# Robust Destination Jamming Aided Secrecy Precoding for an AF MIMO Untrusted Relay System

Authors: Feng Zhou; Rugang Wang; Jiahong Bian
Journal: Wireless Communications and Mobile Computing
 (2019)

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## Abstract

We consider a secrecy dual-hop amplify-and-forward (AF) untrusted relay network, where the relay is willing to forward the signal to the destination while acting as a potential eavesdropper. Assuming that all these nodes are equipped with multiple antennas, we propose a joint destination aided cooperative jamming and precoding at both the source and the relay scheme, with the objective to maximize the worst case secrecy rate. The formulated problem is highly non-convex due to the maximization of the difference of several logarithmic determinant (log-det) functions in the CSI uncertainty region. To handle this challenge, we propose to linearize these log-det terms. After linearization, we tackle the CSI uncertainty based on epigraph reformulation and the sign-definiteness lemma. Finally, an alternating optimization (AO) algorithm is proposed to solve the reformulated problem. Numerical results are provided to demonstrate the performance of the proposed scheme.


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## Body

## 1. Introduction

Due to the openness of wireless transmission medium, wireless information is susceptible to eavesdropping. Thus, the secure communication is a critical issue for wireless systems [1]. Physical layer security (PLS) is a newly emerging secrecy communication technique [2]. Relay is considered as a popular approach to extend network coverage and provide spatial degrees of freedom (DoF) [3]. PLS method in relay networks has raised great attention [4].In practice, relay is not always helpful for secure communication. A misbehaving relay may try to eavesdrop the confidential information from the source node, which is termed as untrusted relay [5]. The untrusted multiple-input multiple-output (MIMO) relay was first studied in [6], in which a joint beamforming design was proposed. Furthermore, the untrusted two-way MIMO relay systems are studied in [7, 8], while, in [9], the authors investigated a novel joint destination-aided cooperative jamming (CJ) and precoding scheme at both the source and the untrusted relay node, which was further extended to untrusted energy harvested relay network in [10].Recently, in [11], the authors proposed a joint beamforming alignment method with suboptimal power allocation for a untrusted two-way relay. In [12], the authors investigated the secrecy performance of untrusted relay systems with a full-duplex jamming destination, while, in [13], the authors investigated the energy-efficient secure transmission design for the internet of things (IoT) with an untrusted relay. In [14], the authors investigated the secure transmission in untrusted wireless-powered two-way relay networks.All these above literatures consider perfect channel state information (CSI) case. However, in practice, perfect CSI is hard to obtain due the channel estimation and quantization errors. To overcome this obstacle, in [15], the authors investigated the untrusted amplify-and-forward (AF) relay design with imperfect CSI, while, in [16], the authors proposed an artificial noise (AN) aided secure precoding for untrusted two-way MIMO relay systems with imperfect CSI.Based on these observations, in this paper, we investigated the robust secrecy design in a untrusted AF MIMO relay network. Specifically, assuming that all the nodes are equipped with multiple antennas and only imperfect CSI can be obtained, we propose a robust destination-aided CJ scheme to maximize the worst-case secrecy rate. However, the formulated problem is highly non-convex and challenging to solve. To handle this challenge, we propose to linearize the difference of multiple logarithmic determinant (log-det) functions. After linearization, we tackle the channel uncertainty based on epigraph reformulation and the sign-definiteness lemma. Finally, an alternating optimization (AO) method is proposed to solve the reformulated problem. Simulation results demonstrate the performance of our proposed design.The rest of this paper is organized as follows. The system model and problem formulation are described in Section2. The AO based method is discussed in Section 3. Numerical results are provided in Section 4, and Section 5 concludes the paper.Notations. Throughout this paper, boldface lowercase and uppercase letters denote vectors and matrices, respectively. The transpose, conjugate transpose, and trace of matrix A are denoted as AT, AH, and TrA, respectively. a=vecA denotes stacking the columns of matrix A into a vector a. A⪰0 indicates that A is a positive semidefinite matrix. · and ⊗ represent the Frobenius norm and the Kronecker product, respectively. Rea denotes the real part of a complex variable a. x+ and E. indicates max⁡0,x and the statistical expectation, respectively. 0n denotes n by n zeros matrix. 0m×n denotes m by n zeros matrix. In denotes n by n identity matrix. Im×n denotes m by n identity matrix. CN0,I denotes a circularly symmetric complex Gaussian random vector with mean 0 and covariance I.


## 2. System Model and Problem Statement

### 2.1. System Model

Let us consider a secure MIMO relay system, which consists of one source node (S), one relay (R), and one destination (D), as shown in Figure1. It is assumed that S, R, and D are equipped with Ns, Nr, and Nd antennas, respectively. The channel coefficients between S to R, D to R, and R to D are denoted as F∈CNr×Ns, H∈CNr×Nd, and G∈CNd×Nr, respectively.Figure 1
System model.In this paper, R is assumed to forward the signal to D and, at the same time, acts as an eavesdropper (Eve) which interprets the signal from S. Besides, we assume that only imperfect S, D, and R’s CSIs can be obtained. Based on the bounded CSI model in [15, 16], the imperfect CSIs are modeled as(1a)F=F∣F=F-+ΔF,ΔF≤χF,(1b)H=H∣H=H-+ΔH,ΔH≤χH,(1c)G=G∣G=G-+ΔG,ΔG≤χG,where F¯, H¯, and G¯ denote the estimates of F, H, and G, respectively; ΔF, ΔH, and ΔG are their respective channel uncertainties; χF, χH, and χG denote the respective sizes of the bounded channel uncertainties region.In the first phase, the signal vectors∈Cd×1 (d≤Ns) is precoded by the precoding matrix Ws∈CNs×d, and the jamming vector z∈Cz×1 (j≤Nd) is emitted by D with precoding matrix Wd∈CNd×z to protect the confidential information. Both s and z satisfy CN0,I. Thus, the received signal at R can be expressed as(2)yr=FWss+HWdz+nr,where nr∈CNr×1 is the noise vector at the relay with nr~CN0,σr2I.In the second phase, R utilizes precoding matrixWr to forward the information; thus, the received signal at D can be expressed as(3)yd=GWrFWss+HWdz+GWrnr+nd,where nd∈CNr×1 is the noise vector at the destination with nd~CN0,σd2I.Furthermore,yd can be expressed as(4)yd=GWrFWss+G¯WrH¯Wdz︸selfinterference+ΔGWrH¯Wdz+G¯WrΔHWdz+ΔGWrΔHWdz︸highorder+GWrnr+nd.Here, we assume that D can partially remove the self interference signal based on the estimated CSI, and the high-order term with respect to (w.r.t) the uncertain CSI is small enough compared with the channel strength. Thus, the received signal at D can be approximately given by(5)y¯d=GWrFWss+ΔGWrH¯Wdz+G¯WrΔHWdz+GWrnr+nd,Accordingly, the worst case secrecy rate can be expressed as(6)Rs=min∀F∈F,∀H∈H,∀G∈GRd-Rr+,where Rd and Rr denote the mutual information at R and D, respectively, and are given by(7a)Rd=12ln⁡Id+WsHFHWrHGHD-1GWrFWs,(7b)Rr=12ln⁡Id+WsHFHR-1FWs,with D and R being given as(8)D=ΔGWrH¯WdWdHH¯HWrHΔGH+G¯WrΔHWdWdHΔHHWrHG¯H+σr2GWrWrHGH+σd2INd,and(9)R=σr2INr+HWdWdHHH.


### 2.2. Problem Statement

In this paper, we investigated the joint precoding matrices design to maximize the worst case secrecy rate, under the power constraints at the respective nodes. Mathematically, the problem can be formulated as(10a)maxWs,Wr,Wdmin∀F∈F,∀H∈H,∀G∈GRd-Rr,(10b)s.t.Ws2≤Ps,Wd2≤Pd,(10c)WrFWs2+WrHWd2+σr2Wr2≤Pr,where Ps, Pd, and Pr are the maximum achievable powers for S, D, and R, respectively.


## 2.1. System Model

Let us consider a secure MIMO relay system, which consists of one source node (S), one relay (R), and one destination (D), as shown in Figure1. It is assumed that S, R, and D are equipped with Ns, Nr, and Nd antennas, respectively. The channel coefficients between S to R, D to R, and R to D are denoted as F∈CNr×Ns, H∈CNr×Nd, and G∈CNd×Nr, respectively.Figure 1
System model.In this paper, R is assumed to forward the signal to D and, at the same time, acts as an eavesdropper (Eve) which interprets the signal from S. Besides, we assume that only imperfect S, D, and R’s CSIs can be obtained. Based on the bounded CSI model in [15, 16], the imperfect CSIs are modeled as(1a)F=F∣F=F-+ΔF,ΔF≤χF,(1b)H=H∣H=H-+ΔH,ΔH≤χH,(1c)G=G∣G=G-+ΔG,ΔG≤χG,where F¯, H¯, and G¯ denote the estimates of F, H, and G, respectively; ΔF, ΔH, and ΔG are their respective channel uncertainties; χF, χH, and χG denote the respective sizes of the bounded channel uncertainties region.In the first phase, the signal vectors∈Cd×1 (d≤Ns) is precoded by the precoding matrix Ws∈CNs×d, and the jamming vector z∈Cz×1 (j≤Nd) is emitted by D with precoding matrix Wd∈CNd×z to protect the confidential information. Both s and z satisfy CN0,I. Thus, the received signal at R can be expressed as(2)yr=FWss+HWdz+nr,where nr∈CNr×1 is the noise vector at the relay with nr~CN0,σr2I.In the second phase, R utilizes precoding matrixWr to forward the information; thus, the received signal at D can be expressed as(3)yd=GWrFWss+HWdz+GWrnr+nd,where nd∈CNr×1 is the noise vector at the destination with nd~CN0,σd2I.Furthermore,yd can be expressed as(4)yd=GWrFWss+G¯WrH¯Wdz︸selfinterference+ΔGWrH¯Wdz+G¯WrΔHWdz+ΔGWrΔHWdz︸highorder+GWrnr+nd.Here, we assume that D can partially remove the self interference signal based on the estimated CSI, and the high-order term with respect to (w.r.t) the uncertain CSI is small enough compared with the channel strength. Thus, the received signal at D can be approximately given by(5)y¯d=GWrFWss+ΔGWrH¯Wdz+G¯WrΔHWdz+GWrnr+nd,Accordingly, the worst case secrecy rate can be expressed as(6)Rs=min∀F∈F,∀H∈H,∀G∈GRd-Rr+,where Rd and Rr denote the mutual information at R and D, respectively, and are given by(7a)Rd=12ln⁡Id+WsHFHWrHGHD-1GWrFWs,(7b)Rr=12ln⁡Id+WsHFHR-1FWs,with D and R being given as(8)D=ΔGWrH¯WdWdHH¯HWrHΔGH+G¯WrΔHWdWdHΔHHWrHG¯H+σr2GWrWrHGH+σd2INd,and(9)R=σr2INr+HWdWdHHH.


## 2.2. Problem Statement

In this paper, we investigated the joint precoding matrices design to maximize the worst case secrecy rate, under the power constraints at the respective nodes. Mathematically, the problem can be formulated as(10a)maxWs,Wr,Wdmin∀F∈F,∀H∈H,∀G∈GRd-Rr,(10b)s.t.Ws2≤Ps,Wd2≤Pd,(10c)WrFWs2+WrHWd2+σr2Wr2≤Pr,where Ps, Pd, and Pr are the maximum achievable powers for S, D, and R, respectively.


## 3. An Alternating Optimization (AO) Method

The formulated problem is highly non-convex due to the maximization of the difference of several log-det functions in the CSI uncertainty region. In this section, we will propose to linearize these log-det terms and tackle the CSI uncertainty based on epigraph reformulation.Firstly, we introduce the following Lemma.Lemma 1 (see [17]).
Definem by m matrix function,(11)EU,V≜UHNU+Im-UHMVIm-UHMVH,where N is any positive definite matrix. Then, the following three equations hold true.Equation 1. For any positive definite matrix S∈Cm×m, we have (12)S-1=argmaxT≻0⁡ln⁡T-TrTS,and (13)-ln⁡S=maxT≻0⁡ln⁡T-TrTS+m.Equation 2. For any positive definite matrix T, we have (14)U~≜argminU⁡TrTEU,V=N+MVVHMH-1MV,and(15)EU~,V=Im-U~HMV=Im+VHMHN-1MV-1.Equation 3. We have (16)ln⁡Im+MVVHMHN-1=argmaxT≻0,U⁡ln⁡T-TrTE+m.Equations 1 and 2 can be proven by the first-order optimality condition, while Equation 3 directly follows from Equations 1 and 2 and the identitylnIm+AB=ln⁡In+BA.Equations (10a), (10b), and (10c) are hard to handle due to the coupled variables and non-convex objective and constraints. In the following, we will decouple (10a), (10b), and (10c) based on these above equations.To utilize the above equations, we rewrittenRs as(17)Rs=12ln⁡Id+GWrFWsGWrFWsHD-1︸≜f1+12ln⁡Id+σr-2HWdWdHHH︸≜f2-12ln⁡Id+σr-2FWsWsHFH+σr-2HWdWdHHH︸≜f3,where(18a)f1=maxV1≻0,U1⁡ln⁡V1-TrV1E1U1,Ws,Wd,Wr+d,(18b)f2=maxV2≻0,U2⁡ln⁡V2-TrV2E2U2,Wd+Nr,(18c)f3=maxV3≻0⁡ln⁡V3-TrV3INr+σr-2FWsWsHFH+σr-2HWdWdHHH+Ne.Furthermore, the matrix functionsE1 and E2 are as follows:(19a)E1U1,Ws,Wd,Wr≜INd×d-U1HGWrFWsINd×d-U1HGWrFWsH+U1HDU1,(19b)E2U2,Wd≜INr×z-σr-1U2HHWdINr×z-σr-1U2HHWdH+U2HU2.Based on the above relationships, (10a), (10b), and (10c) can be rewritten as(20a)maxWs,Wr,Wd,V1,V2,V3≻0,U1,U2,⁡min∀F∈F,∀H∈H,∀G∈G⁡ln⁡V1+ln⁡V2+ln⁡V3-β1-β2-β3,(20b)s.t.TrV1E1U1,Ws,Wd,Wr≤β1,(20c)TrV2E2U2,Wd≤β2,(20d)TrV3INr+σr-2FWsWsHFH+σr-2HWdWdHHH≤β3,(20e)10b,10cFurthermore, by introducing auxiliary variablesαii=09, βkk=13 and denoting S1⪰0Nd,S1=V11/2, Si⪰0Nr,Si=Vi1/2i=23, we obtain the following relationships:(21a)TrS1S1HINd×d-U1HGWrFWsINd×d-U1HGWrFWsH≤α0,(21b)TrS1S1HU1HU1≤α1,(21c)TrS1S1HU1HΔGWrH-WdWdHH-HWrHΔGHU1≤α2,(21d)TrS1S1HU1HG-WrΔHWdWdHΔHHWrHG-HU1≤α3,(21e)TrS1S1HU1HGWrWrHGHU1≤α4,(21f)α0+1+σd2α1+α2+α3+σr2α4≤β1,(21g)TrS2S2HINr×z-σr-1U2HHWdINr×z-σr-1U2HHWdH≤α5,(21h)TrS2S2HU2HU2≤α6,(21i)α5+α6≤β2,(21j)TrS3S3H≤α7,(21k)TrS3S3HFWsWsHFH≤α8,(21l)TrS3S3HHWdWdHHH≤α9,(21m)α7+σr-2α8+σr-2α9≤β3,Equations (21a)–(21m) are still hard to handle due to the CSI uncertainty. Next, we will transform (21a)–(21m) into a more convenient reformulation and employ the following sign-definiteness lemma to handle the CSI uncertainty.Lemma 2 (the sign-definiteness Lemma [18]).
Given a Hermitian matrixA and arbitrary matrices Pi,Qii=1N, the semi-infinite linear matrix inequality (LMI)(22)A⪰∑i=1NPiHXiQi+QiHXiHPi,∀Xi:Xi≤χiholds if and only if there exist real numbers λi≥0i=1N such that(23)A-∑i=1NλiQiHQi-χ1P1H…-χNPNH-χ1P1λ1I…0⋮⋮⋱⋮-χNPN0…λNI⪰0
Firstly, via the Schur complement [19], (21a) can be transformed into(24)TrS1S1HINd×d-U1HGWrFWsINd×d-U1HGWrFWsH≤α0⇒α0INdS1HINd×d-U1HG¯+ΔGWrF¯+ΔFWsINd×d-U1HG¯+ΔGWrF¯+ΔFWsHS1Id⪰0Nd+d⇒α0INdS1HINd×d-U1HG¯WrF¯WsINd×d-U1HG¯WrF¯WsHS1Id︸N1⪰0NdS1HU1HG¯WrΔFWsWsHΔFHWrHG¯HU1S10d+0NdS1HU1HΔGWrF¯WsWsHF¯HWrHΔGHU1S10d=0Nd×NsWsHΔFHWrHG¯HU1S10Nr×d+S1HU1HG¯Wr0d×NrΔF0Ns×NdWs︸N2+0Nd×NrWsHF¯HWrHΔGHU1S10Nd×d+S1HU1H0d×NdΔG0Nr×NdWrF¯Ws︸N3
To handle the uncertainties with respect toΔF and ΔG, we denote(25)A=α0INdS1HINd×d-U1HG¯WrF¯WsId×Nd-WsHF¯HWrHG¯HU1S1Idand P1=0Ns×NdWs, P2=0Nr×NdWrF¯Ws, Q1=WrHG¯HU1S10Nr×d, Q2=U1S10Nd×d.Then, based on Lemma2, N1⪰N2+N3 can be reformulated as(26)A-λ1Q1HQ1-λ2Q2HQ2-χFP1H-χGP2H-χFP1λ1INs0Ns×Nr-χGP20Nr×Nsλ2INr⪰0d+Ns+Nd+Nrwhere λ1≥0,λ2≥0 are introduced auxiliary variables.By substitutingA, P1, P2, Q1, and Q2 into (25), (26) can be recast as(27)α0INd-λ2S1HU1HU1S1-λ1S1HU1HG¯WrWrHG¯HU1S1S1HINd×d-U1HG¯WrF¯Ws0Nd×Ns0Nd×NrId×Nd-WsHF¯HWrHG¯HU1S1Id-χFWsH-χGWsHF¯HWrH0Ns×Nd-χFWsλ1INs0Ns×Nr0Nr×Nd-χGWrF¯Ws0Nr×Nsλ2INr⪰0d+Ns+Nd+NrSimilarly, the constraints (21b)–(21e), (21g), (21h), and (21j)–(21l) can be transformed into the following LMIs:(28)α1INdS1HU1HU1S1INd⪰02Nd(29)α2Iz0z×Nd-χGWdHH¯HWrH0Nd×zINd-λ3S1HU1HU1S10Nd×Nr-χGWrH¯Wd0Nr×Ndλ3INr⪰0Nd+Nr+z(30)α3INd0Nd×z-χHWdH0z×NdIz-λ4S1HU1HG¯WrWrHG¯HU1S10Nd-χHWd0Ndλ4INd⪰02Nd+z(31)α4INd-λ5S1HU1HU1S1S1HU1HG¯Wr0Nd×NrWrHG¯HU1S1INrχGWrH0Nr×NdχGWrλ5INr⪰0Nd+2Nr(32)α5INr-λ6S2HU2HU2S2S2HINr×z-σr-1U2HH¯Wd0Nr×NdIz×Nr-σr-1WdHH¯HU2S2Iz-χHWdH0Nd×Nr-χHWdλ6INd⪰0Nr+z+Nd(33)α6INrS2HU2HU2S2INr⪰02Nr(34)α7INrS3HS3INr⪰02Nr(35)α8INr-λ7S3HS3S3HF¯Ws0Nr×NsWsHF¯HS3IdχFWsH0Ns×NrχFWsλ7INs⪰0Nr+d+Ns(36)α9INr-λ8S3HS3S3HH¯Wd0Nr×dWdHH¯HS3IzχHWdH0d×NrχHWdλ8INd⪰0Nr+z+Ndwhere λm≥0m=38 are introduced auxiliary variables.Lastly, the robust relay power constraint (e.g., (10c)) can be turned into the following LMIs:(37)α10Id-λ9WsHWsWsHF¯HWrH0d×NrWrF¯WsINrχFWr0Nr×dχFWrHλ9INr⪰02Nr+d(38)α11Iz-λ10WdHWdWdHF¯HWrH0z×NrWrF¯WdINrχHWr0Nr×zχHWrHλ10INr⪰02Nr+z(39)α12INrWrWrHINr⪰02Nrwhere α10,α11,α12 and λ9≥0,λ10≥0 are introduced auxiliary variables.Bases on these operations, we obtain the following problem:(40a)maxWs,Wr,Wd,S1,S2,S3≻0,U1,U2,⁡min∀F∈F,∀H∈H,∀G∈G⁡2ln⁡S1+2ln⁡S2+2ln⁡S3-β1-β2-β3,(40b)s.t.TrV1E1U1,Ws,Wd,Wr≤β1,(40c)TrV2E2U2,Wd≤β2,(40d)TrV3I+σr-2FWsWsHFH+σr-2HWdWdHHH≤β3,(40e)10b,37,38,39,To this end, we have turned (10a), (10b), and (10c) into an equivalent problem (40a)–(40e). Equations (40a)–(40e) are still non-convex with respect to all these optimization variables, but is convex with respect to given variables when other variables are fixed. Specifically, these variables can be divided into four groups, Ws,Wd, Wr, S1,S2,S3, U1,U2. Then, (40a)–(40e) can be decoupled into four subproblems with respect to these variables. Both these subproblems can be effectively solved by the convex optimization tool CVX [20], and the optimal local solution to (10a), (10b), and (10c) can be achieved in an alternating method.


## 4. Simulation Results

In this section, we evaluate the performance of our design through Monte Carlo simulations. The simulation settings are assumed as follows:Ps=10dBW, Pr=10dBW, Pd=10dBW, σd2=σr2=10-6. Each entry of F¯, H¯, and G¯ is randomly generated by CN0,10-2 and the channel uncertainties are χF2=χH2=χG2=2×10-4. In addition, we compare our algorithm with the following methods: (1) the perfect CSI case, which can be treated as the upper bound of our robust design; (2) the no jamming method, e.g., setting Wd=0 with only optimize Ws and Wr. The three methods are legend as “the proposed design”, “the perfect CSI case”, and “the no jamming case”, respectively.Firstly, we investigated the convergency performance of our proposed AO method by comparing the worst case secrecy rateRs with the iterative numbers. Figure 2 shows four examples of the convergence behavior with random channel realizations. From this figure, we can see that the proposed AO algorithm can always converge to the optimal solution in limited iterative numbers.Figure 2
The convergency of the proposed method.In Figure3, we show the worst case secrecy rate Rs versus the source transmit power Ps. From this figure, we can see that Rs increase with the increase of Ps for our proposed design and the perfect CSI case. On the other hand, for the no jamming case, Rs firstly increase with the increase of Ps to some point, then decrease. This is a typical phenomenon in untrusted relay case.Figure 3
The worst case secrecy rate versus the source transmit power.In Figure4, we illustrate the impact of the destination transmit power Pd on Rs. From this figure, we can see that Rs increase with the increase of Pd for all these methods. Since in this case, the more jamming signal used to interfere the untrusted relay, the high secrecy performance can be obtained.Figure 4
The worst case secrecy rate versus the destination transmit power.In Figure5, we show the worst case secrecy rate Rs versus the relay transmit power Pr. From this figure, we can see that Rs increase with the increase of Pr for our proposed design and the perfect CSI case. On the other hand, for the no jamming case, Rs firstly increase with the increase of Pr, then decrease at some point.Figure 5
The worst case secrecy rate versus the relay transmit power.In Figure6, we illustrate the impact of the relay antenna numbers Nr on the worst case secrecy rate Rs. It is an interesting result that as the number of antennas of the relay increases, Rs firstly increase to a certain point and then decrease. This is because that as the DoF of the relay increases, the untrusted relay has more ability to cancel the jamming signal and further intercepts the confidential message successfully.Figure 6
The worst case secrecy rate versus the relay antenna numbers.


## 5. Conclusion

In this paper, we investigated an AF MIMO relay system where the relay is untrusted. Assuming that only imperfect CSI can be obtained, we aim to maximize the worst case secrecy rate via designing the precoding matrices. To solve the formulated highly non-convex problem, we propose to linearize the difference of multiple log-det functions. After linearization, we handle the CSI uncertainty based on epigraph reformulation and the sign-definiteness lemma. Finally, an AO method was proposed to solve the reformulated problem. Simulation results demonstrated the performance of our proposed design.



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*Source: 1081682-2019-06-09.xml*