## 4 Interference Analysis
### Mathematical Models for Interference Calculation
#### 4.2.2 Adjacent Satellite Interference (C/ASI)  
In satellite communication, adjacent satellite interference can significantly impact the quality of satellite signals. This discussion aims to shed light on both aspects of adjacent interference to estimate their cumulative effect.  
ITU Recommendation ITU-RS.465-6 offers valuable guidelines for determining the off-axis gain pattern [19; 20]. This gain pattern is represented by the equation G(\(\theta\)) = 29-25 log(\(\theta\)) dBi. For example, considering the case of ES1 and for \(\theta\) = 2\({}^{\circ}\), G(2\({}^{\circ}\)) will correspond to the gain value at 2\({}^{\circ}\) angle from the primary line joining ES1 to S2.  
In our case, the ASI on S2 comes from similar systems on both sides (ES1 and ES3). We will derive the formula for \(\left[\frac{\mathrm{C}}{\mathrm{ASI}}\right]\) that takes this particular condition into account.  
Formula (2) can be used to calculate the value of the EIRP uplink from ES1 toward the operating satellite S2 in off-axis conditions, as given in Equation (12):  
\[\left(\mathrm{EIRP}\right)_{\mathrm{up\_ESI}}=\mathrm{P}_{\mathrm{ES1}}+ \mathrm{G}_{\mathrm{Offaxis\_ESI}} \tag{12}\]  
Considering \(\mathrm{G}_{\mathrm{Offaxis\_ES1}}=\mathrm{G}(\theta)\), the interfering signal power from ES1 towards S2, \(\left(\mathrm{EIRP}\right)_{\mathrm{up\_ESI}}\), is given by Equation (13):  
\[\left(\mathrm{EIRP}\right)_{\mathrm{up\_ESI}}=\mathrm{P}_{\mathrm{ES1}}+29-25 \mathrm{Log}(\theta) \tag{13}\]  
Note that, since we are considering identical ES characteristics, the signal arriving at S2 from ES3 has the same EIRP value as \(\left(\mathrm{EIRP}\right)_{\mathrm{up\_ESI}}\). However, the signal arriving at S2 from ES2 will be calculated with Equation (14):  
\[\left(\mathrm{EIRP}\right)_{\mathrm{up\_ES2}}=\mathrm{P}_{\mathrm{ES2}}+ \mathrm{G}_{\mathrm{ES}} \tag{14}\]Now, \(\text{ASI}_{\text{0,ES1}}\) due to the interference from ES1 is determined by \(\left(\text{EIRP}\right)_{\text{up\_ES1}}\) and the path loss experienced by the signal all the way to S2 transmitted from ES1. On the other hand, the desired signal at S2 is determined by \(\left(\text{EIRP}\right)_{\text{up\_ES2}}\) and the path loss experienced by the signal all the way to S2 transmitted from ES2. Considering this, \(\left[\frac{\text{C}}{\text{AS}_{\text{0,ES1}}}\right]\) can be calculated as in Equation (15):  
\[\begin{split}\left[\frac{\text{C}}{\text{AS}_{\text{0,ES1}}} \right]&=\text{C}_{\text{ES2}}-\text{ASI}_{\text{ES1}}=\left( \text{EIRP}\right)_{\text{up\_ES2}}-\text{PL}_{\text{ES2}\to \text{S2}}-\left[\left(\text{EIRP}\right)_{\text{up\_ES1}}-\text{PL}_{\text{ES1 }\to\text{S2}}\right]\\ &=\text{P}_{\text{ES2}}+\text{G}+\text{PL}_{\text{ES2}\to \text{S2}}-\left[\text{P}_{\text{ES1}}+\text{G}(\theta)-\text{PL}_{\text{ES1 }\to\text{S2}}\right]\end{split} \tag{15}\]  
Since the distance from the ES to the satellite is large, we can consider equal path loss values. Considering the assumptions made, \(\text{P}_{\text{ES2}}=\text{P}_{\text{ES1}}\) and \(\text{PL}_{\text{ES2}\to\text{S2}}=\text{PL}_{\text{ES1}\to\text{S2}}\). Equation (15) becomes:  
\[\left[\frac{\text{C}}{\text{ASI}_{\text{0,ES1}}}\right]=\text{G}-\left[\text{ G}(\theta)\right]=\text{G}-\left[29-25\text{Log}(\theta)\right] \tag{16}\]  
Adding the interference from ES3, Equation (16) becomes:  
\[\left[\frac{\text{C}}{\text{ASI}_{\text{0}}}\right]_{\text{total}}=\text{G}- \left[29-25\text{Log}(\theta)\right]-3\text{dB}=23.03\text{ dB-Hz} \tag{17}\]  
Using Equation (8) to calculate the total carrier-to-ASI ratio per bandwidth of B = 30 MHz:  
\[\left[\frac{\text{C}}{\text{ASI}}\right]=\text{ 25.7 }+\text{ 10 Log }\left(30\text{ }\times\text{ 106}\right)=\text{ 97.8 dB} \tag{18}\]  
In the following section, we will evaluate the interference from the MVDS broadcaster on S2.