Performance Analysis of Reconfigurable Intelligent Surface (RIS)-Assisted Satellite Communications: Passive Beamforming and Outage Probability

Abstract

Reconfigurable intelligent surfaces (RISs), which consist of numerous passive reflecting elements, have emerged as a prominent technology to enhance energy and spectral efficiency for future wireless networks. RISs have the capability to intelligently reconfigure the incident wave, reflecting it towards the intended target without requiring energy for signal processing. Consequently, they have become a promising solution to support the demand for high-throughput satellite communication (SatCom) and enhanced coverage for areas inaccessible to terrestrial networks. This paper presents an asymptotic analysis of an RIS-assisted SatCom system. In this system, an unmanned aerial vehicle equipped with an RIS operates as a mobile reflector between a satellite and users. In particular, a passive beamformer is designed with the aim of asymptotically attaining optimal performance, considering the limitations imposed by practical SatCom systems. Moreover, the closed-form expressions for the ergodic achievable rate and outage probability are derived considering the proposed passive beamforming technique. Furthermore, we extend the system model to a multicast system and asymptotically analyze the optimality of the proposed scheme, leveraging the derived asymptotic results in the unicast system. The results of the simulations confirm that our analyses can precisely and analytically assess the performance of the RIS-assisted SatCom system, confirming the asymptotic optimality of the proposed scheme.

1. Introduction

The demand for high-throughput satellite communication (SatCom) and enhanced coverage for areas inaccessible to terrestrial and nonterrestrial networks has been increasing rapidly, driven mainly by the Internet of Things and upcoming 6G applications, such as drones, vehicles, and terrestrial/nonterrestrial sensors [1,2,3]. Accordingly, reconfigurable intelligent surfaces (RISs) have been investigated to support this growing demand for SatCom systems [4,5,6,7]. RISs intelligently manipulate the amplitude, phase, and frequency of the incident signals and reflect them to a target destination using passive reflecting elements [8,9,10]. In particular, an RIS provides the capability to control the frequency of the incident electromagnetic wave, allowing a controlled frequency shift to the incident RF signal [11]. As RISs are composed of massive reflecting elements, typically constructed from passive components like inductors, capacitors, and resistors, they have the capability to control the phase of the incident signal without requiring additional energy consumption. In this regard, the utilization of an unmanned aerial vehicle (UAV) equipped with RIS has garnered considerable attention in recent years because most UAVs are battery-limited and must be operated energy-efficiently to be power-sustainable in the network [12,13]. A UAV equipped with an RIS can complement existing SatCom systems and enhance the network capacity by covering inaccessible terrestrial/nonterrestrial areas and providing additional capacity to satisfy the steep increase in data requirements [14].

Prior studies [15,16,17] have investigated the challenges of SatCom systems in which a UAV operates as a mobile relay between the satellite and users. In [15], a relay structure for a UAV between the satellites and the ground is examined. A blind beam-tracking approach for SatCom systems was proposed in [16] when a UAV is equipped with a large-scale antenna array and operates as a relay between the satellite and terrestrial networks. Another study [17] analyzed the performance of SatCom systems in which a UAV is equipped with a multi-antenna array and operates as a moving decode-and-forward relay. However, in [15,16,17], a UAV was considered as a mobile active relay that consumes additional energy due to the weight of active elements and signal processing, which is an impractical assumption because UAVs are battery-limited and must be operated energy-efficiently. Also, the authors in [7,18] analyzed the performance of SatCom systems with a UAV equipped with an RIS. However, these works focus on a scenario where all users share the same frequency and consider only the unicast SatCom system. In contrast, this paper examines a scenario where the earth station (ES) allocates different frequencies to each user and analyzes the performance of both unicast and multicast systems.

Meanwhile, in [4], the authors proposed a joint optimization problem that considers the beamforming weights at the satellite and the phase shift at the RIS. In this work, the authors proposed an alternative optimization scheme because the optimized beamforming weight vectors at the satellite and BS and phase shifters at the RIS are coupled. In [5], the authors focused on designing an RIS-assisted SatCom system for Internet of Things (IoT) networks. The primary objective of their study was to improve energy efficiency within the RIS-assisted SatCom system. Furthermore, in [6], the authors tackled the challenge of maximizing the channel capacity in an RIS-assisted SatCom system. They formulated a joint optimization problem that involves power allocation and RIS phase shift to achieve this goal and proposed a sub-optimal algorithm because the problem was formulated as nonconvex due to coupled variables. However, those studies [4,5,6] did not fully address the practical limitations associated with estimating individual channels between the satellite and the RIS, as well as between the RIS and the UE, which remain significant challenges in practical SatCom scenarios. Therefore, it is essential to analyze the performance bounds achievable when utilizing only the practically available information in an RIS-assisted SatCom system.

The main contribution of this work lies in conducting an asymptotic analysis of the data rate within a SatCom system assisted by RIS. Also, the asymptotic analysis takes into consideration the practical RIS environment and its limitations. To address the aforementioned limitations, we present an asymptotic analysis of an RIS-assisted SatCom system that employs a UAV equipped with an RIS. In this system, the UAV acts as a mobile reflector between a satellite and users, while relying solely on the practically available information. In particular, we first derive the asymptotic upper bound of the achievable rate and propose a passive beamformer that can asymptotically achieve the optimality by considering the presence of numerous elements on the RIS. We then derive the closed-form expressions for the ergodic achievable rate and outage probability using the proposed passive beamforming technique. Our analyses can accurately determine the performance of the SatCom system without requiring extensive simulations. Leveraging the derived asymptotic results, we extend our results to cover an additional scenario, such as an RIS-assisted SatCom system that enables a UAV to serve multiple user equipments (UEs) simultaneously. From the simulation results, we can observe that the proposed scheme asymptotically achieves optimal performance and also achieves its upper bound, verifying its asymptotic optimality.

The remainder of this paper is organized as follows. The system model is described in Section 2. The asymptotic performance and outage probability are discussed in Section 3, and the performance of a SatCom system with an RIS serving multiple UEs is analyzed in Section 4. The simulation results are presented in Section 5 and the concluding remarks are presented in Section 6.

Notations: Throughout the paper, matrices are represented by boldface uppercase symbols, while vectors are denoted by boldface lowercase symbols, and ${\mathit{I}}_N$ and ${\mathbf{1}}_{M\times N}$ are, respectively, a size-N identity matrix and an $M\times N$ matrix of ones. The transpose, Hermitian transpose, and conjugate operators are ${\left(·\right)}^{\mathrm{T}}$, ${\left(·\right)}^{\mathrm{H}}$, and ${\left(·\right)}^{*}$, respectively. The operator $\mathrm{diag}\left(x_1,\cdots ,x_n\right)$ is a diagonal operator that constructs a diagonal matrix from a vector, ${\left[x_1,\cdots ,x_n\right]}^{\mathrm{T}}$. The phase and amplitude of a complex number, y, are denoted by $\angle y$ and $\left|y\right|$, respectively, and $mod$ is the modulo operation. $\mathcal{O}\left(·\right)$ is the big O notation, and $\mathrm{E}\left[·\right]$ and $\mathrm{Var}\left[·\right]$ denote, respectively, expectation and variance operators. $\mathcal{CN}\left(m,{\sigma }^2\right)$ denotes a complex Gaussian distribution with mean m and variance ${\sigma }^2$, and $\mathcal{L}\mathcal{N}\left(m,{\sigma }^2\right)$ and ${\mathcal{L}}^2\mathcal{N}\left(m,{\sigma }^2\right)$ denote a lognormal distribution and double-lognormal distribution, respectively.

2. System Model

Consider a SatCom system comprising a single ES serving K single-antenna user equipments (UEs), a single multibeam satellite, and K UAVs with individual RISs (i.e., K RISs) equipped with N reflecting elements. In our system model, a UAV equipped with an RIS operates as a mobile reflector between the satellite and the UEs (even though UAV-based communications may have flight-time issues caused by battery constraints, the proposed scheme can also apply to an RIS attached to manned aircraft or building surfaces, which can be relatively free from battery constraints), as shown in Figure 1. The satellite is equipped with a reflective array antenna consisting of K feeds (i.e., a single feed per beam) in which a transparent satellite payload is considered. The data for the K UEs are multiplexed in frequency at the ES and transmitted to the satellite through the feeder link operating in the optical band. Subsequently, at the satellite side, the received signal is demultiplexed, amplified, translated to RF streams, and then transmitted to the UEs via the corresponding UAVs through multiple beams. Each UAV can passively reflect the incident signal from the satellite using an RIS, and each reflecting element induces a phase shift of the incident wave (i.e., passive beamforming). We assume that the ES allocates a suitable RIS to each UE (i.e., one RIS per UE) and sends a control signal, including the reflection phase shift information, to each RIS through a dedicated wireless control link. In Section 3, we consider the unicast system, i.e., each RIS serves each corresponding UE. Subsequently, in Section 4, the unicast system is extended to the multicast system, i.e., each RIS serves multiple UEs in a corresponding multicast group.

Note that multibeam SatCom systems can use the advantages of beamspace isolation and resource allocation for improved spectral efficiency [19]. Depending on the location of the UAV, the UAVs located far away from each other will not suffer from interbeam interference due to the beamspace isolation, whereas the UAVs with similar locations can suffer from interbeam interference, which can degrade the performance. To tackle this challenge, we make the assumption that the ES allocates orthogonal frequencies among UAVs/UEs located in similar positions, thereby supporting dynamic spectrum and power allocation [20], as shown in Figure 1. Consequently, multibeam satellite systems can be an interbeam interference-free environment. The received signal at UE k is then obtained as

$$y_k=\sqrt{{\rho }_k}{\mathit{f}}_k^{\mathrm{H}}{\mathbf{\Phi }}_k{\mathit{g}}_k{x}_k+n_k,$$

(1)

where ${\rho }_k$ is the signal-to-noise ratio (SNR) for UE k and ${\mathit{f}}_k\in {\mathbb{C}}^{N\times 1}$ and ${\mathit{g}}_k\in {\mathbb{C}}^{N\times 1}$ are, respectively, the fading channels between UE k and RIS k and between the satellite and RIS k. Moreover, $x_k$ and $n_k$ are the transmit signal and additive complex Gaussian noise with zero mean and unit variance, respectively, and ${\mathbf{\Phi }}_k\in {\mathbb{C}}^{N\times N}$ is a diagonal reflection matrix functioning as a passive beamformer. The reflection matrix is determined from the control signal transmitted by the ES, and ${\mathbf{\Phi }}_k$ can be obtained as follows:

$${\mathbf{\Phi }}_k=\mathrm{diag}\left(\sqrt{{\alpha }_{k,1}}{e}^{j{\theta }_{k,1}},\sqrt{{\alpha }_{k,2}}{e}^{j{\theta }_{k,2}},\cdots ,\sqrt{{\alpha }_{k,N}}{e}^{j{\theta }_{k,N}}\right),$$

(2)

where $\sqrt{{\alpha }_{k,n}}\in \left[0,1\right]$ and ${\theta }_{k,n}\in [0,2\pi )$ are, respectively, the amplitude and phase shifting response of the reflection element n in RIS k. Assuming lossless reflection power at all the RISs (i.e., ${\alpha }_{k,n}=1,\forall k,n$), the reflection matrix is given by

$${\mathbf{\Phi }}_k=\mathrm{diag}\left(e^{j{\theta }_{k,1}},e^{j{\theta }_{k,2}},\cdots ,e^{j{\theta }_{k,N}}\right),\forall k.$$

(3)

In (1), the signals directly transmitted from the satellite to the UEs are neglected because of severe path loss and shadowing. For a practical air-to-ground (A2G) channel, ${\mathit{f}}_k=\sqrt{{\zeta }_k}{\widehat{\mathit{f}}}_k$ is generated by correlated Rician fading, which consists of both line-of-sight (LoS) and non-line-of-sight (NLoS) paths, where ${\zeta }_k$ is the path loss of the RIS–UE link and ${\widehat{\mathit{f}}}_k$ is Rician fading, as follows:

$$\begin{array}{cc}\hfill {\zeta }_{k,\mathrm{dB}}& =-10{log}_{10}{\left({\displaystyle \frac{4\pi d_k^{\mathrm{ru}}}{\lambda }}\right)}^2-{\eta }_{\mathrm{L}}{P}_k^{\mathrm{L}}-{\eta }_{\mathrm{NL}}{P}_k^{\mathrm{NL}},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\widehat{\mathit{f}}}_k& =\sqrt{{\displaystyle \frac{{\kappa }_k}{{\kappa }_k+1}}}{\overline{\mathit{f}}}_k+\sqrt{{\displaystyle \frac{1}{{\kappa }_k+1}}}{\tilde{\mathit{f}}}_k.\hfill \end{array}$$

(5)

Here, ${\zeta }_{k,\mathrm{dB}}$ is the path loss in dB scale, $d_k^{\mathrm{ru}}$ is the distance from RIS k to UE k, $\lambda$ represents the wavelength of a signal, and the average additional losses for LoS and NLoS conditions are denoted as ${\eta }_{\mathrm{L}}$ and ${\eta }_{\mathrm{NL}}$, respectively. Based on [21], the presence of a LoS path is contingent upon the distance between the RIS and the UE. Then, the probability of LoS is expressed as follows:

$$P_k^{\mathrm{L}}={\displaystyle \frac{1}{1+c_{1}exp\left[-c_2\left(arctan\left(\frac{d_k^{\mathrm{v}}}{d_k^{\mathrm{h}}}\right)\right)\right]}},$$

(6)

where $d_k^{\mathrm{v}}$ and $d_k^{\mathrm{h}}$ are, respectively, vertical and horizontal distances from UAV k and UE k, and $c_1$ and $c_2$ are determined by the particular configuration of the environment [21]. Also, $P_k^{\mathrm{NL}}$ is the probability of NLoS as given by $P_k^{\mathrm{NL}}=1-P_k^{\mathrm{L}}$. In (5), ${\kappa }_k$ is the Rician factor between UE k and RIS k, and ${\overline{\mathit{f}}}_k\in {\mathbb{C}}^{N\times 1}$, which is composed of deterministic values whose entries are determined depending on the locations of RIS k and UE k. By defining ${\overline{f}}_{k,n}$ as the n-th element of ${\overline{\mathit{f}}}_k$, we have

$${\overline{f}}_{k,n}=exp\left(-j2\pi d_{k,n}^{\mathrm{ru}}/\lambda \right),$$

(7)

where $d_{k,n}^{\mathrm{ru}}$ is the distance between UE k and RIS element n of RIS k [22]. Considering the spatial correlation in the RIS–UE channel, we define ${\tilde{\mathit{f}}}_k={\mathit{G}}_k^{1/2}{\breve{\mathit{f}}}_k$, where ${\mathit{G}}_k\in {\mathbb{C}}^{N\times N}$ is the spatial correlation matrix at RIS k and ${\breve{\mathit{f}}}_k\in {\mathbb{C}}^{N\times 1}$ is composed of NLoS components, with each entry being an i.i.d. complex Gaussian random variable with a mean of zero and a variance of one. For notational simplicity, we define ${\overline{\beta }}_k=\sqrt{{\zeta }_k{\kappa }_k/\left({\kappa }_k+1\right)}$ and ${\tilde{\beta }}_k=\sqrt{{\zeta }_k/\left({\kappa }_k+1\right)}$. In (1), the channel gain vector between the satellite and the RIS, ${\mathit{g}}_k$, includes a multibeam pattern, atmospheric fading, and phase rotations, which can be decomposed as follows [23]:

$${\mathit{g}}_k={\mathit{D}}_k{\mathit{b}}_k,$$

(8)

where ${\mathit{D}}_k\in {\mathbb{C}}^{N\times N}$ is a rain fading matrix and ${\mathit{b}}_k\in {\mathbb{C}}^{N\times 1}$ is the coefficient vector that represents the beam radiation pattern gain with path loss component. As the channel gains between the satellite and N elements on RIS k are highly correlated, we have the followings [24]:

$$\begin{array}{cc}\hfill {\mathit{b}}_k& ={\xi }_k{\sqrt{a}}_k{\mathbf{1}}_{N\times 1},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\mathit{D}}_k& =\left|c_k\right|exp\left(j{\psi }_k\right){\mathbf{I}}_N,\hfill \end{array}$$

(10)

where ${\xi }_k$ and $a_k$ are the coefficients representing, respectively, the effects of the path loss and beam pattern as follows:

$$\begin{array}{c}{\xi }_k={\displaystyle \frac{\lambda }{4\pi d_k^{\mathrm{sr}}}},\end{array}$$

(11)

$$\begin{array}{c}{a}_k=G_{\max }{\left({\displaystyle \frac{J_1\left(u_k\right)}{2u_k}}+36{\displaystyle \frac{J_3\left(u_k\right)}{u_k^3}}\right)}^2.\end{array}$$

(12)

Here, $d_k^{\mathrm{sr}}$ is the distance from the satellite to RIS k, $G_{\max }$ is the maximum gain of the satellite beam, $J_n\left(·\right)$ is the first kind Bessel function of order n, and $u_k=2.07123{\displaystyle \frac{sin{\varphi }_k}{sin{\varphi }_{3\mathrm{dB}}}}$. Also, ${\varphi }_k$ and ${\varphi }_{3\mathrm{dB}}$ are the angle between the beam center and RIS k and the 3 dB angle of the beam, respectively. ${\psi }_k$ is the phase of the fading channel at RIS k that is assumed to be distributed uniformly over $[0,2\pi )$ as in [23]. $\left|c_k\right|$ is the rain fading channel gain that follows a double-lognormal distribution, such as $\left|c_k\right|\sim {\mathcal{L}}^2\mathcal{N}\left({\eta }_k,{\sigma }_k^2\right)$, whose probability density function (PDF) and cumulative distribution function (CDF), respectively, follow [25]

$$\begin{array}{cc}\hfill f_{\left|c_k\right|}\left(x\right)& =-{\displaystyle \frac{exp\left(-\frac{{\left(ln\left(-lnx\right)-{\eta }_k\right)}^2}{2{\sigma }_k^2}\right)}{\sqrt{2\pi }{\sigma }_{k}xlnx}},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill F_{\left|c_k\right|}\left(x\right)& =Q\left({\displaystyle \frac{ln\left(-lnx\right)-{\eta }_k}{{\sigma }_k}}\right),\hfill \end{array}$$

(14)

where $0<x<1$ and $Q\left(·\right)$ is a Gaussian Q-function. Rain fading and LoS components in satellite communications are the main channel factors distinguishing them from terrestrial communication systems.

The cascaded channel from the satellite to UE k via RIS k is given by

$$\begin{array}{cc}\hfill h_k& ={\mathit{f}}_k^{\mathrm{H}}{\mathbf{\Phi }}_k{\mathit{g}}_k\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & ={\xi }_k{\sqrt{a}}_k\left|c_k\right|exp\left(j{\psi }_k\right){\mathit{f}}_k^{\mathrm{H}}{\mathbf{\Phi }}_k{\mathbf{1}}_{N\times 1}.\hfill \end{array}$$

(16)

Then, the SNR at UE k and the achievable sum-rate can be obtained, respectively, as follows:

$$\begin{array}{c}{\gamma }_k={\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left|{\sum }_{n=1}^N{f}_{k,n}^{*}{e}^{j{\theta }_{k,n}}\right|}^2,\end{array}$$

(17)

$$\begin{array}{c}R={\sum }_{k=1}^{K}log\left(1+{\gamma }_k\right),\end{array}$$

(18)

where $f_{k,n}$ represents the n-th element of a vector ${\mathit{f}}_k$. We can observe from (17) that, when ${\theta }_{k,n}=-\angle f_{k,n}^{*}\forall n$, ${\gamma }_k$ attains its maximum value, ${\tilde{\gamma }}_k$, which is given as follows:

$${\tilde{\gamma }}_k={\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left({\sum }_{n=1}^N\left|f_{k,n}\right|\right)}^2.$$

(19)

In this case, the achievable rate of UE k is given by ${\tilde{R}}_k=log\left(1+{\tilde{\gamma }}_k\right).$ Note that RISs are generally passive; moreover, they are typically not equipped with RF chains and are installed on UAVs with limited batteries. Thus, it is hard to accurately estimate RIS-related channel information, such as ${\mathit{f}}_k$, in the considered SatCom system, and UE k cannot practically achieve the data rate of ${\tilde{R}}_k$. Given that it is practically feasible to estimate the partial channel state information (CSI) of RIS-related channels [23], we propose a practical passive beamformer that is solely based on the partial CSI while achieving optimal performance, i.e., ${\tilde{R}}_k$, asymptotically.

3. Asymptotic Performance Analysis in Unicast System

We analyze the asymptotic data rate considering the practical SatCom system according to the following steps. Considering the achievable rate ${\tilde{R}}_k$, we first analyze the upper bound of ${\tilde{R}}_k$ as N increases to infinity. We then propose a practical passive beamformer and verify the asymptotic optimality of the proposed scheme. We finally derive the mean and variance of the achievable data rate of the proposed scheme.

From (19), the achievable rate of UE k, ${\tilde{R}}_k$, and its upper bound, ${\widehat{R}}_k$, are obtained as

$$\begin{array}{cc}\hfill {\tilde{R}}_k& =log\left(1+{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left({\sum }_{n=1}^N\left|f_{k,n}\right|\right)}^2\right)\hfill \\ \hfill & \underset{\left(a\right)}{\le }log\left(1+N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left({\overline{\beta }}_k+{\tilde{\beta }}_k{\displaystyle \frac{{\sum }_n\left|{\tilde{f}}_{k,n}\right|}{N}}\right)}^2\right)\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & \stackrel{\Delta }{=}{\widehat{R}}_k,\hfill \end{array}$$

(21)

where ${\tilde{f}}_{k,n}$ is the n-th element of a vector, ${\tilde{\mathit{f}}}_k$, and (a) is obtained from the following inequality: ${\sum }_n\left|f_{k,n}\right|\le {\sum }_n{\overline{\beta }}_k\left|{\overline{f}}_{k,n}\right|+{\sum }_n{\tilde{\beta }}_k\left|{\tilde{f}}_{k,n}\right|$, where $\left|{\overline{f}}_{k,n}\right|=1$ from (7). As N increases to infinity, we derive the asymptotic upper bound of ${\widehat{R}}_k$ as follows:

$$\begin{array}{cc}\hfill {\widehat{R}}_k& \underset{\left(b\right)}{\le }log\left(1+N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left({\overline{\beta }}_k+{\tilde{\beta }}_k\right)}^2\right)\hfill \\ \hfill & \approx log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\overline{\beta }}_k^2\right)+2log\left(1+\sqrt{{\displaystyle \frac{1}{{\kappa }_k}}}\right)\hfill \\ \hfill & \underset{\left(c\right)}{\approx }log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\overline{\beta }}_k^2\right),\hfill \end{array}$$

(22)

where (b) results from Jensen’s inequality: ${\left({\sum }_n\left|{\tilde{f}}_{k,n}\right|/N\right)}^2\le {\sum }_n{\left|{\tilde{f}}_{k,n}\right|}^2/N$. Here, the sample mean of $|{\tilde{f}}_{k,n}{|}^2$ can be obtained from the asymptotic channel hardening effect [26], under the assumption that the spectral norm of ${\mathit{G}}_k$ is bounded as

$${\displaystyle \frac{{\sum }_{n=1}^N{\left|{\tilde{f}}_{k,n}\right|}^2}{N}}\to 1,$$

(23)

almost surely as $N\to \infty$. Also, (c) results from a large N and a small $\sqrt{1/{\kappa }_k}$ because the Rician factor, ${\kappa }_k$, will be a large value (in the A2G channel model, the maximum value and the average value of the Rician factor introduced in [27] are 20 dB and 10 dB, respectively), considering a practical A2G channel model [27,28]. We can observe from (22) that, as N increases, the upper bound of the SNR increases with $\mathcal{O}\left(N^2\right)$, and the upper bound of the achievable rate increases with $\mathcal{O}\left(logN\right)$.

Note that an ES can leverage global navigation satellite systems (GNSSs) to accurately estimate the three-dimensional positions of UAVs and UEs [23]. Thus, the ES can estimate the deterministic LoS components, denoted as ${\overline{\mathit{f}}}_{\mathit{k}}$, by using the distance between RIS k and UE k, which is partial CSI; however, this is the only information available at the ES side. Hence, we design a suboptimal passive beamformer exclusively based on ${\overline{\mathit{f}}}_{\mathit{k}}$, as follows:

$${\overline{\theta }}_{k,n}=-\angle {\overline{f}}_{k,n}^{*}.$$

(24)

Using (24), the achievable rate is given as follows:

$${\overline{R}}_k=log\left(1+N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left|{\overline{\beta }}_k+{\displaystyle \frac{{\sum }_n{\tilde{\beta }}_k{\tilde{f}}_{k,n}{e}^{j{\overline{\theta }}_{k,n}}}{N}}\right|}^2\right).$$

(25)

Then, we derive the following lemma from the asymptotic analyses.

Lemma 1. 

On the basis of the scaling law for N, the rate ratio between the achievable rate of the proposed passive beamformer and its performance bound becomes one (i.e., ${\displaystyle \frac{{\widehat{R}}_k}{{\overline{R}}_k}}=1$) as N increases to infinity.

Proof. 

In (25), the term ${\sum }_n{\tilde{f}}_{k,n}{e}^{j{\overline{\theta }}_{k,n}}$ represents the sum of N samples, where each sample is an individual random variable whose mean value is equal to zero because $\mathrm{E}\left[{\tilde{f}}_{k,n}\right]=0,\forall n$. Hence, as N increases, ${\sum }_n{\tilde{f}}_{k,n}{e}^{j{\overline{\theta }}_{k,n}}/N$ approaches zero, and we can then approximate ${\overline{R}}_k$ as follows:

$$\begin{array}{cc}\hfill {\overline{R}}_k& \approx log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left|{\overline{\beta }}_k+{\tilde{\beta }}_k{\displaystyle \frac{{\sum }_n{\tilde{f}}_{k,n}{e}^{j{\overline{\theta }}_{k,n}}}{N}}\right|}^2\right)\hfill \\ \hfill & \approx log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\overline{\beta }}_k^2\right)\stackrel{\Delta }{=}{\dot{R}}_k,\hfill \end{array}$$

(26)

where ${\dot{R}}_k$ is defined as the asymptotic achievable rate for a large N. From (22) and (26), the rate ratio between ${\overline{R}}_k$ and ${\widehat{R}}_k$ becomes one as N increases to infinity, which completes the proof. □

Lemma 1 shows that the channel fading of an RIS-assisted SatCom system exclusively depends on the rain fading channel gain, $\left|c_k\right|$, and the impact of the Rician fading on the achievable rate becomes negligible as N increases. Lemma 1 also proves that the proposed passive beamformer achieves the upper bound of the achievable rate and asymptotically achieves the SNR order of $\mathcal{O}\left(N^2\right)$. Moreover, since ${\tilde{R}}_k\le {\widehat{R}}_k$ from (21), it is proven that the proposed passive beamformer achieves the optimal performance, ${\tilde{R}}_k$, which is only achievable using the full CSI of ${\mathit{f}}_{\mathit{k}}$ when ${\theta }_{k,n}=-\angle f_{k,n}^{*}$. Note that the proposed ${\overline{\theta }}_{k,n}$ can be readily obtained from the estimated value of $d_{k,n}^{\mathrm{ru}}$, leading to a low computational complexity (i.e., constant complexity) to calculate all phase shift values of the RIS. Additionally, the proposed scheme exclusively relies on deterministic LoS channel information (i.e., large-scale information), which exhibits slower variation compared to NLoS channels. Consequently, the proposed scheme does not require frequent feedback for CSI and location information, effectively reducing feedback/signaling overhead and rendering the proposed scheme practically feasible.

Next, we analyze the mean (i.e., ergodic achievable rate) and variance of the asymptotic achievable rate for the proposed passive beamformer. From (26), the asymptotic achievable rate, ${\dot{R}}_k$, can be obtained using (13) as follows:

$$\begin{array}{cc}\hfill {\dot{R}}_k& =log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\overline{\beta }}_k^2\right)+2log\left|c_k\right|\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & =log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\overline{\beta }}_k^2\right)-{\displaystyle \frac{{log}_{2}10}{10}}{z}_1,\hfill \end{array}$$

(28)

as N increases, where $z_1\sim \mathcal{L}\mathcal{N}\left({\mu }_k,{\sigma }_k^2\right)$ and ${\mu }_k={\eta }_k-log(log10)+log20$ [25]. As N increases, the mean and variance of the asymptotic achievable rate are obtained, respectively, as follows:

$$\begin{array}{cc}\hfill \mathrm{E}\left[{\dot{R}}_k\right]& =log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\overline{\beta }}_k^2\right)-{\displaystyle \frac{{log}_{2}10}{10}}{e}^{{\mu }_k+{\displaystyle \frac{{\sigma }_k^2}{2}}},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \mathrm{Var}\left[{\dot{R}}_k\right]& ={\left({\displaystyle \frac{{log}_{2}10}{10}}\right)}^2\left(e^{{\sigma }_k^2}-1\right)e^{2{\mu }_k+{\sigma }_k^2}.\hfill \end{array}$$

(30)

Then, the ergodic sum-rate is asymptotically derived as

$$\mathrm{E}\left[\dot{R}\right]={\displaystyle \frac{2K}{ln2}}\left(lnN-{\displaystyle \frac{ln10}{20}}{e}^{{\mu }_0+{\displaystyle \frac{{\sigma }_0^2}{2}}}\right)+{\sum }_{k}log\left({\rho }_k{\xi }_k^2{a}_k{\overline{\beta }}_k^2\right),$$

(31)

when ${\mu }_k={\mu }_0$ and ${\sigma }_k={\sigma }_0,\forall k$. For notational simplicity, we define ${\dot{m}}_k$ and ${\dot{s}}_k^2$ as the asymptotic mean and variance, respectively, of ${\dot{R}}_k$ (i.e., ${\dot{m}}_k\triangleq \mathrm{E}\left[{\dot{R}}_k\right]$ and ${\dot{s}}_k^2\triangleq \mathrm{Var}\left[{\dot{R}}_k\right]$). On the basis of the scaling law for N, we can observe from (29) that the SNR increases with $\mathcal{O}\left(N^2\right)$ and the ergodic achievable rate increases with $\mathcal{O}\left(logN\right)$. Moreover, (30) shows that the variance of ${\overline{R}}_k$ remains constant regardless of N, indicating the reduced channel hardening effect [29]. Using (29) and (30), we can characterize the outage probability, which is defined as the probability that the instantaneous data rate falls below a target rate, as follows.

Lemma 2. 

As N increases, the probability of the outage event $\left\{{\dot{R}}_k<R_0\right\}$ tends to zero, where $R_0$ represents the target desired rate given by $R_0=\delta ·{\dot{m}}_k$ for $0<\delta \le 1$.

Proof. 

From (28), the outage probability can be obtained as follows:

$$\begin{array}{cc}\hfill \mathrm{Pr}\left\{{\dot{R}}_k<R_0\right\}& =\mathrm{Pr}\left\{{\ddot{m}}_k-{\displaystyle \frac{{log}_{2}10}{10}}{z}_1<R_0\right\}\hfill \\ \hfill & =\mathrm{Pr}\left\{z_1>{\displaystyle \frac{10\left({\ddot{m}}_k-R_0\right)}{{log}_{2}10}}\right\}\hfill \\ \hfill & =Q\left({\displaystyle \frac{ln\left(\frac{10\left({\ddot{m}}_k-R_0\right)}{{log}_{2}10}\right)-{\mu }_k}{{\sigma }_k}}\right),\hfill \end{array}$$

(32)

where ${\ddot{m}}_k=log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\overline{\beta }}_k^2\right)$. Consider that $R_0=\delta {\dot{m}}_k$ for $0<\delta \le 1$, we then have

$${\ddot{m}}_k-R_0=\left(1-\delta \right)log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\overline{\beta }}_k^2\right)+{\displaystyle \frac{\delta {log}_{2}10}{10}}{e}^{{\mu }_k+{\displaystyle \frac{{\sigma }_k^2}{2}}}.$$

(33)

It is observed in (33) that ${\ddot{m}}_k-R_0$ increases as N increases. Therefore, the outage probability $\mathrm{Pr}\left\{{\dot{R}}_k<R_0\right\}$ eventually approaches zero according to the increase in N, which completes the proof. □

Lemma 2 demonstrates that, as N increases, the outage probability decreases, verifying that an RIS can offer reliable SatCom for a large N. Moreover, we can observe from (32) that the outage probability of an RIS-assisted SatCom system can be readily estimated using the deterministic values. Thus, we can assess the outage probability of a SatCom system without requiring extensive simulations.

4. Asymptotic Performance Analysis in a Multicast System

In this section, we extend the system model to a multicast system and analyze the asymptotic optimality of the proposed scheme, leveraging the derived asymptotic results in the unicast system. In particular, we will demonstrate that achieving asymptotic optimality in the multicast system is readily attainable through a simple yet practical passive beamforming approach. Consider a SatCom system consisting of a single multibeam satellite, with K RISs, each capable of simultaneously serving $M_k$ UEs. As the satellite is equipped with a reflective array antenna comprising K feeds, we examine K multicast groups ($M_k$ UEs per group), with each multicast group k being served by RIS k. On the basis of the multicast nature, the achievable rate of multicast group k is obtained as follows [30]:

$$R_{\mathrm{MC},k}=\underset{m\in {\mathcal{M}}_k}{min}{R}_{k,m},$$

(34)

where

$$R_{k,m}=log\left(1+{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left|{\sum }_{n=1}^N{f}_{k,n,m}^{*}{e}^{j{\theta }_{k,n}}\right|}^2\right).$$

(35)

Here, ${\mathcal{M}}_k$ is a set of UEs in multicast group k where $\left|{\mathcal{M}}_k\right|=M_k$, and $f_{k,n,m}$ is the fading channel between reflecting element n of RIS k and UE m. As discussed in (19), optimal reflection performance can be achieved by aligning all phase shift values of RIS with the channel phases from RIS to UE. However, in the multicast system, it is impossible to design the optimal passive beamforming to simultaneously match the phases of channels for multiple UEs.

Based on (34), we can observe that maximizing the achievable rate in the multicast system is synonymous with maximizing the individual rate of each UE. Hence, we introduce a practical passive beamforming approach for the multicast system, wherein the reflecting elements of the RIS are evenly divided and allocated to each UE within a multicast group. In particular, N reflecting elements of each RIS are divided into $M_k$ reflecting groups, and the reflection phases of reflecting group m are obtained as follows:

$${\overline{\theta }}_{k,n}=-\angle {\overline{f}}_{k,n,m}^{*},$$

(36)

for $n\in {\mathcal{N}}_{k,m}$. Here, ${\overline{f}}_{k,n,m}$ is the LoS component of $f_{k,n,m}$ and ${\mathcal{N}}_{k,m}$ is a set of reflecting elements in reflecting group m of RIS k, where ${\sum }_{m=1}^{M_k}\left|{\mathcal{N}}_{k,m}\right|=N,\forall k$. For notational simplicity, we assume that $Nmod{M}_k=0,\forall k$. In order to achieve the SNR order of $\mathcal{O}\left(N^2\right)$ for a large N, N reflecting elements are equally divided into $M_k$ reflecting groups (i.e., ${\displaystyle \frac{N}{M_k}}$ elements per each reflecting group) and, similarly in (26), we have:

$$\begin{array}{cc}\hfill {\overline{R}}_{k,m}& \approx log\left({\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2|{\overline{\beta }}_{k,m}\left({\displaystyle \frac{N}{M_k}}+\sum _{n\notin {\mathcal{N}}_{k,m}}{\overline{f}}_{k,n,m}^{*}{e}^{j{\overline{\theta }}_{k,n}}\right){|}^2\right)\hfill \\ \hfill & \approx log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\overline{\beta }}_{k,m}^2/M_k^2\right),\hfill \end{array}$$

(37)

where ${\overline{\beta }}_{k,m}=\sqrt{{\zeta }_{k,m}{\kappa }_{k,m}/\left({\kappa }_{k,m}+1\right)}$ and ${\kappa }_{k,m}$ and ${\zeta }_{k,m}$ are, respectively, the Rician factor and the path loss between RIS k and UE m. From (34) and (37), we can observe that ${\overline{R}}_{\mathrm{MC},k}$ is determined exclusively depending on ${\overline{\beta }}_{k,m}$. Define the minimum ${\overline{\beta }}_{k,m}$ for all $m\in {\mathcal{M}}_k$ as ${\overline{\beta }}_{k,m^{\prime }}$, then we have:

$${\overline{R}}_{\mathrm{MC},k}=log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\overline{\beta }}_{k,m^{\prime }}^2/M_k^2\right).$$

(38)

Next, we derive a theoretical ideal performance for the multicast system under consideration, with the goal of capturing the asymptotic optimality of the multicast system. In this regard, we define an ideal RIS capable of reflecting signals to multiple UEs simultaneously with perfect phase alignment for all UEs. Then, similarly to (20) and (21), the theoretical ideal performance, denoted as ${\tilde{R}}_{k,m}$, and its upper bound, denoted as ${\widehat{R}}_{k,m}$, can be obtained as follows:

$$\begin{array}{cc}\hfill & {\tilde{R}}_{k,m}=log\left(1+{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left({\sum }_{n=1}^N\left|f_{k,n,m}\right|\right)}^2\right)\hfill \\ \hfill & \le log\left(1+N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left({\overline{\beta }}_{k,m}+{\tilde{\beta }}_{k,m}{\displaystyle \frac{\sum _n\left|{\tilde{f}}_{k,n,m}\right|}{N}}\right)}^2\right)\hfill \\ \hfill & \stackrel{\Delta }{=}{\widehat{R}}_{k,m},\hfill \end{array}$$

(39)

where ${\tilde{\beta }}_{k,m}=\sqrt{{\zeta }_{k,m}/\left({\kappa }_{k,m}+1\right)}$ and ${\tilde{f}}_{k,n,m}$ is the NLoS component of $f_{k,n,m}$. As N increases to infinity, we derive the asymptotic upper bound of ${\widehat{R}}_{k,m}$ following a similar approach as in (22), as follows:

$$\begin{array}{cc}\hfill {\widehat{R}}_{k,m}& \le log\left(1+N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\left({\overline{\beta }}_{k,m}+{\tilde{\beta }}_{k,m}\right)}^2\right)\hfill \\ \hfill & \approx log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\overline{\beta }}_{k,m}^2\right).\hfill \end{array}$$

(40)

From (34) and (40), we can observe that the minimum value of (40) for all $m\in {\mathcal{M}}_k$ will be:

$${\widehat{R}}_{\mathrm{MC},k}=log\left(N^2{\rho }_k{\xi }_k^2{a}_k{\left|c_k\right|}^2{\overline{\beta }}_{k,m^{\prime }}^2\right).$$

(41)

From (38) and (41), it is evident that the proposed passive beamformer achieves a theoretical ideal performance with an SNR order of $\mathcal{O}\left(N^2\right)$, establishing the optimality of the proposed scheme. Note that the proposed passive beamformer in the multicast system can achieve near-ideal performance if the value of N is sufficient relative to $M_k$. This holds true even when the number of passive elements is not infinite, as demonstrated in the simulation results presented in Section 5.

Similarly to (29) and (32), the ergodic achievable rate and its outage probability can be obtained, respectively, as follows:

$$\begin{array}{cc}\hfill & \mathrm{E}\left[{\dot{R}}_{\mathrm{MC},k}\right]=log\left({\left(N/M_k\right)}^2{\rho }_k{\xi }_k^2{a}_k{\overline{\beta }}_{k,m^{\prime }}^2\right)-{\displaystyle \frac{{log}_{2}10}{10}}{e}^{{\mu }_k+{\displaystyle \frac{{\sigma }_k^2}{2}}},\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & \mathrm{Pr}\left\{{\dot{R}}_{\mathrm{MC},k}<R_0\right\}=Q\left({\displaystyle \frac{ln\left(\frac{10\left({\ddot{m}}_{\mathrm{MC},k}-R_0\right)}{{log}_{2}10}\right)-{\mu }_k}{{\sigma }_k}}\right),\hfill \end{array}$$

(43)

where ${\dot{R}}_{\mathrm{MC},k}$ is the asymptotic achievable rate of multicast UE k and

$${\ddot{m}}_{\mathrm{MC},k}=log\left({(N/M_k)}^2{\rho }_k{\xi }_k^2{a}_k{\overline{\beta }}_{k,m^{\prime }}^2\right).$$

(44)

Similarly to the unicast system, (43) shows that the outage probability in the multicast system decreases as N increases, verifying that an RIS can offer reliable multicast SatCom for a large N.

5. Simulation Results

In this section, we conduct comprehensive simulations in terms of the achievable rate ratio, ergodic rate, and outage probability. We consider a practical RIS-assisted SatCom system in which a low earth orbit (LEO) satellite serves a cluster of seen UEs via seven RISs (i.e., one RIS per UE and $K=7$). The parameters used for the simulations are detailed in

Table 1. Simulation parameters.

Parameter	Value
Orbit	LEO (2520 km)
Carrier frequency	2 GHz (L-band)
Bandwidth per beam, $P_k$	20 MHz, 50 dBW
K, $G_{\max }$, ${\varphi }_{3\mathrm{dB}}$	7, $49.5$ dB, $3.2^{\circ }$
UE antenna gain	$38.16$ dB
Boltzmann’s constant	$1.38\times {10}^{-23}$ J/K
Noise temperature at UE	207 K
RIS element layout	2D square lattice (spacing $0.01$ m)
Average rain attenuation	3 dB $({\eta }_k=-1.5617,{\sigma }_k=1,\forall k)$
${\kappa }_k$, $d_k^{\mathrm{v}}$, $d_k^{\mathrm{h}},\forall k$	10, 1 km, $0.1$ km
${\eta }_{\mathrm{L}}$, ${\eta }_{\mathrm{NL}}$, $c_1$, $c_2$	$1,20,15,0.4$

. Statistical averaging is performed over a considerable number of independent runs for all simulations. Here, the label “Estimation” represents the estimated result obtained from our analyses, whereas the label “Simulation” represents the simulated results within the practical RIS-assisted SatCom system. In Table 1, the transmission power for UE k is defined as $P_k$; thus, we have ${\rho }_k=P_k/N_0$, where $N_0$ is the noise power and equal power allocation is assumed for all UEs (i.e., $P_1=P_2=\cdots =P_K$).

In Figure 2, Lemma 1 is verified when the optimal achievable rate and its upper bound, respectively, follow (20) and (21). As shown in Figure 2, as N increases, the achievable rate ratio obtained using the proposed scheme approaches 1, confirming the optimality of the proposed scheme, as proven in Lemma 1. Also, the rate ratio of the optimal performance is larger than that of the proposed scheme. In particular, the proposed scheme achieves approximately 89% of the upper bound, while the optimal scheme achieves approximately 90% of the upper bound when $N=64$. However, the optimal performance is only achievable using the full CSI (i.e., ${\mathit{f}}_k$), which is an impractical assumption in the considered RIS-assisted SatCom system. By contrast, the proposed scheme only requires partial CSI (i.e., ${\overline{\mathit{f}}}_k$), which is deterministic LoS channel information and is achievable at the ES using the GNSSs.

In Figure 3 and Figure 4, the asymptotic ergodic sum-rate of the proposed scheme analyzed in (31) is verified. In these figures, the label “Random” represents the case of the random passive beamforming, in which ${\theta }_{k,n}$ is determined randomly. Both figures show that the proposed scheme utilizes partial CSI, but it can achieve a performance of optimal one, which is operated based on full CSI. Evidently, the asymptotic ergodic sum-rates derived from (31) align with the simulation results across the entire range of ${\rho }_k$ and N. In Figure 3 and Figure 4, the random passive beamforming technique requires no CSI; however, its ergodic sum-rate is much lower than that of the proposed scheme and the optimal case. As shown in (31), the ergodic sum-rate increases with $\mathcal{O}\left(logN\right)$ and $\mathcal{O}\left(log{\rho }_k\right)$ as N and ${\rho }_k$ increase, respectively. Accordingly, the ergodic sum-rates linearly increase with increasing ${\rho }_k$ in the dBW scale, as shown in Figure 3, and they increase following the logarithm shape as N increases, as shown in Figure 4, verifying (31).

In Figure 5, the optimality of the proposed scheme in the multicast system is verified. This figure shows the achievable rate ratios between the ergodic rate derived from the proposed scheme and the theoretical ideal performance for various values of $M_k$. In this simulation, $M_k$ UEs are distributed randomly and uniformly within a circle with a radius of 10 m, centered around the beam’s midpoint. As shown in Figure 5, the achievable rate ratios using the proposed scheme approaches 1 as N increases, regardless of $M_k$. In particular, when $M_k=16$ and $N=256$, the proposed scheme achieves approximately 91% of the theoretical ideal performance. Therefore, it is confirmed that the considered RIS-assisted SatCom system enables a UAV to simultaneously serve multiple UEs with optimal performance.

In Figure 6, (32) and (43) are verified. Figure 6 presents a comparison between the outage probabilities derived from simulations and our analytical estimates from (32) and (43). In this figure, we assume $\delta =0.8$ to analyze the outage event when $R_0=0.8{\dot{m}}_k$ or $0.8{\dot{m}}_{\mathrm{MC},k}$. As shown in Figure 6, the outage probabilities derived in (32) and (43) are consistent with the simulation results for a large N. Furthermore, the outage probabilities in both unicast and multicast systems approach zero with increasing N, confirming that an RIS can provide reliable SatCom.

6. Conclusions

In this study, we analyzed the achievable data rate of an RIS-assisted SatCom system asymptotically, considering limitations such as practical CSI acquisition. In particular, we derived the asymptotic upper bound of the achievable rate and designed a passive beamformer that can asymptotically achieve the upper bound using numerous reflecting elements on the RIS. Moreover, we asymptotically derived closed-form expressions for the ergodic achievable rate and outage probability of the proposed scheme. We proved that the derived asymptotic analyses can accurately determine the performance of the RIS-assisted SatCom system concerning the ergodic achievable rate and outage probability without requiring extensive simulations. Leveraging the derived asymptotic results in the unicast system, we extended our results to cover the multicast system, such as enabling a UAV to serve multiple UEs simultaneously. From the simulation results, we can observe that the estimated ergodic rate and outage probability were in agreement with the simulation results. Moreover, the proposed scheme asymptotically achieved the optimal performance, verifying the asymptotic optimality of the proposed scheme. We expect that the proposed scheme can significantly reduce signal overheads and the amount of feedback, rendering it a promising technique in practical SatCom systems.