## 2 The Proposed Non-Stationary Model for High-Speed Train mmWave Band Channel  
We consider the architecture which consists of a mobile relay station on the vehicle and the base station on the side of the track. The two stations for outdoor communication consists of MIMO antennas with S receive and U transit antenna elements, respectively. The BS and MRS are separated by a time-varying distance of \(D_{s}(t)\). The GBSM suggested in [15] as shown in Figure 3 is made up of components from the line of sight, the single bounce from the sphere, and the multiple confocal rays from the elliptic cylinder. The model consists of a sphere and an elliptic cylinder representing the environment scatterers. We adopt a \(2\times 2\) MIMO channel model as an illustration for clarification as an example. The proposed 3D model could be extended to a massive MIMO in a different plane. The first and other taps are represented using the tapped delay line (TDL) Structure. For the first tap, we suppose that around the sphere with radius \(R_{r}\), the receiver is located in the center. There are \(n_{1,1}th(n_{1,1}=1,2,\ldots,N_{1,1})\) scatterers on the sphere, whereas an effective scatterer is defined by \(s^{n_{1,1}}\). The elliptic cylinder is represented by the multiple confocal elliptic-cylinder models using a time delay line structure.The transmitter Bs and the receiver MRs are located at the two foci, respectively.The scatterers are located on the \(1 th\) elliptic-cylinder, where \(l=1,2,\ldots,L(t)\). \(L(t)\) represents the time-variant total number of taps. The \(n_{1,2}th\) scatterer on the semi major axis \(a_{l}\) of the \(lth\) elliptic cylinder is defined by \(s^{n_{2}}\). The time-varying semi-minor axis of the \(l\)th tap (elliptic cylinder) is denoted by \(b_{l}(t)=\sqrt{a_{l}^{2}(t)-f_{l}^{2}(t)}\). In this case, the half length between the two foci is represented by \(f_{l}^{2}(t)\), which is also equal to \(D_{s}(t)/2\).The MRs and the BS have been tilted at an angle \(\theta_{R}\) and \(\theta_{T}\), whereas the elevation of the antenna array is represented by \(\varphi_{R}\) and \(\varphi_{T}\), respectively.The moving direction and the velocity of the MRs denoted by\(\gamma_{R}\) and \(v_{R}\) respectively. The elevation angle of arrival and the azimuth angle of arrival of the LOS component are represented by \(\beta_{R}^{Los}\) and \(\alpha_{R}^{Los}\), respectively. The azimuth angle of departure and elevation angle of departure are denoted by \(\alpha_{T}\) and \(\beta_{T}\), respectively. Figure 2: The cutting environment. Based on [12], the channel impulse response \(h_{pq}(t,\tau)=\sum_{l=1}^{L}(t)h_{l,pq}(t)\delta(\tau-\tau_{l})\) where the complex space-time-variant tap coefficients \(h_{l,pq}(t)\) and discrete propagation delay \(\tau\) of the \(l\)th tap have been represented. This is obtained from the superposition of the components from the LoS, sphere and the elliptic cylinder. The multipath components from the sphere and elliptic cylinder are also known as single bounce components (SB) represented by \(\sum_{i=1}^{l}h_{1,pq}^{SB_{i}}(t)\), where \(i\) is the component from either the sphere \(i=1\) or from the elliptic cylinder \(i=2\). The spacing of the transmitter (Tx) and receiver (Rx) antenna elements are denoted by \(\delta_{T}\) and \(\delta_{R}\), respectively. The half length distance between the foci of the ellipse is denoted by \(f\). Therefore, \(2f\) represents the distance \(D\) between the Tx and Rx. In this theoretical model, some assumptions, such as \(\min\{R_{r},a-f\}\gg\), \(\max\{\delta_{T},\delta_{R}\}\) and \(D\gg R_{R}\), have been used. From Equation (1), the first tap consists of the LOS component and the summation of the single-bounce component from the sphere and the first single-bounce component from the elliptic cylinder.