## 2 System Model
### Multi-User UAV Communication Channel Model with RIS
After several iterations, \(v^{(1)}\) and \(g^{(1)}\) can be defined by the following equations, respectively:  
\[v^{(1)}=\delta_{v^{(1)}}\sum_{j=1}^{R}d_{j}^{2k+1}\overline{V}_{:,j}\overline{ U}_{:,j}^{T}g^{(1)} \tag{46}\]  
\[g^{(1)}=\delta_{g^{(1)}}\sum_{j=1}^{R}d_{j}^{2k}\overline{U}_{:,j}\overline{ U}_{:,j}^{T}g^{(1)} \tag{47}\]  
where \(\delta_{g^{(1)}}\) and \(\delta_{v^{(1)}}\) are the corresponding normalization factors. Thus, after several iterations of PM, \(v^{(1)}\) and \(g^{(1)}\) satisfy \(\frac{\left\|v^{(1)}\right\|_{2}^{2}}{\left\|g^{(1)}\right\|_{2}^{2}}=1\) by the convergence property, and the \(\frac{\delta^{(1)}}{g^{(1)}}\) converges to the maximum singular value \(d_{1}\), at \(Y_{(1)}^{T}\)\(g^{(1)}=\left(\frac{\delta_{g^{(1)}}}{g^{(1)}}\right)v^{(1)}\), \(\left\|Y_{(1)}^{T}\)\(g^{(1)}-d_{1}v^{(1)}\right\|_{2}=0\). From the previous rank-negative-one Tensor-train decomposition, the \(n\)th factor vector \(g^{(n)}\) depends on the previously computed right singular vector \(v^{(n-1)}\), while \(v^{(n-1)}\) is obtained from the computed \(n-1\) factor vector \(g^{(n-1)}\), because the matrix decomposition of each factor vector is performed independently. These properties can lead to problems with local optimization, resulting in a loss of critical performance in RCTPM. To enhance the connectivity between the decomposition factors, a circular update method is proposed, which changes the order in which the left singular vectors are computed. According to the decomposition method of Tensor-train of rank-negative-one and the nature of PM, \(v^{(n)}\) can be obtained from \(g^{(n+1)}\) and \(v^{(n+1)}\), which are reshaped to form \(v^{(n)}\). The relationship \(g^{(n)}\) is updated in the following way:\[v^{(n)}\approx\vartheta^{(n)}=\textit{reshape}\Big{(}d^{(n+1)}g^{(n+1)}v^{(n+1) }{}^{T},[I_{n+1}\cdots I_{N}]\Big{)} \tag{48}\] \[g^{(n)}=\frac{\mathcal{Y}_{(n)}\vartheta^{(n)}}{\left\|\mathcal{Y }_{(n)}\vartheta^{(n)}\right\|_{2}}\]  
The updated factor vector \(g^{(2)}\) can be derived as \(g^{(2)}=\frac{\mathcal{Y}_{(2)}\vartheta^{(2)}}{\left\|\mathcal{Y}_{(2)} \vartheta^{(2)}\right\|_{2}}\). The channel parameter is estimated for the UAV with millimeter-wave massive MIMO based on the FCTPM. The number of factor matrix time slots is associated with the structure of the transmission frame, using a downlink transmission frame structure. It consists of an AOA/AOD estimation phase (\(T_{1}\) time slot), a fast-fading factor estimation phase (\(T_{2}\) time slot), and the data transmission phase (\(T_{3}\) time slot). Using Equations (32) and (33), the received signal at the terrestrial user is represented as the following equation:  
\[\mathcal{Y}^{T_{2}}=\mathcal{T};U_{1},U_{2},U_{3}+\mathcal{N}^{T_{2}} \tag{49}\]  
Matrix \(\{U_{1},U_{2},U_{3}\}\) is the received signal at the user. \(\mathcal{Y}\) corresponds to the matrix of three factors. \(U_{2}\) is the corresponding fast-fading factor matrix. From the Tensor-train decomposition's _Kruskal_ representation, \(U_{n}\) and \(\mathcal{G}_{n}\), the correspondence can be expressed by the following equation:  
\[ \begin{array}{l}U_{n}=\left[\mathcal{G}_{n}\right]_{(2)}\Big{(}1^{T}_{R_{>( n)}}\bigotimes I_{R_{n-1}}R_{n}\bigotimes I^{T}_{R_{<n-1}}\Big{)},n=1,2,3\\ =\Big{[}g^{(n)}\Big{]}_{(2)}\Big{(}1^{T}_{R_{>(n)}}\bigotimes I_{R_{n-1}}R_{n} \bigotimes I^{T}_{R_{<n-1}}\Big{)},n=1,2,3\end{array}  \tag{50}\]  
The factor matrix \(U_{2}=\Big{[}g^{(n)}\Big{]}_{(2)}\Big{(}1^{T}_{R_{>(2)}}\bigotimes I_{R_{1}}R _{n}\bigotimes I^{T}_{R_{1}}\Big{)}\). From Equation (35), we can find that the factor matrix \(U_{2}\) has the fast-fading channel power fading factor associated with the Doppler shift in its elements. We can derive the fast-fading channel power factor. We can obtain \(U_{2}=\Big{[}\overset{\sim}{\kappa}_{1},\cdots\overset{\sim}{\kappa}_{Q}\Big{]} e\mathbb{C}^{T_{2}\times Q},\overset{\sim}{\kappa}_{Q}=\Big{[}\overset{\sim}{ \kappa}_{1}(1),\cdots\overset{\sim}{\kappa}_{Q}(T_{2})\Big{]}^{T}e\mathbb{C}^ {T_{2}\times 1}\).