# Robust Secure Beamforming Design for Cooperative Cognitive Radio Nonorthogonal Multiple Access Networks

Authors: Quanzhong Li; Sai Zhao
Journal: Security and Communication Networks
 (2021)

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## Abstract

By the integration of cooperative cognitive radio (CR) and nonorthogonal multiple access (NOMA), cooperative CR NOMA networks can improve the spectrum efficiency of wireless networks significantly. Due to the openness and exposure of wireless signals, secure communication is an important issue for cooperative CR NOMA networks. In this paper, we investigate the physical layer security design for cooperative CR NOMA networks. Our objective is to achieve maximum secrecy rate of the secondary user by designing optimal beamformers and artificial noise covariance matrix at the multiantenna secondary transmitter under the quality-of-service at the primary user and the transmit power constraint at the secondary transmitter. We consider the practical case that the channel state information (CSI) of the eavesdropper is imperfect, and we model the imperfect CSI by the worst-case model. We show that the robust secrecy rate maximization problem can be transformed to a series of semidefinite programmings based onS-procedure and rank-one relaxation. We also propose an effective method to recover the optimal rank-one solution. Simulations are provided to show the effectiveness of our proposed robust secure algorithm with comparison to the nonrobust secure design and traditional orthogonal multiple access schemes.


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## Body

## 1. Introduction

Cooperative cognitive radio (CR) is a promising technique for easing the strain on the spectrum resources, where the secondary user (SU) works as a relaying station and assists the primary user (PU) to transmit the signals, and as a return, the SU can use the spectrum occupied by the PU to serve its own cognitive user [1–5]. Nonorthogonal multiple access (NOMA) is an effective and new technique over conventional orthogonal multiple access (OMA) to enhance the spectrum efficiency, through transmitting the superposition of the weighted version of the strong (central) and weak (edge) users’ signals and employing successive interference cancellation (SIC) at the strong user [6–11]. By the integration of cooperative CR and NOMA techniques, cooperative CR NOMA networks can improve spectrum utilization efficiency and information rate significantly [12–14].In the early days, information security was mainly focused on wired communication, where gateway selection algorithm [15], multilayer botnet detection technique [16], and intelligent spam e-mail detection [17] could be used to improve the communication security. For wireless communication, because of the openness of the wireless transmission medium, wireless information is susceptible to eavesdropping. Then, more attention is paid to the information security of wireless communication where clone node detection [18], machine learning [19], physical layer security [20], and so on can be employed to enhance the information security. Secure communication is also a critical issue for cooperative CR NOMA networks. Recently, physical layer security has been proposed in [21–27] for cooperative CR NOMA networks, where the security performance of communication networks can be greatly improved by the cooperation of the relay node (i.e., SU). The authors in [21] studied the physical layer security in a cooperative CR NOMA network where all the nodes had a single antenna and the SU employed amplify-and-forward (AF) relaying to forward the PU’s signal, aiming to analyze the security performance of the PU and throughput of the SU. In the study by Chen et al. [22], the authors considered a cooperative CR NOMA network with decode-and-forward (DF) relaying and designed the relay beamforming to improve the security of the PU. In the study by She et al. [23], the authors investigated the optimization of power allocation and transmit covariance matrix to improve the security energy efficiency of the SU for a DF-relaying cooperative CR NOMA network. The works [25–27] also focused on the DF-relaying cooperative CR NOMA network in order to analyze the security performance of the network.Most existing works for physical layer security in cooperative CR NOMA networks consider the analysis of the security performance, e.g., [21, 24–27], but only few works consider the physical layer security optimization, e.g., [22, 23]. Besides, physical layer security optimization for conventional cooperative CR NOMA networks is investigated in [28]. It is worth noting that the optimization design to improve the physical layer security of cooperative CR NOMA networks highly depends on the channel state information (CSI) of the eavesdropper. However, the works [22, 23, 28] consider that the CSI of the eavesdropper is perfect, which is impractical. In fact, getting the perfect CSI of the eavesdropper is very hard or even impossible. Under such cases, the imperfect CSI of the eavesdropper may be obtained in practice, based on the past channel observations or a priori knowledge of the particular propagation environment [29].In this paper, we focus on secure beamforming and artificial noise (AN) covariance matrix design problem for a cooperative CR NOMA network, where the CSI of the eavesdropper is considered to be imperfect. We investigate AF relaying in the cooperative CR NOMA network, where the multiantenna secondary transmitter (ST) helps relaying the information from the primary transmitter (PT) to the PU and serves its own SU, while an eavesdropper located near the SU intends to overhear the signal intended for the SU. By employing the worst model to model the imperfect CSI of the eavesdropper, we aim to achieve maximum secrecy rate of the SU under the quality-of-service (QoS) at the PU and the transmit power constraint at the ST. We transform the robust secrecy rate maximization to a series of semidefinite programmings (SDPs) with the help ofS-procedure and rank-one relaxation. An effective method is also proposed to recover the optimal rank-one solution. Simulation results show the superiority of our proposed robust secure algorithm with comparison to the existing secure design schemes. The main contributions of this paper are summarized as follows:(i)
We propose a robust secure optimization algorithm for a cooperative CR NOMA network based on the semidefinite relaxation scheme, where the CSI of the eavesdropper is imperfect.(ii)
An effective method is designed to find an optimal rank-one solution for the proposed semidefinite relaxation scheme, which offers a global optimal solution for the robust secrecy rate maximization problem.(iii)
We provide simulation results to show the effectiveness of our proposed robust secure algorithm with comparison to the nonrobust secure design and traditional orthogonal multiple access schemes.Note that our work is very different from the previous works in [21–28]. First, we assume that the CSI of the eavesdropper is imperfect, which is more practical than the assumption of the perfect CSI in [21–28]. Second, we focus on the optimization design to improve the security performance, while [21] and [24–27] just provide the analysis of the security performance. Third, we propose a global optimal solution for the secure optimization problem, while only suboptimal solutions are provided in [22, 23]. Last, we employ NOMA transmission to improve the spectrum efficiency, while the conventional OMA scheme is adopted in [28].The rest of this paper is organized as follows. In Section2, the system model is described. In Section 3, the robust secrecy rate maximization problem is formulated, and an efficient optimization algorithm and an optimal rank-one solution recovering method are proposed. Simulation results are provided in Section 4, and we summary this paper in Section 5.Notations. Boldface lowercase and uppercase letters denote vectors and matrices, respectively. AT, A∗, A†, A, and trA denote the transpose, conjugate, conjugate transpose, Frobenius norm, and trace of the matrix A, respectively. ⊗ denotes the Kronecker product. ⊙ denotes the Hadamard product. vecA denotes to stack the columns of a matrix A into a single vector a. Rea denotes the real part of a. By A≽0, we mean that A is positive semidefinite. ek denotes an elementary vector with the kth element being one and others zero. 1 denotes the vector/matrix with all elements being one. E denotes diag1,1,…,1.


## 2. System Model

As shown in Figure1, we consider an AF-relaying cooperative CR NOMA network. The primary network consists of a PT and a PU, where the PT wants to transmit the signal to the PU. Due to impairments such as long distance and obstacles, the direct communication between the PT and PU cannot satisfy the rate demand. They thus need the help of the secondary user to meet the PU’s QoS requirement. The secondary network has a secondary transmitter (ST) with N antennas, which assists the transmission of the PT and serves an SU by using the principle of NOMA. Meanwhile, an eavesdropper (EV) which is located near the SU intends to overhear the signal sent for the SU. The potential application scenarios include device-to-device communications [30], where two mobile phones directly communicate with the help of a femto cell or a laptop, which also transmits its private information to another mobile phone simultaneously.Figure 1
The model for secure communication in an AF-relaying cooperative CR NOMA network.The transmission consists of two consecutive equal-duration time slots. In the first time slot, the PT transmits symbolxp to the ST, and the received signal at the ST is given by(1)yr=hpxp+nr,where hp∈ℂN×1 denotes the channel response between the PT and ST, nr∼CN0,σr2I is the Gaussian noise at the ST, and the transmit power of xp is Exp2=Pt.In the second time slot, the ST employs the NOMA principle to transmit the signals, which multiplies the received signalyr by a beamforming matrix, denoted as F∈ℂN×N, and superimposes Fyr with its own signal xs using the cognitive beamforming vector w∈ℂN×1. To ensure the secure communication of the SU, the ST employs the artificial noise to interfere the eavesdropper. Thus, from (1), the transmit signal at the ST is expressed as(2)xr=Fhpxp+Fnr+wxs+v,where v∈ℂN×1 is the artificial noise with covariance Evv†=V and the signal xs is normalized to Exs2=1. From (2), the average transmit power at the ST is(3)Pxr=Exr2=PtFhp2+σr2F2+w2+trV.The received signal at the PU, SU, and EV is, respectively,(4)yp=gpTFhpxp+gpTFnr+gpTwxs+gpTv+np,(5)ys=gsTFhpxp+gsTFnr+gsTwxs+gsTv+ns,(6)ye=geTFhpxp+geTFnr+geTwxs+geTv+ne,where gp,gs,andge∈ℂN×1 denote the channel response from the ST to PU, SU, and EV, respectively. np∼CN0,σp2, ns∼CN0,σs2, and ne∼CN0,σe2 is the Gaussian noise at the PU, SU, and EV, respectively.To support the QoS requirement at the PU, it is natural to treat the PU as a strong user and the SU as a weak user [12, 13]. At the PU, the signal xs intended for the SU is first detected, and from (4), the signal-to-interference-and-noise ratio (SINR) for detecting xs is given by(7)Γp,s=gpTw2PtgpTFhp2+σr2gpTF2+gpTVps∗+σp2.Suppose1/2log21+Γp,s≥rs, where rs denotes the transmission rate requirement for correctly decoding xs, i.e., Γp,s≥γs where γs=22rs−1, and the PU is able to detect the signal xs. From the NOMA principle, the PU first removes xs by successive interference cancellation to obtain the following signal:(8)y˜p=gpTFhpxp+gpTFnr+gpTv+np.Then, the PU can detect its own signalxp with the following signal-to-noise ratio (SNR):(9)Γp,p=PtgpTFhp2σr2gpTF2+gpTVgp∗+σp2.From (5) and (6), the SINR at the SU and EV to detect signal xs is, respectively, given by(10)Γs=gsTw2PtgsTFhp2+σr2gsTF2+gsTVgs∗+σs2,(11)Γe=geTw2PtgeTFhp2+σr2geTF2+geTVge∗+σe2.Using (10) and (11), the achievable secrecy rate for the SU, which is denoted by Rs, is given by [31]:(12)Rs=12log21+Γs−12log21+Γe.


## 3. Robust Secure Design with Imperfect CSI

In this paper, we assume that the perfect CSI of the eavesdropper is not available, which is the practical case in cooperative CR NOMA networks. As in [32, 33], we assume that the actual channel is within the neighborhood of a nominal channel which, specifically, only the noisy version of actual channel response ge is known, is denoted as g¯e:(13)ge=g¯e+Δge,where g¯e is the estimated channel vector and Δge is the CSI uncertainty and is norm-bounded by the elliptical region:(14)Ge≜Δge|Δge†TΔge≤1,where the matrix T≻0, assumed to be known, determines the quality of CSI.Based on (3) and (7)–(14), the robust secure design with aiming to maximize the worst-case achievable secrecy rate for the SU subject to the QoS constraints at the PU and the transmit power constraint at the ST can be expressed as(15a)maxF,w,V≽0minΔge∈Ge12log21+Γs−12log21+Γe,(15b)s.t.Γp,s≥γs,(15c)Γp,p≥γp,(15d)PtFhp2+σr2F2+w2+trV≤Pr,where γp=22rp−1 with rp being the rate requirement at the PU and Pr is the transmit power constraint at the ST. The robust optimization (15a)–(15d) is nonconvex, and the global optimal solution is very difficult to find. In the following, we propose an effective scheme to solve (15a)–(15d) globally.


### 3.1. Convex Reformulation

Introducing the slack variableβ such that 1/2log21+Γe≤β, problem (15a)–(15d) can be rewritten as(16)maxF,w,V≽0,β12log21+Γs−β,s.t.12log21+Γe≤β,∀Δge∈Ge15b,15c,15d.,To proceed, we need the following result.Lemma 1.
Definef=vecF, and we have(17)qTFp=qTpT⊗If,(18)qTFF†p∗=qT1T⊗IE⊙ff†1⊗Ip∗.Proof.
Please see the Appendix.
Let us define(19)f=vecF,(20)X=ff†,(21)Y=ww†,(22)μX=PthpT⊗IXhp∗⊗I+σr21T⊗IE⊙X1⊗I.
Using Lemma1 and equations (17)–(22), we can rewrite problem (16) with a given β as(23a)maxX,Y,VtrB1YtrA1X+trB1V+σs2,(23b)s.t.ΔgeTDΔge∗+2Red†Δge∗+d≤0,∀Δge∈Ge,(23c)trB2Y−γstrA2X−γstrB2V−γsσp2≥0,(23d)trA3X−γptrB2V−γpσp2≥0,(23e)trA4X+trY+trV≤Pr,(23f)X≽0,Y≽0,V≽0,(23g)RankX=1,RankY=1,where β¯=22β−1 and(24)A1=Pthp∗hpT⊗gs∗gsT+σr2I⊗gs∗gsT,A2=Pthp∗hpT⊗gp∗gpT+σr2I⊗gp∗gpT,A3=Pthp∗hpT⊗gp∗gpT−σr2γpI⊗gp∗gpT,A4=Pthp∗hpT⊗I+σr2I,B1=gs∗gsT,B2=gp∗gpT,D=Y−β¯μX−β¯V,d†=g¯eTD,d=g¯eTDg¯e∗−β¯σe2.
Dropping the rank-one constraints in (23g) and employing the Charnes–Cooper transformation [14], i.e.,(25)t=1trA1X+trB1V+σs2,X^=tX,Y^=tY,V^=tV,we can convert the robust problem (23a)–(23g) equivalently to(26a)maxX^,Y^,V^,ttrB1Y^,(26b)s.t.ΔgeTD^Δge∗+2Red^†Δge∗+d^≤0,∀Δge∈Ge,(26c)trA1X^+trB1V^+σs2t=1,(26d)trB2Y^−γstrA2X^−γstrB2V^−γsσp2t≥0,(26e)trA3X^−γptrB2V^−γpσp2t≥0,(26f)trA4X^+trY^+trV^≤Prt,(26g)X^≽0,Y^≽0,V^≽0,t≥0,where(27)D^=Y^−β¯μX^−β¯V^,(28)d^†=g¯eTD^,(29)d^=g¯eTD^g¯e∗−β¯σe2t.
Due to semi-infinite constraints caused by the channel uncertainty, the robust optimization (26a)–(26g) is difficult to solve. Thus, we need to eliminate the semi-infinite constraints. To make the problem (26a)–(26g) tractable, we convert the semi-infinite constraints into linear matrix inequalities (LMIs) equivalently, using the following S-procedure [34].Lemma 2.
Define the functions:(30)fjx=x†Ajx+2Rebj†x+cj,j=1,2,where Aj=Aj†∈ℂn×n, bj∈ℂn, and cj∈ℝ. The implication f1x≤0⟹f2x≤0 holds if and only if there exists λ≥0 such that(31)λA1b1b1†c1−A2b2b2†c2≽0,provided that there exists a point x0 such that f1x0<0.Using Lemma2, the robust optimization (26a)–(26g) can be converted into the following convex SDP:(32)maxX^,Y^,V^,t,λ≥026a,s.t.λTT−D^−d^−d^†−λ−d^≽0,26c−26g.It is easy to see that if the optimalX^∗,Y^∗ to the SDP (32) is rank-one, it is also optimal to the problem (16) with a given β. However, the solution to the SDP (32) is not always rank-one. In the next subsection, we propose an effective method to construct the optimal rank-one solution to the problem (16) based on X^∗,Y^∗,V^∗,t∗,λ∗, which is optimal to (32).


### 3.2. Rank-One Solution Recovering

First, the worst-caseΔge, denoted as Δge∗, can be computed by solving the following feasibility problem:(33)FindΔges.t.ΔgeTD^∗Δge∗+2Red^∗†Δge∗+d^∗≤0,Δge†TΔge≤1.where D^∗, d^∗, and d^∗ are obtained by replacing X^,Y^,V^,t by X^∗,Y^∗,V^∗,t∗ in (27)–(29). The feasibility problem (33) is also a nonconvex problem; however, its optimal solution can be found by the following Lemma.Lemma 3.
The optimal solution to problem (33) can be found by solving the following convex SDP:(34)minZ≽00,s.t.trC1Z=1,trCiZ≤0,i=2,3,where(35)C1=0001,C2=D^∗d^∗d^∗†d^∗,C3=TT00−1.Proof.
Introducing the slack variablet, we can equivalently rewrite (33) as a homogeneous quadratically constrained quadratic program (QCQP) [35]:(36)FindΔge,ts.t.ΔgeT,tC1ΔgeT,tT=1,ΔgeT,tCiΔgeT,tT≤0,i=2,3.
LetZ=ΔgeT,tTΔgeT,t. The rank-one relaxation of (36) is(37)minZ≽00,s.t.trC1Z=1,trCiZ≤0,i=2,3.
Since there are three linear constraints in (37), according to the rank-one decomposition theorem in [36], there is a rank-one optimal solution to problem (37).
From Lemma3, the optimal rank-one solution to the SDP (32) can be constructed by considering the following convex SDP:(38)maxX^,Y^,V^,t26a,s.t.Δge∗TD^Δge∗∗+2Red^†Δge∗∗+d^≤0,26c−26g,with the optimal solution X^∗,Y^∗,V^∗,t∗, since for the SDP (38), we have Theorem 1.Theorem 1.
There exists a rank-one optimal solution to the SDP (38).Proof.
Since there are five linear constraints and three matrix variables in the SDP (38), according to Theorem 3.2 in [37], there exists an optimal solution Xo,Yo,Vo to the SDP (38) such that(39)Rank2Xo+Rank2Yo+Rank2Vo≤5.
Since the optimal solution is nonzero, we haveRankXo=RankYo=RankVo=1.


### 3.3. Algorithm and Complexity

Based on Theorem1, if the optimal solution X∗,Y∗ we have found has a higher rank than one, we can find another optimal rank-one solution Xo,Yo. Therefore, the optimal solution to robust problem (15a)–(15d) can be found by 1-D search over β, where during each search, the SDP (32) is solved, which is summarized in Algorithm 1.Algorithm 1: Find the optimal solution to robust secure problem (15a)–(15d).
(1)
Choose some largeL. Define Δβ=βu−βl/L. Initialize φ⋆=0.(2)
Forj=0:LSetβ=βl+jΔβ;Solve SDP (32);If the optimal valueφo>φ∗Updateφ∗=φo;Save theoptimal solution as X^∗,Y^∗,V^∗,t∗,λ∗;End(3)
End(4)
IfrankX^∗≥2 or rankY^∗≥2Employ the proposed method in section 3.B to find an optimal rank-one solution to SDP (32).EndTheoptimal value and optimal solution in Algorithm 1 refer to the optimal value and solution to SDP (32), respectively. The upper and lower bounds of β in Algorithm 1 can be chosen as βu=1/2log21+Prg¯e2+1/ν and βl=0, where ν is the minimum eigenvalue of T.The computational complexity of Algorithm1 is mainly from the computation of the SDP (32). From [38], the computational complexity for solving an SDP within a tolerance ϵ is Omsdpnsdp3.5+msdp2nsdp2.5+msdp3nsdp0.5·log1/∈, where nsdp is the dimension of the semidefinite cone and msdp is the number of linear constraints. Thus, the computational complexity of the SDP (32) is about ON7log1/∈, which leads to the complexity of the proposed algorithm which is about OLN7log1/∈. Similarly, the computational complexity of the nonrobust design scheme proposed in [22] is about OLKN3.5log1/∈ with K SUs, and that of the conventional OMA scheme proposed in [28] is about OLN6log1/∈.


## 3.1. Convex Reformulation

Introducing the slack variableβ such that 1/2log21+Γe≤β, problem (15a)–(15d) can be rewritten as(16)maxF,w,V≽0,β12log21+Γs−β,s.t.12log21+Γe≤β,∀Δge∈Ge15b,15c,15d.,To proceed, we need the following result.Lemma 1.
Definef=vecF, and we have(17)qTFp=qTpT⊗If,(18)qTFF†p∗=qT1T⊗IE⊙ff†1⊗Ip∗.Proof.
Please see the Appendix.
Let us define(19)f=vecF,(20)X=ff†,(21)Y=ww†,(22)μX=PthpT⊗IXhp∗⊗I+σr21T⊗IE⊙X1⊗I.
Using Lemma1 and equations (17)–(22), we can rewrite problem (16) with a given β as(23a)maxX,Y,VtrB1YtrA1X+trB1V+σs2,(23b)s.t.ΔgeTDΔge∗+2Red†Δge∗+d≤0,∀Δge∈Ge,(23c)trB2Y−γstrA2X−γstrB2V−γsσp2≥0,(23d)trA3X−γptrB2V−γpσp2≥0,(23e)trA4X+trY+trV≤Pr,(23f)X≽0,Y≽0,V≽0,(23g)RankX=1,RankY=1,where β¯=22β−1 and(24)A1=Pthp∗hpT⊗gs∗gsT+σr2I⊗gs∗gsT,A2=Pthp∗hpT⊗gp∗gpT+σr2I⊗gp∗gpT,A3=Pthp∗hpT⊗gp∗gpT−σr2γpI⊗gp∗gpT,A4=Pthp∗hpT⊗I+σr2I,B1=gs∗gsT,B2=gp∗gpT,D=Y−β¯μX−β¯V,d†=g¯eTD,d=g¯eTDg¯e∗−β¯σe2.
Dropping the rank-one constraints in (23g) and employing the Charnes–Cooper transformation [14], i.e.,(25)t=1trA1X+trB1V+σs2,X^=tX,Y^=tY,V^=tV,we can convert the robust problem (23a)–(23g) equivalently to(26a)maxX^,Y^,V^,ttrB1Y^,(26b)s.t.ΔgeTD^Δge∗+2Red^†Δge∗+d^≤0,∀Δge∈Ge,(26c)trA1X^+trB1V^+σs2t=1,(26d)trB2Y^−γstrA2X^−γstrB2V^−γsσp2t≥0,(26e)trA3X^−γptrB2V^−γpσp2t≥0,(26f)trA4X^+trY^+trV^≤Prt,(26g)X^≽0,Y^≽0,V^≽0,t≥0,where(27)D^=Y^−β¯μX^−β¯V^,(28)d^†=g¯eTD^,(29)d^=g¯eTD^g¯e∗−β¯σe2t.
Due to semi-infinite constraints caused by the channel uncertainty, the robust optimization (26a)–(26g) is difficult to solve. Thus, we need to eliminate the semi-infinite constraints. To make the problem (26a)–(26g) tractable, we convert the semi-infinite constraints into linear matrix inequalities (LMIs) equivalently, using the following S-procedure [34].Lemma 2.
Define the functions:(30)fjx=x†Ajx+2Rebj†x+cj,j=1,2,where Aj=Aj†∈ℂn×n, bj∈ℂn, and cj∈ℝ. The implication f1x≤0⟹f2x≤0 holds if and only if there exists λ≥0 such that(31)λA1b1b1†c1−A2b2b2†c2≽0,provided that there exists a point x0 such that f1x0<0.Using Lemma2, the robust optimization (26a)–(26g) can be converted into the following convex SDP:(32)maxX^,Y^,V^,t,λ≥026a,s.t.λTT−D^−d^−d^†−λ−d^≽0,26c−26g.It is easy to see that if the optimalX^∗,Y^∗ to the SDP (32) is rank-one, it is also optimal to the problem (16) with a given β. However, the solution to the SDP (32) is not always rank-one. In the next subsection, we propose an effective method to construct the optimal rank-one solution to the problem (16) based on X^∗,Y^∗,V^∗,t∗,λ∗, which is optimal to (32).


## 3.2. Rank-One Solution Recovering

First, the worst-caseΔge, denoted as Δge∗, can be computed by solving the following feasibility problem:(33)FindΔges.t.ΔgeTD^∗Δge∗+2Red^∗†Δge∗+d^∗≤0,Δge†TΔge≤1.where D^∗, d^∗, and d^∗ are obtained by replacing X^,Y^,V^,t by X^∗,Y^∗,V^∗,t∗ in (27)–(29). The feasibility problem (33) is also a nonconvex problem; however, its optimal solution can be found by the following Lemma.Lemma 3.
The optimal solution to problem (33) can be found by solving the following convex SDP:(34)minZ≽00,s.t.trC1Z=1,trCiZ≤0,i=2,3,where(35)C1=0001,C2=D^∗d^∗d^∗†d^∗,C3=TT00−1.Proof.
Introducing the slack variablet, we can equivalently rewrite (33) as a homogeneous quadratically constrained quadratic program (QCQP) [35]:(36)FindΔge,ts.t.ΔgeT,tC1ΔgeT,tT=1,ΔgeT,tCiΔgeT,tT≤0,i=2,3.
LetZ=ΔgeT,tTΔgeT,t. The rank-one relaxation of (36) is(37)minZ≽00,s.t.trC1Z=1,trCiZ≤0,i=2,3.
Since there are three linear constraints in (37), according to the rank-one decomposition theorem in [36], there is a rank-one optimal solution to problem (37).
From Lemma3, the optimal rank-one solution to the SDP (32) can be constructed by considering the following convex SDP:(38)maxX^,Y^,V^,t26a,s.t.Δge∗TD^Δge∗∗+2Red^†Δge∗∗+d^≤0,26c−26g,with the optimal solution X^∗,Y^∗,V^∗,t∗, since for the SDP (38), we have Theorem 1.Theorem 1.
There exists a rank-one optimal solution to the SDP (38).Proof.
Since there are five linear constraints and three matrix variables in the SDP (38), according to Theorem 3.2 in [37], there exists an optimal solution Xo,Yo,Vo to the SDP (38) such that(39)Rank2Xo+Rank2Yo+Rank2Vo≤5.
Since the optimal solution is nonzero, we haveRankXo=RankYo=RankVo=1.


## 3.3. Algorithm and Complexity

Based on Theorem1, if the optimal solution X∗,Y∗ we have found has a higher rank than one, we can find another optimal rank-one solution Xo,Yo. Therefore, the optimal solution to robust problem (15a)–(15d) can be found by 1-D search over β, where during each search, the SDP (32) is solved, which is summarized in Algorithm 1.Algorithm 1: Find the optimal solution to robust secure problem (15a)–(15d).
(1)
Choose some largeL. Define Δβ=βu−βl/L. Initialize φ⋆=0.(2)
Forj=0:LSetβ=βl+jΔβ;Solve SDP (32);If the optimal valueφo>φ∗Updateφ∗=φo;Save theoptimal solution as X^∗,Y^∗,V^∗,t∗,λ∗;End(3)
End(4)
IfrankX^∗≥2 or rankY^∗≥2Employ the proposed method in section 3.B to find an optimal rank-one solution to SDP (32).EndTheoptimal value and optimal solution in Algorithm 1 refer to the optimal value and solution to SDP (32), respectively. The upper and lower bounds of β in Algorithm 1 can be chosen as βu=1/2log21+Prg¯e2+1/ν and βl=0, where ν is the minimum eigenvalue of T.The computational complexity of Algorithm1 is mainly from the computation of the SDP (32). From [38], the computational complexity for solving an SDP within a tolerance ϵ is Omsdpnsdp3.5+msdp2nsdp2.5+msdp3nsdp0.5·log1/∈, where nsdp is the dimension of the semidefinite cone and msdp is the number of linear constraints. Thus, the computational complexity of the SDP (32) is about ON7log1/∈, which leads to the complexity of the proposed algorithm which is about OLN7log1/∈. Similarly, the computational complexity of the nonrobust design scheme proposed in [22] is about OLKN3.5log1/∈ with K SUs, and that of the conventional OMA scheme proposed in [28] is about OLN6log1/∈.


## 4. Simulation Results

In this section, we present the simulation results of our proposed robust secure design scheme, where the average worst-case achievable secrecy rate of the SU will be evaluated under various system parameters. We consider a scenario that, in the AF-relaying cooperative CR NOMA network, all the entries in the channel responseshp, gp, gs, and g¯e are independent and identically distributed complex Gaussian random variables with zero mean and unit variance. We assume that the number of antennas equipped at the ST is N=3 and all the noise variances are equal, i.e., σi2=σ2, i∈r,p,s,e, the transmission rate requirement is rs=0.5 bps/Hz, and the transmit power of the PT is fixed to be Pt/σ2=15 dB. For simplicity, the uncertainty region of the eavesdropper’s CSI is assumed to be the norm-bounded, i.e., T=1/ωI, where ω determines the quality of the CSI of the eavesdropper. In simulations, we compare the proposed robust secure design scheme, denoted as “Proposed Robust,” with the nonrobust design scheme proposed in [22], denoted as “NonRobust,” and the conventional OMA scheme proposed in [28], denoted as “OMA.”In Figure2, we present the average worst-case achievable secrecy rate of the SU versus the transmit power of the ST to the noise power ratio, i.e., Pr/σ2, where the QoS target of the PU is rp=1 bps/Hz and the channel error is ω=0.001. The average achievable secrecy rate using the perfect CSI is also presented, denoted as “Perfect CSI.” From Figure 2, it is observed that when the channel error is small, i.e., ω=0.001, the average achievable secrecy rate of the SU by the proposed robust scheme is very close to that by the scheme of “Perfect CSI.” We can also see from Figure 2 that the proposed robust secure design scheme outperforms the nonrobust and OMA schemes.Figure 2
Average worst-case achievable secrecy rate of the SU versus the transmit power of the ST to the noise power ratioPr/σ2.In Figure3, we present the average worst-case achievable secrecy rate of the SU versus the QoS target of the PU rp, where the transmit power of the ST to the noise power ratio is Pr/σ2=15 dB and the channel error is ω=0.001. From Figure 3, it is seen that the average achievable secrecy rate of the SU by the proposed robust scheme performs close to that by the scheme of “Perfect CSI” and is better than those by the nonrobust and OMA schemes under different values of rp. We can also see from Figure 3 that, as the QoS target of the PU becomes higher, the average achievable secrecy rate by all the schemes decreases.Figure 3
Average worst-case achievable secrecy rate of the SU versus the QoS target of the PUrp.In Figure4, we present the average worst-case achievable secrecy rate of the SU versus the channel error ω, where the transmit power of the ST to the noise power ratio is Pr/σ2=15 dB and the QoS target of the PU is rp=1 bps/Hz. From Figure 4, we see that when the channel error ω is small, the average achievable secrecy rate of the SU by the proposed robust scheme performs close to that by the scheme of “Perfect CSI.” As ω increases, the average achievable secrecy rate decreases and the gap between the proposed robust scheme and the scheme of “Perfect CSI” becomes larger. From Figure 4, we also see that the proposed robust scheme performs better than the nonrobust and OMA schemes for different values of ω.Figure 4
Average worst-case achievable secrecy rate of the SU versus the channel errorω.


## 5. Conclusions

In this paper, we have proposed a robust physical layer security design algorithm for AF-relaying cooperative CR NOMA networks, where imperfect CSI of the eavesdropper is considered. We transform the robust secrecy rate maximization optimization problem to a series of convex semidefinite programmings with the help ofS-procedure and rank-one relaxation and propose an effective method to recover the optimal rank-one solution. Simulation results have shown the effectiveness of our proposed secure design scheme. One possible limitation of the proposed robust physical layer security design algorithm is that it is designed for single-antenna users and eavesdropper. Thus, a generalization of the proposed robust algorithm for cooperative CR NOMA networks with multiantenna users and eavesdropper is an interesting topic for future investigation.



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*Source: 5526485-2021-03-19.xml*