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Hybrid Relay-Reflecting Intelligent Surface-Aided Covert Communications

In this paper, we propose to use a hybrid relay-reflecting intelligent surface (HR-RIS) to enhance the performance of a covert communication system. Specifically, the HR-RIS consists of passive reflecting elements and active relay elements to manipulate the wireless signals from a transmitter to a desired receiver while ensuring the covertness of the transmission via avoiding such signals being detected by a warden. To fully explore the benefits offered by the HR-RIS, we first formulate the joint design of the transmit power and relay/reflection coefficients of the HR-RIS as an optimization problem to maximize the covert rate subject to a covertness constraint. To tackle the solution to this optimization problem, we then derive a closed-form expression for an upper bound on covert rate, based on which we develop an alternate algorithm to solve the formulated optimization problem. Our examination shows that the HR-RIS outperforms the traditional RIS in term of achieving a higher covert rate. Interestingly, we also observe the major part of the performance gain brought by the HR-RIS can be obtained by a small number of active relay elements (e.g., 5) and further increasing this number does not improve the covert communication performance.


I. INTRODUCTION
Covert communication, or low probability of detection (LPD) communication, has emerged as a cutting-edge secure communication technique aiming at hiding communications from a watchful warden [1]. Considering different practical constraints and limitations, covert communication in different scenarios was widely studied in the past few years. For example, covert communication in relay networks was studied, and it was proved that the relay can stealthily transmit its own information to the desired destination in the process of forwarding data [2]. In [3], the authors presented the potentials of using the recently emerged intelligent reflecting surface (IRS), also known as reflecting intelligent surface (RIS), to improve covert communication performance.
Following [3], several RIS-aided covert communication approaches have been developed. Specifically, RIS is a flat surface composed of a large number of reconfigurable and lowcost passive reflective elements, each of which is capable of controlling the phase and amplitude of the incident signal for optimal reflection, making the wireless channel between the transmitter and receiver more favorable for communication [4]. For example, the authors of [5] investigated the multiple-inputmultiple-output (MIMO) covert communication assisted by RIS, where the covert rate was maximized by jointly designing the transmit covariance matrix and phase shift matrix. In [6], the authors considered the design of a latency constrained covert communication system with the assistance of RIS with global CSI and without the warden's instantaneous CSI, respectively.
It should be emphasized that the existing studies on RISassisted covert communication adopted the fully-passive beamforming strategies. A main limitation of the traditional RIS compared to relays is the that the passive reflection limits the degrees of freedom in the beamforming. Thus, the traditional RIS cannot outperform a half-duplex relay when the number of the elements in the RIS is not sufficiently large [7]. This implies that if a few passive elements of the RIS are replaced by active ones, the traditional RIS becomes a hybrid relay reflecting intelligent surface (HR-RIS) [8], which may lead to that the HR-RIS is able to significantly improve the assisted system performance. Therefore, in this work we propose a HR-RIS-based covert communication scheme, which applies active beamforming by adding some relaying elements at the traditional RIS to mitigate the limitation of the passive reflecting.
The main contributions of this paper are summarized as follows. We consider the covert communication from a transmitter (Alice) to a receiver (Bob) with the aid of an HR-RIS. Our goal is to jointly optimize the transmit power and the HR-RIS reflection matrix, including its phase shifts and amplitudes, to maximize the covert rate at Bob. In the optimization problem of interest, the coefficient matrix of HR-RIS is optimized by an alternate optimization (AO) method and then the optimal transmit power is obtained under the covertness constraint in terms of the Kullback-Leibler (KL) divergence. Our study shows that the HR-RIS can significantly improve the covert rate compared to conventional RIS.

A. Considered Scenario and Adopted Assumptions
As shown in Fig. 1, we propose a covert communication transmission scheme assisted by an HR-RIS, where a transmitter (Alice) intends to send confidential information to a legitimate receiver (Bob) with the aid of the HR-RIS, while a warden (Willie) attempts to detect the existence of this transmission. Specifically, there are two links from Alice to Bob, i.e., the direct link from Alice to Bob, and the reflection/relaying link form Alice to the HR-RIS and then to Bob. Similarly, there are two paths from Alice to Willie. Alice, Bob and Willie are assumed to be equipped with N a , N b , and N w antennas, respectively. In addition, it is assumed that the signals reflected by the HR-RIS twice or more are ignored due to the significant path loss [9]. The HR-RIS is assumed to be equipped with N elements, including M passive reflecting elements and K active relaying elements (i.e., M + K = N ). The passive reflecting elements are implemented by phase shifter, while the active relaying elements can tune the phase and amplitude of the incident signals. We assume that the active elements work in the AF mode. Therefore, for K = 0, HR-RIS converges to a traditional RIS. For K = N , by contrast, it becomes an AF relay station equipped with N antennas. Hence, in this work we have 1 ≤ K ≤ N . Furthermore, similar to the conventional RIS, we assume that each (active/passive) element of HR-RIS can independently reflect the received signals.

B. Transmission from Alice to Bob
When Alice transmits confidential message, the received signal at Bob is given by where n bt = H rb Ψn r + n b represents the total effective noise at the Bob. H ar and H ab are the steering vectors from Alice to HR-RIS and the steering vector from Alice to Bob, respectively, and H rb denotes the steering vector from HR-RIS to Bob. In addition, n r ∼ CN (0, σ 2 r I K ) and n b ∼ CN (0, σ 2 b I N b ) are the complex additive white Gaussian noise (AWGN) space vectors at the K active elements of the HR-RIS and at the Bob, respectively. For simplicity, we assume that σ 2 ). x is the signal vector transmitted by Alice, with each elements following CN (0, 1), and P a is Alice's transmit power.
Following (4), the transmit power of the active elements at the HR-RIS can be expressed as which should meet the constraint condition P r ≤ P max r , where P max r is the maximum transmit power of all the K active elements. We assume that Bob's channel state information (CSI) is perfectly known by himself through channel estimation [10]. Following (4), Bob's covert rate in the HR-RIS-aided covert communication system can be expressed as [8] f where

C. Binary Hypothesis Testing at Willie
In this work, we focus on the delay-constrained covert communication, that is, the number of channel uses L is finite. In order to detect the existence of a transmission, Willie attempts to distinguish the following two hypotheses: where H aw and H rw are the steering vector from Alice to Willie and the steering vector from HR-RIS to Willie, respectively, and n w ∼ CN (0, σ 2 w I Nw ) is the AWGN space vector at Willie. H 0 denotes the null hypothesis in which Alice does not transmit, and H 1 denotes the alternative hypothesis where Alice transmits signals. Similarly, we assume that σ 2 w = σ 2 r . Therefore, the total noise power at Willie is Considering the worst-case scenario for covert communications, we assume that Willie knows the channels perfectly. The probability density function of y w under H 0 and H 1 can be respectively given by ). As such, the signal-to-interference-plus-noise ratio (SINR) at Willie under H 1 is given by where U w = (H rw ΘH ar + H aw )(H rw ΘH ar + H aw ) H and M = (I Nw + H rw ΨΨ H H H rw ). We note that U w and M are Hermitian matrices. Then, we perform the eigenvalue decomposition (EVD) on the above two matrices, which can be written as U w = GΞG −1 and M = JΛJ −1 , where G and J are matrices of eigenvectors of U w and M, respectively, and G and J ∈ C Nw×Nw , Ξ = diag {ω 1 , ω 2 , · · · , ω Nw }, ω n is the n-th eigenvalue of U w , Λ = diag {κ 1 , κ 2 , · · · , κ Nw }, κ n is the n-th eigenvalue of M. As such, we have In covert communications, D 01 ≤ 2ǫ 2 is generally adopted as the covertness constraint, where ǫ is a small value to determine the required covertness level and the KL divergence D 01 is given by [11]

D. Problem Formulation
In this part, we jointly design the transmit power at Alice and relay/reflection coefficients of the HR-RIS to maximize the covert rate at Bob subject to the covertness and other constraints, of which the optimization problem can be formulated as (11d)

III. COVERT COMMUNICATION DESIGN
In this section, our goal is to maximize the covert rate at Bob by jointly designing P a and Θ. We propose an alternate algorithm to optimize P a and Θ. Specifically, we first optimize Θ for a given P a and the object function is transformed into a form that is easy to handle. Then, we optimize P a for a given Θ.
A. Optimizing Θ for a Given P a First, we randomly generate the coefficient of HR-RIS and use D 01 = 2ǫ 2 to get the feasible P a . The object function f (Θ, P a ) is non-convex with respect to Θ. In addition, the feasible set of (P1) is non-convex due to the unit-modulus constraint (11c). Therefore, (P1) is difficult to be tackled. Thus, we approximate the objective function f (Θ, P a ) by using its upper bound f 0 (Θ, P a ), which can be written as where ρ = P a /σ 2 b and a is achieved by set Q = ∅. We note that this upper bound becomes tighter as log 2 |R| decreases.
Considering the case where the active elements employed at the HR-RIS are few, we know that Ψ is sparse, which means log 2 |R| is small as per its definition given (6). Therefore, for a given P a the optimization problem (P1) can be rewritten as Although f 0 (Θ, P a ) is still non-convex, the optimal reflection coefficient of HR-RIS can be obtained by our proposed method detailed in the following subsection.

B. Transformation of the Objective Function
Generally, the proposed solution is a sequential procedure where in each iteration, a specific coefficient of HR-RIS is updated when the others are fixed. Specifically, we let a H n ∈ C N b ×1 denote the n-th row of H ab , and b n ∈ C N b ×1 denote the n-th column of H rb , i.e., H ar = [a 1 , a 2 , · · · , a N ] H , and H rb = [b 1 , b 2 , · · · , b N ]. Since Ψ and Θ are diagonal matrices, we have H rb ΘH ar = N n=1 θ n b n a H n and H rb Ψ = n∈Q θ n b n . Hence, we can rewrite f 0 (Θ, P a ) as Consequently, f 0 (Θ, P a ) can be rewritten as f 0 (Θ, P a ) = log 2 A n + |θ n | 2 B n + θ n C n + θ * n C H n . (15) For n ∈ Q, A n , B n and C n can be respectively written as For n / ∈ Q, A n , B n and C n are defined as It can be observed that the matrices A n , B n and C n do not contain the variable θ n , which means that these matrices can be obtained if all the elements {θ i } N i=1,i =n are fixed. Similarly, the relationship between the transmit power of the relay and Ψ can be determined as , which is a constant due to that the variables i∈Q,i =n ψ i are fixed. Therefore, (16) can be rewritten as Here, we notice that |ψ n | 2 = |β n | 2 for n ∈ Q.
C. An Efficient Algorithm to Solve (P2) 1) Problem of Updating Θ: In our proposed algorithm, ∈Q is fixed when optimizing θ n in each iteration, the objective function f 0 (Θ, P a ) can be rewritten as where A n is an invertible matrix satisfying rank(A n ) = N b . Moreover, log 2 (|A n |) is a constant, and f 1 (Θ, P a ) is given by Following the above transformation, the problem of updating Θ, denoted by (P 3 ), is given by 2) Solution to (P3): (P3) in (20) admits a closed-form solution, and, thus, it is efficient for practical implementation. In order to derive it, the objective function f 1 (Θ, P a ) can be rewritten as where D n = I N b + |θ n | 2 A −1 n B n and E n = A n D n . We next analyse the objective function f 1 (Θ) by considering the first term in (21), i.e., log 2 |D n |. Specifically, for |D n |, we note that rank(A −1 n B n ) < rank(B n ) = 1. Moreover, the probability of rank(A −1 n B n ) is close to zero (it only happens when A −1 n B n = 0). Thus, we have rank(A −1 n B n ) = 1.
Similarly, we find that A −1 n B n is not diagonalizable when rank(A −1 n B n ) = 0, which usually rarely happens. Based on this, we have (A −1 n B n ) = 0 with a high probability and A −1 n B n is diagonalizable. Hence, we can rewrite A −1 n B n = W n Σ n W −1 n based on EVD, where Σ n = diag{ι n , 0, · · · , 0}, ι n is the only non-zero eigenvalue of (A −1 n B n ). Finally, since both A n and B n are positive semidefinite, ι n is nonnegative and real. Thus, we have We are now focusing on the second term of (21). By a similar argument for the first term, we have E −1 n C n diagonalizable with a high probability as well. Thus, we have E −1 n C n = T n Γ n T −1 n based on the EVD, where T n ∈ C N b ×N b , Γ n = diag {λ n , 0, · · · , 0}, λ n is the sole non-zero eigenvalue of E −1 n C n . Let V n = T n A n T −1 n , and v n denote first element of the first column of V −1 n and v ′ n denote first element of the first row of V n . Note that it follows that v ′ n v n = 1. So, according to the [12], we can write where R denotes the real part of a complex number. We note that the additional coefficient |θ n | 2 is related to the active relay elements in HR-RIS, which does not exist in traditional RIS.
In the HR-RIS, Q is available to determine {|β n |} n∈Q . Therefore, from (20), we obtain As a result, the optimal solution to (P3) is given as + P a ||b n || 2 ], n ∈ Q to (26), it is observed that a larger K results in a smaller |β n |. Therefore, increasing the number of active elements (i.e., K) does not always guarantee the covert rate improvement of HR-RIS over traditional RIS. In particular, with a limited power budget P max r , the HR-RIS can have |β n | < 1, which attenuates the signal and degrades the covert rate. In this case, HR-RIS with a smaller K is more likely to attain a covert rate gain than those with more active elements. This conclusion will be further demonstrated numerically. Proof: In order to determine the monotonicity of D 01 with respect to P a , we derive its first derivative as For a given Θ, following (28), we can find optimal transmit power of Alice P * a by solving D 01 = 2ǫ 2 . Since P a does not affect the optimization of Θ, P * a is the global optimal solution.

IV. NUMERICAL RESULTS AND DISCUSSIONS
In this section, the numerical results will validate the performance of our proposed scheme. We assume that uniform linear arrays (ULAs) are deployed at the Alice, Bob and Willie, respectively. In contrast, HR-RIS uses a unified plane array (UPA) with N elements. Furthermore, assuming that there is a half-wavelength distance between Alice, Bob, Willie and HR-RIS arrays. We consider a two-dimensional coordinate system, Alice, HR-RIS, Bob, and Willie are respectively located at (0m, 0m), (51m, 0m), (50m, 2m) and (30m, 5m). All channel realizations are drawn from Rician fading. The path loss of a link distance d is given by [9], [12], χ(d) = χ 0 ( d 1m ) αij , where χ 0 is the path loss at the reference distance of 1 meter, and α ij is the path loss exponent. Specifically, the path loss exponents are set as α ar = 2.2, α rb = 2.8, α ab = 4.2, α aw = 4.2, and α rw = 2.8.
In Fig. 2, we plot the covert rate of Bob versus the total reflection elements N of the HR-RIS under different numbers of the active elements K. In this figure, we first observe that Bob's covert rate increases as N increases. It can also be seen that Bob's covert transmission rate increases as K increases. As the number of active elements increases, we can see that the increase in the covert rate becomes smaller. This is due to the power limit of active elements, too many active elements can cause |β n | < 1, which is explained in Remark 1. Therefore, we can get the optimal number of active elements from this figure. Based on which, we can conclude that a small amount of active elements is sufficient for HR-RIS to achieve a significant improvement in term of the covert rate when compared to conventional RIS-aided covert communication schemes.

V. CONCLUSION
In this work, we proposed a covert communication scheme with the help of the HR-RIS, where several elements are active relay elements and the remaining ones as passive reflecting elements. We first used KL divergence to represent Willie's detection capability, and proved that the KL divergence adopted in the covertness constraint is a monotonically increasing function of the transmit power at Alice. Based on this, we obtained the optimal value of P a . Then, we used an alternate optimization algorithm to identify the optimal reflection coefficients of HR-RIS to improve the covert communication performance. The numerical results demonstrated that the proposed scheme can significantly outperform the conventional RIS-aided covert communication schemes in term of covert rate by using a small number of active elements.