# Quantification and Analysis of Carrier-to-Interference Ratio in High-Throughput Satellite Systems
## 3 Quantification Method of CIR
### Multi-Carrier Scenarios  
The above general quantification method of the CIR is based on the assumption that there is only one continuous frequency band in each beam. For very-high-throughput satellite (VHTS) communication systems, the frequency bands within each beam may not be continuous, and the two multiplexed beams do not belong to the absolute same color. The quantification method can be improved adaptively as follows. Each sub-carrier is treated as a \"beam\". It is worth noting that the possible area where users are located for each sub-carrier within the same beam is the same, that is, the entire beam area. In order to help understand this conclusion better, we illustrate the equivalent representation of an interfering beam with multi-carriers in Figure 7.  
In Figure 7, the green line, which points to a beam with a yellow color, represents the useful signal for the target user. The transmitting power and the frequency band of the useful signal are denoted as \(P_{1}\) and \([f_{1},f_{2}]\), respectively. The other beams with a yellow color are interferers naturally. As mentioned above, the interfering power is the sum of that coming from every interfering beam, i.e., \(I=\sum_{i}I_{i}\). However, in multi-carrier scenarios, it should be emphasized that beams with the same color do not occupy the identical frequency band, although their frequency bands overlap to a certain degree. That is to say, part of the transmitting power of an interfering beam, although not all, will cause the interference in the beam where the target user is located. Taking the i-th interfering beam for example, as shown in Figure 7, we denote the transmitting power, the total bandwidth, and the number of carriers in this beam as \(P_{i}\), \(B_{i}\), and \(N_{i}\), respectively. Specifically, the frequency band of the j-th carrier (\(j\leq N_{i}\)) is denoted as \([f_{ij}^{start},f_{ij}^{stop}]\) in a generic way. It is easy to conclude that:  
\[B_{i}=\sum_{j=1}^{N_{i}}{(f_{ij}^{stop}-f_{ij}^{start})}. \tag{18}\]  
If there are no guard bands between the frequency bands of these carriers, then \(B_{i}=f_{iN_{i}}^{stop}-f_{i1}^{start}\). It is worth noting that there is no difference in the spatial orientation among all the carriers. Each carrier can serve a user located at any position belonging to the beam area. Therefore, a carrier can be modelled as a virtual beam, and an interfering real beam with multi-carriers can be modelled as the addition of multiple virtual beams. As a result, the interference coming from the \(i\)-th interfering beam is equal to the sum of the contribution of all the virtual beams: \(I_{i}=\sum_{j}I_{ij}\). The transmitting power of the j-th virtual beam (\(j\leq N_{i}\)) is denoted as \(P_{ij}\), and \(P_{i}=\sum_{j}P_{ij}\). Furthermore, because the multi-carriers in a real beam shall share the same travelling wave tube amplifier (TWTA), the power spectral density (PSD) of them are equal,i.e., the transmitting power of each carrier, or each virtual beam, is proportional to its own bandwidth. Hence, \(P_{ij}\) can be calculated by:  
\[P_{ij}=\frac{(f_{ij}^{stop}-f_{ij}^{start})P_{i}}{B_{i}}. \tag{19}\]  
Furthermore, a fraction of \(P_{ij}\) would create an interference in the target beam and the proportion depends on the overlapping of \([f_{1},f_{2}]\) and \([f_{ij}^{start},f_{ij}^{stop}]\). Thus, the weighted power, \(\widetilde{P}_{ij}\), would be:  
\[\widetilde{P}_{ij}=\frac{[(f_{ij}^{stop}-f_{ij}^{start}+f_{2}-f_{1})-(\max(f_{ ij}^{stop},f_{2})-\min(f_{ij}^{start},f_{1}))]P_{ij}}{f_{ij}^{stop}-f_{ij}^{start}}. \tag{20}\]  
As described in Section 3.2, the interfering power coming from the \(j\)-th virtual beam would be:  
\[I_{ij}=\widetilde{P}_{ij}G_{i}(\theta). \tag{21}\]  
Figure 7: The equivalent representation of an interfering beam with multi-carriers. To summarize, in multi-carrier scenarios, Equation (17) should turn into:  
\[