Secure Outage Analysis of RIS-Assisted Communications with Discrete Phase Control

Reconfigurable intelligent surface (RIS) is a metasurface that consists of a large number of passive reflecting elements with integrated low power electronics [1][2]. A main feature of an RIS is that the amplitude and phase of each reflecting element can be independently controlled through software, thereby realizing passive beamforming (BF) for improving the signal quality at intended receivers. Due to these merits, RISs have been considered for various wireless applications, e.g., in millimeter-wave (mmWave) [3] and Terahertz (THz) [4] communications, to enhance spectral and energy efficiencies [5].

In recent years, physical layer security (PLS) has gained considerable interest for securing wireless communications. As a complement to conventional cryptographic methods, PLS ensures secure communications by exploiting the dynamics of propagation channels. Since RISs have the ability of adjusting the propagation channels, their deployment empowers the design of PLS with an additional dimension by exploiting passive BF. In order to maximize the theoretical secrecy, literature [6, 7, 8] investigated joint optimization of the active and passive BFs at the transmitter and RIS, respectively. In [9], the average secrecy rate (SR) was characterized for an RIS-assisted two-way communication system through a lower bound. More recently in [10], the SR was further analyzed for a system where the RIS reflection is utilized as a multiplicative randomness against a wiretapper.

Besides the analysis of SR, secrecy outage probability (SOP) is another relevant performance measure to quantify the performance of PLS, especially for systems undergoing slow-varying channels. The SOP is defined as the probability that the instantaneous secrecy capacity falls below a target SR. The SOPs of RIS-aided communication systems have been investigated in [11, 12, 13]. Both analytical and asymptotic analyses have been provided to reveal the impacts of key system parameters on SOP. In particular, the work in [12] considered the SOP of an RIS-aided unmanned aerial vehicle (UAV) relay system. In [13], the authors analyzed the SOP as well as the probability of non-zero secrecy capacity of an RIS-aided device-to-device (D2D) communication system. However, most studies on SOP analysis considered RIS with continuous phase shifts, which leads to unaffordable high complexity in practice. Even though discrete phase shifts have been considered for RIS reflection optimizations, e.g., in [14][15], few studies have been conducted on theoretical performance analysis with discrete RIS phase shifts especially in terms of SOP. Quantitative insight on RIS design has only been discovered for some non-security scenarios [16]. This is because discrete phase shifts make the performance expressions of the cascaded RIS channels much less tractable. In particular for secure communications, it can be even challenging to directly derive the corresponding cascaded channel distributions of both the legitimate user and eavesdropper.

In this work, we investigate the SOP of an RIS-assisted secure communication system and quantitatively characterize the impacts of discrete phase shifts in closed-form expressions. Concretely, we first derive the exact distributions of the received signal-to-noise-ratios (SNRs) at the legitimate user and the eavesdropper. Then, we present a closed-form expression for a tight upper bound of the SOP. Based on the obtained expressions, the asymptotic scaling law of SOP is characterized in high-SNR regimes. In particular, the SOP decreases with the slope of ${\rm e}^{-0.8N}$ for large $N$ and $b\!\geq\!3$, where $N$ is the number of RIS elements and $b$ is the number of quantization bits. Compared with the ideally continuous phase control, we further obtain that the performance loss in terms of SOP caused by using binary phase shifts, i.e., $b\!=\!1$, can be compensated by deploying a larger RIS with size $1.6N$.

The remainder of this paper is organized as follows. The system model is introduced in Section II. In Section III, we derive the distribution of the received SNRs at the legitimate user and at the eavesdropper. In Section IV, we provide a closed-form expression for an upper bound of the SOP, and we study it in notable asymptotic regimes. Simulation results and conclusions are given in Section V and VI, respectively.

We consider an RIS-assisted secure communication system consisting of a source ($S$), an RIS with $N$ reflecting elements, a legitimate user ($D$), and an eavesdropper ($E$), as illustrated in Fig. 1. The direct link between $S$ and $D$ is assumed to be blocked by obstacles, such as buildings, which is likely to occur at high frequency bands. In this scenario, the data transmission between $S$ and $D$ is ensured by the RIS. The eavesdropper is at a location where it can overhear the information from both $S$ and the RIS.¹¹1The direct link between $S$ and $E$ exists when the eavesdropper is not blocked by obstacles [11][13]. Nodes $S$, $D$, and $E$ are equipped with a single antenna for transmission and reception and all links experience Rayleigh fading.²²2The RIS-related links can be modeled as Rayleigh fading, also as [11][13], when the RIS is not optimally deployed to ensure strong LoS links. The channel coefficients of the $S$-RIS, RIS-$D$, RIS-$E$, and $S$-$E$ links are respectively denoted by $h_{i}$, $g_{i}$, $p_{i}$, and $h_{S\!E}\sim\mathcal{CN}(0,1)$, where ${\cal{CN}}$ is the complex Gaussian distribution. By applying channel estimation methods in [3, 4] and the references therein, we assume that the channel coefficients of $h_{i}$ and $g_{i}$ are perfectly known to $S$. However, the channel gains of $p_{i}$ and $h_{S\!E}$ are not available to $S$, as the eavesdropper is usually a passive device that does not emit signals.

By assuming quasi-static flat fading channels, the signal received at $D$ is expressed as

where $P$ denotes the transmit power at $S$, $x$ is the transmit signal, $\eta\in(0,1]$ is the RIS amplitude reflection coefficient with $\eta=1$ corresponding to lossless reflection, $\left\{\phi_{i}\right\}_{i=1}^{N}$ represents the phase shift of the $i$th reflecting element of the RIS, and $n_{D}\sim\mathcal{CN}(0,N_{0})$ is the additive white Gaussian noise (AWGN) with zero mean and variance $N_{0}$. Without loss of generality, the signal power is normalized, i.e., $\mathbb{E}[\left|x\right|^{2}]=1$ where $\mathbb{E}\left[\cdot\right]$ denotes the expectation of a random variable (RV). In addition, $d_{S\!R}$ and $d_{R\!D}$ are the distances of the $S$-RIS and RIS-$D$ links, respectively, and $\upsilon$ is the path loss exponent.

From (1), the received SNR at $D$ is calculated as

In the case of ideally continuous phase shifts, $\gamma_{D}$ is maximized by setting the phases of the RIS elements equal to $\phi_{i}\!=\!\phi_{i}^{\rm opt}\!\triangleq\!-\angle\left({h_{i}}{g_{i}}\right)$, where $\angle$ returns the phase of the complex number. This optimized phase compensates the phase shift introduced by fading channels. Due to hardware limitations, however, the phase shifts $\left\{\phi_{i}\right\}_{i=1}^{N}$ of the RIS elements are usually limited to a finite number of controllable discrete values. In particular, the set of discrete phase shifts is denoted by $\mathcal{A}\triangleq\left\{0,\frac{2\pi}{2^{b}},...,\frac{(2^{b}-1)2\pi}{2^{b}}\right\}$, where $b$ is the number of quantization bits. In this case, usually, the phase shift of the $i$th RIS element, $\phi_{i}^{\rm sub}\in\mathcal{A}$, is chosen as

Then, the received SNR in the presence of the discrete phase shifts is rewritten as

where ${\bar{\gamma}}_{S\!R\!D}\triangleq\frac{{\eta}^{2}P}{N_{0}d_{S\!R}^{\upsilon}d_{R\!D}^{\upsilon}}$ denotes the average SNR.

The eavesdropper receives signals from the direct link from $S$ and the reflected link from the RIS. Then, the received signal at $E$ is written as

where $d_{R\!E}$ and $d_{S\!E}$ denote the distances of the RIS-$E$ and $S$-$E$ links, respectively, and $n_{E}\sim\mathcal{CN}(0,N_{0})$ is the AWGN at $E$. Then, the received SNR at $E$ is

where ${\bar{\gamma}}_{S\!R\!E}\triangleq\frac{{\eta}^{2}P}{N_{0}d_{S\!R}^{\upsilon}d_{R\!E}^{\upsilon}}$ and ${\bar{\gamma}}_{S\!E}\triangleq\frac{{\eta}^{2}P}{N_{0}d_{S\!E}^{\upsilon}}$ represent the average SNRs of the $S$-RIS-$E$ and $S$-$E$ links, respectively.

In order to analyze the SOP of the system, we need to first characterize the distributions of $\gamma_{D}$ and $\gamma_{E}$.

In this section, Monte-Carlo simulations are illustrated to validate our analysis. The tested parameters are set to ${\bar{\gamma}}_{S\!E}\!=\!-5~{}\rm{dB}$ and ${\bar{\gamma}}_{S\!R\!E}\!=\!0~{}\rm{dB}$.

Fig. 2 shows the impact of $b$ on the SOP when $N=30$ and $C_{\rm th}=0.05$. We observe that the analytical results in (15) match well with the numerical curves. Monte-Carlo simulations are illustrated for values of the SOP no smaller than $10^{-3}$ due to the limited number of channel realizations simulated. As stated in Case 3, the gap between $b=3$ and $b=+\infty$ in Fig. 2 does appear negligible for high SNRs.

In Fig. 3, we consider a larger $C_{\rm th}=0.2$ to verify the effectiveness of the asymptotic analysis. We plot the SOP for $N=30$ by setting $b=1$ and $b=+\infty$. As expected, the asymptotic expressions in (20) and (21) are quite tight in the high-SNR regime. Then, we plot the SOP for $b=1$ by setting $N=48$ (i.e., which is equal to $1.6\times 30$) and $N=60$ ($2\times 30$). We see that the setup $N=48$ with $b=1$ provides, in the high-SNR regime, the same SOP as the setup $N=30$ with $b=+\infty$, which validates the obtained guideline in Remark 3.

This paper studied the SOP of an RIS-assisted communication system with discrete phase shifts. The main contribution is to unveil the achievable scaling law of SOP with respect to $N$ and $b$. Specifically, the increased number of RIS elements was quantified to compensate the performance loss caused by binary phase shifts.