# 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)
In _case 2_, we considered increasing the number of floors per building from four in case 1 to 20, which is typical in dense urban environments, so that the number of apartments per building becomes 160. Now, considering the same minimum distances between co-channel SBSs of \({d_{\mathrm{planet}}}^{*}\geq 12\) m and \({d_{\mathrm{linear}}}^{*}\geq 7.07\) m or equivalently \({d_{\mathrm{ver}}}^{*}\geq 5\) m like case 1, the number of 3D clusters per building becomes \((160/8)=20\) (i.e., \(\xi_{\mathrm{reuse}}=20\)). Figure 4 shows the capacity performance response of aTMS for both OSS and nOSS for case 1 and case 2 with SBSs for a single building (i.e., \(L=1\)). From Figure 4, it can be found that the achievable capacities using nOSS in both cases considerably outperformed the corresponding capacities when using OSS. This is due to the fact that the same satellite spectrum can be reused four times in case 1 and 20 times in case 2 to the SBSs per building when using nOSS compared to when using OSS. Furthermore, from Figure 4, it can be found that due to reusing the same 40 MHz satellite spectrum for both case 1 and case 2, there was no change in capacity and hence spectral efficiency performances using OSS (Figure 5a). However, energy efficiency using OSS in case 2 decreased significantly as shown in Figure 5b when compared to that in case 1, due to a greater consumption of power by more SBSs in case 2 than that by the SBSs in case 1. A noticeable observation was that due to considering more realistic modeling (i.e., non-collocated deployment of SBSs and CGC stations in an apartment), OSS suffered from obtaining high spectral efficiency even though the same satellite spectrum was reused to SBSs deployed in multiple buildings (Figure 5a). This drawback can be overcome easily by adopting the nOSS with SBSs within the same building, as shown in Figure 5a. Figure 4: Capacity responses of OSS and nOSS of aTMS for \(L=1\). From Figure 5, it can be found that, in general, nOSS outperformed OSS irrespective of the number of floors per building in terms of both spectral efficiency and energy efficiency. More specifically, due to extensive reuse of the same spectrum vertically to SBSs per building, both spectral efficiency and energy efficiency performances using nOSS for case 2 improved significantly when compared to the corresponding spectral efficiency and energy efficiency performances using OSS in case 1 and case 2 as well as using nOSS for case 1. Furthermore, even though energy efficiency improved negative exponentially for low values of \(L\) and became almost steady for high values of \(L\), spectral efficiency improved proportionally as \(L\) increased due to having had a significant vertical spectrum reuse gain

Figure 5: Spectral efficiency and energy efficiency responses of OSS and nOSS of aTMS for the variation of \(L\). (**a**) Average spectral efficiency versus \(L\); (**b**) Average energy efficiency versus \(L\). per building, resulting in a substantial gap between spectral efficiency gains due to nOSS and OSS (Figure 5a). Note that the steadiness in energy efficiency performances using OSS and nOSS in both cases for higher values of \(L\) can be clarified by Equation (22) for energy efficiency where it can be found that the aggregate transmission power due to the macrocell BS and picocell BSs had a considerable impact on that of the SBSs due to the low transmission power of each SBS. However, as \(L\) increased considerably, the impact of aggregate transmission power as well as the aggregate capacity of the macrocell BS and picocell BSs became insignificant when compared to that of SBSs, resulting in canceling the presence of \(L\) in both the denominator and the numerator, leaving energy efficiency performance with \(L\) almost constant. On the contrary, Equations (20) and (21) imply that spectral efficiency varies proportionally with \(L\).