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Robust AN-Aided Secure Transmission Scheme in MISO Channels with Simultaneous Wireless Information and Power Transfer Maoxin Tian, Xiaobin Huang, Qi Zhang, and Jiayin Qin
Abstract—In this letter, considering the simultaneous wireless information and power transfer scheme, we study the robust artificial noise (AN)-aided secure transmission design in multiple-input-single-output channels where the channel uncertainties are modeled by worst-case model. Our objective is to maximize the worst-case secrecy rate with respect to both the worst-case channel uncertainties and the worst-case eavesdropper among multiple eavesdroppers, under the transmit power constraint and the worst-case energy harvesting constraint. The optimal solution to the problem can be found by two-dimensional (2-D) search. Since the 2-D search algorithm has high computational complexity, we propose to neglect the correlation of the channel uncertainties from the transmitter to the information-decoding receiver and reformulate the problem as a sequence of convex semidefinite programming (SDP) which is solved efficiently by SDP based one-dimensional line search method. It is shown through computer simulations that the proposed robust AN-aided secure transmission schemes have significant performance gain over the non-robust AN-aided secure transmission scheme and the robust secure transmission scheme without the aid of AN. Index Terms—Artificial noise (AN), energy harvesting (EH), secure transmission, simultaneous wireless information and power transfer (SWIPT), worst-case model.
I. INTRODUCTION
T
HE simultaneous wireless information and power transfer (SWIPT) scheme for single-input-single-output (SISO) channel was investigated in [1], [2]. Motivated by the benefits of multi-antenna techniques, the SWIPT schemes for multiple-input-multiple-output (MIMO) and multiple-input-single-output (MISO) broadcast channels were studied in [3]–[5]. The SWIPT schemes for MIMO relay networks were proposed in [6], [7].
Manuscript received August 12, 2014; revised October 12, 2014; accepted November 02, 2014. Date of publication November 07, 2014; date of current version November 14, 2014. This work was supported in part by the National Natural Science Foundation of China under Grants 61472458, 61202498, and 61173148. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Yan Lindsay Sun. The authors are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, Guangdong, China (e-mail:
[email protected];
[email protected];
[email protected]. edu.cn;
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2014.2368695
Because of the broadcast nature of radio propagation and the inherent randomness of wireless channel, radio transmission is vulnerable to attacks from unexpected eavesdroppers [8]–[10]. For conventional MISO broadcast systems, the robust secure transmission schemes were studied in [8], [9] where the channel uncertainties are modeled by the worst-case model. The robust secrecy transmission schemes with artificial noise (AN) and generalized AN were proposed in [9], [10]. Considering the SWIPT scheme, secure communications in MISO SWIPT systems were investigated in [11]–[13] where the perfect channel state information (CSI) was considered. In practice, it is difficult to obtain perfect CSI because of channel estimation and quantization errors. Considering the worst-case channel uncertainties, the robust secure beamforming in the multiuser MISO SWIPT system, which consists of a transmitter, multiple co-located information-decoding (ID) and energy-harvesting (EH) receivers, and multiple eavesdroppers, was proposed in [14]. In [15], the robust secure beamforming in the MISO SWIPT system, which includes an ID receiver and multiple co-located EH receivers and eavesdroppers, was investigated. In this letter, considering the separated ID receiver, EH receiver and eavesdroppers and the SWIPT scheme, we investigate the robust AN-aided secure transmission design problem in the MISO channels, where the channel uncertainties are modeled by the worst-case model as in [8], [9]. Unlike the system in [15], the channel uncertainties from the transmitter to the ID receiver are also considered. Our objective is to design the robust AN-aided secure transmission scheme which maximizes the worst-case secrecy rate with respect to both the worst-case channel uncertainties and the worst-case eavesdropper among multiple eavesdroppers, under the transmit power constraint and the worst-case EH constraint. The optimal solution to the problem can be found by two-dimensional (2-D) search. Since the 2-D search algorithm has high computational complexity, we propose to neglect the correlation of channel uncertainties from the transmitter to the ID receiver and reformulate the problem as a sequence of convex semidefinite programming (SDP) which is solved by SDP based one-dimensional (1-D) line search method. Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. The and denote the conjugate transpose and trace of the matrix , respectively. By , we mean that is positive semidefinite. The denotes the Euclidean norm of the vector .
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II. SYSTEM MODEL AND PROBLEM FORMULATION
(7)
Consider a wireless broadcast system which consists of a transmitter, a legitimate information-decoding (ID) receiver, an EH receiver and eavesdroppers. The transmitter is equipped with antennas. Each of the other nodes is equipped with a single antenna. Denote the channel responses from the transmitter to the legitimate ID receiver, to the EH receiver and to the th, , eavesdropper as , and , respectively. When the transmitter transmits confidential signal, , the received signals at the legitimate ID receiver and the th eavesdropper, denoted as and , respectively, are
(8)
and
where , , and , denote the estimates of the channels , , and , respectively; , , and denote the channel uncertainties of , , and , respectively; , , , denote the radii of the uncertainty regions for , , and , respectively. Our objective is to design the transmit covariance matrix and the AN covariance matrix which maximize the worstcase secrecy rate subject to the transmit power constraint at the transmitter and the worst-case EH constraint at the EH receiver, which is formulated as follows
(1)
denotes the signal vector transmitted from where the transmitter and , denote the additive Gaussian noises at the legitimate ID receiver and the eavesdropper, respectively. Without loss of generality, we assume that the noise variance is in this letter. To enhance physical layer security, the transmitter employs the AN-aided secure communication scheme [9], [10] for transmission. Specifically, the transmitting signal vector, , consists of both the message-bearing signal and the AN, i.e., , where denotes the message-bearing signal and denotes the AN. For the secure MISO communication with the existence of multiple eavesdroppers, the achievable secrecy rate, denoted as , is characterized as [9]
(9) where
denotes the worst-case EH constraint.
III. ROBUST AN-AIDED SECURE TRANSMISSION SCHEME Introducing the slack variable , the objective function of problem (9) is equivalently rewritten as [9]
(10) Introducing the slack variable , (10) is recast as (11a)
(2)
(11b)
and denote the mutual informawhere tion at the legitimate destination and the th eavesdropper, respectively,
(11c)
(3) (4) The transmit power constraint at the transmitter is , where denotes the maximum transmit power. The harvested energy at the EH receiver is (5)
is omitted since it is monotonically where the function increasing function which has no effect on the optimization problem. Given and , the problem (11) can be transformed into a convex SDP. Thus, the problem (11) is solved by the line search over and the bisection search over . Since the aforementioned 2-D search algorithm has high computational complexity, we propose a suboptimal solution to the problem (11) in the following. Let and
(12)
We assume that the transmitter knows the imperfect CSI on , and . This assumption is valid when the eavesdroppers are active. When the eavesdroppers are passive, it is shown in [16] that a high signal-to-noise ratio (SNR), the additional knowledge of the eavesdropper CSI does not yield any gains in terms of the secrecy rate for slow-fading channels. In this letter, we model the channel uncertainties by the worst-case model as in [8], [9]. The channel uncertainties of the channel responses , , and are bounded by the regions, denoted as , , and , respectively, are
in the numerator We propose to neglect the correlation of and denominator of the left-hand side of (11b) and derive the suboptimal solution. Employing Charnes-Cooper transformation [17], we transform (11) into
(6)
(13d)
(13a) (13b) (13c)
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(13e) (13f) Denoting the optimal solution to problem (13) as and the corresponding optimal objective value as the following proposition. Proposition 1: The optimal solution problem (13) should satisfy the equation
cone, and is the accuracy of solving the SDP. Compare the SDP (19) with the standard form in [21], we have and . Thus, the computational complexity of our proposed scheme is
, we have to (14)
Proof: See Appendix A. By applying the -Procedure [18] and introducing the slack variable , we convert (13b) into a linear matrix inequality (LMI) as follows (15) and . Similarly, the constraints where (13c)-(13e) are also equivalently expressed as follows
(16) (17) (18) where , , and are introduced slack variables, , and . Combining (13) with (15)-(18), the optimization problem (13) is recast as (19) It is noted that given , the problem (19) is convex SDP which can be solved efficiently using the interior-point method [20]. We find the maximum worst-case secrecy rate by performing 1-D line search over . It is noted that the lower bound of is 1. The upper bound of is obtained as follows (20) We have the following proposition. Proposition 2: The rank of the optimal solution to the problem (19), denoted as , is less than or equal to 2. Proof: See Appendix B. It is noted that the conclusion of Proposition 2 is unlike those in [10] and [19]. From (25), this is because of the additional worst-case EH constraint. If rank , we obtain the beamforming vector, denoted as , by decomposing . If rank , we employ the Gaussian randomization method to generate the suboptimal rank-one solution. Remark: From [21], the complexity of solving an SDP problem is where denotes the number of semidefinite cone constraints, denotes the dimension of the semidefinite
(21) where
is the number of 1-D line search. IV. SIMULATION RESULTS
In this section, we evaluate the performance of the proposed robust AN-aided secure transmission scheme through computer simulations. In simulations, we consider that the distances from the transmitter to the legitimate ID receiver, the EH receiver and the eavesdroppers are comparable such that the normalized path loss assumption is reasonable. The channels from the transmitter to the legitimate ID receiver, the EH receiver and the eavesdroppers undergo the flat Rayleigh fading. Therefore, all the entries in the channel estimates , , and , , are assumed to be independent and identically distributed (i.i.d.) complex Gaussian random variables with zero-mean and unit variance. The radii of the uncertainty regions for , , and , , are . We produce 1000 randomly generated channel realizations and compute the average worst-case secrecy rate. In Fig. 1, we present the average worst-case secrecy rate comparison of the proposed 2-D and 1-D robust AN-aided secure transmission scheme (denoted as “Robust, 2-D” and “Robust, 1-D” in the legend, respectively), the proposed 1-D robust scheme with Gaussian randomization (denoted as “Robust, 1-D, GR”), the proposed 2-D robust scheme without the worst-case EH constraint (denoted as “Robust, 2-D, w/o EH”), the non-robust AN-aided secure transmission scheme (denoted as “Non-Robust”), and the robust secure transmission scheme without the aid of AN (denoted as “Robust w/o AN”) for different worst-case EH constraints at the EH receiver, , where , , and dB. The non-robust AN-aided secure transmission scheme is obtained by solving (9) where , , and , . After obtaining and , if the worst-case EH constraint at the EH receiver is not satisfied, the secrecy rate is 0. If otherwise, the worst-case secrecy rate is computed by using the similar method proposed in Section III. The robust secure transmission scheme without the aid of AN is obtained by solving (19) with . From Fig. 1, it is observed that the proposed robust AN-aided secure transmission scheme has significant performance gain over the “Non-Robust” scheme and the “Robust w/o AN” scheme. It is also found that the performance gap between “Robust, 1-D” and “Robust, 2-D” schemes is narrow. In Fig. 2(a)–(c), we present the percentage of consumed AN power, , of the proposed 1-D robust AN-aided secure transmission scheme with and without the worst-case EH constraint versus the number of antennas at the transmitter, , the number of eavesdroppers, , and , respectively, where is defined as (22)
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without the aid of AN. Future work may be carried out on the robust AN-aided secure transmission scheme with SWIPT in the MIMO channels. APPENDIX PROOF OF PROPOSITION 1 If
is optimal solution to (13), we have (23)
Fig. 1. The average worst-case secrecy rate versus different worst-case EH ; performance comparison of different constraints at the EH receiver, , , , and dB. schemes where
We prove Proposition 1 by reductio ad absurdum. Consider the following two situations. If and , we can always find , , which satisfies (13c)-(13f). Thus, we can find the larger objective value which satisfies (23). If and , , , we can find a solution , which makes the constraint tighter while satisfying and (13c). These contradict that is optimal solution to (13). APPENDIX PROOF OF PROPOSITION 2
Fig. 2. (a) The percentage of consumed AN power, , versus and dB; (b) versus , where and versus , where and .
, where dB; (c)
In Fig. 2(a), we assume that and dB. In Fig. 2(b), we assume that and dB. In Fig. 2(c), we assume that and . From Fig. 2(a), it is found that with the increase of , the percentage of consumed AN power decreases. This is because more transmit antennas allow the transmitter to steer its antenna beam towards the legitimate ID receiver and the EH receiver. Thus, the transmitter requires relatively less aid from AN. From Fig. 2(c), it is found that when dB dB, the percentage of consumed AN power increases while when dB dB, the percentage of consumed AN power decreases. This is because that when is low, the SNRs at ID receiver and eavesdroppers are low. High consumed AN power is not necessary. With the increase of , the percentage of consumed AN power should increase which ensures the secure communications and satisfies the worst-case EH constraint. When is high, to further increase the achievable secrecy rate requires that more power should be allocated to the message-bearing signals.
Assume that the dual variables , are corresponding to the constraints (15)-(18), , , respectively. Taking partial derivative of the Lagrangian dual function of (19) with respect to and applying the KarushKuhn-Tucher (KKT) conditions, we have (24) is the dual variable corresponding to the constraint . It is noted that from KKT conditions. Since the size of and is , we have rank rank . It is noted that if , rank . We prove by reductio ad absurdum. If , the constraint is not active because from (15), is its dual variable. If where is the worst channel uncertainty which minimizes , we can always find a scaler which satisfies . Substituting into , we obtain the lower value than that obtained by . It is contradictory to the assumption that minimizes . Thus, and rank . Furthermore, rank . We have rank . Similarly, rank , . Multiplying both sides of (24) with , we have where
V. CONCLUSIONS In this letter, we propose 2-D and 1-D robust AN-aided secure transmission scheme with SWIPT in the MISO channels. It is shown through computer simulations that the proposed robust AN-aided secure transmission scheme has significant performance gain over the non-robust AN-aided secure transmission scheme and the robust secure transmission scheme
where it is noted that . If rank, since rank . If rank . Thus, we also have rank that
(25) has full , we have rank , it can be verified .
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