# A Framework for Assessing the Interference from NGSO Satellite Systems to a Radio Astronomy System
## 3 Framework of the Interference Analysis
### Space-Time Distribution of Satellite EIRP
#### 3.1.1 Generating the CDFs of EIRP  
To generate the CDFs for satellites' EIRP, we need to model the spatial distribution of user terminals. There are different approaches that can be used to model the user distribution, depending on the specific characteristics of the distribution. In order to facilitate the interference analysis, this study assumed that users were uniformly distributed within the satellite coverage, which meant that the satellite beam was directed towards any point within the coverage area with equal probability. The coverage diagram of the satellite is depicted in Figure 4, where the boundary corresponds to the minimum elevation angle \(e_{min}\) and \(E\) represents a point on the edge. Figure 3: The flowchart for evaluating the harmful interference to an RAS telescope from an NGSO satellite system. The beam pointing of the NGSO satellite is subject to a uniform distribution on the cap. For a point A inside the cap, it can be expressed using \(\vartheta_{A}\) and \(\varphi_{A}\). Then, the probability that the geocentric angle between the NGSO satellite and point \(A\) is less than \(\vartheta_{A}\) can be expressed as the following formula:  
\[P(\vartheta\leq\vartheta_{A})=\frac{2\pi R_{e}^{2}(1-cos\vartheta_{A})}{2\pi R _{e}^{2}(1-cos\vartheta_{E})}. \tag{22}\]  
Since \(\varphi\) is subject to a uniform distribution in the range \([0,2\pi]\), its probability density function is \(\frac{1}{2\pi}\). According to Equation (22), the probability density function is derived by taking its derivative. Therefore, the expression for the joint probability density function of \(\vartheta_{A}\) and \(\varphi_{A}\) is as follows:  
\[p(\vartheta_{A},\varphi_{A})=\frac{sin\vartheta_{A}}{2\pi(1-cos\vartheta_{E} )}. \tag{23}\]  
According to the geometry in Figure 4, the expressions of the elevation and azimuth angles regarding point \(A\) are derived below,  
\[El_{A}^{sat}=\frac{\pi}{2}-arccos(\frac{sin\vartheta_{A}cos\varphi_{A}}{\sqrt {sin^{2}\vartheta_{A}+\left[(1-cos\vartheta)+\frac{h}{R_{e}}\right]^{2}}}), \tag{24}\]  
\[Az_{A}^{sat}=arccos(\frac{(1-cos\vartheta)+\frac{h}{R_{e}}}{\sqrt{sin^{2} \vartheta_{A}sin^{2}\varphi_{A}+\left[(1-cos\vartheta)+\frac{h}{R_{e}}\right] ^{2}}}). \tag{25}\]  
For the cell \(G_{ij}^{sat}\), the elevation and azimuth angles corresponding to the centre of the cell are \((i+0.5)\Delta El-El_{max}\) and \((j+0.5)\Delta Az-Az_{max}\), respectively. If the satellite beam is directed towards point A, the EIRP expression at the centre of \(G_{ij}^{sat}\) is denoted below,  
\[EIRP_{ij}^{A}=P_{RAS}G_{t}(\omega_{ij}(El_{A}^{sat},Az_{A}^{sat}))\Delta G(El_ {A}^{sat},Az_{A}^{sat}) \tag{26}\]  
Figure 4: The coverage diagram of an NGSO satellite. where \(\omega_{ij}(El_{A}^{sat},Az_{A}^{sat})\) is the off-axis angle with respect to cell \(G_{ij}^{sat}\) and \(\Delta G(El_{A}^{sat},Az_{A}^{sat})\) is the compensation of the beam, which is expressed as  
\[\Delta G(El_{A}^{sat},Az_{A}^{sat})=\frac{Re^{2}(sin^{2}\theta_{A}+\left[(1- cos\theta)+\frac{h}{K_{e}}\right]^{2})}{h^{2}}. \tag{27}\]  
The expression of the off-axis angle \(\omega_{ij}(El_{ij}^{sat},Az_{ij}^{sat})\) is shown as follows:  
\[\omega_{ij}(El_{ij}^{sat},Az_{ij}^{sat})=arccos(sinEl_{ij}^{sat}sinEl_{A}^{ sat}+cosEl_{ij}^{sat}cosEl_{A}^{sat}cos(Az_{A}^{sat}-Az_{ij}^{sat})). \tag{28}\]  
By taking a large number of samples within the cap of the satellite coverage according to the probability density function, we were able to obtain a cumulative probability density function of the satellite's radiated power within each cell. \(\theta_{A}\) and \(\varphi_{A}\) of the NGSO satellite can be seen as mutually independent random variables. Samples could be drawn according to their probability density functions. The samples of \(\varphi_{A}\) were drawn subject to a uniform distribution in \([0,2\pi]\), denoted as \(\{\varphi_{A,1},\varphi_{A,2},\cdots,\varphi_{A,N}\}\), where \(N\) is the number of samples.