# 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  
Typically, geometric modeling methods based on ellipsoids, hemispheres, and cylinders are used [33; 34; 35]. However, the ellipsoid model is only suitable for situations where the receiver is higher than the scatterers. The channel model based on hemispheres usually assumes that the elevation and azimuth angles of the propagation path are correlated. These models can only be applied to specific scenarios and are not perfectly applicable to HAP environments. Nevertheless, the model based on cylinders can adapt to open channel environments and is particularly suitable for situations with high transmitting ends. Therefore, we choose the cylinder model for the HAP channel. This paper considers a downlink channel with \(n_{T}\) transmit antennas and \(n_{R}\) receive antennas in SBS and TMS, respectively. As shown in Figure 1, we propose a 3D GBSM for HAP-MIMO channel and demonstrate its line-of-sight (LoS) and non-line-of-sight (NLoS) paths, which is the foundation for constructing a uniform linear array with any number of antennas. The two connecting lines between HAP and points \(m\) as well as \(n\), represent the LoS path. The two lines that reach the midpoint \(S\) represent the NLoS path. The height of the cylinder is \(H\), and the radius is \(R\). We assume that there is no scatterer in the SBS, and the SBS moves at a speed of \(v_{a}\). \(S_{n}\) (\(n=1,2,\ldots,N\)) denotes scatterers located on the surface of the cylinder. Scatterers and the TMS move together at a speed of \(v_{b}\). \(\delta_{T}\) and \(\delta_{R}\) denote the distance between two adjacent antenna elements at the TMS and the SBS, respectively, angles \(\psi_{R}\) denotes the elevation angle of the TMS antenna element in the x-y plane, angles \(\theta_{T}\) and \(\theta_{R}\) describe the antenna orientations. The heights of \(O_{T}\) and \(O_{R}\) are \(H_{T}\) and \(H_{R}\), respectively, and the projection distance between them on the x-y plane is \(D\). The elevation angle of \(O_{T}\) relative to \(O_{R}\) is \(\beta_{T}\),which is approximately equal to \(\arctan(H_{T}/D)\). For the ease of reference, we summarize the parameters of the proposed model in Table 1.  
We assume that both the receiving and transmitting ends utilize uniformly linear arrays for their antenna configurations. Therefore, the channel between the HAP and the TMS can be modeled as  
\[H(t;\tau)=\left[ \begin{array}{cccc}h_{1,1}(t;\tau)&\ldots&h_{1,n_{R}}(t;\tau )\\ \vdots&\ddots&\vdots\\ h_{n_{T},1}(t;\tau)&\ldots&h_{n_{T},n_{R}}(t;\tau)\end{array} \right]_{n_{T} \
eq n_{R}}, \tag{1}\]  
where the matrix element \(h_{n_{T}n_{R}}(t;\tau)\) represents the channel impulse response (CIR) of the subchannel between the \(n_{R}\)-th receive antenna and the \(n_{T}\)-th transmit antenna. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Meaning** \\ \hline \(n_{T}\), \(n_{R}\) & The number of transmit antennas and receive antennas, respectively. \\ \hline \(H_{T}\), \(H_{R}\) & The height of TMS and SBS, respectively. \\ \hline \(\delta_{T}\), \(\delta_{R}\) & The spacing between two adjacent antenna elements at the TMS and the SBS, respectively. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definition of the parameters in the 3D HAP-MIMO model. The general form of the S-T communication channel follows a Rice distribution, which effectively describes the statistical characteristics of the channel. Considering the geometric model depicted in Figure 1, it is evident that the channel impulse response is a result of the superposition of LoS components and NLoS components.