## 2 The Proposed Non-Stationary Model for High-Speed Train mmWave Band Channel
Deducing \(\min\{R_{R},a-f\}\gg\max\{\delta_{T},\delta_{R}\}\),\(D\gg R_{R}\) and \(\sqrt{1+x}\approx 1+x/2\), we have  
\[\varepsilon\approx\xi-\frac{\delta_{R}}{2\xi}[\frac{\delta_{T}}{2}\sin\varphi_ {T}\sin\varphi_{R}-Q\cos\varphi_{T}\cos\varphi_{R}] \tag{4a}\] \[\varepsilon_{pn_{l,1}}\approx\xi_{n_{l,1}}-\frac{\delta_{T}}{2\xi_{n_{l,1}}}[R _{r}\sin\beta_{R}^{(n_{l,1})}\sin\varphi_{T}+Q_{n_{l,1}}\cos\varphi_{T}\cos(a _{T}^{(n_{l,1})}-\theta_{T})]\] (4b) \[h_{1,pq}(t)=h_{1,pq}^{Los}(t)+\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t) \tag{1}\]0 (4c) \[h_{1,pq}(t)=h_{1,pq}^{Los}(t)+\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t) \tag{1}\]1 (4d) \[h_{1,pq}(t)=h_{1,pq}^{Los}(t)+\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t) \tag{1}\]2 (4e) \[h_{1,pq}(t)=h_{1,pq}^{Los}(t)+\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t) \tag{1}\]3 (4f) \[h_{1,pq}(t)=h_{1,pq}^{Los}(t)+\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t) \tag{1}\]4 (4g) \[h_{1,pq}(t)=h_{1,pq}^{Los}(t)+\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t) \tag{1}\]5 \[h_{1,pq}(t)=h_{1,pq}^{Los}(t)+\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t) \tag{1}\]6  
with \(b_{l}\) denoting the semi-minor axis of the \(lth\) elliptic-cylinder. The azimuth angle of departure (AAoD)/elevation angle of departure (EAoD) (i.e., \(\alpha_{T}^{(n_{l,j})},\beta_{T}^{(n_{l,j})}\)) and azimuth angle of arrival (AAoA)/elevation angle of arrival (EAoA) (i.e., \(\alpha_{R}^{(n_{l,l})},\beta_{R}^{(n_{l,j})}\)) for the SB rays are correlated. From the sphere model, the derivation of the relationship between angle of arrival (AoA) and the angle of departure (AoD) SB components are: \(\alpha_{T}^{(n_{l,1})}\approx\frac{R_{R}}{D}\sin\alpha_{R}^{n_{l,1}}\), \(\beta_{T}^{(n_{l,1})}\approx\arccos\left(\frac{D+R_{r}\cos\beta_{R}^{(1,1)}\cos a _{R}^{1,1}}{\xi_{(n_{l,1})}}\right)\). The angular relationships for SB rays de rived from the elliptic cylinder model are: \(\alpha_{T}^{(n_{l,2})}=\arcsin\left(\dfrac{b_{i}^{2}\sin\alpha_{R}^{(n_{l,2})}}{a_ {l}^{2}+f^{2}+2a_{i}f\cos\alpha_{R}^{(n_{l,2})}}\right)\), and  
\[h_{1,pq}(t)=h_{1,pq}^{Los}(t)+\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t) \tag{1}\]7  
where \(\alpha,\beta\in(-\pi,\pi)\alpha_{0}\in(-\pi,\pi)\) account for the mean values of the azimuth angle \(\alpha\) and the elevation angle \(\beta\), respectively, and \(k(k\geq 0)\) is a real valued parameter that controls the concentration of the distribution as identified by the mean direction \(\alpha_{0}\) and \(\beta_{0}\). In this case, for angular descriptions, i.e., the AAoA \(\alpha_{R}^{(1,1)}\) and EAoA \(\beta_{R}^{(1,1)}\) for the sphere and the AAoA \(\alpha_{R}^{(l,2)}\) and EAoA \(\beta_{R}^{(l,2)}\) for the multiple elliptic cylinders, the parameters (\(\alpha_{0}\), \(\beta_{0}\), and \(k\)) of the VMF PDF are in \((\alpha_{R0}^{(l,1)}\), \(\beta_{R0}^{(i,1)}\), and \(k^{(l,1)}\)) and \((\alpha_{R0}^{(l,2)}\), \(\beta_{R0}^{(l,2)}\), and \(k^{(l,2)}\)), respectively. By considering the scenario of the line of sight (LoS) component, the time varying AAoA \(\left(\alpha_{R}^{LoS}(t)\right)\) and EAoA \(\left(\beta_{R}^{LoS}(t)\right)\) can be expressed as  
\[\alpha_{R}^{LoS}(t)= \begin{cases}\alpha_{R}^{LoS}(t_{0})+\arccos(\frac{D_{s}( t_{0})+v_{R}t\cos\gamma_{R}}{D_{s}(t)}),&-\pi\leq\gamma_{R}\leq 0\\ \alpha_{R}^{LoS}(t_{0})+\arccos(\frac{D_{s}(t_{0})+v_{R}t\cos\gamma_{R}}{D_{s}( t)}),&0\leq\gamma_{R}\leq\pi\end{cases}  \tag{6}\]  
where \(\alpha_{R}^{LoS}(t_{0})=\arcsin(\frac{k_{p}\delta_{T}}{D_{s}(t_{0})}\sin\varphi _{R})\) denotes the initial LoS AOA at time \(t=t_{0}\).