# A Wideband Non-Stationary 3D GBSM for HAP-MIMO Communication Systems at Millimeter-Wave Bands
## 2 Channel Model
### Descriptions of 3D GBSM for HAP-MIMO Channel
i.e., [32]  
\[h_{pn}(t;\tau)=h_{pn,\text{LoS}}(t;\tau)+h_{pn,\text{NLoS}}(t;\tau), \tag{2}\]  
where  
\[h_{pn,\text{LoS}}(t;\tau)=\sqrt{\frac{K}{K+1}}e^{-j\frac{2\pi}{\lambda}d(p,n)} e^{j2\pi tf_{pn}^{\text{LoS}}(t)}\delta(\tau-\tau_{\text{LoS}}), \tag{3}\]  
\[h_{pn,\text{NLoS}}(t;\tau)=\sqrt{\frac{K}{K+1}}\frac{1}{\sqrt{N}}\sum_{i=1}^{ N}e^{-j\frac{2\pi}{\lambda}(d(p,n)+d(s,n))+j\varphi(i)}e^{j2\pi tf_{pn}^{\text{NLoS}}(t)} \delta(\tau-\tau_{\text{NLoS}}^{(n)}), \tag{4}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Meaning** \\ \hline \(v_{a},v_{b}\) & The velocity of TMS and SBS, respectively. \\ \hline \(\beta_{T}\) & The elevation angle of TMS relative to SBS. \\ \hline \(H\) & The height of the cylinder. \\ \hline \(R\) & The radius of the cylinder. \\ \hline \(S_{n}\) & The position of the \(n\)th scatterer. \\ \hline \(\psi_{R}\) & The elevation angle of the TMS antenna element in the x–y plane. \\ \hline \(\theta_{T},\theta_{R}\) & The antenna orientations of TMS and SBS, respectively. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 1: 3D GBSM with LoS and NLoS components for HAP-MIMO channel. where \(K\) and \(\lambda\) denote the Ricean factor and carrier wavelength, respectively; \(d(p,n)\) denotes the distance between \(p\) and \(n\); \(f_{pn}^{\text{LoS}}(t)\) and \(f_{pn}^{\text{NLoS}}(t)\) denote the Doppler frequency of the LoS components and NLoS components, respectively; \(\varphi(n)\) denotes the phase introduced by the \(n\)th scatterer; \(\tau_{\text{LoS}}\) and \(\tau_{\text{NLoS}}^{(n)}\) are the travel times of the LoS and NLoS waves, respectively; \(\delta\) denotes the Dirac delta function. The Doppler frequency \(f_{pn}^{\text{LoS}}(t)\) and \(f_{pn}^{\text{NLoS}}(t)\) can be written as [32]  
\[f_{pn}^{\text{LoS}}(t)=f_{Tmax}cos(\pi-\alpha_{\text{LoS}}(t)-\gamma_{T})+f_{ Rmax}cos(\alpha_{\text{LoS}}(t)-\gamma_{R}), \tag{5}\]  
\[f_{pn}^{\text{NLoS}}(t)=f_{Tmax}cos(\gamma_{T})+f_{Rmax}cos(\alpha_{n}(t)- \gamma_{R})cos\beta_{n}(t), \tag{6}\]  
where \(f_{Tmax}=v_{a}/\lambda\) and \(f_{Rmax}=v_{b}/\lambda\) denote the maximum Doppler frequencies associated with SBS and TMS, respectively. \(\alpha_{\text{LoS}}(t)\) denotes the azimuth angle of arrival (AAoA) of LoS components at the TMS. \(\alpha_{n}(t)\) and \(\beta_{n}(t)\) denote the azimuth angle of departure (AAoD) from the scatterer and the elevation angle of departure (EAoD) from the scatterer. In the HAP channel, during the propagation of signals, electromagnetic waves are reflected, scattered, and diffracted due to factors such as buildings, trees, vegetation, terrain, and sea surface along the propagation path. This results in the signal reaching the receiving end being mixed with echoes propagated from multiple paths. The arrival time of each component of the multipath field at the receiving end is different, and multipath signals of different phases overlap with each other, causing interference to the signal and distorting the original signal. Errors may occur. Therefore, it is necessary to study the impact of this multipath effect on the channel and model the multipath effect accordingly. The number of time-varying multipath is represented by \(N(t)\). The time-varying number of multipath can be represented by a Poisson distribution for modeling the number of multipath. Its probability density function can be written as  
\[P_{\text{Poisson}}(N(t))=\frac{\rho^{N(t)}}{N(t)!}\text{e}^{-\rho}, \tag{7}\]  
where \(\rho\) denotes the mean of \(N(t)\). The delay of the \(k\)-th path is represented by the \(\tau_{k}(t)\), where the LoS path delay is between the HAP and the ground station. The NLoS path component delay can be obtained based on the increment of the LoS path delay, and its distribution is affected by the communication scenario and elevation angle.