## 2 System Model  
We aim to design a IoT network by enabling IRS, NOMA and UAV techniques, where a single antenna source \((S)\) powers the signal transmission by IRS to serve many users at destinations. The IRSs are mounted in UAVs for better transmission from the base station to ground users. These ground users are divided into many groups, and a considered group contains two users \((D_{i};i=1,2)\). The system can maximize the benefits of IRS if multiple \(I\) IRSs are deployed, as shown in Figure 1. It is assumed that there is no direct path between the source \(S\) and ground users (destinations). To reduce the cost of the design, we refer to the first scenario where two IRSs, i.e., \(I_{1}\) and \(I_{2}\), possess reflecting elements \(N_{1}\) and \(N_{2}\). Those IRSs are placed at specific locations (i.e., buildings) to reflect passive beamforming signals towards the destinations (IoT devices). We need to answer how many IRSs are required to achieve the best performance at destinations. Therefore, we move our attention to the second scenario (the benchmark) by assigning three IRSs, i.e., \(I_{1}\), \(I_{2}\) and \(I_{3}\), which are installed with reflecting elements \(N_{1}\), \(N_{2}\) and \(N_{3}\), respectively (it is reasonable to design double-IRS due to cost efficiency in deployment. Although multi-IRS was developed in [31], the numerical result demonstrated a small gap between the double-IRS and three-IRSs case. Therefore, in this study, we emphasize the performance for two scenarios of IRSs).  
We denote the (scalar) channels from the \(S\) to the IRSs, and from the IRSs to \(D_{i}\), respectively, as \(h_{SI_{u},n}\) and \(h_{I_{u}D_{i},n}\). The hardware impairment at the \(S\) is denoted as \(\tau_{S}\sim CN(0,Y_{S}^{2}P)\), where \(Y_{S}\) represents the proportionality coefficients, which describe the severity of the distortion noises at \(S\). The hardware impairments at the \(D_{i}\) is \(\tau_{D_{ui}}\sim CN\left(0,Y_{D_{i}}^{2}P\right|\sum\limits_{u=1}^{I}\sum \limits_{n=1}^{N_{u}}h_{SI_{u},n}h_{I_{u}D_{i},n}\eta_{un}e^{i\theta_{un}^{(D_{ i})}}\right)^{2}\)[32], where \(Y_{D_{i}}\) represents the proportionality coefficients. Here, \(Y_{D_{i}}\) is associated with the distortion noises at \(D_{i}\). Then, the overall received signal at the user \(D_{i}\) with multiple \(I\) IRSs is given as [30; 33]  
\[y_{D_{i}}=\sum\limits_{u=1}^{I}\sum\limits_{n=1}^{N_{u}}h_{SI_{u},n}h_{I_{u}D_{ i},n}\eta_{un}e^{i\theta_{un}^{(D_{i})}}\left(\sqrt{P\chi_{1}}x_{1}+\sqrt{P \chi_{2}}x_{2}+\tau_{S}\right)+\tau_{D_{ui}}+n_{D_{i}}, \tag{1}\]  
Figure 1: The IRS–NOMA–UAV system model with source, two IRSs and ground users.  
where \(P\) is the total transmit power at the source, \(x_{i}\) is the transmitted signal by source, \(\chi_{i}\) denotes the power allocation factor for message \(x_{i}\) with \(\chi_{1}+\chi_{2}=1\), \(\chi_{1}>\chi_{2}\)[34], \(\theta_{in}^{(D_{i})}\) is the adjustable phase applied by the \(n\)-th reflecting element of the \(S-I-D_{i}\), \(\eta_{un}\) is the reflection coefficients of \(I\) with \(\eta_{un}\in(0,1]\), and \(n_{D_{i}}\) stands for the circularly symmetric additive White Gaussian noise (AWGN) with zero mean and variance \(\sigma_{n}^{2}\), i.e., \(n_{D_{i}}\sim\mathcal{CN}\big{(}0,\sigma_{n}^{2}\big{)}\).