## 3 Communication and Channel Modeling
### Channel Modeling
#### 3.1.1 Deployment of UAV-BS and RISeUAV  
After the deployment of a UAV equipped with an aerial RIS as the RISeUAV, the channel modeling is obtained as follows. The same methodology can be applied to the RIS-assisted UAV-BS. The channel gain of the cascaded \(T_{x}-RIS-R_{x}\) LoS path for the RISeUAV, in Figure 3a, is obtained as [39; 40]:  
\[\mathcal{G}_{CLoS}^{TR}\approx\frac{\lambda}{4\pi}~{}\mathcal{G}_{LoS}^{RIS} \Psi^{RIS}\mathcal{G}_{LoS}^{T\_RIS}, \tag{3}\]  
where the notations of (3) is presented in the following: \(\dagger\) denotes conjugate (Hermitian) transpose; \(\mathcal{G}_{CLoS}^{TR}\in\mathbb{C}\) denotes the channel gain of the indirect CLoS link between transmitter and receiver through RIS (i.e., the \(T_{x}-RIS-R_{x}\) link); \(\Psi^{RIS}\in\mathbb{C}^{M\times M}\) denotes the RIS phase shift matrix given as follows:  
\[\Psi^{RIS}=\text{diag}\Big{\{}e^{i\varphi_{1}^{RIS}},e^{i\varphi_{2}^{RIS}}, \ldots,e^{i\varphi_{m}^{RIS}},\ldots,e^{i\varphi_{M}^{RIS}}\Big{\}}\in\mathbb{ C}^{M\times M}, \tag{4}\]  
where \(\psi_{m}^{RIS}\) denotes the phase shift of the \(m^{th}\) RIS element, and \(M\) denotes the number of RIS reflector elements; \(\mathcal{G}_{LoS}^{T\_RIS}\in\mathbb{C}^{M}\) denotes the LoS channel gain of the \(T_{x}-RIS\) link given as follows:  
\[\mathcal{G}_{LoS}^{T\_RIS}\approx\frac{\lambda}{4\pi}\frac{\sqrt{G_{T}}}{ \left\|p^{Tx}-p^{RIS}\right\|_{2}}\Big{[}e^{-j\phi_{(m=1)}^{T\_RIS}},\ldots,e^ {-j\phi_{m}^{T\_RIS}},\ldots,e^{-j\phi_{M}^{T\_RIS}}\Big{]}^{\top} \tag{5}\]  
where \(p^{RIS}= \begin{bmatrix}x^{RIS}&y^{RIS}&z^{RIS}\end{bmatrix} ^{\top}\in\mathbb{ R}^{3}\) denotes the 3D position of the RIS; \(\phi_{m}^{T\_RIS}=\frac{2\pi}{\lambda}d_{m}^{T\_RIS}\) indicates the imposed phase shift in the associated incident path (i.e., the path between the transmitter and \(m^{th}\) RIS element) where \(d_{m}^{T\_RIS}=\left\|p^{Tx}-p_{m}^{RIS}\right\|_{2}\); and \(p_{m}^{RIS}= \begin{bmatrix}x_{m}^{RIS}&y_{m}^{RIS}&z_{m}^{RIS}\end{bmatrix} ^{ \top}\in\mathbb{R}^{3}\) denotes the Cartesian coordinates of the \(m^{th}\) RIS element. \(p_{m}^{RIS},~{}\gamma m\in M\) can be obtained based on the RIS position and its spacing among its elements [41], see Figure 4 and Appendix A. \(\mathcal{G}_{LoS}^{RIS\_R}\in\mathbb{C}^{M}\) denotes the LoS channel gain of the \(RIS-R_{x}\) link given as follows:  
\[\mathcal{G}_{LoS}^{RIS\_R}\approx\frac{\lambda}{4\pi}\underbrace{\frac{\sqrt{ G_{R}}}{\left\|p^{RIS}-p^{Rx}\right\|_{2}}}_{d^{RIS\_R}}\left[e^{-j\phi_{(m=1)}^{ RIS,R}},\ldots,e^{-j\phi_{m}^{RIS,R}},\ldots,e^{-j\phi_{M}^{RIS,R}}\right]^{\top} \tag{6}\]  
where \(\phi_{m}^{RIS\_R}=\frac{2\pi}{\lambda}d_{m}^{RIS\_R}\) indicates the imposed phase shift in the associated reflection path (i.e., the path between \(m^{th}\) RIS element and the receiver), where \(d_{m}^{RIS\_R}=\left\|p_{m}^{RIS}-p^{Rx}\right\|_{2}\). Therefore, the equivalent channel gain of the indirect \(\mathrm{T_{x}-RIS-R_{x}}\) CLoS link can be obtained as follows:  
\[\mathcal{G}_{CLoS}^{TR}\approx\left(\frac{\lambda}{4\pi}\right)^{2}\sqrt{G_{T} }~{}\sqrt{G_{R}}~{}\underbrace{\frac{\sum_{m=1}^{M}e^{-(\psi_{m}^{RIS}+\phi_{m }^{T\_RIS}-\phi_{m}^{RIS,R})}}{\left\|p^{Tx}-p^{RIS}\right\|_{2}}\Big{\|}p^{ RIS}-p^{Rx}\Big{\|}_{2}}_{d^{RIS\_R}} \tag{7}\]  
By assigning the phase shift for the \(m^{th}\) RIS element as follows:  
\[\psi_{m}^{RIS}=\phi_{m}^{RIS\_R}-\phi_{m}^{T\_RIS}, \tag{8}\]  
all reflected signals by the RIS elements arrive at the receiver with the same phase shift so they are inherently accumulated in the receiver. Therefore, the indirect CLoS is obtained as follows:(9)  
Finally, the achievable rate is obtained as follows:  
\[\mathcal{R}=\log_{2}\!\left(1+\frac{P_{\mathrm{T}}\big{|}\mathcal{g}_{LoS}^{TR}+ \mathcal{g}_{CLoS}^{TR}\big{|}^{2}}{\sigma^{2}}\right) \tag{10}\]  
which is a concave function. There are critical points given from the equivalent channel gain of the CLoS link which are highlighted in the following remarks. **Remark 3**.