# Outage Constraint Robust Cooperative Beamforming and Jamming Design for Secure AF Relay Networks

Authors: Jinxin Zhu; Jun Shao
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This paper investigates a joint robust scheme in a secrecy relay network where distributed relays perform cooperative beamforming and a friendly jammer transmits jamming signal to enhance the information security. Specifically, we consider the outage constraint secrecy rate maximization design with imperfect channel state information. Through semidefinite relaxation and one-dimensional search, we propose a two-layer optimization method to solve the nonconvex problem. In addition, the Bernstein-type inequality and large deviation inequality are utilized to convert the probabilistic constraint. Simulation results demonstrate the performance of the proposed design.


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## Body

## 1. Introduction

Physical layer security is considered as a promising technique in wireless networks to prevent illegitimate eavesdropping on the confidential message. Recently, cooperative transmission using relays to improve the physical layer security in wireless network has attracted increasing attention [1], while the cooperative beamforming (CB) and cooperative jamming (CJ) are the most useful ways for improving the security in relay networks [2].Specifically, in [3], the authors proposed a joint CB and jamming design to provide Bob with certain quality of service (QoS). In [4], the authors proposed a joint CB, jamming and power allocation to secure amplify-and-forward (AF) relay system, and was extend in [5] with considering hybrid opportunistic relaying and jamming, while in [6], the authors proposed a jamming scheme in secure AF relay system considering two-slot eavesdropping. Recently in [7], the authors proposed a destination-aided CJ scheme for the AF relay system when the relay is untrusted.The above works assumed that the perfect channel state information (CSI) can be obtained. In practice, it may be difficult to obtain the perfect CSI due to channel estimation and quantization errors. To handle this obstacle, two kinds of robust secrecy design, e.g., the worst case (WC) design and the outage constraint (OC) design have been widely investigated. The WC secrecy design, which assumes that the CSI errors lie in a deterministic region, is an absolute safe design and the performance may be conservative. On the other hand, the OC method is a better choice when the CSI errors have certain probabilistic properties [8]. Specifically, the OC secrecy design was investigated for multiple-input single-output (MISO) system in [9–12] and for multiple-input multiple-output (MIMO) system in [13–15], respectively. Among these works, the Bernstein-type inequality (BTI) and large deviation inequality (LDI) are widely used to convert the probabilistic constraint into convex constraint.However, all of these works investigated the direct communication scenario, and the OC secrecy design in relay network with friendly jammer has not been investigated yet. Motivated by this observation, in this paper, we concentrate on the OC robust design in friendly jammer-assisted secrecy AF relay networks. Specifically, assuming that the CSI error can be modeled as certain probabilistic distribution, we investigate the outage constraint secrecy rate maximization (OCSRM) problem, via jointly optimizing the CB and jamming covariance. Here, we model the CSI uncertainty level as a function of the signal-to-noise ratio (SNR), which rarely appears in related works. To solve the formulated nonconvex problem, we derive a safe approximation of the outage secrecy rate, and then a two-layer optimization algorithm, which consists of one-dimensional search and semidefinite relaxation (SDR), was proposed. Furthermore, the probabilistic constraints are transformed by the BTI and LDI, respectively. Simulation results demonstrate the performance of the proposed method.


### 1.1. Notations

Throughout this paper, boldface lowercase and uppercase letters denote vectors and matrices, respectively. The transpose, conjugate transpose, and trace of matrixA are denoted as AT, AH, and TrA, respectively. a=vecA denotes to stack the columns of matrix A into a vector a. A≽0 indicates that A is a positive semidefinite matrix.  denotes the Frobenius norm. ⊙ denotes elementwise product. Da represents a diagonal matrix with a on the main diagonal. I is an identity matrix with proper dimension. Rea denotes the real part of a complex variable a. CN0,I denotes a circularly symmetric complex Gaussian random vector with mean 0 and covariance I. λmaxA,B denotes the maximum generalized eigenvalue of matrix pair A,B. E denotes the mathematical expectation.


## 1.1. Notations

Throughout this paper, boldface lowercase and uppercase letters denote vectors and matrices, respectively. The transpose, conjugate transpose, and trace of matrixA are denoted as AT, AH, and TrA, respectively. a=vecA denotes to stack the columns of matrix A into a vector a. A≽0 indicates that A is a positive semidefinite matrix.  denotes the Frobenius norm. ⊙ denotes elementwise product. Da represents a diagonal matrix with a on the main diagonal. I is an identity matrix with proper dimension. Rea denotes the real part of a complex variable a. CN0,I denotes a circularly symmetric complex Gaussian random vector with mean 0 and covariance I. λmaxA,B denotes the maximum generalized eigenvalue of matrix pair A,B. E denotes the mathematical expectation.


## 2. System Model and Problem Formulation

### 2.1. System model

We consider a cooperative relay network as shown in Figure1, in which Alice communicates with Bob with the help of M relays and a friendly jammer. The set of Eves is denoted as K≜1,…,K. We assume that no direct link exists between Alice and Bob or between Alice and Eves. Each node is equipped with a single antenna except the jammer, which is equipped with Nj antennas. All the channels are flat-fading Rayleigh distribution. Specifically, the channel between Alice to relays, between relays to Bob and Eves, between the jammer to Bob and Eves are denoted by h∈ℂM, gb∈ℂM, ge,k∈ℂM, gb,j∈ℂNj, and gk,j∈ℂNj, respectively, and we assume that these channels undergo flat Rayleigh fading. Since the relays operate in a half-duplex mode with the AF scheme, one transmission round is composed of two phases.Figure 1
The secrecy relay system model with CB and jamming.In the first phase, Alice broadcasts its informations which satisfies Es2=1 to the relays. Thus, the received signal at the relays is expressed as yr=Pshs+nr, where Ps is the transmit power at Alice and nr is the additive noise at the relays with distribution CN0,σr2I.In the second phase, the relays employ the CB vectorw≜w1,…,wMT∈ℂM to forward the information to the Bob, which grants Eves the opportunity to overhearing. Thus, the signal transmitted by the relays is given by x=Dyrw.To confuse Eves, the jammer emits jamming signalz∈ℂNj with distribution CN0,Q isotropically, where Q is the jamming covariance. Thus, the received signals at Bob and k-th Eve are, respectively, given by(1a)yb=PsgbHDhws+nrTDHgbw+gb,jHz+nb,(1b)ye,k=PsgkHDhws+nrTDHgkw+gk,jHz+nk,where nb and nk are additive noises at Bob and k-th Eve, with variance σb2 and σk2, respectively.The signal-to-interference-plus-noise ratios (SINRs) at Bob andk-th Eve can be written as(2a)Γb≜wHAbwwHBbw+gb,jHQgb,j+σb2,(2b)Γe,k≜wHAkwwHBkw+gk,jHQgk,j+σk2,∀k∈K,where Ab=PsDHhgbgbHDh, Bb=σr2DgbDHgb, Ak=PsDHhgkgkHDh, and Bk=σr2DgkDHgk,∀k∈K, respectively.Accordingly, the achievable secrecy rate for the relay network is given by(3)R=min∀k∈Kfkw,Q,where fkw,Q is the difference between the information rates for Bob and k-th Eve, which can be expressed as(4)fkw,Q=Cbw,Q−Ce,kw,Q,where Cb and Ce are given as(5a)Cbw,Q=12log1+Γb,(5b)Ce,kw,Q=12log1+Γe,k,∀k∈K.


### 2.2. Problem Formulation

Notably, we assume a practical scenario that the relays and jammer have perfect CSI of Bob but only imperfect CSI about Eves. Similar to [11], the channel error vectors are modeled as circularly symmetric complex Gaussian (CSCG) distribution, i.e.,(6a)Gk=gk|gk=g¯k+Δgk,Δgk∼CN0,Ck,(6b)Gk,j=gk,j|gk,j=g¯k,j+Δgk,j,Δgk,j∼CN0,Ck,j,where g¯k and g¯k,j are the estimated channel vectors, respectively, and Δgk and Δgk,j denote the channel uncertainty with covariance Ck∈ℍ+M and Ck,j∈ℍ+Nj, respectively. In addition, gk is independent of gl, for any k≠l. Similarly, gk,j is independent of gl,j, for any k≠l.In this paper, we assume thatCk=εk2I and Ck,j=ζk2I, with εk2 and ζk2 denoting the channel uncertainty level. Besides, according to [16, 17], we consider two types of channel estimation error models:(1)
εk2 and ζk2 are fixed and independent of the average SNR.(2)
εk2 and ζk2 are functions of the average SNR. Take εk2 as example. The variance of the error can be modeled as εk2=Ωk/1+ϖγΩk, where Ωk=d−α is the variance of gk, e.g., EgkgkH=ΩkI, where d denotes the link distance, and α is the attenuation factor. In addition,γ is the transmit SNR, and ϖ>0 depends on the cost of acquiring CSI in light of the training pilot power consumption and reflects the quality of channel estimation.In this paper, we aim to maximize the outage secrecy rate via jointly designing the CB vectorw and jamming covariance Q. Specifically, the OCSRM problem is formulated as(7a)maxw,Q,RsRs,(7b)s.t.Prgk,gk,jmin∀k∈Kfkw,Q≥Rs≥1−δ,(7c)wHFw+TrQ≤PT,(7d)wHFw≤PR,(7e)TrQ≤PJ,where δ is the secrecy outage probability, e.g., a parameter denoting the chance of R falling below the target rate Rs due to CSI uncertainty. Apparently, Rs is the outage secrecy rate. In addition, F=PsDHhDh+σr2I, and PT denotes the sum power budget for the relays and jammer, while PR and PJ stand for the individual power budget for the relays and the jammer, respectively.It should be noted that (7a)–(7e) are nonconvex due to the coupled variables and constraints. Hence, in the following, we will propose an effective way to convert (7a)–(7e) into a solvable problem.


## 2.1. System model

We consider a cooperative relay network as shown in Figure1, in which Alice communicates with Bob with the help of M relays and a friendly jammer. The set of Eves is denoted as K≜1,…,K. We assume that no direct link exists between Alice and Bob or between Alice and Eves. Each node is equipped with a single antenna except the jammer, which is equipped with Nj antennas. All the channels are flat-fading Rayleigh distribution. Specifically, the channel between Alice to relays, between relays to Bob and Eves, between the jammer to Bob and Eves are denoted by h∈ℂM, gb∈ℂM, ge,k∈ℂM, gb,j∈ℂNj, and gk,j∈ℂNj, respectively, and we assume that these channels undergo flat Rayleigh fading. Since the relays operate in a half-duplex mode with the AF scheme, one transmission round is composed of two phases.Figure 1
The secrecy relay system model with CB and jamming.In the first phase, Alice broadcasts its informations which satisfies Es2=1 to the relays. Thus, the received signal at the relays is expressed as yr=Pshs+nr, where Ps is the transmit power at Alice and nr is the additive noise at the relays with distribution CN0,σr2I.In the second phase, the relays employ the CB vectorw≜w1,…,wMT∈ℂM to forward the information to the Bob, which grants Eves the opportunity to overhearing. Thus, the signal transmitted by the relays is given by x=Dyrw.To confuse Eves, the jammer emits jamming signalz∈ℂNj with distribution CN0,Q isotropically, where Q is the jamming covariance. Thus, the received signals at Bob and k-th Eve are, respectively, given by(1a)yb=PsgbHDhws+nrTDHgbw+gb,jHz+nb,(1b)ye,k=PsgkHDhws+nrTDHgkw+gk,jHz+nk,where nb and nk are additive noises at Bob and k-th Eve, with variance σb2 and σk2, respectively.The signal-to-interference-plus-noise ratios (SINRs) at Bob andk-th Eve can be written as(2a)Γb≜wHAbwwHBbw+gb,jHQgb,j+σb2,(2b)Γe,k≜wHAkwwHBkw+gk,jHQgk,j+σk2,∀k∈K,where Ab=PsDHhgbgbHDh, Bb=σr2DgbDHgb, Ak=PsDHhgkgkHDh, and Bk=σr2DgkDHgk,∀k∈K, respectively.Accordingly, the achievable secrecy rate for the relay network is given by(3)R=min∀k∈Kfkw,Q,where fkw,Q is the difference between the information rates for Bob and k-th Eve, which can be expressed as(4)fkw,Q=Cbw,Q−Ce,kw,Q,where Cb and Ce are given as(5a)Cbw,Q=12log1+Γb,(5b)Ce,kw,Q=12log1+Γe,k,∀k∈K.


## 2.2. Problem Formulation

Notably, we assume a practical scenario that the relays and jammer have perfect CSI of Bob but only imperfect CSI about Eves. Similar to [11], the channel error vectors are modeled as circularly symmetric complex Gaussian (CSCG) distribution, i.e.,(6a)Gk=gk|gk=g¯k+Δgk,Δgk∼CN0,Ck,(6b)Gk,j=gk,j|gk,j=g¯k,j+Δgk,j,Δgk,j∼CN0,Ck,j,where g¯k and g¯k,j are the estimated channel vectors, respectively, and Δgk and Δgk,j denote the channel uncertainty with covariance Ck∈ℍ+M and Ck,j∈ℍ+Nj, respectively. In addition, gk is independent of gl, for any k≠l. Similarly, gk,j is independent of gl,j, for any k≠l.In this paper, we assume thatCk=εk2I and Ck,j=ζk2I, with εk2 and ζk2 denoting the channel uncertainty level. Besides, according to [16, 17], we consider two types of channel estimation error models:(1)
εk2 and ζk2 are fixed and independent of the average SNR.(2)
εk2 and ζk2 are functions of the average SNR. Take εk2 as example. The variance of the error can be modeled as εk2=Ωk/1+ϖγΩk, where Ωk=d−α is the variance of gk, e.g., EgkgkH=ΩkI, where d denotes the link distance, and α is the attenuation factor. In addition,γ is the transmit SNR, and ϖ>0 depends on the cost of acquiring CSI in light of the training pilot power consumption and reflects the quality of channel estimation.In this paper, we aim to maximize the outage secrecy rate via jointly designing the CB vectorw and jamming covariance Q. Specifically, the OCSRM problem is formulated as(7a)maxw,Q,RsRs,(7b)s.t.Prgk,gk,jmin∀k∈Kfkw,Q≥Rs≥1−δ,(7c)wHFw+TrQ≤PT,(7d)wHFw≤PR,(7e)TrQ≤PJ,where δ is the secrecy outage probability, e.g., a parameter denoting the chance of R falling below the target rate Rs due to CSI uncertainty. Apparently, Rs is the outage secrecy rate. In addition, F=PsDHhDh+σr2I, and PT denotes the sum power budget for the relays and jammer, while PR and PJ stand for the individual power budget for the relays and the jammer, respectively.It should be noted that (7a)–(7e) are nonconvex due to the coupled variables and constraints. Hence, in the following, we will propose an effective way to convert (7a)–(7e) into a solvable problem.


## 3. An SDR-Based Approach to the OCSRM Problem

In this section, we will derive an SDR-based approach to (7a)–(7e). Firstly, we decouple the probabilistic constraint in (7b). Since gk,gl,l≠k and gk,j,gl,j,l≠k are independent, we obtain(8)7b⇐Prgk,gk,jfkw,Q≥Rs≥1−δ¯,∀k,where δ¯=1−1−δ1/K (see more details in [8]).


### 3.1. Relaxation of the OCSRM Problem

In this part, we will propose a two-layer optimization way to (7a)–(7e). To start with, we rewrite (7a)–(7e) as(9a)Rs∗=maxw,Q≽0,β≥1Cbw,Q−logβ,(9b)s.t.Prgk,gk,jCe,kw,Q≤logβ≥1−δ¯,∀k∈K,(9c)7b−7d,where β is an introduced slack variable to simplify the objective function.By substituting (2a) and (2b) into (9a)–(9c), we reexpress (9a)–(9c) as(10a)Rs∗=maxw,Q≽0,β≥1log1+wHAbwwHBbw+gb,jHQgb,j+σb2−logβ,(10b)s.t.Prgk,gk,j1+wHAkwwHBbw+gk,jHQgk,j+σk2≤β≥1−δ¯,∀k∈K,(10c)7b−7d.Furthermore, utilizing the SDR [18], e.g., defining W=wwH and neglecting the nonconvex constraint rankW=1, we attain the following SDR problem:(11a)Rs∗=maxW≽0,Q≽0,β≥1log1+TrAbWσb2+TrBbW+gb,jHQgb,j−logβ,(11b)s.t.Prgk,gk,jβ−1TrBkW+gk,jHQgk,j+σk2≥TrAkW≥1−δ¯,∀k∈K,(11c)TrFW+TrQ≤PT,(11d)TrFW≤PR,(11e)TrQ≤PJ.


### 3.2. An SDR-Based Line Search Method for (11a)–(11e)

We now focus on solving the relaxation (11a)–(11e). In fact, (11a)–(11e) can be reformulated as a single-variable optimization problem, which can be efficiently handled by SDR. To show this, note that(12)1≤β≤1+wHAbwσb2+wHBbw+gb,jHQgb,j≤1+wHAbwwHBbw≤1+λmaxAb,Bb.Then, we rewrite (11a)–(11e) as(13a)eRs∗=maxαφα,(13b)s.t.1+λmaxAb,Bb−1≤α≤1,where α=1/β and(14a)φα=maxW≽0,Q≽0σb2+TrBbW+gb,jHQgb,j+TrAbWα−1TrBbW+gb,jHQgb,j+σb2,(14b)s.t.Prgk,gk,jα−1−1TrBkW+gk,jHQgk,j+σk2≥TrAkW≥1−δ¯,∀k∈K,(14c)11c−11e.φα does not have a closed form but is numerically tractable. In particular, (14a)–(14c) can be converted into a convex optimization problem. Specifically, via the Charnes–Cooper transformation [19], we introduce a change of variables W=W˜/ξ, Q=Q˜/ξ with ξ>0. Then, (14a)–(14c) can be equivalently expressed as(15a)φα=maxW˜≽0,Q˜≽0,ξ>0ξσb2+TrBbW˜+gb,jHQ˜gb,j+TrAbW˜,(15b)s.t.TrBbW+gb,jHQ˜gb,j+ξσb2=α,(15c)Prgk,gk,j1−αTrBkW˜+gk,jHQ˜gk,j+ξσk2≥αTrAkW˜≥1−δ¯,∀k∈K,(15d)TrFW˜+TrQ˜≤ξPT,(15e)TrFW˜≤ξPR,(15f)TrQ˜≤ξPJ.


### 3.3. Transformation of Probabilistic Constraints (15c)

The remaining challenging is that (15a)–(15f) have probabilistic constraints (15c), which is intractable. In the following, we will handle (15c) via the BTI and LDI, respectively.Firstly, using the following identityaHDb=bHDa, we obtain TrAkW˜=gkHHW˜HHgk and TrBkW˜=gkHσr2I⊙W˜gk.Then, we transform (15a)–(15f) into(16a)φα=maxW˜≽0,Q˜≽0,ξ>0ξσb2+TrBbW˜+gb,jHQ˜gb,j+TrAbW˜,(16b)s.t.Prgk,gk,jgkHXgk+1−αgk,jHQ˜gk,j+1−αξσk2≥0≥1−δ¯,∀k∈K,(16c)15b,15d−15f,where X=1−ασr2I⊙W˜−αPsHW˜HH.It should be noted that the outage probability in (16b) can be characterized by the quadratic inequality with respect to (w.r.t.) gk and gk,j, e.g., (16b) can be safely approximated via the following BTI and LDI, respectively.Lemma 1 (BTI [11]).
For anyA,u,c∈ℍN×ℂN×ℝ, v∼CN0,I, and η∈0,1, the following inequalities hold:(17)PrvvHAv+2ℜvHu+c≥0≥1−η,⇐TrA−−2lnηx+lnηy+c≥0,vecA2u≤x,yI+A≽0,y≥0,where x and y are slack variables. Moreover, (17) is convex w.r.t. all the variables A,u,c,x,y.To use Lemma1, we first define gk=g¯k+Ck1/2ek and gk,j=g¯k,j+Ck,j1/2ek,j, where ek∼CN0,I and ek,j∼CN0,I, respectively.Then, the part in the braces of (16b) can be equivalently rewritten as(18)ekHCk1/2XCk1/2ek+2ℜekHCk1/2Xg¯k+g¯kHXg¯k+1−αξσk2+1−αek,jHCk,j1/2QC˜k,j1/2ek,j+21−αℜek,jHCk1/2Q˜g¯k,j+1−αg¯k,jHQ˜g¯k,j≥0.Furthermore, (18) can be rewritten as(19)vkHAkvk+2ℜvkHuk+ck≥0,where Ak, vk, uk, and ck are given as(20a)Ak=Ck1/2XCk1/2001−αCk,j1/2QC˜k,j1/2,(20b)vk=ekH,ek,jH,(20c)uk=Ck1/2X001−αCk1/2Q˜g¯kg¯k,j,(20d)ck=g¯kH,g¯k,jHX001−αQ˜g¯kg¯k,j+1−αξσk2.Combining all these equations, (16a)–(16c) can be recast as the following convex problem:(21a)φα=maxW˜≽0,Q˜≽0,ξ>0,yk>0ξσb2+TrBbW˜+gb,jHQ˜gb,j+TrAbW˜,(21b)s.t.TrAk−−2lnδ¯xk+lnδ¯yk+ck≥0,∀k∈K,(21c)vecAk2uk≤xk,∀k∈K,(21d)ykI+Ak≽0,yk≥0,∀k∈K,(21e)15b,15d−15f,20,which can be efficiently solved by available solvers such as CVX [20].It should be mentioned that for the obtainedW∗, if rankW∗=1, we can attain the optimal w∗ via matrix eigenvector decomposition. On the other hand, if rankW∗≥1, we employ the Gaussian randomization (GR) method to generate the suboptimal rank-one solution [18].In the above part, we have utilized the BTI to transform the probabilistic constraint. To reduce the computational complexity, in the following we will utilize the LDI to convert the probabilistic constraint.Lemma 2 (LDI [11]).
Letx∼CN0,I, and let A∈ℍn×n, r∈ℂn×1 be given variables. Then, for any v>1/2 and ζ>0, we have(22)PrxHAx+2ℜxHr≤TrA−ζ≤exp−ζ24T2,0<ζ≤2v¯vT,exp−v¯vζT+v¯v2,ζ>2v¯vT,where v¯=1−1/2v2 and T=vA+1/2r.By Lemma2 and following the similar way in [11], (16b) can be transformed into the following relationships:(23)TrAk+ck≥2−lnδ¯xk+yk,∀k∈K,uk≤2xk,∀k∈K,vAk≤yk,∀k∈K,where v>1/2 is the solution to 1−1/2v2v=−lnδ¯ (see more details in reference [11]).Thus, (16a)–(16c) can be recast as the following convex problem:(24a)φα=maxW˜≽0,Q˜≽0,ξ>0,xk,ykξσb2+TrBbW˜+gb,jHQ˜gb,j+TrAbW˜,(24b)s.t.15b,15d−15f,20,23,which can also be solved by CVX. Similar to (21a), GR is utilized to extract the rank-one solution.Following the theory in [11], we obtain the complexity comparison between the BTI and LDI methods, which are shown in Table 1, where ϑ denotes the accuracy requirement. From the comparison, we can see that the LDI method achieves lower complexity than the BTI method.Table 1
Complexity comparison of the BTI and LDI methods.
MethodsComplexity order (ln1/ϑβXC)BTIβX=5+3K+KM,C=n5+3K+KM3+n25+3K+KM2+nKM2+M+Nj2+n3,andn=OM2+Nj2+2K+1.LDIβX=5K+5,C=n2+n5K+5+nKM4+M2+2MNj+Nj2+n3,andn=OM2+Nj2+2K+1.


## 3.1. Relaxation of the OCSRM Problem

In this part, we will propose a two-layer optimization way to (7a)–(7e). To start with, we rewrite (7a)–(7e) as(9a)Rs∗=maxw,Q≽0,β≥1Cbw,Q−logβ,(9b)s.t.Prgk,gk,jCe,kw,Q≤logβ≥1−δ¯,∀k∈K,(9c)7b−7d,where β is an introduced slack variable to simplify the objective function.By substituting (2a) and (2b) into (9a)–(9c), we reexpress (9a)–(9c) as(10a)Rs∗=maxw,Q≽0,β≥1log1+wHAbwwHBbw+gb,jHQgb,j+σb2−logβ,(10b)s.t.Prgk,gk,j1+wHAkwwHBbw+gk,jHQgk,j+σk2≤β≥1−δ¯,∀k∈K,(10c)7b−7d.Furthermore, utilizing the SDR [18], e.g., defining W=wwH and neglecting the nonconvex constraint rankW=1, we attain the following SDR problem:(11a)Rs∗=maxW≽0,Q≽0,β≥1log1+TrAbWσb2+TrBbW+gb,jHQgb,j−logβ,(11b)s.t.Prgk,gk,jβ−1TrBkW+gk,jHQgk,j+σk2≥TrAkW≥1−δ¯,∀k∈K,(11c)TrFW+TrQ≤PT,(11d)TrFW≤PR,(11e)TrQ≤PJ.


## 3.2. An SDR-Based Line Search Method for (11a)–(11e)

We now focus on solving the relaxation (11a)–(11e). In fact, (11a)–(11e) can be reformulated as a single-variable optimization problem, which can be efficiently handled by SDR. To show this, note that(12)1≤β≤1+wHAbwσb2+wHBbw+gb,jHQgb,j≤1+wHAbwwHBbw≤1+λmaxAb,Bb.Then, we rewrite (11a)–(11e) as(13a)eRs∗=maxαφα,(13b)s.t.1+λmaxAb,Bb−1≤α≤1,where α=1/β and(14a)φα=maxW≽0,Q≽0σb2+TrBbW+gb,jHQgb,j+TrAbWα−1TrBbW+gb,jHQgb,j+σb2,(14b)s.t.Prgk,gk,jα−1−1TrBkW+gk,jHQgk,j+σk2≥TrAkW≥1−δ¯,∀k∈K,(14c)11c−11e.φα does not have a closed form but is numerically tractable. In particular, (14a)–(14c) can be converted into a convex optimization problem. Specifically, via the Charnes–Cooper transformation [19], we introduce a change of variables W=W˜/ξ, Q=Q˜/ξ with ξ>0. Then, (14a)–(14c) can be equivalently expressed as(15a)φα=maxW˜≽0,Q˜≽0,ξ>0ξσb2+TrBbW˜+gb,jHQ˜gb,j+TrAbW˜,(15b)s.t.TrBbW+gb,jHQ˜gb,j+ξσb2=α,(15c)Prgk,gk,j1−αTrBkW˜+gk,jHQ˜gk,j+ξσk2≥αTrAkW˜≥1−δ¯,∀k∈K,(15d)TrFW˜+TrQ˜≤ξPT,(15e)TrFW˜≤ξPR,(15f)TrQ˜≤ξPJ.


## 3.3. Transformation of Probabilistic Constraints (15c)

The remaining challenging is that (15a)–(15f) have probabilistic constraints (15c), which is intractable. In the following, we will handle (15c) via the BTI and LDI, respectively.Firstly, using the following identityaHDb=bHDa, we obtain TrAkW˜=gkHHW˜HHgk and TrBkW˜=gkHσr2I⊙W˜gk.Then, we transform (15a)–(15f) into(16a)φα=maxW˜≽0,Q˜≽0,ξ>0ξσb2+TrBbW˜+gb,jHQ˜gb,j+TrAbW˜,(16b)s.t.Prgk,gk,jgkHXgk+1−αgk,jHQ˜gk,j+1−αξσk2≥0≥1−δ¯,∀k∈K,(16c)15b,15d−15f,where X=1−ασr2I⊙W˜−αPsHW˜HH.It should be noted that the outage probability in (16b) can be characterized by the quadratic inequality with respect to (w.r.t.) gk and gk,j, e.g., (16b) can be safely approximated via the following BTI and LDI, respectively.Lemma 1 (BTI [11]).
For anyA,u,c∈ℍN×ℂN×ℝ, v∼CN0,I, and η∈0,1, the following inequalities hold:(17)PrvvHAv+2ℜvHu+c≥0≥1−η,⇐TrA−−2lnηx+lnηy+c≥0,vecA2u≤x,yI+A≽0,y≥0,where x and y are slack variables. Moreover, (17) is convex w.r.t. all the variables A,u,c,x,y.To use Lemma1, we first define gk=g¯k+Ck1/2ek and gk,j=g¯k,j+Ck,j1/2ek,j, where ek∼CN0,I and ek,j∼CN0,I, respectively.Then, the part in the braces of (16b) can be equivalently rewritten as(18)ekHCk1/2XCk1/2ek+2ℜekHCk1/2Xg¯k+g¯kHXg¯k+1−αξσk2+1−αek,jHCk,j1/2QC˜k,j1/2ek,j+21−αℜek,jHCk1/2Q˜g¯k,j+1−αg¯k,jHQ˜g¯k,j≥0.Furthermore, (18) can be rewritten as(19)vkHAkvk+2ℜvkHuk+ck≥0,where Ak, vk, uk, and ck are given as(20a)Ak=Ck1/2XCk1/2001−αCk,j1/2QC˜k,j1/2,(20b)vk=ekH,ek,jH,(20c)uk=Ck1/2X001−αCk1/2Q˜g¯kg¯k,j,(20d)ck=g¯kH,g¯k,jHX001−αQ˜g¯kg¯k,j+1−αξσk2.Combining all these equations, (16a)–(16c) can be recast as the following convex problem:(21a)φα=maxW˜≽0,Q˜≽0,ξ>0,yk>0ξσb2+TrBbW˜+gb,jHQ˜gb,j+TrAbW˜,(21b)s.t.TrAk−−2lnδ¯xk+lnδ¯yk+ck≥0,∀k∈K,(21c)vecAk2uk≤xk,∀k∈K,(21d)ykI+Ak≽0,yk≥0,∀k∈K,(21e)15b,15d−15f,20,which can be efficiently solved by available solvers such as CVX [20].It should be mentioned that for the obtainedW∗, if rankW∗=1, we can attain the optimal w∗ via matrix eigenvector decomposition. On the other hand, if rankW∗≥1, we employ the Gaussian randomization (GR) method to generate the suboptimal rank-one solution [18].In the above part, we have utilized the BTI to transform the probabilistic constraint. To reduce the computational complexity, in the following we will utilize the LDI to convert the probabilistic constraint.Lemma 2 (LDI [11]).
Letx∼CN0,I, and let A∈ℍn×n, r∈ℂn×1 be given variables. Then, for any v>1/2 and ζ>0, we have(22)PrxHAx+2ℜxHr≤TrA−ζ≤exp−ζ24T2,0<ζ≤2v¯vT,exp−v¯vζT+v¯v2,ζ>2v¯vT,where v¯=1−1/2v2 and T=vA+1/2r.By Lemma2 and following the similar way in [11], (16b) can be transformed into the following relationships:(23)TrAk+ck≥2−lnδ¯xk+yk,∀k∈K,uk≤2xk,∀k∈K,vAk≤yk,∀k∈K,where v>1/2 is the solution to 1−1/2v2v=−lnδ¯ (see more details in reference [11]).Thus, (16a)–(16c) can be recast as the following convex problem:(24a)φα=maxW˜≽0,Q˜≽0,ξ>0,xk,ykξσb2+TrBbW˜+gb,jHQ˜gb,j+TrAbW˜,(24b)s.t.15b,15d−15f,20,23,which can also be solved by CVX. Similar to (21a), GR is utilized to extract the rank-one solution.Following the theory in [11], we obtain the complexity comparison between the BTI and LDI methods, which are shown in Table 1, where ϑ denotes the accuracy requirement. From the comparison, we can see that the LDI method achieves lower complexity than the BTI method.Table 1
Complexity comparison of the BTI and LDI methods.
MethodsComplexity order (ln1/ϑβXC)BTIβX=5+3K+KM,C=n5+3K+KM3+n25+3K+KM2+nKM2+M+Nj2+n3,andn=OM2+Nj2+2K+1.LDIβX=5K+5,C=n2+n5K+5+nKM4+M2+2MNj+Nj2+n3,andn=OM2+Nj2+2K+1.


## 4. Simulation Results

In this section, simulation results are presented to evaluate the performance of the proposed design. The channelsh, gb, gk, gb,j, and gk,j are assumed to be independent and identically distributed complex Gaussian random variables with zero mean and unit variance. We set K=3, M=5, Nj=3, Ps=10dB, PT=10dB, σr2=σb2=σe,k2=1, α=4, d=20, ϖ=0.2, and δ=10−2. In addition, we compare our design with the following methods: (1) the proposed design in the case of perfect Eves’ CSI, which can be seen as the upper bound of the robust design; (2) the no jam case, e.g., setting Q=0 while only optimizing w; and (3) the fixed CSI uncertainty model with BTI method, and εk2=ζk2=10−4. These methods are denoted as “BTI method,” “LDI method,” “perfect CSI case,” “no jam method,” and “fixed CSI uncertainty,” respectively.Firstly, we compare the outage secrecy rates versus the total power budgetPT, and the result is shown in Figure 2. From this figure, we can see that the outage secrecy rates increase with the increase of PT, while the BTI method outperforms the LDI method and the no jam method. In addition, the performance gap between perfect and imperfect CSI cases is apparent, especially for relative large PT, which shows the influence of the CSI error on the security. Besides, the fixed CSI uncertainty model suffers certain performance loss, especially for relative large PT, since when PT tends to large, according to the function model, the corresponding CSI tends to smaller. However, the fixed CSI uncertainty model cannot reflect this relationship, thus leading to performance loss.Figure 2
The outage secrecy rate versus the total power budget.Next, we investigate the effect of the number of the relays on the outage secrecy rate. Figure3 shows the relationship between the outage secrecy rate and the number of the relays. From this figure, we can see that for all these methods, higher outage secrecy rate can be achieved with more relays, since higher spatial degrees of freedom can be utilized to improve the secrecy performance.Figure 3
The outage secrecy rate versus the number of the relays.Lastly, Figure4 investigates the relationship between the outage secrecy rate versus the secrecy outage probability. As can be seen in this figure, with the increase of δ, the outage secrecy rate tends to increase. This is mainly due to the fact that with the increase of δ, the relay network permits more chance to outage, and thus the threshold of outage rate tends to increase, e.g., the outage secrecy rate tends to increase.Figure 4
The outage secrecy rate versus the secrecy outage probability.


## 5. Conclusion

In this paper, we have investigated the OCSRM design in secrecy AF relay networks, via jointly optimizing the CB and jamming covariance. To solve the formulated nonconvex problem, we derived a safe approximation of the outage secrecy rate, and then a two-layer optimization algorithm was proposed. Simulation results demonstrated the performance of the proposed method.



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*Source: 5919617-2021-11-05.xml*