## 2 System Model  
In this paper, we consider a downlink UAV-assisted communication system, as illustrated in Figure 1. In practice, RIS-UAV is relevant in various scenarios. A UAV equipped with RIS or relay, for example, could function as a mobile RIS or mobile relay to extend the network coverage in an urban environment where signals suffer from severe shadowing and blocking due to dense high-rise buildings [21]. As another example, the mobile UAV relay and RIS-UAV could fly around to establish favorable condition links for vehicular users [22]. In our considered model, a rotary-wing UAV, equipped with either RIS or active relay, serves as a mobile RIS or relay node to assist the communication between a BS and the UE,  
Figure 1: UAV-assisted communication. where the direct link between BS and UE is blocked. Both the BS and UE are equipped with only a single antenna, and their positions are fixed. In addition, the UAV will hover at a certain altitude. For the RIS elements mounted on the UAV, each of them is half of the size of the wavelength; hence, the elements reflect the signal with constant gain toward the desired direction [12]. The definition of parameters is listed in Tables 12 and 12 in Appendix A. The locations of BS, UAV, and UE in 3D Cartesian coordinates are \(\ell_{BS}=(x_{BS},y_{BS},0)\), \(\ell_{UAV}=(x_{UAV},y_{UAV},z_{UAV})\), and \(\ell_{UE}=(x_{UE},y_{UE},0)\), respectively. Additionally, the locations of BS, UE, and UAV in 2D Cartesian coordinates can be denoted as \(\mathbf{w}_{BS}=(x_{BS},y_{BS})\), \(\mathbf{w}_{UE}=(x_{UE},y_{UE})\), and \(\mathbf{w}_{UAV}=(x_{UAV},y_{UAV})\), respectively. Therefore, the distance between the BS and the UAV, \(d_{1}\) can be defined as:  
\[d_{1}=\sqrt{\mid\mathbf{w}_{UAV}-\mathbf{w}_{BS}\mid^{2}+z_{UAV}^{2}}, \tag{1}\]  
while the distance between the UAV and the UE, \(d_{2}\), can be defined as:  
\[d_{2}=\sqrt{\mid\mathbf{w}_{UAV}-\mathbf{w}_{UE}\mid^{2}+z_{UAV}^{2}}. \tag{2}\]  
The G-A and A-G channel links between BS and the UAV, and between UAV and the UE, respectively, are computed using [PERSON]'s channel model with Rician fading as introduced in [23]. Accordingly, the probability of LoS for each channel link is defined as follows:  
\[P_{LoS}(\theta)=\frac{1}{1+a\exp(-b[\theta-a])}, \tag{3}\]  
where \(a\) and \(b\) are the constant environmental parameters, and \(\theta\) is the elevation angle between the UAV and the ground nodes. The elevation angle between the BS and UAV is given as:  
\[\theta_{bs}=\arctan\biggl{(}\frac{z_{UAV}}{\mathbf{w}_{UAV}-\mathbf{w}_{BS}}\biggr{)}, \tag{4}\]  
whereas the elevation angle between the UAV and the UE is given as:  
\[\theta_{ue}=\arctan\biggl{(}\frac{z_{UAV}}{\mathbf{w}_{UAV}-\mathbf{w}_{UE}}\biggr{)}. \tag{5}\]  
In addition, the probability of non-LoS (NLoS) for each channel link is defined as:  
\[P_{NLoS}(\theta)=1-P_{LoS}(\theta). \tag{6}\]  
Meanwhile, the channel gains between the BS and the UAV, and between the UAV and the UE, are defined as [24]:  
\[\mid h_{sr}\mid^{2}=\frac{A\mid\Omega_{sr}\mid^{2}}{d_{1}^{a_{sr}}}, \tag{7}\]  
\[\mid h_{rd}\mid^{2}=\frac{A\mid\Omega_{rd}\mid^{2}}{d_{2}^{a_{rd}}}, \tag{8}\]  
where \(A\) represents the effect of the antenna gain and operating frequency, \(\mid\Omega_{sr}\mid\) and \(\mid\Omega_{rd}\mid\) are the independent and identically distributed (i.i.d) random variables that follow a Rician distribution, while \(\alpha_{sr}\) and \(\alpha_{rd}\) are the aerial path-loss exponents for the links between BS and UAV, and between UAV and the UE, respectively. In detail, the constant coefficient, \(A\), is defined as:  
Figure 1:0  
where \(f_{c}\) is the operating frequency and \(c\) is the speed of light. The Rician fading channel model is adopted in this paper due to the possible existence of LoS and NLoS paths between the BS and UE. In other words, the received signals at the UE comprise both direct wave and multipath scatterers.