On Secrecy Performance of RIS-Assisted MISO Systems over Rician Channels with
Spatially Random Eavesdroppers

PLS has been studied in diverse RIS-assisted communication scenarios, but rarely considered in the general case of randomly located eavesdroppers. Although the authors of [29] and [30] considered random eavesdropper locations, there are still several research gaps left to be filled. In [29], a single-antenna setting was adopted at the base station to analyze the ESC performance, but the Rician fading assumption and RIS phase shifts optimization were not taken into consideration for the multiple-antenna setting. In [30], a study was developed based on a simplified transmit beamforming design and the analyses of secrecy diversity order and capacity were not conducted. In addition, the works in [29] and [30] both need to be further explored to quantify the impact of the key parameters, e.g., the number of RIS reflecting elements, on the attainable secrecy performance in order to provide insightful guidelines for system design. Table I provides a summary of current studies related to the RIS-assisted secure communication systems and compares our work with them. In this paper, we investigate the SOP and ESC performance of an RIS-assisted multiple-input single-output (MISO) system over Rician fading channels with spatially random eavesdroppers. The main contributions of our work are summarized as follows.

• •

  Assuming maximum ratio transmission (MRT) for the transmit beamforming, the optimal reflect beamforming at the RIS is designed. Using tools from stochastic geometry, we derive accurate closed-form expressions for the distributions of the received signal-to-noise-ratios (SNRs) at the legitimate user and randomly located eavesdroppers located according to a homogeneous PPP.

• •

  We propose a new analytical PLS framework for the RIS-aided MISO secure communication system over Rician fading channels. Novel closed-form expressions are derived to characterize the SOP and ESC of the system. The derived results show that both the SOP and ESC are mainly affected by the number of RIS reflecting elements, and are not strong functions of the number of transmit antennas nor the transmit power at the base station, which means that increasing either of these latter resources will not significantly improve the secrecy performance.

• •

  To obtain more insightful observations, the asymptotic secrecy performance in the high SNR regime is also analyzed to characterize the SOP and ESC. The asymptotic SOP demonstrates that the secrecy diversity order of RIS-aided MISO secure communication systems depends on the path loss exponent of the RIS-to-ground links. In addition, when the locations of spatially random eavesdroppers are unknown, it is more efficient to deploy the RIS closer to the legitimate user than to the base station. Moreover, the impact of the randomly located eavesdropper density, $\lambda_{e}$, on the asymptotic ESC performance is quantitatively evaluated to be additive and proportional to $\log_{2}{\left({1}/{\lambda_{e}}\right)}$.

The rest of this paper is organized as follows. We introduce the system model in Section II, and in Section III, we derive the exact distributions of the received SNRs for the legitimate user and randomly located eavesdroppers. In Section IV, we provide a closed-form expression for the SOP, and we study the secrecy diversity order at high SNR. The ergodic secrecy capacity is investigated in Section V and simulation results are discussed in Section VI. Finally, we draw our conclusions in Section VII.

Notation: Boldface lowercase (uppercase) letters represent vectors (matrices). The set of all complex numbers is denoted by $\mathbb{C}$. The set of all positive real numbers is denoted by $\mathbb{Z}^{+}$. The superscripts $(\cdot)^{T}$, $(\cdot)^{\ast}$, and $(\cdot)^{H}$ stand for the transpose, conjugate, and conjugate-transpose operations, respectively. A circularly symmetric complex Gaussian distribution is denoted by ${\cal{CN}}$. A Rician distribution is denoted by $Rice$. $\mathbb{E}\{\cdot\}$ denotes the expectation of a random variable (RV). ${\rm diag}\left\{\cdot\right\}$ indicates a diagonal matrix. The operators $|\cdot|$ and $\left\|\cdot\right\|$ take the norm of a complex number and a vector, respectively. $\angle$ returns the phase of a complex value. The symbol $\Gamma\left(\cdot\right)$ is the Gamma function. The symbols $\gamma\left(\cdot,\cdot\right)$ and $\Gamma\left(\cdot,\cdot\right)$ denote the lower and upper incomplete Gamma functions, respectively. $\left[x\right]^{+}={\rm max}\left\{0,x\right\}$ returns the maximum between $0$ and $x$.

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

where the cascaded channel $\mathbf{g}_{D}^{H}\triangleq\mathbf{h}_{R\!D}^{H}\mathbf{\Theta}\mathbf{H}_{S\!R}\in\mathbb{C}^{1\times K}$, $\mathbf{f}\in\mathbb{C}^{K\times 1}$ is the normalized beamforming vector, $s$ denotes the transmit signal that satisfies the power constraint $\mathbb{E}\{\left|s\right|^{2}\}=P_{T}$, and $n_{D}\sim\mathcal{CN}(0,\sigma_{D}^{2})$ is additive white Gaussian noise (AWGN) at $D$ with variance $\sigma_{D}^{2}$. Therefore, the received SNR at $D$ is calculated as

where the RV $\left|{\rm A}\right|\triangleq\left|\mathbf{g}_{D}^{H}\mathbf{f}\right|$, and $\rho_{d}\triangleq\frac{P_{T}}{\sigma_{D}^{2}}$ denotes the transmit SNR from $S$ to $D$.

Due to passive eavesdropping, the channel state information of the eavesdropper is not known to the RIS, thus we determine the transmit beamforming at $S$ and the reflect beamforming at the RIS by maximizing the received signal power at $D$.

Theorem 1: When MRT beamforming is adopted, i.e., $\mathbf{f}=\frac{\mathbf{g}_{D}}{\left\|\mathbf{g}_{D}^{H}\right\|}$, the optimal reflection matrix of the RIS is given as

Proof: See Appendix A.$\hfill\blacksquare$

With the optimized RIS phase shifts in Theorem 1, the RV $\left|{\rm A}\right|$ is expressed as $\left|{\rm A}\right|\!=\!\sqrt{K\nu}\sum_{n=1}^{N}\left|h_{R\!D}\left(n\right)\right|$ which follows the distribution characterized in the following lemma.

Lemma 1: The cumulative distribution function (CDF) of $\left|{\rm A}\right|$ is well approximated by

with shape parameter $k=N\frac{\frac{\pi}{4}\left(L_{\frac{1}{2}}\left(-\epsilon\right)\right)^{2}}{1+\epsilon-\frac{\pi}{4}\left(L_{\frac{1}{2}}\left(-\epsilon\right)\right)^{2}}$ and scale parameter $\theta=\sqrt{K}\sqrt{\frac{\mu_{D}\nu}{\epsilon+1}}\frac{1+\epsilon-\frac{\pi}{4}\left(L_{\frac{1}{2}}\left(-\epsilon\right)\right)^{2}}{\frac{\sqrt{\pi}}{2}L_{\frac{1}{2}}\left(-\epsilon\right)}$, in which $L_{q}\left(x\right)$ is the Laguerre polynomial defined in [40, Eq. (2.66)].

Proof: See Appendix B.$\hfill\blacksquare$

By applying Lemma 1, we can obtain the CDF and probability density function (PDF) of $\gamma_{D}$ in (7), respectively, as

Corollary 1 (Asymptotic Analysis): For large $N$, the average received SNR at the legitimate user $D$ is obtained as

Proof: We can infer from (7) and Lemma 1 that $\mathbb{E}\{\gamma_{D}\}={\rho_{d}}\mathbb{E}\{|{\rm A}|^{2}\}={\rho_{d}}k(1+k){\theta}^{2}$ [41]. By substituting the shape and scale parameters, the proof is completed. $\hfill\blacksquare$

Remark 1: From Corollary 1, we see that $\mathbb{E}\{\gamma_{D}\}$ scales with $K$ and $N^{2}$, which implies that deploying more transmit antennas and RIS reflecting elements both increase the average received SNR at the legitimate user, while the impact of $N$ is more dominant. Such a “squared improvement” in terms of $N$ is due to the fact that the optimal reflect beamforming attained in Theorem 1 not only enables the system to achieve a beamforming gain of $KN$ in the $S$-RIS link, but also acquires an additional gain of $N$ by coherently collecting signals in the RIS-$D$ link.

Remark 2: From Corollary 1, it is also found that $\mathbb{E}\{\gamma_{D}\}$ is increasing with respect to (w.r.t.) the Rician factor $\epsilon>0$. In other words, the average received SNR at the legitimate user is greater when the proportion of the LoS component in the RIS-$D$ link is higher. It is worth noting that the average SNR in (12) can be upper bounded as $\mathbb{E}\{\gamma_{D}\}\leq\mathbb{E}\{\gamma_{D}\}^{\rm U}={\rho_{d}}\mu_{D}\nu KN^{2}$ which holds for large $N$ and $\epsilon$ since $\lim_{\epsilon\to\infty}{\frac{\left(L_{1/2}\left(-\epsilon\right)\right)^{2}}{\epsilon+1}}=\frac{4}{\pi}$.

Before calculating the effective SNR of the independent and homogeneous PPP distributed eavesdroppers, we first derive the SNR of the $m$-th eavesdropper $E_{m}$. The signal received at $E_{m}$ is formulated as

where $n_{E_{m}}\sim\mathcal{CN}(0,\sigma_{E}^{2})$ is AWGN at $E_{m}$ with variance $\sigma_{E}^{2}$. The received SNR at $E_{m}$ is given in the following proposition.

Proposition 1: The received SNR at $E_{m}$ is expressed as

where $\rho_{e}\triangleq\frac{P_{T}}{\sigma_{E}^{2}}$ denotes the transmit SNR and we define the RV $Z_{E_{m}}\!\triangleq\!\sum_{n=1}^{N}{h_{R\!E_{m}}^{\ast}\left(n\right){\rm e}^{-j\angle h_{R\!D}^{\ast}\left(n\right)}}$.

Proof: See Appendix C.$\hfill\blacksquare$

According to Proposition 1, we present Lemma 2 before deriving the distribution of $\gamma_{E_{m}}$.

Lemma 2: The RV $Z_{E_{m}}$ follows a complex Gaussian distribution with mean $M_{E_{m}}$ and variance $V_{E_{m}}$, given by

where $\delta_{1}\!=\!\sin{\psi_{R\!D}^{a}}\sin{\psi_{R\!D}^{e}}\!-\!\sin{\psi_{R\!E_{m}}^{a}}\sin{\psi_{R\!E_{m}}^{e}}$ and $\delta_{2}\!=\!\cos{\psi_{R\!D}^{e}}\!-\!\cos{\psi_{R\!E_{m}}^{e}}$.

Proof: See Appendix D.$\hfill\blacksquare$

As disclosed in Lemma 2, we conclude that $\gamma_{E_{m}}$ is a non-central Chi-squared RV with two degrees of freedom. Then, the CDF of $\gamma_{E_{m}}$ is given by

where $Q_{1}(\cdot,\cdot)$ is the first-order Marcum $Q$-function [42], and

In the case of non-colluding eavesdroppers, the eavesdropper with the strongest channel dominates the secrecy performance. Thus, the corresponding CDF of the eavesdropper SNR is derived as

where $({\rm a})$ follows from the i.i.d. characteristic of the eavesdroppers’ SNRs and their independence from the point process ${\Phi_{e}}$, $({\rm b})$ follows from the probability generating functional (PGFL) of the PPP [43, Eq. (4.55)], and $({\rm c})$ is obtained by using $\mu_{E_{m}}=\beta_{0}r^{-\alpha_{2}}$ and defining the parameters

From the characterization in [42, Eq. (2)], we have the following approximation for the Marcum $Q$-function in (20): $Q_{1}\left(\varpi,\Xi\sqrt{x}r^{\frac{\alpha_{2}}{2}}\right)\simeq{\rm exp}\left[-{\rm e}^{v\left(\varpi\right)}\left(\Xi\sqrt{x}r^{\frac{\alpha_{2}}{2}}\right)^{\mu\left(\varpi\right)}\right]$, where $v\left(\varpi\right)$ and $\mu\left(\varpi\right)$ are polynomial functions of $\varpi$ defined as $v\left(\varpi\right)\!=\!-0.840\!+\!0.327\varpi\!-\!0.740\varpi^{2}\!+\!0.083\varpi^{3}\!-\!0.004\varpi^{4}$ and $\mu\left(\varpi\right)\!=\!2.174\!-\!0.592\varpi\!+\!0.593\varpi^{2}\!-\!0.092\varpi^{3}\!+\!0.005\varpi^{4}$. Then, (20) is further calculated as

where the last equality is obtained from [46, Eq. (3.326)] with the definitions $t_{0}=\frac{2\pi\lambda_{e}}{\frac{\alpha_{2}}{2}\mu\left(\varpi\right){\rm e}^{\frac{4v\left(\varpi\right)}{\alpha_{2}\mu\left(\varpi\right)}}\Xi^{\frac{4}{\alpha_{2}}}}$, $t_{1}=\frac{2}{\frac{\alpha_{2}}{2}\mu\left(\varpi\right)}$, $t_{2}={\rm e}^{v\left(\varpi\right)}\Xi^{\mu\left(\varpi\right)}{r_{e}}^{\frac{\alpha_{2}}{2}\mu\left(\varpi\right)}$, $t_{3}=\frac{\mu\left(\varpi\right)}{2}$, and $t_{4}=\frac{2}{\alpha_{2}}$.

Therefore, the PDF of the overall eavesdropper SNR can be further derived from (23) as

A popular metric for quantifying PLS is the SOP, which is defined as the probability that the instantaneous secrecy capacity falls below a target secrecy rate $C_{\rm th}$. Mathematically, the SOP is evaluated by

where $\varphi\triangleq{\rm e}^{C_{\rm th}}$. By substituting (10) and (24) into (25), the SOP can be easily expressed as follows

where $I_{1}$ and $I_{2}$ are defined as

However, it is difficult to directly compute an accurate closed-form expression for (26), because (27) and (28) both involve an intractable integral. In order to analyze the secrecy performance, an approximate closed-form expression for the SOP is presented in Proposition 2.

Proposition 2: When $r_{e}\rightarrow\infty$, the SOP can be approximated as²²2The assumption of a large $r_{e}$, similar to [44][45], only denotes an upper limit of distance and does not mean that the eavesdroppers must be far away from the RIS. Also, we will illustrate that the insights we obtained under this assumption are still applicable when $r_{e}$ is relatively small in the simulations.

where $G_{s,t}^{m,n}\left(z\right)$ is Meijer’s $G$ function [46], $p,q\in\mathbb{Z}^{+}$, ${p}/{q}=\alpha_{2}$, and $\Delta=\left[0,\frac{1}{p},\ldots,\frac{p-1}{p},\frac{k}{4q},\frac{k+1}{4q},\ldots,\frac{k+4q-1}{4q}\right]$.

Proof: See Appendix E.$\hfill\blacksquare$

Proposition 2 provides an explicit relation between the secrecy outage probability and various system parameters. A number of interesting points can be made based on this expression.

Remark 3: The SOP in (29) is not a function of the transmit power $P_{T}$ at the base station, which implies that increasing $P_{T}$ does not enhance the system’s secrecy performance. This is intuitive since an increase in $P_{T}$ would yield a proportional increase in the transmit SNRs at both the legitimate user and eavesdroppers.

Proof: According to Proposition 2, we see that the transmit power $P_{T}$ affects only the term $\frac{t_{0}^{p}}{\rho_{d}^{2q}}$ in (29), which is given by

This equality shows that the SOP depends on the ratio of the transmit SNRs at the legitimate user and the eavesdroppers, i.e., $\frac{\rho_{e}}{\rho_{d}}$, regardless of the specific values of $P_{T}$. $\hfill\blacksquare$

Remark 4: The SOP in (29) is mainly affected by the number of RIS reflecting elements, $N$, while the impact of the number of transmit antennas, $K$, is marginal. This is readily checked by calculating $\frac{t_{0}^{p}}{\theta^{4q}}$ in (29) because the other terms are obviously irrelevant to $K$. We obtain $\frac{t_{0}^{p}}{\theta^{4q}}\!\!=\!\!\left(\frac{2\pi\lambda_{e}}{\frac{\alpha_{2}}{2}\mu\left(\varpi\right){\rm e}^{\frac{4v\left(\varpi\right)}{\alpha_{2}\mu\left(\varpi\right)}}}\right)^{p}\!\!\!\!\frac{1}{\left(\Xi\theta\right)^{4q}}$, where $\Xi\theta\!=\!\frac{\chi}{\sqrt{N}}$ depends only on $N$, since the coefficient $\chi$ is independent of $N$ and $K$.

In addition, Proposition 2 is a general analysis of the SOP for any path loss exponent $\alpha_{2}$. Some specific case studies are reported below. Note that other values of $\alpha_{2}$ also admit closed-form expressions using (29).

Corollary 2: For the special case of $\alpha_{2}=2$, i.e., $p=2$ and $q=1$, which corresponds to free space propagation [47], the SOP in (29) reduces to

Corollary 3: For the special case of $\alpha_{2}=4$, i.e., $p=4$ and $q=1$, which is a common practical value for the path-loss exponent in outdoor urban environments [47], the SOP in (29) simplifies to the following expression

where $K_{\nu}\left(\cdot\right)$ denotes the $\nu$-th-order modified Bessel function of the second kind [46, Eq. (8.407)].

From (32), it is obvious that the SOP is a monotonically increasing function w.r.t. $\frac{t_{0}}{\sqrt{\rho_{d}}\theta}=\sqrt{\frac{\rho_{e}}{\rho_{d}}}\lambda_{e}d_{R\!D}^{2}\beta\left(N,\epsilon\right)$ with fixed $k$, where $\beta\left(N,\epsilon\right)$ consists of the parameters $N$, $\epsilon$, and constant terms. Some new observations can thus be obtained in addition to the results presented in Remark 3 and Remark 4. We evince that the SOP increases with the density parameter $\lambda_{e}$, which implies that a larger density of randomly located eavesdroppers leads to a negative effect on the secrecy performance. Moreover, we can also see that the SOP is only related to the distance of the RIS-$D$ link, i.e., $d_{R\!D}$. Therefore, when the locations of the eavesdroppers are unknown, this suggests that the RIS should be deployed closer to the legitimate user than to the base station.

In order to derive the secrecy diversity order and gain further insights, we adopt the analytical framework proposed in [48] where the secrecy diversity order is defined as follows

where ${\rm SOP}^{\infty}$ represents the asymptotic value of the SOP in (29) for $\rho_{d}\rightarrow\infty$, and the transmit SNR $\rho_{e}$ is fixed.

According to [49, Eq. (07.34.06.0006.01)], the SOP in (29) can be expanded as

where $x\!=\!\!\frac{\left(t_{0}\Gamma\left(t_{1}\right)\varphi^{t_{4}}\right)^{p}}{p^{p}\left(4q\sqrt{\rho_{d}}\theta\right)^{4q}}\!\rightarrow\!0$, and $\mathcal{O}$ denotes higher order terms.

When the transmit SNR from $S$ to $D$ is sufficiently large, i.e., $\rho_{d}\rightarrow\infty$, only the dominant terms $l=0$ and $l=1$ in the summation of (34) are retained, which yields the asymptotic SOP as expressed in (35),

where the last step is calculated by applying Gauss’ multiplication formula [50, Eq. (6.1.20)].

Remark 5: By substituting (35) into (33), the secrecy diversity order is obtained as $\frac{2}{\alpha_{2}}$, which only depends on the path loss exponent of the RIS-to-ground links. This implies that the secrecy diversity order of this system improves when the RIS is deployed to provide better LoS links to the terminals.

The ESC is an alternative fundamental metric that denotes the statistical average of the secrecy rate over fading channels, which is mathematically expressed as

Given the received SNRs at both the legitimate user and eavesdroppers in Section III, we first derive an approximate expression for the ESC in the following proposition.

Proposition 3: The ESC for the RIS-assisted system is evaluated as

where $R_{D}$ and $R_{E}$ are the ergodic rates of the legitimate user $D$ and the eavesdroppers $E$, respectively, and are expressed as

Proof: Using Jensen’s inequality, an effective approximation of the ESC can be written as [51][52]

where $R_{D}$ and $R_{E}$ are respectively calculated as follows