# On Developing Techniques for Sharing Satellite Spectrum with Indoor Small Cells in 5G +
## 3 Mathematical Analysis
### Estimation of an Optimal Number of Almost Blank Subframes (ABSs)
Hence, to make a trade-off between these efficiencies with reasonable performances, a value of SBS density in terms of \(L\) can be set up based on the operators' requirements. #### Impact of Co-Channel Interference on System Performances

We considered a set of arbitrary scenarios shown in Table 2 to show the impact of both intra-floor and inter-floor co-channel interference thresholds on achieving a certain target for capacity, spectral efficiency, and energy efficiency. The data for both intra-and inter-floor levels for four floors per building, as shown in Figure 5, were considered as the baseline. By increasing the number of floors per building to 20 without changing the number of apartments (i.e., SBSs, per floor), the size of each 3D cluster of SBSs per building was estimated. Table 2 shows the size of 3D cluster sizes for a number of scenarios and the corresponding resource reuse times per building by varying the interference thresholds both intra- and inter-floor levels with respect to the baseline data. It can be found that relaxing (i.e., increasing either intra-floor or inter-floor or both) results in a corresponding decrease in 3D cluster size for the same number of apartments per building and increasing \(\xi_{\text{reuse}}\) and vice versa. Due to greater reuse of the same spectrum vertically within a building to SBSs, an increase in \(\xi_{\text{reuse}}\) per building decreased the required number of buildings \(L\) to deploy in real 2-dimensional (2D) space on Earth in order to achieve a certain target value of capacity, spectral efficiency, and energy efficiency for next-generation mobile communication systems. Note that both the intra-floor and inter-floor co-channel interference effects at UEs within a building can be managed by applying techniques such as transmission power control mechanisms to in-building SBSs. ### Performance Comparison

We evaluated the spectral efficiency and energy efficiency performances of both the orthogonal and non-orthogonal spectrum sharing techniques with respect to that required for 5G mobile networks. It is expected that 5G mobile networks will be able to deliver an average spectral efficiency of 24-37 bps/Hz [32; 33] and energy efficiency of 3\(\mu\)J/b [34; 35]. Now using Figure 5, the values of \(L\) required to meet both spectral efficiency and energy efficiency requirements for 5G networks as aforementioned using OSS, nOSS with four floors per building, and nOSS with 20 floors per building are 94, 24, and 5, respectively. This implies that nOSS with more vertical reuse of spectrum (i.e., the higher upper limit of aggregate co-channel interference effect at a UE both intra-floor and inter-floor levels) \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Scenario** & \(\alpha_{\text{tra,glamer}}\) & \(\alpha_{\text{tra,linear}}\) & \(\alpha_{\text{tra,linear}}\) & \(\alpha_{\text{linear}}\) & \(\alpha_{\text{linear}}\) & \(\alpha_{\text{linear}}\) & \(\alpha_{\text{linear}}\) & \(\alpha_{\text{linear}}\) & \(\alpha_{\text{3D,cluster}}\) & \(\delta_{\text{F}}\) & \(\delta_{\text{reuse}}\) \\ \hline
1 & 0.578 & 0.044 & 12 & 7.07 & 5 & 2 & 4 & 2 & 8 & 32 & 4 \\ \hline
2 & 0.05 & 0.03 & 27.14 & 7.81 & 6 & 4 & 16 & 2 & 32 & 160 & 5 \\ \hline
3 & 0.05 & 0.044 & 27.14 & 7.07 & 5 & 4 & 16 & 2 & 32 & 160 & 5 \\ \hline
4 & 0.578 & 0.03 & 12 & 7.81 & 6 & 2 & 4 & 2 & 8 & 160 & 20 \\ \hline
5 & 0.578 & 0.1 & 12 & 4.24 & 3 & 2 & 4 & 1 & 4 & 160 & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Co-channel interference effect on 3D clustering and resource reuse times. requires less density of small cells than that required by OSS and nOSS with lower aggregate interference thresholds to meet both the spectral efficiency and energy efficiency demands for 5G networks. This in turn, results in reducing costs as well as complexity from the deployment, maintenance, and operation of in-building small cells.