## 2 System Model
### Multi-User UAV Communication Channel Model with RIS
\(F_{BB,k}\epsilon\mathbb{C}^{N_{RF}\times N_{k,k}}\) and \(F_{RF}\epsilon\mathbb{C}^{N_{T}\times N_{RF}^{T}}\) represent the baseband beamforming matrix and the analog beamforming matrix, respectively. \(n_{k}\sim\mathcal{CN}(0,\delta^{2}I)\) is the \(k\) additive Gaussian white noise at the user. \(\tilde{M}_{k}(t)\) is the millimeter-wave large-scale channel matrix from the RIS to the user. \(\tilde{R}(t)\) is the millimeter-wave large-scale channel matrix from the UAV to the RIS. \(\tilde{M}_{k}(t)\) and \(\tilde{R}(t)\) are defined as follows:  
\[\tilde{M}_{k}(t)=\sqrt{\frac{\Upsilon N_{R}}{Q_{\tilde{M}_{k}}}}\sum_{q=1}^{Q _{\tilde{M}_{k}}}\tilde{\kappa}_{km,q}(t)\tilde{\pi}_{R}\Big{(}\varphi_{km,q}{} ^{R}(t)\Big{)}\tilde{\pi}_{P_{1}}^{H}\Big{(}\varphi_{km,q}{}^{T}(t),\eta_{km,q}{ }^{T}(t)\Big{)}\epsilon\mathbb{C}^{N_{R}\times\Upsilon} \tag{20}\]  
\[\tilde{R}(t)=\sqrt{\frac{\Upsilon N_{T}}{Q_{R}}}\sum_{q=1}^{Q_{\tilde{R}}} \tilde{\kappa}_{r,q}(t)\tilde{\pi}_{P_{2}}\Big{(}\varphi_{r,q}{}^{R}(t),\eta_{r,q}{}^{R}(t)\Big{)}\tilde{\pi}_{T}^{H}\Big{(}\
u_{r,q}{}^{T}(t)\Big{)}\epsilon \mathbb{C}^{\Upsilon^{\prime}\times N_{T}} \tag{21}\]  
The UAV and the user are equipped with uniform linear arrays. RIS is equipped with a uniform planar array of size \(T^{{}^{\prime}}_{y}\times T^{{}^{\prime}}_{z}\). \(Q_{\tilde{M}_{k}}\) and \(Q_{\tilde{R}}\) are the numbers of channel paths from the RIS to the \(k\) user and the number of channel paths between the UAV and the RIS. \(\tilde{\kappa}_{km,q}(t)\) denotes the path gain from the RIS to the \(k\) user, and the path gain \(\tilde{\kappa}_{r,q}(t)\) denotes the fast fading path gain between the UAV and the RIS. \(\tilde{\pi}_{R}\Big{(}\varphi_{km,q}{}^{R}(t)\Big{)}\), \(\tilde{\pi}_{T}\big{(}\
u_{r,q}{}^{T}(t)\big{)}\), \(\tilde{\pi}_{P_{1}}\Big{(}\varphi_{km,q}{}^{T}(t)\Big{)}\), \(\eta_{km,q}{}^{T}(t)\) and \(\tilde{\pi}_{P_{2}}\big{(}\varphi_{r,q}{}^{R}(t),\eta_{r,q}{}^{R}(t)\big{)}\) are the steering vectors corresponding to the uniform linear array and the uniform planar array, respectively. \[\tilde{\pi}_{R}\Big{(}\varphi_{km,q}{}^{R}(t)\Big{)}=\frac{1}{\sqrt{N_{R}}} \Big{[}1,\ldots,e^{i(N_{R}-1)dsin(\varphi_{km,q}{}^{R}(t))\frac{2\pi}{N}} \Big{]} \tag{22}\]  
\[\tilde{\pi}_{T}\Big{(}\
u_{r,q}{}^{T}(t)\Big{)}=\frac{1}{\sqrt{N_{T}}}\Big{[}1,\ldots,e^{i(N_{T}-1)dsin(\varphi_{km,q}{}^{T}(t))\frac{2\pi}{N}}\Big{]} \tag{23}\]  
\[ \begin{array}{l}\tilde{\pi}_{P_{1}}\Big{(}\varphi_{km,q}{}^{T}(t),\eta_{km,q}{ }^{T}(t)\Big{)}\\ =\frac{1}{\sqrt{M_{y}M_{z}}}\Big{[}1,\ldots,e^{idsin((M_{y}-1)sin(\varphi_{km,q} {}^{T}(t))sin(\eta_{km,q}{}^{T}(t))+(M_{z}-1)cos(\varphi_{km,q}{}^{T}(t))) \frac{2\pi}{N}}\Big{]}\end{array}  \tag{24}\]  
where \(\widetilde{\alpha}_{R}\Big{(}\varphi_{km,q}{}^{R}(t)\Big{)}\) and \(\widetilde{\alpha}_{P}\Big{(}\eta_{km,q}{}^{T}(t)\Big{)}\) represent the receive vector and the send vector, respectively. Equation (28) is expressed as the following equation:  
\[\mathrm{Y}(t)=\sum_{q=1}^{Q_{\widetilde{N}_{R}}}\widetilde{\kappa}_{km,q}(t) \widetilde{\alpha}_{R}\Big{(}\varphi_{km,q}{}^{R}(t)\Big{)}\widetilde{\alpha}_ {P}\Big{(}\eta_{km,q}{}^{T}(t)\Big{)}+(W_{RF}W_{BB})^{H}\overset{\leftarrow}{N }(t) \tag{31}\]  
The received signal tensor for \(T\) consecutive time slots as a rank-one vector accumulation pattern is expressed as the following equation:  
\[\mathcal{Y}=\sum_{q=1}^{Q}\mathcal{T}\widetilde{\alpha}_{R}\Big{(}\beta_{q}{ }^{R}(t)\Big{)}\cdot\widetilde{\alpha}_{T}\Big{(}\gamma_{q}{}^{T}(t)\Big{)} \cdot\widetilde{\kappa}_{q}(t)+\mathcal{N} \tag{32}\]  
where \(\mathcal{T}\) is the weight of the multiplication of the rank-one factors. \(\mathcal{Y}\)\(\in\)\(\mathbb{C}^{N^{R}_{RF}\times N^{T}_{RF}\times T}\) is the received signal tensor. \(\widetilde{\kappa}_{q}(t)=\left[\kappa_{q}(1),\cdots,\kappa_{q}(T)\right]^{T} \)\(\in\)\(\mathbb{C}^{T\times 1}\) is the vector containing the fast-fading coefficients from \(T\) consecutive time slots. \(\mathcal{N}\)\(\in\)\(\mathbb{C}^{N^{R}_{RF}\times N^{T}_{RF}\times T}\) is the noise tensor.