Analysis With Deep Learning of Robust UAV-Mounted Active IRS NOMA Networks With Imperfections

This paper introduces a robust cooperative network where an active intelligent reflecting surface (A-IRS) mounted on an unmanned aerial vehicle (UAV) is employed in order to significantly enhance the air-to-ground communications. By utilizing advanced maneuver control and intelligent reflection, the network optimizes wireless channels, substantially improving spectrum efficiency through a non-orthogonal multiple access (NOMA) scheme. We consider non-ideal system imperfections, such as co-channel interference, hardware impairments, and imperfect successive interference cancellation. We derive the expressions for users’ outage probability (OP), ergodic capacity, and system throughput in both delay-limited and delay-tolerant modes under Nakagami fading channels, reflecting realistic channel variations. Additionally, we present an asymptotic OP analysis to gain useful insights into the high signal-to-noise ratio regime and diversity order, which are useful in optimizing network parameters for maximal reliability. Our study advances complex optimization problems for deep neural network (DNN) hyperparameters, power allocation, and UAV positioning, which are crucial for the dynamic aerial communication environment. We also introduce a new method to evaluate the robustness of our system, the analysis reveals that the system performs well with fewer IRS elements, optimizing the balance between energy efficiency and outage performance. Given the significant complexity of the proposed system model, directly deriving closed-form expressions for the OP and the ergodic sum capacity is a challenge. We develop a DNN framework that predicts OP and ergodic sum capacity in real-time scenarios to overcome this issue. Extensive simulations validate the derived expressions and demonstrate that a UAV-mounted A-IRS NOMA network outperforms both passive IRS NOMA setups and traditional relaying methods. These results affirm notable enhancements in reliability and performance, establishing the network’s superiority in modern wireless communication scenarios and underscoring its potential to enhance both service quality and economic viability in deploying advanced communication infrastructures.

SECTION I. Introduction

As the demand for higher data rates and broader user connectivity grows, efforts to advance wireless networks towards beyond 5G and 6G technologies are intensifying. Key innovations being explored include intelligent reflecting surfaces (IRSs) or reconfigurable intelligent surfaces (RISs), non-orthogonal multiple access (NOMA), and unmanned aerial vehicles (UAVs). These technologies aim to improve the quality of service (QoS) by increasing data rates, enhancing spectral efficiency, and expanding coverage [1], [2].

IRS-assisted communication systems have attracted growing interest in both academia and industry, thanks to their affordability, adaptability, and energy efficiency. An IRS leverages meta-material elements and programmable phase shifts for enhanced communication through smart reflection. It uses surfaces such as microstrip patches to adjust the phase and amplitude of reflected signals. Unlike conventional transceivers, IRS manipulates the wireless medium directly, improving signal propagation without generating signals itself [3]. However, the conventional IRS faces the challenge of double path loss due to its pure passive reflecting gain [4]. A recently introduced concept called active IRS (A-IRS), where the incident signal can be amplified without significantly affecting the power budget [5], has emerged as a promising solution to overcome the double path-loss challenge. An A-IRS provides significant advantages over passive IRS (P-IRS) in scenarios where signal amplification, broader coverage, and dynamic signal shaping are needed. However, this comes with higher power consumption and complexity. Thus, the choice between A-IRS and P-IRS depends on the specific application requirements, with A-IRS being better suited for high-demand and large-area networks. Reflecting-only IRS designs have limited coverage when users are on the opposite side of the IRS with blocked or no communication paths. As a consequence, a novel simultaneously transmitting and reflecting RIS (STAR-RIS) has been introduced [6], [7]. Li et al. [8] examined the secrecy performance of STAR-RIS assisted NOMA networks, analyzing how configurable elements and hardware impairments impact security outcomes. In parallel, UAVs are envisioned as pivotal technology in future networks due to their low cost and controllable mobility. Thus, UAVs are deployed in applications like aerial base stations, relay, and Internet-of-Things (IoT) sensor data collection [9], [10]. In particular, UAV-aided systems play an important role in emergency communications and natural disaster scenarios where traditional terrestrial networks fail. However, the performance of UAV-aided systems is constrained by limited energy budget or battery capacity [11]. More recently, the concept of aerial IRS or UAV-mounted IRS systems has attracted extensive research interest. This system reaps the benefits of both IRS and UAV in extending network coverage, as well as improving the spectral and energy efficiency of the system.

In the realm of multiple user communications, NOMA has emerged as a prominent multiple access scheme for the next-generation wireless networks. In NOMA deployed systems, users are multiplexed primarily in the power domain, rather than the time-frequency domain used in conventional orthogonal multiple access (OMA) systems. This is achieved through superposition coding at the transmitter and successive interference cancellation (SIC) at the receiver. By enabling simultaneous resource sharing among multiple users, NOMA enhances spectral efficiency, connectivity, and reduces latency, thereby outperforming the traditional OMA scheme [12]. The use of NOMA in IRS-enhanced UAV communications is highly appealing due to several reasons. Firstly, NOMA provides more flexible and efficient resource allocation than OMA, meeting diverse user requirements and enhancing spectrum efficiency. Secondly, in conventional NOMA, SIC decoding orders are fixed by channel conditions. However, UAVs and IRSs can modify these conditions through mobility and adjustable reflection coefficients, enabling smarter NOMA operations [13], [14].

A. Related Works

Recent studies have extensively explored the use of IRS in wireless communication networks [4], [15], [16], [17], [18], [19], [20], [21]. Zhi et al. [4] demonstrated the superiority of an A-IRS assisted system over a P-IRS assisted system, under the same power budget constraint, in terms of achievable rate. The reliability and robustness of an A-IRS assisted physical layer security communication systems was studied in terms of outage and secrecy outage performance in [15] In [16], a joint design of transmit beamforming for the source node and 3D passive beamforming for the aerial IRS was proposed to maximize the worst-case SNR of the system. Shafique et al. [17] examined the performance of a hybrid UAV-IRS relaying system under IRS only mode, UAV only mode, and UAV-mounted IRS mode. Similarly, Solanki et al. [18] studied a UAV-mounted IRS-assisted single-user network alongside an IoT system, showing that the IRS’s deployment notably improves the performance of both the considered systems. On the other hand, Tyrovolas et al. [19] utilized the UAV-mounted IRS system for data collection and analyzed the system performance using average throughput and average collected data per flight. Qin et al. [20] proposed a covert communication scheme facilitated by a relaying UAV equipped with an IRS. In particular, Sun et al. [21] introduced a novel framework using a multi-functional IRS to enhance mobile edge computing-assisted integrated aerial-ground networks (MEC-IAGN) for 6G. On the other hand, An et al. [22] explored the integration of IRS and simultaneous wireless information and power transfer (SWIPT) in high-altitude platform (HAP) networks. In [23], a cost-effective, active-passive cascaded IRS-aided receiver architecture using a large-scale antenna array for enhanced anti-jamming communications was studied. Sun et al. in [24] enhanced anti-jamming in IoT networks using an IRS and robust beamforming optimization, demonstrating improved performance and reduced power consumption in simulations compared to conventional methods.

Several studies indicate that integrating NOMA with IRS in a multiuser setup can enhance the spectral and energy efficiency of the system. In particular, Ahmed et al. [25] combined IRS with NOMA backscatter communication (BC) systems and highlighted the performance gain of the system in terms of outage probability (OP), data rate, and energy efficiency. In [26], [27], [28], the performance of NOMA with P-IRS system was analyzed in terms of users’ OP and ergodic capacity. Gu et al. [29] examined an P-IRS-aided cooperative NOMA (C-NOMA) system for two power-domain users, where the near user acts as a decode-and-forward (DF) half-duplex (HD) relay to perform device-to-device (D2D) communications for the far user. In [30], the performance of a power domain (PD)-NOMA-enabled base station in a multi-user IRS-assisted ambient BC (AmBC) system was studied by analyzing the users’ outage and asymptotic OP.

In the context of multiple users, Do et al. [31] demonstrated the advantages of P-IRS NOMA networks with ordered users, showing improved outage and ergodic performance compared to traditional relaying methods. Further, Mu et al. [32] comprehensively analyzed the capacity and rate regions of P-IRS NOMA networks with multiple users. Guo et al. [33] analyzed the outage behavior in non-terrestrial networks assisted by RIS, incorporating cognitive radio and NOMA. Additionally, Liu et al. [34] emphasized on the improvements in two-way NOMA networks facilitated by P-IRS. In [35], an algorithm was proposed to obtain the optimal IRS beamforming vectors to maximize the throughput of an A-IRS assisted multi-user system with NOMA scheme. A STAR-RIS assisted NOMA network for IoT network was studied in [36], [37], [38], under the impact of imperfect SIC (iSIC). Yue et al. [39] presented the novel active STAR-RIS for NOMA communications, deriving new metrics for OP and data rates. Singh and Upadhyay [40] considered a full-duplex (FD)/HD NOMA in an underlay cognitive spectrum sharing networks under the impact of imperfect SIC (iSIC). In [41], a double IRS-aided network, i.e., both PRIS-ARIS-assisted NOMA network was studied, under cascaded Rician and Nakagami-m fading channels.

Radio frequency (RF) transceivers in wireless networks suffer from hardware impairments (HIs), including amplifier nonlinearities and phase noises [42]. Despite the use of cost-effective devices and compensation algorithms, residual impairments persist. Li et al. [43] analyzed the impact of HIs on cooperative and non-cooperative NOMA networks, deducing that cooperative NOMA performs better under high signal-to-noise conditions. Authors in [44], [45] investigated A-RIS assisted NOMA networks factoring in HIs. The impact of HIs on simultaneous wireless information and power transfer enabled cooperative NOMA in massive IoT systems, excluding spectrum-sharing considerations, was recently analyzed in [46]. Vu et al. [47] studied UAV-aided NOMA systems with iSIC by deriving closed-form expression for the key performance metrics of the system. Besides HIs, co-channel interference (CCI) is another detrimental effect that significantly degrades signal quality in a wireless communication systems. With this motivation, Hussein et al. [48] examined how CCI and primary transmitter interference affect cognitive radio networks in Nakagami-m channels. Similarly, in [49], [50], the impact of CCI on the users’ outage and capacity performance in a downlink IRS-aided system was examined.

In recent years, deep learning (DL) has increasingly captivated the interest of researchers in the field of communications for predicting the performance metrics of analytically complex networks because of its low computational complexity [51]. It is effective at identifying complex patterns and relationships in data, making it appropriate for performance analysis. Within the framework of DL, deep neural networks (DNNs) can be trained on historical outage and capacity data, network performance metrics, and other system parameters for precise modeling and prediction of OP and ergodic capacity. By processing large datasets, DNNs can find intricate patterns and relationships that conventional analytical approach often lack, leading to better accuracy and faster performance predictions. Moreover, a DNN based approach was deployed in [52] to improve the productivity of a cognitive NOMA-based network, proposing relay selection methods. A sum-rate optimization problem for an IRS-NOMA system was explored through deep reinforcement learning for IRS phase shift tuning in [53].

B. Motivations and Key Contributions

As discussed in the previous subsections, the synergistic integration of IRS with UAV relay, along with advanced technologies like NOMA, can significantly enhance system performance compared to the individual fixed IRS or traditional UAV relaying based systems. In real-world applications, especially in challenging environments like disaster-hit high terrains or extreme weather conditions, this type of relay communication could be highly effective for ensuring continuous end-to-end communication. In situations where P-IRS is limited by heavy double path-loss and fails to meet quality of service standards, employing A-IRS transmission can be remarkably beneficial. It is worth recognizing that Authors in [18], [26] considered a P-IRS assisted UAV system in their analysis. In [36], [37], [38], a P-IRS aided NOMA system with fixed IRS deployment was considered. On the other hand, the impact of iSIC and HIs on a cooperative NOMA system was considered in [39], [44]. However, their analysis was restricted to fixed RIS relaying systems and did not account for the direct link and CCI signals. In contrast, the work in [47] considered a UAV-aided NOMA system for two pairs of ground users. However, they did not explore the potential of IRS for relaying, nor did they consider the inevitable presence of CCI in practical systems. A cooperative NOMA system with conventional relays for spectrum sharing cooperation was considered in [40], without incorporating IRS and UAVs. In the above studies, the authors did not include a direct signal link from the base station to the users. Moreover, they focused on ideal hardware conditions at both the transmitter and receiver, and did not consider the interference from neighboring cells, which are typically included in more practical scenarios.

Therefore, inspired by the above discussion and aiming to address the gaps in the existing literature, we propose a UAV-mounted A-IRS/P-IRS NOMA network over generalized Nakagami-m fading channels for enhanced robustness and reliability of the communication systems. Additionally, we consider non-ideal system imperfections such as CCI, HIs, iSIC in addition to perfect SIC (pSIC) to reflect a more realistic scenario. The key contributions of this work are outlined as follows:

• We propose a UAV-mounted A-IRS/P-IRS NOMA communication systems that utilizes both direct and IRS-assisted links to cater to ground users. By employing a coherent phase shifting strategy, we derive the optimal signal-to-interference-plus-distortion-plus-noise ratio (SIDNR) for NOMA users, considering the effects of non-ideal system imperfections like CCI, HIs, iSIC.

• The SIDNR expressions allow us to derive accurate OP and ergodic capacity expressions for the users subject to iSIC and pSIC, as well as system throughput in both delay-tolerant and delay-limited modes, thereby enabling the performance analysis. We further carry out the asymptotic OP analysis at high SNR, which allows us to evaluate the diversity order of the system and examine the impact of key parameters on the users’ OP.

• We formulate two optimization problems to determine (a) UAV’s optimal 3D position for minimizing the users’ OP, (b) Optimal power allocation factors for each user with focus on minimizing the users’ OP, (c) Optimal DNN hyperparameters for both OP and ergodic capacity of the users.

• Due to the challenges involved in deriving complex analytical expressions, existing studies have not provided the expressions for key performance metrics of UAV-mounted A-IRS/P-IRS NOMA networks. Thus, we develop a DNN model that offers a high-accuracy, low-latency data-driven alternative for OP and ergodic capacity prediction. This model aims to enable real-time implementation of analytically complex networks, such as UAV-mounted IRS NOMA networks. We use execution time, loss, and root-mean-square error (RMSE) parameters to validate the effectiveness and precision of the designed DNN framework.

• We introduce a novel approach to define and quantify the robustness of the proposed system. It demonstrates that the system maintains high performance and adaptability with fewer IRS elements, highlighting its resilience against energy constraints. Through this analysis, our paper contributes significantly to optimizing the balance between energy efficiency and outage performance of the considered system.

C. Mathematical Notations

$\Upsilon (\cdot,\cdot)$ and $\Gamma (\cdot)$ denote the lower incomplete Gamma function and Gamma function, as defined in [54, (8.350.1) and (8.310.1)], respectively. The notation $\mathcal {C N} (\mu, \sigma ^{2})$ is used for the complex Gaussian distribution with mean $\mu$ and variance $\sigma ^{2}$ . $F_{X}(\cdot)$ and $f_{X}(\cdot)$ are used to represent the cumulative distribution function (CDF) and the probability density function (PDF) of a random variable (RVs) X, respectively. $\mathbb {E}{[\cdot]}$ signifies the expectation.

D. Organization of the Paper

This paper is structured as follows: Section II details the system and channel models of a UAV-mounted A-IRS/P-IRS NOMA network and provides the SIDNR expressions. In Section III, we present the expressions for OP, asymptotic OP, and throughput followed by the optimization problems. Comparison of the proposed system with UAV P-IRS NOMA is detailed in Section IV. Section V introduces a DNN framework for real-time system assessment. Numerical and simulation results are provided in Section VI, with conclusions in Section VII. Finally, the proofs of key theorems and lemmas are included in the appendices.

SECTION II. System Model

We consider a UAV-mounted A-IRS/P-IRS relaying network depicted in Fig. 1, where a base station $(B_{s})$ communicates with two NOMA users ($U_{1}$ and $U_{2}$ ) with the assistance of a UAV-mounted IRS relay $(R)$ . Each user receives the signal from the $B_{s}$ through both a direct link and a UAV-mounted IRS relay link. Notably, both the base station and users utilize single-antenna devices, which are presumed to have cost-effective RF transceiver components, thus subjecting them to HIs.1 Moreover, we take into account a heterogeneous network configuration where all users are subject to independent and identically distributed (i.i.d) CCI caused by I neighboring base stations $S_{i}$ , for $i\in \{1,2, {\dots },I\}$ [48], [56]. The IRS has N reflecting elements, which can intelligently modify the phase and amplitude of the incident signals to enhance the users’ SIDNRs. Considering the existing channel estimation techniques as discussed in [57], we assume perfect the channel state information availability throughout this paper.

FIGURE 1.

A UAV-mounted A-IRS/P-IRS NOMA network.

FIGURE 2.

DNN deployment architecture.

FIGURE 3.

OP comparison of (a) UAV A-IRS NOMA vs. UAV P-IRS NOMA and (b) UAV A-IRS NOMA vs. UAV A-IRS OMA of the primary network.

FIGURE 4.

Comparison plots for (a) UAV A-IRS NOMA vs. UAV P-IRS NOMA and (b) UAV A-IRS NOMA vs. UAV A-IRS OMA of the secondary network.

FIGURE 5.

Impact of number of reflecting elements $\text {(}N\text {)}$ on OP of the primary user.

FIGURE 6.

(a) System robustness (${\mathcal {R}}_{S}$ ) vs. SNR for different target OPs (${\mathcal {P}}_{T}$ ) and (b) Impact of SNR and N on the robustness index (${\mathcal {R}}_{I}$ ).

FIGURE 7.

Impact of (a) UAV height $\text {(}{\mathcal {H}}_{r}\text {)}$ , (b) radius $\text {(}R_{r}\text {)}$ , and (c) amplification factor $\text {(}\eta _{a}\text {)}$ , CCI power $\text {(}\rho _{iu_{2}}\text {)}$ , and iSIC $\text {(}\xi \text {)}$ on the users’ OP.

A. Airborne Path-Loss Model

UAV-mounted A-IRS/P-IRS communication network follows an airborne channel model that considers both large-scale and small-scale fading effects on the overall channel gain [18]. Specifically, for the UAV, large-scale fading depends on distance, altitude, and elevation angle, which are the primary factors affecting airborne propagation. Ground nodes $B_{s}$ , $S_{i}$ , $U_{1}$ , and $U_{2}$ are represented in three-dimensional (3-D) Cartesian space by their coordinates as $\mathbf {v}_{b}=({\mathcal {X}}_{b},{\mathcal {Y}}_{b},{\mathcal {Z}}_{b})$ , $\mathbf {v}_{s_{i}}=(\mathcal {X}_{s_{i}},\mathcal {Y}_{s_{i}},\mathcal {Z}_{s_{i}})$ , $\mathbf {v}_{u_{1}}=(\mathcal {X}_{u_{1}},\mathcal {Y}_{u_{1}},0)$ , and $\mathbf {v}_{u_{2}}=(\mathcal {X}_{u_{2}},\mathcal {Y}_{u_{2}},0)$ , respectively. The UAV is expected to follow a circular path with radius ${\mathcal {R}}_{r}$ , at an altitude ${\mathcal {H}}_{r}$ within $[{\mathcal {H}}_{r}^{\min }, {\mathcal {H}}_{r}^{\max }]$ , where ${\mathcal {H}}_{r}^{\min }$ and ${\mathcal {H}}_{r}^{\max }$ represent the minimum and maximum allowed altitudes. Let $\phi _{r}$ be the angle representing the UAV’s location relative to the x-axis. Therefore, the UAV’s location can be represented as $\mathbf {v}_{r}=({\mathcal {R}}_{r} \cos \phi _{r}, \, {\mathcal {R}}_{r} \sin \phi _{r}, \, {\mathcal {H}}_{r})$ . The positions of $B_{s}$ , $S_{i}$ , $U_{1}$ , $U_{2}$ , and the UAV are represented using two-dimensional Cartesian coordinates as $\mathbf {w}_{b}=({\mathcal {X}}_{b},{\mathcal {Y}}_{b})$ , $\mathbf {w}_{s_{i}}=(\mathcal {X}_{s_{i}},\mathcal {Y}_{s_{i}})$ , $\mathbf {w}_{u_{1}}=(\mathcal {X}_{u_{1}},\mathcal {Y}_{u_{1}})$ , $\mathbf {w}_{u_{2}}=(\mathcal {X}_{u_{2}},\mathcal {Y}_{u_{2}})$ , and $\mathbf {w}_{r}=({\mathcal {R}}_{r} \cos \phi _{r}, \, {\mathcal {R}}_{r} \sin \phi _{r})$ , respectively. Let $\theta _{br}=\arctan \left ({{{}\frac {{\mathcal {H}}_{r}}{|\mathbf {w}_{r}-\mathbf {w}_{b}|}}}\right)$ and $\theta _{u_{j}r}=\arctan \left ({{{}\frac {{\mathcal {H}}_{r}}{|\mathbf {w}_{r}-\mathbf {w}_{u_{j}}|}}}\right)$ , where $j \in \{1,2\}$ , denote the elevation angles in radians between $B_{s}$ and R, and j-th user and R, respectively.

The UAV’s mobility increases the probability of LoS establishment with ground nodes [58]. The LoS probability between the UAV and nodes $B_{s}$ , $U_{1}$ , $U_{2}$ is given as $$\begin{equation*} \mathbb {P}^{L}\left ({{\theta _{{X}r}}}\right)=\Big (1+\epsilon _{{X}r}\, \exp \big (- \iota _{{X}r}\left ({{\theta _{{X}r}-\epsilon _{{X}r}}}\right)\big)\Big)^{-1}, \tag {1}\end{equation*}$$ where $\epsilon _{{X}r}$ and $\iota _{{X}r}$ $({X} \in \{b, u_{1}, u_{2} \})$ represent environment constants. The associated path-loss exponent is given as $$\begin{equation*} \alpha _{{X}r} \left ({{\theta _{{X}r}}}\right)=\mathbb {P}^{L}\left ({{\theta _{{X}r}}}\right)\varsigma _{{X}r} +\upsilon _{{X}r}, \tag {2}\end{equation*}$$ where $\varsigma _{{X}r}$ and $\upsilon _{{X}r}$ are constants related to the uplink and downlink propagating environment.

For the small-scale fading model, we assume that all airborne and terrestrial channel coefficients are independent RVs and follow Nakagami-m distribution [59]. The Nakagami-m distribution is a versatile model that encompasses various fading scenarios. For instance, it can also be used to characterize Rician fading by setting its parameter m as $m = (1-(K/(K+1))^{2})^{-1}$ , where K represents the Rician factor [60]. Let us represent the $N\times 1$ airborne channel coefficient vector from $B_{s}$ to R and R to $U_{j}$ as $\mathbf {g}_{b} = [g_{b,1}, {\dots }, g_{b,N}]^{\scriptscriptstyle \textsf {H}}$ and $\mathbf {g}_{u_{j}} = [g_{1,u_{j}}, {\dots }, g_{N,u_{j}}]^{\scriptscriptstyle \textsf {H}}$ , respectively. The channel coefficients for the communication links are denoted as follows: $g_{b,n}$ for the links from $B_{s}$ to n-th A-IRS element at the UAV $(n \in \mathcal {N} \triangleq {1, {\dots }, N})$ , $g_{n,u_{j}}$ from the n-th A-IRS element to the j-th user, $h_{b,u_{j}}$ from $B_{s}$ to j-th user, and $h_{i,u_{j}}$ from $S_{i}$ to j-th user. The Nakagami-m distribution parameters for these links are $g_{b,n} \sim Nakagami(m_{br}, \Omega _{br})$ , $g_{n,u_{j}} \sim Nakagami(m_{ru_{j}}, \Omega _{ru_{j}})$ , $h_{bu_{j}} \sim Nakagami(m_{bu_{j}}, \Omega _{bu_{j}})$ , and $h_{iu_{j}} \sim Nakagami(m_{iu_{j}}, \Omega _{iu_{j}})$ . Furthermore, we assume perfect channel state information (CSI) acquisition, throughout the paper.2

Additionally, the ${N} \times {N}$ diagonal reflection/amplification matrix at the IRS is denoted by $\boldsymbol {\Theta } = \textrm {diag} (\eta _{1} e^{j\varphi _{1}}, {\dots }, \eta _{n} e^{j\varphi _{n}}, {\dots }, \eta _{N} e^{j \varphi _{N}})$ , where $\varphi _{n} \in [0, 2\pi$ ) and $\eta _{n}$ represent the phase shift and reflection coefficient amplitude of the n-th IRS reflecting element, respectively.3 In the case of a P-IRS, the reflection coefficient amplitude for the n-th element is $\eta _{n} \in [{0, 1}]$ , we assume a lossless reflection scenario where $\eta _{n}=1~\forall n$ . For the A-IRS case, $\eta _{n}\gt 1~\forall n$ , representing its signal amplification capability. Further, for simplicity, we assume $\eta _{n}=\eta _{a}~\forall n$ , where $\eta _{a}$ is the amplification factor, and then define $\boldsymbol {\Theta }= \eta _{a} \, \textrm {diag} (e^{j\varphi _{1}}, {\dots }, e^{j\varphi _{n}}, {\dots }, e^{j\varphi _{N}})= \eta _{a} \boldsymbol {\Phi }$ .

B. Formulation of SIDNR of the UAV-Mounted A-IRS NOMA Networks

In a downlink NOMA aided transmission, the $B_{s}$ transmits a superimposed signal as $\sqrt {\delta _{1}P_{b}}x_{1}+\sqrt {\delta _{2}P_{b}}x_{2}$ , where $x_{1}$ and $x_{2}$ are unit-power bearing message signals intended for $U_{1}$ and $U_{2}$ , respectively, with transmit power $P_{b}$ and power allocation factors $\delta _{1}$ and $\delta _{2}$ , such that $\delta _{1} + \delta _{2}=1$ . Note that due to the higher priority assigned to the primary network $U_{1}$ , more power is directed towards primary network than secondary network $U_{2}$ , i.e., $\delta _{1}\gt \delta _{2}$ , [40]. Let us assume that users $U_{1}$ and $U_{2}$ are contaminated by I CCI symbols $x_{iu_{j}}$ with $\mathbb {E}[|x_{iu_{j}}|^{2}]=1$ , each having an average power of $P_{i}$ at the I-th interfering base station $(S_{i})$ . Thus, the received signal at the j-th user, including direct and UAV-mounted A-IRS links in the presence of CCI, is expressed as $$\begin{align*}& y_{u_{j}}^{a} = \underbrace {\big (a_{j}h_{bu_{j}} + b_{j} \mathbf {g}_{u_{j}}^{\scriptscriptstyle \textsf {H}}\boldsymbol {\Theta } \mathbf {g}_{b}\big) \big (\sqrt {\delta _{1}P_{b}}x_{1} + \sqrt {\delta _{2}P_{b}}x_{2} + \zeta ^{t}_{b}\big) + \zeta ^{r}_{bu_{j}} }_{\textrm {Desired signal}} \\& \;{}+ \underbrace {\hat {b_{j}} \mathbf {g}_{u_{j}}^{\scriptscriptstyle \textsf {H}}\boldsymbol {\Theta } \psi _{t}}_{\textrm {Thermal noise}} + \underbrace { \sum _{i=1}^{I} c_{j} h_{iu_{j}} \big (\sqrt {P_{i}} x_{iu_{j}} + \zeta ^{t}_{i} \big) + \zeta ^{r}_{iu_{j}} }_{\textrm {CCI signal}} + \underbrace {\psi _{u_{j}}}_{\textrm {AWGN}}, \tag {3}\end{align*}$$ where $a_{j}=(d_{bu_{j}}/d_{0})^{{}\frac {-\alpha _{bu_{j}}}{2}}$ , $b_{j}=(d_{br}/d_{0})^{{}\frac {-\alpha _{br}(\theta _{br})}{2}} \times (d_{ru_{j}}/d_{0})^{{}\frac {-\alpha _{ru_{j}}(\theta _{ru_{j}})}{2}}$ , $\hat {b_{j}}= (d_{ru_{j}}/d_{0})^{{}\frac {-\alpha _{ru_{j}}(\theta _{ru_{j}})}{2}}~c_{j}=(d_{s_{i}u_{j}}/d_{0})^{{}\frac {-\alpha _{iu_{j}}}{2}}$ , with $d_{0}$ as the reference distance. The Euclidean distances between $B_{s}$ to R, $B_{s}$ to j-th user, R to j-th user, and $S_{i}$ to j-th user are $d_{br}=||\mathbf {v}_{r}-\mathbf {v}_{b}||$ , $d_{bu_{j}}=\sqrt {|\mathbf {w}_{u_{j}}-\mathbf {w}_{b}|^{2}+{\mathcal {Z}}_{b}^{2}}$ , $d_{ru_{j}}=\sqrt {|\mathbf {w}_{u_{j}}-\mathbf {w}_{r}|^{2}+{\mathcal {H}}_{r}^{2}}$ , and $d_{s_{i}u_{j}}=\sqrt {|\mathbf {w}_{u_{j}}-\mathbf {w}_{s_{i}}|^{2}+\mathcal {Z}_{s_{i}}^{2}}$ , respectively; $\alpha _{bu_{j}}$ and $\alpha _{iu_{j}}$ represent the path-loss exponents for the terrestrial links. Further, $\psi _{t} \in \mathbb {C}^{N \times 1}$ is the thermal noise generated at the A-IRS $\psi _{t} \sim \mathcal {CN}(0,\sigma _{a}^{2} \mathbf {I}_{N})$ and $\psi _{u_{j}} \sim \mathcal {CN}(0,\sigma _{j}^{2})$ is the additive white Gaussian noise (AWGN) at the j-th user; $\zeta ^{t}_{b} \sim \mathcal {CN}(0,\lambda _{tb}^{2}P_{b})$ and $\zeta ^{t}_{i} \sim \mathcal {CN}(0,\lambda _{ti}^{2}P_{i})$ represent distortion noise arising from HIs at the transmitter section of $B_{s}$ and $S_{i}$ , respectively. Furthermore, $\zeta ^{r}_{bu_{j}} \sim \mathcal {CN}(0,\lambda _{rbu_{j}}^{2}P_{b}|a_{j}h_{bu_{j}} + b_{j} \mathbf {g}_{u_{j}}^{H}\boldsymbol {\Theta } \mathbf {g}_{b}|^{2})$ and $\zeta ^{r}_{iu_{j}} \sim \mathcal {CN}(0,\lambda _{riu_{j}}^{2}P_{i}|h_{iu_{j}}|^{2})$ denote the distortion noise at the receiving node $U_{j}$ , resulting from HIs on the $B_{s}$ to $U_{j}$ and $S_{i}$ to $U_{j}$ channels, respectively. The levels of HIs, $\lambda _{tb}$ , $\lambda _{ti}$ , $\lambda _{rbu_{j}}$ , and $\lambda _{riu_{j}}$ , can be quantified as error vector magnitudes (EVMs) [59].

In accordance with the NOMA protocol, $U_{1}$ directly decodes its own message since it has a higher power allocation factor $\delta _{1}$ . Consequently, the SIDNR for $U_{1}$ to decode its own information is expressed as $$\begin{equation*} \gamma _{u_{1},x_{1}}^{a}=\frac {\delta _{1}\, \rho _{b}\mathcal {Z}_{1}}{ \varrho _{1} \mathcal {Z}_{1} + \Xi _{1} \|\mathbf {g}_{u_{1}}^{H}\boldsymbol {\Phi } \|^{2} + \beta _{1} |h_{iu_{1}}|^{2} + 1}, \tag {4}\end{equation*}$$ where $\mathcal {Z}_{1}=|a_{1}h_{bu_{1}} + b_{1} \eta _{a} \mathbf {g}_{u_{1}}^{H}\boldsymbol {\Phi } \mathbf {g}_{b}|^{2}$ , $\varrho _{1}=(\delta _{2} + \lambda _{bu_{1}}^{2}) \rho _{b}$ , $\Xi _{1}=\hat {b_{1}} \eta _{a}^{2} \sigma _{a}^{2}$ , $\beta _{1}=\sum _{i=1}^{I} (1 + \lambda _{iu_{1}}^{2}) c_{1} \rho _{i}$ . Moreover, $\lambda _{bu_{1}}^{2}=\lambda _{tb}^{2}+\lambda _{ru_{1}}^{2}$ and $\lambda _{iu_{1}}^{2}=\lambda _{ti}^{2}+\lambda _{ru_{1}}^{2}$ elucidate the aggregate HIs level at the user $U_{1}$ from the $B_{s}$ to $U_{1}$ and $S_{i}$ to $U_{1}$ links, respectively. Further, $\rho _{b}={}\frac {P_{b}}{\sigma ^{2}}$ and $\rho _{i}={}\frac {P_{i}}{\sigma ^{2}}$ denote the transmit signal-to-noise ratio (SNR) for $B_{s}$ and $S_{i}$ , respectively. Conversely, $U_{2}$ uses SIC as per NOMA, first decoding $U_{1}$ ’s signal with its own signal as noise, leading to SIDNR expressions for decoding $x_{1}$ at $U_{2}$ as $$\begin{equation*} \gamma _{u_{2},x_{1}}^{a}=\frac {\delta _{1}\, \rho _{b}\mathcal {Z}_{2}}{ \varrho _{2} \mathcal {Z}_{2} + \Xi _{2} \|\mathbf {g}_{u_{2}}^{H}\boldsymbol {\Phi } \|^{2} + \beta _{2} |h_{iu_{2}}|^{2} + 1}, \tag {5}\end{equation*}$$ where $\mathcal {Z}_{2}=|a_{2}h_{bu_{2}} + b_{2} \eta _{a} \mathbf {g}_{u_{2}}^{H}\boldsymbol {\Phi } \mathbf {g}_{b}|^{2}$ , $\varrho _{2}=(\delta _{2} + \lambda _{bu_{2}}^{2}) \rho _{b}$ , $\Xi _{2}=\hat {b_{2}} \eta _{a}^{2} \sigma _{a}^{2}$ , and $\beta _{2}=\sum _{i=1}^{I} (1 + \lambda _{iu_{2}}^{2}) c_{2} \rho _{i}$ . Further, the expressions $\lambda _{bu_{2}}^{2}=\lambda _{tb}^{2}+\lambda _{ru_{2}}^{2}$ and $\lambda _{iu_{2}}^{2}=\lambda _{ti}^{2}+\lambda _{ru_{2}}^{2}$ signify the combined HIs levels for user $U_{2}$ , encompassing both the links from the $B_{s}$ to $U_{2}$ and the $S_{i}$ to $U_{2}$ . Subsequently, after eliminating the signal $x_{1}$ using SIC, $U_{2}$ decodes its own message $x_{2}$ . The corresponding SIDNR is expressed as $$\begin{align*} \gamma _{u_{2},x_{2}}^{a} = \frac {\delta _{2}\, \rho _{b} \mathcal {Z}_{2}}{\delta _{1}\, \rho _{b}|\hbar _{2}|^{2}+\hat {\varrho }_{2}\mathcal {Z}_{2} + \Xi _{2} \|\mathbf {g}_{u_{2}}^{H}\boldsymbol {\Phi } \|^{2} + \beta _{2} |h_{iu_{2}}|^{2} + 1}, \tag {6}\end{align*}$$ where $\hat {\varrho }_{2}= \lambda _{bu_{2}}^{2} \rho _{b}$ , $\hbar _{2}$ represents the channel coefficient of the residual interference signal (IS) at $U_{2}$ . It experiences Nakagami-m fading with fading severity $\textit {m}_{u_{2}}$ and an average power of $\mathbb {E}[|h_{u_{2}}|^{2}]=\xi \Omega _{u_{2}}$ . Here, $\xi ~(0\leq \xi \leq 1)$ denote residual IS from iSIC, with $\xi =0$ indicating pSIC.

SECTION III. Performance Analysis of UAV-Mounted A-IRS NOMA Networks

In this section, we start by evaluating the OP performance of the system. Following this, we present an asymptotic OP analysis to provide valuable insights, including the optimal power allocation factor and optimal UAV positioning for reliable communication. Furthermore, we explore metrics such as the ergodic rate and system throughput to evaluate the overall effectiveness of the system under consideration.

A. Outage Probability

A user is said to be in outage when its SIDNR falls below a predefined threshold level, and the likelihood of an outage event is termed as OP.

1) Outage Probability for User $U_{1}$

With the help of SIDNR given in (4), we can mathematically define the OP for $U_{1}$ with a target rate of $r_{\textrm {th}}^{1}$ as $$\begin{align*} \mathcal {P}_{u_{1}}^{a}=& \mathbb {P}_{r} \left [{{ \gamma _{u_{1},x_{1}}^{a} \lt \gamma _{\textrm {th}}^{1} }}\right ] \\=& \mathbb {P}_{r} \left [{{ \mathcal {Z}_{1} \lt \frac {\gamma _{\textrm {th}}^{1} \left \{{{ 1+\beta _{1} |h_{iu_{1}}|^{2} + \Xi _{1} \|\mathbf {g}_{u_{1}}^{H}\boldsymbol {\Phi } \|^{2} }}\right \}}{\delta _{1}\, \rho _{b}-\left ({{\delta _{2}+\lambda _{bu_{1}}^{2}}}\right) \rho _{b}\gamma _{\textrm {th}}^{1}} }}\right ], \tag {7}\end{align*}$$ where $\gamma _{\textrm {th}}^{1}=2^{r_{\textrm {th}}^{1}}-1$ is the SIDNR threshold. Thus, to compute $\mathcal {P}_{u_{1}}^{a}$ in (7), we can proceed by applying Theorem 1.

Theorem 1:

The $\mathcal {P}_{u_{1}}^{a}$ for the proposed UAV-mounted A-IRS NOMA network is given as (17), shown at the bottom of the next page, for $\gamma _{\textrm {th}}^{1}\lt {}\frac {\delta _{1}}{\delta _{2}+\lambda _{bu_{1}}^{2}}$ and $u\gt 0$ . In this context, $\mathcal {T}_{1}={}\frac {\gamma _{\textrm {th}}^{1}}{\delta _{1}\, \rho _{b}-(\delta _{2}+\lambda _{bu_{1}}^{2}) \rho _{b}\gamma _{\textrm {th}}^{1}}$ and $\tilde {b}_{1}=b_{1} \eta _{a}$ . The shape and scale parameters of the Gamma distributed RV are $m_{z}={}\frac {[{\mathcal {M}}_{\hat { \mathcal {Z}}}(1)]^{2} }{{\mathcal {M}}_{\hat { \mathcal {Z}}}(2) - [{\mathcal {M}}_{\hat { \mathcal {Z}}}(1)]^{2}}$ and $\Omega _{z}={}\frac {{\mathcal {M}}_{\hat { \mathcal {Z}}}(2) - [{\mathcal {M}}_{\hat { \mathcal {Z}}}(1)]^{2}}{{\mathcal {M}}_{\hat { \mathcal {Z}}}(1)}$ , respectively. The extended Fox-H function $(\bar {\textrm {H}})$ uses coefficient sets $\Delta _{u,1}=\left \{{{\left ({{1-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{iu_{1}, a\}}$ and $\Delta _{u,2}=\left \{{{\left ({{-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{iu_{1}, a\}}$ , for $\{a_{t}\}_{t\in \{1, 2\}}=\{a_{1},a_{2}\}$ . The expression for $\varpi _{l}={}\frac {1+\varPsi _{l}}{1-\varPsi _{l}}$ , where $\varPsi _{l}=\cos \left ({{{}\frac {2l-1}{2L}\pi }}\right)$ , and L is the complexity-accuracy tradeoff parameter, with $$\begin{align*} {\mathcal {M}}_{\hat { \mathcal {Z}}}(1)=& {\mathcal {M}}_{\hat {\mathcal {X}}}(1) + {\mathcal {M}}_{\hat { \mathcal {Y}}_{n}}(1), \tag {8}\\ {\mathcal {M}}_{\hat { \mathcal {Z}}}(2)=& {\mathcal {M}}_{\hat {\mathcal {X}}}(2) + {\mathcal {M}}_{\hat { \mathcal {Y}}_{n}}(2) + 2 {\mathcal {M}}_{\hat {\mathcal {X}}}(1) {\mathcal {M}}_{\hat { \mathcal {Y}}_{n}}(1), \tag {9}\\ {\mathcal {M}}_{\hat {\mathcal {X}}}(1)=& \frac {\Gamma \left ({{m_{bu_{1}}+1/2}}\right)}{\Gamma \left ({{m_{bu_{1}}}}\right)} \sqrt {\frac {\tilde {\Omega }_{bu_{1}} }{m_{bu_{1}}}}, \tag {10}\\ {\mathcal {M}}_{\hat {\mathcal {X}}}(2)=& \frac {\Gamma \left ({{m_{bu_{1}}+1}}\right)}{\Gamma \left ({{m_{bu_{1}}}}\right)} \frac {\tilde {\Omega }_{bu_{1}} }{m_{bu_{1}}} = \tilde {\Omega }_{bu_{1}}, \tag {11}\\ {\mathcal {M}}_{\hat { \mathcal {Y}}_{n}}(1)=& \sum _{n=1}^{N} {\mathcal {M}}_{{\mathcal {Y}}_{n}}(1), \tag {12}\\ {\mathcal {M}}_{\hat { \mathcal {Y}}_{n}}(2)=& { \sum _{n=1}^{N} {\mathcal {M}}_{{\mathcal {Y}}_{n}}(2) +2 \sum _{n=1}^{N} {\mathcal {M}}_{{\mathcal {Y}}_{n}}(1) \sum _{n^{\prime }=n+1}^{N} {\mathcal {M}}_{\mathcal {Y}_{n^{\prime }}}(1). } \tag {13}\end{align*}$$

Proof:

Please refer to Appendix A.■

Note that when $\gamma _{\textrm {th}}^{1}\geq {}\frac {\delta _{1}}{\delta _{2}+\lambda _{bu_{1}}^{2}}$ , $\mathcal {P}_{u_{1}}^{a}$ in (7) becomes unity, indicating that user $U_{1}$ experiences a ceiling effect.

2) Outage Probability for User $U_{2}$

Following the NOMA protocol, $U_{2}$ first decodes the message of $U_{1}$ and then applies SIC to decode its own message. An outage occurs if either of these decoding steps fails. Thus, the OP for $U_{2}$ is given as $$\begin{align*} \mathcal {P}_{u_{2}}^{a}=& \mathbb {P}_{r}\left \{{{\gamma _{u_{2},x_{1}}^{a}\lt \gamma _{\textrm {th}}^{1}, \gamma _{u_{2},x_{2}}^{a}\lt \gamma _{\textrm {th}}^{2} }}\right \} \\=& 1-\underbrace {\mathbb {P}_{r}\left \{{{\gamma _{u_{2},x_{1}}^{a}\gt \gamma _{\textrm {th}}^{1}, \gamma _{u_{2},x_{2}}^{a}\gt \gamma _{\textrm {th}}^{2} }}\right \} }_{\mathcal {P}_{2}}, \tag {14}\end{align*}$$ where $\gamma _{\textrm {th}}^{2}=2^{r_{\textrm {th}}^{2}}-1$ is the SIDNR threshold and $r_{\textrm {th}}^{2}$ is the target rate at $U_{2}$ to detect $x_{2}$ . To compute (14), we need the joint probability term $\mathcal {P}_{2}$ , which is derived in the following subsections for iSIC and pSIC cases.

(i) iSIC: For the case of iSIC, the expression for $\mathcal {P}_{2}$ is derived in the Lemma 1.

Lemma 1:

The probability $\mathcal {P}_{2}$ in (14) can be calculated as $$\begin{align*} \mathcal {P}_{2}=\begin{cases} \mathcal {P}_{21}^{iSIC}\times \mathcal {P}_{22}^{iSIC},& \textrm {if}~\gamma _{\textrm {th}}^{1} \geq \frac {\delta _{1}}{\delta _{2} + \lambda _{bu_{2}}^{2}}, \\ \mathcal {P}_{22}^{iSIC}, & \textrm {if}~\gamma _{\textrm {th}}^{1} \lt \frac {\delta _{1}}{\delta _{2} + \lambda _{bu_{2}}^{2}}. \end{cases} \tag {15}\end{align*}$$ Now, the expressions for $\mathcal {P}_{21}^{iSIC}$ and $\mathcal {P}_{22}^{iSIC}$ are given in (18) and (19), shown at the bottom of the page, respectively, where $\Delta _{u^{\prime },1}=\left \{{{\left ({{1-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{iu_{2}, a\}}$ , $\Delta _{u^{\prime },2}=\left \{{{\left ({{-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{iu_{2}, a\}}$ , $\Delta _{u^{^{\prime \prime }},1}=\left \{{{\left ({{1-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{u_{2}, a, iu_{2}\}}$ and $\Delta _{u^{^{\prime \prime }},2}=\left \{{{\left ({{-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{u_{2}, a, iu_{2}\}}$ . The formula for $\varpi _{g}={}\frac {1+\varPsi _{g}}{1-\varPsi _{g}}$ and $\varpi _{u}={}\frac {1+\varPsi _{u}}{1-\varPsi _{u}}$ , where $\varPsi _{g}=\cos \left ({{{}\frac {2g-1}{2G}\pi }}\right)$ and $\varPsi _{u}=\cos \left ({{{}\frac {2u-1}{2U}\pi }}\right)$ , with G and U being the complexity-accuracy trade-off parameters, with $\mathcal {T}_{2}={}\frac {\gamma _{\textrm {th}}^{1}}{\delta _{1}\, \rho _{b}-(\delta _{2}+\lambda _{bu_{2}}^{2}) \rho _{b}\gamma _{\textrm {th}}^{1}}$ and $\mathcal {T}_{3}= {}\frac {\gamma _{\textrm {th}}^{2}}{\delta _{2}\, \rho _{b}-\lambda _{bu_{2}}^{2} \rho _{b}\gamma _{\textrm {th}}^{2}}$ .

Proof:

Please refer to Appendix B.■

Finally, on substituting (15) into (14), we obtain the OP expression for $\mathcal {P}_{u_{2}}^{a}$ using iSIC scenario.

(ii) pSIC: In the case of pSIC, the OP for user $U_{2}$ can be calculated using (14). To begin, calculate the SIDNRs as specified in (14), incorporating from (5) and (6). Specifically, ensure that $\xi =0$ in (6). Afterward, the probability term $\mathcal {P}_{22}^{pSIC}$ can be calculated as $$\begin{align*} \mathcal {P}_{22}^{pSIC}=\begin{cases} \bar {\mathcal {P}}_{22}^{pSIC},& \textrm {if}~\gamma _{\textrm {th}}^{2} \lt \frac {\delta _{2}}{\lambda _{bu_{2}}^{2}}, \\ 0,& \textrm {if}~\gamma _{\textrm {th}}^{2} \geq \frac {\delta _{2}}{\lambda _{bu_{2}}^{2}}. \end{cases} \tag {16}\end{align*}$$ $$\begin{align*} &\mathcal {P}_{u_{1}}^{a}= \sum _{l=1}^{L}\frac {2\pi \sqrt {1-\varPsi _{l}^{2}}}{L\left ({{1-\varPsi _{l}}}\right)^{2}} \times \left [{{ \frac {1}{\Gamma \left ({{m_{z}}}\right)} \gamma \left ({{m_{z}, \frac {\sqrt {\mathcal {T}_{1}+\varpi _{l}} }{\Omega _{z}} }}\right) \prod _{t\in \{iu_{1}, a\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{2,2}^{0,2} \left [{{e^{\varpi _{l}}\Bigg \lvert \begin{array}{c} \Delta _{u,1} \\ \Delta _{u,2} \end{array} }}\right ] }}\right ]. \tag {17}\\\mathcal {P}_{21}^{iSIC}=& 1 - \sum _{g=1}^{G}\frac {2\pi \sqrt {1-\varPsi _{g}^{2}}}{G\left ({{1-\varPsi _{g}}}\right)^{2}} \times \left [{{ \frac {1}{\Gamma \left ({{m_{z}^{\prime }}}\right)} \gamma \left ({{m_{z}^{\prime }, \frac {\sqrt {\mathcal {T}_{2}+\varpi _{g}} }{\Omega _{z}^{\prime }} }}\right) \prod _{t\in \{iu_{2}, a\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{2,2}^{0,2} \left [{{e^{\varpi _{g}}\Bigg \lvert \begin{array}{c} \Delta _{u^{\prime },1} \\ \Delta _{u^{\prime },2} \end{array} }}\right ] }}\right ]. \tag {18}\\ \mathcal {P}_{22}^{iSIC}=& 1 - \sum _{u=1}^{U}\frac {2\pi \sqrt {1-\varPsi _{u}^{2}}}{U\left ({{1-\varPsi _{u}}}\right)^{2}} \times \left [{{ \frac {1}{\Gamma \left ({{m_{z}^{\prime \prime }}}\right)} \gamma \left ({{m_{z}^{\prime \prime }, \frac {\sqrt {\mathcal {T}_{3}+\varpi _{u}} }{\Omega _{z}^{\prime \prime }} }}\right) \prod _{t\in \{u_{2}, a, iu_{2}\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{\varpi _{u}}\Bigg \lvert \begin{array}{c} \Delta _{u^{\prime \prime },1} \\ \Delta _{u^{\prime \prime },2} \end{array} }}\right ]}}\right ]. \tag {19}\end{align*}$$ Now, the expression for $\bar {\mathcal {P}}_{22}^{pSIC}$ is given in (20), shown at the bottom of the page, $$\begin{align*}\bar {\mathcal {P}}_{22}^{pSIC}= 1 - \sum _{s=1}^{S}\frac {2\pi \sqrt {1-\varPsi _{s}^{2}}}{S\left ({{1-\varPsi _{s}}}\right)^{2}} \times \left [{{ \frac {1}{\Gamma \left ({{m_{z}^{\prime \prime }}}\right)} \gamma \left ({{m_{z}^{\prime \prime }, \frac {\sqrt {\mathcal {T}_{3}+\varpi _{s}} }{\Omega _{z}^{\prime \prime }} }}\right) \prod _{t\in \{iu_{2}, a\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{2,2}^{0,2} \left [{{e^{\varpi _{s}}\Bigg \lvert \begin{array}{c} \Delta _{\bar {u},1} \\ \Delta _{\bar {u},2} \end{array} }}\right ] }}\right ]. \tag {20}\end{align*}$$ where coefficient sets are given by $\Delta _{\bar {u},1}=\left \{{{\left ({{1-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{iu_{2}, a\}}$ and $\Delta _{\bar {u},2}=\left \{{{\left ({{-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{iu_{2}, a\}}$ . Additionally, $\varpi _{s}={}\frac {1+\varPsi _{s}}{1-\varPsi _{s}}$ , where $\varPsi _{s}=\cos \left ({{{}\frac {2s-1}{2S}\pi }}\right)$ and S is the complexity-accuracy trade-off parameter.

Now, by substituting (16) into (15) using a similar process and conditions as iSIC, and then taking that outcome and inserting it into (14), one can derive the necessary OP expression for $U_{2}$ for pSIC scenario.

B. Asymptotic Outage Probability Evaluation

To gain deeper insights into the system performance, we analyze the asymptotic OP behavior of each user at high SNR $(\rho _{b}\rightarrow \infty)$ . Initially, we utilize the approximation of the lower incomplete gamma function as $\Upsilon ({m},z)\mathop {\approx }\limits _{z \to 0} z^{m}/{m}$ [54, (8.354.1)]. The analytical expression for asymptotic OP of the user is expressed in Lemma 2.

Lemma 2:

The asymptotic OP $\mathcal {P}_{u_{1}}^{a,\infty }$ of $U_{1}$ , $\mathcal {P}_{u_{2},{pSIC}}^{a,\infty }$ and $\mathcal {P}_{u_{2},{iSIC}}^{a,\infty }$ of $U_{2}$ with both pSIC and iSIC, are given in (21), (22), and (23), shown at the bottom of the previous page, $$\begin{align*}\mathcal {P}_{u_{1}}^{a,\infty }=& \sum _{a=0}^{{m}_{iu_{1}}-1} \sum _{b=0}^{{m}_{z}/2} \frac {\left ({{-1}}\right)^{a} \left ({{{m}_{iu_{1}}+{m}_{a}-1+b}}\right)! \left ({{\mathcal {T}_{1}}}\right)^{\frac {{m}_{z}}{2}-b}}{\Gamma \left ({{{m}_{a}}}\right) \Gamma \left ({{{m}_{iu_{1}}}}\right) \Gamma \left ({{{m}_{z}}}\right) \left ({{{m}_{a}+a}}\right) \left ({{{m}_{z}}}\right) \left ({{\Omega _{z}}}\right)^{{m}_{z}} } \binom {{m}_{iu_{1}}-1}{a} \binom {\frac {{m}_{z}}{2}}{b} \left ({{\frac {{m}_{a}}{\Omega _{a}\mathcal {T}_{1}\Xi _{1}} }}\right)^{{m}_{a}} \left ({{\frac {{m}_{iu_{1}}}{\Omega _{iu_{1}} \mathcal {T}_{1}\beta _{1}}}}\right)^{-\left ({{{m}_{a}+b}}\right)}. \tag {21}\\ \mathcal {P}_{u_{2},{pSIC}}^{a,\infty }=& \sum _{a=0}^{{m}_{iu_{2}}-1} \sum _{b=0}^{{m}_{z}/2} \frac {\left ({{-1}}\right)^{a} \left ({{{m}_{iu_{2}}+{m}_{a}-1+b}}\right)! \left ({{\mathcal {T}_{2}}}\right)^{\frac {{m}_{z}}{2}-b} }{\Gamma \left ({{{m}_{a}}}\right) \Gamma \left ({{{m}_{iu_{2}}}}\right) \Gamma \left ({{{m}_{z}}}\right) \left ({{{m}_{z}}}\right) \left ({{\Omega _{z}}}\right)^{{m}_{z}} \left ({{{m}_{a}+a}}\right) } \binom {{m}_{iu_{2}}-1}{a} \binom {\frac {{m}_{z}}{2}}{b} \left ({{\frac {{m}_{a}}{\Omega _{a}\mathcal {T}_{2}\Xi _{2}} }}\right)^{{m}_{a}} \left ({{\frac {{m}_{iu_{2}}}{\Omega _{iu_{2}} \mathcal {T}_{2}\beta _{2}}}}\right)^{-\left ({{{m}_{a}+b}}\right)}. \qquad \tag {22}\\ \mathcal {P}_{u_{2},{iSIC}}^{a,\infty }=& 1-\sum _{a=0}^{{m}_{iu_{2}}-1} \sum _{b=0}^{{m}_{iu_{2}}+{m}_{a}-1} \sum _{c=0}^{{m}_{z}/2} \frac {\left ({{-1}}\right)^{a+b} \left ({{{m}_{iu_{2}}+{m}_{a}+{m}_{u_{2}}-1+c}}\right)! \left ({{\mathcal {T}_{3}}}\right)^{\frac {{m}_{z}}{2}-c} \left ({{\frac {{m}_{a}}{\Omega _{a}\mathcal {T}_{3}\Xi _{2}} }}\right)^{{m}_{a}} }{\Gamma \left ({{{m}_{a}}}\right) \Gamma \left ({{{m}_{iu_{2}}}}\right) \Gamma \left ({{{m}_{u_{2}}}}\right) \Gamma \left ({{{m}_{z}}}\right) \left ({{{m}_{a}+a}}\right)\left ({{{m}_{u_{2}}+b}}\right) \left ({{{m}_{z}}}\right) \left ({{\Omega _{z}}}\right)^{{m}_{z}}} \left ({{\frac {{m}_{u_{2}}}{\Omega _{u_{2}}\mathcal {T}_{3}\delta _{1}\, \rho _{b}} }}\right)^{{m}_{u_{2}}} \\& {}\times \binom {{m}_{iu_{2}}-1}{a} \binom {{m}_{iu_{2}}+{m}_{a}-1}{b} \left ({{\frac {{m}_{iu_{2}}}{\Omega _{iu_{2}} \mathcal {T}_{3}\beta _{2}}}}\right)^{-\left ({{{m}_{a}+{m}_{u_{2}}+c}}\right)} \binom {\frac {{m}_{z}}{2}}{c}. \tag {23}\end{align*}$$ respectively.

Proof:

Please refer to Appendix C.■

C. Diversity Analysis

The diversity order is a crucial metric for evaluating performance in wireless networks, as it directly influences the network’s robustness and ability to withstand fading. A system with a higher diversity order experiences a faster reduction in OP, making it more resilient to fading, especially at high SNRs [63]. Analyzing diversity order is vital for improving network performance and developing more effective diversity strategies. The formula for diversity order is expressed as $$\begin{equation*} D_{order} = - \lim \limits _{\rho _{b} \to \infty } \frac {\log \left ({{\Pr \left ({{\rho _{b} }}\right)}}\right)}{\log \left ({{\rho _{b} }}\right)}, \tag {24}\end{equation*}$$ where $\rho _{b}={}\frac {P_{b}}{\sigma ^{2}}$ represents the transmit SNR, while $\Pr (\rho _{b})$ denotes the asymptotic OP at high SNR levels, which is detailed in Section III-B.

Remark 1:

Based on (21) and (22), the diversity order for user $U_{1}$ and user $U_{2}$ in the pSIC case is $m_{z}/2$ which is influenced by the number of reflecting elements N and Nakagami-m distribution parameters. In the case of iSIC the asymptotic OP in (23) is constant with respect to $\rho _{b}$ . This suggests that the diversity order for user $U_{2}$ in a UAV-mounted A-IRS NOMA network using iSIC is zero, attributed to the prevailing residual interference from iSIC. This observation is further supported by numerical results, which will be demonstrated in Section VI.

D. Optimal Power Allocation Factors

In a UAV-mounted A-IRS NOMA network, the OP for each user depends on the power allocation factors, i.e., $\mathcal {P}_{u_{j}}^{a}(\delta _{1}, \delta _{2})$ . Thus, we focus on obtaining the optimal values of $\delta _{1}$ and $\delta _{2}$ to achieve a balance between the OPs of the users. The optimization problem is formulated below as: $$\begin{align*} \textbf {P1}~:~\mathcal {P}_{u_{j}}^{a*}=& \underset {\delta _{1}, \delta _{2}}{\textrm {min}} \, \, \mathcal {P}_{u_{j}}^{a}\left ({{\delta _{1}, \delta _{2}}}\right), \tag {25}\\ \textrm {s.t.}~\left ({{0+ \tau }}\right) \, \le \delta _{i} \,\le & \, \left ({{1-\tau }}\right), \, \, \, \, \, \,\, \, \, \, \tau \approx 0, \tag {26}\\ \delta _{1}+\delta _{2}=& 1. \tag {27}\end{align*}$$ Considering that $\delta _{2} = 1 - \delta _{1}$ , the derivatives of $U_{1}$ ’s OP, $\mathcal {P}_{u_{1}}^{a}$ , and $U_{2}$ ’s OP, $\mathcal {P}_{u_{2}}^{a}$ , with respect to $\delta _{1}$ from (21), (22), and (23), respectively, at high SNR are ${}\frac {\partial \mathcal {P}_{u_{1}}^{a}}{\partial \delta _{1}}\lt 0$ and ${}\frac {\partial \mathcal {P}_{u_{2}}^{a}}{\partial \delta _{1}}\gt 0$ . As a result, $\mathcal {P}_{u_{1}}^{a}$ decreases while $\mathcal {P}_{u_{2}}^{a}$ increases as a function of $\delta _{1}$ (this observation is also verified in Fig. 8(a). Therefore, to minimize $\mathcal {P}_{u_{j}}^{a*}$ , the optimal $\delta _{1}^{*}$ can be determined by solving $\mathcal {P}_{u_{2}}^{a}=\mathcal {P}_{u_{1}}^{a}$ . This allows us to simplify problem P1 as $$\begin{align*} \bar {\textbf {P}}\textbf {1}~:~\underset {\delta _{1}}{\textrm {min}}& \left ({{\mathcal {P}_{u_{1}}^{a}-\mathcal {P}_{u_{2}}^{a}}}\right), \tag {28}\\ \textrm {s.t.}& 0.5+\tau \le \delta _{1} \, \le \, \left ({{1-\tau }}\right). \tag {29}\end{align*}$$ Given that $\mathcal {P}_{u_{1}}^{a}$ and $\mathcal {P}_{u_{2}}^{a}$ at high SNR involve the sum of multiple complex terms, deriving the exact closed-form expression for $\mathcal {P}_{u_{j}}^{a*}$ is challenging. Hence, we introduce a straightforward search method to accurately determine the optimal $\delta _{1}^{*}$ , as described in Algorithm 1 within a specified accuracy $\mathbb {T}_{s}$ .

FIGURE 8.

(a) OP vs. power allocation factor $\delta _{1}$ and (b) OP performance comparison of UAV A-IRS NOMA with FD/HD DF relaying schemes.

Algorithm 1:

Optimal Power Allocation Factor $(\delta _{1}^{*})$

E. Optimal UAV Position

In this subsection, we focus on obtaining the optimal UAV positioning to enhance the reliability by minimizing the user’s OP $(\mathcal {P}_{u_{j}}^{a})$ . Based on this, we develop an optimization framework to identify the optimal 3-D coordinates $\{{\mathcal {R}}_{r}^{*}, \phi _{r}^{*}, {\mathcal {H}}_{r}^{*} \}$ of the UAV as follows: $$\begin{align*}& \textbf {P2}~:~\left \{{{{\mathcal {R}}_{r}^{*}, \phi _{r}^{*}, {\mathcal {H}}_{r}^{*} }}\right \}=\textrm {arg} \underset {{\mathcal {R}}_{r}, \phi _{r}, {\mathcal {H}}_{r}}{\textrm {min}} \mathcal {P}_{u_{j}}^{a}\left ({{{\mathcal {R}}_{r}, \phi _{r}, {\mathcal {H}}_{r}}}\right), \tag {30}\\& \; \textrm {s.t.}\quad \mathcal {R}_{r,{min}} \leq {\mathcal {R}}_{r} \leq \mathcal {R}_{r,{max}},\phi _{r,{min}} \\& \quad {}\leq \phi _{r} \leq \phi _{r,{max}}, \,\mathcal {H}_{r,{min}} \leq {\mathcal {H}}_{r} \leq \mathcal {H}_{r,{max}}. \tag {31}\end{align*}$$ The variables ${\mathcal {R}}_{r}$ , $\phi _{r}$ , and ${\mathcal {H}}_{r}$ are interrelated in (17), (18), (19) and (20), creating a nonconvex optimization problem. To address this, an Alternating Optimization (AO) strategy is used, optimizing one variable at a time while keeping others constant. This simplifies the complex problem, for example, optimal value of ${\mathcal {R}}_{r}$ for minimizing the OP of the user can be obtained by varying ${\mathcal {R}}_{r}$ with fixed $\phi _{r}$ and ${\mathcal {H}}_{r}$ , given by $$\begin{align*}& \mathcal {R}_{r,{so}}= \textrm {arg} \, \underset {{\mathcal {R}}_{r}}{\textrm {min}} \, \mathcal {P}_{u_{j}}^{a}\left ({{{\mathcal {R}}_{r}}}\right), \, \,\, \tag {32}\\& \;\textrm {s.t.}\quad \mathcal {R}_{r,{min}} \leq {\mathcal {R}}_{r} \leq \mathcal {R}_{r,{max}}. \tag {33}\end{align*}$$ On similar lines, by varying the other UAV coordinates one by one while fixing the rest, the optimal values of the 3-D coordinates can be obtained. Thus, we introduce Algorithm 2, which iteratively optimizes each UAV coordinate within its range, keeping the other two coordinates fixed, and repeats until convergence. The computation is mainly intensive in steps 3, 5, and 7, with complexities $\mathcal {O}({\mathcal {S}}_{{\mathcal {R}}_{r}})$ , $\mathcal {O}({\mathcal {S}}_{\phi _{r}})$ , and $\mathcal {O}({\mathcal {S}}_{{\mathcal {H}}_{r}})$ , respectively, resulting in an overall complexity of $k_{max}\mathcal {O}({\mathcal {S}}_{{\mathcal {R}}_{r}}+{\mathcal {S}}_{\phi _{r}}+{\mathcal {S}}_{{\mathcal {H}}_{r}})$ , where ${\mathcal {S}}_{{\mathcal {R}}_{r}}$ , ${\mathcal {S}}_{\phi _{r}}$ , and ${\mathcal {S}}_{{\mathcal {H}}_{r}}$ are the size of $\mathcal {R}$ , $\phi$ , and $\mathcal {H}$ , respectively.

Algorithm 2:

Optimal UAV Position $({\mathcal {R}}_{r}^{*}, \phi _{r}^{*}, {\mathcal {H}}_{r}^{*})$

F. Ergodic Capacity

The ergodic capacities for $U_{1}$ and $U_{2}$ with iSIC and pSIC are analyzed as follows.

1) Ergodic Capacity for $U_{1}$

The ergodic capacity for decoding desired message $x_{1}$ at $U_{1}$ can be expressed as $$\begin{equation*} \mathcal {C}_{u_{1}}^{a}=\mathbb {E}\left [{{\log _{2}\left ({{1+\gamma _{u_{1},x_{1}}^{a}}}\right)}}\right ]. \tag {34}\end{equation*}$$ We derive the expression for the ergodic capacity of $U_{1}$ , i.e., $\mathcal {C}_{u_{1}}^{a}$ in Theorem 2.

Theorem 2:

The ergodic capacity of $U_{1}$ , i.e., $\mathcal {C}_{u_{1}}^{a}$ is expressed in (36), shown at the bottom of the page, where $\Delta _{q,1}=\left \{{{\left ({{1-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{z, a, iu_{1}\}}$ , $\Delta _{q,2}=\left \{{{\left ({{-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{z, a, iu_{1}\}}$ , $\Delta _{v,1}=\left \{{{\left ({{1-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{z, a, iu_{1}\}}$ , and $\Delta _{v,2}=\left \{{{\left ({{-{}\frac {m_{t}}{\Omega _{t}},1,m_{t}}}\right)}}\right \}_{t\in \{z, a, iu_{1}\}}$ . The formula for $\varpi _{q}={}\frac {1+\varPsi _{q}}{1-\varPsi _{q}}$ and $\varpi _{v}={}\frac {1+\varPsi _{v}}{1-\varPsi _{v}}$ , where $\varPsi _{q}=\cos \left ({{{}\frac {2q-1}{2Q}\pi }}\right)$ and $\varPsi _{v}=\cos \left ({{{}\frac {2v-1}{2V}\pi }}\right)$ , with Q and V are complexity-accuracy tradeoff parameters.

Proof:

Please refer to Appendix D.■

2) Ergodic Capacity for $U_{2}$

The ergodic capacity for decoding the targeted message $x_{2}$ at $U_{2}$ is expressed as $$\begin{equation*} \mathcal {C}_{u_{2}}^{a}=\mathbb {E}\left [{{\log _{2}\left ({{1+\gamma _{u_{2},x_{2}}^{a}}}\right)}}\right ]. \tag {35}\end{equation*}$$ To compute (35), we continue by deriving the expressions for the ergodic capacity at $U_{2}$ under iSIC and pSIC scenarios, denoted as $\mathcal {C}_{u_{2},{iSIC}}^{a}$ and $\mathcal {C}_{u_{2},{pSIC}}^{a}$ , respectively, in Lemma 3.

Lemma 3:

The ergodic capacities $\mathcal {C}_{u_{2},{iSIC}}^{a}$ and $\mathcal {C}_{u_{2},{pSIC}}^{a}$ of user $U_{2}$ for the iSIC and pSIC cases, respectively, are detailed in (37) and (38), shown at the bottom of the page. $$\begin{align*} \mathcal {C}_{u_{1}}^{a}=& \sum _{q=1}^{Q}\frac {2\pi \sqrt {1-\varPsi _{q}^{2}}}{Q\left ({{1-\varPsi _{q}}}\right)^{2}} \log _{2}\left ({{1+\varpi _{q}}}\right) \prod _{t\in \{z, a, iu_{1}\}} \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{\varpi _{q}}\Bigg \lvert \begin{array}{c} \Delta _{q,1} \\ \Delta _{q,2} \end{array} }}\right ] \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} - \sum _{v=1}^{V} \frac {2\pi \sqrt {1-\varPsi _{v}^{2}}}{V\left ({{1-\varPsi _{v}}}\right)^{2}} \\& {}\times \log _{2}\left ({{1+\varpi _{v}}}\right) \prod _{t\in \{z, a, iu_{1}\}} \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{\varpi _{v}}\Bigg \lvert \begin{array}{c} \Delta _{v,1} \\ \Delta _{v,2} \end{array} }}\right ] \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}}. \tag {36}\\\mathcal {C}_{u_{2},{iSIC}}^{a}=& \sum _{a=1}^{A}\frac {2\pi \sqrt {1-\varPsi _{a}^{2}}}{A\left ({{1-\varPsi _{a}}}\right)^{2}} \log _{2}\left ({{1+\varpi _{a}}}\right) \prod _{t\in \{u_{2}, z, a, iu_{2}\}} \bar {\textrm {H}}_{4,4}^{0,4} \left [{{e^{\varpi _{a}}\Bigg \lvert \begin{array}{c} \Delta _{a^{\prime },1} \\ \Delta _{a^{\prime },2} \end{array} }}\right ] \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{ m_{t}} - \sum _{b=1}^{B} \frac {2\pi \sqrt {1-\varPsi _{b}^{2}}}{B\left ({{1-\varPsi _{b}}}\right)^{2}} \log _{2}\left ({{1+\varpi _{b}}}\right) \\& {}\times \prod _{t\in \{u_{2}, z, a, iu_{2}\}} \bar {\textrm {H}}_{4,4}^{0,4} \left [{{ e^{\varpi _{b}} \Bigg \lvert \begin{array}{c} \Delta _{b^{\prime },1} \\ \Delta _{b^{\prime },2} \end{array} }}\right ] \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{ m_{t}}. \tag {37}\\ \mathcal {C}_{u_{2},{pSIC}}^{a}=& \sum _{a=1}^{A} \frac {2\pi \sqrt {1-\varPsi _{a}^{2}}}{A\left ({{1-\varPsi _{a}}}\right)^{2}} \log _{2}\left ({{1+\varpi _{a}}}\right) \prod _{t\in \{z, a, iu_{2}\}} \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{\varpi _{a}} \Bigg \lvert \begin{array}{c} \Delta _{a^{\prime \prime },1} \\ \Delta _{a^{\prime \prime },2} \end{array} }}\right ] \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} - \sum _{b=1}^{B} \frac {2\pi \sqrt {1-\varPsi _{b}^{2}}}{B\left ({{1-\varPsi _{b}}}\right)^{2}} \log _{2}\left ({{1+\varpi _{b}}}\right) \\& {}\times \prod _{t\in \{z, a, iu_{2}\}} \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{\varpi _{b}}\Bigg \lvert \begin{array}{c} \Delta _{b^{\prime \prime },1} \\ \Delta _{b^{\prime \prime },2} \end{array} }}\right ] \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{ m_{t}}. \tag {38}\end{align*}$$

Proof:

By substituting the relevant PDFs for $\hat {\mathcal {C}}$ and $\tilde {\mathcal {C}}$ , and applying the same approach as used for (67) and (68) in Appendix D, we obtain the results as presented in (37) and (38) for the iSIC and pSIC cases, respectively.■

Remark 2:

Using (36)–(38), we obtain the ergodic sum capacity for the UAV-mounted A-IRS NOMA network with iSIC as $\mathcal {C}_{{S},{iSIC}}^{a}=\mathcal {C}_{u_{1}}^{a}+\mathcal {C}_{u_{2},{iSIC}}^{a}$ and with pSIC as $\mathcal {C}_{{S},{pSIC}}^{a}=\mathcal {C}_{u_{1}}^{a}+\mathcal {C}_{u_{2},{pSIC}}^{a}$ .

G. System Throughput

In this subsection we derive the system throughput for both delay-limited and delay-tolerant transmission modes.

1) Delay-Limited Transmission Mode

In the delay-limited transmission mode, the source sends messages at a constant rate, subject to potential outages based on the wireless channels’ fading conditions. It can be calculated as the total of the achievable individual target rates for users over these fading channels as $$\begin{equation*} {\mathcal {S}}_{T,DL}=\left ({{1-\mathcal {P}_{u_{1}}^{a}}}\right) r_{\textrm {th}}^{1}+\left ({{1-\mathcal {P}_{u_{2}}^{a}}}\right) r_{\textrm {th}}^{2}. \tag {39}\end{equation*}$$ Setting $r_{\textrm {th}}^{1}=r_{\textrm {th}}^{2}=R$ in (39), the maximum throughput is identified as $2R$ , achievable under ideal conditions with negligible OP.

2) Delay-Tolerant Transmission Mode

In the delay-tolerant transmission mode, the source transmits the information at any fixed rate less than or equal to the ergodic capacity. Hence, the system throughput for both the users under delay-tolerant mode can be evaluated as $$\begin{equation*} {\mathcal {S}}_{T,DL}=\mathcal {C}_{u_{1}}^{a}+\mathcal {C}_{u_{2}}^{a}. \tag {40}\end{equation*}$$ From (40), one can observe that the maximum achievable throughput for the delay-tolerant mode depends on the evaluated ergodic capacity, and hence, yields higher value than the delay-limited mode, as illustrated through numerical results in Section VI.

SECTION IV. Performance Comparison With UAV-Mounted P-IRS NOMA Network

This section compares the performance of the proposed UAV-mounted A-IRS NOMA network with its UAV-mounted P-IRS NOMA counterpart. Mathematically, P-IRS scenario can be expressed as a specific case of the A-IRS. For P-IRS, it is considered that $\eta _{a}=1$ , and $\sigma _{a}^{2}=0$ , as there is no amplification of incident signal and thermal noise at the P-IRS. Hence, in the case of a P-IRS, the received signal at the j-th user is given as $$\begin{align*} y_{u_{j}}^{p}=& \underbrace {\big (a_{j}h_{bu_{j}} + b_{j} \mathbf {g}_{u_{j}}^{H}\boldsymbol {\Theta } \mathbf {g}_{b}\big) \big (\sqrt {\delta _{1}P_{b}}x_{1} + \sqrt {\delta _{2}P_{b}}x_{2} + \zeta ^{t}_{b}\big) + \zeta ^{r}_{bu_{j}} }_{\textrm {Desired signal}} \\& {}+ \underbrace { \sum _{i=1}^{I} c_{j} h_{iu_{j}} \big (\sqrt {P_{i}} x_{iu_{j}} + \zeta ^{t}_{i} \big) + \zeta ^{r}_{iu_{j}} }_{\textrm {CCI signal}} + \underbrace {\psi _{u_{j}}}_{\textrm {AWGN}}. \tag {41}\end{align*}$$ The SIDNR for $U_{1}$ to decode its own message $x_{1}$ is $$\begin{equation*} \gamma _{u_{1},x_{1}}^{p}=\frac {\delta _{1}\, \rho _{b}\mathcal {Z}_{1}}{\left ({{\delta _{2}+\lambda _{bu_{1}}^{2}}}\right) \rho _{b}\mathcal {Z}_{1} + \beta _{1} |h_{iu_{1}}|^{2} + 1}, \tag {42}\end{equation*}$$ where $\mathcal {Z}_{1}=|a_{1}h_{bu_{1}} + b_{1} \mathbf {g}_{u_{1}}^{H}\boldsymbol {\Phi } \mathbf {g}_{b}|^{2}$ . After SIC, $U_{2}$ decodes the $U_{1}$ ’s signal, treating the $U_{2}$ ’s signal as noise. Thus, the corresponding SIDNR is given as $$\begin{equation*} \gamma _{u_{2},x_{1}}^{p}=\frac {\delta _{1}\, \rho _{b}\mathcal {Z}_{2}}{\left ({{\delta _{2}+\lambda _{bu_{2}}^{2}}}\right) \rho _{b}\mathcal {Z}_{2} + \beta _{2} |h_{iu_{2}}|^{2} + 1}, \tag {43}\end{equation*}$$ where $\mathcal {Z}_{2}=|a_{2}h_{bu_{2}} + b_{2} \mathbf {g}_{u_{2}}^{H}\boldsymbol {\Phi } \mathbf {g}_{b}|^{2}$ . After removing the signal $x_{1}$ through SIC, $U_{2}$ proceeds to decode its own message $x_{2}$ . The corresponding SIDNR is expressed as $$\begin{equation*} \gamma _{u_{2},x_{2}}^{p}=\frac {\delta _{2}\, \rho _{b} \mathcal {Z}_{2}}{\delta _{1}\, \rho _{b}|\hbar _{2}|^{2}+\lambda _{bu_{2}}^{2} \rho _{b}\mathcal {Z}_{2} + \beta _{2} |h_{iu_{2}}|^{2} + 1}. \tag {44}\end{equation*}$$ Now, the OP and ergodic capacity expressions for $U_{1}$ in the UAV-mounted P-IRS NOMA network are derived by substituting (42) into (7) and (34), respectively and following a similar procedure as used in the A-IRS system for OP and ergodic capacity calculation. Similarly, for calculating the OP and ergodic capacity of $U_{2}$ in the UAV-mounted P-IRS network, (43) and (44) are substituted into (14) and (35), respectively or both iSIC and pSIC scenarios. These results can be used as benchmark comparisons with the proposed UAV-mounted A-IRS NOMA network.

SECTION V. Deep Neural Network Framework

This section introduces a DNN framework designed to efficiently estimate OP and ergodic capacity. This approach significantly reduces computational complexity and accelerates runtime, offering a practical alternative to the demanding nature of mathematical analysis and time-consuming Monte-Carlo simulations. The computation of OP and ergodic capacity is influenced by several parameters, as detailed in Section V-B. By training a neural network, we aim to learn and understand these relationships effectively.

A. Overview of the Deep Neural Network (DNN)

To tackle the challenge of estimating the system’s OP and ergodic capacity through regression in supervised learning, we have designed a DNN using a multiple output DNN (M-DNN). This approach is illustrated in Fig. 2. The M-DNN consists of one input layer with ${\mathcal {N}}_{n}$ neurons (representing 14 parameters from Section V-B), multiple hidden layers $({\mathcal {L}}_{h}=1, {\dots },{\mathcal {L}}_{H})$ , each with $({\mathcal {N}}_{h}=1, {\dots },{\mathcal {N}}_{H})$ neurons, and a one output layer. Each hidden layer employs activation functions such as exponential linear unit (eLU), rectified exponential linear unit (ReLU), and scaled exponential linear unit (SeLU). The output layer, with five neurons, uses Sigmoid activation to estimate values of $\mathcal {P}_{u_{1}}^{a}$ , $\mathcal {P}_{u_{2}}^{a}$ , $\mathcal {C}_{u_{1}}^{a}$ , $\mathcal {C}_{u_{2}}^{a}$ , and $\mathcal {C}_{u_{s}}^{a}$ .

The DNN is structured around three principal phases: Data generation, model training, and prediction. The initial phase involves generating a dataset through Monte Carlo simulations. Training involves two steps: first, fine-tuning hyperparameters like number of hidden layers $({\mathcal {L}}_{H})$ , activation functions $({\mathcal {A}}_{\mathcal {F}})$ , weight initialization (Random, Normal, and He (Kaiming)), neuron count, and ${\mathcal {N}}_{h}$ for optimal DNN performance; second, training the model with these hyperparameters. The final phase focuses on real-time predictions.

B. Dataset Generation Methodology

The dataset was generated using Monte-Carlo simulations, drawing from OP and ergodic capacity formulas and influenced by various parameters as $\rho _{b}\in [{0, 40}]$ , $\rho _{iu_{j}}\in [{0, 10}]$ , $\lambda _{0}\in [{0, 0.3}]$ , $\delta _{1}\in [{0.51, 0.99}]$ , $\xi \in [{0, 1}]$ , $N \in [{1, 50}]$ , $r_{\textrm {th}}^{1}=r_{\textrm {th}}^{2} \in [{0.1, 1}]$ , $\eta _{a} \in [{3, 7}]$ , ${\mathcal {R}}_{r} \in [{0.1, 3}]$ , ${\mathcal {H}}_{r} \in [{0.1, 1}]$ , $\phi _{r} \in [0, 2\pi]$ , $({\mathcal {X}}_{b}, {\mathcal {Y}}_{b}, {\mathcal {Z}}_{b}) \in [{-}3, 0]$ , $(\mathcal {X}_{u_{1}}, \mathcal {Y}_{u_{1}}, \mathcal {Z}_{u_{1}}) \in [{-}3, 3]$ , and $(\mathcal {X}_{u_{2}}, \mathcal {Y}_{u_{2}}, \mathcal {Z}_{u_{2}}) \in [{-}2, 0]$ . These parameters form the input variables for the training samples, uniformly generated and structured as a $\mathbf {I}=[\rho _{b}, \rho _{iu_{j}}, \lambda _{0}, \delta _{1}, {\dots }]$ . Consequently, p-th training sample, denoted as $\mathbf {O}_{p}=(\mathbf {I}_{p}, \{V_{1,p}, V_{2,p}, {\dots }, V_{5,p}\})$ , where $\{V_{1,p}, V_{2,p}, {\dots }, V_{5,p}\}$ corresponds to OP and ergodic capacity values for the p-th sample, creating a training set $\mathbf {O}=\{\mathbf {O}_{1}, \mathbf {O}_{2}, {\dots }\}$ .

The dataset, comprising over $10^{6}$ samples as specified above, is divided randomly into training $(70\%)$ , testing $(15\%)$ , and validation $(15\%)$ subsets. This division prevents overfitting and maintains test accuracy, normalizing each sample $\mathbf {S}_{p}^{k}$ $$\begin{equation*} \mathbf {S}_{p}^{k}=\frac {\mathbf {O}_{p}^{k}- \mathbb {E}\left [{{\mathbf {O}_{p}}}\right ]}{\textrm {max}\left ({{\mathbf {O}_{p}}}\right)-\textrm {min}\left ({{\mathbf {O}_{p}}}\right)}, k=\overline {1,|\mathbf {O}_{p}|} \tag {45}\end{equation*}$$

C. Design Methodology for DNN

In DL models, customizing hyperparameters to fit a dataset is crucial but challenging. To address this, we use hyperparameter optimization (HPO) to systematically select hyperparameters that provide the best performance, like highest accuracy or lowest error, for a given dataset. These optimized hyperparameters are then employed in DNN models and training. To enhance OP and ergodic capacity prediction performance, we refine the DNN’s hyperparameters using a random search algorithm (RSA), a commonly employed method. Hyperparameters, which remain fixed during the DNN’s training phase, include the number of hidden layers, neuron count per hidden layer, activation functions, and key aspects crucial for training efficiency and accuracy, such as batch size, learning rate, optimizer, and weight initialization. HPO marks the final step in model design and the initial phase of DNN training. This process, which includes training and testing of the DNN, is detailed in Algorithm 3. During the HPO phase, the RSA randomly initializes hyperparameters within predefined limits and iteratively minimizes the mean-square error (MSE) of the OP and ergodic capacity analysis model. The DNN, acting as the fitness function, evaluates the MSE using a loss function that compares actual and predicted values as $$\begin{equation*} \textrm {Loss}\left ({{V^{(t)}_{p}, \hat V^{(t)}_{p}}}\right) = \frac {1}{5 {\mathcal {T}}_{s}}\sum _{t = 1}^{{\mathcal {T}}_{s}} \sum _{p = 1}^{5} \big (V^{(t)}_{p} - \hat V^{(t)}_{p}\big)^{2}, \tag {46}\end{equation*}$$ where ${\mathcal {T}}_{s}$ denotes the total number of training samples, $V^{(t)}_{p}$ and $\hat {V}^{(t)}_{p}$ are the actual and predicted values (i.e., OP and ergodic capacity), respectively. The performance of the designed DNN is assessed using the RMSE, defined as $\textrm {RMSE}=\sqrt {\textrm {MSE}}$ . We use the Adam optimization algorithm to minimize the loss function in backpropagation due to its quicker training speed relative to methods like stochastic gradient descent and RMSProp. The DNN leverages optimal hyperparameters from an earlier phase for learning input-output mappings.

Algorithm 3:

Training and Testing in DNN

D. Outage Probability and Ergodic Capacity Real-Time Prediction

Once the offline training is completed, involving iterative loss function calculations, backpropagation weight updates, and biases, the DNN model undergoes a transformation. It effectively becomes a concise mapping function ${\mathcal {M}}_{F}(.)$ , composed of its weights and biases. This allows it to rapidly and accurately predict the OP and ergodic capacity with new data, displaying the predictions as $$\begin{equation*} \left [{{\hat V^{\mathrm { pre}}_{1}, \hat V^{\mathrm { pre}}_{2}, \hat V^{\mathrm { pre}}_{3}, \hat V^{\mathrm { pre}}_{4}, \hat V^{\mathrm { pre}}_{5}}}\right ] = {\mathcal {M}}_{F}\left ({{\mathbf {I}}}\right). \tag {47}\end{equation*}$$

E. Computational Complexity of the DNN Model

In the proposed DNN model, the complexity largely stems from generating the training dataset. Additionally, the computational cost during estimation is dictated by the total number of parameters $({\mathcal {N}}_{Par})$ and the number of floating-point operations (FLOPs) $({\mathcal {N}}_{FLOPs})$ . The parameters and FLOPs within the DNN are computed as $$\begin{align*} {\mathcal {N}}_{Par}=& 14 \times {\mathcal {N}}_{n} + {\mathcal {N}}_{n} + \cdots + {\mathcal {N}}_{h}^{\left [{{{\mathcal {L}}_{h}}}\right ]} \times {\mathcal {N}}_{h}^{\left [{{{\mathcal {L}}_{h}+1}}\right ]} \\& {}+ {\mathcal {N}}_{h}^{\left [{{{\mathcal {L}}_{h}+1}}\right ]}+\cdots + {\mathcal {N}}_{h}^{\left [{{{\mathcal {L}}_{H}}}\right ]}\times 5 + 5, \tag {48}\\ {{\mathcal {N}}_{FLOPs}}=& 14 \times {\mathcal {N}}_{n} \times 2 + \cdots + {\mathcal {N}}_{h}^{\left [{{{\mathcal {L}}_{h}}}\right ]} \times {\mathcal {N}}_{h}^{\left [{{{\mathcal {L}}_{h}+1}}\right ]} \times 2 \\& {}+ {\mathcal {N}}_{h}^{\left [{{{\mathcal {L}}_{h}+1}}\right ]}+\cdots + {\mathcal {N}}_{h}^{\left [{{{\mathcal {L}}_{H}}}\right ]}\times 5\times 2 + 5. \tag {49}\end{align*}$$ where ${\mathcal {N}}_{n}={\mathcal {N}}_{h}=160$ and ${\mathcal {L}}_{h}=\{1, {\dots }, 4\}$ . The findings show that the designed DNN model comprises $80,485$ parameters and $160,325$ FLOPs.

SECTION VI. Numerical And Simulation Results

In this section, we validate the derived analytical results via Monte Carlo simulations implemented in MATLAB version R2023a. Unless explicitly stated otherwise, we present the simulation parameters in Table 1. Additionally, we define $\lambda _{tb}=\lambda _{ti}=\lambda _{ru_{1}}=\lambda _{ru_{2}}=\lambda _{0}$ as the levels of HIs, with $\lambda _{bu_{1}}=\lambda _{bu_{2}}=\lambda _{iu_{1}}=\lambda _{iu_{2}}=\sqrt {2}\lambda _{0}$ determined by the EVMs [59]. To construct a DNN model, we utilized Python $3.10.12$ in combination with Keras $2.14.0$ and TensorFlow $2.14.0$ . It executed on hardware equipped with a computer featuring an i7-1255 processor, an 8 GB GeForce GTX 1080 GPU, and 32 GB of RAM. In all numerical studies, the DNN predictions align well with simulations and analysis from Section III, detailed in Algorithm 3 for DNN training and testing.

TABLE 1 Simulation Parameters

A. Performance Analysis of Primary Network

Fig. 3(a) and Fig. 3(b) present the performance of user $U_{1}$ in UAV-mounted A-IRS and P-IRS NOMA networks and UAV-mounted A-IRS NOMA and A-IRS OMA schemes, respectively, highlighting the OP versus SNR curves at different target rates. In Fig. 3(a), the A-IRS NOMA networks exhibit superior outage performance compared to P-IRS NOMA networks at fixed target rates, owing to active elements amplifying signals by $\eta _{a}$ . However, as the target rate increases, $U_{1}$ ’s performance declines due to higher SNR demands, leading to greater vulnerability to channel impairments and interference, thereby degrading the outage performance. The outage performance further degrades due to HIs with $\lambda _{0} = 0.2$ , particularly affecting P-IRS networks and more so at higher target rates. Further, in Fig. 3(b), the UAV-mounted A-IRS NOMA network initially outperforms the A-IRS OMA network for the same target rate, as OMA requires two time slots, leading to a higher SIDNR threshold. But, with increasing target rates, the degradation in $U_{1}$ ’s performance is more pronounced in OMA due to the need for stronger signals, indicating a higher susceptibility to outages at insufficient SNR levels. The impact of HIs is notably significant in the A-IRS OMA network at higher target rates, illustrating the UAV-based A-IRS NOMA network’s resilience to hardware imperfections.

B. Performance Analysis of Secondary Network

In the study illustrated by Fig. 4(a) and Fig. 4(b), the OP of user $U_{2}$ is analyzed in UAV-mounted A-IRS and P-IRS NOMA networks, as well as A-IRS OMA networks, considering both iSIC and pSIC cases. The curves in Fig. 4(a) show that iSIC scenarios exhibit a significantly higher OP compared to pSIC, especially at fixed SNR levels, and demonstrate an error floor with zero diversity order in high SNR environments due to residual interference from imperfect decoding. The OP worsens due to HIs with $\lambda _{0} = 0.2$ , particularly affecting iSIC performance. In Fig. 4(b), we illustrate the OP at $U_{2}$ under iSIC for NOMA. Compared to OMA, as the transmit SNR increases, NOMA with pSIC achieves a lower OP, while the OP under iSIC conditions becomes saturated. Additionally, the OP of the NOMA-based network with pSIC remains consistently better than that of the OMA-based network across the entire SNR range. However, the NOMA-based network exhibits a higher OP under iSIC compared to OMA, particularly in the high SNR region (i.e., for SNR values exceeding 27.5 dBm). This is because with iSIC, the residual interference persists in NOMA, which becomes more significant at high SNR (also known as interference limited regime). In contrast, OMA scheme inherently avoids inter-user interference, which allows it to outperform NOMA-based networks at high SNR regime. Furthermore, the OP of the UAV-mounted A-IRS NOMA network decrease as the target rates decrease.

C. Robustness of the System

Constantly activating all IRS elements in a UAV-mounted A-IRS network can quickly deplete energy, posing risks of power loss, especially when renewable energy sources are unavailable. To address this challenge, we also explore the impact of selectively activating and deactivating the reflecting elements on the OP. This can help in reducing the energy consumption of the IRS when one needs to run the UAV on battery power as the renewable energy source for that time is not available. For instance, under a low power budget scenario the UAV-mounted A-IRS running on the battery, activating only a portion of the reflecting elements can achieve the desired outage performance target, which can help to prolong the battery life of the UAV-mounted A-IRS. In view of this, the robustness of the A-IRS system, can be defined as the ratio of the all available IRS elements $(N_{max})$ used in order to achieve the target OP $({\mathcal {P}}_{T})$ to the minimum number of IRS elements $(n)$ required to achieve ${\mathcal {P}}_{T}$ . Let ${\mathcal {R}}_{S}$ denote the robustness of the A-IRS system, expressed as ${\mathcal {R}}_{S}= {}\frac {N_{max}}{n}$ .

Thus, utilizing only n out of $N_{max}$ IRS elements to attain the desired ${\mathcal {P}}_{T}$ signifies $N_{max}/n$ times robustness against energy dissipation associated with the IRS components. Subsequently, we also use a quantitative metric termed as robustness index $({\mathcal {R}}_{I})$ , used to assess robustness of the considered system. ${\mathcal {R}}_{I}$ highlights the system’s adaptability and efficiency in achieving the specified QoS targets under given parameters, such as varying the number of IRS elements and SNR. Mathematically, we define ${\mathcal {R}}_{I}$ in terms of ${\mathcal {P}}_{T}$ and achieved OP $({\mathcal {P}}_{A})$ as $$\begin{align*} {\mathcal {R}}_{I}=\begin{cases} \frac {{{\mathcal {P}}_{T}}-{{\mathcal {P}}_{A}}}{{\mathcal {P}}_{T}}, & \textrm {if}~ {\mathcal {P}}_{T} \ge {\mathcal {P}}_{A}, \\ 0, & \textrm {if}~{\mathcal {P}}_{T} \lt {\mathcal {P}}_{A}. \end{cases} \tag {50}\end{align*}$$

Following the above discussion on system robustness, we delve into insights derived from evaluating the OP. The OP analysis in UAV-mounted A-IRS NOMA networks (Fig. 5) indicates performance enhancement with an increase in reflecting elements $(N)$ , at the expense of higher energy usage. This improvement, linked to IRS-induced flexibility and wireless link enhancement in NOMA networks (Section III-B), is quantified by SNR gains: a notable gain of 7.35 dBm is observed when N increases from 3 to 6 for an OP of $10^{-2}$ . However, subsequent increments in N show diminishing SNR gains, with 4.15 dBm SNR gain for $N = 6$ to $N = 9$ , 3 dBm SNR gain for $N = 9$ to $N = 12$ , and 2.2 dBm SNR gain for $N = 12$ to $N = 15$ , highlighting more significant performance gains at within fewer reflecting elements.

Fig. 6(a) and Fig. 6(b) collectively analyze the system robustness $({\mathcal {R}}_{S})$ and robustness index $({\mathcal {R}}_{I})$ of a UAV-mounted A-IRS network. From Fig. 6(a), it is observed that at an SNR of 25 dBm, the resulting ${\mathcal {R}}_{S}$ values are 2.2 and 1.5 for target outages of $10^{-1}$ and $10^{-3}$ , respectively. For a fixed $N_{max} = 16$ , using the relation ${\mathcal {R}}_{s} = {}\frac {N_{max}}{n}$ , the minimum number of required elements are $n \approx 8$ and $n \approx 11$ for $10^{-1}$ and $10^{-3}$ , respectively. Similarly, at a higher SNR of 30 dBm, to achieve target outages of $10^{-1}$ and $10^{-3}$ , we require $n \approx 4$ and $n \approx 8$ , respectively. We may observe that to achieve a desired QoS, either the SNR (which depends on the transmit power) or n must be increased. If the transmit power is limited (i.e., low SNR), A-IRS can aid in achieving the desired QoS, conversely, with sufficient transmit power, the minimum number of IRS elements required can be significantly lower. Further, Fig. 6(b) plots the variation in ${\mathcal {R}}_{I}$ with N and SNR. We note that as N increases, ${\mathcal {R}}_{I}$ first grows and then saturates to a maximum value of 0.9975. For example, at an SNR of 14 dBm, ${\mathcal {R}}_{I}$ rapidly grows up to $N = 16$ elements before saturating. Similarly, at 26 dBm SNR, maximum ${\mathcal {R}}_{I}$ is achieved with only $N = 6$ . This indicates that the system achieves maximum ${\mathcal {R}}_{I}$ within fewer reflecting elements, and additional increment in N has minimal impact on ${\mathcal {R}}_{I}$ signifying the ability of the system to maintain effectiveness and resilience, even with fewer reflecting elements in operation.

D. Comparative Analysis of the System

Fig. 7(a) and Fig. 7(b) analyze the impact of UAV height $({\mathcal {H}}_{r})$ and radius $(R_{r})$ on user OP performance in a UAV-mounted A-IRS NOMA network. As the UAV height increases (at $0.2, 0.6, 1$ km with a fixed radius of 2 km), user outage performance worsens, particularly at higher altitudes due to increased path-loss. Conversely, increasing the UAV’s radius ($1, 2, 2.5$ km at a constant height of 0.5 km) improves user performance and coverage, though transmission power limitations may affect connectivity with ground users beyond 2 km, exacerbated by hardware degradation and increasing user distance. Fig. 7(c) explores the impact of amplification factor $(\eta _{a})$ , CCI power $(\rho _{iu_{2}})$ , and iSIC $(\xi)$ on the secondary network’s OP. The network’s OP decreases with an increase in $\eta _{a}$ to 3, 5, 7 dB, due to enhanced signal strength and SNR. Conversely, reducing CCI power to 25, 20, 15 dBm, while keeping $\eta _{a}$ and $\xi$ constant, also decreases OP. Additionally, OP significantly decreases with lower iSIC levels (1, 0.6, 0), with constant $\eta _{a}$ and $\rho _{iu_{2}}$ .

Fig. 8(a) demonstrates how power allocation coefficients $\delta _{1}$ and $\delta _{2}$ affect the OP of primary and secondary networks at various SNR levels. An increase in $\delta _{1}$ consistently improves the primary network’s OP while adversely affecting the secondary network’s OP, with only minor OP variations at high SNRs. A performance trade-off between the two networks is evident. To optimize $\delta _{1}$ and balance the OPs of both networks, Algorithm 1 is introduced, targeting an equitable primary-secondary network performance relationship. Fig. 8(b) compares the performance of UAV-mounted A-IRS NOMA with FD/HD DF relaying NOMA schemes. A-IRS NOMA with pSIC outperforms both FD and HD DF relaying in higher SNR regimes, primarily due to loop interference in FD DF relaying and the more efficient spectrum use in FD NOMA. While FD-NOMA with pSIC excels in lower to mid SNR ranges, the plot also shows that nearby users in these systems have better outage performance than distant ones, benefiting from lower interference and higher diversity order.

The optimal UAV positioning for both users is essential for minimizing their OP, as illustrated in Fig. 9(a) and Fig. 9(b). We vary the UAV’s radial distance $(R_{r})$ within 0 to 4 km and angular position $(\phi _{r})$ from $-\pi /2$ to $\pi /2$ radians at a fixed height of 0.5 km and SNR 30 dBm to plot the variation in the users’ OP. It may be observed that the OP for $U_{1}$ is minimum when the UAV is positioned at $R_{r}^{*}\approx 1.9, \phi _{r}^{*}\approx \pi /107.588$ . Similarly, it can be observed that the OP of $U_{2}$ is minimum at $R_{r}^{*}\approx 3.1, \phi _{r}^{*}\approx -\pi /4.682$ . Moreover, the coordinates obtained from Algorithm 2 closely match these optimal points, validating the algorithm’s effectiveness. Thus, positioning the UAV near these obtained coordinates is key to minimizing the users’ OP.

FIGURE 9.

Impact of $\phi _{r}$ and $R_{r}$ on the OP performance of $U_{1}$ and $U_{2}$ .

E. Evaluation of the Proposed Algorithms

We recall from Fig 8(a), that a trade-off exists between the performance of $U_{1}$ and $U_{2}$ . To achieve a balance between their OPs, we propose Algorithm 1 to find the optimal value of $\delta _{1}$ . In Fig. 10(a), we compare the OP difference between $U_{1}$ and $U_{2}$ using the optimal $\delta _{1}$ (determined via Algorithm 1) against arbitrary values of $\delta _{1}$ . As shown, the optimal $\delta _{1}^{*}$ yields the smallest OP difference, demonstrating the effectiveness of the proposed algorithm.

FIGURE 10.

Evaluation of (a) Algorithm 1 and (b), (c) Algorithm 2.

In Fig. 10(b) and Fig. 10(c), we compare the OP performance using the optimum UAV parameters (determined via Algorithm 2) against three arbitrary settings of UAV parameters. It can be observed that the OP with optimum UAV parameters outperforms all other parameter combinations at a fixed UAV height of 0.5 km, demonstrating the effectiveness of the proposed algorithm.

F. Overall System Performance Analysis

From Fig. 11 we observe that UAV-mounted A-IRS NOMA outperforms UAV-mounted P-IRS NOMA and UAV-mounted A-IRS OMA in both delay-limited and delay-tolerant modes. The throughput of both the UAV-A-IRS NOMA and UAV-A-IRS OMA networks converges at high SNR in delay-limited mode. This occurs because, in delay-limited transmission mode, the sum target rate for NOMA users is set to be the same as for OMA users. Moreover, delay-tolerant mode provide better system throughput by leveraging flexible timing for data transfer compared to stricter timing constraints of delay-limited mode.

FIGURE 11.

System throughput vs. SNR in delay-tolerant and delay limited modes.

In Fig. 12(a), the ergodic capacity is plotted against SNR and $\delta _{1}$ . We observe that at a constant $\delta _{1}=0.7$ , the crossover between the ergodic capacities of $U_{1}$ and $U_{2}$ occurs around an SNR of 26 dBm. Conversely, in the range $0.8 \lt \delta _{1} \lt 1$ , the ergodic capacity of $U_{1}$ remains higher than that of $U_{2}$ across all SNR regimes. Thus, the relative ergodic capacity of both users varies depending on the operating SNR and power allocation factors. Next, in Fig. 12(b), the ergodic sum capacity system is plotted against the SNR. It is observed that the UAV-mounted A-IRS NOMA network achieves the highest capacity across all SNR levels in ideal conditions. However, the HIs with $\lambda =0.3$ significantly reduces the ergodic sum capacity, especially at higher SNRs. For instance, at 20 dBm, the ergodic sum capacity of the UAV-mounted A-IRS NOMA decreases by 34%. This reduction is more pronounced at 40 dBm, where the ergodic sum capacity of the system decreases by 62.6%. Moreover, the impact of CCI is less severe than that of HIs at higher SNR, with a 55% reduction in the capacity at 20 dBm and 14% at 40 dBm. Thus, at lower SNRs, CCI mainly limits system capacity, while at higher SNRs, HIs become the primary constraint. Moreover, the ergodic sum capacity of UAV-mounted P-IRS NOMA network is consistently lower compared to UAV-mounted A-IRS NOMA network.

FIGURE 12.

(a) Impact of $\delta _{1}$ on users’ ergodic capacity at different SNRs and (b) Impact of CCI and HIs on the ergodic sum capacity of UAV A-IRS NOMA and UAV P-IRS NOMA networks.

G. Deep Learning Approach

Fig. 13(a) and Fig. 13(b) demonstrate that increasing hidden layers in DNN and CNN models lowers RMSE, thereby improving data generalization and network capacity. Optimal performance is achieved with four hidden layers, with the DNN model outperforming the CNN in terms of RMSE, highlighting the limitations of single-layer models in complexity management. Fig. 13(c) shifts focus to RMSE in the logarithmic domain, comparing training and validation sets over iterations with different learning rates. A rate of 0.001 proves optimal, striking a balance between learning rate and avoiding local optima or loss oscillation, with RMSE stabilizing after about 30 iterations. Fig. 13(d) then examines the impact of batch sizes $(20, 50, 100, 200)$ on RMSE during DNN convergence, finding that a batch size of 50 yields the smallest RMSE, indicating a slower convergence rate for larger sizes. Comparing DNN and CNN models in execution time for prediction, DNN calculates OP (0.058 s) and ergodic capacity (0.077 s) faster than CNN (0.094 s, 0.133 s). Analytical analysis and Monte Carlo simulation take longer, at 24.065 s/58.359 s and 157.35 s/182.26 s, respectively. RMSE values for OP $(0.109)$ and ergodic capacity $(0.121)$ validate the accuracy of these methods.

FIGURE 13.

RMSE vs. number of hidden neurons and epoch.

SECTION VII. Conclusion

In this work, we investigated the robustness and reliability of a UAV-mounted A-IRS and P-IRS relaying NOMA network under non-ideal system imperfections such as CCI, HIs, and iSIC. We derived the analytical expressions for OP, ergodic capacity, and system throughput in both delay-limited and delay-tolerant transmission modes over Nakagami-m fading channels. The analytical results closely aligned with the Monte-Carlo simulations and demonstrated that UAV-mounted A-IRS NOMA networks outperform the benchmark UAV-mounted P-IRS NOMA networks, UAV-mounted A-IRS/P-IRS-OMA networks, and traditional relay schemes such as HD-DF and FD-DF in terms of OP, ergodic capacity, and system throughput. To gain a better understanding of the system design, we presented an asymptotic outage approximation for high SNR regions to determine the diversity order of users. It was revealed that the diversity order of users depends on the number of reflecting elements and the Nakagami-m fading parameters and is zero for the iSIC case. The network’s performance improves with higher A-IRS amplification gain but is hindered by thermal noise, CCI power, and non-ideal system imperfections like iSIC and HIs. Specifically, the impact of CCI and HIs is severe in the low and high SNR regime, respectively. The robustness analysis provided the minimum number of reflecting elements required to attain a given target outage and revealed that specific QoS can be achieved with fewer reflecting elements. Further, we designed algorithms for optimal UAV position, power allocation factors for each user, and DNN hyperparameters to minimize users’ OP. We also presented a DNN framework for metrics prediction providing better computational efficiency and accuracy compared to conventional analytical methods. It was shown that the DNN prediction for OP and ergodic capacity closely matched the simulations. Also, the designed DNN model surpassed the CNN in terms of RMSE and runtime, offering a viable, low-latency approach for future networks.

This paper can be generalized along several potential research directions. The proposed approach can be generalized to account for imperfections at the IRS, such as imperfect phase cancellation arising due to imperfect CSI and practical phase-dependent amplitude.

Appendix A Proof of Theorem 1

With perfect channel state information, we set the n-th element’s phase shift to $\varphi _{n}=-\varphi _{bu_{j}}+\varphi _{b,n}+\varphi _{n,u_{j}}$ for optimal channel gain. Let $\mathcal {X}=|h_{bu_{1}}|$ , $\hat {\mathcal {X}}= a_{1}\mathcal {X}$ , ${\mathcal {Y}}_{n}= \tilde {b}_{1} |g_{n,u_{1}}|\,|g_{b,n}|$ , $\tilde {b}_{1}=b_{1} \eta _{a}$ , $\hat {\mathcal {Y}}_{n}= \sum _{n=1}^{N}{\mathcal {Y}}_{n}$ , $\hat {\mathcal {Z}}=(\hat {\mathcal {X}} + \hat {\mathcal {Y}}_{n})$ , $\mathcal {W}_{1}=\sum _{n=1}^{N}|g_{n,u_{1}}|^{2}$ , $\mathcal {R}_{1}=|h_{iu_{1}}|^{2}$ , and $\mathcal {Z}_{1}=(\hat {\mathcal {Z}})^{2}=(\hat {\mathcal {X}} + \hat {\mathcal {Y}}_{n})^{2}$ . Thus, the OP for $U_{1}$ can be expressed as $$\begin{align*} \mathcal {P}_{u_{1}}^{a}=& \mathbb {P}_{r}\left [{{\mathcal {Z}_{1}\lt z }}\right ]= F_{\mathcal {Z}_{1}}(z) = {F_{\hat {\mathcal {Z}}}\left ({{\sqrt {z}}}\right)} \\=& \mathbb {P}_{r}\left [{{\hat {\mathcal {Z}}\lt \sqrt {\left ({{\mathcal {T}_{1}+\mathcal {T}_{1} \beta _{1} \mathcal {R}_{1}+ \mathcal {T}_{1} \Xi _{1} \mathcal {W}_{1}}}\right)} }}\right ]. \tag {51}\end{align*}$$ Let, $\mathcal {U}_{1}=\mathcal {T}_{1} \beta _{1} \mathcal {R}_{1}+ \mathcal {T}_{1} \Xi _{1} \mathcal {W}_{1}$ , where $\mathcal {T}_{1}={}\frac {\gamma _{\textrm {th}}^{1}}{\delta _{1}\, \rho _{b}-(\delta _{2}+\lambda _{bu_{1}}^{2}) \rho _{b}\gamma _{\textrm {th}}^{1}}$ , $\mathcal {R}_{1}$ and $\mathcal {W}_{1}$ are independent but not necessarily identical distributed (i.n.i.d.) Gamma random variables (RVs). Therefore, $\mathcal {T}_{1} \beta _{1} \mathcal {R}_{1}\sim Gamma(m_{iu_{1}},m_{iu_{1}} \tilde {\Omega }_{iu_{1}})$ and $\mathcal {T}_{1} \Xi _{1} \mathcal {W}_{1}\sim Gamma(m_{a}, m_{a}/\tilde {\Omega }_{a})$ are the scaled Gamma distributed RV, with $\tilde {\Omega }_{iu_{1}}= \mathcal {T}_{1} \beta _{1} \Omega _{iu_{1}}$ and $\tilde {\Omega }_{a}= \mathcal {T}_{1} \Xi _{1} \Omega _{a}$ , respectively. The PDF of $\mathcal {U}_{1}$ [64] is given as $$\begin{align*} {f}_{\mathcal {U}_{1}}({u}) = \prod _{t\in \{iu_{1}, a\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{2,2}^{0,2} \left [{{e^{u}\Bigg \lvert \begin{array}{c} \Delta _{u,1} \\ \Delta _{u,2} \end{array} }}\right ]. \tag {52}\end{align*}$$ Now, using joint RVs theory concept, we can rewrite (51) as $$\begin{equation*} \mathcal {P}_{u_{1}}^{a}=\int _{0}^{\infty }F_{\hat {\mathcal {Z}}}\left ({{\sqrt {\mathcal {T}_{1}+u} }}\right)f_{\mathcal {U}_{1}}(u)du. \tag {53}\end{equation*}$$ To solve the above mentioned integral (53), we initially characterize the distribution of the RV $\hat {\mathcal {Z}}$ by moment matching approach. Since $\mathcal {X} \sim Nakagami(m_{bu_{1}}, \Omega _{bu_{1}})$ , the p-th moment of scaled Nakagami distributed RV $\hat {\mathcal {X}}=a_{1}\mathcal {X}\sim Nakagami(m_{bu_{1}}, \tilde {\Omega }_{bu_{1}})$ with $\tilde {\Omega }_{bu_{1}}=a_{1}^{2}\Omega _{bu_{1}}$ , ${\mathcal {M}}_{\hat {\mathcal {X}}}(p)\triangleq \mathbb {E}[\hat {\mathcal {X}}^{p}]$ , can be derived as $$\begin{equation*} {\mathcal {M}}_{\hat {\mathcal {X}}}(p) = \frac {\Gamma \left ({{m_{bu_{1}}+p/2}}\right)}{\Gamma \left ({{m_{bu_{1}}}}\right)} \sqrt {\left ({{\frac { \tilde {\Omega }_{bu_{1}} }{m_{bu_{1}}}}}\right)^{p}}. \tag {54}\end{equation*}$$ From (54), the first and second moments of $\hat {\mathcal {X}}$ can be obtained and are given in (10) and (11), respectively. Since $g_{b,n} \sim Nakagami(m_{br}, \Omega _{br})$ and $g_{n,u_{1}} \sim Nakagami(m_{ru_{1}}, \Omega _{ru_{1}})$ . To obtain the p-th moment of ${\mathcal {Y}}_{n}= \tilde {b}_{1} |g_{n,u_{1}}|\,|g_{b,n}|$ , the PDF of ${\mathcal {Y}}_{n}$ [31] is given as $$\begin{equation*} f_{{\mathcal {Y}}_{n}}(y) = \frac {4 \mu _{n}^{m_{br}+m_{ru_{1}}} }{\Gamma \left ({{m_{br}}}\right) \Gamma \left ({{m_{ru_{1}}}}\right) } y^{ m_{br} + m_{ru_{1}}-1} \mathcal {K}_{m_{ru_{1}} - m_{br}} \left ({{2 \mu _{n} y}}\right), \tag {55}\end{equation*}$$ where $\mu _{n}=\sqrt {{}\frac {m_{br} m_{ru_{1}} }{ (\tilde {b}_{1})^{2} \Omega _{br} \Omega _{ru_{1}}}}$ . Following this, the p-th moment of ${\mathcal {Y}}_{n}$ is derived as $$\begin{equation*} {\mathcal {M}}_{{\mathcal {Y}}_{n}}(p) =\mathbb {E}\left [{{{\mathcal {Y}}_{n}^{p}}}\right ] = \int _{0}^{\infty } y^{p} f_{{\mathcal {Y}}_{n}}(y) dy, \tag {56}\end{equation*}$$ utilizing [54], ${\mathcal {M}}_{{\mathcal {Y}}_{n}}(p)$ can be written as $$\begin{equation*} {\mathcal {M}}_{{\mathcal {Y}}_{n}}(p)=\mu _{n}^{-p} \frac {\Gamma \left ({{m_{br}+p/2}}\right) \Gamma \left ({{m_{ru_{1}}+p/2}}\right)}{\Gamma \left ({{m_{br}}}\right) \Gamma \left ({{m_{ru_{1}}}}\right)}. \tag {57}\end{equation*}$$ Using the p-th moment of ${\mathcal {Y}}_{n}$ , we model $\hat { \mathcal {Y}}_{n}=\sum _{n=1}^{N} {\mathcal {Y}}_{n}$ via a Gamma distribution and define its CDF as $$\begin{align*}& F_{\hat { \mathcal {Y}}_{n}}(y)=\frac {1}{\Gamma \left ({{\frac {N \left [{{{\mathcal {M}}_{{\mathcal {Y}}_{n}}(1)}}\right ]^{2} }{{\mathcal {M}}_{{\mathcal {Y}}_{n}}(2) - \left [{{{\mathcal {M}}_{{\mathcal {Y}}_{n}}(1)}}\right ]^{2}} }}\right)} \\& \;{}\times \gamma \left ({{\frac {N \left [{{{\mathcal {M}}_{{\mathcal {Y}}_{n}}(1)}}\right ]^{2} }{{\mathcal {M}}_{{\mathcal {Y}}_{n}}(2) - \left [{{{\mathcal {M}}_{{\mathcal {Y}}_{n}}(1)}}\right ]^{2}}, \frac {{\mathcal {M}}_{{\mathcal {Y}}_{n}}(1) y}{{\mathcal {M}}_{{\mathcal {Y}}_{n}}(2) - \left [{{{\mathcal {M}}_{{\mathcal {Y}}_{n}}(1)}}\right ]^{2}} }}\right). \tag {58}\end{align*}$$ Note that despite using Gamma approximations in (58), the disparity between these approximations and the exact results is minimal [18]. Using the multinomial expansion [65], the p-th moment of $\hat { \mathcal {Y}}_{n}$ is given as $$\begin{align*}& {\mathcal {M}}_{\hat { \mathcal {Y}}_{n}}(p) = \mathbb {E}\left [{{\hat { \mathcal {Y}}_{n}^{p}}}\right ] = \sum _{p_{1}=0}^{p} \sum _{p_{2}=0}^{p_{1}} \cdots \\& {} \sum _{p_{N -1}=0}^{p_{N -2}} {\binom {p }{ p_{1}}} {\binom {p_{1} }{ p_{2}}} \cdots {\binom {p_{N -2} }{ p_{N -1}}} \\& \;\quad {}\times {\mathcal {M}}_{\mathcal {Y}_{1}}\left ({{p -p_{1}}}\right) {\mathcal {M}}_{\mathcal {Y}_{2}}\left ({{p_{1} -p_{2}}}\right) \cdots {\mathcal {M}}_{{\mathcal {Y}}_{N}}\left ({{p_{N -1}}}\right). \tag {59}\end{align*}$$ On substituting (57) into (59), the first and second moments of $\hat { \mathcal {Y}}_{n}$ and given in (12) and (13), respectively. Additionally, as $\hat {\mathcal {X}}$ and $\hat { \mathcal {Y}}_{n}$ are independent, the p-th moment of $\hat {\mathcal {Z}}=(\hat {\mathcal {X}} + \hat {\mathcal {Y}}_{n})$ can be obtained as $$\begin{align*} {\mathcal {M}}_{\hat { \mathcal {Z}}}(p)=& \mathbb {E} \left [{{\left ({{\hat {\mathcal {X}} + \hat {\mathcal {Y}}_{n}}}\right)^{p}}}\right ] = \mathbb {E} \left [{{ \sum _{k=0}^{p} {\binom {p }{ k}} \hat {\mathcal {X}}^{k} \hat {\mathcal {Y}}_{n}^{p-k} }}\right ] \tag {60}\\=& \sum _{k=0}^{p} {\binom {p }{ k}} {\mathcal {M}}_{\hat {\mathcal {X}}}(k) {\mathcal {M}}_{\hat { \mathcal {Y}}_{n}}(p-k). \tag {61}\end{align*}$$ Using equation (61) the first and second moments of $\hat {\mathcal {Z}}$ can be obtained as (8) and (9). Now, the CDF of $\hat {\mathcal {Z}}$ can be written as $$\begin{equation*} F_{\hat {\mathcal {Z}}}\left ({{\sqrt {\mathcal {T}_{1}+u} }}\right)=\frac {1}{\Gamma \left ({{m_{z}}}\right)} \gamma \left ({{m_{z}, \frac {\sqrt {\mathcal {T}_{1}+u} }{\Omega _{z}} }}\right). \tag {62}\end{equation*}$$ Finally, inserting the PDF and CDF from (52) and (62) into (53), we find that obtaining $\mathcal {P}_{u_{1}}^{a}$ in closed form is intractable. To address this issue, we use a Gaussian-Chebyshev quadrature method [59]. This method leads to an accurate closed form expression of $\mathcal {P}_{u_{1}}^{a}$ as in (17).

Appendix B Proof of Lemma 1

Using $\mathcal {P}_{2} =\mathbb {P}_{r}\{\gamma _{u_{2},x_{1}}^{a}\gt \gamma _{\textrm {th}}^{1}, \gamma _{u_{2},x_{2}}^{a}\gt \gamma _{\textrm {th}}^{2} \}$ and substituting the SIDNR expressions from (5) and (6) along with the condition on $\mathcal {Z}_{2}$ , we can examine the scenarios where $\gamma _{\textrm {th}}^{1}\gt \gamma _{u_{2},x_{1}}^{a}$ and $\gamma _{\textrm {th}}^{1}\lt \gamma _{u_{2},x_{1}}^{a}$ , leading to the derivation of the expression in (15). In this context, $\mathcal {P}_{21}^{iSIC}=1- \mathbb {P}_{r}[\gamma _{u_{2},x_{1}}^{a}\lt \gamma _{\textrm {th}}^{1}]$ is derived by following the same approach as used in deriving $\mathcal {P}_{u_{1}}^{a}$ , leading to the result in (18). Besides, $\mathcal {P}_{22}^{iSIC}=\mathbb {P}_{r}[\gamma _{u_{2},x_{2}}^{a} \geq \gamma _{\textrm {th}}^{2}] = 1- F_{\gamma _{u_{2},x_{2}}^{a}}(x)$ is determined as $$\begin{equation*} \mathcal {P}_{22}^{iSIC}=1-\int _{0}^{\infty }F_{\hat {\mathcal {Z}}}\left ({{\sqrt {\mathcal {T}_{3}+u} }}\right)f_{\mathcal {U}_{2}^{\prime \prime }}(u)du, \tag {63}\end{equation*}$$ where $\mathcal {U}^{^{\prime \prime }}_{2}$ is the sum of three Gamma RVs. Their PDFs are derived using the sum of i.n.i.d. Gamma RVs with non-integer parameters, similar to (52). Thus, the PDF of $\mathcal {U}^{^{\prime \prime }}_{2}$ is given as $$\begin{align*} f_{\mathcal {U}_{2}^{\prime \prime }}(u)= \prod _{t\in \{u_{2}, a, iu_{2}\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{u}\Bigg \lvert \begin{array}{c} \Delta _{u^{\prime \prime },1} \\ \Delta _{u^{\prime \prime },2} \end{array} }}\right ]. \tag {64}\end{align*}$$ By substituting the relevant PDFs from (64) into (63) and applying the same method as used for obtaining (62) in Appendix A, we get the expression for $\mathcal {P}_{22}^{iSIC}$ as in (19).

Appendix C Proof of Lemma 2

Utilizing (51), we represent the asymptotic OP for user $U_{1}$ as $$\begin{equation*} \mathcal {P}_{u_{1}}^{a,\infty }=F_{\hat {\mathcal {Z}}^{2}}(z)=\int _{0}^{\infty }F_{\hat {\mathcal {Z}}}\left ({{\sqrt {\mathcal {T}_{1}+u} }}\right)f_{\mathcal {U}_{1}}(u)du. \tag {65}\end{equation*}$$ Given that $\mathcal {U}_{1}=\mathcal {T}_{1} \beta _{1} \mathcal {R}_{1}+ \mathcal {T}_{1} \Xi _{1} \mathcal {W}_{1}$ , the PDF of ${f}_{\mathcal {U}_{1}}({u}) ={}\frac {1}{(\mathcal {T}_{1})^{2}\beta _{1}\Xi _{1}}\int _{0}^{u} f_{\mathcal {W}_{1}}\left ({{{}\frac {w}{\mathcal {T}_{1} \Xi _{1}}}}\right)f_{\mathcal {R}_{1}}\left ({{{}\frac {u-w}{\mathcal {T}_{1} \beta _{1}}}}\right) dw$ . By substituting the relevant PDFs and CDF from (62) into (65) and solving the associated integral using [54, (1.111)] and [54, (3.351.3)], we achieve the expression for $\mathcal {P}_{u_{1}}^{a,\infty }$ as in (21).

Appendix D Proof of Theorem 2

Substituting $\gamma _{u_{1},x_{1}}^{a}$ from (4) into (34), $\mathcal {C}_{u_{1}}^{a}$ can be rewritten as $$\begin{equation*} \mathcal {C}_{u_{1}}^{a}=\underbrace {\mathbb {E}\left [{{\log _{2}\left ({{1+\mathcal {Q}}}\right)}}\right ]}_{\left ({{\hat {\mathcal {C}}}}\right)}- \underbrace {\mathbb {E}\left [{{\log _{2}\left ({{1+\mathcal {V}}}\right)}}\right ]}_{\left ({{\tilde {\mathcal {C}}}}\right)}, \tag {66}\end{equation*}$$ where $\mathcal {Q}={ (\varrho _{1}+\delta _{1}\, \rho _{b})\mathcal {Z}_{1} + \Xi _{1} \|\mathbf {g}_{u_{1}}^{H}\boldsymbol {\Phi } \|^{2} + \beta _{1} |h_{iu_{1}}|^{2}}$ and $\mathcal {V}={ \varrho _{1}\mathcal {Z}_{1} + \Xi _{1} \|\mathbf {g}_{u_{1}}^{H}\boldsymbol {\Phi } \|^{2} + \beta _{1} |h_{iu_{1}}|^{2}}$ . Now $\hat {\mathcal {C}}$ and $\tilde {\mathcal {C}}$ can be expressed as $$\begin{align*} \hat {\mathcal {C}}=& {\mathbb {E}\left [{{\log _{2}\left ({{1+\mathcal {Q}}}\right)}}\right ]} =\int _{0}^{\infty }\log _{2}(1+q)f_{\mathcal {Q}}(q)dq, \tag {67}\\ \tilde {\mathcal {C}}=& {\mathbb {E}\left [{{\log _{2}\left ({{1+\mathcal {V}}}\right)}}\right ]}=\int _{0}^{\infty }\log _{2}(1+v)f_{\mathcal {V}}(v)dv. \tag {68}\end{align*}$$ Note that $\mathcal {Q}$ and $\mathcal {V}$ are each the sum of three Gamma RVs. To define their PDFs, we use the sum of i.n.i.d. Gamma RVs with non-integer parameters. Following a similar approach to (52), the PDFs of $\mathcal {Q}$ and $\mathcal {V}$ are expressed as $$\begin{align*} {f}_{\mathcal {Q}}({q})=& \prod _{t\in \{z, a, iu_{1}\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{q}\Bigg \lvert \begin{array}{c} \Delta _{q,1} \\ \Delta _{q,2} \end{array} }}\right ] \tag {69}\\ {f}_{\mathcal {V}}({v})=& \prod _{t\in \{z, a, iu_{1}\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{v}\Bigg \lvert \begin{array}{c} \Delta _{v,1} \\ \Delta _{v,2} \end{array} }}\right ]. \tag {70}\end{align*}$$ On substituting the relevant PDFs from (69) into (67) and (70) into (68), we obtained $\hat {\mathcal {C}}$ and $\tilde {\mathcal {C}}$ as $$\begin{align*} \hat {\mathcal {C}}=& \int _{0}^{\infty } \log _{2}(1+q) \prod _{t\in \{z, a, iu_{1}\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{q}\Bigg \lvert \begin{array}{c} \Delta _{q,1} \\ \Delta _{q,2} \end{array} }}\right ] dq, \tag {71}\\ \tilde {\mathcal {C}}=& \int _{0}^{\infty } \log _{2}(1+v) \prod _{t\in \{z, a, iu_{1}\}} \left ({{ \frac {m_{t}}{\tilde {\Omega }_{t}} }}\right)^{m_{t}} \times \bar {\textrm {H}}_{3,3}^{0,3} \left [{{e^{v}\Bigg \lvert \begin{array}{c} \Delta _{v,1} \\ \Delta _{v,2} \end{array} }}\right ] dv. \tag {72}\end{align*}$$ Finally, by following the same approach as in Appendix A for solving the (53), we obtain $\hat {\mathcal {C}}$ and $\tilde {\mathcal {C}}$ . Then, on substituting these values into (66), we arrive at (36). The proof is completed.