A UAV-Assisted STAR-RIS Network with a NOMA System

Abstract

In this paper, we investigate a simultaneous transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-assisted non-orthogonal multiple access (NOMA) communication system where the STAR-RIS is mounted on an unmanned aerial vehicle (UAV) with adjustable altitude. Due to severe blockages in urban environments, direct links from the base station (BS) to users are assumed unavailable, and signal transmission is realized via the STAR-RIS. We formulate a joint optimization problem that maximizes the system sum rate by jointly optimizing the UAV’s altitude, BS beamforming vectors, and the STAR-RIS phase shifts, while considering Rician fading channels with altitude-dependent Rician factors. To tackle the maximum achievable rate problem, we adopt a block-wise optimization framework and employ semidefinite relaxation and gradient descent methods. Simulation results show that the proposed scheme achieves up to 22% improvement in achievable rate and significant reduction in bit error rate (BER) compared to benchmark schemes, demonstrating its effectiveness in integrating STAR-RIS and UAV in NOMA networks.

1. Introduction

Reconfigurable intelligent surfaces (RISs) have attracted considerable attention in recent years as a novel and fascinating area. As a developing technology, RIS can enhance wireless channel quality by adjusting the amplitude and phase of incident signals via low-cost and independently controllable passive elements on a surface. In the past few years, considerable research works have been conducted in response to the new research direction of RIS. For instance, the authors of [1] studied a RIS-aided single-cell wireless system, which minimizes the energy consumption of the system by optimizing the transmit beamforming at the base station (BS) and reflect beamforming at the RIS. In addition, the authors in [2,3,4] proposed several spectral efficient (SE) and energy efficient (EE) wireless solutions under various channel conditions. For example, the authors of [2] proposed two cost-effective approaches for optimizing the phase shifters of a RIS in a downlink multi-user communication system by using alternating maximization and non-convex optimization to improve SE and EE. These methods can also be used to reduce the bit error rate (BER) and enhance the achievable rate. However, the major limitation is that RISs can only reflect the signal, and the coverage area is constrained due to the physical characteristics. As a solution to this issue, the simultaneous transmitting and reflecting RISs (STAR-RIS) is proposed, which not only solves the challenge of omnidirectional coverage but also provides more degrees of freedom for controlling the STAR-RIS. Following this, several studies have been conducted on STAR-RIS-aided wireless communication systems. Liu et al. in [5] applied STAR-RIS to unmanned aerial vehicle (UAV) networks to maximize achievable rates through the joint optimization of UAV trajectory and phase shifters. The authors in [6] investigated three operational protocols for STAR-RIS, and simulation results confirmed that the performance of STAR-RIS using the proposed protocols surpasses that of traditional RIS. Based on [6], the authors in [7] proposed an innovative emergency communication network for UAVs utilizing both STAR-RIS and non-orthogonal multiple access (NOMA). NOMA provides a significant advantage compared to conventional orthogonal multiple access schemes [8]. Additionally, because of STAR-RIS’s capability for both transmission and reflection, users in the STAR-RIS scenario can be divided into two categories as follows: transmission users and reflection users. Several works have combined the NOMA and STAR-RIS in light of this. For example, the authors in [9] jointly optimized both the position of the UAV and the passive beamforming of STAR-RIS to obtain a sum-rate maximization solution in STAR-RIS-aided NOMA networks. The authors in [10] proposed a NOMA system empowered by mobile STAR-RIS, which outperforms various other schemes in terms of average achievable rate. The authors in [11] developed a two-layer iterative algorithm to jointly optimize user pairing, decoding order, beamforming, and resource allocation in a STAR-RIS-assisted NOMA framework, significantly improving the minimum user rate compared to conventional schemes. The authors in [12] proposed a STAR-RIS-assisted NOMA system that jointly optimizes decoding order, power allocation, and beamforming to maximize system sum rate. A two-layer iterative algorithm was developed to efficiently solve the non-convex problem, yielding superior performance compared to conventional RIS-assisted NOMA and RIS-assisted orthogonal multiple access schemes. A detailed comparison of related works can be found in

Table 1. Comparison with existing works.

Ref. and the Proposed Scheme	RIS Type	NOMA Type	UAV Type	Channel Model	Mathematical Approach
[11]	STAR-RIS	Hybrid NOMA	No UAV	Rayleigh fading	Alternating optimization
[12]	STAR-RIS	NOMA	No UAV	Rician fading	Successive convex approximation and sequential constraint relaxation
[7]	RIS	NOMA	Fix UAV	Rician fading	Lagrange relaxation and gradient descent
[13]	No RIS	NOMA	UAV can alter its altitude	Rician fading	Path-following algorithms
The proposed scheme	STAR-RIS	NOMA	UAV can alter its altitude	Rician fading	Gradient descent and semidefinite relaxation

. However, most of the works considering NOMA face the following bottleneck. In a scenario with a large number of users, the successive interference cancellation (SIC) limits the capacity of a single STAR-RIS in the proposed STAR-RIS-aided NOMA system to handle multiple users. A common approach is to allocate a specific region to each STAR-RIS. Therefore, each STAR-RIS is responsible for managing communication within its designated area. The problem we address is how to maximize system utilization and SE for each user by optimizing the deployment of STAR-RIS-equipped UAVs while ensuring compliance with SIC limitation.

Based on this analysis, we consider a STAR-RIS-aided NOMA network where the STAR-RIS is equipped in a UAV, and we focus on constructing the system model and conducting an approach to maximize the SE. Our work is novel and contributive in the following aspects:

• We jointly optimize the UAV’s flying altitude, antenna beamforming, and the phase shifts of the reflective elements, demonstrating significant performance gains from this integrated design.
• We formulate an SE maximization problem under constraints including the total power, UAV altitude, and coverage. To address its complexity, we adopt gradient descent and semidefinite relaxation (SDR) techniques and propose an efficient algorithm for its solution.
• Numerical results demonstrate that the proposed gradient descent-based algorithm significantly outperforms baseline schemes in terms of SE and BER.

Organization: The remainder of this paper is organized as follows. Section 2 presents the system model, and Section 3 presents the problem formulation. Section 4 introduces the proposed algorithm and provides a solution to the problem. Section 5 presents the simulation results. Finally, Section 6 concludes the paper.

2. System Model

In this section, we introduce the proposed STAR-RIS-aided NOMA system model which is shown in Figure 1. Without loss of generality, a two-dimensional (2D) coordinate system is considered. In this scenario, direct communications between the two users and the BS are blocked. To overcome the blocking, a UAV equipped with STAR-RIS establishes a communication link between the users and the BS.

The coordinates for user 1 (U1), user 2 (U2), and STAR-RIS are $(L,0)$, $(-L,0)$, and $(r,h_{uav})$. We use $h_{uav}$ to denote the height of the UAV and r to represent the horizontal coordinate of the UAV. In the system, the BS is located far away, and its coordinate is $(h_{bs},r_{bs})$. The O point which is located in $(0,0)$ denotes the origin point of the 2D coordinate system. $d_{BS}$ denotes the distance from the BS to the STAR-RIS, and $d_r$ and $d_t$ represent the distances from the STAR-RIS to each user, which are, respectively, shown as

$$d_r=\sqrt{{\left(L-r\right)}^2+{h_{uav}}^2},$$

(1a)

$$d_t=\sqrt{{\left(L+r\right)}^2+{h_{uav}}^2},$$

(1b)

$$d_{BS}=\sqrt{{\left(r_{bs}-r\right)}^2+{\left(h_{bs}-h_{uav}\right)}^2},$$

(1c)

$$r\in (0,L].$$

(1d)

As shown in Figure 1 and Equations (1a)–(1d), the system model simplifies the position by ignoring the y-axis component. This simplification is reasonable because the horizontal position of the UAV is not involved in the optimization process. Since the focus of this work is on optimizing the UAV altitude and STAR-RIS configuration, modeling in two dimensions suffices, without loss of generality.

The wireless signal incident on a particular STAR-RIS element is split into transmitted and reflected signals, as shown in Figure 1. The signal incident on the mth element of the STAR-RIS is represented as $s_m$, where $m\in \left\{1,2,\dots ,M\right\}$ with M representing the total number of elements. In addition, the signals transmitted and reflected by the mth element can be, respectively, denoted as $t_m=\sqrt{{\beta }_m^t}{e}^{j{\varphi }_m^t}{s}_m$ and $r_m=\sqrt{{\beta }_m^r}{e}^{j{\varphi }_m^r}{s}_m$, where $\sqrt{{\beta }_m^t},\sqrt{{\beta }_m^r}\in (0,1)$ and ${\varphi }_m^t,{\varphi }_m^r\in [0,2\pi )$. ${\beta }_m^r$ and ${\beta }_m^t$ denote the transmission and reflection coefficients for the mth element, respectively; ${\varphi }_m^t$ and ${\varphi }_m^r$ denote the phase shift for the mth element, respectively. All STAR-RIS elements are assumed to work in the energy splitting protocol [6], which means the signal arriving on each element is divided into the transmitting signal and reflection signal, with an energy splitting ratio of ${\beta }_m^t:{\beta }_m^r$. Since the total signal energy equals the transmitted and reflected energies, the transmission and reflection coefficients always satisfy ${\beta }_m^t+{\beta }_m^r=1$ for each element. In this case, the phase shift matrix of the transmission link ${\mathbf{\Theta }}_t=diag(\sqrt{{\beta }_1^t}{e}^{j{\varphi }_1^t},\dots ,\sqrt{{\beta }_M^t}{e}^{j{\varphi }_M^t})$ and the phase shift matrix of the reflection link ${\mathbf{\Theta }}_r=diag(\sqrt{{\beta }_1^r}{e}^{j{\varphi }_1^r},\dots ,\sqrt{{\beta }_M^r}{e}^{j{\varphi }_M^r})$ and $diag(·)$ represents a diagonal matrix.

2.1. Channel Modeling

In the proposed system, N denotes the number of antennas in BS [1]. $\mathbf{G}\in {\mathbb{C}}^{M\times N}$ is the channel between BS and STAR-RIS; ${\mathbf{F}}_r\in {\mathbb{C}}^{M\times 1}$ and ${\mathbf{F}}_t\in {\mathbb{C}}^{M\times 1}$ represent the channel between STAR-RIS and U1 and the channel between STAR-RIS and U2, respectively. In dense urban scenarios, the direct link between the BS and users is often obstructed by tall buildings and other physical structures. Therefore, in this paper, the channels are modeled as Rician fading channels, where the line-of-sight (LoS) and non-line-of-sight (NLoS) components are jointly considered through the Rician factor. The path loss coefficient of the proposed model is $\chi ={\chi }_1{d}_0^{-ϵ}$, where $ϵ$ denotes the path loss exponent, $d_0$ denotes the distance between the transmitter and receiver, and ${\chi }_1$ denotes the path loss coefficient at 1 m. Then, the channels are modeled as

$$\begin{array}{cc}\hfill & {\mathbf{F}}_k={\chi }_1{d}_k^{-{ϵ}_k}\times \left(\sqrt{{\displaystyle \frac{K_k}{K_k+1}}}{\mathbf{F}}_k^{\left(LoS\right)}+\sqrt{{\displaystyle \frac{1}{K_k+1}}}{\mathbf{F}}_k^{\left(NLoS\right)}\right),\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill & \mathbf{G}={\chi }_1{d}_{BS}^{-{ϵ}_G}\times \left(\sqrt{{\displaystyle \frac{K_G}{K_G+1}}}{\mathbf{G}}^{\left(LoS\right)}+\sqrt{{\displaystyle \frac{1}{K_G+1}}}{\mathbf{G}}^{\left(NLoS\right)}\right),\hfill \end{array}$$

(2b)

$$\begin{array}{cc}\hfill & k\in \left\{\begin{array}{c}t,r\end{array}\right\}.\hfill \end{array}$$

(2c)

where $K_t$, $K_r$, and $K_G$ denote the Rician factors for ${\mathbf{F}}_t$, ${\mathbf{F}}_r$, and $\mathbf{G}$, respectively. ${ϵ}_t$, ${ϵ}_r$, and ${ϵ}_G$ denote the exponents of path loss for ${\mathbf{F}}_t$, ${\mathbf{F}}_r$, and $\mathbf{G}$, respectively. ${\mathbf{F}}_t^{\left(LoS\right)}$, ${\mathbf{F}}_r^{\left(LoS\right)}$, and ${\mathbf{G}}^{\left(LoS\right)}$ are the LoS components for ${\mathbf{F}}_t$, ${\mathbf{F}}_r$, and $\mathbf{G}$, respectively. ${\mathbf{F}}_t^{\left(NLoS\right)}$, ${\mathbf{F}}_r^{\left(NLoS\right)}$, and ${\mathbf{G}}^{\left(NLoS\right)}$ represent the NLoS components for ${\mathbf{F}}_t$, ${\mathbf{F}}_r$, and $\mathbf{G}$, respectively.

2.2. Path Loss Exponent and Rician Factor Modeling

Unlike simplified models that assume a fixed LoS or complete blockage, the Rician factor in the proposed scheme is dynamically changed based on the UAV altitude, the elevation angle between the UAV and users. Intuitively, when the altitude of the UAV decreases, the channels suffer severe fading due to the obstacles between the UAV and the users. In addition, the elevation angle of the UAV relative to the ground user is identified as a key factor in determining the Rician factor [14]. Therefore, by introducing the non-decreasing function $KE\left({\theta }_X\right)$, where $X\in \left\{\begin{array}{c}t,r,G\end{array}\right\}$ and ${\theta }_X\in \left[0,{\displaystyle \frac{\pi }{2}}\right]$, the Rician factor is modeled as a function of the elevation angle $K_X=KE\left({\theta }_X\right)$. The elevation angles between the UAV and the ground users are denoted as ${\theta }_r$ and ${\theta }_t$, and the elevation angle between the UAV and the BS is ${\theta }_G$. In fact, a large ${\theta }_X$ leads to a higher LoS contribution and less multipath scatters at the receiver, resulting in a larger $K_X$ [15]. According to [14], the Rician factors can be modeled by the following function:

$$\begin{array}{cc}\hfill & K_X=KE\left({\theta }_X\right),\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill & KE\left({\theta }_X\right)=a_1^X·e^{b_1^X{\theta }_X},\hfill \end{array}$$

(3b)

$$\begin{array}{cc}\hfill & a_1^X={\kappa }_0^X,b_1^X={\displaystyle \frac{2}{\pi }}ln\left({\displaystyle \frac{{\kappa }_{\pi /2}^X}{{\kappa }_0^X}}\right),\hfill \end{array}$$

(3c)

$$\begin{array}{cc}\hfill & X\in \left\{\begin{array}{c}t,r,G\end{array}\right\},{\theta }_X\in \left[0,{\displaystyle \frac{\pi }{2}}\right].\hfill \end{array}$$

(3d)

where ${\kappa }_0^X=KE\left({\theta }_X\right)$ is the minimum Rician factor for ${\theta }_X=0$ and ${\kappa }_{\pi /2}^X=KE\left({\theta }_X\right)$ is the maximum Rician factor for ${\theta }_X={\displaystyle \frac{\pi }{2}}$. ${\kappa }_0^X$ and ${\kappa }_{\pi /2}^X$ are fixed values related to the environment, and as the environmental parameters change, the shape of $KE\left({\theta }_X\right)$ will also vary accordingly. Furthermore, (3b) reflects the influence of the elevation angle on the LoS component and the reduction in multipath effects as ${\theta }_X$ increases.

Similarly, the path loss exponent is affected by the elevation angle, with $ϵ$ potentially decreasing as the UAV’s elevation angle ${\theta }_X$ increases. In this approach, ${ϵ}_0^X$ represents the maximum path loss exponent at ${\theta }_X=0$, while ${ϵ}_{\pi /2}^X$ denotes the minimum path loss exponent at ${\theta }_X={\displaystyle \frac{\pi }{2}}$. Generally, denser environments, with more obstacles between the transmitter and receiver, typically result in higher path loss exponents. To account for this, function ${ϵ}_X\left({\theta }_X\right)$ is described based on the LoS function, $P_{\mathrm{LoS}}\left({\theta }_X\right)$, between the UAV and the ground user [16], which is given by the following:

$$\begin{array}{cc}\hfill & {ϵ}_X\left({\theta }_X\right)=a_2^X·{\mathcal{P}}_{LoS}\left({\theta }_X\right)+b_2^X,\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill & {\mathcal{P}}_{\mathrm{LoS}}\left({\theta }_X\right)={\displaystyle \frac{1}{a_3^X{e}^{-b_3^X{\theta }_X}+1}},\hfill \end{array}$$

(4b)

$$\begin{array}{cc}\hfill & X\in \left\{\begin{array}{c}t,r,G\end{array}\right\},{\theta }_X\in \left[0,{\displaystyle \frac{\pi }{2}}\right].\hfill \end{array}$$

(4c)

where the parameters $a_2^X$, $b_2^X$, $a_3^X$, and $b_3^X$ are influenced by the environmental conditions and the transmission frequency. $a_2^X$ and $b_2^X$ can be modeled as below   

$$\begin{array}{cc}\hfill & a_2^X={\displaystyle \frac{{ϵ}_{\pi /2}^X-{ϵ}_0^X}{{\mathcal{P}}_{LoS}\left(\frac{\pi }{2}\right)-{\mathcal{P}}_{\mathrm{LoS}}\left(0\right)}}\cong {ϵ}_{\pi /2}^X-{ϵ}_0^X,\hfill \end{array}$$

(5a)

$$\begin{array}{cc}\hfill & b_2^X={ϵ}_0^X-a_2^X·{\mathcal{P}}_{\mathrm{LoS}}\left(0\right)\cong {ϵ}_0^X,\hfill \end{array}$$

(5b)

$$\begin{array}{cc}\hfill & X\in \left\{\begin{array}{c}t,r,G\end{array}\right\}.\hfill \end{array}$$

(5c)

where the approximations arise from the fact that ${\mathcal{P}}_{\mathrm{LoS}}\left(0\right)\to 0$ and ${\mathcal{P}}_{\mathrm{LoS}}(\pi /2)\to 1$, with detailed proofs provided in [15].

2.3. Received Signal and NOMA Decoding

The BS is equipped with a uniform linear array (ULA) consisting of N transmit antennas, and we consider a uniform rectangular array (URA) at the STAR-RIS. The BS sends the superposition coding to those two users according to the working principle of NOMA. As a result, each user’s received signal can be represented as

$$\begin{array}{cc}\hfill & y_k={\mathbf{F}}_k^H{\mathbf{\Theta }}_k\mathbf{G}\mathbf{x}+n_k,\hfill \\ \hfill & k\in \left\{\begin{array}{c}t,r\end{array}\right\}.\hfill \end{array}$$

(6)

where $n_k$ denotes the additive white Gaussian noise (AWGN) satisfying $n_k\sim \mathcal{C}\mathcal{N}\left(0,{\sigma }_k^2\right)$, and $\mathbf{x}$ is the transmitted signal from the BS which is shown as

$$\begin{array}{c}\hfill \mathbf{x}=\sqrt{a_t}{\mathbf{w}}_t{s}_t+\sqrt{a_r}{\mathbf{w}}_r{s}_r,\end{array}$$

(7)

where $s_k,k\in \left\{t,r\right\}$ is the transmitted symbol sent to U1 and U2, respectively, with $\mathbb{E}\left\{{\left|s_k\right|}^2\right\}=1$, ${\mathbf{w}}_k\in {\mathbb{C}}^{N\times 1},k\in \left\{t,r\right\}$ denoting the beamforming vector for each user. Moreover, $a_r$ and $a_t\in \left(0,1\right)$ with $a_r+a_t=1$ denote the power coefficient allocated to the U1 and U2, respectively. Because of the constraint imposed by (1d), the UAV is always closer to U1. To balance the achievable rates between the two users and mitigate significant rate disparity, a larger power allocation coefficient is assigned to U2. Therefore, $a_t$ is set to be greater than $a_r$ (i.e., $a_t>a_r$). At U1 and U2, SIC is used, and the signals are decoded in decreasing order of their channel gains.

The achievable rate of U1 and U2 is given by

$$\begin{array}{cc}\hfill & R_r={log}_2\left(1+{\displaystyle \frac{{\left|{\mathbf{F}}_r^H{\mathbf{\Theta }}_r\mathbf{G}{\mathbf{w}}_r\right|}^2}{{\left|{\mathbf{F}}_t^H{\mathbf{\Theta }}_t\mathbf{G}{\mathbf{w}}_r\right|}^2+{\sigma }_r^2}}\right),\hfill \end{array}$$

(8a)

$$\begin{array}{cc}\hfill & R_t={log}_2\left(1+{\displaystyle \frac{{\left|{\mathbf{F}}_t^H{\mathbf{\Theta }}_t\mathbf{G}{\mathbf{w}}_t\right|}^2}{{\sigma }_t^2}}\right).\hfill \end{array}$$

(8b)

3. Problem Formulation

The objective of the proposed system is to maximize the achievable rate of the users. Therefore, the problem can be formulated as below  

$$\begin{array}{cc}\hfill & \underset{{\mathbf{w}}_k,h_{uav},{\mathbf{\Theta }}_k}{max}{log}_2\left(1+{\displaystyle \frac{{\left|{\mathbf{F}}_r^H{\mathbf{\Theta }}_r\mathbf{G}{\mathbf{w}}_r\right|}^2}{{\left|{\mathbf{F}}_t^H{\mathbf{\Theta }}_t\mathbf{G}{\mathbf{w}}_r\right|}^2+{\sigma }_r^2}}\right)+{log}_2\left(1+{\displaystyle \frac{{\left|{\mathbf{F}}_t^H{\mathbf{\Theta }}_t\mathbf{G}{\mathbf{w}}_t\right|}^2}{{\sigma }_t^2}}\right),\hfill \end{array}$$

(9a)

$$\begin{array}{c}\hfill \mathrm{s}.\mathrm{t}.\sum _{k\in \left\{t,r\right\}}{∥{\mathbf{w}}_k∥}^2\le P_{max}\end{array}$$

(9b)

$$\begin{array}{c}\hfill R_k\ge R_k^{min}\forall k\in \{t,r\},\end{array}$$

(9c)

$$\begin{array}{c}\hfill 0<{\beta }_m^t,{\beta }_m^r<1,{\beta }_m^t+{\beta }_m^r=1,{\beta }_m^k\in \left(0,1\right),\forall k\in \{t,r\}\forall m\in \mathcal{M},\end{array}$$

(9d)

$$\begin{array}{c}\hfill {\beta }_m^k\in \left(0,1\right),\forall k\in \{t,r\},\forall m\in \mathcal{M},\end{array}$$

(9e)

$$\begin{array}{c}\hfill {\varphi }_k^m\in [0,2\pi ),\forall k\in \{t,r\},\forall m\in \mathcal{M},\end{array}$$

(9f)

where (9b) denotes that the maximum total transmit power is $P_{max}$; $R_r$ and $R_t$ denote the achievable rate of U1 and U2, respectively; (9c) denotes that the minimum required communication rate of each user is $R_k^{min}$; and $\mathcal{M}\triangleq \left\{1,2,\dots ,M\right\}$, ${\alpha }_k$, and ${\beta }_m^k$ are fixed parameters that remain constant throughout the optimization process. The UAV altitude $h_{uav}$ plays a critical role in determining the characteristics of the channels. Specifically, it influences both the path loss exponents and the Rician factors of the channels ${\mathbf{F}}_k$ and $\mathbf{G}$. As the UAV descends to a lower altitude, it is more likely to experience blockages and scattering from surrounding buildings and obstacles in the urban environment. This leads to a higher path loss exponent and a lower Rician factor. Therefore, the UAV altitude is crucial for the overall system performance and is explicitly considered in our joint optimization. Ultimately, $R_t+R_r$ is collectively determined by these three variables through their joint effects on channel characteristics.

4. Solution of the Problem

To solve the original joint optimization problem involving UAV altitude, beamforming vectors, and STAR-RIS phase shifts, we adopt an optimization framework. The original problem is challenging and includes coupled variables and multiple constraints, making direct joint optimization intractable. Therefore, we decompose the problem into two subproblems. This approach is a widely used technique in solving non-convex optimization problem [17]. In particular, when the UAV altitude is fixed, the beamforming and STAR-RIS phase optimization becomes more tractable. The decomposition significantly reduces computational burden and simplifies implementation. Although a theoretical convergence guarantee for the proposed scheme is difficult, our simulation results in Section 5 confirm that the proposed algorithm converges consistently across different sets of simulation parameters.

4.1. UAV Deployment Optimization

In the proposed system, the UAV hovers at a point in a 2D region, while the STAR-RIS is attached on it for better communication for the entire system. ${SIC}_{max}$ denotes the maximum demodulation capability of SIC in the system, which means that the BS can transmit the superposition coding signal from ${SIC}_{max}$ users in maximum one transmission. If the number of users exceeds the SIC limit, the communication mode switches to time division multiple access plus non-orthogonal multiple access (TDMA + NOMA) mode [18]. For example, there are four users and ${SIC}_{max}=2$. In this circumstance, the BS transmits the superposition coding signal containing the two users’ symbols in each transmission. If there are four users and ${SIC}_{max}=4$, the BS transmits the superposition coding signal containing the four users’ symbols in each transmission. To guarantee that the STAR-RIS exactly serves one user on each side, we set ${SIC}_{max}=2$ in the proposed scheme. Setting the ${SIC}_{max}=2$ is essential to avoid mutual interference between more than one user served on the same side of the STAR-RIS. Without this constraint, the signals intended for these users would interfere with each other, significantly degrading system performance. Moreover, scenarios with more than one user on the same side of the STAR-RIS are beyond the scope of our paper.

We define the user density as $U_{density}$, denoting the number of users per square meter. The radius $R_{uav}$ of the coverage area is given by

$$\begin{array}{c}\hfill R_{uav}=h_{uav}tan({\theta }_{uav}/2),\end{array}$$

(10)

where ${\theta }_{uav}\in (0,\pi )$ is the angle, and the coverage area is calculated as

$$\begin{array}{c}\hfill {Area}_{uav}=\pi ·{\left(h_{uav}tan({\theta }_{uav}/2)\right)}^2.\end{array}$$

(11)

The expected number of users in each transmission is

$$\begin{array}{c}\hfill \iota =\left\{\begin{array}{c}{U}_{density}\pi {\left(h_{uav}tan({\theta }_{uav}/2)\right)}^2(if{U}_{density}\pi {\left(h_{uav}tan({\theta }_{uav}/2)\right)}^2\le {SIC}_{max}),\\ {SIC}_{max}\left(else\right),\end{array}\right.\end{array}$$

(12)

where ${SIC}_{max}$, $U_{density}$, and ${\theta }_{uav}$ are fixed constants.

Given ${\mathbf{w}}_k$ and ${\mathbf{\Theta }}_k$, the problem (9a) can be simplified to the following problem formulation:

$$\underset{h_{uav}}{max}{\displaystyle \frac{\iota }{{SIC}_{max}}}\sum _{k\in \left\{t,r\right\}}{R}_k,$$

(13a)

$$\mathrm{s}.\mathrm{t}.h_{min}\le h_{uav}\le h_{max},\left(9\mathrm{c}\right).$$

(13b)

Since (13a) is a continuous function, gradient descent can be applied to optimize $h_{uav}$. Before designing the iterative procedure, we first derive the slope function of (13a) as below

$$\begin{array}{c}\hfill f_{gd}^{\prime }\left(h_{uav}\right)={\left({\displaystyle \frac{\iota }{{SIC}_{max}}}\sum _{k\in \left\{t,r\right\}}{R}_k\right)}^{\prime }.\end{array}$$

(14)

First, we set the initial value $h_{uav}=h_{min}$, and then, the value of $h_{uav}$ is updated in each iteration using the formula given as

$$\begin{array}{c}\hfill h_{uav}^{new}=h_{uav}+\Delta ·f_{gd}^{\prime }\left(h_{uav}\right).\end{array}$$

(15)

where $\Delta$ is the learning rate and $h_{uav}^{new}$ is the new value of $h_{uav}$. Next, we calculate the difference between the new and old values of $h_{uav}$. If this difference is smaller than the convergence threshold $\eta$ or $h_{uav}^{new}$ exceeds the operational range of the UAV, the algorithm terminates, indicating convergence. Otherwise, we set $h_{uav}=h_{uav}^{new}$, and the algorithm continues. The details of the algorithm used to realize the above optimization are given in Algorithm 1.

Algorithm 1 Gradient descent for UAV deployment optimization

Require: Learning rate $\Delta >0$, initial value $h_{min}$ and $h_{max}$, convergence threshold $\eta >0$, maximum iterations ${\eta }_{\max }\mathrm{iter}$   1: Initialize environment.   2: Set $h_{uav}\leftarrow h_{min}$.   3: for $i=1$ to ${\eta }_{\max \mathrm{iter}}$ do   4:    Compute gradient: $f_{gd}^{\prime }\left(h_{uav}\right)$.   5:    Update variable: $h_{uav}^{new}=h_{uav}+\Delta ·f_{gd}^{\prime }\left(h_{uav}\right)$.   6:    if $|h_{uav}^{new}-h_{uav}|<\eta$ then   7:      break: Convergence reached.   8:    end if   9:    if $f_{gd}^{\prime }\left(h_{uav}\right)<0$, $h_{uav}^{new}\le h_{min}$ or $f_{gd}^{\prime }\left(h_{uav}\right)>0$, $h_{uav}^{new}\ge h_{max}$ then 10:      break: Minimum height or maximum height reached. 11:    end if 12:    Set $h_{uav}\leftarrow h_{uav}^{new}$. 13: end for 14: output: Optimal solution $h_{uav}^{*}$.

4.2. Beamforming and Phase Shift Optimization

In this optimization, both users’ quality of service requirements should be met. SDR is a widely used method in wireless communication optimization problems, especially for dealing with non-convex quadratic constraints [1]. In this section, our objective is to maximize the achievable rate of the system by applying SDR to solve problem (9a). We assume the channel state information is available at the STAR-RIS. Then, we propose a centralized algorithm for this solution. Given any phase shifts ${\varphi }_k^m,\forall m\in \mathcal{M}$, the following principle is valid. According to [19], maximum ratio transmission ${\mathbf{w}}_k^{\mathrm{MRT}}\triangleq \sqrt{P_{max}}{\displaystyle \frac{{\left({\mathbf{F}}_k^H{\mathbf{\Theta }}_k\mathbf{G}\right)}^H}{∥{\mathbf{F}}_k^H{\mathbf{\Theta }}_k\mathbf{G}∥}}$ is the optimal transmit beamforming solution to problem (9a). So we have the optimal beamforming vector ${\mathbf{w}}_k^{*}$, where ${\mathbf{w}}_k^{*}=\sqrt{P_{max}}{\displaystyle \frac{{\left({\mathbf{F}}_k^H{\mathbf{\Theta }}_k\mathbf{G}\right)}^H}{∥{\mathbf{F}}_k^H{\mathbf{\Theta }}_k\mathbf{G}∥}}\triangleq {\mathbf{w}}_k^{\mathrm{MRT}}$. Given $h_{uav}$ and ${\mathbf{w}}_k^{*}$, the problem (9a) can be simplified to the following equivalent problem

$$\begin{array}{cc}\hfill & \underset{{\mathbf{\Theta }}_k}{max}{log}_2\left(1+{\displaystyle \frac{{∥{\mathbf{F}}_r^H{\mathbf{\Theta }}_r\mathbf{G}∥}^2}{{∥{\mathbf{F}}_t^H{\mathbf{\Theta }}_t\mathbf{G}∥}^2+{\sigma }_r^2}}\right)+{log}_2\left(1+{\displaystyle \frac{{∥{\mathbf{F}}_t^H{\mathbf{\Theta }}_t\mathbf{G}∥}^2}{{\sigma }_t^2}}\right),\hfill \end{array}$$

(16a)

$$\begin{array}{cc}\hfill & \mathrm{s}.\mathrm{t}.k\in \left\{t,r\right\},\left(9\mathrm{c}\right),\left(9\mathrm{d}\right),\left(9\mathrm{e}\right),\left(9\mathrm{f}\right).\hfill \end{array}$$

(16b)

We assume ${\mathit{v}}_k={\left[v_k^1,\cdots ,v_k^m,\cdots ,v_k^M\right]}^H$ where $v_k^m=\sqrt{{\beta }_m^k}{e}^{j{\varphi }_m},\forall k\in \{t,r\},\forall m\in \mathcal{M}$. Then, we have ${\mathbf{F}}_k^H{\mathbf{\Theta }}_k\mathbf{G}={\mathit{v}}_k^H{\mathbf{\Phi }}_k$, where ${\mathbf{\Phi }}_k=diag\left({\mathbf{F}}_k^H\right)\mathbf{G}$. Thus, problem (16a) is equivalent to

$$\begin{array}{cc}\hfill & \underset{{\mathbf{\Theta }}_k}{max}{log}_2\left(1+{\displaystyle \frac{{∥{\mathit{v}}_r^H{\mathbf{\Phi }}_r∥}^2}{{∥{\mathit{v}}_t^H{\mathbf{\Phi }}_t∥}^2+{\sigma }_r^2}}\right)+{log}_2\left(1+{\displaystyle \frac{{∥{\mathit{v}}_t^H{\mathbf{\Phi }}_t∥}^2}{{\sigma }_t^2}}\right),\hfill \end{array}$$

(17a)

$$\begin{array}{cc}\hfill & \mathrm{s}.\mathrm{t}.k\in \left\{t,r\right\},\left(9\mathrm{c}\right),\left(9\mathrm{d}\right),\left(9\mathrm{e}\right),\left(9\mathrm{f}\right).\hfill \end{array}$$

(17b)

Since ${\sigma }_k$ is already given, problem (17a) depends only on the ${∥{\mathit{v}}_k^H{\mathbf{\Phi }}_k∥}^2$. For variable ${∥{\mathit{v}}_k^H{\mathbf{\Phi }}_k∥}^2$, problem (17a) is a monotonically increasing function. The proof of monotonicity is given in Appendix A. Therefore, problem (17a) can be expressed as follows

$$\begin{array}{cc}\hfill & \underset{{\mathit{v}}_k}{max}\sum _{k\in \left\{t,r\right\}}{\mathit{v}}_k^H{\mathbf{\Phi }}_k{\mathbf{\Phi }}_k^H{\mathit{v}}_k\hfill \end{array}$$

(18a)

$$\begin{array}{cc}\hfill & \mathrm{s}.\mathrm{t}.\left(9\mathrm{c}\right),\left(9\mathrm{d}\right),\left(9\mathrm{e}\right),\left(9\mathrm{f}\right),\hfill \end{array}$$

(18b)

where problem (18a) is a non-convex quadratically constrained quadratic program (QCQP), and can be reformulated as a homogeneous QCQP as follows:

$$\begin{array}{cc}\hfill & \underset{{\overline{\mathit{v}}}_k}{max}\sum _{k\in \left\{t,r\right\}}{\overline{\mathit{v}}}_k^H\mathcal{H}{\overline{\mathit{v}}}_k\hfill \end{array}$$

(19a)

$$\begin{array}{cc}\hfill & \mathrm{s}.\mathrm{t}.\left(9\mathrm{c}\right),\left(9\mathrm{d}\right),\left(9\mathrm{e}\right),\left(9\mathrm{f}\right),\hfill \end{array}$$

(19b)

where ${\overline{\mathit{v}}}_k={\mathit{v}}_k$, $\mathcal{H}={\mathbf{\Phi }}_k{\mathbf{\Phi }}_k^H$, and ${\overline{\mathit{v}}}_k^H\mathcal{H}{\overline{\mathit{v}}}_k=Tr\left(\mathcal{H}{\overline{\mathit{v}}}_k{\overline{\mathit{v}}}_k^H\right)$. Problem (19a) is a non-deterministic polynomial that is hard in general [20]. We assume ${\mathbf{V}}_k={\overline{\mathit{v}}}_k{\overline{\mathit{v}}}_k^H$, which needs to satisfy $Rank\left({\mathbf{V}}_k\right)=1$ and ${\mathbf{V}}_k⪰0$. Then, we apply SDR to relax the rank-one constraint ${\mathbf{V}}_k$. As a result, problem (19a) is simplified to

$$\begin{array}{cc}\hfill & \underset{{\mathbf{V}}_k}{max}\sum _{k\in \left\{t,r\right\}}Tr\left(\mathcal{H}{\mathbf{V}}_k\right)\hfill \end{array}$$

(20a)

$$\mathrm{s}.\mathrm{t}.Rank\left({\mathbf{V}}_k\right)=1,{\mathbf{V}}_k⪰0,\left(9\mathrm{c}\right),\left(9\mathrm{d}\right),\left(9\mathrm{e}\right),\left(9\mathrm{f}\right).$$

(20b)

It is clear that problem (20a) is a standard convex semidefinite program that can be solved by convex optimization solvers such as CVX. However, the solution of the relaxed problem may not be a rank-one solution, i.e., $Rank\left({\mathbf{V}}_k\right)\ne 1$, which indicates that the solution of (20a) only stands for an upper bound of problem (19a). In order to construct a rank-one solution from the optimal higher-rank solution to problem (20a). We also need a couple of additional steps. First, we obtain the eigenvalue decomposition of ${\mathbf{V}}_k$ as ${\mathbf{V}}_k={\mathbf{U}}_k{\Sigma }_k{\mathbf{U}}_k^H$, where ${\mathbf{U}}_k=\left[e_1^k,\cdots ,e_M^k\right]$ and ${\Sigma }_k=diag\left({\lambda }_1^k,\cdots ,{\lambda }_M^k\right)$ are a unitary matrix and diagonal matrix, respectively. Second, we obtain a suboptimal solution to (19a) as ${\overline{\mathit{v}}}_k={\mathbf{U}}_k\sqrt{{\Sigma }_k}{\mathbf{r}}_k$, where ${\mathbf{r}}_k\in {\mathbb{C}}^{M\times 1}$ is a random vector generated according to ${\mathbf{r}}_k\sim \mathcal{C}\mathcal{N}\left(0,{\mathbf{I}}_M\right)$ with $\mathcal{C}\mathcal{N}\left(0,{\mathbf{I}}_M\right)$ denoting the the circularly symmetric complex Gaussian distribution with zero mean and covariance matrix ${\mathbf{I}}_M$. With independently generated Gaussian random vectors ${\mathbf{r}}_k$, the objective value of (19a) is approximated as the maximum one attained by the best ${\overline{\mathit{v}}}_k$ among all ${\mathbf{r}}_k$ values. Finally, the solution ${\overline{\mathit{v}}}_k$ to problem (19a) can be recovered by ${\overline{\mathit{v}}}_k=e^{jarg\left({\left[{\displaystyle \frac{{\overline{\mathit{v}}}_k}{{\overline{v}}_M}}\right]}_{(1:M)}\right)}$, where ${\left[\mathbf{z}\right]}_{(1:M)}$ denotes the vector that contains the first M elements in $\mathbf{z}$. It has been shown that such an SDR approach followed by a sufficiently large number of randomizations of ${\mathbf{r}}_k$ guarantees a ${\displaystyle \frac{\pi }{4}}$-approximation of the optimal objective value of problem (19a) [16].

The following algorithm is proposed to implement the above solution. First, we initialize ${\theta }_{uav}$, $h_{uav}$, $U_{density}$, ${\mathbf{w}}_k$, ${\mathbf{\Theta }}_k$, and $h_{uav}$. Second, we solve problem (13a) for given ${\mathbf{w}}_k$ and ${\mathbf{\Theta }}_k$, respectively. The optimal beamforming vector ${\mathbf{w}}_k^{*}$ can be found by setting ${\mathbf{w}}_k={\mathbf{w}}_k^{\mathrm{MRT}}$. After updating ${\mathbf{w}}_k$ and $h_{uav}$, ${\mathbf{\Theta }}_k$ can be computed by solving problem (16a). The details of the algorithm to realize the above optimization are given in Algorithm 2.

Algorithm 2 Process for solving problem (9a)

Require: Minimum required rate of each user $R_k^{min}$, maximum total transmit power $P_{max}$, learning rate $\Delta >0$, initial value $h_{min}$ and $h_{max}$, convergence threshold $\eta >0$, maximum iterations ${\eta }_{\max \mathrm{iter}}$, ${\theta }_{uav}$,$U_{density}$.   1: Initialize ${\mathbf{w}}_k$, ${\mathbf{\Theta }}_k$, and $h_{uav}$.   2: With given ${\mathbf{w}}_k$, ${\mathbf{\Theta }}_k$, $\Delta$, $h_{min}$, $h_{max}$, $\eta$, ${\eta }_{\max \mathrm{iter}}$, solve the problem (13a) and calculate the optimal value $h_{uav}^{*}$ with Algorithm 1.   3: Update $h_{uav}$ with optimal solution $h_{uav}^{*}$.   4: Update ${\mathbf{w}}_k$ with optimal solution ${\mathbf{w}}_k^{*}=\sqrt{P_{max}}{\displaystyle \frac{{\left({\mathbf{F}}_k^H{\mathbf{\Theta }}_k\mathbf{G}\right)}^H}{∥{\mathbf{F}}_k^H{\mathbf{\Theta }}_k\mathbf{G}∥}}$.   5: For given ${\mathbf{w}}_k$ and $h_{uav}$, solve problem (16a) by applying SDR.   6: Update ${\mathbf{\Theta }}_k$ with optimal solution in (20a).   7: output: Optimal solution $h_{uav}^{*}$, ${\mathbf{w}}_k^{*}$ and ${\mathbf{\Theta }}_k$.

5. Numerical and Simulation Results

5.1. System Setup

In this subsection, we present the Monte Carlo simulation results obtained from the proposed system under various conditions. The simulation parameters of the proposed model are shown in

Table 2. Simulation parameters.

Description of Parameter	Parameter	Value
Path loss exponent for ${\theta }_r,{\theta }_t=\pi /2$	${ϵ}_{\pi /2}^r$, ${ϵ}_{\pi /2}^t$	2.2
Path loss exponent for ${\theta }_r,{\theta }_t=0$	${ϵ}_0^r$, ${ϵ}_0^t$	3.5
Path loss exponent for ${\theta }_G=0$	${ϵ}_0^G$	2.8
Path loss exponent for ${\theta }_G=\pi /2$	${ϵ}_{\pi /2}^G$	2.2
Rician factor for ${\theta }_r,{\theta }_t=\pi /2$	${\kappa }_{\pi /2}^r$, ${\kappa }_{\pi /2}^t$	15 dB
Rician factor for ${\theta }_r,{\theta }_t=0$	${\kappa }_0^r$, ${\kappa }_0^t$	5 dB
Rician factor for ${\theta }_G=0$	${\kappa }_0^G$	15 dB
Rician factor for ${\theta }_G=\pi /2$	${\kappa }_{\pi /2}^G$	10 dB
Path loss at 1 meter	${\chi }_1$	−30 dB
Maximal height of UAV	$h_{max}$	100 m
Minimal height of UAV	$h_{min}$	10 m
Distance between U1/U2 and O point	L	10 m
BS position	$(r_{bs},h_{bs})$	$(50,10)$
U1 position	$(L,0)$	$(10,0)$
U2 position	$(-L,0)$	$(-10,0)$
Number of antennas in BS	N	2
Power coefficient of the U1	$a_r$	0.3
Power coefficient of the U2	$a_t$	0.7
Transmission coefficients	${\beta }_m^t,\forall m\in \mathcal{M}$	0.5
Reflection coefficients	${\beta }_m^r,\forall m\in \mathcal{M}$	0.5
Learning rate	$\Delta$	0.1
Convergence condition	$\eta$	$1\times {10}^{-5}$
Maximum number of iterations	${\eta }_{\max \mathrm{iter}}$	1000

.

5.2. Baseline Schemes

5.2.1. Baseline 1 Scheme (RIS)

The objective of this scheme is to provide a conventional UAV-assisted RIS network with a NOMA system model and prove that using STAR-RIS can enhance performance. Therefore, the STAR-RIS is replaced by the RIS in the conventional system model. We assume that the power consumption of this system is the same as that of the proposed system. Apart from this replacement, the other elements will remain consistent.

5.2.2. Baseline 2 Scheme (Random Phase Shift)

In this scheme, the elements of STAR-RIS work in random phase mode. Apart from this difference, the other elements will remain consistent with the proposed scheme.

5.3. Simulation Results

Figure 2 demonstrates the product of the achievable rate and the expected number of users versus the number of iterations for the proposed scheme when the UAV’s height $h_{uav}$ is set to 10 m initially. The figure compares the performance under three different signal-to-noise ratio (SNR) conditions. It shows that in higher SNR, the system converges within more iterations, whereas in lower SNR, convergence requires fewer iterations. This fast convergence can be explained as follows: In low-SNR scenarios, the communication link between the STAR-RIS and users suffers from severe fading, leading to limited channel gains. As a result, the optimization space becomes effectively constrained, and the algorithm quickly reaches a point where further iterations yield negligible improvement.

In Figure 3, we investigate the achievable rate of all three schemes versus the number of reflection elements when the position of the UAV is fixed. From this figure, it is first observed that the proposed scheme achieves the best performance for all cases of M. For instance, when $M=24$, the achievable rate of the proposed scheme can achieve 2.7 bps/Hz, which is almost triple that of the baseline 2 scheme and slightly better than that in the baseline 1 scheme. The baseline 2 scheme lacks the optimization of several STAR-RIS parameters, including the beamforming vector and phase shifts. In contrast, the proposed scheme optimizes these parameters, resulting in superior performance. This further validates that the proposed me can effectively adjust the STAR-RIS phase shifters to find an optimal solution. In addition, the scheme with RIS performs worse than the proposed scheme because the STAR-RIS has both reflective and transmissive elements, allowing the proposed scheme to adjust the phases of its elements more precisely.

In Figure 4, we analyze the BER of the three schemes as the number of reflection elements varies. The baseline 2 scheme, using a random phase approach, shows that the BER remains relatively high for both users. As M increases, the BER of user 1 and user 2 slightly decreases. This decreased performance is partly due to the complexity of adjusting multiple STAR-RIS parameters, including the beamforming vector and phase shifts, in the baseline 2 scheme. In contrast, the proposed scheme demonstrates a significant reduction in BER for both users as M increases. Meanwhile, user 1 and user 2 also experience a rapid decline in BER, indicating the overall superior performance of the proposed scheme. For the baseline 1 scheme, which involves phase shift adjustment, the BER performance is better than that of baseline 2 but worse than that of the proposed scheme. This performance gap is mainly due to the physical characteristics of RIS and STAR-RIS, in which the STAR-RIS allows for more parameters to be adjusted, providing greater flexibility and finer control, which leads to enhanced performance. Overall, the proposed scheme outperforms the baseline 1 and 2 schemes, highlighting the benefits of using beamforming and phase shift optimization to reduce BER.

6. Conclusions

In this paper, we have proposed a STAR-RIS-aided NOMA system by deploying STAR-RISs on the UAV. The active beamforming at the BS and the passive beamforming at the STAR-RISs are optimized to maximize the achievable rate for the users. Rician fading was adopted to model the channel with altitude-dependent Rician factors. Based on the model, two baseline schemes were proposed and discussed. By leveraging the SDR techniques, simulation results showed that the BER and achievable rate performance improvements are achieved by applying the proposed scheme when compared to the baseline schemes. Despite these good results, the proposed scheme has several limitations. First, it considered only a two-user scenario and assumed perfect channel state information, which may not be practical. Furthermore, although convergence was verified in simulations, a formal theoretical analysis remains to be discussed. Future work could investigate developing robust optimization methods that account for imperfect channel state information and environmental uncertainty, improving the resilience of the system in practical deployments. The insights presented in this work are expected to contribute for the future and practical deployments of RIS-assisted systems to address the implementation challenges in future networks.