated the resource allocation technique for RIS-assisted wireless powered systems. In [7], the authors studied the RIS-assisted mobile edge computing (MEC) networks, where a joint design of the downlink/uplink phase beamforming, transmission power, time slot assignment, and the local computing frequencies was proposed to maximize the total computation bits. Then, the authors in [8] studied the RIS-assisted IoT network. Moreover, in [9, 10], the authors investigated the multilayer RIS structure and suggest that multilayer RIS can further improve the communication performance. Nowadays, RIS-assisted transmission has been investigated in various scenarios such as the simultaneous wireless information and power transfer (SWIPT) network [11], the physical layer security network [12, 13], the millimeter-wave (mmWave) networks [14, 15], the nonorthogonal multiple access (NOMA) networks [16], wireless cashing network [17], hybrid terrestrial-aerial networks [18], and the antijamming communication scenario [19].Meanwhile, some works have studied the RIS-aided URLLC networks. Specifically, [20] studied the user grouping and reflecting BF for RIS-aided URLLC multiuser network. Then, [21] investigated the joint BF and phase shift optimization in RIS-aided orthogonal frequency division multiple access- (OFDMA-) URLLC networks. Also, [22] studied the average rate of RIS-aided short packet communication in URLLC systems. While [23] studied the RIS-aided URLLC in a factory automation scenario. Furthermore, RIS has been demonstrated to reduce the latency for MEC-URLLC networks by improving the channel gain for the users that are far away from the MEC node [24]. While in [25], a deep reinforcement learning based method was proposed in a RIS-assisted URLLC network. These works suggest the huge potential of RIS in URLLC networks.On the other hand, security is an important aspect in wireless network, since the confidential information sent to the legitimate user (Bob) maybe eavesdropped by the eavesdropper (Eve). The physical layer security (PLS) technique, which exploits the random characteristic of wireless channel such as fading and noise, has been considered as a promising technique to improve the security in wireless networks [26]. Moreover, [27, 28] studied the secrecy beamforming (BF) and precoding design in RIS-aided multiple-input single-output (MISO) and multiple-input multiple-output (MIMO) network, where an alternating direction method of multipliers (ADMM) technique and a majorization-minimization (MM) optimization method were developed, respectively. Then, in [29], the authors proposed an RIS-assisted secure wireless communications in MIMO network, where an element wise optimization algorithm was developed to optimize the reflecting coefficient (RC). In [30], the authors investigated the RIS-assisted secure MEC networks, where the max-min computation efficiency problem under the secure computation rate requirements was solved by the Dinkelbach-type method and block coordinate descent technique.However, most work about PLS assume that the confidential information have infinite block-length. In particular, the confidential information may have finite block-length and need to meet the reliability and latency requirements. Thus, [31] investigated the nonasymptotic fundamental limits of wiretap channel, while [32] studied the secure short-packet communication for IoT application. However, the secrecy transmission design in RIS-assisted URLLC network has not been studied yet. Motivated by this observation, in this work, we exploit the potential of RIS in improving the security in URLLC network. The main contributions are summarized as follows:
(i)
We propose a secrecy URLLC network with the assistance of a RIS, where the RIS reflect the incident signal to multiple Bobs, under the eavesdropping threat of an Eve. Specifically, we formulate the average effective secrecy rate (AESR) maximization problem, via jointly optimizing the BF, the artificial noise covariance, as well as the phase shifters. In fact, the AESR in URLLC network contains not only the difference of the logarithmic functions but also two terms about the decoding error probability introduced by finite block-length coding. Thus, the AESR is a nonconcave objective, which is difficult to address(ii)
To solve the formulated highly nonconvex problem, we first decouple the complicated objective into a linear function by introducing several slack variables. Then, we approximate the nonconvex constraints by the first order expression and formulate two subproblems with respect to the BF/AN, and the phase shifters. Moreover, we handle the nonconvex unit modulus constraint (UMC) for the phase shifters by using the penalty concave-convex procedure (PCCP) technique. Finally, an alternating optimization (AO) technique was proposed to solve the reformulated problem. In fact, the PCCP method can be utilized to other problems with respect to the phase shifters, with just some necessary modification about the objective and some constraints(iii)
We proof the convergence of the proposed AO scheme by rigorous derivation and show that the proposed algorithm enjoys polynomial time computational complexity with respect to the main system parameters, which is beneficial to implementation. Finally, the simulation results show that our proposed scheme achieves better AESR performance than other schemes and provide some insights: (1) RIS can significantly improve the AESR of secure URLLC network; (2) RIS plays more important role than AN in enhancing the secure performance; (3) it is beneficial to deploy the RIS closer to the Tx or Bobs to improve the secure performanceThe rest of this work is organized as follows. A system model and problem formulation is given in Section2. Section 3 investigates the joint design problem, where an AO approach is established. Simulation results are illustrated in Section 4. Section 5 concludes this work.Notations: throughout this work, boldface lowercase and uppercase letters denote vectors and matrices, respectively. The superscripts ∗, T, and H denote the conjugate, transpose, and conjugate transpose, respectively. Diaga1,⋯,aN is a diagonal matrix with diagonal elements a1,⋯,aN, while diagA denotes a vector consists of the main diagonal elements of A, respectively. A±0 indicates that A is a positive semidefinite matrix. · denotes the Frobenius norm. I is an identity matrix with proper dimension. Ra and a denote the real part and the modulus of a complex variable a, respectively. In addition, CN0,I denotes a circularly symmetric complex Gaussian random vector with mean 0 and covariance I.


## 2. System Model and Problem Formulation

### 2.1. System Model

Let us consider a RIS-aided secure URLLC system as shown in Figure1, which consists of one Tx, one RIS, K Bobs, and one Eve. The Tx has Nt antennas, and the RIS has M elements, respectively. In addition, each Bob and Eve is equipped with single antenna, and the set of the Bobs is denoted as KΔ=1,⋯,K. In the nth slot, the channel between Tx and the kth Bob and the Eve is denoted by gkn∈ℂNt×1 and gen∈ℂNt×1, respectively. Besides, the channel between Tx and the RIS is denoted as Fn∈ℂM×Nt, between the RIS and the kth Bob, and the Eve is denoted by hkn∈ℂM×1 and hen∈ℂM×1, respectively. All channels are assumed to be frequency-flat, and the channel-state-information (CSI) is perfectly obtained by the Tx. Different from the common used uniform linear array (ULA), we utilize the uniform plane array (UPA) at Tx since the Bobs/Eve maybe not move on a line in practice. Therefore, a UPA may be a better choice.Figure 1
The system model.Letskn∈ℂ1×1 be the information symbol sent to the kth Bob in the nth slot. The transmit signal x can be written as xn=∑k=1Kwknskn+vn, where wkn∈ℂNt×1 is the transmit BF for the kth Bob in the nth slot, and vn∈ℂNt×1 denotes the artificial noise (AN) which is modeled as a circularly symmetric complex Gaussian (CSCG) vector with vn~CN0,Ωn, where Ωn±0∈ℍN is the covariance matrix of the AN. In the following part, we replace vn by w0n to simplify the natation.The phase shifter matrix of the RIS in thenth slot is denoted by Θn=Diagθ1n,⋯,θMn, with θmn denoting the RC of the mth reflection element in the nth slot. Since only the phase of the received signal is changed, the strength of the reflection signal is thus maximized, i.e., θmn2=1.Hence, the signals received by thekth Bob and Eve in the nth slot are, respectively, given by
(1)ykn=gkHn+hkHnΘHnFnxn+nkn,yen=geHn+heHnΘHnFnxn+nen,where nkn and nen denote the antenna noise at the kth Bob and Eve with nk~CN0,σk2n and ne~CN0,σe2n, respectively.In fact, by definingθn=θ1n,⋯,θMnT, we have
(2)gkHn+hkHnΘHnFn=gkHn+θHnDiaghkHnFn⏟Hkn=θHn1H⏟θ^nHkngkHn⏟H¯kn.Similarly, we have(3)geHn+heHnΘHnFn=θ^nHengeHn⏟H¯en.Thus, the signal-to-interference-noise-ratio (SINR) for thekth Bob can be rewritten as
(4)Γkn=θ∧HnH¯knwkn2∑j=0,j≠kKθ∧HnH¯knwjn2+σk2n.Similarly, when Eve try to eavesdrop the confidential information sent to thekth Bob, the SINR at Eve is given by
(5)Γen=θ∧HnH¯enwkn2∑j=0,j≠kKθ∧HnH¯enwjn2+σe2n.According to [31, 32], for the given total information block length N, the constraint on the decoding error probability of εk at the kth Bob, and the secrecy constraint on the information leakage of δ, a lower bound on the secrecy rate for the kth Bob in the nth time slot can be approximated expressed as
(6)Rkn=log21+Γkn−log21+Γen−λknNQ−1εkln2−λenNQ−1δln2,where λkn and λen denote the channel dispersions for the kth Bob and Eve in the nth time slot, respectively, given by
(7)λkn=1−1+Γkn−2,λen=1−1+Γen−2,and Q−1 is the inverse of the Gaussian Q-function Qx=∫x∞1/2πe−t2/2dt.It should be noted thatλkn and λen measure the stochastic variability of the channel relative to a deterministic channel with the same capacity. In addition, (6) implies that when compared with the traditional secrecy capacity, εk and δ incur two penalty terms on the achievable secrecy rate with finite block-length coding. Besides, when the information block-length N approaches infinity, these penalty terms diminish, and the achievable secrecy rate in (6) asymptotically coincides with the secrecy capacity. Thus, the AESR over all N time slots is given by
(8)Raesr=1N∑n=1Nmink1−εkRkn.


### 2.2. Problem Formulation

Here, we aim to maximize the AESR via jointly designingwkn and θ^n, subject to the Tx power budget and the UMC of the RC. Mathematically, the problem is formulated as
(9a)maxwkn,θ^nRaesr,(9b)s.t.∑k=0KwkHnwkn≤Psn,∀n,(9c)θ^nm,m=1,∀n,∀m,where Psn denotes the power budget for Tx at the nth time slot. It should be noted that (9a), (9b), and (9c) are nonconvex due to the coupled variables and multiple nonconvex constraints [33]. In the following, we will propose an effective way to convert (9a), (9b), and (9c) into a solvable problem. To simplify the notation, we neglect the slot index n in related variables in the rest part.


## 2.1. System Model

Let us consider a RIS-aided secure URLLC system as shown in Figure1, which consists of one Tx, one RIS, K Bobs, and one Eve. The Tx has Nt antennas, and the RIS has M elements, respectively. In addition, each Bob and Eve is equipped with single antenna, and the set of the Bobs is denoted as KΔ=1,⋯,K. In the nth slot, the channel between Tx and the kth Bob and the Eve is denoted by gkn∈ℂNt×1 and gen∈ℂNt×1, respectively. Besides, the channel between Tx and the RIS is denoted as Fn∈ℂM×Nt, between the RIS and the kth Bob, and the Eve is denoted by hkn∈ℂM×1 and hen∈ℂM×1, respectively. All channels are assumed to be frequency-flat, and the channel-state-information (CSI) is perfectly obtained by the Tx. Different from the common used uniform linear array (ULA), we utilize the uniform plane array (UPA) at Tx since the Bobs/Eve maybe not move on a line in practice. Therefore, a UPA may be a better choice.Figure 1
The system model.Letskn∈ℂ1×1 be the information symbol sent to the kth Bob in the nth slot. The transmit signal x can be written as xn=∑k=1Kwknskn+vn, where wkn∈ℂNt×1 is the transmit BF for the kth Bob in the nth slot, and vn∈ℂNt×1 denotes the artificial noise (AN) which is modeled as a circularly symmetric complex Gaussian (CSCG) vector with vn~CN0,Ωn, where Ωn±0∈ℍN is the covariance matrix of the AN. In the following part, we replace vn by w0n to simplify the natation.The phase shifter matrix of the RIS in thenth slot is denoted by Θn=Diagθ1n,⋯,θMn, with θmn denoting the RC of the mth reflection element in the nth slot. Since only the phase of the received signal is changed, the strength of the reflection signal is thus maximized, i.e., θmn2=1.Hence, the signals received by thekth Bob and Eve in the nth slot are, respectively, given by
(1)ykn=gkHn+hkHnΘHnFnxn+nkn,yen=geHn+heHnΘHnFnxn+nen,where nkn and nen denote the antenna noise at the kth Bob and Eve with nk~CN0,σk2n and ne~CN0,σe2n, respectively.In fact, by definingθn=θ1n,⋯,θMnT, we have
(2)gkHn+hkHnΘHnFn=gkHn+θHnDiaghkHnFn⏟Hkn=θHn1H⏟θ^nHkngkHn⏟H¯kn.Similarly, we have(3)geHn+heHnΘHnFn=θ^nHengeHn⏟H¯en.Thus, the signal-to-interference-noise-ratio (SINR) for thekth Bob can be rewritten as
(4)Γkn=θ∧HnH¯knwkn2∑j=0,j≠kKθ∧HnH¯knwjn2+σk2n.Similarly, when Eve try to eavesdrop the confidential information sent to thekth Bob, the SINR at Eve is given by
(5)Γen=θ∧HnH¯enwkn2∑j=0,j≠kKθ∧HnH¯enwjn2+σe2n.According to [31, 32], for the given total information block length N, the constraint on the decoding error probability of εk at the kth Bob, and the secrecy constraint on the information leakage of δ, a lower bound on the secrecy rate for the kth Bob in the nth time slot can be approximated expressed as
(6)Rkn=log21+Γkn−log21+Γen−λknNQ−1εkln2−λenNQ−1δln2,where λkn and λen denote the channel dispersions for the kth Bob and Eve in the nth time slot, respectively, given by
(7)λkn=1−1+Γkn−2,λen=1−1+Γen−2,and Q−1 is the inverse of the Gaussian Q-function Qx=∫x∞1/2πe−t2/2dt.It should be noted thatλkn and λen measure the stochastic variability of the channel relative to a deterministic channel with the same capacity. In addition, (6) implies that when compared with the traditional secrecy capacity, εk and δ incur two penalty terms on the achievable secrecy rate with finite block-length coding. Besides, when the information block-length N approaches infinity, these penalty terms diminish, and the achievable secrecy rate in (6) asymptotically coincides with the secrecy capacity. Thus, the AESR over all N time slots is given by
(8)Raesr=1N∑n=1Nmink1−εkRkn.


## 2.2. Problem Formulation

Here, we aim to maximize the AESR via jointly designingwkn and θ^n, subject to the Tx power budget and the UMC of the RC. Mathematically, the problem is formulated as
(9a)maxwkn,θ^nRaesr,(9b)s.t.∑k=0KwkHnwkn≤Psn,∀n,(9c)θ^nm,m=1,∀n,∀m,where Psn denotes the power budget for Tx at the nth time slot. It should be noted that (9a), (9b), and (9c) are nonconvex due to the coupled variables and multiple nonconvex constraints [33]. In the following, we will propose an effective way to convert (9a), (9b), and (9c) into a solvable problem. To simplify the notation, we neglect the slot index n in related variables in the rest part.


## 3. An AO-PCCP-Based Approach to the AESR Maximization Problem

In this section, we will propose an AO- and PCCP-based iterative algorithm to solve (9a), (9b), and (9c).


### 3.1. Optimization ofwkk=0K

Firstly, via introducing several slack variablesuk,ue,vb,ve,rb,re, we transform (9a), (9b), and (9c) into the following reformulation
(10a)maxwk,θ^∑n=1N1−εkrb−re−vb+veQ−1εkNln2,(10b)s.t.uk≤θ∧HHkwk2∑j=0,j≠kKθ∧HHkwj2+σk2,∀k,(10c)ue≥θ∧HHewk2∑j=0,j≠kKθ∧HHewj2+σe2,∀k,(10d)vb2≥1−1+uk−2,∀k,ve2≥1−1+ue−2,(10e)1+uk≥2rb,∀k,1+ue≤2re,(10f)vb≥0,ve≥0,9b,9c,where the logarithm objective is linearized due to the monotonically property of exponential function.In the following, we propose an AO method to solve (10a), (10b), (10c), (10d), (10e), and (10f). We first reformulate (10a), (10b), (10c), (10d), (10e), and (10f) into an approximated with respect to (w.r.t) wk, with fixed θ~. Then, we optimize θ^ with fixed w~k.Firstly, (10b) and (10c) can be rewritten as
(11a)∑j=0,j≠kKθ~HHkwj2+σk2≤θ~HHkwk2uk,(11b)∑j=0,j≠kKθ~HHewj2+σe2≥θ~HHewk2ue.It is known that the right-hand side of (11a) is a quadratic-overlinear function [33], which is convex. Thus, (11a) is nonconvex. Following the idea of the PCCP [33], we approximate the term with the corresponding first-order Taylor expansion. Then, (19) can be approximated as
(12)∑j=0,j≠kKθ~HHkwj2+σk2≤2Rθ~HHkwkw~kHHkHθ~uk−ukθ~HHkw~k2u~k2,where w~k and u~k are given points achieved by the previous iteration.Similarly, (11b) can be approximated as
(13)∑j=0,j≠kK2Rθ~HHewkw~kHHeHθ~−θ~HHew~kw~kHHeHθ~+σe2≥θ~HHew~k2ue,where u~e is a given point achieved by the previous iteration.Similarly, the nonconvex (10e) can be approximated as
(14)1+ue≤2r~ere−r~eln2+1,where r~e is a given point achieved by the previous iteration.Then, we turn our attention to (10d). In fact, the term vb2≥1−1+uk−2 is in the form of two superlevels of convex functions, which can be approximated as [32]
(15)v~b2+2v~bvb−v~b≥1−1+u~k−2+21+u~k−3uk−u~k,where v~b and u~k are the fixed points achieved by the previous iteration. It is easily known that ve2≥1−1+ue−2 can be handled in a similar way.At last, combining these steps, we obtain the following approximated problem around given pointw~k,θ~,u~k,u~e,v~b,v~e,r~e:
(16a)maxwk∑n=1N1−εkrb−re−vb+veQ−1εNln2,(16b)s.t.1+uk≥2rb,10f,12−15,which is convex and can be effectively solved by available optimization solvers such as CVX [34].


### 3.2. Optimization ofθ^

Then, we focus on the optimization ofθ^ with fixed w~k. Firstly, by denoting Ω=∑j=0,j≠kKw~jw~jH, we have ∑j=0,j≠kKθ∧HHkw~j2=θ∧HHkΩHkHθ^. Thus, we have the following approximations:
(17)θ∧HHkΩHkHθ^+σk2≤2Rθ~HHkw~kw~kHHkHθ^uk−ukθ~HHkw~k2u~k2,∑j=0,j≠kK2Rθ~HHew~kw~kHHeHθ^−θ∧HHew~kw~kHHeHθ^+σe2≥θ∧HHew~k2ue.Then, the only nonconvex constraint in (16a) is the UMC. To handle this, a PCCP method is utilized. The constraint of (9c) is equivalent to θ∧m2≤1 and θ∧m2≥1, where the former is convex and the latter is not convex. Since the global lower-bound of a convex function θ∧m2 can be expressed by the first-order Taylor expansion at any feasible point, then nonconvex constraint θ∧m2≥1 can be transformed into an affine constraint. Thus, (16a) and (16b) become
(18a)maxθ^∑n=1N1−εkrb−re−vb+veQ−1εkNln2−η∑m=12Mbm,(18b)s.t.θ~m∗θ~m−2Rθ~m∗θ^m≤bm−1,m=1,⋯,M+1,(18c)θ^m∗θ^m≤1+bm+M+1,m=1,⋯,M+1,(18d)b±0,where b=b1,⋯,b2M+2T are slack variables introduced for the UMC, ∑m=12Mbm is the penalty term, and η is employed as the regularization factor to control the feasibility of the constraints [35].


### 3.3. Convergence and Computational Complexity

Combining the above steps, we obtain the integrated AO approach in Algorithm1.Algorithm 1: The AO algorithm.
1: Setl=1 and initialize wk1k=0K,θ∧1.2: repeata) Obtainwklk=0K via solving (16)b) Obtainθ∧l via solving (18)c) Updatewkl−1k=0K,θ∧l−1⟵wklk=0K,θ∧l.d)l⟵l+1.3:until Convergence.4: Outputwk★k=0K,θ∧★.Here, we provide a brief proof for the convergence of Algorithm1. Firstly, we note that the convergence of the PCCP method can be guaranteed by the rigorous analysis in [36]. As for the whole AO procedure, we denote the objective value sequence of (9a), (9b), and (9c) as Rwklk=0K,θ∧l, and the objective function of (10a), (10b), (10c), (10d), (10e), and (10f) and (18a), (18b), (18c), and (18d) as f1wklk=0K,θ∧l, and f2wklk=0K,θ∧l, respectively. Then, we have
(19)Rwkl+1k=0K,θ∧l+1=f2wkl+1k=0K,θ∧l+1≥f2wkl+1k=0K,θ∧l=f1wkl+1k=0K,θ∧l≥f1wklk=0K,θ∧l=Rwklk=0K,θ∧l.From (19), we can see that Algorithm 1 generates a monotonic increasing sequence Rwklk=0K,θ∧l. In addition, Rwklk=0K,θ∧l is upper bounded due to the power constraint (9c), thus is guaranteed to converge.We now analyze the complexity of Algorithm1, which is determined by the complexities of each subproblem and the number of iterations. According to [33], for problems (16a), (16b) and (18a), (18b), (18c), and (18d), the complexities are given by OK+2Nt2 and OM2+M, respectively. Thus, the total computational complexity of Algorithm 1 is
(20)C=OTmaxK+2Nt2,M2+M,with T being the number of iterations. Therefore, Algorithm 1 has the polynomial time complexity.In contrast, when adopting the semidefinite relaxation (SDR) method to solve (10a), (10b), (10c), (10d), (10e), and (10f) and then select the optimal w through Gaussian randomization (GR), according to [33], the complexity of SDR at each iteration is OKNt3.5, and the GR has the complexity of ONt3. Similarly, when using the SDR and GR methods to optimize θ^, the complexity is OM3.5+OM3. Thus, the SDR-GR method leads to higher complexity than the proposed method.


## 3.1. Optimization ofwkk=0K

Firstly, via introducing several slack variablesuk,ue,vb,ve,rb,re, we transform (9a), (9b), and (9c) into the following reformulation
(10a)maxwk,θ^∑n=1N1−εkrb−re−vb+veQ−1εkNln2,(10b)s.t.uk≤θ∧HHkwk2∑j=0,j≠kKθ∧HHkwj2+σk2,∀k,(10c)ue≥θ∧HHewk2∑j=0,j≠kKθ∧HHewj2+σe2,∀k,(10d)vb2≥1−1+uk−2,∀k,ve2≥1−1+ue−2,(10e)1+uk≥2rb,∀k,1+ue≤2re,(10f)vb≥0,ve≥0,9b,9c,where the logarithm objective is linearized due to the monotonically property of exponential function.In the following, we propose an AO method to solve (10a), (10b), (10c), (10d), (10e), and (10f). We first reformulate (10a), (10b), (10c), (10d), (10e), and (10f) into an approximated with respect to (w.r.t) wk, with fixed θ~. Then, we optimize θ^ with fixed w~k.Firstly, (10b) and (10c) can be rewritten as
(11a)∑j=0,j≠kKθ~HHkwj2+σk2≤θ~HHkwk2uk,(11b)∑j=0,j≠kKθ~HHewj2+σe2≥θ~HHewk2ue.It is known that the right-hand side of (11a) is a quadratic-overlinear function [33], which is convex. Thus, (11a) is nonconvex. Following the idea of the PCCP [33], we approximate the term with the corresponding first-order Taylor expansion. Then, (19) can be approximated as
(12)∑j=0,j≠kKθ~HHkwj2+σk2≤2Rθ~HHkwkw~kHHkHθ~uk−ukθ~HHkw~k2u~k2,where w~k and u~k are given points achieved by the previous iteration.Similarly, (11b) can be approximated as
(13)∑j=0,j≠kK2Rθ~HHewkw~kHHeHθ~−θ~HHew~kw~kHHeHθ~+σe2≥θ~HHew~k2ue,where u~e is a given point achieved by the previous iteration.Similarly, the nonconvex (10e) can be approximated as
(14)1+ue≤2r~ere−r~eln2+1,where r~e is a given point achieved by the previous iteration.Then, we turn our attention to (10d). In fact, the term vb2≥1−1+uk−2 is in the form of two superlevels of convex functions, which can be approximated as [32]
(15)v~b2+2v~bvb−v~b≥1−1+u~k−2+21+u~k−3uk−u~k,where v~b and u~k are the fixed points achieved by the previous iteration. It is easily known that ve2≥1−1+ue−2 can be handled in a similar way.At last, combining these steps, we obtain the following approximated problem around given pointw~k,θ~,u~k,u~e,v~b,v~e,r~e:
(16a)maxwk∑n=1N1−εkrb−re−vb+veQ−1εNln2,(16b)s.t.1+uk≥2rb,10f,12−15,which is convex and can be effectively solved by available optimization solvers such as CVX [34].


## 3.2. Optimization ofθ^

Then, we focus on the optimization ofθ^ with fixed w~k. Firstly, by denoting Ω=∑j=0,j≠kKw~jw~jH, we have ∑j=0,j≠kKθ∧HHkw~j2=θ∧HHkΩHkHθ^. Thus, we have the following approximations:
(17)θ∧HHkΩHkHθ^+σk2≤2Rθ~HHkw~kw~kHHkHθ^uk−ukθ~HHkw~k2u~k2,∑j=0,j≠kK2Rθ~HHew~kw~kHHeHθ^−θ∧HHew~kw~kHHeHθ^+σe2≥θ∧HHew~k2ue.Then, the only nonconvex constraint in (16a) is the UMC. To handle this, a PCCP method is utilized. The constraint of (9c) is equivalent to θ∧m2≤1 and θ∧m2≥1, where the former is convex and the latter is not convex. Since the global lower-bound of a convex function θ∧m2 can be expressed by the first-order Taylor expansion at any feasible point, then nonconvex constraint θ∧m2≥1 can be transformed into an affine constraint. Thus, (16a) and (16b) become
(18a)maxθ^∑n=1N1−εkrb−re−vb+veQ−1εkNln2−η∑m=12Mbm,(18b)s.t.θ~m∗θ~m−2Rθ~m∗θ^m≤bm−1,m=1,⋯,M+1,(18c)θ^m∗θ^m≤1+bm+M+1,m=1,⋯,M+1,(18d)b±0,where b=b1,⋯,b2M+2T are slack variables introduced for the UMC, ∑m=12Mbm is the penalty term, and η is employed as the regularization factor to control the feasibility of the constraints [35].


## 3.3. Convergence and Computational Complexity

Combining the above steps, we obtain the integrated AO approach in Algorithm1.Algorithm 1: The AO algorithm.
1: Setl=1 and initialize wk1k=0K,θ∧1.2: repeata) Obtainwklk=0K via solving (16)b) Obtainθ∧l via solving (18)c) Updatewkl−1k=0K,θ∧l−1⟵wklk=0K,θ∧l.d)l⟵l+1.3:until Convergence.4: Outputwk★k=0K,θ∧★.Here, we provide a brief proof for the convergence of Algorithm1. Firstly, we note that the convergence of the PCCP method can be guaranteed by the rigorous analysis in [36]. As for the whole AO procedure, we denote the objective value sequence of (9a), (9b), and (9c) as Rwklk=0K,θ∧l, and the objective function of (10a), (10b), (10c), (10d), (10e), and (10f) and (18a), (18b), (18c), and (18d) as f1wklk=0K,θ∧l, and f2wklk=0K,θ∧l, respectively. Then, we have
(19)Rwkl+1k=0K,θ∧l+1=f2wkl+1k=0K,θ∧l+1≥f2wkl+1k=0K,θ∧l=f1wkl+1k=0K,θ∧l≥f1wklk=0K,θ∧l=Rwklk=0K,θ∧l.From (19), we can see that Algorithm 1 generates a monotonic increasing sequence Rwklk=0K,θ∧l. In addition, Rwklk=0K,θ∧l is upper bounded due to the power constraint (9c), thus is guaranteed to converge.We now analyze the complexity of Algorithm1, which is determined by the complexities of each subproblem and the number of iterations. According to [33], for problems (16a), (16b) and (18a), (18b), (18c), and (18d), the complexities are given by OK+2Nt2 and OM2+M, respectively. Thus, the total computational complexity of Algorithm 1 is
(20)C=OTmaxK+2Nt2,M2+M,with T being the number of iterations. Therefore, Algorithm 1 has the polynomial time complexity.In contrast, when adopting the semidefinite relaxation (SDR) method to solve (10a), (10b), (10c), (10d), (10e), and (10f) and then select the optimal w through Gaussian randomization (GR), according to [33], the complexity of SDR at each iteration is OKNt3.5, and the GR has the complexity of ONt3. Similarly, when using the SDR and GR methods to optimize θ^, the complexity is OM3.5+OM3. Thus, the SDR-GR method leads to higher complexity than the proposed method.


## 4. Simulation Results

The simulation deployment is shown in Figure2, where there exist one Tx, one RIS, 4 Bobs, and one Eve. The coordinates of Tx and the RIS are 0m,20m, 50m,20m, respectively. Then, the coordinate for Eve is 50m,0m, while each Bob is randomly placed in a circle centered at the Eve, with radius of 2m, respectively.Figure 2
The deployment of the simulation.Unless otherwise specified, the simulation settings are assumed as follows:N=200, M=40, Nt=4, Psn=30dBW, σk2n=σe2n=−80dBm,∀n, εk=0.01, and δ=0.01. The large-scale path loss is given by PL=PL0−10αlog10d/d0, where PL0=−30dB is the path loss at the reference distance d0=1m, d is the link distance, and α is the path loss exponent. Similar to [27], the path loss exponent between Tx and the Bobs/Eve is given by αT=4, and the small-scale fading is assumed to be Rayleigh fading. The path loss exponents of the RIS-related links is set as αR=2.2. As for the small-scale fading, take F as example, which is modeled as F=κ/κ+1FLoS+1/κ+1FNLoS, where κ=3dB is the Rician factor, FLoS is the line-of-sight (LoS) component, and FNLoS is the non-LoS (NLoS) component that is Rayleigh fading. Since we assume that UPA is employed at Tx, FLoS is given by FLoS=arψr,βratψt,βtH, and the transmit array response vectors atψt,βt is given as
(21)atψt,βt=1Nt1,⋯,ej2πd/λmsinψtsinβt+ncosβt,⋯,ej2πd/λW−1sinψtsinβt+H−1cosβtT,where ψtβt represents the azimuth and (elevation) angles of departures, λ is the signal wavelength, d is the distance between the antennas or RIS elements, and 0≤m<W and 0≤n<H denote the horizontal and vertical antenna element indices of the antenna plane, respectively. Besides, the whole antenna array size is Nt=WH. Other array response vectors can be similarly defined. In the following, we set d=λ/2.Firstly, we assess the convergence behavior of the proposed algorithm, the results are shown in Figure3. From this figure, we can see that the propose method always converge in 10 iteration, which confirm the practice of the proposed design.Figure 3
The convergence behavior.Moreover, we compare our design with the following benchmarks: (1) the proposed design in the case of infinite block-length, which can be seen as the upper bound of the proposed design; (2) the proposed design in the case of no AN; (3) the random RIS case, e.g., choose the RC randomly; (4) no RIS-aided case. These methods are labelled as “Proposed method,” “Infinite case,” “No AN,” “Random RIS,” and “No RIS,” respectively.Firstly, we compare the AESR of different schemes versusPsn; the result was shown in Figure 4. From this figure, we can see that the AESR of all these methods increase with Ps, while the proposed method outperforms the other benchmarks, and the performance gap between the proposed method and the infinite case is relatively small, which suggests the superiority of the proposed design. In addition, by comparing the curves of the proposed design with the no AN case and the no RIS case, we can find that RIS plays a more important role than AN in improving the AESR.Figure 4
The AESR versusPsn.Then, we compare the AESR of different schemes versusN; the result was shown in Figure 5. As expected, we can see that with the increase of N, the gap between the AESR and secrecy capacity tends to become small. Besides, the performance gap between the finite and infinite N is not negligible for relatively small N, which further confirms the illustration below (6).Figure 5
The AESR versusN.Next, we compare the AESR of different schemes versusM; the result was shown in Figure 6. From this figure, we can see that for all these methods, higher AESR can be achieved with more elements. The performance gain mainly comes from the fact that with larger M, more signals can reach the RIS, and the sum of the reflected signals increases, provided that the phase shifts are appropriately optimized. This result shows that a larger RIS can further improve the secrecy performance when optimizing the phase shifts properly.Figure 6
The AESR versusM.Finally, to show the impact of the location of the RIS on the AESR performance, Figure7 illustrates the obtained AESR for these schemes versus different Tx-RIS horizontal distances. It is observed that for all RIS-aided methods, when RIS moves along the x-axis from the Tx to the Bob-Eve area (about xRIS≤50m), the AESR first decreases than increases. Then, when RIS moves away from the Bob-Eve area (about xRIS>50m), the AESR decreases obviously due to the severe channel fading. Actually, the large-scale channel gain of the cascaded link can be estimated by PL=PL02d/d0−αRD−d/d0−αR, where d and D denote the distances from Tx to the RIS, and that from Tx to the center of the Bob/Eve area, respectively. Thus, the cascaded channel gain obtains the minimum value when d=D/2, e.g., the RIS is located close to the middle point. Hence, to improve the performance, it is beneficial to deploy the RIS close to Tx or Bobs.Figure 7
The AESR versus the Tx-RIS horizontal distance.


## 5. Conclusion

In this work, we have studied the transmission design in secure URLLC network aided by a RIS. Specifically, we aimed to maximize the AESR by jointly designing the active BF, the AN covariance and the phase shifter, subjected to the constraint of Tx power and the UMC. An AO- and PCCP-based method was proposed to obtain the solution. Simulation results verified the performance of the proposed algorithm and showed the effectiveness of RIS in improving the security in URLLC network.



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*Source: 8718279-2022-06-01.xml*