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Low-complexity Joint Phase Adjustment and Receive Beamforming for Directional Modulation Networks via IRS

Intelligent reflecting surface (IRS) is a revolutionary and low-cost technology for boosting the spectrum and energy efficiencies in future wireless communication network. In order to create controllable multipath transmission in the conventional line-of-sight (LOS) wireless communication environment, an IRS-aided directional modulation (DM) network is considered. In this paper, to improve the transmission security of the system and maximize the receive power sum (Max-RPS), two alternately optimizing schemes of jointly designing receive beamforming (RBF) vectors and IRS phase shift matrix (PSM) are proposed: Max-RPS using general alternating optimization (Max-RPS-GAO) algorithm and Max-RPS using zero-forcing (Max-RPS-ZF) algorithm. Simulation results show that, compared with the no-IRS-assisted scheme and the no-PSM optimization scheme, the proposed IRS-assisted Max-RPS-GAO method and Max-RPS-ZF method can significantly improve the secrecy rate (SR) performance of the DM system. Moreover, compared with the Max-RPS-GAO method, the proposed Max-RPS-ZF method has a faster convergence speed and a certain lower computational complexity.

secure communication in the presence of eavesdroppers [12]. Directional modulation (DM), as an advanced and promising PLS communications technique, has been regarded as a useful method for fifth generation (5G) millimeter-wave wireless communications [13], [14]. DM employs signal processing technologies like beamforming and artificial noise (AN) in radio frequency frontend or baseband, so that the signal in the desired direction can be recovered as fully as possible, while the signal constellation diagram in the undesired direction is distorted [15]- [17].
In [18], a DM scheme using the phased arrays to generate modulation was presented, and the secure transmission was achieved since the signal was direction-dependent and purposely distorted in the undesired directions. In [19], a robust synthesis method for multi-beam DM in broadcasting systems was proposed, a robust maximum signal-to-leakage-noise ratio and maximum the signal-to-AN ratio scheme for the desired and eavesdroppers directions were presented, and an obvious bit error rate (BER) improvement over the existing orthogonal projection method along the desired direction for a given signal-to-noise ratio (SNR) was achieved. The authors in [20] developed a multi-carrier based DM framework using antenna arrays, which achieved simultaneous data transmission over multiple frequencies, and a higher data rate was achieved. In [21], the impact of imperfect angle estimation on spatial and directional modulation system was investigated, with the help of the union bound and statistics theory, the average BERs for the legitimate user and eavesdropper were derived. A scenario for DM network with a full-duplex malicious attacker was considered in [22], and three receive beamforming methods were proposed for enhancing the security performance. In [23], the authors investigated the performance of a hybrid analog and digital DM with mixed phase shifters. The closed-form expressions of signal-to-interference-plus-noise ratio, secrecy rate (SR), and BER were derived based on the law of large numbers.
The rapid development of wireless networks will lead to serious energy consumption. Different from the relay [24]- [26], intelligent reflecting surface (IRS) has been considered to be a promising green and cost-effective solution to improve the performance of wireless communication in recent years [27]- [31]. The IRS can change the phase shift of the incident electromagnetic wave, thereby intelligently reconfiguring the signal propagation environment, enhancing the power of the required received signal or suppressing interference signals [32]. A challenging scenario was considered in [33], where the eavesdropping channel was stronger than the legitimate user channel and they were highly correlated. The access point (AP) transmit beamforming and IRS reflect beamforming were jointly designed to maximize the SR. In [34], an IRS-aided single-cell wireless system was investigated. To minimize the total transmit power at the AP, based on semidefinite relaxation and alternating optimization methods, efficient schemes were proposed to make a tradeoff between system performance and computational complexity. The authors in [35] employed an IRS at the cell boundary to enhance the cell-edge user performance in multi-cell communication systems, in order to maximize the weighted sum rate of all users, the block coordinate descent algorithm was proposed for alternately optimizing the transmit precoding matrices at the base stations (BSs) and the passive beamforming at the IRS. In [36], an IRS-aided decode-and-forward relay network was proposed, three high-performance beamforming methods were designed to maximize receive power. In [37], the authors investigated an IRS aided millimeter wave (mmWave) communication system using hybrid precoding at the BS. Based on the rank-one property of mmWave channels, the closed-form solution of the approximated maximum received power of user was derived.
To overcome the limitation that only one bit stream can be transmitted between the BS and user in the conventional DM networks and create controllable multipath transmission in the line-of-sight (LOS) scenario, employing IRS in DM network has been considered. An IRS-assisted DM system was proposed in [38] to utilize the multipath propagation environment for enhancing the PLS, and the closed-form expression for the SR was derived. The authors in [39] considered a double-IRS-aided two-way DM network, two transmit beamforming methods were proposed to enhance the secrecy sum rate (SSR), and an effective power allocation scheme was designed to maximize the SSR performance. In [40], in order to maximize the SR performance of IRS-assisted DM system, two alternating iterative methods, called general alternating iterative and null-space projection, were proposed. The former was of high-performance and the latter was of low-complexity.
However, the authors in [40] aimed to maximize SR of the system by designing the transmit beamforming at the transmitter and the phase shift matrix (PSM) at the IRS, without considering the receive beamforming. Therefore, in this paper, the design of the receive beamforming is focused on and two receive beamforming (RBF) methods are proposed to improve the SR performance by taking the phase optimization of IRS into account. The main contributions of this paper are summarized as follows: 1) To enhance the SR performance in the traditional DM networks, an IRS-aided DM system is considered. To improve the transmission security of the system and reduce the detection complexity of receiver, a general alternating optimization (GAO) of maximizing the receive power sum (Max-RPS) algorithm, called Max-RPS-GAO, is proposed firstly to attain two RBF vectors and the PSM of IRS by making use of the Rayleigh-Ritz theorem and derivative operation. Its basic idea is to alternatively optimize the IRS PSM and the two RBF vectors. 2) To receive confidential messages (CMs) from the direct path and the IRS reflected path independently, a Max-RPS using zero-forcing (Max-RPS-ZF) method is proposed. Here, the first RBF vector forces the signal directly from Alice to zero, and the second one forces the signal from IRS to zero. Simulation results show that, compared with the no-IRS-assisted scheme and no-PSM optimization scheme, the proposed IRS-assisted Max-RPS-GAO algorithm and Max-RPS-ZF algorithm can improve the SR performance of the DM system. Furthermore, compared to the the Max-RPS-GAO, the proposed Max-RPS-ZF algorithm converges faster with a lower computational complexity. The remainder of this paper is organized as follows. Section II describes the IRS-based DM system model. The Max-RPS-GAO and Max-RPS-ZF methods are proposed in Section III and Section IV, respectively. Section V presents the simulation results and related analysis. Finally, we draw conclusions in Section VI.
Notations: throughout this paper, scalar, vector, and matrix are denoted by letters of lower case, bold lower case, and bold upper case, respectively. Symbols (·) T , (·) H , (·) −1 , (·) † , and det{·} are transpose, conjugate transpose, inverse, pseudoinverse, and matrix determinant, respectively. The notation I N denotes the N ×N identity matrix. The sign 0 N ×M represents the N × M matrix of all zeros. As shown in Fig. 1, an IRS-aided DM communication network is considered in this paper, where the transmitter (Alice) is equipped with N A antennas, IRS is equipped with M low-cost passive reflecting elements, legitimate user (Bob) and eavesdropper (Eve) are equipped with N B and N E antennas, respectively. In the following, it is assumed that the signals reflected more than once by the IRS are omitted due to the significant path loss [34], and the channels from Alice to IRS, Alice to Bob, Alice to Eve, IRS to Bob, and IRS to Eve are the LOS channels.

A. System Model
The transmit baseband signal is where P s is the total transmit power, β 1 , β 2 and β 3 denote the power allocation parameters of CMs and AN, respectively, and represent the beamforming vector of forcing the two CMs to the desired user Bob, where v H 1 v 1 = 1 and v H 2 v 2 = 1. P AN denotes the projection matrix for controlling the direction of AN. x 1 and x 2 are CMs which satisfy E x 1 2 = 1 and E x 2 2 = 1. z represents the AN vector with complex Gaussian distribution, i.e., z ∼ CN (0, I NA ).
The received signal at Bob is given by where u Bi ∈ C NB ×1 represents the receive beamforming vec- denotes the complex additive white Gaussian noise (AWGN) at Bob. g AIB = g AI g IB denotes the equivalent path loss coefficient of Alice-to-IRS channel and IRS-to-Bob channel, and g AB is the path loss coefficient between Alice and Bob. The normalized steering vector is given by where θ is the direction angle of arrival or departure, d denotes the antenna spacing, n represents the index of antenna, and λ is the wavelength. Similarly, the received signal at Eve is given by where u Ei ∈ C NE ×1 is the receive beamforming vector, represents the Alice-to-Eve channel, and n E ∼ CN (0, σ 2 E I NE ) denotes AWGN at Eve. g AIE = g AI g IE represents the equivalent path loss coefficient of Alice-to-IRS channel and IRS-to-Eve channel, and g AE denotes the path loss coefficient of Alice-to-Eve channel. In what follows, we assume that σ 2 B = σ 2 E = σ 2 . Assuming that the AN is only transmitted to Eve for interference, then P AN should satisfy Let us define a large virtual CM channel as follows then P AN can be casted as

B. Problem Formulation
Since the channels from Alice to IRS, Alice to Bob, and IRS to Bob are LOS channels, we have rank(H AI )=1 and rank(H AB )=1. This means that rank(H CM ) in (7) is equal to or smaller than 2. There are at least N A − 2 degrees of freedom for AN projection matrix P AN .
In this case, substituting (8) back into (2) and (5), we can obtain and Let us define and Then the received signal in (9) can be rewritten as Then, the achievable rate at Bob is given by Similarly, we define and Then (10) can be recasted as The achievable rate at Eve is The achievable SR R s can be given by Then, the SR optimization problem is given as follows For simplification of the objective function and computational convenience, we convert the SR optimization problem in (20) to the optimization problem of Max-RPS at Bob as follows max uB1,uB2,Θ Solving this problem is a challenge since the unit modulus constraint is difficult to handle. In this case, two optimal alternating methods are proposed to design the receive beamforming vectors and IRS PSM.

III. PROPOSED MAX-RPS-GAO SCHEME
In this section, the transmit beamforming vectors are designed firstly. Then, we will propose a GAO-based Max-RPS method to obtain the confidential message RBF vectors u B1 , u B2 and IRS PSM Θ by alternately optimizing one and fixing another.

A. Design of the transmit beamforming vectors
Firstly, to fix CM precoding vectors v 1 and v 2 , channel matrix H CM in (7) is first decomposed as the singular-value decomposition (SVD) where U and V are unitary matrices, and Σ CM is a matrix containing the singular values of H CM and along its main diagonal. Let us define the transmit beamforming vector v 1 = v 1 , v 2 = v 2 , where v 1 and v 2 can be obtained from the eigenvectors corresponding to the first two largest eigenvalues in Σ CM , respectively.

B. Optimize RBF vectors u B1 and u B2 given IRS PSM Θ
Let us define a large virtual receive channel as follows To obtain the initial values u where U and V are unitary matrices, and Σ BR represents a matrix containing the singular values of H BR and along its main diagonal. We define the RBF u (0) Bi = u i , where u i can be derived from the eigenvectors corresponding to the first two largest eigenvalues in Σ BR .
To simplify the expression of RPS related to the receive beamforming vectors, we regard Θ as a given constant matrix, and the optimization problem of Max-RPS at Bob related to RBF u B1 can be simplified to According to the Rayleigh-Ritz theorem [41], the optimal u B1 can be obtained from the eigenvector corresponding to the largest eigenvalue of the matrix β 1 P s H B v 1 v H 1 H H B . Similarly, given the determined or known u B1 and Θ, the subproblem to optimize u B2 can be expressed as follows: In accordance with the Rayleigh-Ritz theorem, the optimal u B2 can be obtained from the eigenvector corresponding to the largest eigenvalue of the matrix

C. Optimize IRS PSM Θ given the RBF vectors u B1 and u B2
To simplify the expression of RPS in this subsection, we regard u B1 and u B2 as the given constant vectors and define the IRS phase-shift vector θ containing all the elements on the diagonal of Θ, i.e., where θ = e jϕ1 , · · · , e jϕi , · · · , e jϕM T .
Letting θ i = e jϕi be the i-th element of θ, the IRS phase-shift vector θ should satisfy In what follows, let us define where (a) holds due to the fact that diag{a}b = diag{b}a. Then, the optimization problem of Max-RPS at Bob related to θ can be expressed as follows The objective function in (36) can be rewritten as To obtain the optimal IRS phase-shift vector, we need to compute the derivative of f (θ) with respect to θ, which yields

D. Overall Algorithm
So far, we have completed the design of RBF vectors and PSM. The iterative idea of the proposed Max-RPS-GAO algorithm is summarized as follows: given a fixed IRS PSM Θ, the corresponding RBF vectors can be computed in a closedform expression iteratively; given the RBF vectors u B1 and u B2 , θ can be determined by (39) in a closed-form expression; reform θ = exp{j∠(θ)}, Θ = diag{θ}. The alternative iteration process among u B1 , u B2 , and Θ is repeated until the termination condition is met, i.e., R float-point operations (FLOPs), where D denotes the maximum number of alternating iteration.

IV. PROPOSED LOW-COMPLEXITY MAX-RPS-ZF SCHEME
In this section, a maximizing RPS alternate optimization method is proposed to reduce computational complexity. In accordance with the zero-forcing principle, the receive beamforming vectors u B1 and u B2 can be determined by which means that the RBF vector u B1 is only used to receive the CM reflected from the IRS, and u B2 is used to receive the CM through the direct path. The received signals at Bob and Eve are and respectively. Then, the optimization problem of Max-RPS in (21) can be casted as follows max uB1,uB2,Θ In what follows, we consider to optimal RBF vectors and IRS PSM by alternately calculating u B1 , u B2 , and Θ.
A. Optimize RBF vectors u B1 and u B2 given IRS PSM Θ In this section, we regard Θ as a given constant matrix, and the optimization problem of Max-RPS at Bob related to RBF u B1 can be simplified to Based on the Rayleigh-Ritz theorem, the optimal RBF vector u B1 can be derived from the eigenvector corresponding to the largest eigenvalue of the matrix Similarly, given the determined or known u B1 and Θ, the subproblem to optimize u B2 can be expressed as follows In accordance with the Rayleigh-Ritz theorem, the optimal u B2 can be obtained from the eigenvector corresponding to the largest eigenvalue of the matrix B. Optimize IRS PSM Θ given the RBF vectors u B1 and u B2 Since the second item of the objective function in (47) are independent of θ, the subproblem to optimize θ can be expressed as follows According to the Rayleigh-Ritz theorem, the optimal θ can be derived from the eigenvector corresponding to the largest eigenvalue of the matrix w hAIB1 w H hAIB1 .

C. Overall Algorithm
First, we can obtain a new objective function according to the zero-forcing criterion. Then, fix IRS PSM Θ and use the Rayleigh-Ritz theorem to obtain the RBF vectors u B1 , u B2 . In the following, fix the u B1 and u B2 , convert the objective variable Θ into the phase shift vector θ, and obtain the optimal θ by Rayleigh-Ritz theorem. Since the IRS phase-shift restriction of (29), we reform θ = exp{j∠(θ)}, Θ = diag{θ}. Finally, loop the above steps, and solve u B1 , u B2 and Θ alternately until the termination condition is satisfied.
The computational complexity of the proposed Max-RPS-ZF algorithm is FLOPs, where L denotes the maximum number of alternating iteration.

V. SIMULATION RESULTS
In this section, simulations are presented to evaluate the performance of the proposed two schemes. System default parameters are set as follows: The distances of Alice-to-IRS, Alice-to-Bob, and Alice-to-Eve are set as d AI = 10 m, d AB = 50 m, and d AE = 50 m, respectively. The angles of departure (AoDs) of each channel are set as θ t AI = 5π/36, θ t AB = 11π/36, and θ t AE = π/3, respectively. Since the AoD and distance of each channel are given, the channel state information (CSI) of each channel in IRS-aided DM system can be determined.
In what follows, there are two benchmark schemes used to compare with our proposed methods: 1) Random Phase: The phase for each reflection element of IRS is uniformly and independently generated from [0, 2π).
2) No-IRS: We assume that the IRS related channel matrices are zero matrices, i.e., H AI = 0, H IB = 0, and H IE = 0.   Fig. 3 demonstrates the curves of SR versus transmit power P s of the two proposed methods and two benchmark schemes. It can be seen from this figure that the SR of four schemes increases gradually with the increases of transmit power, and the SR performances of the proposed Max-RPS-GAO and Max-RPS-ZF methods are approximately double that of the no-IRS and random phase schemes regardless of the transmit power. Moreover, the difference of the SR between the no-IRS scheme and the random phase scheme is negligible. This implies that optimizing the IRS phase shift can bring a significant performance improvement.   It can be seen from the figure that in low SNR region, the difference in SR performance achieved between the Max-RPS-GAO and Max-RPS-ZF algorithms is trivial. However, the difference in SR performance between the two proposed algorithms gradually increases with the increase in SNR, and the difference of SR is about 2 bits/s/Hz when the SNR is equal to 20dB. Fig. 6 shows the SR versus the azimuth angle θ t AE of Eve, where θ t AE changes from 0 to 2π, θ t AI = π/12, and d AB = d AE = 100m. Since the transmitter and receiver are both linear arrays, the SR performance of θ t AE ∈ (π, 2π) and θ t AE ∈ (0, π) are almost symmetrical to each other. Observing Fig. 6, once Eve and Bob have the same direction angle, i.e., θ t AE = θ t AB = 11π/36, the SR performance of the four schemes will all decline sharply. This is because Eve is located on the direct path from the Alice to Bob, enabling Eve to eavesdrop on CMs to the greatest extent. Nevertheless, when θ t AE = θ t AB , the proposed Max-RPS-GAO method still obtains the best SR performance. VI. CONCLUSION In this paper, we have proposed two alternating iteration optimization schemes, called Max-RPS-GAO algorithm and Max-RPS-ZF algorithm, designing the RBF vectors and PSM in an IRS-aided DM network. The proposed Max-RPS-ZF method uses the zero-forcing criterion to separate two RBF vectors and IRS PSM, such that the two-path CMs from the direct path and the IRS reflected path are independently recovered. Simulation results showed that, compared with the no-IRS-assisted scheme and the no-PSM optimization scheme, the proposed Max-RPS-GAO and Max-RPS-ZF method can significantly improve the SR performance of the DM system as the number of IRS phase shift elements tends to large scale. Compared to the the Max-RPS-GAO, a faster convergence speed and a lower computational complexity can be achieved by the proposed Max-RPS-ZF method. The proposed methods may be applied to the future wireless networks like unmanned aerial vehicle network, satellite communications, vehicle-toeverything, even sixth generation.