Due to the broadcast nature of the wireless media, wireless communications are vulnerable to eavesdropping. In order to provide the wireless communications with sound and solid security, physical layer security based technologies, such as the artificial noise (AN), cooperative jamming (CJ) and friendly jamming, have been studied for the recent years [1, 2]. However, these techniques only focus on the signal processing at the transceiver to adaptive the changes of the wireless environments, but cannot eliminate the negative effects caused by the uncontrollable electromagnetic wave propagation environments [3, 4]. Meanwhile, recently, a new technology following the development of the Micro-Electro-Mechanical Systems (MEMS) named as intelligent reflecting surfaces (IRS) has been proposed, which can reconfigure the wireless propagation environment via software-controlled reflection [3, 4, 5] and shows tremendous potentials in enhancing the wireless communication performance, such as the transmission rate and security, with low cost and significant performance gain, and has received considerable attentions.
One the one hand, for the IRS assisted wireless communications, in , the problem of jointly optimizing the access point (AP) active beamforming and IRS passive beamforming with AP transmission power constraint to maximize the received signal power for one pair of transceivers was discussed. Based on the semidefinite relaxation and the alternate optimization, both the centralized algorithm and distributed algorithm were proposed therein. The work 
extended the previous work to the multi-users scenario but with the individual signal-to-noise ratio (SNR) constraints, where the joint optimization of the AP active beamforming and IRS passive beamforming was discussed to minimize the total AP transmission power, and two suboptimal algorithms with different performance-complexity tradeoff were presented. Huang et al. considered the IRS-based multiple-input single-output (MISO) downlink multi-user communications for an outdoor environment, where studied optimizing the base station (BS) transmission power and IRS phase shift with BS transmission power constraint and user signal-to-interference-and-noise-ratio (SINR) constraint to maximize sum system rate. Since the formulated resource allocation problem is non-convex, Majorization-Minimization (MM) and alternating optimization (AO) was jointly used, and the convergence of this algorithm was analyzed. Different from the continuous phase shift assumption of the IRS reflecting elements in existing studies,  considered that each IRS reflecting element can only achieve discrete phase shift and the joint optimization of the multi-antenna AP beamforming and IRS discrete phase shift was discussed under the same scenario as . Then the performance loss caused by the IRS discrete phase shift was quantitatively analyzed via comparing with the IRS continuous phase shift. It is surprised that, the results have shown that as the number of IRS reflecting elements approaches infinity, the system can obtain the same square power gain as IRS with continuous phase shift, even based on 1-bit discrete phase shift. Furthermore,  and  discussed the joint AP power allocation and IRS phase-shift optimization to maximize system energy and spectrum efficiency, where the user has a minimum transmission rate constraint and the AP has a total transmit power constraint. Due to the presented problem is non-convex, the gradient descent based AP power allocation algorithm and fractional programming (FP) based IRS phase shift algorithm were proposed therein. For the IRS assisted wireless communication system, Han and Tang et al.  analyzed and obtained a compact approximation of system ergodic capacity and then, based on statistical channel information and approximate traversal capacity, the optimal IRS phase shift was proved. The authors also derived the required quantized bits of the IRS discrete phase shift system to obtain an acceptable ergodic capacity degradation. In 
, a new IRS hardware architecture was presented and then, based on compressed sensing and deep learning, two reflection beamforming methods were proposed with different algorithm complexity and channel estimation training overhead. Similar to, Huang and Debbah et al.  proposed a deep learning based algorithm to maximize the received signal strength for IRS-assisted indoor wireless communication environment. Some recently studies about the IRS assisted wireless communications could be found in [15, 16, 17, 18], and they were focused on the IRS assisted millimeter band or non-orthogonal multiple access (NOMA) based wireless communications.
On the other hand, for the IRS assisted secure wireless communications, in , the authors studied the problem in jointly optimizing the beamforming at the transmitter and the IRS phase shifts to maximize the system secrecy rate, based on the block coordinate descent (BCD) and the MM techniques, two suboptimal algorithms were proposed to solve the resulted non-convex optimization problem for small- and large-scale IRS, respectively. In , Chen and Liang studied the minimum-secrecy-rate maximizing problem for a downlink MISO broadcast system, based on the AO and the path-following (PF) algorithm, an iterative algorithm was proposed for the joint optimization problem. In addition, the authors also extended the proposed approach to the case with discrete reflecting coefficients at the IRS. To maximize the MISO system secrecy rate subject to the source transmit power constraint and the unit modulus constraints imposed on the phase shifts at the IRS,  proposed an AO algorithm for the scenario that the eavesdropper is configured with single antenna, then the study was extended to the scenario where the eavesdropper is equipped with multiple antennas. 
investigated the secure transmission framework with an IRS to minimize the system energy consumption in cases of rank-one and full-rank AP-IRS links. In particular, since the beamforming vector and phase shift design are independent in the rank-one channel model, thus a closed-form expression of beamforming vector was derived. However, since beamforming and phase shift depend on each other in the full-rank model, then an eigenvalue-based algorithm for conventional wiretap channel was used to obtain beamforming vector. Different from[19, 20, 21, 22],  considered the scenario that the eavesdropping channel is stronger than the legitimate channel and they are also highly correlated in space, then to maximize the secrecy rate of the legitimate communication link, an algorithm based on the AO and semidefinite relaxation was proposed. Moreover, in  and , for the IRS assisted MISO secure communications with AN transmission at the transmitter, an alternate optimization algorithm to jointly optimize active beamforming, AN interference vector and reflection beamforming with the goal of maximizing system secrecy rate was presented. The difference between these two papers is that,  focused on the scenario with a single legitimate user and multiple eavesdroppers, while  considered the scenario with multiple legitimate users but single eavesdropper.
Although lots of research works have been done for the IRS assisted secure communications, they all have assumed that the legitimate receiver is equipped with only one antenna [19, 20, 21, 22, 23, 24, 25]. However, in order to further improve the communication performance of the mobile users in the next generation wireless local networks (WLANs) such as the IEEE 802.11ax, or the fifth generation (5G) mobile communication networks, multi-antenna enabled mobile device designs have been widely adopted in the current mobile terminals, such as the Phones, laptops and the tablets. Therefore, it is necessary to study the IRS assisted secure communications with multiple-transmit and multiple-receive antennas enabled networks. In this paper, IRS assisted secure communications with multiple-transmit and multiple-receive antennas are studied, where, an AP equipped with multiple antennas has the secure communications demands with a multiple-antennas enabled legitimate user in the downlink at the present of an eavesdropper configured with multiple antennas, referring to it as the IRS assisted multi-input, multi-output, multi-eavesdropper (IRS-MIMOME) system. Particularly, we discuss the joint optimization of the transmit covariance matrix at the AP and the reflecting coefficients at the IRS to maximize the secrecy rate for the IRS-MIMOME system, with two different assumptions on the phase shifting capabilities at the IRS, i.e., the IRS has the continuous reflecting coefficients and the IRS has the discrete reflecting coefficients. For the former case, due to the non-convexity of the formulated problem, an AO based algorithm is proposed, i.e., for given the reflecting coefficients at the IRS, the successive convex approximation (SCA)-based algorithm is used to solve the transmit covariance matrix optimization, while given the transmit covariance matrix at the AP, alternative optimization is used again in the individually optimizing of each reflecting coefficient at the IRS with fixing the other reflecting coefficients. For the individual reflecting coefficient optimization, the close-form or an interval of the optimal solution is provided. Then, the overall algorithm was extended to the discrete reflecting coefficient model at the IRS. Finally, some numerical simulations have been done to demonstrate the performance of the proposed algorithms.
The rest parts of this paper are organized as follows. In Section II, the system model and the considered optimization problem are presented. In Section III, we discuss and solve the formulated optimization problem, and an AO based algorithm is proposed. The simulation results are presented in Section IV and then we conclude this paper.
Notation: We use uppercase boldface letters for matrices and lowercase boldface letters for vectors. , , and denote the transpose, conjugate, and conjugate transpose, respectively. and
stand for the trace of a matrix and the statistical expectation for random variables, respectively.and indicate that are positive semidefinite and positive definite matrix. and
denote the identity matrix with appropriate size and the inverse of a matrix, respectively., and stand for the absolute value, the argument and the real part of a complex number, respectively, whereas denotes the determinant of . The notation represents a diagonal matrix where the diagonal elements are from a vector, and represents .
Ii System Model and The Problem
In this section, firstly, we present the system model of the IRS assisted secure communications with multiple antennas at both the legitimate transceiver and the eavesdropper, referring to it as the IRS-MIMOME system. Then, we illustrate the IRS reflecting model and signal model for our considered system. Finally, we formulate the discussed optimization problem.
Ii-a System Model
Consider the IRS assisted MIMOME system, as shown in Fig. 1, where an AP equipped with antennas serves a legitimate user at the present of an eavesdropper. Both the legitimate user and the eavesdropper are equipped with multi-antenna and the number of the antennas at these two users are and , respectively. In addition, an IRS composed of passive elements is installed on a surrounding wall to assist the secure communications between the AP and the legitimate user. The IRS has a smart controller, who has the capability of dynamically adjusting the phase shift of each reflecting element based on the propagation environment learned through periodic sensing . In particular, the IRS controller coordinates the switching between two working modes, i.e., receiving mode for environment sensing (e.g., channel state information (CSI) estimation) and reflecting mode for scattering the incident signals from the AP .
Ii-B IRS Reflecting Coefficient Model
Following , the phase shift matrix of the IRS can be defined as , where and for , and denotes a diagonal matrix whose diagonal elements are given by the corresponding vector and denotes the set of reflecting coefficients of the IRS. In this paper, two different sets of reflecting coefficients are considered as below.
Ii-B1 Continuous Reflecting Coefficients
That is, the reflecting coefficient with the constant amplitude and continuous phase shift is characterized as
Ii-B2 Discrete Reflecting Coefficients
In this model, the reflecting coefficient has constant amplitude but discrete phase shift and is defined as
where is the number of quantized reflection coefficient values of the element of the IRS.
Note that, due to the limitations of the hardware, the realization of the continuous reflecting model is difficult or even impossible . Therefore, the discrete model is more practical from the perspective of application. However, the continuous reflecting model is still discussed herein for the obtained performance can be regarded as the upper bound of the system. Furthermore, our algorithm for the discrete model is based on the algorithm of the continuous model.
Ii-C Signal Model
For our considered system, as , the signals that are reflected by the IRS multi-times are ignored due to significant path loss. Therefore, combined with IRS reflecting coefficient model, the signals received at the legitimate user and the eavesdropper can be expressed as
where and represent the complex baseband channels from AP to the legitimate user and the eavesdropper, respectively, and denote the complex baseband channels from the IRS to the legitimate user and the eavesdropper, respectively, and defines the complex baseband channel from AP to the IRS. and are used to characterize the equivalent channel from AP to the legitimate user and the eavesdropper, respectively. and denote the independent circularly symmetric complex Gaussian (CSCG) noise vectors at the legitimate user and the eavesdropper, respectively. In which, and denote the average noise power at the legitimate user and the eavesdropper, respectively. and represent the identity matrix with and dimensions, respectively. As in , the quasi-static flat-fading channel model is adopted herein and all the CSI are perfectly known at the AP.
where . And and represent the achievable transmission rates from AP to the legitimate user and from AP to the eavesdropper, respectively, and they are defined as follows,
where is the transmit signal covariance matrix at the AP and . Hence, the achievable secrecy rate for the legitimate user is characterized by
Note that dropping the operator has no impact on the optimization of the secrecy rate, thus this operator is removed in the sequel analysis.
Ii-D Problem Formulation
As mentioned earlier, in this paper, we discuss the joint optimization of the transmit covariance matrix at the AP and the reflection coefficients at the IRS to maximize the system secrecy rate subjected to the transmit power constraint at the AP and the reflection coefficient constraint at the IRS. Thus we have the following optimization problem OP1,
In which, C1 characterizes the total transmit power constraint at the AP, C2 defines the positive semi-define constraint on transmit covariance matrix, and C3 represents the IRS reflecting coefficient model. It is obvious that OP1 is a non-convex nonlinear programming with non-convex objective function and the uni-modular constraint on each reflection coefficient , which makes it difficult to be solved. Therefore, in the sequel, we pursue the suboptimal approach to handle OP1.
Iii Alternating Optimization based Joint optimization Algorithm
In this section, a suboptimal algorithm is proposed to solve OP1. As aforementioned that, our formulated problem OP1 is a non-convex nonlinear programming. However, our analysis indicates that, given the reflecting coefficients at the IRS and by leveraging the SCA , a convex approach can be used to solve the transmit covariance matrix optimization at the AP, while for given the transmit covariance matrix at the AP, we can use the alternative optimization to find the suboptimal solution for the reflecting coefficients at the IRS. Based on that, we present an alternative suboptimal algorithm for OP1. In addition, we also discuss the extension of the algorithm to the case with discrete reflecting coefficients at the IRS at the end of this section.
Iii-a Optimization of the transmit covariance matrix
In this subsection, we discuss the transmit covariance matrix optimization at the AP for given the reflecting coefficients at the IRS. Hence, we have the following problem OP2,
Herein, given the reflection coefficient matrix at the IRS, and are simplistically denoted as and , respectively. One may note that, now, the formulated problem OP2 is the secrecy rate maximization problem for the MIMOME system which has been discussed in ,  and , and various algorithms have been proposed therein. In this paper, following , the SCA-based suboptimal algorithm is used to handle OP2. And the key point is to obtain a tight concave lower bound of , which can be achieved by retaining the concave part in (10) and linearizing the concave function [35, 36]. That is, at , we have the concave approximation of as follows,
where . Based on the above approximation and given , the problem OP2 can be transformed into the following formulation,
Then this problem is convex and can be easily solved using standard interior-point methods . That is, the Karush CKuhn CTucker (KKT) conditions  for the above convex approximation problem are, namely,
Herein, . and are the dual variables associated with the transmission power constraint and the positive semi-definite constraint on the , respectively. Correspondingly, the Lagrangian function of (12) can be written as
Since problem (12) is convex and satisfies the Slater’s condition, the duality gap is zero between (12) and its dual problem. Thus, the optimal solution of (12) can be determined via solving the following Lagrange dual problem
To sum up, we have the SCA based suboptimal algorithm for OP2 which is summarized as the Algorithm 1 as below.
|Algorithm 1: Optimize transmit covariance matrix|
|S1: Initialize: and ;|
|a) Solve the problem in (16) with given and ,|
|Obtain the optimal transmit covariance ;|
|b) Update based on the subgradient method;|
|S4: Until the required accuracy;|
|S5: Update , and reset ;|
|S6: Until the required accuracy;|
|S7: Output .|
To meet the transmission power constraint at the AP and the positive semi-definite constraint of the transmit covariance matrix at the beginning of the algorithm, we set .
Iii-B Optimize the IRS reflecting coefficients
In this subsection, given the transmit covariance matrix at the AP, the optimization the reflecting coefficient matrix at IRS with the continuous model is discussed. Particularly, we have the following problem OP3.
It is obvious that OP3 is a non-convex programming with both non-convex constraints and non-concave objective function, which makes it is difficult to be solved. However, we prove that, given , the formulated optimization problem with respect to can be solved with the close-form optimal solution or to have an interval about the optimal solution. Therefore, the alternative optimization approach is used here again to solve OP3, i.e., we alternatively solve OP3 in variable with given until the procedure is converged. The details are illustrated as follows.
1) Objective function transformation: In order to use the alternative optimization approach to solve OP3, we should first make an objective function transformation for OP3. Note that, the relationship of the objective function with is rather implicit. Thus, we rewrite the objective function as an explicit function over . That is ,
Herein, let as the eigenvalue decomposition (EVD) of , and
are unitary matrix and diagonal matrix, respectively, and all the diagonal elements inare non-negative real numbers. Also, in (18), we define , , , , , and , , . Now, is represented in an explicit form of the reflection coefficients . Therefore, given and , can be rewritten as a function of as,
We denote , , and . Since both and are the sum of identity matrix and the two positive semi-define matrixes, thus we have , , and . Moreover, for and we have and , respectively. Therefore, can be rewritten as
Herein, and . Hence, the maximization of is equivalent to maximize the , namely,
Herein, and . In addition, due to both and are full-rank, we have and .
2) Deriving the tractable expressions for and : Following the above, herein, according to the value of (or ), i.e., () or (), we separately derive the tractable expressions of and which are then used to analyze the corresponding optimal solution of .
Case : At first, we present a lemma as below.
Lemma 1 (): is diagonalizable if and only if .
Based on the Lemma 1, then we can derive the expression of under and , separately.
If , namely, is non-diagonalizable, with and due to . Hence, the expression of can be transformed into
in which, the last equation is hold with for .
If , the EVD of can be expressed as , where, and with denoting the sole non-zero eigenvalue of . Set and it is a Hermitian matrix with . Let and denote the first column of and the first row of . Note that it follows that ; moreover, let and denote the first element in and , respectively, we have since both and are Hermitian matrices. Hence, can be further simplified as ,
Case : Similarly, if , we have
And if , the EVD of can be expressed as , where, and with denoting the sole non-zero eigenvalue of . Set , let and denote the first column of and the first row of and let and denote the first element in and , respectively. Hence, can be further simplified as,