Energy Harvesting Fairness in AN-aided Secure MU-MIMO SWIPT Systems with Cooperative Jammer

Transcription

1 Energy Harvesting Fairness in AN-aided Secure MU-MIMO SWIPT Systems with Cooperative Jammer arxiv: v1 [cs.it] 1 Feb 2018 Zhengyu Zhu, Zheng Chu, Ning Wang, Zhongyong Wang, Inkyu Lee Schoo of Information Engineering, Zhengzhou University, China Department of Eectrica and Computer Engineering, McMaster University, Canada 5G Innovation Center, Institute of Communication Systems, University of Surrey, UK Schoo of Eectrica Engineering, Korea University, Korea Emai: zhuzhengyu6@gmai.com, andrew.chuzheng7@gmai.com, iezywang@zzu.edu.cn, inkyu@korea.ac.kr Abstract In this paper, we study a muti-user mutipe-inputmutipe-output secrecy simutaneous wireess information and power transfer (SWIPT) channe which consists of one transmitter, one cooperative jammer (CJ), mutipe energy receivers (potentia eavesdroppers, ERs), and mutipe co-ocated receivers (CRs). We expoit the dua of artificia noise (AN) generation for faciitating efficient wireess energy transfer and secure transmission. Our aim is to imize the imum harvested energy among ERs and CRs subject to secrecy rate constraints for each CR and tota transmit power constraint. By incorporating norm-bounded channe uncertainty mode, we propose a iterative agorithm based on sequentia parametric convex approximation to find a near-optima soution. Finay, simuation resuts are presented to vaidate the performance of the proposed agorithm outperforms that of the conventiona AN-aided scheme and CJaided scheme. I. INTRODUCTION Recenty, wireess power transfer (WPT) has been a promising paradigm to scavenge energy from the radio frequency (RF) signas [1]. As a key technoogy for reay perpetua communications, simutaneous wireess information and power transfer (SWIPT) has been an promising of interests in RF-enabed signa to provide power suppies for wireess networks, which have been studied in various scenarios [2]-[5]. On the other hand, secrecy transmission, especiay physicaayer security (PLS), has extracted more and more attentions in 5G wireess networks [6]. Specificay, PLS has been recognized as a important issue for SWIPT system due to its inherent characteristics make the wireess information more vunerabe to eavesdropping [7]- [11]. Moreover, the secure transmission with SWIPT schemes has aso been investigated in the muti-user mutipe-inputmutipe-output (MU-MIMO) broadcasting [9]. It is noted that the assumption that perfect channe state information (CSI) is avaiabe at the transmitter in [2]-[4] [7]-[9]. In practice, it is not aways possibe to obtain perfect CSI at the transmitter due to the channe errors. Secure communication with SWIPT woud be more chaenging with imperfect CSI at the transmitter. Some robust optimization techniques have been constructed to secrecy SWIPT transmission under imperfect channe reaization in [10][11] [13]-[18]. Considering the SWIPT scheme, an optima transmit covariance matrix robust design has been proposed for MIMO secure channes with muti-antenna eavesdroppers [11]. In addition, some of state-of-art techniques have been deveoped to introduce more interference to the eavesdroppers [12]- [21]. Artificia noise (AN) technique has been used to embed the transmit beamforg to confuse the eavesdropper [12]. In secrecy SWIPT systems, AN pays both the roes of an energycarrying signa for WPT and protecting the secrecy information transmission, which has been considered as interfering the eavesdropper and harvesting power simutaneousy in [13]- [15]. In addition, to further increase the secrecy rates, jamg node has been introduced in the secrecy networks, which has the capabiity to improve the egitimate user s performance and prevent the eavesdroppers from intercepting the intended messages [16]-[20]. Based on the worst-case scheme, cooperative jamg signa was generated by a externa cooperative jammer (CJ) node to interfere the eavesdropper and improve the secrecy rate in mutipe-input-singe-output (MISO) secure SWIPT system with wiretap channes[18]. When information receivers (IRs) and energy-harvesting receivers (ERs) are paced in a same ce, the ERs are normay assumed to be coser to the transmitter compared with IRs. This gives rise to a new information security issue in the SWIPT systems. In such a situation, ERs have a possibiity of eavesdropping the information sent to the IRs, and thus can become potentia eavesdroppers [8] [14] [15]. In this paper, considering SWIPT, we investigate a secure transmission design probem in MU-MIMO secrecy system with one muti-antenna transmitter, one muti-antenna CJ, mutipe singe-antenna co-ocated receiver (CR), and mutipe muti-antenna ER. The CR empoys power spitting (PS) scheme to spit the received signas into two streams for ID and energy harvesting (EH) simutaneousy. Unike [17] [18], our objective is to imize the imum of harvested energy (- HE) of both ERs and CRs subject to secrecy rate constraints for each CR and tota transmit power constraint. Assug the imperfect CSI case, we seek to the optima

2 transmission strategy to jointy optimize the AN-aided beamforg, the AN vector, the CJ vector and the PS ratio design. Considering the worst-case scheme by incorporating the norm-bounded channe uncertainty mode, we derive equivaent forms of secrecy rate constraint and the imum of harvested energy. An iterative agorithm based on sequentia parametric convex approximation (SPCA) is aso addressed to recover a high quaity rank-one beamforg soution to the origina probem. Finay, the performance anaysis are provided to verify that the proposed agorithm outperforms the conventiona scheme. Notation: defines the Kronecker product. C M L and H M L describe the space of M L compex matrices and Hermitian matrices, respectivey. H + equas the set of positive semi-definite Hermitian matrices, and R + denotes the set of a nonnegative rea numbers. For a matrix A, A 0 means that A is positive semi-definite, and A F, tr(a), A and rank(a) denote the Frobenius norm, trace, deterant, and the rank, respectivey. vec(a) stacks the eements of A in a coumn vector. 0 M L is a nu matrix with M L size. E{ } describes expectation, and R{ } stands for the rea part of a compex number. [x] + represents {x,0} and λ (A) indicates the imum eigenvaue of A. II. SYSTEM MODEL AND PROBLEM FORMULATION In this section, we consider a MU-MIMO secrecy channe, which consists of one muti-antenna transmitter, one mutiantenna CJ, K singe-antenna CRs, and L muti-antenna ERs, where the CR empoys the PS scheme to decode information and expoit power simutaneousy. It is assumed that the transmitter and the CJ are equipped with N T and N J transmit antennas, and each ER has N E receive antennas. We denote h k C NT as the channe vector between the transmitter and the k-th CR, H C NT NE as the channe matrix between the transmitter and the -th ER, g k C NJ as the channe vector between the CJ and the k-th CR, G C NJ NE as the channe matrix between the CJ and the -th ER, respectivey. In order to improve the reiabe transmission, the transmitter empoys the transmit beamforg with AN, which acts as interference to the ERs and simutaneousy provides energy to the CRs and ERs. Thus, the transmitter sends the confidentia message x k by using transmit beamforg with AN to the k-th CR as x k = w k s k + z, (1) where w k C NT denotes the inear beamforg vector for the k-th CR at the transmitter, s k C represents the information-bearing signa intended for the k-th CR satisfying E{ s k 2 } = 1, and z C NT is the energy-carrying AN vector, which can aso be composed by mutipe energy beams. The received signa at the k-th CR and the -th ER can be expressed as y k = h H k y =H H k=1 w ks k +h H k z+g H k qs J +n k, k = 1,...,K, k=1 w ks k +H H z+g H qs J +n, = 1,...,L, (2) where s J is the cooperative jamg signa introduced by the CJ satisfying E{ s J 2 } = 1, q C NJ indicates the CJ vector, n k CN(0,σk 2) and n CN(0,σ 2 I) stand for the additive Gaussian noise by the receive antenna at the k-th CR and the -th ER. In addition, each CR considers the PS scheme to manage the processes of ID and EH simutaneousy. Based on this reason, the received signa at the k-th CR is divided into ID and EH by PS ratio ρ k (0,1]. Thus, the received signa for ID at the k-th CR can be given as y ID k = ρ k y k +n p,k = ρ k ( h H k k=1 w ks k +h H k z+g H k qs J +n k ) +np,k (3) where n p,k CN(0,δk 2 ) is the antenna noise introduced by signa process of the ID at the k-th CR [2]. We denotew k = w k wk H as the transmit covariance matrix, Z = zz H as the AN covariance matrix, and Q = qq H as the CJ covariance matrix. Hence, the achieved secrecy rate can be cacuated by [ ] ( ) + ˆR k = og 2 1+SINRk C,k, k, (4) where ρ k h H k SINR k = W kh k ρ k (σk 2 +hh k ( j k W j +Z)h k +gk HQg k)+δk 2, C,k =og 2 I+(H H ZH +G H QG +σ 2 I) 1 H H W k H. Thus, the harvested power at the k-th CR and the -th ER are written as E c,k =η c,k (1 ρ k ) ( ( h H k W k+z ) h k +g H j=1 k Qg k+σk 2 ), ( E e, =η e, tr(h H ( W k+z)h )+tr(g H k=1 QG )+N E σ 2 ), (5) where η c,k and η e, denote the energy conversion efficiency of the k-th CR and the -th ER. Due to channe estimation and quantization errors, it may not be possibe to achieve the perfect CSI at the transmitter in practice. In this section, our aim is to jointy optimize the - worst-case HE formuation at the imperfect CSI case. Now, we adopt the imperfect CSI case under the normbounded channe uncertainty mode [11][13][14]. Specificay, the actua channes h k, H, g k, and G can be given as h k = h k +e k H = H +E,, g k = ḡ k +ẽ k G = Ḡ+Ẽ,, where h k,ḡ k, H, and Ḡ denote the estimated channe avaiabe at the transmitter and the CJ, respectivey, and e k,ẽ k,e, and Ẽ are the channe errors, bounded as e k 2 ε k, ẽ k 2 ε k, E F θ, and Ẽ F θ, respectivey. In this paper, our aim is to imize the imum of the tota harvested power among the a CRs and ERs subject to the secrecy rate constraint and the tota transmit power constraint at the transmitter and the CJ power constraint. By taking the (6)

3 above channe mode into account, the transmit beamforg design is formuated as a muti-object optimization probem which can be given by s.t. ρ k,{w k },Z,Q e k ε k ẽ k ε k C k Ê c,k +Ê e, E F θ Ẽ F θ C R s (7a) (7b) k=1 tr(w k)+tr(z) P T, tr(q) P J, (7c) 1 ρ k > 0, W k 0, Z 0, Q 0, rank(w k ) = 1, (7d) where Ê c,k Ê e, τe k, k, e k ε k, ẽ k ε k c,k E e, F θ, Ẽ F θ (1 τ)e e,,, τ is the priority parameter, Rs stands for a given secrecy rate threshod, and P T and P J denote the avaiabe power budget at the transmitter and the CJ, respectivey. Variabe τ 0 refects the preference of the system operator. Probem (7) is non-convex due to the secrecy rate constraint and the objective function, and thus cannot be soved directy. III. PROPOSED ROBUST DESIGN METHOD In order to circumvent the roust - HE probem, we transform probem (7) by introducing two sack variabes Ēs and Ēe into τēs +(1 τ)ēe (8a) ρ k,{w k },Z,Q,Ēs,Ēe s.t. e k ε k, ẽ k ε k k E c,k Ēs, k, (8b) E F θ, Ẽ F θ (7b), (7c), (7d). E e, Ēe,, (8c) Probem (8) is sti non-convex in terms of (8b), (8c) and (7b). Now, et us consider another formuation of probem (7) based on SPCA method [22]. The optimization framework can aso be recast as a convex form by incorporating channe uncertainties. First, by appying the matrix inequaity I+A 1+tr(A) [23], the robust secrecy rate constraint (7b) can be reaxed as tr C k og 2 (1+ ( H H W ) ) kh σ 2+tr( H H ZH +G H QG ) R s. (9) To make the constraint (9) tractabe, we introduce two sack variabes r 1 > 0 and > 0, the robust secrecy rate (9) can be equivaenty reaxed as og(r 1 ) R s, (10a) ρ k h H k 1+ W kh k ρ k ( σk 2+hH k ( j k W j+z)h k +gk HQg k)+δk 2 r 1,(10b) tr ( ) H H W k H 1+ σ 2 +tr( H H ZH +G H QG ) 1,. (10c) Then, we can be further simpify (10) as r 1 2 R s, (11a) h H k W kh k r 1 1,,(11b) σk 2 +hh k ( j k W j+z)h k +gk HQg k + δ2 k ρ k σ 2 +tr( H H ZH +G H QG ) σ 2 +tr( H H (Z+W k )H +G H QG ), k. (11c) The inequaity constraint (11a) is equivaent to 2 R s+2 + (r 1 ) 2 (r 1 + ) 2, which can be converted into a conic quadratic-representabe function form as [ ] 2 R s +2 r 1 r1 +. (12) Design H k h H h k k, Ḡ k ḡk Hḡk, Ĥ H H H, and Ĝ Ḡ H Ḡ. The inequaities in (11b) and (11c) can be rearranged, respectivey, which give σ 2 k+ j k wh j ( H k + k )w j +z H ( H k + k )z+q H (Ḡk+Φ k )q wh k ( H k + k )w k (13a) ρ k r 1 1 σ 2 +z H (Ĥ+Υ )z+wk H (Ĥ+Υ )w k +q H (Ĝ+Ψ )q + δ2 k σ2 +zh (Ĥ+Υ )z+q H (Ĝ+Ψ )q,, (13b) where k = h k e H k +e k h H k +e ke H k, Φ k = ḡ k ẽ H k +ẽ kḡ H k + ẽ k ẽ H k, Υ = H E H + E HH + E E H, and Ψ = ḠẼH + Ẽ Ḡ H + ẼẼH. which stand for the CSI uncertainty. It is straightforward to show that k F h k e H k F + e k hh k F + e k e H k F h k e H k + e k h H k + e k 2 = ε 2 k +2ε k h k, Φ k F ḡ k ẽ H k F + ẽ k ḡ H k F + ẽ k ẽ H k F ḡ k ẽ H k + ẽ k ḡ H k + ẽ k 2 = ε 2 k +2 ε k ḡ k, Υ F H E H F + E HH F + E E H F H F E H F +E F H H F+ E 2 F = θ 2 +2θ H F, Ψ F ḠẼH F + ẼḠH F + ẼẼH F Ḡ F ẼH F + Ẽ F ḠH F + Ẽ 2 F = θ 2 +2 θ Ḡ F. (14) (15) (16) (17) Note that k, Φ k, Υ, and Ψ are norm-bounded matrices as k F ξ k, Φ k F ξ k, Υ F α, and Ψ F α, where ξ k = ε 2 k + 2ε k h k, ξk = ε 2 k + 2 ε k ḡ k, α = θ 2 + 2θ H F, and α = θ 2 +2 θ Ḡ F. According to [24], we can imize constraint (11b) by imizing the eft-hand side (LHS) of (13a) whie imiz-

4 ing its the right-hand side (RHS). Then (13a) and (13b) can be approximatey rewritten as, respectivey, k F ξ k, Φ k F ξ k σ 2 k + j k wh j ( H k + k )w j +z H ( H k + k )z +q H (Ḡk+Φ k )q+ δ2 k wk H ( H k + k )w k ρ k k F ξ k r 1 1 (18) Υ F α, Ψ F α σ 2 +z H (Ĥ+Υ )z+w H k (Ĥ +Υ )w k +q H (Ĝ +Ψ )q σ 2 +zh (Ĥ+Υ )z+q H (Ĝ+Ψ )q,. Υ F α, Ψ F α (19) In order to imize the RHS of (18) and (19), a oose approximation [24] is appied, which gives wk H ( H k + k )w k k F ξ k r 1 1 wh k ( H k ξ k I)w k r 1 1 σ 2 +zh (Ĥ+Υ )z+q H (Ĝ+Ψ )q Υ F α, Ψ F α σ2 +zh (Ĥ α I)z+q H (Ĝ α I)q. (20) Using simiar technique to the LHS of (18) and (19) yieds k F ξ k, Φ k F ξ k σ 2 k+ j k wh j ( H k + k )w j +z H ( H k + k )z+q H (Ḡk+Φ k )q+ δ2 k ρ k σ 2 k+ j k wh j ( H k +ξ k I)w j +z H ( H k +ξ k I)z+q H (Ḡk+ ξ k I)q+ δ2 k ρ k Υ F α, Ψ F α σ 2 +zh (Ĥ+Υ )z+w H k (Ĥ+Υ )w k +q H (Ĝ+Ψ )q σ 2 +zh (Ĥ+α I)z (21) +w H k (Ĥ+α I)w k +q H (Ĝ+ α I)q,. (22) From (18)-(22), (13a) and (13b) can be given as, respectivey, σ 2 k + j k wh j ( H k +ξ k I)w j +z H ( H k +ξ k I)z +q H (Ḡk+ ξ k I)q+ δ2 k wh H k ξs,kw k ρ k r 1 1 σ 2 +z H (Ĥ+α I)z+w H k (Ĥ+α I)w k +q H (Ĝ+ α I)q σ2 +zh Ĥ ξe,z+q H Ĝ ξe, q, (23) (24) where H ξs,k= H k ξ k I, Ĥ ξe,=ĥ α I and Ĝ ξe, =Ĝ α I. We observe that these two constraints are non-convex, but the RHS of both (23) and (24) have the function form of quadratic-over-inear, which are convex functions [25]. Based on the idea of the constrained convex procedure [22], these quadratic-over-inear functions can be repaced by their firstorder expansions, which transforms the probem into convex programg. Specificay, we define f A,a (w,t) = wh Aw t a, (25) where A 0 and t a. At a certain point ( w, t), the firstorder Tayor expansion of (25) is given by F A,a (w,t, w, t) = 2R{ wh Aw} t a wh A w ( t a) 2(t a). (26) By using the above resuts of Tayor expansion, for the points ( w k, r 1 ), ( z, ) and ( q, ), we can transform (23) and (24) into convex forms, respectivey, as σ 2 k + j k wh j ( H k +ξ k I)w j +z H ( H k +ξ k I)z+q H (Ḡk + ξ k I)q+ δ2 k F Hξs,k ρ,1(w k,r 1, w k, r 1 ), (27a) k σ 2 +z H (Ĥ+α I)z+wk H (Ĥ+α I)w k +q H (Ĝ+ α I)q σ( 2 2 r 2 2 )+FĤξe,,0 (z,, z, )+FĜ ξe,,0 (q,, q, ).(27b) In order to approximate the EH constraint (8b) and (8c) to convex one, we appy an SCA-based method. First, by using a oose approximation approach for (8b) and (8c), we have η c,k (1 ρ k )( j=1 wh j ( Hk + k )w j +z H ( H k + k )z +q H (Ḡk+Φ k )q+σ 2 k) Ē s (28a) z H (Ĥ+Υ )z+ Ěe N E σ 2,, j=1 wh k (Ĥ+Υ )w k +q H (Ĝ+Ψ )q (28b) where Ěe = Ēe η e,. Using oose approximation [24] to the LHS of (28a) and (28b) yieds η c,k (1 ρ k ) ( j=1 wh j ( H k ξ k I)w j +z H ( H k ξ k I)z +q H (Ḡk ξ k I)q+σ 2 k) Ē s (29a) z H (Ĥ α I)z+ j=1 wh k (Ĥ α I)w k +q H (Ĝ+ α I)q Ěe N E σ,. 2 (29b) It is observed that w H j ( H k ξ k I)w j, z H ( H k ξ k I)z and q H (Ḡk ξ k I)q are the concave part of constraints (29a) and (29b). In order to make (29a) and (29b) more tractabe, we empoy the SCA technique for (29a) and (29b) to obtain convex approximations. Firsty, we take z H ( H k ξ k I)z as an exampe. Let z be an initia feasibe point. We substitute z= z+ z into z H ( H k ξ k I)z as foows z H ( H k ξ k I)z =( z+ z) H Hξs,k( z+ z) z H Hξs,k z+2r{ z H Hξs,k z}, (30)

5 where (30) are derived by dropping the quadratic form z H Hξs,k z. Then, defining w k and q as initia feasibe point. Substituting w k = w k + w k z = z+ z, and q = q+ q into the LHS of (29a) and (29b). Then, we can use the the simiar method with (30) to achieve inear approximations of the concave constraints (29a) and (29b), respectivey, as j=1 η c,k a k (1 ρ k ) Ēs (31a) [ w j H Ĥξ e, w j +2R{ w j H H ] ξe, w j } +2R{ z H Hξe, z} + z H Hξe, z+ q H Ĝ ξe, q+2r{ qh Ĝ ξe, q} Ěe N E σ, 2 (31b) where a k = H j=1 [ w j Hξs,k w j +2R{ w j H H ξs,k w j } ] + z H Hξs,k z + 2R{ z H Hξs,k z} + q HḠ ξs,k q + 2R{ q H Ḡ ξs,k q} + σ2 k and Ḡ ξs,k = Ḡk+ ξ k I. In addition, it is noted that (31a) is sti non-convex in its current form since it invove couped a k s and 1 ρ k s. The first-order Tayor expansion of Ē s is e s = Ẽ s +0.5Ẽ 1 2 s (Ēs Ẽs). Thus, (31a) can be rewritten to a convex second-order cones (SOC) constraint as [ / 2e s ηc,k,a k +ρ k 1 ] a k ρ k +1. (32) In addition, (7c) can be reformuated to two SOC forms as [ w1 T,...,wT K,zT] P T, q T P J. (33a) (33b) Eventuay, probem (7) is converted into the foowing convex second order cone programg (SOCP) probem as τēs ρ k,{w k },z,q,ẽs,ēe, r1,r2 +(1 τ)ēe s.t. (12), (27a), (27b), (31b), (32), (33a), (33b), 0 < ρ k 1. (34) Given { w k }, q, z, r 1,, and Ẽs probem (34) is convex and can be soved by empoying convex optimization software toos such as CVX [26]. Based on the SPCA method, an approximation with the current optima soution can be updated iterativey, which impies that (7) is optimay soved. IV. SIMULATION RESULTS In this section, we provide the simuation resuts to vaidate the performance of our proposed schemes. We set that K = 2, L = 2, M = 1, N T = 4, N J = 4, and N E = 2. We assume the estimated channe h k, H,g k, and G are respectivey modeed as h k =H(d k )h I, H = H(d )H I, g k = H(f k )g I, and G = H(f )G I, where h I CN(0,I), H I CN(0,I), g k = CN(0,I), and G = CN(0,I), H(d k ) = c 4πf c ( 1 d k ) κ 2. We define dk = 100 m and f k = 100 m meters as the distance between the transmitter as we as the CJ and a the CR, and d = 9 m and f = 9 m meters as the distance between the transmitter as we as the CJ and a the ER, uness otherwise specified. Moreover, c = ms 1 is the speed of ight; f c = 900 MHz is the carrier frequency; and κ = 2.7 is the path oss exponent. In addition, the noise power at the CR is set to be σk 2 = 90 dbm and δ2 k = 50 dbm. Aso the noise power of a the ERs is σk 2 = 90 dbm, k. Aso we set the channe error bound for the deteristic mode as ε s = ε k = ε k, k and ε e = θ = θ, k. The EH efficiency coefficients are set to η c,k = η e, = 0.3 and the priority parameter τ is 0.5. In our simuations, we compare the foowing transmit designs: the perfect CSI case, the proposed SOCP-SPCA agorithm, the no-cj scheme which is the robust design w/o CJ by setting Q = 0 [15], the no-an scheme which means the robust design w/o CJ by setting Z = 0 [18], and the nonrobust method which is a scheme that assumes no uncertainty in the CSI. Average harvested energy (dbm) SOCP SPCA (P T = 60 dbm) SOCP SPCA (P T = 50 dbm) SOCP SPCA (P T = 40 dbm) Iteration Number Fig. 1. Average harvested power versus iteration numbers Fig. 1 iustrates the convergence of the SOCP-SPCA method with respect to iteration numbers for P T = 40 dbm, P J = 40 dbm, Rs = 0.5 bps/hz, and ε = 0.01, respectivey. It is easiy seen that convergence of the SOCP-SPCA method is achieved for a cases within just 5 iterations. Average harvested energy (dbm) Perfect CSI SOCP SPCA (ε = 0.01) no AN no CJ non robust SOCP SPCA (ε = 0.05) Target secrecy rate (bps/hz) Fig. 2. Average harvested power versus target secrecy rate Fig. 2 shows the average harvested power in terms of different target secrecy rates with P T = 30 dbm and P J = 30 dbm, respectivey. It is observed that the harvested power of a schemes decine with the increase of the secrecy rate target. Aso, the performance gain of the scheme with ǫ = 0.01 over the scheme with ǫ = 0.05 is 0.8 db at a the target secrecy rate region. Compared with the no-an scheme and the no-cj scheme, the harvested power of the SOCP-SPCA agorithm are

6 6 db and 9 db higher. Moreover, we can check that the SOCP- SPCA agorithm perform better than the non-robust scheme, and the performance gap increases as the target secrecy rate becomes arge. Average harvested energy (dbm) Perfect CSI SOCP SPCA (ε = 0.01) SOCP SPCA (ε = 0.05) no AN non robust P J Fig. 3. Average harvested power versus the power budget at the CJ Fig. 3 depicts the average harvested power versus the power budget at the CJ with P T = 10 dbm and R s = 0.5 bps/hz, respectivey. It is easiy observed that the achieved harvested power increases with P J, and the curves of the perfect CSI case and the SOCP-SPCA agorithm increase with the same sope. Moreover, we can check that as P J increases, the performance gap between the proposed agorithms and the no- AN scheme becomes arger and the performance oss of the non-robust scheme grows. V. CONCLUSION In this paper, we have studied the robust secure beamforg design for a MU-MIMO SWIPT secrecy system with the PS scheme by incorporating the norm-bounded channe uncertainties. We aim to imize the imum of harvested energy by jointy optimizing the AN-aided beamforg, the AN vector, the CJ vector and the PS ratio design. To sove the non-convex probem, we use the SPCA method, oose approximation and SCA-based method to reformuate the origina probem as an convex SOCP probem. Aso, an SPCAbased iterative agorithm is aso addressed. Finay, simuation resuts have been provided to vaidate the performance of our proposed agorithm. In addition, the proposed robust design methods outperforms the non-robust schemes. REFERENCES [1] P. Grover and A. Sahai, Shannon meets tesa: Wireess information and power transfer, in Proc. IEEE ISIT, pp , Jun [2] R. Zhang and C. Ho, MIMO broadcasting for simutaneous wireess information and power transfer, IEEE Trans. Wireess Commun., vo. 12, no. 5, pp , May [3] Z. Zhu, Z. Wang, Z. Chu, X. Gao, Y. Zhang, and J. Cui, Robust beamforg based on transmit power anaysis for mutiuser mutipeinput-singe-output interference channes with energy harvesting, IET Commun., vo. 10, no. 10, pp , Ju [4] H. Lee, S.-R. Lee, K.-J. Lee, and I. Lee, Optima beamforg designs for wireess information and power transfer in MISO interference channes, IEEE Trans. Wireess Commun., vo. 14, no. 9, pp , Sep [5] Z. Zhu, Z. Wang, K.-J. Lee, Z. Chu, and I. Lee, Robust transceiver designs in mutiuser MISO broadcasting with simutaneous wireess information and power transmission, Journa of Commun. and Networks, vo. 18, no. 2, pp , Apr [6] J. Zhu, R. Schober and V. K. Bhargava, Secure transmission in mutice massive MIMO systems, IEEE Trans. Wireess Commun., vo. 13, no. 9, pp , Sep [7] Z. Zhu, Z. Chu, Z. Wang, and J. Cui, Robust beamforg design for mutipe-input-singe-output secrecy muticasting systems with simutaneous wireess information and power transmission, IET Commun., vo. 10, no. 15, pp , Oct [8] Z. Zhu, Z. Chu, F. Zhou, H. Niu, Z. Wang, and I. Lee, Secure beamforg designs for secrecy MIMO SWIPT systems, IEEE Wireess Commun. Lett., accepted for pubished. [9] Q. Shi, W. Xu, J. Wu, E. Song, and Y. Wang, Secure beamforg for MIMO broadcasting with wireess information and power transfer, IEEE Trans. Wireess Commun., vo. 14, no. 5, pp , May [10] Z. Zhu, Z. Chu, Z. Wang, and I. Lee, Outage constrained robust beamforg for secure broadcasting systems with energy harvesting, IEEE Trans. Wireess Commun., vo. 15, no. 11, pp , Nov [11] D. W. K. Ng, E. S. Lo, and R. Schober, Robust beamforg for secure communication in systems with wireess information and power transfer, IEEE Trans. Wireess Commun., vo. 13, no. 8, pp , Aug [12] S. Goe and R. Negi, Guaranteeing secrecy using artificia noise, IEEE Trans. Wireess Commun., vo. 7, no. 6, pp , Jun [13] Z. Chu, Z. Zhu, M. Johnston, and S. L. Goff, Simutaneous wireess information power transfer for MISO secrecy channe, IEEE Trans. Vehicuar Techno., vo. 15, no. 1, pp , Jan [14] M. R. A. Khandaker and K.-K. Wong, Masked beamforg in the presence of energy-harvesting eavesdroppers, IEEE Trans. Inf. Forensics Security, vo. 10, pp , Jan [15] Z. Zhu, Z. Chu, N. Wang, S. Huang, Z. Wang, and I. Lee, Beamforg and power spitting designs for AN-aided secure muti-user MIMO SWIPT systems, IEEE Trans. Inf. Forensics Security, vo. 12, no. 12, pp , Dec [16] J. Moon, H. Lee, C. Song, and I. Lee, Secrecy performance optimization for wireess powered communication networks with an energy harvesting jammer, IEEE Trans. Commun., vo. 65, no. 2, pp , Feb [17] Z. Zhu, Z. Chu, Z. Wang, I. Lee, Joint optimization of AN-aided beamforg and power spitting designs for MISO secrecy channe with SWIPT, in Proc. IEEE ICC, pp. 1-6, May [18] Q. Zhang, X. Huang, Q. Li, and J. Qin, Cooperative jamg aided robust secure transmission for wireess information and power transfer in MISO channes, IEEE Trans. Commun., vo. 63, no. 3, pp , Mar [19] W. Liu, X. Zhou, S. Durrani and P. Popovski, Secure communication with a wireess-powered friendy jammer, IEEE Trans. Wireess Commun., vo. 15, no. 1, pp , Jan [20] Z. Chu, K. Cumanan, Z. Ding, M. Johnston, and S. Le Goff, Secrecy rate optimizations for a MIMO secrecy channe with a cooperative jammer, IEEE Trans. Veh. Techno., vo. 64, no. 5, pp , May [21] Q. Li and W.-K. Ma, Spatiay seective artificia-noise aided transmit optimization for MISO muti-eves secrecy rate imization, IEEE Trans. Signa Process., vo. 61, no. 10, pp , May [22] A. S. Vishwanathan, A. J. Smoa, and S. V. N. Vishwanathan, Kerne methods for missing variabes, in Proc. 10th Int. Workshop Artif. Inte. Stat., pp , Jan [23] K. B. Petersern and M. S. Pedersern, The Matrix Cookbook. Cambridge, MA, USA: MIT Press, Nov [Onine]. Avaiabe: matrixcookbook.com [24] M. Bengtsson and B. Ottersten, Optimum and suboptimum transmit beamforg, in Handbook of Antennas in Wireess Communications, L. C. Godara, Ed., CRC Press, Aug [25] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, [26] M. Grant and S. Boyd., Cvx: Matab software for discipined convex programg, Avaiabe: Sep