 Research
 Open Access
Joint distributed beamforming and jamming schemes in decodeandforward relay networks for physical layer secrecy
 Chengmin Gu^{1} and
 Chao Zhang^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s136380170997x
© The Author(s) 2017
Received: 9 June 2017
Accepted: 24 November 2017
Published: 6 December 2017
Abstract
In this paper, we propose joint cooperative beamforming and jamming schemes in decodeandforward (DF) relay networks for physical layer secrecy. In DF relay networks, only the relays decoding the message from the source correctly have to join in the forwarding phase and the relays with decoding error are not utilized sufficiently. Taking this property into consideration, we let the relays decoding successfully transmit information signals and the relays decoding in error transmit jamming signal to improve the secrecy capacity of system. For this purpose, we design a bilevel optimization algorithm to search the optimal beamforming vector and jamming vector for the relays via the semidefinite relaxation (SDR). In addition, for balancing the cost of system and secrecy performance, we also study some suboptimal schemes to improve information secrecy. Finally, the simulation results show that the optimal scheme outperforms all other simulated schemes and the suboptimal schemes achieve good tradeoff between secrecy performance and computational complexity.
Keywords
 Physical layer secrecy
 Distributed beamforming
 Jamming
 Relay networks
1 Introduction
For openness and broadcast properties in wireless communications, information carried by radio waves is vulnerable to be intercepted by the unintended users. Traditional secrecy mechanisms, which depend on data encryption in application layer, mainly utilize unaffordable computation complexity to prevent the eavesdroppers to obtain the encrypted messages. However, with the rapid development of computing apparatuses, the traditional encryption technology confronts the unprecedented challenge. Thus, to achieve information security in wireless transmission, physical layer secrecy, which takes advantage of intrinsic characteristics of the wireless channels to achieve transmission security without applying encryption technology, has drawn much attention and been applied into many scenarios [1–4].
On the other hand, relayaided cooperative communication technology has been applied in wireless scenarios, since it can extend the coverage and improve the reliability of signal transmission [5]. Additionally, relayaided cooperative networks have been proved to be able to enhance the transmission security [6, 7]. In the multiplerelay networks without the direct link from the source to the destination, Cooperative beamforming (CB), singlerelay and singlejammer (SRSJ) scheme, singlerelay and multiplejammer (SRMJ) scheme, multiplerelay and single jammer (MRSJ) scheme, and multiplerelay and multiplejammer (MRMJ) scheme are five main transmission schemes for physical layer secrecy. In the CBbased relay systems, relays just perform distributed beamforming directly to the legal destination in order to enlarge the capacity of legal channel as much as possible [8–11]. In the SRSJ scheme, for relaxing the requirement of signal synchronization, how to select the best pair of forwarding relay and jammer was addressed in [12]. In the SRMJ scheme, the best relay is picked out to forward the information and the left relays transmit jamming signals to confuse the eavesdropper [13]. Alternatively, the MRSJ scheme selects the best jammer from all relays and let left relays perform cooperative beamforming to the legal user [14]. As the relay with amplifyandforward (AF) scheme can always transmit signals to the destination [15], the MRMJ scheme, in which some relays are used to transmit information to the legal destination and left ones produce jamming signals to improve the secrecy capacity of the information transmission, is usually investigated in AF relay networks [16–18]. In DF relay networks, a MRMJ scheme was proposed in [19], where the relays without decoding error are divided into two groups, one for information beamforming and the other one for cooperative jamming. However, it does not make the most of the relays that cannot decode the message from the source successfully to enhance the secrecy of information transmission. In [20], a simultaneous beamforming and jamming scheme was proposed in DF relay networks. However, the relays deocoding in error are not sufficiently exploited. A MRMJ scheme with fixed jamming and beamforming set was proposed in [21]. Similarly, a multirelay secrecy transmission scheme with a dedicated multiantenna jammer was proposed in [22]. Both [21] and [22] have deployed dedicated jamming nodes which have no ability of forwarding information to the legal destination. If there exists the direct link from the source to the destination, cooperative jamming (CJ) and transmission switching between CB and CJ were also proposed to improve the system secrecy [23, 24]. Moreover, joint CB and CJ schemes in colocated multiantenna scenarios were also investigated in [25] and [26]. In [27], we have investigated the optimal joint CB and CJ scheme for the DF relay networks with direct links from the source to destination and eavesdropper. However, the assumptions on relay set partition and the inference cancelation are difficult to be implemented. In this paper, we consider a more practical scenario for the joint beamforming and jamming scheme in the DF relay networks and provide optimal scheme and suboptimal schemes with lower complexity to investigate the proposed idea of joint beamforming and jamming well. In [28], Guo et al. also addressed the power allocation in a joint beanforming and jamming scheme in DF relay networks to achieve the maximum secrecy rate.
In this paper, we intend to employ joint cooperative beamforming and jamming to improve the secrecy capacity in DF relay networks without direct link between the source and the destination. In DF relay networks, only the relays decoding the message from the source correctly have to forward information to the legitimate destination during the relaying phase. To efficiently utilize all relays, the relays decoding successfully consist of the beamforming set, where relays perform distributed beamforming to transmit information signals. At the same time, the relays decoding in error belong to the jamming set, where the relays transmit jamming to disturb the eavesdropper. Herein, our goal is to design the beamforming vector and jamming vector in order to maximize the achievable secrecy capacity under the constraint of total relay power. For this purpose, we design a based bilevel optimization algorithm to search the optimal beamforming vector and jamming vector for the relays via the semidefinite relaxation. In addition, for balancing the cost of system and secrecy performance, we also study some suboptimal schemes to improve information secrecy. Finally, by our numerical results, the optimal scheme outperforms all existing schemes and the proposed suboptimal schemes. In addition, some suboptimal schemes with low computational complexity also have better secrecy performance than existing schemes.

In most existing work, the relays decoding in error are not usually utilized during the cooperative transmission phase. In this paper, taking the distinguishing feature of DF relay into consideration, we employ the relays decoding in error to transmit jamming signals to improve the secrecy of cooperative transmission.

How to design the beamforming vector and jamming vector in order to maximize the achievable secrecy capacity under the constraint of total relay power is provided. To reduce the computation complexity, we also propose four suboptimal designs.

Through simulations, we compare the proposed designs with existing CB, SRMJ, MRSJ and SRSJ schemes and prove that the optimal design and some suboptimal designs outperform the existing schemes. The suitable scenarios of these proposed designs are also addressed.
We adopt the following notations. Bold uppercase letters denote matrices and bold lowercase letters denote column vectors. Transpose and conjugate transpose are represented by (·)^{ T } and (·)^{ H } respectively; I _{ K } is the identity matrix of K×K; \({\mathbb C}^{n}\) denotes the space of n×1 column vector with complex entries; Tr(·) is the trace of the matrix; Rank(·) denotes the rank of a matrix; ∥·∥^{2} is the twonorm of a vector; A≽0 and A≻0 mean that A is positive semidefinite and definite matrix, respectively; x⊥y denotes vector x and vector y are orthogonal; x∥y denotes vector x and vector y are parallel; \(\mathbb E\{\cdot \}\) denotes expectation.
2 System model and transmission scheme
2.1 System model
2.2 Transmission scheme
The whole cooperative transmission is separated into two phases: source broadcasting and relay transmitting.
2.2.1 Source broadcasting
where P _{ s } is the transmit power of the source, h _{ si } is the channel coefficient from source to the ith relay, and n _{ si } is the receiver noise. Also, x is the normalized information symbol, i.e., \(\mathbb {E}\{{x}^{2}\}=1\). If the ith relay can decode the received message correctly, it will join the information transferring in the relay transmitting phase. Otherwise, the ith relay is assigned to jam the eavesdropper. We assume all transmitted information blocks are protected by an ideal error control coding. In other words, if the received signaltonoise ratio (SNR) of the ith relay, i.e., γ _{ si }=P _{ s }h _{ si }^{2}/N _{0}, is larger than the threshold γ _{ th }, where \(\phantom {\dot {i}\!}\gamma _{th}=2^{2R_{0}}1\) and R _{0} is the target information transmission rate, the receiver can decode the information packet correctly. Thus, if γ _{ si }≥γ _{ th }, then the ith relay belongs to \(\mathcal {D}\). Otherwise, it is in the set \(\mathcal {J}\). As a result, each relay knows whether it can decode the message from the source successfully and which set it belongs to. Without loss of generality, we set \(\mathcal {D}=\{R_{1},R_{2},\cdots,R_{M}\}\) and \(\mathcal {J}=\{J_{1},J_{2},\cdots,J_{K}\}\), where 1≤R _{ i },J _{ i }≤N are the indexes of these relays (see Fig. 1) and there is M+K=N. Additionally, we assume that all receiver noises have the same noise power N _{0}.
We define the weight coefficient vector of set \(\mathcal {D}\) for beamforming as \(\mathbf {w}_{R}\,=\,\left [\!w_{R_{1}}^{*}\!,\!w_{R_{2}}^{*}\!,\!\cdots,\!w_{R_{M}}^{*}\!\right ]^{T}\!\) and weight coefficient vector of set \(\mathcal {J}\) for jamming as \({\mathbf {w}}_{J}=\left [w_{J_{1}}^{*},w_{J_{2}}^{*},\cdots,w_{J_{K}}^{*}\right ]^{T}\). Moreover, \({\mathbf {h}}_{rd}=\left [h_{R_{1}D},h_{R_{2}D},\cdots,h_{R_{M}D}\right ]^{T}\) represents the channel vector from the beamforming set to the destination and \({\mathbf {h}}_{je}=\left [h_{J_{1} E},h_{J_{2} E},\cdots,h_{J_{K} E}\right ]^{T}\) denotes the channel vector from the jamming set to the eavesdropper. Before relay transmitting, the destination can transmit a training signal to let each relay estimate the channel coefficient from the destination to itself. Note that the eavesdropper is often a wireless user unauthorized to access the message for the destination [29]. Hence, the eavesdropper is able to cooperatively transmit training signal to all relays. Then, each relay can obtain its channel coefficient h _{ id }, i=1,2,...,N. Moreover, we set up a central control node (CCN), which can be a relay node or a dedicated node. After that, each relay reports its related channel information to the CCN. Therefore, the CCN can compute the beamforming vector w _{ R } for the relays in set \(\mathcal {D}\) and jamming vector w _{ J } for the relays in set \(\mathcal {J}\). In the following analysis, we assume that all receivers can estimate their received channel coefficients perfectly to exploit the ideal performance and check up the theoretical feasibility of our proposals. The effect of estimation error will be discussed in future work. Whereas, the source node has no knowledge of its transmitting channel state information due to practical constraint.
2.2.2 Relay transmitting
Note that C _{ s }(M)=0 means the eavesdropper can obtain no less correct information than the destination, which is called completely unsafe transmission and should be avoided. Our aim herein is to maximize the secrecy capacity C _{ s }(M) as much as possible with the power constraint of relays.
3 Optimal design for maximizing secrecy capacity
Since secrecy capacity is the difference of two concave functions, it is a nonconvex optimization in general. To deal with this issue, we will conduct a series of transformations to turn it into a convex problem, which can be solved by some available solvers.
where W=w w ^{ H } is a rankone square matrix. To solve the problem of (11), we employ the idea of SDR in [30] to drop the rankone nonconvex constraint. After the relaxation transformation, the problem of (11) is still a nonconvex optimization problem owing to the presence of auxiliary variable τ. However, the problem of (11) can be treated as the quasiconvex problem for each fixed τ [31]. Therefore, it can be treated as a bilevel optimization problem: the outerlevel optimization is about auxiliary variable τ and the innerlevel optimization is a quasiconvex problem.
3.1 Innerlevel optimization
Note that there is Z=ψ W. Using the wellknown CVX toolbox [31], we can solve problem (12) easily.
3.2 Outerlevel optimization
where \({\phi (\tau)}=\text {Tr}({\widetilde {\mathbf {H}}}_{jd}{\mathbf {Z}^{\star }(\tau)})+{\psi ^{\star }(\tau)}N_{0}\) and Z ^{⋆}(τ) and ψ ^{⋆}(τ) are solutions of the innerlevel optimization problem (12) given a τ. After onedimension searching during [τ _{min},τ _{max}], the optimal τ ^{⋆} that makes ϕ(τ) minimum can be found.
As a result, we finally obtain the solution of problem (11), i.e., τ ^{⋆} and W ^{⋆}=Z ^{⋆}/ψ ^{⋆}. If Rank(W ^{⋆})=1, we can obtain w ^{⋆} via singular value decomposition (SVD) [34] from W ^{⋆}. If the rank of W ^{⋆} is larger than one, we can extract an approximate solution from W ^{⋆} via using the Gaussian Randomization Procedure (GRP) in [30]. To make it more clearly, we draw the procedure of solving the optimization problem in Algorithm 1.
4 Suboptimal designs with low complexity
It means that we can firstly determine the optimal directions of w _{ R } and w _{ J } and then seek the optimal power allocation between w _{ R } and w _{ J }. Furthermore, given a fixed w _{ R }, we just need to search the optimal w _{ J } in a space with lower dimension than w and vice versa. From this point, we propose the following suboptimal designs in which w _{ R } or w _{ J } is directly determined by related channel information and the other weight vector is optimized to maximize the secrecy capacity.
4.1 Information beam determined schemes
4.1.1 w _{ R }⊥h _{ re } scheme
4.1.2 w _{ R }∥h _{ rd } scheme
where \({\mathbf {W}}_{J}={\mathbf {w}}_{J}{\mathbf {w}}_{J}^{H}\) is a rankone matrix. Observe (25) and (11), we can also apply Algorithm 1 to solve the problem (25). Then, we can obtain the optimal jamming vector \(\mathbf {w}_{J}^{\star }\) for a given P _{ j }.
In order to achieve the optimal power allocation between P _{ r } and P _{ j }, we appeal to the onedimension searching on P _{ j } over the interval [0,P _{ t }]. So that, we can finally obtain the optimal \(P_{j}^{\star }\), \(P_{r}^{\star }\), \(\mathbf {w}_{R}^{\star }\) and \(\mathbf {w}_{J}^{\star }\).
4.2 Jamming beam determined schemes
4.2.1 w _{ J }⊥h _{ jd } scheme
in which \(\zeta _{max}({\widetilde {\mathbf {F}}}^{1}{\widetilde {\mathbf {E}}})\) represents the eigenvector corresponding to the maximum eigenvalue of the matrix \({\widetilde {\mathbf {F}}}^{1}{\widetilde {\mathbf {E}}}\). Similarly, we also need to search the optimal \(P_{j}^{\star }\) during the interval [0,P _{ t }] to maximize C _{ s }(M).
4.2.2 w _{ J }∥h _{ je } scheme
where \(\phi _{max}({\widetilde {\mathbf {H}}}^{1}{\widetilde {\mathbf {G}}})\) denotes the eigenvector corresponding to the maximum eigenvalue of matrix \({\widetilde {\mathbf {H}}}^{1}{\widetilde {\mathbf {G}}}\).
Thus, for a P _{ j }∈[0,P _{ t }] we can obtain \(\mathbf {w}_{J}^{\star }(P_{j})\) and \(\mathbf {w}_{R}^{\star }(P_{j})\). After checking enough power configurations, we ultimately obtain the global optimal solution \(({\mathbf {w}}_{R}^{\star }, \mathbf {w}_{J}^{\star }, P_{j}^{\star })\) that makes C _{ s }(M) become maximum.
4.3 Computational complexity analysis
In this subsection, we intend to compare the computational complexities of these proposed schemes. Since these designed schemes have different beamforming and jamming patterns in the relaying phase, we just need to analyze the computational complexity of computing the beamforming and jamming vectors given the beamforming set \(\mathcal {D}=M\) and jamming set \(\mathcal {J}=K\). Note that there is N=M+K.
4.3.1 Computational complexity of the optimal scheme
By Algorithm 1, we first need to determine τ _{ min } and τ _{ max }, which means we need to solve the Rayleigh quotient problem to get λ _{ min }(B ^{−1} A) in (15) and λ _{ max }(D ^{−1} C) in (16). Due to [33], the computational complexity of Rayleigh quotient problem is \(\mathcal {O}(22N^{2})\). So, the computational complexity of determining τ _{ min } and τ _{ max } is \(\mathcal {O}(44N^{2})\). During the searching scope [τ _{ min },τ _{ max }], there are τ _{ i }, i=1,2,…,L so that the problem (12) will be operated L times. According to [25] and [36], the computational complexity of an inner level optimization (12) is \(\mathcal {O}\left ((N+1)^{0.5}(2(N+1)^{3}+4(N+1)^{2}+8)\right)\log (1/\epsilon)\) in which ε is the accuracy of solving the SDP. Please note that in Algorithm 1, running L times of problem (12) is activated after τ _{ min } and τ _{ max } are determined. It means computing τ _{ min } and τ _{ max } is only run once and the above two steps are performed sequentially. In addition, as the GRP is activated in a very slight probability and most of results from SDP meet the rankone constraint, we only consider the computational complexity of SVD herein. Due to [36], the computational complexity of SVD is \(\mathcal {O}(N^{3})\). Therefore, the total computational complexity of the proposed optimal scheme is \(\mathcal {O}\left (44N^{2}\right)+\mathcal {O}(N^{3})+\mathcal {O}\left (L((N+1)^{0.5}(2(N+1)^{3}+4(N+1)^{2}+8))\right)\log (1/\epsilon)\).
4.3.2 Computational complexity of w _{ R }⊥h _{ re } scheme
In w _{ R }⊥h _{ re } scheme, we just need to calculate (20). Therefore, the computational complexity is \(\mathcal {O}(M^{2})\) [33].
4.3.3 Computational complexity of w _{ R }∥h _{ rd } scheme
4.3.4 Computational complexity of w _{ J }⊥h _{ jd } Scheme
In w _{ J }⊥h _{ jd } scheme, we just need to calculate (27) and (29). Thus, in the light of [33], the computational complexity of w _{ J }⊥h _{ jd } scheme is \(\mathcal {O}\left (L_{p}K^{2}\right)+\mathcal {O}\left (22L_{p}M^{2}\right)\).
4.3.5 Computational complexity of w _{ J }∥h _{ je } scheme
Similar to w _{ J }⊥h _{ jd } scheme, the computational complexity is also \(\mathcal {O}\left (L_{p} K^{2}\right)+\mathcal {O}\left (22L_{p}M^{2} \right)\).
4.3.6 Comparison

The optimal scheme has the most computational complexity among all proposed schemes.Table 1
Computational complexity of all schemes
Scheme
Complexity
Optimal
\(\mathcal {O}\left (44N^{2}\right)+\mathcal {O}(N^{3})+\mathcal {O}\left (L\left ((N+1)^{0.5}\left (2(N+1)^{3}\right.\right.\right.\) +4(N+1)^{2}+8))) log(1/ε)
w _{ R }⊥h _{ re }
\(\mathcal {O}(M^{2})\)
w _{ R }∥h _{ rd }
\(\mathcal {O} \left (L_{p}M^{2}\right)+ \mathcal {O}\left (44L_{p}K^{2}\right)+\mathcal {O}(L_{p}K^{3})\) \(+\mathcal {O}\left (L_{p}L((K+1)^{0.5}(2(K+1)^{3}+4(K+1)^{2}+8)\log (1/\epsilon))\right)\)
w _{ J }⊥h _{ jd }
\(\mathcal {O}\left (L_{p}K^{2}\right)+\mathcal {O}\left (22L_{p} M^{2}\right)\)
w _{ J }∥h _{ je }
\(\mathcal {O}\left (L_{p} K^{2}\right)+\mathcal {O}\left (22L_{p} M^{2} \right)\)

w _{ R }⊥h _{ re } scheme incurs least computational complexity among all proposed schemes.
5 Numerical results
6 Conclusions
In this paper, we proposed an optimal scheme and four suboptimal schemes with low computational complexity in the DF relay networks to enhance the transmission security. Unlike the prior works, the proposed transmission schemes utilize the property of DF relays to let the relays decoding incorrectly transmit jamming signals to confound the eavesdropper and the relays decoding correctly transmit information beamforming to the destination. By our numerical results, the optimal scheme outperforms all existing schemes and the proposed suboptimal schemes. In addition, some suboptimal schemes with low computational complexity also have better secrecy performance than existing schemes. Moreover, we found that our proposed schemes are more suitable for the largescale relay networks and the scenarios where relays are near the middle position between the source and destination.
Declarations
Funding
The work was supported in part by the National Natural Science Foundation of China under Grant No. 61431011, NSFC, China, and the Fundamental Research Funds for the Central Universities, XJTU, China.
Authors’ contributions
CG and CZ proposed the ideas in this paper. CG performed the analysis and simulations and wrote the paper. CZ reviewed and edited the manuscript. All authors read and approved the manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 M Bloch, J Barros, PhysicalLayer Security: From Information Theory to Security Engineering (Cambridge University Press, Cambridge, 2011).View ArticleMATHGoogle Scholar
 A Mukherjee, S Fakoorian, J Huang, AL Swindlehurst, Principles of physical layer security in multiuser wireless networks: a survey. IEEE Commun. Surveys Tuts. 16(3), 1550–1573 (2014).View ArticleGoogle Scholar
 N Yang, HA Suraweera, IB Collings, C Yuen, Physical layer security of TAS/MRC with antenna correlation. IEEE Trans. Inf. Forensic Secur. 8(1), 254–259 (2013).View ArticleGoogle Scholar
 J Zhang, C Yuen, CK Wen, S Jin, KK Wong, H Zhu, Large system secrecy rate analysis for SWIPT MIMO wiretap channels. IEEE Trans. Inf. Forensic Secur. 11(1), 74–85 (2015).View ArticleGoogle Scholar
 X Chen, C Zhong, C Yuen, HH Chen, Multiantenna relay aided wireless physical layer security. IEEE Commun. Mag. 53(12), 40–46 (2015).View ArticleGoogle Scholar
 R Bassily, E Ekrem, X He, E Tekin, J Xie, MR Bloch, S Ulukus, A Yener, Cooperative security at the physical layer: a summary of recent advances. IEEE Signal Process. Mag. 30(5), 16–28 (2013).View ArticleGoogle Scholar
 A Kuhestani, A Mohammadi, M Noori, Optimal power allocation to improve secrecy performance of nonregenerative cooperative systems using an untrusted relay. IET Commun. 10(8), 962–968 (2016).View ArticleGoogle Scholar
 L Dong, Z Han, AP Petropulu, HV Poor, Improving wireless physical layer security via cooperating relays. IEEE Trans. Signal Process. 58(3), 1875–1888 (2010).MathSciNetView ArticleGoogle Scholar
 C Jeong, I Kim, DI Kim, Joint secure beamforming design at the source and the relay for an amplifyandforward MIMO untrusted relay system. IEEE Trans. Signal Process. 60(1), 310–325 (2012).MathSciNetView ArticleGoogle Scholar
 Y Yang, Q Li, W Ma, J Ge, PC Ching, Cooperative secure beamforming for AF relay networks with multiple eavesdroppers. IEEE Signal Process. Lett. 20(1), 35–38 (2013).View ArticleGoogle Scholar
 X Gong, H Long, H Yin, F Dong, B Ren, Robust amplifyandforward relay beamforming for security with mean square error constraint. IET Commun. 9(8), 1081–1087 (2015).View ArticleGoogle Scholar
 L Wang, Y Cai, Y Zou, W Yang, L Hanzo, Joint relay and jammer selection improves the physical layer security in the face of CSI feedback delays. IEEE Trans. Veh. Technol. 65(8), 6259–6274 (2016).View ArticleGoogle Scholar
 C Wang, H Wang, X Xia, Hybrid opportunistic relaying and jamming with power allocation for secure cooperative networks. IEEE Trans. Wireless Commun. 14(2), 589–605 (2015).View ArticleGoogle Scholar
 H Wang, M Luo, X Xia, Q Yin, Joint cooperative beamforming and jamming to secure AF relay systems with individual power constraint and no eavesdropper’s CSI. IEEE Signal Process. Lett. 20(1), 39–42 (2013).View ArticleGoogle Scholar
 X Chen, L Lei, H Zhang, C Yuen, Largescale MIMO relaying techniques for physical layer security: AF or DF?IEEE Trans. Wirel. Commun. 14(9), 5135–5146 (2015).View ArticleGoogle Scholar
 M Lin, J Ge, Y Yang, An effective secure transmission scheme for af relay networks with twohop information leakage. IEEE Commun. Lett. 17(8), 1676–1679 (2013).View ArticleGoogle Scholar
 S Vishwakarma, A Chockalingam, in Proc.IEEE Int. Conf. Commun. (ICC). Amplifyandforward relay beamforming for secrecy with cooperative jamming and imperfect CSI (IEEEBudapest, 2013), pp. 1640–1645.Google Scholar
 H Hui, A Swindlehurst, G Li, J Liang, Secure relay and jammer selection for physical layer security. IEEE Signal Process. Lett. 22(8), 1147–1151 (2015).View ArticleGoogle Scholar
 L Wang, C Cao, M Song, Y Cheng, in Proc.IEEE Int. Conf. Commun. (ICC). Joint cooperative relaying and jamming for maximum secrecy capacity in wireless networks (IEEESydney, 2014), pp. 4448–4453.Google Scholar
 X Guan, Y Cai, Y Wang, W Yang, in Proc of IEEE PIMRC, Toronto, Canada. Increasing secrecy capacity via joint design of cooperative beamforming and jamming (IEEEToronto, 2011), pp. 1279–1283.Google Scholar
 N Kolokotronis, A Manos, in Proc.IEEE Signal Process. Conf. (EUSIPCO). Improving physical layer security in DF relay networks via twostage cooperative jamming, (2016), pp. 1173–1177.Google Scholar
 ER Alotaibi, KA Hamdi, Optimal cooperative relaying and jamming for secure communication. IEEE Wireless Commun. Lett. 6:, 689–692 (2015).View ArticleGoogle Scholar
 J Li, AP Petropulu, S Weber, On cooperative relaying schemes for wireless physical layer security. IEEE Trans. Signal Process. 59(10), 4985–4997 (2011).MathSciNetView ArticleGoogle Scholar
 Z Lin, Y Cai, W Yang, L Wang, Robust secure switching transmission in multiantenna relaying systems: cooperative jamming or decodeandforward beamforming. IET Commun. 10(13), 1673–1681 (2016).View ArticleGoogle Scholar
 B Li, Z Fei, Robust beamforming and cooperative jamming for secure transmission in DF relay systems. EURASIP J. Wireless Commun (2016). Networking. doi:10.1186/s1363801605601.
 J Myung, H Heo, J Park, Joint beamforming and jamming for physical layer security. ETRI. 37(6), 898–905 (2015).View ArticleGoogle Scholar
 C Gu, C Zhang, in Proc.of IEEE International Conference on Communication Systems (ICCS). Adaptive distributed beamforming and jamming in DF relay networks for physical layer secrecy (IEEEShenzhen, 2016), pp. 1–5.Google Scholar
 H Guo, Z Yang, L Zhang, J Zhu, Y Zou, in Proc.of IEEE ICC. Optimal power allocation for joint cooperative beamforming and jamming assisted wireless networks (IEEEParis, 2017).Google Scholar
 M Bloch, J Barros, MRD Rodrigues, SW McLaughlin, Wireless informationtheoretic security. IEEE Trans. Inf. Theory. 54(6), 2515–2534 (2008).MathSciNetView ArticleMATHGoogle Scholar
 Z Luo, W Ma, A So, Y Ye, S Zhang, Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27(3), 20–34 (2010).View ArticleGoogle Scholar
 S Boyd, L Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004).View ArticleMATHGoogle Scholar
 A Charnes, W Cooper, Programming with linear fractional functionals. Naval Res. Logist Quarter. 9:, 181–186 (1962).MathSciNetView ArticleMATHGoogle Scholar
 GH Golub, CF Van Loan, Matrix Computations, 3rd edn. (The John Hopkins Univ. Press, Baltimore, 1996).MATHGoogle Scholar
 RA Horn, CR Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985).View ArticleMATHGoogle Scholar
 A Goldsmith, Wireless Communications (Cambridge University Press, Cambridge, 2004).Google Scholar
 M Lobo, L Vandenberghe, S Boyd, H Lebret, Applications of secondorder cone programming. Linear Algebra Appl. 284:, 193–228 (1998).MathSciNetView ArticleMATHGoogle Scholar