## 2 The Proposed Non-Stationary Model for High-Speed Train mmWave Band Channel
Therefore, from the equation  
\[h_{1,pq}(t)=h_{1,pq}^{Los}(t)+\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t) \tag{1}\]  
where  
\[h_{1,pq}^{Los}(t)=\sqrt{\frac{K_{pq}}{K_{pq}+1}}e^{-2\pi f_{c}\tau_{pq}(t)}e^{ 2\pi f_{max}t\cos(\kappa_{R}^{Los}-\tau_{R})}\cos\beta_{R}^{Los} \tag{2a}\] \[h_{1,pq}(t)=\sqrt{\frac{\eta_{SB_{1,l}}}{K_{pq}(t)+1}}\lim_{N_{1,l}\to \infty}\sum_{n_{1,l}=1}^{N_{1,l}}\frac{1}{\sqrt{N_{1,l}}}e^{j(\varphi_{n_{1}}- 2\pi f_{c}\tau_{pq}\varphi_{n_{1}(l)})}x^{2\pi f_{max}t\cos(\kappa_{R}^{H}(t) -\tau_{R})\cos\beta_{R}^{(\mu_{ij})}} \tag{2b}\]  
The parameters \(\tau_{pq}(t)\) and \(\tau_{pq,n_{ij}}(t)(l=1,\dots,L)\), are time varying. This makes the underlying GBSM a non-stationary one. \(\eta_{SB_{1,l}}\) designates the mean power for the \(lth\) tap, \(\tau_{pq}(t)=\epsilon_{pq}(t)/c\), and \(\tau_{pq,n_{1,l}}(t)=(\epsilon_{pn1,i}(t)+\epsilon_{n,l_{iq}}(t))/c\) links \(T_{p}\to R_{q}\) and \(T_{p}\to s^{n_{1,l}}\to R_{q}\), respectively. The \(k\) and \(c\) represent the Rician K-factor and speed of light, respectively. The parameters related to energy are \(n_{SB_{1,l}}\), which define the percentage contribution of single-bounce rays to the overall scattered power of the first tap. It is worth noting that these energy-related characteristics have been standardized to meet the requirements \(\sum_{l=1}^{l}n_{SB_{1,l}}=1\). Figure 3: Proposed GBSM for a one sphere and multiple elliptic cylinder model. The green solid line represents tap one, the blue solid line represents tap two, and the red dashed line represents the MRS sphere. For other taps, the complex tap coefficients \((i>1)\) of only SB components make up the link \(T_{p}\to R_{q}\). The other taps consist of the single-bounce component from the elliptic cylinder, which is also the multiple confocal elliptic cylinder. This may be written as  
\[h_{l,pq}(t)=h_{l,pq}^{SB_{2}}(t) \tag{3a}\] \[=\sqrt{\eta_{SB_{l,2}}}\lim_{N_{l,2}\to\infty}\sum_{n_{l,2}=1}^{N_{l,2}}\frac{1 }{\sqrt{N_{l,2}}}e^{(\psi_{n_{l,2}}-2\pi f_{\zeta}\tau_{p}\psi_{n_{l,2}})} \times e^{j2\pi f_{max}t\cos(a_{R}^{(n_{l,2})}(t)-\gamma_{R})\cos\bar{\rho}_{R} ^{n_{l,2}}} \tag{3b}\]  
where \(\tau_{pq,n_{l,2}}=(\varepsilon_{pn_{l,2}}+\varepsilon_{n_{l,2}})/c\) is the time it takes for waves to travel via the link \(T_{p\to s}s^{(n_{l,2})}\to T_{q}\). For other taps, parameters related to power are \(\eta_{SB_{l,2}}\) which define the percentage contribution of single-bounce rays to the overall scattered power of the other taps, and it satisfies \(\eta_{SB_{l,2}}=1\). The phases \(\psi_{n_{l,2}}\) and \(\psi_{n_{l,j}}\), i.i.d. are random variables that are uniformly distributed over \([-\pi,\pi]\), while the maximum Doppler frequency relative to the mobile station is denoted by \(f_{max}\). In the case of the first tap, SB rays are generated from the scatterers on either the sphere or the first elliptic cylinder on addition to the los component, as shown in Figure 3. For additional taps, the SB rays components are assumed to be generated from the scatterers located on the corresponding elliptic cylinder. The underlying assumptions as in [17] are based on the application of the cosine laws in triangles (i.e., \(\sin x\approx x\) and \(\cos x\approx 1\) for small \(x\)).