Robust secrecy beamforming for wireless information and power transfer in multiuser MISO communication system
 Wei Wu^{1}Email author and
 Baoyun Wang^{1, 2}
https://doi.org/10.1186/s1363801503782
© Wu and Wang; licensee Springer. 2015
Received: 16 December 2014
Accepted: 10 May 2015
Published: 9 June 2015
Abstract
In this paper, we study the maxmin fairness for robust secrecy beamforming design in a multiuser MISO communication system with simultaneous wireless information and power transfer (SWIPT). In particular, the imperfect channel state information (CSI) and power splitting information receiver (IR) are taken into account. We model the design as an optimization problem which maximizes the minimum harvested energy among the multiantenna energy receivers (ERs). Besides, both the secure communication requirement and the lowest electrical energy storage threshold at IR must be guaranteed in our formulated optimization problem. The considered maxmin problem is nonconvex and hard to tackle. By introducing the technique of semidefinite relaxation (SDR), we prove the tightness of this relaxation and obtain the global optimal solution of our original optimization problem. Moreover, a suboptimal beamforming design scheme is proposed with low computational complexity. Simulation results show that our proposed robust schemes are more efficient than other schemes in terms of energy harvesting and transmit power saving.
Keywords
1 Introduction
The rapid growth of wireless communication requirement brings abundant energy consumption, which leads to the conventional batterypowered wireless communication devices encountering a bottleneck in providing sufficient power. Traditionally, we can harvest plentiful renewable energy from the solar, wind and tide, etc. However, the technique which we employed in collecting these energy may not be suitable for our portable communication devices. On the other hand, simultaneous wireless information and power transfer (SWIPT), which is regarded as a potential feasible technique to overcome this bottleneck, has attracted great interest of the researchers in recent years [1–5]. Using this technology, the batterypowered wireless communication devices can harvest energy from the ambient radio frequency (RF) signals to prolong their lifetime. In particular, for a threenode multiple antenna SWIPT system, the optimal precoder was designed to achieve information and energy transmission tradeoff [1]. The similar secrecy rate and power transfer tradeoff problem between the IR and ERs was investigated in [3].
It is worth noting that all above works which considered the systems with SWIPT are under the assumption of perfect channel state information (CSI). However, in practice, it is hard to obtain perfect CSI at the transmitter due to the signal transmission delay, quantization errors, and channel breakdowns. Furthermore, as we all know, the perfect CSI methods are quite sensitive to the channel uncertainties. Therefore, some systems are constructed under imperfect channel realization (e.g., [6–8]). In particular, [6] and [7] studied the secure transmission with wiretap channel for SWIPT in the MISO broadcast system. Moreover, two kinds of suboptimal solutions are proposed to achieve the lower bound of robust secrecy rate since the optimal solution is hard to acquire in [7]. Both the system with perfect and imperfect CSIs are studied in [8], where the optimal power allocation scheme under the dual use of ERs are derived. In [9], the authors investigated the optimal resource allocation design for the multiuser MISO system with SWIPT, the artificial noise [10] is incorporated to facilitate the energy harvesting at ER. As far as the authors know, most of the previous works on power transfer either focus on the optimization of transmit power of transmitter or secrecy capacity of IR. And they always assume that the receivers are equipped with single antenna or the CSIs are perfectly known at transmitter. These motivate us to research the rarely considered problem of fair power harvesting among all ERs which are equipped with multiantenna. Besides, the CSIs of transmittertoER links are imperfect.
In this paper, we concentrate on the optimal secrecy beamforming design for SWIPT in a multiuser MISO communication system with multiple multiantenna ERs. The CSIs between transmitter and ERs are assumed to be imperfect. Besides, the power splitting receivers [11] and the ANaided transmit strategy are taken into account. We aim to, under the constraints of secure communication requirement and the lowest electrical energy storage threshold at IR, maximize the minimum harvested energy among the multiantenna ERs by jointly optimizing the transmit beamforming vectors, AN covariance, and power splitting ratio. The design of interest results in a nonconvex problem, which can be solved through the ‘separable SDR’ [12], and the resulting solution is proved to be the global optimal solution. Besides, we propose a suboptimal robust beamforming design scheme with smaller available solution set for power transfer and information decoding.

We studied the problem of fair energy harvesting among all ERs by considering imperfect CSI and power splitting IR in the multiuser MISO communication system. An optimal secrecy beamforming design scheme was proposed to achieve this fair energy transfer to ERs.

We proved the tightness of the adopted SDR technique in our optimal design scheme by exploiting the structure of resulting solution and obtained the global optimal solution of our original optimization problem.

We proposed a suboptimal secrecy beamforming design scheme by simplifying the analysis of the solution of original problem. Also, we proved the tightness of the adopted SDR programming in our suboptimal scheme.
Notations: In this paper, the bold capital and lowercase letters are used to denote matrices and vectors, respectively. ε{·}, (·)^{ H }, T r(·), rank(·), indicate the expectation, Hermitian transpose, trace, and rank, respectively. I _{ N } denotes the N×N identity matrix; \({\mathbb {C}^{N \times M}}\) and \({\mathbf {\mathbb {H}}}_ +^{N}\) denote the N×M complex matrices and N×N Hermitian matrices, respectively. ∥ ·∥_{2} means the Euclidean norm of a vector while ∥ ·∥_{ F } means the Frobenius norm of a matrix; null(·) denotes the null space of a vector or matrix. \({\mathbf {x}} \sim \mathcal {CN}\left ({{\boldsymbol {\mu }}, {\mathbf {\Lambda }}} \right)\) means the vector x is a complex Gaussian variable with mean μ and covariance Λ.
2 System model and problem formulation
2.1 System model
2.2 Problem formulation
Remark 1.
The dual use of artificial noise is advocated to facilitate the energy transfer and provide security.
where η is the same as η _{ k }.
Remark 2.
For the case that the energy beams carry no information but only serve as the pseudorandom signals which can be cancelled with a cancellation operation at each ER [14], thus the energy beams cause no interference at ER besides the AN.
3 Solution of the optimization problem
In this section, we aim to turn problem (3) to a tractable convex problem and to find a rankone optimal solution W by studying the solution structure of (3).
Problem (3) is intractable for robust downlink beamforming design because of the semiinfiniteness of constraints C1, C2, C3, and C5. To facilitate the solution, we consider to turn these constrains into linear matrix inequalities (LMIs) [16] by using the Sprocedure method:
Lemma 1.
We note that the relaxed SDP problem (7) is a common convex optimization problem which can be solved efficiently by the existing solvers such as SDPT3 [18] and SeDuMi [19]. If the obtained solution W is rank one, then the SDR optimization problem (7) shares the same optimal solution and objective value with the original optimization problem (1). As a result, we work out the information beam w by performing eigenvalue decomposition on W. However, as we all know, the obtained solution W for the SDR problem (7) may not be rank one, so the resulting solution is not the optimal solution of the original optimization problem. In the following, we will introduce a theorem to reveal the tightness of the relaxed SDP problem (7). Then, by analyzing the structure of the resulting solution, we construct an optimal solution for problem (7) with rank (W)=1. Furthermore, we propose a suboptimal secrecy beamforming design scheme with lower computational complexity by simplifying the optimality condition of constructing the rankone solution.
3.1 Optimal solution
In this subsection, we will reveal the tightness of our proposed relaxed SDP problem (7) by introducing a theorem which shares the similar idea as the Proposition 4.1 in [3].
and \({r_{1}} = {\text {rank}}\left ({{\mathbf {B}}_{1}^{\ast }} \right)\). Where \({\mathbf {Z}}_{e,k}^{*\left ({l,l} \right)} \in \mathbb {H}_ +^{N}\) and \({\mathbf {U}}_{e,k}^{*\left ({l,l} \right)} \in \mathbb {H}_ +^{N}\) are the lth entry matrices on the diagonal of \({\tilde {\mathbf { G}}_{k}}{{\mathbf {X}}_{k}}\tilde {\mathbf { G}}_{k}^{H} \in {\mathbf {\mathbb {H}}}_ +^{NM}\) and \({\tilde {\mathbf { G}}_{k}}{{\mathbf {Y}}_{\mathbf {k}}}\tilde {\mathbf { G}}_{k}^{H} \in {\mathbf {\mathbb {H}}}_ +^{NM}\), respectively; where \({\tilde {\mathbf { G}}_{k}} = \left [ {\begin {array}{*{20}{c}} {{{\mathbf {I}}_{\textit {NM}}}}&{{{\bar {\mathbf { g}}}_{k}}}\end {array}} \right ]\), and X _{ k } and Y _{ k } are the Lagrange dual variables with respect to M _{ER,k }(W,V,V _{0},λ _{ k }) and \({{\mathbf {T}}_{\text {ER},k}}\left ({{\mathbf {W}},{{\mathbf {V}}_{0}},{{\tilde \lambda }_{k}}} \right)\), respectively. Furthermore, we model the orthogonal basis of the null space of \({\mathbf {B}}_{1}^{\ast } \) as \({{\mathbf {N}}_{1}} \in {\mathbb {C}^{N \times \left ({N  {r_{1}}} \right)}}\), and \({{\boldsymbol {\pi }}_{1,n}} \in {\mathbb {C}^{N \times 1}}, 1 \le n \le N  {r_{1}}\) as the nth column of N _{1}. Upon that, we have the following theorem.
Theorem 1.
 1.The optimal solution W ^{∗} can be expressed as$$ {{\mathbf{W}}^{\ast}} = \sum\limits_{n = 1}^{N  {r_{1}}} {{\mu_{n}}} {{\boldsymbol{\pi }}_{1,n}}{\boldsymbol{\pi }}_{1,n}^{H} + f{{\boldsymbol{\tau }}_{1}}{\boldsymbol{\tau }}_{1}^{H}, $$(9)
where μ _{ n }≥0,∀n,f≥0 and \({{\boldsymbol {\tau }}_{1}} \in {\mathbb {C}^{N \times 1}}, \parallel {{\boldsymbol {\tau }}_{1}}{\parallel _{2}} = 1\) satisfies \({\boldsymbol {\tau }}_{1}^{H}{{\mathbf {N}}_{1}} = 0\).
 2.If the optimal solution W ^{∗} given in (9) has rank(W ^{∗})>1, i.e., μ _{n}>0,∃n. Then, we have another solution$$ {\widetilde {\mathbf{W}}^{\ast}} = {{\mathbf{W}}^{\ast}}  \sum\limits_{n = 1}^{N  {r_{1}}} {{\mu_{n}}} {{\boldsymbol{\pi }}_{1,n}}{\boldsymbol{\pi }}_{1,n}^{H} = f{{\boldsymbol{\tau }}_{1}}{\boldsymbol{\tau }}_{1}^{H}, $$(10)$$ {\widetilde {\mathbf{V}}^{\ast}} = {{\mathbf{V}}^{\ast}} + \sum\limits_{n = 1}^{N  {r_{1}}} {{\mu_{n}}{{\boldsymbol{\pi }}_{1,n}}{\boldsymbol{\pi }}_{1,n}^{H}}, $$(11)$$ \widetilde {\mathbf{V}}_{0}^{\ast} = {\mathbf{V}}_{0}^{\ast},~{\tilde \rho^{\ast}} = {\rho^{\ast}}, ~{\tilde t^{\ast}} = {t^{\ast} }, $$(12)
with \({\widetilde {\mathbf {W}}^{\ast }}\) serve as the new optimal solution and has \({\text {rank}}\left ({{{\widetilde {\mathbf {W}}^{\ast }}}} \right)= 1\).
Proof.
: Please refer to Appendix 1.
With Theorem 1, the global optimal solution of problem (1) is achieved. First, we solve the SDR problem (7) via CVX and obtain the solution \(\left \{ {{{\mathbf {W}}^{\ast } }, {{\mathbf {V}}^{\ast } }, {\mathbf {V}}_{0}^{\ast }, {\rho ^{\ast }}, {t^{\ast } }} \right \}\). If the information beamforming matrix satisfies rank (W ^{∗})=1, the solution turns out to be optimal. If not, i.e., rank (W ^{∗})>1. Then, we format an alternative optimal solution \(\left \{ {{\widetilde {\mathbf {W}}^{\ast }}, {\widetilde {\mathbf {V}}^{\ast }}, {\widetilde {\mathbf {V}}_{0}^{\ast }}, {\tilde \rho ^{\ast }}, {{\tilde t}^{\ast } }} \right \}\) in accordance with (10)–(12) with rank\(({\widetilde {\mathbf {W}}^{\ast }}) = 1\) and achieve the same objective value.
3.2 Suboptimal downlink beamforming design
As discussed in Appendix 1, it involves solving the complex dual optimization problem (18) to construct an optimal solution set \(\left \{ {{{\mathbf {W}}^{\ast } }, {{\mathbf {V}}^{\ast }}, {\mathbf {V}}_{0}^{\ast }, {\rho ^{\ast }},{t^{\ast }}} \right \}\) of the relaxed SDP problem (7) with rank (W ^{∗})=1. Besides, the uncertainty of the value of rank(\(B_{1}^{\ast }\)) results in an obscure value of rank(W ^{∗}). Therefore, an additional procedure is inevitable to construct an alternative optimal solution set \(\left \{ {{\widetilde {\mathbf {W}}^{\ast }},{\widetilde {\mathbf {V}}^{\ast }},{\widetilde {\mathbf {V}}_{0}^{\ast }},{\tilde \rho ^{\ast }},{\tilde t^{\ast }}} \right \}\) with rank\(({\widetilde {\mathbf {W}}^{\ast }}) = 1\) when rank (W ^{∗})>1. In this case, we propose a suboptimal secrecy downlink beamforming design scheme which achieves a rankone optimal solution W with lower computational complexity.
Proposition 1.
Consider \({\mathbf {X}}_{k}^{\ast }\) and \({\mathbf {Y}}_{k}^{\ast }\) as the optimal Lagrange dual multiplier matrixes of the relaxed SDP problem (7) associated with constraints C5 and C2, respectively. The condition \(\sum \limits _{k = 1}^{K} {\sum \limits _{l = 1}^{M} {\left ({{\mathbf {U}}_{e,k}^{*\left ({l,l} \right)}  {\mathbf {Z}}_{e,k}^{*\left ({l,l} \right)}} \right)}} \succeq {\mathbf {0}}\) must be held to ensure rank (W ^{∗})=1.
Proof.
Please refer to Appendix 2.
Compared to constraint C5, it is obvious that \(\overline {C5} \) neglects the contribution of information beam in terms of energy harvesting, i.e., W is wiped out. As a result, the new optimization problem (13) performs worse compared to problem (7). We note that the reformulated constraint \(\overline {C5} \) is convex, thus problem (13) can be solved efficiently through the aforementioned numerical solvers. Furthermore, it always has a rankone optimal beamforming solution, i.e., rank (W)=1, which proves the tightness of the SDP relaxation.
4 Simulation results
In this section, we evaluate the performance of our proposed robust downlink beamforming design scheme for multiuser MISO communication system with imperfect CSI via simulation. We set the simulation parameters as N=5, K=3, M=3, r = 10 dB, r _{ k }=0 dB, P=10 mW, P _{min}=1 mW, η = 0.5 and \({\sigma _{I}^{2}} = {\sigma _{E}^{2}} = {10^{ 3}}\). The channel entries associated with our system are randomly generated i.i.d. complex Gaussian variables which obey \(\mathcal {CN}\left ({0,1}\right)\). All simulation results were achieved by an average of 1000 channel realizations.
4.1 Average total harvested power
To the imperfect CSI between the transmitter and ERs, we define \({\alpha _{e,k}} = \frac {{{\varepsilon _{e,k}}}}{{\sqrt {E\left \{ {\parallel {{\overline G }_{k}}{\parallel _{F}^{2}}} \right \}} }},k = 1,\ldots, K\) as channel uncertainty ratio to evaluate the kth channel estimate error. We will set α _{ e,1}=…=α _{ e,K }=α _{ e } and choose α _{ e }=0.05, unless specified.
4.2 Average total transmit power
4.3 Secrecy capacity
5 Conclusions
An optimal robust secrecy beamforming design for MISO communication system with SWIPT was investigated in this paper. In order to solve the formulated original optimization problem efficiently, we converted it into a convex SDR problem and proved the tightness of this adopted SDR. As a result, the obtained power splitting ratio and the transmit beamforming matrices based on the worstcase maxmin fair energy harvesting among K ERs are the global optimal solution. In addition, a suboptimal scheme was proposed with lower computational complexity. Simulation results demonstrated the superior performance of our proposed schemes compared to the other schemes.
6 Appendix 1
7 Proof of Theorem 1
where \({\mathbf {X}} = {\mathrm {\{ }}{\mathbf {W}},{\mathbf {V}},{{\mathbf {V}}_{0}},{\lambda _{k}},{\tilde \lambda _{k}},{{\mathbf {X}}_{k}},{{\mathbf {Y}}_{k}},\alpha,\beta,\gamma,{\mathbf {\Phi }},{\mathbf {\Xi }},{\mathbf {\Omega }}{\mathrm {\} }}\) includes all the primal and dual variables, and \({{\mathbf {X}}_{k}} \in \mathbb {H}_ +^{NM + 1}, \forall k\), \({{\mathbf {Y}}_{k}} \in \mathbb {H}_ +^{NM + 1}, \forall k, \alpha \in {\mathbb {R}_ + }\), \(\beta \in {\mathbb {R}_ + }\) and \(\gamma \in {\mathbb {R}_ + }\) are the dual variables with respect to M _{ER,k }(W,V,V _{0},λ _{ k }), \({{\mathbf {T}}_{\text {ER},k}}\left ({{\mathbf {W}},{{\mathbf {V}}_{0}},{{\tilde \lambda }_{k}}} \right)\), C4, C1, and C3, respectively. \({\mathbf {\Phi }} \in \mathbb {H}_ +^{N}\), \({\mathbf {\Xi }} \in \mathbb {H}_ +^{N}\), and \({\mathbf {\Omega }} \in \mathbb {H}_ +^{N}\) are the dual variables regard to W, V, and V _{0}, respectively.
i.e., N _{1} lies in the null space of both \({\mathbf {A}}_{1}^{\ast }\) and H. Since rank(N _{1})=N−r _{1}, it deduces that \({\text {rank}}\left ({{\mathbf {A}}_{1}^{\ast }} \right) \le N  \left ({N  {r_{1}}} \right) = {r_{1}}\). What is more, according to (20), we achieve another inequality: \({\text {rank}}\left ({{\mathbf {\mathrm {A}}}_{1}^{\ast }} \right) \ge {\text {rank}}\left ({{\mathbf {B}}_{1}^{\ast }} \right)  {\text {rank}}\left ({\mathbf {H}} \right) = {r_{1}}  1\). Finally, \({\text {rank}}\left ({{\mathbf {A}}_{1}^{\ast }} \right)\) is bounded by \({r_{1}}  1 \le {\text {rank}}\left ({{\mathbf {A}}_{1}^{\ast }} \right) \le {r_{1}}.\) Following with this, the rank of W ^{∗} can be bounded between N−r _{1}≤rank(W ^{∗})≤N−r _{1}+1.
where μ _{ n }≥0, ∀n and f>0. The first part of Theorem 1 is thus proved.
The properties from (25) to (30) demonstrate that the alternative solution \(\left \{ {{\widetilde {\mathbf {W}}^{\ast }}, {\widetilde {\mathbf {V}}^{\ast }}, {\widetilde {\mathbf {V}}_{0}^{\ast } }, {\tilde {\rho }^{\ast }}, {{\tilde t}^{\ast } }} \right \}\) not only achieves the same optimal value as \(\left \{ {{{\mathbf {W}}^{\ast } }, {{\mathbf {V}}^{\ast } }, {\mathbf {V}}_{0}^{\ast }, {{\rho ^{\ast }}}, {t^{\ast } }} \right \}\) but also satisfies all the constraints of our primal optimization problem with \({\text {rank}}\left ({{\widetilde {\mathbf {W}}^{\ast }}} \right) = 1\). Thus, we finish the proof of the second part of Theorem 1.
8 Appendix 2
9 Proof of Proposition 1
We note that rank (H)=1 and β ^{∗}+γ ^{∗}>0, so we have rank((β ^{∗}+γ ^{∗})H)=1. On the other hand, as discussed in Appendix 1, we know that W ^{∗}≠0. Thus, we conclude that rank (W ^{∗})=1 when \(\sum \limits _{k = 1}^{K} {\sum \limits _{l = 1}^{M} {\left ({{\mathbf {U}}_{e,k}^{*\left ({l,l} \right)}  {\mathbf {Z}}_{e,k}^{*\left ({l,l} \right)}} \right)}} \succeq {\mathbf {0}}\) holds.
Declarations
Acknowledgements
This paper was supported by the National Natural Science Foundation of China (No. 61271232, 61372126); the Open research fund of National Mobile Communications Research Laboratory, Southeast University (No. 2012D05); the Priority Academic Program Development of Jiangsu Province (Smart Grid and Control Technology).
Authors’ Affiliations
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