# A Wideband Non-Stationary 3D GBSM for HAP-MIMO Communication Systems at Millimeter-Wave Bands
## 2 Channel Model
### Birth-Death Process  
The clusters in the HAP-MIMO channel represent groups of signal paths that result from reflections, diffractions, and scattering phenomena. These clusters exhibit dynamic characteristics due to the movement of the SBS and the changing positions of scatterers in the environment. The Markov birth-death process is a stochastic model that describes the evolution of a system where elements can either \"birth\" or \"die\" over time, with transition probabilities between different states. To be able to talk about a Markov process, the three properties of stationarity, regularity, and independence, on which the derivation of the characteristics is based, should be satisfied. Stability refers to the fact that the probability of a system's state transition remains constant over time. We assume that the statistical properties of the HAP channel are stable over a long time scale, meaning that the statistical properties of the channel do not change over time. Regularity requires that the state transition probability of the system is finite and non-zero. In HAP channel modeling, we assume that there is a non-zero probability of transitions between channel states, and these probabilities are finite. Independence refers to the fact that the future state of a system depends only on the current state and is independent of past states. In HAP channel modeling, we assume that the transition of channel states is independent of previous states, meaning that future channel states only depend on the current channel state. In the context of HAP-MIMO channel modeling, the clusters are represented as states in the Markov process, and the birth and death events correspond to the creation or disappearance of clusters as the wireless environment changes [36]. The variability of the wireless channel over time primarily arises from the mobility of the TMS and the SBS. As these entities move, the wireless channel experiences change due to variations in path loss, shadowing, and interference conditions. Such dynamic characteristics contribute to the time-varying nature of the wireless channel in this context. Therefore, the time-varying channel fluctuations caused by the motion of the above two during the period from \(t\) to \(t+\Delta t\) can be expressed as [19]  
\[\delta_{P}(t,\Delta t)=\delta_{\text{SBS}}(t,\Delta t)+\delta_{\text{TMS}}(t, \Delta t), \tag{10}\]  
where \(\delta_{\text{SBS}}(t,\Delta t)\) and \(\delta_{\text{TMS}}(t,\Delta t)\) represent time variation caused by SBS and TMS, respectively, and can be defined as  
\[\delta_{\text{SBS}}(t,\Delta t)=\int_{t}^{t+\Delta t}v_{\text{SBS}}(t)dt, \tag{11}\]  
\[\delta_{\text{TMS}}(t,\Delta t)=\int_{t}^{t+\Delta t}v_{\text{TMS}}(t)dt. \tag{12}\]  
Due to the short time interval, we assume that SBS and TMS are in uniform motion. The above two equations can be simplified as [19]  
\[\delta_{\text{SBS}}(t,\Delta t)=v_{\text{SBS}}(t)\times\Delta t, \tag{13}\]  
\[\delta_{\text{TMS}}(t,\Delta t)=v_{\text{TMS}}(t)\times\Delta t. \tag{14}\]  
Therefore, \(\delta_{P}(t,\Delta t)\) gives the degree of correlation between CIR at different time instants. We introduce a Markov birth-death description that leads to the time-varying number \(N(t)\) of clusters for CIR realizations. It can be divided at any time into newly generated and previously existing clusters. The Markov process can be described by the generation rate of clusters (\(\lambda_{G}\)) and the recombination rate of clusters (\(\lambda_{R}\)). The expectation of the total scatterer number is given by [19]  
\[E\{N(t)\}=N(t_{0})=\frac{\lambda_{G}}{\lambda_{R}}. \tag{15}\]  
Each cluster survives from one CIR at \(t\) to the next one at \(t+\Delta t\) with the following probability:  
\[P_{sur}(\Delta t)=e^{-\lambda_{R}\frac{\delta_{P}(t,\Delta t)}{D_{c}}}, \tag{16}\]  
where \(D_{c}\) is the related factor that changes with the scenario.