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Joint active and passive beamforming optimization for intelligent reflecting surface assisted proactive eavesdropping

2021; Institution of Engineering and Technology; Volume: 15; Issue: 8 Linguagem: Inglês

10.1049/cmu2.12144

1751-8636

Jie Yang, Kaizhi Huang, Xiaoli Sun, Yi Wang,

Advanced Antenna and Metasurface Technologies

IET CommunicationsVolume 15, Issue 8 p. 1085-1095 ORIGINAL RESEARCH PAPEROpen Access Joint active and passive beamforming optimization for intelligent reflecting surface assisted proactive eavesdropping Jie Yang, orcid.org/0000-0002-5848-0954 Information Engineering University, Zhengzhou, ChinaSearch for more papers by this authorKaizhi Huang, Corresponding Author huangkaizhi@tsinghua.org.cn Information Engineering University, Zhengzhou, China Correspondence Kaizhi Huang, Information Engineering University, Zhengzhou, 450002, China. Email: huangkaizhi@tsinghua.org.cnSearch for more papers by this authorXiaoli Sun, Information Engineering University, Zhengzhou, ChinaSearch for more papers by this authorYi Wang, Zhengzhou University of Aeronautics, Zhengzhou, ChinaSearch for more papers by this author Jie Yang, orcid.org/0000-0002-5848-0954 Information Engineering University, Zhengzhou, ChinaSearch for more papers by this authorKaizhi Huang, Corresponding Author huangkaizhi@tsinghua.org.cn Information Engineering University, Zhengzhou, China Correspondence Kaizhi Huang, Information Engineering University, Zhengzhou, 450002, China. Email: huangkaizhi@tsinghua.org.cnSearch for more papers by this authorXiaoli Sun, Information Engineering University, Zhengzhou, ChinaSearch for more papers by this authorYi Wang, Zhengzhou University of Aeronautics, Zhengzhou, ChinaSearch for more papers by this author First published: 13 March 2021 https://doi.org/10.1049/cmu2.12144AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinked InRedditWechat Abstract This paper investigates a three-node proactive eavesdropping system with the aid of intelligent reflecting surface (IRS), where a monitor tries to wiretap information and transmit jamming signals simultaneously to interfere with the suspicious link. The IRS is deployed around the suspicious users to reconstruct propagation channel. Meanwhile, with large low-cost passive reflecting elements, it can provide more spatial degrees of freedom to moderate the link quality. In order to degrade exposure risk of eavesdropping behavior, the authors' objective is to minimize the jamming power by carefully designing jamming beamforming vectors and the phase shifts under the constraint of reliable intercepted information. The formulated problem is non-trivial to solve due to the coupling variables and unit-modulus constraints. Fortunately, by using alternating optimization and successive convex approximation techniques, the original problem is transformed into convex form and a sub-optimal solution is achieved. Numerical results show that the proposed scheme can effectively reduce the jamming power and the number of the jamming antennas over benchmark schemes, which is quite suitable for covertly overhearing signal. 1 INTRODUCTION Recently, the ubiquitous accessibility of wireless communication system has greatly changed our daily life. However, due to the broadcast nature of wireless communication, safeguarding wireless communications system has become an important issue. Physical layer security technology has been ascertained as a promising approach, which could provide lightweight protection compared with traditional encryption technology. In [1], secure communication is studied in multi-user massive multiple-input multiple-output system. Mamaghani et al. [2, 3] investigate a two-way secure communication scheme where legitimate communication parties exchange confidential information via a wireless-powered untrusted relay. In [4], two cooperative unmanned aerial vehicles (UAVs) are exploited to assist secure communication and energy harvesting. In these studies, many designs have been carried out for the purpose that makes the transmitted messages indecipherable to eavesdroppers. While, in some areas, wireless communication may be used for unkind purposes, such as commercial crime. Specially, malicious users can send out secret data via infrastructure-free communication means, such as device-to-device communications, which is hard to access for authorized parties. Therefore, as a new paradigm shift in secure communication, wireless information surveillance is proposed and has garnered increased attention for its capability in dealing with malicious users[5]. Since the monitor is usually located away from the suspicious users, the eavesdropping link is inferior to the communication link and it is hard to decode the suspicious signal clearly for passive eavesdropping. Thus, jamming-assisted and relay-assisted proactive eavesdropping are widely recognized solutions [6-9]. Xu et al. [6] considers a three-node surveillance model, where the monitor moderates the suspicious communication rate by transmitting jamming signal, and derives the optimal jamming strategy for the maximum eavesdropping non-outage probability. In [7], to facilitate covertly distant eavesdropping, multiple intermediate nodes switching between jamming and monitoring mode are introduced and deployed around a pair of suspicious users. And the optimal transmit power allocation and mode selection strategy is developed to achieve the maximum eavesdropping rate. In [8], power splitting technique is utilized by the legitimate monitor and the received signal is split into two parts for information eavesdropping and spoofing relaying, respectively. Then, the relay is designed to act as different roles, including jamming, constructive relaying and so on, to improve eavesdropping performance. Furthermore, UAVs acted as amplify-and-forward (AF) relays with the advantage of less interference air-to-ground relay channels are utilized to improve information surveillance[9]. Despite such striking performance improvements brought by these methods, there exists a risk of exposing eavesdropping behavior. Relatively high jamming power and active relay nodes are easy to be detected by the suspicious users. Therefore, it is worth paying attention to considering how to realize reliable eavesdropping on the basis of concealment. Meanwhile, intelligent reflecting surface (IRS), which can reconstruct wireless propagation environment in real time, has been viewed as a key emerging technology for 6G network[10]. IRS is composed of lots of low-cost reflecting units, and each unit can adjust the reflection coefficient on the incident signal independently whereby the reflected signal can be strengthened or weakened at specified users. Moreover, with the characteristic of low profile and conformal geometry, it is easy to mount on surrounding objects [11]. Thus, as a novel promising solution, IRS has been investigated in many applications such as coverage extension[12], physical layer security[13, 14], energy efficiency improvement[15]. In [16], the author studies the secrecy performance gains of non-orthogonal multiple access system via IRS and derives the closed-form average secrecy capacity expressions. In [17], a novel adverse application of IRS is proposed, where the IRS controlled by malicious users is utilized to sabotage the legitimate communication system without any energy footprint. In fact, IRS is naturally applicable for covertly eavesdropping due to its inherent features of passive enhanced communication and easy-to-deploy. Compared with the existing schemes, more design parameters brought by IRS is more challenge and usually intractable. By far, there are few work considering IRS in proactive eavesdropping system. Motivated by this, we study an IRS-aided three-node surveillance model with the goal to minimize the jamming power. Our main contributions are summarized as follows: (1) An IRS-assisted three-node proactive eavesdropping scheme is proposed. In this model, IRS is deployed covertly around the suspicious users and is remotely controlled by the monitor. The monitor works in full-duplex mode and transmits jamming signals to moderate the suspicious communication rate. Meanwhile, more reflected channel paths are brought by the IRS and the received signals are re-analyzed. The phase shifts of the IRS are properly tuned to adjust both the eavesdropping and communication channel quality. (2) A resource allocation problem aimed at jamming power minimization is established by jointly designing jamming beamforming and the phase shifts. The original problem is non-convex with a complicated form. To tackle it, a sub-optimal iterative algorithm is proposed. First, the problem is decomposed into two sub-problems by applying an alternating optimization algorithm. Then each sub-problem is transformed into convex with successive convex approximation (SCA) and semi-definite relaxation (SDR) techniques. Moreover, the convergence of the algorithm is analyzed. Finally, we prove that when the number of reflection units approaches infinity, the monitor can overhear the signal reliably without transmitting jamming signal. (3) Simulation results show the impact of parameters, such as the location of the monitor, jamming antenna numbers, on system performance. Compared with the scheme without IRS and the scheme with AF relay, our proposed scheme can achieve better performance and dramatically reduce jamming power and jamming antenna numbers. Moreover, when the number of low cost reflecting units increases, the required jamming power can be further cut down, which leads to higher power efficiency and is beneficial for proactive eavesdropping. The remainder of this paper is organized as follows. Section 2 introduces the system model and formulates the optimal problem. Section 3 focuses on the algorithms for the active and passive beamforming design. Sections 4 and 5 give simulation results and conclusions, respectively. 2 SYSTEM MODEL AND PROBLEM FORMULATION 2.1 System model As depicted in Figure 1, we consider a three-node point-to-point surveillance model. The suspicious transmitter Alice exchanges the secret data with the suspicious receiver Bob via device-to-device communications without going through cellular infrastructures. The monitor Mon tries to overhear the suspicious signal. In order to avoid being detected, Mon is deployed far away from Alice, which leads to that the eavesdropping channel condition is worse than the communication channel condition. Moreover, Mon exploits full-duplex technology to simultaneously eavesdrop and jam[18]. Meanwhile, an intelligent reflecting surface, which is composed of N reflecting units, is covertly deployed on surrounding buildings near Bob and is controlled by Mon through a private wireless channel. Alice is equipped with M a antennas, and Bob is deployed with a single antenna. While, Mon is equipped with two kinds of antennas, that is single antenna for receiving and M e antennas for jamming. FIGURE 1Open in figure viewerPowerPoint IRS-assisted proactive eavesdropping model We have considered a block Rician fading environment[6,7], where the channel condition remains unchanged over each transmission block but changes from one block to another. Each communication link is described as follows: the suspicious communication link from Alice to Bob is composed of a direct link and a reflected link, where the direct link is denoted by h a b H and the reflected link from Alice to IRS and IRS to Bob are represented as H a r and h r b H , respectively. The eavesdropping link from Alice to Mon is also consisted of a direct link and a reflected link, where the direct link is denoted by h a e H and the reflected link from Alice to IRS and IRS to Mon are denoted by H a r and h r e H . Mon performs co-channel jamming to Bob while eavesdropping. The jamming link from Mon to Bob is also composed of a direct link and a reflected link, where the direct link is denoted by h e b H , the reflected link from Mon to IRS and from IRS to Bob are represented as H e r and h r b H . Then the phase shift matrix at the IRS is represented as Φ = diag ( e j θ 1 , e j θ 2 , … e j θ N ) , where θ n ∈ [ 0 , 2 π ] represents the phase shift at its nth reflecting element. Compared with the model without IRS, the signal received at each user is the combination of reflected signal and direct signal, which needs to be re-analyzed. The transmitted suspicious signal from Alice is expressed as x = P t w s , (1)where s ∼ C N ( 0 , 1 ) , P t denotes the transmit power, w ∈ C M a × 1 denotes the transmit precoding vector. The received signal at Bob is the combination of the transmitted signal from Alice and the jamming signal from Mon, which can be expressed as y b = P t ( h a b H + h r b H Φ H a r ) w s + ( h e b H + h r b H Φ H e r ) f a + n 1 , (2)where n 1 ∼ C N ( 0 , σ 0 2 ) denotes the additive Gaussian white noise(AWGN) at Bob, a ∼ C N ( 0 , 1 ) denotes the jamming signal from Mon, and f denotes the jamming beamforming vector from Mon. Then the signal-to-interference-plus noise ratio (SINR) at Bob can be given by γ b = P t ( h a b H + h r b H Φ H a r ) w 2 ( h e b H + h r b H Φ H e r ) f 2 + σ 0 2 . (3) Let v H = [ v 1 , v 2 , · · · , v N ] and v n = e j θ n , then we can have h r b H Φ H a r = v H H a r b , h r b H Φ H e r = v H H e r b , (4)where H a r b = diag ( h r b H ) H a r , H e r b = diag ( h r b H ) H e r . By denoting H b 1 = [ H a r b h a b H ] , H b 2 = [ H e r b h e b H ] , v ¯ H = [ v H , 1 ] , the SINR at Bob can be simplified as γ b = P t v ¯ H H b 1 w 2 v ¯ H H b 2 f 2 + σ 0 2 . (5) From (5), we can observe that v ¯ H H b 1 and v ¯ H H b 2 can be seen as the equivalent communication link from Alice to Bob and the equivalent jamming link from Mon to Bob, respectively. Thus, the suspicious communication rate at Bob is written as R b = log 2 ( 1 + γ b ) = log 2 1 + P t v ¯ H H b 1 w 2 v ¯ H H b 2 f 2 + σ 0 2 . (6) Similarly, the received signal at Mon is given by y e = P t ( h a e H + h r e H Φ H a r ) w s + ( ρ h e e H + h r e H Φ H e r ) f a + n 2 , (7)where n 2 ∼ C N ( 0 , σ 0 2 ) denotes the AWGN at Mon. As shown in [19][20], the residual self-interference link is described as ρ h e e H , where h e e H ∈ C 1 × M e is the fading loop channel with complex Gaussian random variables (RV) with zero-mean and variance λ. Thus, the SINR at Mon is derived as γ e = P t ( h a e H + η r e H Φ H a r ) w 2 ( ρ h e e H + h r e H Φ H e r ) f 2 + σ 0 2 . (8) Further, let us define the equivalent eavesdropping link as v ¯ H H e = v ¯ H [ H a r e h a e H ] , the equivalent jamming link as v ¯ H H e 2 = v ¯ H [ H e r e ρ h e e H ] , where H a r e = diag ( h r e H ) H a r , H e r e = diag ( h r e H ) H e r , and then (8) can be transformed into γ e = P t v ¯ H H e w 2 v ¯ H H e 2 f 2 + σ 0 2 . (9) 2.2 Problem formulation According to [21], when the monitor has a better link than the suspicious receiver, the suspicious signal can be decoded with an arbitrarily small error probability, and the effective eavesdropping rate is defined as R e v = log 2 ( 1 + γ b ) . Otherwise, the suspicious signal cannot be decoded without error, and the effective eavesdropping rate is defined as R e v = 0 . In fact, since the monitor is usually located far away from the suspicious users, the eavesdropping link is worse than the communication link with high probability, which results in poor eavesdropping performance. Although jamming-assisted scheme can compensate this gap, excessive jamming power has a negative effect on the concealment of the monitor. Meanwhile, from (5)-(9), one can find that IRS could provide more spatial degrees of freedom with little energy. Naturally, we consider a strategy that Mon works on passive eavesdropping mode and only deploys a large number of reflection units to ensure reliable information reception. And the problem is similar to the adverse application of [13], where the closed-form solution is hard to derive. Fortunately, when the number of reflection units approaches infinity, we have the following proposition. Proposition 1.When Mon works on passive eavesdropping mode and the reflection coefficients at the IRS are adjusted to enhance the received signal at Mon, the average SINR at Mon approaches infinity as the number of reflection units approaches infinity. Proof.see Appendix 8.1. □ Then, based on this simple phase shifts design strategy, since the channel h a b H and h a e H are independent, the phase shifts is random with respect to Bob. And it is easy to know that as the number of reflection units approaches infinity, the SINR at Mon must be higher than that of Bob, which means Mon can overhear signal reliably without transmitting jamming signal. Thus, when the difference between the communication link and the eavesdropping link is small, Mon can deploy passive IRS to reliably overhearing, which is verified in the subsequent simulation. Otherwise, jamming power is still needed since too large units are difficult to be fabricated in practice. Based on above analysis, our goal is to minimize the jamming power by jointly designing the reflection coefficients and jamming beamforming vector while ensuring monitoring effectively. Hence, the optimization problem can be established by min v ¯ H , f Tr ( f H f ) s . t . γ e ≥ γ b v n = 1 , n = 1 , 2 ⋯ N . v N + 1 = 1 (10) From problem (10), channel state information (CSI) is necessary for Mon to obtain the optimal solution. In this model, the CSI of each link can be perceived as follows. First, the CSI of the equivalent links can be acquired similar to that of conventional system without IRS. The equivalent jamming link v ¯ H H b 2 and eavesdropping link v ¯ H H e can be estimated by overhearing the public pilot signal and control signal[22]. The equivalent loop link v ¯ H H e 2 can be estimated beforehand by Mon. The equivalent suspicious communication link v ¯ H H b 1 is relatively difficult to obtain. Fortunately, it could be perceived by listening the channel feedback sent from Bob to Alice[8, 22]. Secondly, to obtain the CSI of the direct link, such as h a e H , h a b H , the reflected link should be estimated first. The estimation approach of the reflected link at the IRS is also the hotspot of current research[23]. Semi-passive IRS and Fully-passive IRS have been recognized two estimation methods. We consider the former approach where there exists some low-power sensing devices integrated into the IRS and the reflected link can be probed by these sensors. How to recover all the CSI of the reflected link can refer to [24, 25], which is beyond the scope of our discussion. As the IRS is controlled by Mon, the phase shifts and the CSI of the reflected link are known by Mon via privacy channel. Therefore, the direct link can be easily derived from formula (4). To study the fundamental performance limit of our proposed scheme, similar to [18], we assume perfect CSI of all links are available at Mon. Furthermore, by substituting (5) and (9) into problem (10), the original problem can be retransformed as: min v ¯ H , f Tr ( f H f ) (11a) s . t . P t v ¯ H H e w 2 v ¯ H H e 2 f 2 + σ 0 2 ≥ P t v ¯ H H b 1 w 2 v ¯ H H b 2 f 2 + σ 0 2 (11b) v n = 1 , n = 1 , 2 ⋯ N (11c) v N + 1 = 1 . (11d) Kindly note that for problem (11), v ¯ H and f are designed by Mon, while transmit beamforming vector w and transmit power P t are set by Alice, rather than Mon. Fortunately, from (11b), variable P t has no effect on the problem. Different from conventional physical secure transmission model, of which Alice jointly designs transmit beamforming vector w and phase shifts v ¯ H to maximize the secrecy rate at Bob [26], suspicious users have no knowledge about the existence of Mon and IRS. However, it can be predicted that the design criterion of vector w by Alice is still to maximize communication rate and then w is set based on the Maximal Ratio Transmission criterion (MRT). The CSI of the suspicious link estimated by Alice is denoted as h Alice − Bob H . From (5), we have h Alice − Bob H = v ¯ 0 H H b 1 , where v ¯ 0 H denotes the phase shifts at the IRS during channel estimation stage. Then we have: w ̂ H = v ¯ 0 H H b 1 v ¯ 0 H H b 1 . (12) It should be emphasized that since Mon has no knowledge about the instantaneous CSI of any links during channel estimation stage, the phase shifts cannot be designed in advance. Thus, at this time, a simple scheme with low energy consumption can be considered that all the reflecting units at the IRS are simply turned off, namely v ¯ 0 H = [ 0 N , 1 ] and h Alice − Bob H = h ab H . Actually, from the follow-up simulation, the performance of random phase shifts is almost equal to the case without IRS. Moreover, in the slow-varying channel, statistical CSI can be long-term observed [27, 28] and the phase shifts v ¯ 0 H during channel estimation stage can be preset based on statistical CSI, which is left for future work. Thus, substituting (12) into the constraint (11b), we have γ 0 v ¯ H H b 2 f 2 + 1 γ 0 v ¯ H H e 2 f 2 + 1 ≥ v ¯ H H b 1 h a b 2 v ¯ H H e h a b 2 , (13)where γ 0 = 1 / σ 0 2 . It is worth noting that problem (11) is non-convex and intractable due to the non-concave fractional constraints, unit-modulus constraints and the coupling variables. In the following section, an efficient algorithm is developed to find a sub-optimal solution. 3 JOINT DESIGN OF BEAMFORMING AND PHASE SHIFTS In this section, alternating optimization is firstly utilized to separate the original problem into two sub-problems. Then, with SDR and SCA techniques, non-convex sub-problem is transformed to convex form approximately, which can be effectively solved. 3.1 Optimizing f for given v ¯ H When v ¯ H is fixed, the right-hand side of (13) is independent of f, which can be regarded as a constant. Let , , F = ff H , and we have Rank ( F ) = 1 . Due to the non-convex constraint of Rank-1, SDR technique is utilized to drop it. Then with the properties of matrix traces, the problem can be equivalently rewritten as: min F Tr ( F ) (14a) s . t . γ 0 Tr ( H ̂ b 2 H F ) + 1 ≥ λ ( γ 0 Tr ( H ̂ e 2 H F ) + 1 ) (14b) F ≥ 0 , (14c)where λ = | v ¯ H H b 1 h a b | 2 | v ¯ H H e h a b | 2 is a constant. Then, We can observe that problem (14) is a convex semi-definite programming (SDP) problem with linear constraints, which can be directly solved via the CVX solver. Since the constraint of Rank-1 is dropped in problem (14), it is necessary to verify the Rank of the obtained solution. Fortunately, we can conclude Proposition 1. Then, the optimal f can be directly recovered based on the eigenvalue decomposition. Proposition 2.The solution of F always satisfies Rank ( F ) ≤ 1 Proof.see Appendix 8.2. □ 3.2 Optimizing v ¯ H for given f When f is fixed, the original problem is transformed into a check feasible problem. The convergence speed is slow when solving the problem directly. Thus, a slack variable α ≥ 0 is integrated into the right-hand side of the constraint (13), and the problem of finding feasible solution v ¯ H is equivalent to finding the maximum α, which can accelerate the convergence[29]. Note that since the constraints (11c) and (11d) are not consistent, which cannot be dealt with in a unified form, we introduce slack variable w ¯ ∈ [ 0 , 2 π ] and denote v ∼ H = e j w ¯ v ¯ H . Then the constraint (11b-11d) can be rewritten as: γ 0 v ∼ H H b 2 f 2 + 1 γ 0 v ∼ H H e 2 f 2 + 1 ≥ 2 α v ∼ H H b 1 h a b 2 v ∼ H H e h a b 2 (15a) v n = 1 , n = 1 , 2 ⋯ N + 1 . (15b) From (15), we can see that the constraint (11d) can be relaxed without effect on the optimal solution. And the original problem can be transformed into finding the optimal v ∼ H . Next, let H ¯ b 2 = H b 2 f , H ¯ ̂ b 2 = H ¯ b 2 H ¯ b 2 H , H ¯ e 2 = H e 2 f , H ¯ ̂ e 2 = H ¯ e 2 H ¯ e 2 H , H ¯ b 1 = H b 1 h a b , H ¯ ̂ b 1 = H ¯ b 1 H ¯ b 1 H , H ¯ e 1 = H e h a b , H ¯ ̂ e 1 = H ¯ e 1 H ¯ e 1 H , V ̂ = v ∼ v ∼ H , and we have Rank ( V ̂ ) = 1 . Similarly, the Rank-1 constraint is firstly discarded. Then with the properties of matrix traces, the original problem can be equivalently given by: max V ̂ α (16a) s . t . γ 0 Tr ( H ¯ ̂ b 2 V ̂ ) + 1 γ 0 Tr ( H ¯ ̂ e 2 V ̂ ) + 1 ≥ 2 α Tr ( V ̂ H ¯ ̂ b 1 ) Tr ( V ̂ H ¯ ̂ e 1 ) (16b) V ̂ n , n = 1 ( n = 1 , 2 ⋯ N + 1 ) , V ̂ ≥ 0 (16c) α ≥ 0 . (16d) Clearly, (16a), (16c) and (16d) are convex constraints while (16b) remains non-convex following from the fact that both sides of the constraint contain fractional operations. To tackle it, slack variables φ, t1, t2, t3, t4 are introduced, and after simple exponential operations, the problem can be equivalently derived as: γ 0 Tr ( H ¯ ̂ b 2 V ̂ ) + 1 ≥ 2 t 1 (17a) γ 0 Tr ( H ¯ ̂ e 2 V ̂ ) + 1 ≤ 2 t 2 (17b) t 1 − t 2 ≥ α + φ (17c) Tr ( V ̂ H ¯ ̂ b 1 ) ≤ 2 t 3 (17d) Tr ( V ̂ H ¯ ̂ e 1 ) ≥ 2 t 4 (17e) 2 t 3 − t 4 ≤ φ . (17f) Further, (17b) and (17d) are still non-convex because both sides of the constraints are convex. To deal with it, SCA algorithm based on first-order Taylor expansion is used to approximate the exponential function of the right-hand side (17b) and (17d) in an iterative manner[30]. At the kth iteration, t 2 [ k ] , t 3 [ k ] are denoted as the value solved at current iteration, and t ¯ 2 [ k ] , t ¯ 3 [ k ] are denoted as a feasible point, then we can have the lower bound of the exponential function: 2 t 2 [ k ] ≥ 2 t ¯ 2 [ k ] [ 1 + ( t 2 [ k ] − t ¯ 2 [ k ] ) ln 2 ] (18a) 2 t 3 [ k ] ≥ 2 t ¯ 3 [ k ] [ 1 + ( t 3 [ k ] − t ¯ 3 [ k ] ) ln 2 ] . (18b) With (18), (17b) and (17d) can be written as the following linear inequality γ 0 Tr ( H ¯ ̂ e 2 V ̂ ( k ) ) + 1 ≤ 2 t ¯ 2 [ k ] [ 1 + ( t 2 [ k ] − t ¯ 2 [ k ] ) ln 2 ] (19a) Tr ( V ̂ ( k ) H ¯ ̂ b 1 ) ≤ 2 t ¯ 3 [ k ] [ 1 + ( t 3 [ k ] − t ¯ 3 [ k ] ) ln 2 ] , (19b)where V ̂ ( k ) is the solution at the kth iteration. Finally, the approximated form of problem (16) is given by problem (20), which is a convex SDP problem and can be solved via the CVX solver. max V ̂ ( k ) α (20a) s . t . ( 16 c ) , ( 16 d ) , ( 17 a ) , ( 17 c ) , ( 17 e ) (20b) ( 17 f ) , ( 19 a ) , ( 19 b ) . (20c) It is worth mentioning that due to relaxing the Rank-1 constraint in (16), there is no guarantee that the Rank of the obtained matrix V ̂ ∗ is one and the vector v ∼ H may not be directly obtained. If the obtained solution is not Rank-1, Gaussian randomization is utilized for recovering vector approximately [13]. Then according to the definition of v ∼ H , we can easily derive the phase shift matrix Φ. Convergence Analysis: We can obtain the approximation with the current solution by iteratively updating until constraints (19a) and (19b) hold with equality, which indicates that problem (20) is optimally solved. Moreover, from (19a) and (19b), the ( k + 1 ) th iteration admits a larger feasible region than the kth iteration, since the ( k + 1 ) th iterative solution cannot be inferior to the kth iterative solution. Furthermore, since the power of jamming signal is limited and v ∼ H obeys unit-modulus constraint, the convergence of SCA-based algorithm can be guaranteed. 3.3 Overall algorithm To summarize, the outline of solving problem (11) is listed in Algorithm 1, where L and K represent the maximum iteration number for problem (11) and problem (20), respectively. ALGORITHM 1. Proposed algorithm for solving problem (11) 1: Initialize the phase shifts at the IRS as v ∼ ( 0