Sum Rate Maximization for Intelligent Reflecting Surface-Assisted UAV-Enabled NOMA Network

Abstract

In the next-generation network, intelligent reflecting surface (IRS), non-orthogonal multiple access (NOMA), and simultaneous wireless information and power transfer (SWIPT) are promising wireless communication techniques to effectively improve system sum rates. In traditional unmanned aerial vehicles (UAV) communication systems, the sum rate and coverage are greatly affected when there is an occlusion on the direct transmission link. To solve this problem, the IRS technology is introduced to improve the poor channel conditions. However, most of the previous research on IRS-assisted UAV to optimize system sum rate only considers frameworks that utilize the partially joint-combining techniques of IRS, NOMA, and SWIPT. In this paper, in order to further improve the sum rate of the system, we simultaneously integrate IRS, NOMA, and SWIPT technologies and establish a sum rate maximization optimization problem when the direct link is blocked. Then, an alternative optimization (AO) algorithm based on the maximizing system sum rate is proposed to solve the non-convex optimization problem, in which the IRS location and phase, the reflecting amplitude coefficient, UAV forwarding altitude, and power splitting factor are considered. To let the non-convex and non-linear function be transformed into a convex one, we first use an iterative approach to optimize the position of the IRS. After that, an optimization problem is constructed to maximize the system sum rate with the constraints of the IRS phase shifts, successful successive interference cancellation (SIC), maximum transmit power of base station (BS), and UAV. Numerical results show that the proposed algorithm outperforms the traditional orthogonal multiple access (OMA) and algorithms without IRS-assisted links in terms of the system sum rate.

1. Introduction

Unmanned aerial vehicles (UAV) communication, which has become a popular choice for quickly establishing networks in areas with limited communication capabilities, is playing an increasingly important role in today’s wireless communication networks. Moreover, the integration of key technologies from next-generation communication networks into UAV networks has also become a research hotspot. These newly introduced technologies have the potential to improve UAV communication performance in aspects such as channel quality and resource distribution strategies.

UAV communication systems are receiving more and more interest in research. Yang et al. propose a UAV-aided relaying system with energy harvesting in [1] while they analyze the outage probability in different environmental parameters. In [2], Yang et al. use a UAV as a relay and optimize its trajectory to achieve the maximum throughput. In [3], a general link framework for ground-to-air (G2A) systems is presented, which combines highly correlated path loss exponent and small-scale fading. An analytic expression for optimum UAV height to minimize the interruption probability of any air-to-groud (A2G) link is derived. In [4], the UAV relay assistant interference coordination technology is proposed, which jointly optimizes throughput and UAV energy consumption to enhance data transmission capability.

Intelligent reflecting surface (IRS) is the surface of electromagnetic material (EM) that can be electronically controlled via integrated electronics [5]. Composed of a multitude of passive reflective components, it possesses unique reflective characteristics distinct from conventional materials. Each individual element has the ability to independently control the amplitude and phase of incoming signals. The IRS is regarded as an emerging technology that has shown tremendous potential for communication applications and is expected to play a key role in next-generation networks. In [6], the paper compares the IRS technology with traditional decoding-forwarding (DF) relay and discusses the availability of the IRS. In [7,8], Basar et al. describe the fundamental differences between the IRS and other technologies, and introduce some complex solutions that include IRS. This paper demonstrates how to reconstruct the communication channel model when using IRS in wireless networks and the necessity of adding corresponding physical performance constraints when formulating optimization problems. In [9], Björnson et al. review the basic principles of the IRS and then explain three specific features that are easily misinterpreted. IRS is introduced to enhance the throughput of a UAV-enabled communication network in [10,11], and then they, respectively, suggest optimization algorithms aimed at maximizing the average achievable rate. The upper bound of traversal spectral efficiency for an IRS-assisted large-scale antenna is evaluated in [12]. In various propagation conditions, the study investigates the effects of phase shift on the overall spectral efficiency. The top bound of total spectral efficiency and statistical channel state data are used to optimize the phase shifts. In [13], Swami et al. studied a multi-IRS-assisted two-phase communication system to jointly optimize the location and size of IRS and minimize the upper bound of system outage probability under various constraints. In [14], Zheng et al. pursue a theoretical performance comparison between non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) in the IRS-assisted downlink communication, in which the minimizing transmit power problem is formulated, taking into account the discrete unit-modulus reflection constraint applied to each IRS element. In [15], Yangs, et al. provide an IRS-assisted UAV scheme to improve the coverage and effectiveness of UAV communication systems. In this scheme, the IRS mounted on a building is utilized to reflect signals originating from a ground source towards a UAV. Moreover, analytic equations for the probability density function (PDF), system interruption probability, and average capacity of the instantaneous signal-to-noise ratio(SNR) are derived.

Wireless energy harvesting (EH) stands as a significant technology that can prolong the lifespan of wireless nodes and enhance energy efficiency in situations with limited energy resources. It achieves this by harnessing energy from radio frequency signals, as discussed in [3]. In this regard, wireless EH has been conducted in half-duplex relaying systems to boost energy efficiency. Moreover, for a full-duplex (FD) relaying system, the self-interference can be viewed as an extra power source to facilitate wireless EH. Integrating wireless-powered FD technology with NOMA augments the Spectral Efficiency (SE) of the system. It also delivers energy to nodes with energy constraints, thereby extending their operational duration. This synergy merits substantial consideration and exploration. In [16], Yang et al. considered a decode-and-forward (DF) wireless relay network with self-energy recycling and analyzed the outage probability performance of this system. In [17], Liu et al. proposed a new cooperative NOMA scheme with simultaneous wireless information and power transfer(SWIPT), where the user close to the transmitter worked as a wireless-powered relay to help the far user.

There is already some research on the combination of UAV, NOMA, and IRS. For example, Refs. [18,19] assume that IRS is mounted on a UAV. Reference [18] derives the outage probability for the proposed system and obtains the upper bound of the ergodic spectral efficiency. In [19], Liu et al. consider scenarios with multiple IRSes and UAVs. They derived and analyzed the maximum coverage probability achievable by two users. In [20], Mu et al. present a novel framework for multi-UAV communication networks in which UAV serves as BS in the sky. Under this framework, they jointly optimize the UAV transmission power, IRS reflection matrix, and NOMA user decoding orders to maximize system capacity. Reference [21] focuses primarily on the resource allocation problem in an IRS-assisted UAV-enabled system, and Cai et al. minimize system energy consumption by jointly designing resource allocation strategies, three-dimensional trajectories for drones, and phase control for IRS. It is worth noting that none of these articles consider combining energy harvesting with their scenarios. Motivated by related works, we propose a scenario in which the IRS is used to reflect the signal transmitted from the BS source to the UAV, considering EH. Unlike the [18,19], we assume that the IRS can be installed on the building, which separates the IRS from UAV. Both the position of the IRS and the UAV can be optimized for dynamic adjustments. In contrast to [20], we assume the UAV serves as a relay rather than BS. Moreover, we incorporate the optimization of the IRS’s position as part of the joint optimization process. This consideration is vital because it can allow our proposed model to effectively adapt to the rapidly changing environments characteristic of next-generation communication networks.

In particular, the main contributions of this work include the following: (i) We proposed a framework of IRS-assisted UAV-enabled NOMA network, considering non-line-of-sight (NLOS) links between users and BS with NOMA and OMA. (ii) We establish an optimization goal for system sum rate maximization under SWIPT. (iii) We proposed an algorithm to jointly optimize the reflecting amplitude coefficient of beam shaping design, UAV altitude, and the IRS location based on sum-rate maximization. Considering the computational complexity, we transform the non-convex problem established above into a convex problem using the alternative optimization (AO) iteration optimization algorithm. During the solution process, we use stochastic gradient descent (SGD) to achieve local optimal convergence. (iv) We present the effect of the transmission SNR and the number of elements on the IRS on the system sum rate.

2. System Model and Problem Formulation

An IRS-assisted UAV-enabled communication system is considered. As shown in Figure 1, it includes a base station (BS), an intelligent reflector IRS mounted on a building, a UAV working in the context of wireless energy harvesting, and a group of NOMA users waiting for transmission. The NOMA users have NLOS channel conditions and cannot receive signals transmitted directly from the BS. The transmission process consists of two phases. In the first phase, BS sends signals to the UAV through an IRS-assisted G2A link. The BS first sends the signals to the IRS; the IRS adjusts the reflection phase according to the channel state information of BS to IRS and IRS to UAV. Then, IRS uses the optimized phase to reflect the signals to the UAV. The UAV decodes the received signal through successive interference cancellation (SIC) decoding, and it also collects energy, using the harvested energy for transmitting signals in the second phase.

In the second phase, the decoded signal is sent to the NOMA user $U_1$ and $U_2$ through an A2G link using NOMA access to the pair of users in the service area. The sending power of the UAV is limited by the energy collected in the previous phase, and the users receive the signal through successive interference SIC decoding. We assume that both the $U_1$ and $U_2$ have the SIC ability and ignore the influence of the transmission and processing delay.

2.1. IRS-Assisted BS-UAV Link

As in [6], in the first phase, BS sends the signal to the UAV through an IRS-assisted G2A link. Then, the received signal at the UAV can be written as

$$y_u={\mathbf{H}}_{\mathrm{BI}}{}^{\mathrm{T}}\Theta {\mathbf{H}}_{\mathbf{I}\mathbf{U}}\sqrt{P_s}{x}_s+n_1$$

(1)

The ${\mathbf{H}}_{\mathrm{BI}}\in C^{N\times 1}$, ${\mathbf{H}}_{\mathrm{IU}}\in C^{N\times 1}$ are the channel matrix from BS to IRS, IRS to UAV, respectively. $h_{BI}^i$, $h_{IU}^i$ are, respectively, the $ith$ element of ${\mathbf{H}}_{\mathrm{BI}}$, ${\mathbf{H}}_{\mathrm{IU}}$. while the reflection properties are determined by the diagonal matrix $\Theta =\mathbf{W}·\phi$, $\phi =diag\left(e^{j{\phi }_1},\dots ,e^{j{\phi }_N}\right)$, $\mathbf{W}=diag\left(w_1,\dots ,w_N\right)$. ${\phi }_1,\dots ,{\phi }_N\in [0,2\pi ]$ and $w_1,\dots ,w_N\in [0,1]$, respectively, are the phase shift and reflecting amplitude of the $ith$ IRS element. $n_1\sim \mathcal{CN}\left(0,{\sigma }^2\right)$ is a complex additive Gaussian noise with zero mean and variance ${\sigma }^2$. The received signal at the UAV can be further represented as

$$y_u=\left[\sum _{i=1}^N{h}_{BI}^i{e}^{j{\phi }_i}{h}_{IU}^i{w}_i\right]\sqrt{P_s}{x}_s+n_1$$

(2)

where $P_s$ is the transmission power of BS and $x_s$ represents a transmitted signal, it can be expressed as

$$x_s=\sqrt{{\alpha }_1}{x}_1+\sqrt{{\alpha }_2}{x}_2$$

(3)

where $x_1$ and $x_2$ are transmitted signal for $U_1$ and $U_2$, respectively, ${\alpha }_1$ and ${\alpha }_2$ are power allocation factors for $U_1$, $U_2$, respectively, satisfying ${\alpha }_1<{\alpha }_2$ and ${\alpha }_1+{\alpha }_2=1$. N represents the reflector element number of the IRS. i refers to the ith element of the IRS, so ${\phi }_i$ is the $ith$ element to apply IRS for reflective phase shifting, $h_{BI}^i=\frac{1}{\sqrt{L_{BI}}}{\epsilon }_i{e}^{-j{\psi }_i}$ and $h_{IU}^i=\frac{1}{\sqrt{L_{IU}}}{ϵ}_i{e}^{-j{\varphi }_i}$ represents channel gain of BS-IRS and IRS-UAV. ${\mathcal{E}}_i$ is a Rician exponential variable, its mean is $\sqrt{\pi }/2$, variance is $(4-\pi )/4$, ${ϵ}_i$ is also a Rician exponential variable. Substituting the channel variables into (2)

$$y_u=\frac{1}{\sqrt{L_{BI}}}\frac{1}{\sqrt{L_{IU}}}\left[\sum _{i=1}^N{\epsilon }_i{ϵ}_i{e}^{-j\left({\psi }_i+{\varphi }_i\right)}{e}^{j{\phi }_i}{w}_i\right]\sqrt{P_s}{x}_s+n_1$$

(4)

Path Loss from BS to IRS $L_{BI}$ and path Loss from IRS to UAV $L_{IU}$, respectively, can be written as $L_{BI}=10log10\left(l_{BI}^{\alpha }\right)+A$ and $L_{IU}=10log10\left(l_{IU}^{\alpha }\right)+A$. Where A is a constant depending on the signal frequency and transmission environment. $l_{BI}^{\alpha },l_{RU}^{\alpha }$ are, respectively, the distance between the BS and the IRS and the distance between the IRS and the UAV. $\alpha$ is path loss exponent. From [7], the maximum signal-to-noise ratio can be obtained by setting ${\phi }_i={\psi }_i+{\varphi }_i$. Therefore, the received signal can be rewritten as

$$y_u=\left(\sum _{i=1}^N{\epsilon }_i{ϵ}_i{w}_i\right)\sqrt{P_s}{x}_s+n_1$$

(5)

From (4), the maximum instantaneous signal-to-noise ratio at the UAV can be expressed as:

$${\gamma }_1=\frac{{\left({\sum }_{i=1}^N{\epsilon }_i{ϵ}_i{w}_i\right)}^2{P}_s}{N_0{L}_1}$$

(6)

where $L_1=L_{SR}{L}_{RU}$, $P_s$ is the transmission power of the BS, PDF of ${ϵ}_i$ can be approximated as a mixed gamma distribution

$$f_{{ϵ}_i}\left(r\right)=\sum _{j=1}^{M}2a_j{r}^{2b_j-1}{e}^{-cr}$$

(7)

where M is the number of terms of the approximation. $a_j=\frac{s_j}{{\sum }_{m=1}^M{s}_m\Gamma \left(b_m\right)c^{-bm}},b_j=j,c=1+K_1,s_j=\frac{K_1^{j-1}{\left(1+K_1\right)}^j}{e^{K_1{[(j-1)!]}^2}}$ are the parameters of the $jth$ addend in (5). $\Gamma (·)$ is the gamma function, and $K_1$ is the Rician factor.

In the first phase, the BS emits superposed signals to the UAV. UAV harvests power from the received signal using $P_s$ architecture, i.e., the part of $\beta P_s(0⩽\beta ⩽1)$ for EH and the rest part of $(1-\beta )P_s$ for decoding. As we know, for the FD relaying system, the self-interference can be viewed as an extra power source. Set the self-interference cancellation factor of the UAV to be $k_R$; by substituting the specific parameters, the received signal of the UAV is

$$y_u=\sqrt{(1-\beta )}\left(\sum _{i=1}^N{\epsilon }_i{ϵ}_i{w}_i\right)\left(\sqrt{{\alpha }_1{P}_s}{x}_1+\sqrt{{\alpha }_2{P}_s}{x}_2\right)+\sqrt{P_u}{h}_r{x}_{1,u}+n_1$$

(8)

where $P_u$ is the transmission power of UAV, $h_r$ is self-interference channel, $x_{1,u}$ is the transmission signal after UAV decoding. The harvested energy at UAV in the first phase is given by

$$E_H=T\eta \left(\left({\alpha }_1\beta +{\alpha }_2\beta \right)P_{\mathrm{s}}{\left(\sum _{i=1}^N{ϵ}_i{\epsilon }_i{w}_i\right)}^2+k_R{P}_u{\left|h_r\right|}^2\right)$$

(9)

$\eta (0<\eta <1)$ is the EH efficiency, T is taken as transmission time period. Assume transmission time is the unit time, $k_R$ approximates to 0. The transmit power from UAV can be expressed as $P_u={\left({\sum }_{i=1}^N{ϵ}_i{\epsilon }_i{w}_i\right)}^2{P}_S\eta \beta$. $h_{BI}^i$ and $h_{IU}^i$ are the $ith$ element of ${\mathbf{H}}_{\mathrm{BI}}$ and ${\mathbf{H}}_{\mathrm{IU}}$.

2.2. UAV Forwards Signals to NOMA-User

In the second phase, the UAV sends the decoded signal to the NOMA user system through an A2G link with harvested energy from the source and transmitted signals. Accordingly, the signal received at user $U_1$ and $U_2$ are given by $y_1=\sqrt{P_u}{h}_{u_1}{x}_s+n_{u_1}$ and $y_2=\sqrt{P_u}{h}_{u_2}{x}_s+n_{u_2}$. Further considering power constraints, the received signals at two users can be given as

$$\begin{array}{cc}\hfill & y_1=\sqrt{\frac{P_s{\left({\sum }_{i=1}^N{\epsilon }_i{ϵ}_i{w}_i\right)}^2}{\frac{1}{\eta (1-\beta )}-k_R{\left|h_r\right|}^2}}{h}_{u_1}\left(\sqrt{{\alpha }_1}{x}_1+\sqrt{{\alpha }_2}{x}_2\right)+n_{u_1}\hfill \\ \hfill & y_2=\sqrt{\frac{P_s{\left({\sum }_{i=1}^N{\epsilon }_i{ϵ}_i{w}_i\right)}^2}{\frac{1}{\eta (1-\beta )}-k_R{\left|h_r\right|}^2}}{h}_{u_2}\left(\sqrt{{\alpha }_1}{x}_1+\sqrt{{\alpha }_2}{x}_2\right)+n_{u_2}\hfill \end{array}$$

(10)

$n_{u_1},n_{u_2}\sim \mathcal{CN}\left(0,{\sigma }^2\right)$ are also complex additive Gaussian noise with zero mean and variance ${\sigma }^2$. $h_{u_1}$ and $h_{u_2}$ are the channel gain from UAV to $U_1$, $U_2$, In this phase, $U_2$ first decodes $x_2$ and subtracts it by SIC for decoding $x_1$. The corresponding SINR of $U_1$ for decoding $x_1$ and $x_2$ are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\gamma }_{u_1}^{x_2}=\frac{{\left|h_{u_1}\right|}^2{P}_u{\alpha }_1}{{\left|h_{u_2}\right|}^2{P}_u{\alpha }_2+{\sigma }^2}\hfill \\ \hfill & {\gamma }_{u_1}^{x_1}=\frac{{\left|h_{u_1}\right|}^2{P}_u{\alpha }_1}{{\sigma }^2}\hfill \end{array}\end{array}$$

(11)

$U_2$ treats $x_2$ as the interference and decodes $x_2$ by using the following SINR

$${\gamma }_{u2}^{x_2}=\frac{{\left|h_{u2}\right|}^2{P}_u{\alpha }_2}{{\sigma }^2}$$

(12)

In a dual-hop DF system, the capacity can be calculated as

$$R={\int }_0^{\infty }{log}_2(1+\gamma )f_{\gamma }\left(\gamma \right)d\gamma$$

(13)

$\gamma$ is the minimum SNR between two links. $f_{\gamma }\left(\gamma \right)$ is the PDF of $\gamma$. We ignore the impact of delay on communication capacity; the signal-to-noise ratio of the two signals demodulated at the UAV are

$$\begin{array}{cc}\hfill & {\gamma }_1={\gamma }_{sr}^{x_1}=\frac{{\alpha }_1{P}_s}{{\alpha }_2{P}_s+{\sigma }^2/(1-\beta ){\left({\sum }_{i=1}^N{\epsilon }_i{ϵ}_i{w}_i\right)}^2}\hfill \\ \hfill & {\gamma }_2={\gamma }_{sr}^{x_2}=\frac{{\theta }^2{\alpha }_2{P}_s}{{\sigma }^2{\theta }^2/(1-\beta ){\left({\sum }_{i=1}^N{\epsilon }_i{ϵ}_i{w}_i\right)}^2}=\frac{{\alpha }_2{P}_s}{{\sigma }^2}(1-\beta ){\left(\sum _{i=1}^N{\epsilon }_i{ϵ}_i{w}_i\right)}^2\hfill \end{array}$$

(14)

where $\theta$ is given as $\sqrt{(1-\beta )}\left({\sum }_{i=1}^N{\epsilon }_i{ϵ}_i{w}_i\right)/\left(1-\sqrt{P_u}{h}_r\right)$, the system bandwidth is B. According to the Shannon capacity formula, comprehensive consideration (11) and (12), the transmission rate at users in bits/second/Hz (bps/Hz) can be expressed as

$$\begin{array}{cc}\hfill & R_1={log}_2\left(1+{\gamma }_1\right)={log}_2\left(1+\frac{{\left|h_1\right|}^2{P}_u{\alpha }_1}{{\left|h_2\right|}^2{P}_s{\alpha }_2+{\sigma }^2}\right)\hfill \\ \hfill & R_2={log}_2\left(1+{\gamma }_2\right)={log}_2\left(1+\frac{{\left|h_2\right|}^2{P}_u{\alpha }_2}{{\sigma }^2}\right)\hfill \end{array}$$

(15)

Since the system model in this paper is a two-phase-relay communication, the maximum achievable rate at the user’s location is theoretically affected by the demodulation SNR at the UAV at the same time, so the maximum achievable rate can be expressed as

$$\begin{array}{c}{C}_{u_1}^{x_1}=min\left({log}_2\left(1+{\gamma }_{u_1}^{x1}\right),\right.\left.R_1\right)\hfill \\ \\ C_{u_2}^{x_2}=min\left({log}_2\left(1+{\gamma }_{u_2}^{x2}\right),\right.\left.R_2\right)\hfill \end{array}$$

(16)

However, when we consider the rates of both links, only the SNR at the user’s location needs to be considered. So, the system maximization sum rate is

$$R_{\mathrm{sum}}=R_1+R_2={log}_2\left(1+\frac{{\left|h_1\right|}^2{P}_u{\alpha }_1}{{\left|h_2\right|}^2{P}_u{\alpha }_2+{\sigma }^2}\right)+{log}_2\left(1+\frac{{\left|h_2\right|}^2{P}_u{\alpha }_2}{{\sigma }^2}\right)$$

(17)

We aim to maximize the system sum rate through appropriate reflecting amplitude coefficients, location, and phase shifts of IRS and UAV height. Therefore, the formulated joint optimization problem is given by

$$\begin{array}{cc}\hfill & max{R}_{\mathrm{sum}}\hfill \end{array}$$

(18a)

$$\begin{array}{cc}\hfill s.t.& 0\le {\alpha }_1,{\alpha }_2\le 1\hfill \end{array}$$

(18b)

$$\begin{array}{cc}\hfill & P_s\le P_{max}\hfill \end{array}$$

(18c)

$$\begin{array}{cc}\hfill & P_{u}T\le E_H\hfill \end{array}$$

(18d)

$$\begin{array}{cc}\hfill & \mathbf{W}=diag\left(w_1,\dots ,w_N\right)w_1,\dots ,w_N\in [0,1]\hfill \end{array}$$

(18e)

$$\begin{array}{cc}\hfill & \left|{(\Theta )}_{nn}\right|=1,n=1,\dots ,N\hfill \end{array}$$

(18f)

In the optimization problem, we optimize the position and the phase matrix of the IRS, the reflecting amplitude coefficient, and the altitude of the UAV. The variations in the position of the IRS and the altitude of the UAV have an impact on the channel conditions, as reflected in the (18a). (18b) is the power allocation factor constraints for NOMA transmission. (18c), (18d), respectively, represent the constraints on the BS and the UAV transmit power, in which $P_{max}$ is the maximum transmission power of BS. (18e) represents the diagonal constraint. Constraint in (18f) is the constant modulus constraint of the intelligent reflectarray matrix. Obviously, the problem is non-convex and mathematically intractable, which cannot be solved directly. Therefore, we decompose the problem into multiple subproblems. In the following sections, we will develop an alternative optimization algorithm to separate the optimization variables.

3. Proposed Algorithm

In this section, expanding on the previous section, we optimize the sum rate of the proposed system model under corresponding constraints. We construct the optimization objective function and apply the semidefinite relaxation method to relax the optimization problem. The AO iterative algorithm is designed by considering the IRS location, the IRS reflecting amplitude factor, and UAV altitude. First, we optimize the position of the IRS. In the joint consideration of the UAV altitude in the A2G link framework, we establish a two-dimensional matrix to represent the IRS positions. Then, a grid iteration algorithm is employed to obtain the optimal position of the IRS, which directly determines the corresponding ${\mathbf{H}}_{\mathbf{B}\mathbf{I}}$ and ${\mathbf{H}}_{\mathbf{I}\mathbf{U}}$. After the optimization of the position, problem (18a) can be transformed into a linearly solvable problem with $\mathbf{W}$. We employ an interference minimization-based linear approach to solve $\mathbf{W}$.

3.1. Placement and Phase Shifts Optimization of the IRS

In the scenario designed above, not optimizing the position of the IRS is not well-considered in practice. In the proposed IRS-assisted communication system, the performance is affected by two major factors, including the distance from IRS to BS and the distance from the selected IRS to the blocked users.

Figure 2 illustrates a two-dimensional position optimization trajectory diagram of the IRS. We assume that during the optimization process of the IRS position, there exists an area in which the IRS has line-of-sight(LOS) links to both the BS and the UAV. The IRS placement is a basis to further optimize the phase matrix in order to find the IRS placement that provides the best assistance to the UAV. Once the position optimization is completed, the position no longer moves. We set the two-dimensional coordinates $\left(C_{\mathrm{hor}}^n,C_{\mathrm{ver}}^n\right)$ to represent the IRS location.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & C_{\mathrm{hor}}^n=c_0+{\lambda }_{n}n\hfill \\ \hfill & C_{ver}^n=c_0+V_{n}n\hfill \end{array}\end{array}$$

(19)

where $c_0$ is the initial optimization position, $C_{\mathrm{hor}}$ and $C_{\mathrm{ver}}$ are horizontal and vertical distances from the origin point to the current optimization position. ${\lambda }_n$, $V_n$ represent the distance interval, which can be expressed as $V_n={\lambda }_n=R_{op}/N_{iter}$. The optimized area of the intelligent reflection surface is a circle with a radius of $R_{op}$, and the number of position optimization iterations is $N_{iter}$. We adopt a discrete position iteration method to optimize the location of the IRS. Thus, n is the iteration step value, $n\in \left[1,2,\dots ,N_{iter}\right]$. When the IRS is placed in $\left(C_{hor}^i,C_{ver}^j\right)$, through substituting the position coordinates into the distance variable in (4), we can obtain $l_{BI}^{\alpha }=\sqrt{C_{\mathrm{hor}}^i{}^2+C_{\mathrm{ver}}^j{}^2+H_{IRS}{}^2}$, and $l_{IU}^{\alpha }=\sqrt{{\left(L_{UAV}-C_{hor}^i\right)}^2+C_{ver}^j{}^2+{\left(H-H_{IRS}\right)}^2}$, we can obtain the distances from IRS to the BS $l_{BI}^{\alpha }$ and UAV $l_{IU}^{\alpha }$, $L_{UAV}$ represent the horizontal projection position of UAV, $H_{IRS}$ and H, respectively, are the height of IRS and the UAV.

In traditional UAV communication, the height of the UAV is usually fixed. However, the deduction shows that the sum rate of the system will change with the altitude of the UAV. As the altitude of the UAV changes, the system sum rate will also change. This is because in different forwarding altitudes, the distance between the UAV and the BS, the UAV, and users both change, which can improve the LOS channel quality. Moreover, the probability of LOS links between the UAV and the users should also be considered. Compared to conventional static relays, we offer a new degree of freedom for performance enhancement via a careful relay altitude design. Introducing a propagation model of wireless signals [22] transmitted by low-altitude aerial platforms (LAP) base stations in free space, the resulting A2G mean pathloss can be modeled as

$$P{L}_i=FSPL+{\psi }_i$$

(20)

where $FSPL$ represents the free space pathloss between the UAV and a ground receiver, and i is used to represent different channel states. When i is 1, it indicates that the link is an LOS link, and when i is 2, the link is an NLOS link. Noticing that, the excessive pathloss $\psi$ affecting the A2G link depends largely on the propagation group rather than the elevation angle.

In order to find the spatial expectation of the pathloss denoted as $\Lambda$ (measured in dB), we will apply the following expectation rule

$$\Lambda =\sum _{i=1}^{2}p(i,\rho )P{L}_i$$

(21)

where $p(i,\rho )$ represents the probability of occurrence of a certain propagation user, which is strongly dependent on the elevation angle, while $\rho$ is the elevation angle. In our study, we are following the assumption of the two dominant propagation users that strictly correspond to the LoS condition; the NOMA users’ probabilities are linked as the following

$$p(\mathrm{NLoS},\rho )=1-p(LoS,\rho )$$

(22)

Referring to the International Telecommunication Union (ITU) standard, this probability is dependent on statistical parameters $\xi$ related to the environment. Parameter $\xi$ is a scale parameter that describes the buildings’ height distribution according to the Ricain probability density function. We set it as a constant 50 m.

$$f\left(H\right)=\left(H/{\xi }^2\right)exp\left(-H^2/2{\xi }^2\right)$$

(23)

H is the altitude of UAV, we can write the resulting LoS probability in a single equation as

$$p(LoS)=\prod _{n=0}^m\left[1-exp\left(-\frac{{\left[H-\kappa \right]}^2}{2{\xi }^2}\right)\right]$$

(24)

where $h_{\mathrm{RX}}$ is the elevation of users. m is determined by the average number of buildings per unit area v; it can be expressed as $m=$ floor $(d\sqrt{v}-1)$. $\frac{\left(n+\frac{1}{2}\right)\left(H-h_{\mathrm{RX}}\right)}{m+1}=\kappa$, d is the distance between UAV and user. So (21) can be rewritten as (26)

$$\begin{array}{cc}\hfill & P{L}_{\mathrm{LoS}}=20logd+20logf+20log\left(\frac{4\pi }{c}\right)+{\eta }_{\mathrm{LoS}}\hfill \\ \hfill & P{L}_{\mathrm{NLOS}}=\underset{\mathrm{FSPL}}{\underbrace{20logd+20logf+20log\left(\frac{4\pi }{c}\right)}}+\underset{{\eta }_{\mathrm{s}}}{\underbrace{{\eta }_{\mathrm{NLOS}}}}\hfill \end{array}$$

(25)

$$\Lambda =p\left(\mathrm{LoS}\right)\times P{L}_{\mathrm{LOS}}+p\left(\mathrm{NLOS}\right)\times P{L}_{\mathrm{NLOS}}$$

(26)

where f is the system frequency and c is the speed of light. ${\eta }_{\mathrm{LoS}}$ and ${\eta }_{\mathrm{NLoS}}$ are the excessive pathloss of LOS and NLOS link. By taking the second derivative of Equation (26), we can obtain the maximum value of $\Lambda$ and the corresponding UAV height H at that maximum value. The $h_{BI}^i$ and $h_{IU}^i$ can be rewritten as

$$\begin{array}{cc}\hfill & h_{BI}^i=\frac{1}{\sqrt{10log10\left(\sqrt{{\left(C_{hor}^i\right)}^2+{\left(C_{ver}^j\right)}^2+h_{IRS}^2}\right)+A}}{\epsilon }_i{e}^{-j{\psi }_i}\hfill \\ \hfill & h_{IU}^i=\frac{1}{\sqrt{10log10\left(\sqrt{{\left(L_{UAV}-C_{hor}^i\right)}^2+{\left(C_{ver}^j\right)}^2+{\left(H-h_{IRS}\right)}^2}\right)+A}}{ϵ}_i{e}^{-j{\varphi }_i}\hfill \end{array}$$

(27)

To this end, we solve the altitude IRS horizontal position alternatively, i.e., at each iteration, we first optimize the IRS horizontal position and UAV altitude by simulated annealing algorithm. The phase factor in (8) is adjusted based on the current channel state to obtain the maximum SNR at the location of the UAV in the current iteration. After optimizing the altitude of the UAV using (26), the position of the IRS can be dynamically determined.This implies that for the optimization problem (18a), we can further obtain the matrices ${\mathbf{H}}_{\mathrm{BI}}$ and ${\mathbf{H}}_{\mathrm{IU}}$. Then, using ${\mathbf{H}}_{\mathrm{BI}}$ and ${\mathbf{H}}_{\mathrm{IU}}$, we can optimize the power allocation sub-problem through reflecting amplitude factor design in the next subsection.

3.2. Design of Reflecting Amplitude Matrix $\mathbf{W}$

After determining the location and phase shifts of the IRS through iterative optimization, as discussed in the previous section, we obtain the deterministic channel state information and calculate the corresponding distance vectors to determine the optimal height of the UAV. It can be seen that the optimization objective in (18a) is related to the transmission power of the UAV, which depends on the energy collected by the UAV. According to (9) and (27), the $L_{BI}$ and $L_{IU}$ can be determined by the position optimization of the IRS. Adjusting the phase of the IRS according to ${\phi }_i={\psi }_i+{\varphi }_i$, the energy collection of the UAV is proportional to ${\sum }_{k=1}^N{ϵ}_k{\epsilon }_k{w}_k$. The optimization problem can be reformulated as

$$\begin{array}{cc}\hfill & \underset{w_N}{max}\sum _{k=1}^N{ϵ}_k{\epsilon }_k{w}_k\hfill \end{array}$$

(28a)

$$\begin{array}{cc}\hfill s.t.& w_k⩾\frac{\left(P_s{\sum }_{j=k+1}^N{w}_j{h}_j+{\sigma }^2\right)}{P_R{h}_k}\hfill \end{array}$$

(28b)

$$\begin{array}{cc}\hfill & P_R=\frac{E_H}{T}\hfill \end{array}$$

(28c)

k represents the $kth$ element of IRS. From the expression, it can be observed that the above problem is convex and can be solved using standard convex algorithms. Since the near element does not experience interference from other elements, the minimum power constraint is given by $w_N⩾\frac{{\sigma }^2}{P_s{h}_{BI}^N}=w_N^{\mathrm{LB}}$. Setting a lower bound on the reflecting amplitude coefficient can reduce interference with other elements $w_{N-1}⩾\frac{\left(w_N{P}_{\mathrm{R}}{h}_{BI}^N+{\sigma }^2\right)}{P_{\mathrm{R}}{h}_{BI}^{N-1}}⩾\frac{{\sigma }^2}{P_{\mathrm{R}}{h}_{BI}^{N-1}}=w_{N-1}^{\mathrm{LB}}$. For first element, we can set the optimal power reflection coefficient to 1, while as for next element $w_2^{*}=min\left\{1,w_1^{\mathrm{UB}}\right\}$ Where $w_1^{\mathrm{UB}}=\left\{\frac{1}{h_{BI}^1}\left(\frac{h_{BI}^1}{r^{min}}-{\tilde{h}}_{BI}^1-\frac{{\sigma }^2}{P_s}\right)\right\}$ and ${\tilde{h}}_{BI}^1=w_2^{\mathrm{LB}}{h}_{BI}^2$.

4. Numerical Results

In this section, we present some numerical results to verify our analysis. The parameters setup used in the figures is $\mathrm{A}=1,\mathrm{K}\left(0\right)=5\mathrm{dB},\mathrm{K}(\pi /2)=15\mathrm{dB},\mathrm{A}1=-1.5,\mathrm{B}1=3.5,\mathrm{p}=1,\mathrm{q}=1,{\theta }_1=\pi /3,{\theta }_2=\pi /3,{\mathrm{R}}_{\mathrm{op}}=100\mathrm{m}$. Three conventional algorithms are compared with our proposed algorithm in the section. They are IRS with orthogonal frequency division multiple access(OFDMA), DF with NOMA, and amplify and forward(AF) with NOMA strategies, respectively. For IRS with OFDMA algorithm, the system uses IRS to assist UAV, and the user uses OFDMA algorithm to access the network. In DF with the NOMA algorithm and AF with the NOMA algorithm, a relay adopting the DF or AF forwarding mode is placed at the same location as the substitution of IRS, and NOMA is used on the user side to access the network. We analyze the proposed algorithm in two aspects, including system sum rate and system interrupt probability.

In Figure 3, we illustrate the impact of the IRS placement. Two main factors affecting the performance of IRS are the distance between IRS and BS and the distance between IRS and UAV. We depict the variation trend of the sum rate for two users under different IRS placement locations, with the BS serving as the reference point. From Figure 3, it can be seen that different positions of IRS have an impact on the system sum rate, as the location of IRS has an impact on ${\mathbf{H}}_{\mathrm{BI}}$, ${\mathbf{H}}_{\mathrm{IU}}$, which can further determine the transmission of UAV.

Figure 4 illustrates the sum rate performance under different energy harvesting factors ${\beta }_n$ and UAV altitudes $H_n$. As expected, the sum rate first decreases and then increases when the altitude of UAV increases. This verifies our analysis in Equation (21), where the increase in UAV altitude leads to a higher LOS link probability. However, the actual link quality deteriorates due to the increased distance. Conversely, reducing the UAV altitude decreases the LOS link probability but improves link quality due to reduced distance. Hence, there always exists an optimal altitude where the system’s sum rate is maximized in each iteration.

Figure 5 shows the system average capacity of the considered system model with and without the IRS. It is worth noting that there is good consistency between the analysis results and simulation results, which indicates that our proposed signal and link analysis for two phases are accurate. It can be seen from Figure 5 that by increasing the transmission SNR of the BS, the total transmission rate of the system has been increased in varying degrees. In the stage of low SNR of transmission, the four schemes have achieved a proportional growth effect, and the scheme with IRS assistance has a faster growth rate. At the stage of relatively high SNR, the two schemes(IRS with NOMA, IRS with OFDMA) continue to show a proportional growth effect, but the system capacity of the latter two schemes tends to stabilize with the increase of SNR. That is because, compared to traditional relay methods, the IRS only requires a small amount of energy to adjust the phase of its components. When the BS has high transmission power, this helps to reduce interference.

In order to analyze the effect of the UAV’s altitude on the provided service, Figure 6 is plotted. It can be seen that in a reasonable range of UAV altitude, the capacity of the system decreases with the increase of forwarding altitude. After analyzing the three-hop link, we can find that the system sum rate mainly depends on the channel condition of the second-hop dual-user. As the forwarding altitude increases, the channel condition of the user deteriorates, the fading degree increases gradually, and finally tends to a larger constant value. Simultaneously, when simulating the interrupt probability of users, we find that the altitude of the UAV will also affect the communication quality of far users through the change of coverage. Therefore, it is most reasonable to select the altitude by considering both the average capacity and coverage.

Figure 7 illustrates the outage probability of proposed strategy and the corresponding benchmark schemes’ system under different SNR conditions. As shown in the figure, the SNR at both the UAV and user sides gradually improves with an increase in the transmit power. This leads to a decrease in the outage probability as the transmit SNR increases for all four strategies. Moreover, the proposed access strategy utilizes an IRS-assisted G2A link for the first hop, where the various reflecting elements on the IRS enhance the received signal through beamforming, resulting in low outage probability. Therefore, as the transmit power increases, the diversity gain becomes more notable, leading to a larger advantage of the proposed scheme under high SNR conditions.

5. Conclusions

In this paper, an IRS-assisted UAV-enabled NOMA transmission network is proposed. We analyze the IRS-assisted G2A link and the link between UAV and NOMA users under SWIPT. We use the iterative algorithm to jointly optimize the parameters, including the location and phase matrix of the IRS, the altitude of UAV, and the power reflecting factors in order to maximize the sum rate of the UAV-enabled network. The iterative optimization approach of IRS location was firstly proposed to transform the non-convex and non-linear problem into a convex problem, in which we can directly determine the ${\mathbf{H}}_{\mathrm{BI}}$, ${\mathbf{H}}_{\mathrm{IU}}$. Then, we perform linear convex optimization on the amplitude reflecting factor. The simulation results show that the IRS technology can significantly improve the sum rate of the system than the traditional algorithms. Moreover, the simulation results demonstrate that the proposed algorithm also reduces the system’s interruption probability to a certain extent.