# A Framework for Assessing the Interference from NGSO Satellite Systems to a Radio Astronomy System
## 2 Model Description
### EPFD at the RAS Telescope  
To calculate the interference of an NGSO fixed satellite service to an RAS telescope, the ITU-R M.1583 [26] gives a method based on the EPFD. Ensuring that the RAS telescope is not affected by harmful interference from NGSO satellite systems in any observation direction, the EPFD needs to be evaluated when the radio telescope is pointed in any direction. The model of the interference from the NGSO constellation to the RAS telescope is shown in Figure 1.  
In broadband LEO constellation systems, satellites often use phased-array technology to provide internet access to ground users. These satellite beams provide access to users in a time-division manner. In order to simplify the analysis, we assumed that the number of NGSO satellite beams with the same frequency was 1, and that the user terminals were uniformly distributed on the ground, with their beams providing service to users in their coverage in an equal probability manner. The instantaneous EPFD for a single NGSO satellite can be expressed as follows:  
\[EPFD_{i}(t)=\frac{P_{t}G_{t}(\theta_{i}(t))G_{t}(\phi_{i}(t))\Delta G(d^{ \prime}_{i}(t))}{4\pi d^{2}_{i}(t)} \tag{1}\]  
where \(P_{t}\) denotes the transmitting power of the NGSO satellite when its beam is pointed at the nadir, and \(\theta_{t}\) and \(\phi_{t}\) denote the off-axis angle of the satellite beam along the RAS telescope direction and the angle between the RAS observation direction and the direction of the NGSO satellite \(S_{t}\), respectively. \(d_{i}(t)\) and \(d^{\prime}_{i}(t)\) represent the distance between the NGSO satellite \(S_{i}(t)\) and the RAS telescope, and the distance from the NGSO satellite \(S_{i}(t)\)  
Figure 1: Interference from NGSO constellation to RAS telescope.  
to its corresponding terminal, respectively. \(G_{t}\) and \(G_{r}\) denote the NGSO satellite transmit antenna gain and the RAS receive antenna gain, respectively. \(\Delta G(d_{t}^{{}^{\prime}}(t))\) is the compensation gain that maintains the same receive power of the terminal for longer paths as in the case of the subsatellite, which is given by the following equation:  
\[G(d_{t}^{{}^{\prime}}(t))=\frac{d_{t}^{{}^{\prime}}(t)}{h^{2}} \tag{2}\]  
where \(h\) represents the orbit height of the NGSO satellite. If we assume that the number of interfering NGSO satellites visible to the RAS telescope at time step t is \(N_{t}\), and denote the set they constitute as \(C_{t}\), then the total EPFD that the RAS telescope experiences due to interference from all of these satellites within its field of view can be expressed as follows:  
\[EPFD(t)=\sum_{i=1}^{N_{t}}\frac{P_{i}G_{t}(\theta_{i}(t))G_{r}(\phi_{i}(t)) \Delta G(d_{t}^{{}^{\prime}}(t))}{4\pi d_{t}^{2}(t)}. \tag{3}\]  
After the EPFD at time \(t\) has been calculated, it can be integrated over a period of \(\Delta t\). The resulting average EPFD corresponding to the RAS observation can then be calculated as follows:  
\[\overline{EPFD}(t)=\frac{1}{\Delta t}\int_{t_{s}}^{t_{s}+\Delta t}\sum_{S_{i} (t)\in C_{t}}\frac{P_{t}G_{t}(\theta_{i}(t))G_{r}(\phi_{i}(t))\Delta G(d_{t}^ {{}^{\prime}}(t))}{4\pi d_{t}^{2}(t)}dt. \tag{4}\]  
where \(\Delta t\) is the observation time of the RAS telescope. For a single observation, we need to randomly select multiple starting times of the integration for the average EPFD calculations. The more times we pick, the more accurate the simulation results will be. Then, the average EPFD cumulative probability density distribution of the system under this observation is counted. Assume that after \(K\) iterations, the probability of the average EPFD is less than \(l\) and can be approximated by the following expression,  
\[P(\overline{EPFD}(t)<l)\approx\frac{1}{K}\sum_{k=1}^{K}I(\overline{EPFD}_{k}( t)<l), \tag{5}\]  
where \(\overline{EPFD}_{k}(t)\) is the average EPFD in the \(k\)th simulation and \(l\) is an indicator function. The value of \(l\) is 1, if \(\overline{EPFD}_{k}(t)\) is less than \(l\). Otherwise, the value of \(l\) is 0.