Secure Wireless Transmission for Intelligent Reflecting Surface-Aided Millimeter-Wave Systems

Due to the broadcast nature of millimeter wave (mmWave) communications, physical layer security has always been a fundamental but challenging concern. Fortunately, the recent advance of intelligent reflecting surface (IRS) introduces another dimension for mmWave secure communications by reconfiguring the transmission environments. In this article, we investigate the physical layer security issue for IRS-aided mmWave communications. Specifically, two scenarios are considered, i.e., with and without the knowledge of the eavesdropper’s channel. When the eavesdropper’s channel is known, under the unit modulus constraints at the IRS and the transmit power constraint at the access point (AP), the secrecy rate is maximized by jointly designing the beamforming vector of the AP and the phase shifts of the IRS. We propose an alternating optimization algorithm for improving the secrecy rate to solve the non-convex optimization problem. In particular, the phase shift matrix of the IRS is firstly optimized based on the manifold optimization algorithm. Under the given phase shift matrix of the IRS, the original problem is transformed into a difference of convex (DC) programming problem. To solve this problem, we use the successive convex approximation (SCA)-based method to transform the original problem into a convex approximation problem, and the transmit beamforming vector is obtained by solving the convex approximation problem. In addition, when the eavesdropper’s channel is unknown, under the transmit power constraint and the minimum user rate constraint, an artificial-noise (AN)-aided scheme is proposed to jam the eavesdropper by maximizing the AN power. Numerical results evaluate that the performance of our proposed schemes is better than that of the conventional scheme in terms of secrecy rate. Moreover, simulations also demonstrate that the secrecy behaviors of the IRS-aided mmWave communication system are superior to the mmWave communications without IRS.


I. INTRODUCTION
Intelligent reflecting surface (IRS) has recently attracted a lot of research interests from academia as one candidate technology for the fifth-generation (5G) cellular networks [1], [6], [8]. An IRS consists of many reflecting elements whose reflecting phase shifts can adaptively adjust to change the signal propagation environment for improving the rate of the communication system. For instance, the IRS can adequately reflect the signals sent from the access point (AP) towards desirable directions, to efficiently enhance received signal The associate editor coordinating the review of this manuscript and approving it for publication was Pietro Savazzi . power at intended users and suppress interference at unintended users. Furthermore, the IRS is passive antenna arrays with significantly reduced energy consumption. Based on the above-mentioned advantages of the IRS, IRS technology has been extensively applied in many communication scenarios recently [9]- [15].

A. RELATED WORK
Recently, there is a lot of literature on the design and optimization of IRS-aided wireless systems [1]- [7]. [1] provided an overview of the IRS-aided wireless systems, including the applications, hardware structure, beamforming optimization, channel state information (CSI) acquisition, and network deployment. In [2], to maximize the received signal-to-noise ratio (SNR) for an IRS-aided singleuser multiple-input single-output (MISO) system, the phase shift of IRS and transmit beamforming are jointly optimized. In [3], under minimum user rate constraints, the minimization power problem of IRS-aided MISO systems was investigated.
In [4] and [5], an alternating optimization algorithm for achieving higher energy efficiency and sum-rate was proposed by joint allocating the user's power and designing the phase shift matrix at the IRS. In [6] and [7], compared with the above works, these works took into account the IRS with discrete phase shits instead of the IRS with continuous phase shifts, IRS with discrete phase shifts was investigated for MISO systems with a single user and multiple users. In IRS-aided systems with wireless networks, in addition to sum-rate optimization of systems addressed in above these papers, many other problems have also been investigated in the literature, including information rate maximization in [14], [16]- [21], channel estimation in [15], [36], and positioning in [37], [37], [38]. To summarize, these studies show that the IRS is beneficial for enhancing the performance of various wireless communication systems, e.g., sum-rate, channel estimation accuracy, and energy efficiency. However, to the best of our knowledge, in these works, the optimization problem of the secrecy rate of IRS-aided mmWave communication systems rarely is investigated, which is a vital issue for mmWave communication systems.
• When the channel of the eavesdropper is perfectly known, under the transmit power constraints of the access point (AP) and unit-modulus constraint, our objective is to maximize the secrecy rate. However, the non-convexity of unit-modulus constraints of the IRS renders the considered problem very difficult to solve. As such, we propose an alternating optimization algorithm. Specifically, we design the phase shift matrix of the IRS by resorting to the manifold optimization technique while the transmit beamforming vector is considered as a known parameter. Given the obtained phase shift matrix, the original optimization problem is reformulated as a difference of convex (DC) programming. Then we propose a successive convex approximation (SCA)-based algorithm to solve it.
• When the channel of the eavesdropper is unknown, to jam the eavesdropper, we introduce an artificial noise (AN). Under the transmit power constraints, by maximizing the AN power to restrain the eavesdropper.
• Finally, what is presented by simulation results is that IRS is capable of upgrading the secrecy rate of mmWave greatly, but a point that can not be neglected is the phase shift of the IRS has to be designed reasonably.
Simulation results also show that more transmit power is beneficial to security. Moreover, to realize the full potential of the IRS, it is important to select the IRS location.
Organization: The rest of the paper is organized as follows. We describe the system model and problem formulation in Section II. In Section III, we propose an alternating optimization algorithm for the secure beamforming problem and VOLUME 8, 2020 also give an AN-based scheme for without eavesdropper's channel. In Section IV, numerical simulations are provided to show the algorithm efficiency and the secrecy rate enhancement. Section V concludes this article. Notations: Vectors and matrices are denoted as boldface lowercase, and uppercase letters, respectively. The transpose, conjugate, conjugate transpose, Frobenius norm, and trace of the matrix X are denoted X T , X * , X H , X , and tr(X), respectively. The is Hadamard product. The vec(X) denotes to stack the columns of the matrix X into a single vector. Let Re{a} be the real part of a complex variable a. The positive semidefinite and positive definite of the matrix X are denoted as X 0 and X 0, respectively.

II. SYSTEM MODEL AND PROBLEM FORMULATION
As shown in Fig.1, we consider an IRS-aided mmWave system, which consists of one AP, one IRS, one user and one eavesdropper. The AP and IRS are respectively equipped with N t antennas and N r reflecting elements. The user and eavesdropper are each equipped with a single antenna. In this article, it is assumed that the direct link between the AP and the user or eavesdropper is blocked by obstacles [13], [42]. In addition, we assume that the eavesdropper is active, thus the eavesdropper's channel is known at the AP [41].
Denote the AP-to-IRS mmWave channel, the IRS-to-user mmWave channel, and IRS-to-eavesdropper mmWave channel as G AI ∈ C N r ×N t , h IU ∈ C N r ×1 , and h IE ∈ C N r ×1 , respectively. The received signal at the IRS is expressed as where r ∈ C N r ×1 . s ∈ C 1×1 and w ∈ C N t ×1 denote the transmit data and the transmit beamforming vector at the AP with E[ss H ] = 1; Then, the IRS reflects it with a phase shift matrix = diag(θ ) ∈ C N r ×N r , where θ = [θ 1 , · · · , θ N r ] T ∈ C N r ×1 and θ j = e jφ j , j = 1, · · · , N r with φ j being the reflection phase shift. The received signal at the user can be written as where n is the additive white Gaussian noise (AWGN) and n ∼ CN (0, σ 2 ). Similarly, the received signal at the eavesdropper is expressed as where n e is also the additive white Gaussian noise (AWGN) and n e ∼ CN (0, σ 2 e ). From (2) and (3), the achievable rate at the user is The achievable rate at the eavesdropper is Therefore, the secrecy rate I s is denoted as [22], where [x] + = max(0, x). The transmit power constraint at the AP is where P is the maximum transmit power at the AP. Based on (6) and (7), the secure beamforming optimization problem for the IRS-aided mmWave system with power constraint is formulated as max w, s.t. tr ww H ≤ P, Due to the non-convexity of the unit-modulus of the IRS and the objective function, the problem in (8) is non-convex.

III. SECURE BEAMFORMING FOR IRS-AIDED mmWave SYSTEMS A. JOINT TRANSMIT BEAMFORMING AND PHASE SHIFT OF THE IRS DESIGN WITH FULL EAVESDROPPER's CHANNEL
Since there is no general method to solve the non-convex problem in (8) efficiently and optimally, we develop an alternating optimization algorithm to solve (8) in this article. In the following, an alternating optimization algorithm is proposed to solve this problem. Specifically, first, we optimize the phase shift matrix by using manifold optimization technology. Then the phase shift matrix is fixed, we transform the problem in (8) into a DC programming problem [35]. To solve this problem, an SCA-based algorithm is proposed to obtain the local optimal solution of transmit beamforming of the AP.
For resolving the problem in (8), the auxiliary variables x and v i , i = 1, 2 are introduced and they satisfy the following equalities 192926 VOLUME 8, 2020 Using (9)-(11), the problem in (8) can be rewritten as In problem (12), since − ln(1 + v 1 ) is convex, the objective function is the difference of two convex functions. It is not difficult to find that (12e) is convex. Thus, except constraints (12b)-(12d) and (12f), the problem in (12) has a DC form. Now, we deal with these non-convex constraints (12b)-(12d) and have: Theorem 1: We can equivalently transform constraints (12b), (12c), and (12d) into tr The equations in (15)- (17) can be further transformed into a compact form The proof is given in Appendix A. In (13) and (14), constraints are linear and convex. Based on Theorem 1, it is not difficult to find that constraint (18) is a standard DC form. Then, we replace (12b), (12c), and (12d) with (13), (14), and (18). Problem (12) can be rewritten as a DC programming problem To cope with the problem in (19), an alternating optimization algorithm is proposed for solving problem (19). Next, we firstly use manifold optimization to deal with the constraint condition in (19f).

B. MANIFOLD OPTIMIZATION ALGORITHM
To simplify the expression of the problem in (19), we define the functions as follows Employing the exact penalty method [35], (19) can be rewritten as where λ > 0 is a sufficiently large weight. Since (19f) is non-convex, problem (22) is difficult to solve for traditional methods. Hence, we adopt the alternating optimization algorithm and divide (22) into two subproblems, i.e.
wherex,w,v i ,¯ are fixed values. We use manifold optimization technology to resolve problem (23). Based on the notion of manifold optimization, (23) is denoted as where M is the manifold space defined in the unit-modulus constraints in (23b) and M is expressed as  (29) ; 5 Determine the step size δ (t+1) based on [34] ; 6 Perform gradient descent algorithm over the current tangent space using θ (t) − δ∇ θ (t) f (θ (t) ); 7 Update (t+1) according to (30) ; The optimal solution * .
The principle of manifold optimization method is to operate the gradient descent algorithm in manifold space. It is not difficult to find that gradient descent algorithm on Riemannian manifold is similar to that working in Euclidean space. But manifold optimization needs to calculate the Riemannian gradient as search direction. The Riemannian gradient of (25) at the current point θ is defined as a projection of search direction in Euclidean space onto the tangent space T θ M, which can be expressed as Then, the Euclidean gradient of (25) at θ can be computed by where . Based on the Euclidean gradient, the Riemannian gradient of (25) is expressed as Thus, the current point θ in the tangent space T θ M is updated by θ − δ∇ θ f (θ ), where δ > 0 is the step size. It should be noticed that the update point may leave the manifold space. Thus, a retraction operation is used to make the point stay in the manifold. Finally, the retraction operation is expressed as Via these operations, we can get the solution θ * and * = diag(θ * ). The details are summarised in Algorithm 1. Because Algorithm 1 is a gradient descent algorithm, in each iteration, the (25) is monotonically decreasing. Hence, Algorithm 1 will converge to a stationary point of (25) [34].

C. SCA-BASED OPTIMIZATION ALGORITHM
In this subsection, we optimize the w, x, and v i when = . (24a) is a standard DC programming problem. However, it is still non-convex. To deal with the non-convex objective function, we use the SCA method in [35] to transform Algorithm 2 Proposed SCA-Based Algorithm for Problem (43) 1 i , accuracy 1 . 2 Repeat: 3 Solving the convex problem (43) by using CVX [35] and the optimal solution is w (t+1) , − w (t) ≤ 1 . 7 Output: the optimal solution w * . the non-convex part of the objective function into a convex function, and then iteratively solve the convex approximation problem. In the following, we focus on finding the convex bounds of the concave functions f (v 2 ) and −h(w,¯ , x). We give the first-order Taylor expansion on point (w,x,v 2 ), and they are written as It is not difficult to find that the first-order Taylor expansions of (20) and (21) are the linear function, and they are convex.
Since f (v 2 ) and −h(w,¯ , x) are concave, according to [35], we obtain convex upper bounds of f (v 2 ) and −h(w,¯ , x) as Therefore, we have the following convex approximate problem tr ww H ≤ P s .
Selecting a point that satisfies the constraints of the problem (22), then the proposed SCA algorithm solve problem (43) in each iteration. The proposed SCA algorithm for problem (43) is summarized in Algorithm 2. According to Algorithm 1 and Algorithm 2, the alternating optimization algorithm for problem (19) is summarized in Algorithm 3.

Algorithm 3
Proposed Alternating Optimization Algorithm for Problem (19) 1 2 Repeat: 3 Executive the Algorithm 1 to get * 4 Executive the Algorithm 2 to get w * 5 Until: The objective function value is stable. 6 Output: the optimal solution w * and * .

D. BEAMFORMING AND PHASE SHIFT OPTIMIZATION WITHOUT EAVESDROPPER's CHANNEL
In this subsection, we consider the scenario when the eavesdropper's channel is unknown. Since the eavesdropper's channel is unknown, we cannot optimize the secrecy rate.
To solve this problem, we adopt the AN-aided scheme in [24]. In this scheme, AP transmits AN to jam the eavesdropper. Therefore, the received signal at the IRS is expressed as where z ∈ C N t ×1 is the AN. Then, the received signal at the user is denoted as The received signal at the eavesdropper is expressed as where Z = E[zz H ] denotes the covariance matrix of z. The achievable rate at the user is expressed as The achievable rate at the eavesdropper is given bỹ We hope that the achievable rate of user satisfies the minimum user rate requirement, the power in information transmission is minimized, and more power is used to transmit AN, further confusing potential eavesdropper to optimize security. Therefore, our objective is to maximize AN power under the transmit power constraint and minimum user rate constraint. The problem can be expressed as |θ n | = 1, n = 1, · · · , N r .
The optimization problem in (49) is also a non-convex problem, and the optimal solution is difficult to find. Similar to the problem in (12), The auxiliary variables x, Y , and u are introduced According to Theorem 1, we can equivalently transform (50), (51), and (52) into The third term of left hand side of (54) is convex and the proof is given in Appendix B. We define functionh( , x, u, Y , Z) as Hence, employing the exact penalty method [35], problem (55) can be rewritten as Similarly, we first optimize the phase shift matrix by resolving the following optimization problem The Euclidean gradient of (57) is denoted as  (29). 5 Determine the step size δ (t+1) based on [34]. 6 Perform gradient descent algorithm over the current tangent space using θ (t) − δ∇ θ (t) f (θ (t) ). 7 Update (t+1) according to (30).
10 Output: The optimal solution * . fixed value. Then we can compute the tangent space, Riemannian gradient and retraction according to (29) and (30). The algorithm is summarized in Algorithm 4. In the following we optimize w, Z, x, Y , u by setting =ˆ and the suboptimal problem is finally denoted as whereh(ˆ , x, u, Y , Z|ˆ ,x,û,Ŷ ,Ẑ) is Taylor series expansion and they are given by Then by using the proposed algorithm in the previous section, we can obtain the suboptimal solution of (59). The scheme is summarized in Algorithm 5. The numerical simulations are employed to evaluate the performance of the proposed algorithms in this article.

Repeat:
3 Executive the Algorithm 4 to get * . 4 Executive the Algorithm 5 to get w * . 5 Until: The objective function value is stable. 6 Output: the optimal solution w * and * . respectively, where and 3 are estimation accuracy. According to [34], the computation complexity of solving problem (23) and (57) is I t1 (N 2 r N t +N 2 r +2N r ) and I t3 (N 3 r +N 2 r N t + N 2 t N r +2N r ), respectively, where I t1 and I t3 are the number of iterations of Algorithm 1 and 4. Therefore, the computation complexity of the proposed Algorithm 3 and 6 is O(I t2 ((6N 2 t + 4N 2 r + 2N t N r ) 3.5 log(1/ ) + I t1 (N 2 r N t + N 2 r + 2N r ))) (61) and O(I t4 ((6N 2 t + 4N 2 r + 2N t N r ) 3.5 log(1/ 3 ) + I t3 (N 3 r + N 2 r N t + N 2 t N r + 2N r ))), (62) where I t2 and I t4 are the number of iterations of Algorithm 3 and 6.
According to [43], the algorithm using the manifold optimization is guaranteed to converge to the point where the gradient of the objective function is 0. Therefore, Algorithm 1 and 4 ensure the decrease of the objective function and obtains a local optimal solution in each iteration. In addition, according to [35], the SCA algorithm also can converge to a stable point. Therefore, the alternating optimization algorithm converge.

IV. NUMERICAL RESULTS
In this section, we provide simulation results to validate the effectiveness of the proposed scheme. As shown in Fig.2, the AP, equipped with a ULA of N t = 16 antennas, is located in the center of the IRS-aided mmWave system. The number of reflecting units of IRS N r = 16. The distance between the AP and the obstacle r = 10 m. The distances between the eavesdropper and the UE, the AP and the IRS, the AP and the UE are respectively set to d EI = 20 m, d = 80 m, and d IA = 100 m. The transmit power is set as P = 30 dBm. In our simulations, the IRS-user and IRS-eavesdropper mmWave channel are generated according to the following geometric channel model [18] and where L 1 and L 2 are the number of paths, α l and β l are the complex gain associated with the lth path. According to [44]- [47], φ l ∈ [0, 2π ) and ϕ l ∈ [0, 2π ) are the associated azimuth angle of departure. a I and a E are the array response vectors and they are respectively denoted as and where λ is the wavelength and d = λ 2 denotes the antenna spacing. The complex gains α l and β l are generated according to CN (0, 10 −0.1PL(d) ) [17], where PL(d) = η a + 10η b log 10 (d) + κ is the log-normal shadowing variance, where η a = 72, η b = 2.92, σ κ = 8.7 dB, and κ ∼ CN (0, σ 2 κ ). It is assumed that the IRS with N r reflecting elements is installed on some high-rise buildings around the AP. Therefore, the LoS path is dominant for the AP-IRS mmWave channel and the rank-one channel model is adopted, i.e., where ϑ 1 ∈ [0, 2π ) and ϑ 2 ∈ [0, 2π ) denote the azimuth angle of arrival and elevation angle of arrival associated with the BS-IRS path, respectively. ε ∈ [0, 2π ) is the associated angle of departure [47]. a r (ϑ 1 , ϑ 2 ) and a t (ε) represent the receive and transmit array response vectors, respectively, where a r (ϑ 1 , ϑ 2 ) is denoted as where ⊗ stands for the Kronecker product. a x (ϑ 2 ) and a y (ϑ 1 ) are expressed as where M x and M y denote denote the number of antennas in the horizontal direction and the number of antennas in the vertical direction, respectively. a t (ε) is denoted as Similarly, the complex gain γ is generated according to CN (0, 10 −0.1PL(d) ), where η a = 61.4, η b = 2, and σ κ = 5.8 dB. The noise power σ 2 e = σ 2 = −110 dBm and the transmit power P = 1 W.
The convergence performance of the proposed algorithm is investigated. The iterations of the manifold optimization algorithm are termed as inner-layer iterations, and the iterations of the SCA-based algorithm are termed as the outer-layer iteration. In Fig. 3 and Fig. 4, we investigate the convergence behavior of the proposed algorithm with  versus the number of reflection elements N r . It is observed that the average secrecy rate of the manifold optimization algorithm converges with the 25 iterations. The numerical results demonstrate the efficiency of the proposed algorithm. In addition, we find the slower convergence speed with more reflection elements. The reason is that more optimal variables bring heavier computation. Fig. 5 and Fig. 6 show that the convergence performance of the SCA-based algorithm used for optimizing transmit beamforming. With the increase in the number of iterations, the secrecy rate increases and finally converges to a stable value. Similar to the inner-layer iteration, the convergence behavior of the outer-layer iteration can also get a similar conclusion, and the secrecy rate increases with the increase of the number of IRS phase shifts. In addition, it is not difficult to find that the SCA-based algorithm converges when the number of in the inner-layer iterations is about 30.
In Fig. 7 and Fig. 8, we evaluate the average secrecy rate as a function of the transmit power for various P and  UE-to-eavesdropper distance d when the eavesdropper's CSI is unknown. In Fig. 7 and Fig. 8, with the decrease of P and the increase of distance d, the average secrecy rate increases. The main reason is that the more transmit power P will be used to suppress the eavesdropper. In addition, Fig. 7 and Fig. 8 also show that with the increase of the number of IRS phase shifts, the secrecy rate increases.
In Fig. 9, we evaluate the average secrecy rate as a function of the horizontal distance between the AP and user when the eavesdropper's CSI is unknown. These results demonstrate that the average secrecy rate gradually degrades with the decrease of the distance. It is because of high propagation loss caused by transmission between AP and the user that the reflected gain of IRS decreases a lot,which means only when the user adequately approaches IRS shall the reflected gain of IRS be easier to be utilized. Also, we observe that the proposed scheme outperforms the conventional algorithms. The reason is that, for the SCA scheme, the optimal objective is closer to the performance upper bound than others.  In addition, we see that the proposed IRS-aided mmWave system can achieve a higher average secrecy rate than the conventional mmWave system without IRS. This is because the phase shifts of the IRS are properly designed to enhance the received signal and suppress the eavesdropper. Finally, we find that the secrecy rate obviously increases with N r . The result further confirms that more security improvement can be achieved by using a large IRS with more reflect elements.
In Fig.10, we investigate the transmit power on secrecy outage probability. Under different the number of phase shift of the IRS is plotted. The predefined threshold of achievable rate at the destination is γ = 4 bps/Hz. In Fig. 10, it is observed that when γ = 4 bps/Hz is fixed, as the increase of P, the secrecy outage probability of proposed scheme increases. This is because more power is allocated for AN from AP which reduces outage probability. It is not difficult to find when the transmit power is enough and the N r is big enough, the proposed scheme can perfectly suppress the eavesdropper. The simulation results demonstrate that the IRS can get a good effect on achieving secrecy communication as well.  In Fig.11, we investigate the performance gap between the convex approximation problem in (43) and the original problem in (26). From Fig.11, It is not difficult to find that although we use a series of approximation operations to deal with the original problem, the performance gap between the convex approximation and original problem is small enough. The result demonstrates that the proposed approximation algorithm can achieve very good effect on secrecy communication.

V. CONCLUSION
We proposed alternating optimization algorithms for the IRS-aided mmWave system, i.e., the SCA and manifold optimization-based alternating optimization algorithm in this article. When the eavesdropper's channel is known, we transformed the original non-convex secrecy rate optimization problem into two sub-problems. First, we designed the phase shift of the IRS based on the manifold optimization algorithm. Then, given the phase shift matrix, we obtain the transmit beamforming of AP by exploiting the SCA algorithm. When the eavesdropper's channel is unknown, since the secrecy rate can not be obtained, we use the AN-aided scheme by maximizing the AN power to suppress the eavesdropper under the minimum user rate requirement and transmit power constraints. Similarly, to resolve the AN power optimization problem, we proposed an SCA and manifold optimization-based alternating optimization algorithm. Finally, we analyzed the computation complexity of the proposed algorithms. Simulation results have shown that the IRS can improve the secrecy rate and proposed alternating optimization algorithms achieve higher secrecy rate than the conventional algorithms.