# On Developing Techniques for Sharing Satellite Spectrum with Indoor Small Cells in 5G +
## 3 Mathematical Analysis
### Estimation for a Single Building
#### 3.1.4 Overall System-Level Performances  
Now, the overall system-level performances due to all SBSs per building, regardless of collocating with CGC stations, are given as follows. The overall capacity of small cells per building due to sharing the whole satellite spectrum orthogonally with all SBSs is given by  
\[\sigma_{\text{cl,sc,orth}}^{\text{all SBSs}}=\sigma_{\text{cl,sc}}^{\text{ SBS, CGC}}+\sigma_{\text{cl,sc}}^{\text{SBS only}} \tag{10}\]  
Therefore, the overall system-level capacity of aTMS due to sharing the satellite spectrum orthogonally with all SBSs per building is given by  
\[\sigma_{\text{aTMS,orth}}^{\text{sys,all SBSs}}=\sigma_{\text{aTMS}}+\sigma_{ \text{cl,sc,orth}}^{\text{all SBSs}} \tag{11}\]  
**Remark 2**: _The capacity achieved due to sharing the satellite spectrum can be interpreted as the additional capacity obtained due to reusing the satellite spectrum to in-building SBSs of aTMS. This is because of the fact that the satellite spectrum is proprietary, which is licensed by the iSMS provider, not by the aTMS operator. Hence, the effective aTMS's spectrum, even after sharing the satellite spectrum, is only its licensed spectrum [22]. This explanation is equally applicable for orthogonal as well as non-orthogonal allocations of the satellite spectrum to in-building small cells._  
Now, the overall spectral efficiency of aTMS due to orthogonal allocation of the satellite spectrum to all SBSs per building in bps/Hz is given by  
\[\sigma_{\text{aTMS,orth}}^{\text{SE,all SBSs}}=\sigma_{\text{aTMS,orth}}^{ \text{sys,all SBSs}}/(M_{\text{aTMS}}\times Q) \tag{12}\]  
Let \(\boldsymbol{T_{\text{non-ABS}}^{\text{eb, SPS}}}\) denote the set of non-ABSs when both SBSs and CGC stations are collocated. Then, using Remark 1, the average system-level energy efficiency of aTMS per building due to all SBSs in joules/bit (J/b) is given by  
\[\sigma_{\text{aTMS,orth}}^{\text{EE,all SBSs}}=\left( \begin{array}{l}S_{ \text{F}}\times(1/4)\times\left(\left|\boldsymbol{T_{\text{non-ABS}}^{\text{sb, SPS}}}\right|/|\boldsymbol{T}|\right)\\ \times P_{\text{PC,SPS}}\\ (S_{\text{F}}/1/4)\times P_{\text{PC,SPS}})+\\ (S_{\text{P}}\times P_{\text{PC}})+(S_{\text{M}}\times P_{\text{MC}})\end{array}  \right)\left((\sigma_{\text{aTMS,orth}}^{\text{sys,all SBSs}}/Q\right) \tag{13}\]  
Now, using (A7) in Appendix B, for the non-orthogonal allocation of the satellite spectrum with small cells per building, the system capacity improves by \(\underline{\kappa}_{\text{reuse}}\) since the same satellite spectrum is reused to the same number of small cells per building by \(\underline{\kappa}_{\text{reuse}}\) times. Hence, the overall system-level capacity, spectral efficiency, and energy efficiency of small cells of aTMS due to sharing the whole satellite spectrum non-orthogonally with all SBSs per building are given by Equations (14)-(16), respectively,  
\[\sigma_{\text{aTMS,non-orth}}^{\text{sys,all SBSs}}=\sigma_{\text{aTMS}}+ \left(\underline{\kappa}_{\text{reuse}}\times\left(\sigma_{\text{cl,sc}}^{ \text{SBS, CGC}}+\sigma_{\text{cl,sc}}^{\text{SBS only}}\right)\right) \tag{14}\]  
\[\sigma_{\text{aTMS,non-orth}}^{\text{SE,all SBSs}}=\sigma_{\text{aTMS,non-orth}}^{ \text{sys,all SBSs}}/(M_{\text{aTMS}}\times Q) \tag{15}\]  
\[\sigma_{\text{aTMS,non-orth}}^{\text{EE,all SBSs}}=\left( \begin{array}{l}S_{ \text{F}}\times(1/4)\times\\ \left(\left|\boldsymbol{T_{\text{non-ABS}}^{\text{sb, SPS}}}\right|/|\boldsymbol{T}|\right)\\ \times P_{\text{PC,SPS}}\\ (S_{\text{F}}/1/4)\times P_{\text{PC,SPS}}\\ +(S_{\text{P}}\times P_{\text{PC}})+\\ (S_{\text{M}}\times P_{\text{MC}})\end{array}