## 3 Analytical Non-Stationary S2A Channel Model
### Time-Frequency ACFs  
With the assumption that the LOS path, the specular reflection path, and the diffuse scattering paths are uncorrelated with each other, the ACFs can be derived through the time variant transfer functions. With time difference denoted as \(\Delta t\) and frequency difference denoted as \(\Delta f\), the time-frequency correlation function can be expressed as Equation (29), where \(R_{\text{LOS}}\left(t,\Delta t;f,\Delta f\right)\), \(R_{\text{SPE}}\left(t,\Delta t;f,\Delta f\right)\) and \(R_{\text{DIF}}\left(t,\Delta t;f,\Delta f\right)\) are the time-frequency ACFs of the LOS component, the specular reflection component, and the diffuse scattering components, respectively, as shown in Equations (30)-(32). \[R\left(t,\Delta t;f,\Delta f\right)=R_{\text{LOS}}\left(t,\Delta t;f,\Delta f \right)+R_{\text{SPE}}\left(t,\Delta t;f,\Delta f\right)+R_{\text{DIF}}\left(t,\Delta t;f,\Delta f\right) \tag{29}\]  
\[R_{\text{LOS}}\left(t,\Delta t;f,\Delta f\right) =E\left\{H_{\text{LOS}}^{\ast}\left(t,f\right)\cdot H_{\text{ LOS}}\left(t+\Delta t,f+\Delta f\right)\right\}\] (30) \[=E\left\{\alpha_{\text{LOS}}\left(t\right)\cdot\exp\left(\mathrm{ j}2\pi f\tau_{\text{LOS}}\left(t\right)\right)\exp\left(-\mathrm{j}k_{0}d_{\text{ LOS}}\  
\[R_{\text{DBF}}\left(t,\Delta t;f,\Delta f\right)=E\left\{H_{\text{DBF}} \left(t,f\right)\cdot H_{\text{DBF}}\left(t+\Delta t,f+\Delta f\right)\right\} \tag{32}\] \[=E\left\{\left[\sum_{l=1}^{L}\frac{\lambda\sqrt{C_{\text{TJ}}\left( t\right)G_{\text{RJ}}}\left(t\right)}{4\pi d\tau_{\text{DJ}}d_{\text{RDJ}}} \right]\exp\left(\left|2\pi f\tau_{\text{DBF}}\left(t\right)\right|\cdot\exp \left(-\mathrm{i}\eta_{l}\left(t\right)\right)\cdot\exp\left(-\mathrm{i}\omega_ {d}d_{\text{DBFJ}}\left(t\right)\right)\right.\] \[\left.\exp\left(-\mathrm{i}\frac{2\pi}{\lambda}\int_{0}^{t} \left(\mathbf{n}_{\text{TDJ}}^{\text{T}}\left(t\right)\cdot\mathbf{v}_{\text {T}}\left(t\right)+\mathbf{n}_{\text{RDJ}}^{\text{T}}\left(t\right)\cdot \mathbf{v}_{\text{R}}\left(t\right)\right)\mathrm{d}t\right)\right]\] \[\left[\sum_{l=1}^{L}\frac{\lambda\sqrt{C_{\text{TJ}}\left(t+ \Delta t\right)G_{\text{RJ}}+\Delta t}}{4\pi d_{\text{TDJ}}d_{\text{RDJ}}} \cdot\exp\left(-\mathrm{i}2\pi\left(f+\Delta f\right)\tau_{\text{DBFJ}}\right) \cdot\exp\left\{\phi_{l}\left(t\right)\cdot\exp\left\{\phi_{l}\left(t+\Delta t \right)\right\}\right.\] \[\left.\exp\left(\mathrm{i}k_{0}d_{\text{DBF}}\left(t+\Delta t \right)\right)\right.\] \[\left.\exp\left(\mathrm{i}\frac{2\pi}{\lambda}\int_{0}^{t+\Delta t }\left(\mathbf{n}_{\text{TDJ}}^{\text{T}}\left(t\right)\cdot\mathbf{v}_{ \text{T}}\left(t\right)+\mathbf{n}_{\text{RDJ}}^{\text{T}}\left(t\right) \cdot\mathbf{v}_{\text{R}}\left(t\right)\right)\mathrm{d}t\right)\right]\right\}\]  
With the assumption that the reflection coefficient \(\boldsymbol{\Gamma}\), delay \(\tau\), distance \(d\), additional phase shift \(\varphi\), antenna gain \(G\), velocity \(\mathbf{v}\), and etc. are time independent, the scattering channel satisfies the feature of wide sense stationary uncorrelated scattering (WSSUS) and the auto-correlation functions of each paths can be simplified as Equations (33)-(35).