Robust and Secure Sum-Rate Maximization for Multiuser MISO Downlink Systems with Self-sustainable IRS

The sixth-generation (6G) networks are expected to serve as a key enabler for realizing the future intelligent digital society in 2030, offering superior communication services compared with the current fifth-generation (5G) networks, such as ultra-high data rate, high energy efficiency, large global coverage, and highly secure communications [2]. To satisfy these requirements, various technologies have been proposed in recent years. In particular, massive multiple-input multiple-output (MIMO), millimeter wave (mmWave) communications, and ultra-dense networks (UDNs) are prominent candidates. Although these technologies are able to enhance the wireless network coverage and capacity to a certain extent, they generally incur increased hardware costs, energy consumption, and signal processing complexity, due to the aggressive deployment of large-scale antennas and active nodes in the systems [3, 4]. As such, it is still a fundamental challenge to achieve a sustainable capacity growth of future wireless networks in terms of the communication cost and energy efficiency.

To fulfill the stringent requirements of 6G networks[2], the emerging intelligent reflecting surface (IRS)-assisted wireless communications [5] have received an unprecedented upsurge of attentions recently. Specifically, an IRS consists of a large number of low-cost passive reflection elements which can independently reflect the incident electromagnetic wave with controlled phase shifts. By intelligently adapting the phase shift of each IRS element to the conditions of communication channels, the reflected signals can be coherently combined at the desired receivers. As such, IRSs are able to reshape the signal propagation and to establish a favorable communication environment for achieving various purposes, such as enhancing the physical layer security, harnessing multiple access interference, and improving the quality of communication in terms of both spectral efficiency and energy efficiency[6]. For example, [5] confirmed that IRS-assisted communication systems can extend the signal coverage compared with conventional systems where the IRS is absent. Furthermore, the results in [7] showed that the introduction of an IRS can significantly improve both the achievable system data rate and the total harvested power in simultaneous wireless information and power transfer (SWIPT) systems.

Despite the fruitful results in the literature, e.g. [8, 9, 10, 5, 6, 11, 12, 13], most of the works idealistically assumed that the energy consumption of the IRS is negligible. However, typical power consumption values of each phase shifter are $1.5$, $4.5$, $6$, and $7.8$ mW for $3$-, $4$-, $5$-, and $6$-bit resolution phase shifters, respectively [14, 15, 16, 17]. Considering a hundred of IRS elements, the operational power of IRS with $3$-bit phase shifters becomes $0.15$ W which is comparable to the radiated communication power. In fact, IRSs are expected to be battery- or grid- powered [18]. However, powering IRSs via conventional powerline networks would not only increase the implementation cost, but would also decrease the flexibility of IRS deployment. On the other hand, battery-powered IRS are usually equipped with a limited energy storage leading to restrictive operational lifetime that creates a bottleneck in communication networks. Moreover, manually replacing batteries of IRS may be costly or even impossible due to environmental hazards. As a result, the amalgamation of energy harvesting with IRSs is a promising alternative for providing self-sustainable and uninterrupted communication services. In practice, the availability of major conventional energy harvesting resources, such as wind, geothermal, and solar, are usually limited by the weather and energy source locations. As a result, harvesting energy from radio frequency (RF) in wireless communication systems is more reliable and suitable than that from natural sources, due to the controllability of wireless power transfer (WPT)[19]. To enable uninterrupted communication services, a self-sustainable IRS enabled by WPT was considered in [20, 21] for improving the system sum-rate and capacity, respectively. However, [20] focused on the resource allocation design of a single-antenna AP and the obtained results cannot be directly extended to the case adopting a multi-antenna transmitter. On the other hand, although a multi-antenna AP system was considered in [21], the obtained results were developed for the case of a single receiver system, which are not applicable to systems serving with multiple receivers. Moreover, all IRS elements in [20, 21] are forced to adopt the same operating mode in every time instant, which is a strictly suboptimal setting decreasing the flexibility in resource allocation. Furthermore, both [20, 21] ideally assumed a linear energy harvesting model for the conversion of the scavenged RF energy to the direct current (DC) power for simplicity. Yet, the conventional linear energy harvesting model fails to capture the non-linear features of practical energy harvesting circuits [22, 23] and resource allocation designs based on the simplified linear model can lead to severe resource allocation mismatches and performance degradation. Additionally, [20, 21] assumed the availability of continuous phase shifters at IRS which can hardly implemented in particle due to high hardware implementation cost. On the other hand, to facilitate effective energy harvesting, transmitters in SWIPT systems usually increase the power of the information-carrying signals, which may also increase the susceptibility to eavesdropping due to the higher potential of information leakage, which is not considered in [20, 21]. Thus, communication security is a fundamental issue in wireless-powered IRS-assisted systems.

Conventionally, security provisioning methods rely on cryptographic techniques applied at the upper layers of wireless networks, which require high computational complexity leading to a large amount of energy consumption. Fortunately, different from cryptographic techniques, physical layer security is computationally-efficient and effective to safe guard communications by exploiting the inherent randomness of wireless channels [24]. In particular, there are several methods to enhance the physical layer security in wireless networks, e.g. cooperative jamming[25] and artificial-noise (AN)-aided beamforming [26]. Indeed, exploiting AN or jamming signals in wireless networks is a double-edged sword. On the one hand, it can facilitate secure communications by degrading the quality of eavesdroppers’ channels. On the other hand, it may also cause inevitable interference to legitimate users. As a remedy, the application of IRS and AN has been investigated to guarantee secure communications, e.g. [27, 28]. In practice, the degrees of freedom (DoF) offered by the IRS can provide a better interference management which alleviates the negative impacts caused by AN transmission. Indeed, to unleash the potential gain brought by the IRS, accurate channel state information (CSI) is needed for effective beamforming design. However, numerous literature, e.g. [27, 28], over optimistically assumed that perfect CSI of eavesdropper channels is available at the AP which can hardly be obtained in practice. Especially, potential eavesdroppers may not interact with the AP and acquiring perfect CSI of the eavesdropping channel at the AP is challenging. In other words, adopting the results from [27, 28] for robust beamforming design may increase the risk of security breach even if an IRS is deployed. To this end, considering imperfect CSI of eavesdroppers, a secure IRS-assisted wireless network for maximizing the system sum-rate in the presence of multiple users and eavesdroppers was presented in [29]. Nevertheless, the design in [20, 21, 29] still required the availability of perfect CSI of IRS communication channels. Unfortunately, due to the nearly passive implementation of the IRS, obtaining perfect CSI for the IRS related links is challenging, if not impossible. Also, the authors in [30] showed that when the joint transmitter precoder and IRS phase shifts design ignores the existence of possible CSI errors, existing non-robust designs, e.g. [27, 28], cannot always guarantee the required target rate which jeopardizes the system performance significantly. As a result, an efficient robust beamforming design to strike a balance between the system sum-rate and the IRS self-sustainability for secure communication is desired which has not been reported in the literature, yet.

Motivated by the aforementioned observations, we consider a robust and secure IRS-assisted multiuser MISO downlink wireless system in the presence of multiple single-antenna potential eavesdroppers. In particular, by exploiting the large number of IRS elements, our design allows a portion of the IRS elements to reflect signals from the AP and the remaining to harvest energy for supporting the energy consumption of the IRS elements. To maximize the system sum-rate, we jointly optimize the beamforming for information signal and AN, the discrete phase shifts of the IRS, and the harvesting schedule at the IRS. The resource allocation design is formulated as a non-convex mixed-integer optimization problem takes into account the imperfect CSI of all wireless channels, the non-linearity of energy harvesting, and the communication security quality of service (QoS) requirement. To address the design problem at hand, this paper proposes a computationally-efficient iterative suboptimal algorithm. In particular, we first propose a series of transformations to resolve the coupling between the optimization variables which paves the way for the development of a computationally efficient iterative algorithm based on the successive convex approximation (SCA) and semi-definite relaxation (SDR). Unlike the conventional alternation optimization (AO) method [20, 21, 11, 5], which can be easily trapped in inefficient stationary points over iterations in practice, our proposed algorithm is able to jointly optimize all these optimization variables in each iteration enjoying a higher system sum-rate. Our results not only show the non-trivial trade-off between the system sum-rate and the self-sustainability of IRS, but also unveil the impact of limited bit resolution of the IRS phase shifters on the system performance. In addition, our results reveal that deploying self-sustainable IRS can significantly enhance the physical layer security in wireless communication systems.

Notations: The scalars, vectors, and matrices are represented by lowercase letter $x$, boldface lowercase letter $\mathbf{x}$, and boldface uppercase letter $\mathbf{X}$, respectively. $\mathbb{B}^{N\times M}$, $\mathbb{R}^{N\times M}$, and $\mathbb{C}^{N\times M}$ denote the space of $N\times M$ matrices with binary, real, and complex entries, respectively. $\mathbb{H}^{N}$ denotes the set of all $N\times N$ Hermitian matrices. The modulus of a complex-valued scalar is denoted by $|\cdot|$. The Euclidean norm, spectral norm, nuclear norm, and Frobenius norm of a matrix are denoted by $\|\cdot\|$, $\|\cdot\|_{2}$, $\|\cdot\|_{*}$, and $\|\cdot\|_{\mathrm{F}}$, respectively. The transpose, conjugate transpose, conjugate, expectation, rank, and trace of a matrix are denoted as $(\cdot)^{\mathrm{T}}$, $(\cdot)^{\mathrm{H}}$, $(\cdot)^{*}$, $\mathbb{E}\{\cdot\}$, $\mathrm{Rank}(\cdot)$, and $\mathrm{Tr(\cdot)}$, respectively. The maximum eigenvalue of a matrix is denoted by ${\lambda}_{\max}(\cdot)$ and the eigenvector associated with the maximum eigenvalue is denoted by $\bm{\lambda}_{\max}(\cdot)$. $[x]^{+}$ represents $\max\{0,x\}$. $\mathbf{X}\succeq\mathbf{0}$ and $\mathbf{X}\preceq\mathbf{0}$ mean that matrix $\mathbf{X}$ is positive semi-definite and negative semi-definite, respectively. $\mathrm{Re}(\cdot)$ stands for the real part of a complex number. $\mathrm{diag(\mathbf{x})}$ denotes a diagonal matrix with its diagonal elements given by vector $\mathbf{x}\in\mathbb{C}^{N\times 1}$ and $\mathrm{vec}(\mathbf{X})$ results in a column vector obtained by sequentially stacking of the columns of matrix $\mathbf{X}$. Operator $[\mathbf{X}]_{nm}$ returns the element at the $n$-th row and the $m$-th column of the matrix $\mathbf{X}$. $j$ denotes the imaginary unit. For a continuous function $f(\mathbf{X})$, $\nabla_{\mathbf{X}}f(\cdot)$ represents the gradient of $f(\cdot)$ with respect to matrix $\mathbf{X}$. The distribution of a circularly symmetric complex Gaussian (CSCG) random variable with mean $\mu$ and variance $\sigma^{2}$ is denoted by $\mathcal{CN}(\mu,\sigma^{2})$ and $\sim$ stands for “distributed as”. $\mathbf{I}_{N}$ denotes an $N\times N$ identity matrix. $\otimes$ stands for the Kronecker product.

In this paper, we aim to maximize the system sum-rate while maintaining the self-sustainability of the IRS by jointly designing the precoding vector $\{\mathbf{w}_{k}\}_{k\in\mathcal{K}}$, the AN covariance matrix $\mathbf{Z}$, the mode selection indicators $\{\alpha_{n}\}_{n\in\mathcal{N}}$, and the discrete phase shifters $\{\theta_{n}\}_{n\in\mathcal{N}}$. The joint design can be formulated as the following optimization problem⁷⁷7The considered problem formulation serves as a generalized optimization framework that subsumes various existing designs as subcases, e.g. robust precoding [30], secure precoding [28], or energy-harvesting IRS [20], etc.:

The objective function in (13) maximizes the minimum value of all the possible sum-rate due to CSI errors. Constraint $\mathrm{C1}$ ensures that the transmit power at the AP does not exceed its maximum transmit power budget $P_{\max}$. Constraint $\mathrm{C2}$ specifies that the phase shift of a $b$-bit resolution IRS reflecting element can only be selected from a discrete set $\mathcal{F}$. Constraint $\mathrm{C3}$ indicates that the total power consumed at the IRS should not exceed its total harvested power⁸⁸8If the harvested energy by the IRS is not even enough to supply one IRS reflection element, the proposed problem (13) can be solved by ignoring constraint C3 and the related IRS cascaded links, which is essentially the case of beamforming design without the IRS. from the AP, $P_{\mathrm{PH}}$, while taking into account the imperfect CSI knowledge. Constraint $\mathrm{C4}$ is imposed to guarantee that each IRS element can only operate in either reflection mode or energy harvesting mode. Constant $\tau_{k,j}$ in $\mathrm{C5}$ is the maximum tolerable information leakage to the $j$-th potential eavesdropper for wiretapping the signal transmitted to the $k$-th legitimate user considering the CSI error in the AP-Eve channel and the cascaded AP-IRS-Eve link. In particular, $\mathrm{C5}$ can guarantee that the system secrecy rate $R_{\mathrm{s}}$ is bounded from below, i.e. $R_{\mathrm{s}}\geq\sum_{k\in\mathcal{K}}\Big{[}R_{\mathrm{s},k}-\tau_{k,j}\Big{]}^{+}$.

The formulated problem is non-convex due to the coupling between variables $\mathbf{w}_{k}$, $\mathbf{Z}$, $\theta_{n}$, and $\alpha_{n}$ in constraints $\mathrm{C3}$, $\mathrm{C5}$, and the objective function in (13), the discrete phase shift in constraint $\mathrm{C2}$, and the binary variable $\alpha_{n}$ in constraint $\mathrm{C4}$. Besides, there are infinitely many possibilities in the objective function, constraint $\mathrm{C3}$, and constraint $\mathrm{C5}$, due to the CSI uncertainties in continuous variable sets. In general, solving such a problem optimally requires the use of the exhaustive search method which is computationally intractable even for a moderate size system. As a compromise approach, we aim to design a computationally-efficient suboptimal algorithm to obtain an effective solution. Fig. 3 summarizes the procedure of the proposed problem solving methodology. In order to address the optimization problem without using the AO method, we firstly introduce a set of slack variables and reformulate the problem in (13) to problem in (24), which paves the way for decomposing the coupling variables $\{\mathbf{w}_{k}$, $\mathbf{Z},\theta_{n},$ $\alpha_{n}\}$. Secondly, we adopt the big-M reformulation to decouple the coupling between the discrete variables, i.e., the mode selection matrix, $\mathbf{S}$, and the continuous variables, i.e., precoder, $\mathbf{W}_{k}$, and AN matrix, $\mathbf{Z}$, in $\overline{\mathrm{C3a}}$ and $\overline{\mathrm{C3b}}$. Besides, decomposing the coupling between the continuous variables in the objective function and constraints $\mathrm{C5},\mathrm{C9}$, and $\mathrm{C10}$ requires exploiting the properties of logarithm and Frobenius norm, respectively. Thirdly, after decomposing all these coupling variables, the $\mathcal{S}$-procedure is adopted to tackle the infinite number of constraints caused by imperfect CSI in $\mathrm{C3}$, $\mathrm{C5}$, $\mathrm{C9}$, and $\mathrm{C3}$. Fourthly, we exploit the successive convex approximation (SCA) to deal with all functions in difference of convex (D.C.) form, i.e.,$\mathrm{C4d}$, $\mathrm{C5a}$, $\overline{\mathrm{C9}}$, $\overline{\mathrm{C10}}$, and $\mathrm{C11b}$. Lastly, we apply the semi-definite relaxation (SDR) method to tackle the rank-one constraint, i.e., $\mathrm{C8}$, for the precoder design. As such, a computationally efficient suboptimal solution can be obtained.