# 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)
The received signal-to-interference-plus-noise ratio for a UE at RB \(=i\) in TTI \(=t\) is given by

\[\rho_{t,i}=\Big{(}P_{t,i}/\big{(}N_{t,i}^{\text{s}}+I_{t,i}\big{)}\Big{)}.\,H_{t,i} \tag{1}\]where \(P_{t,i}\) is the transmit power; \(N_{t,i}^{\text{s}}\) is the noise power; \(I_{t,i}\) is the total interference signal power; and \(H_{t,i}\) is the link loss for a link between a UE and a BS at RB \(i\) in TTI = \(t\). \(H_{t,i}\) can be expressed in dB as

\[H_{t,i}(\text{dB})=(G_{\text{t}}+G_{\text{r}})-(L_{\text{F}}+PL_{t,i})+(LS_{t,i }+SS_{t,i}) \tag{10}\]

where \((G_{\text{t}}+G_{\text{r}})\) and \(L_{\text{F}}\) are total antenna gain and connector loss, respectively. \(LS_{t,i}\), \(SS_{t,i}\), and \(PL_{t,i}\) denote shadowing effect, small scale Rayleigh or Rician fading, and path loss between a BS and a UE at RB = \(i\) in TTI = \(t\), respectively. Using [PERSON]'s capacity formula, a link throughput at RB = \(i\) in TTI = \(t\) in bps per Hz is given by [36; 37]

\[\sigma_{t,i}(\rho_{t,i})=\left\{ \begin{array}{ll}0,&\rho_{t,i}<-10\text{ dB}\\ \beta\log_{2}\!\left(1+10^{(\rho_{t,i}(\text{dB})/10)}\right),&-10\text{ dB}\leq\rho_{t,i}\leq 22\text{ dB}\\ 4.4,&\rho_{t,i}>22\text{ dB}\end{array} \right\} \tag{11}\]

where \(\beta\) is considered as the implementation loss factor. ## Appendix B Modeling In-Building 3D Clusters of Nodes for nOSS

For simplicity in modeling, we assumed that the deployment pattern of CGC stations followed the same as that of the SBSs and the transmit power of each CGC station and each SBS was the same power. Such assumptions help to set a minimum distance between co-channel CGCs and co-channel SBSs resulting in forming clusters (i.e., a group of nodes allocated with orthogonal RBs where a node can be either a CGC station or a SBS) of the same size for both CGC stations and SBSs. We assumed that there was at most one CGC station or one SBS per apartment as shown in Figure 11.

We considered adopting the model proposed by the authors in [25] to define a 3D cluster of nodes in a building under co-channel interference constraints. Note that as shown in Figure 11, each 3D cluster of nodes consist of nodes in the intra-floor level as well as inter-floor levels, so that the minimum distances between co-channel nodes in intra-floor level \(d_{\text{planer}}\)\({}^{*}\) and inter-floor \(d_{\text{linear}}\)\({}^{*}\) level can be given by [25]

\[d_{\text{planer}}^{*}\geq d_{\text{min}}\left(y_{\text{max,planer}}/\alpha_{ \text{thr,planer}}\right)^{1/3} \tag{12}\]

\[d_{\text{linear}}^{*}\geq d_{\text{min}}\left(10^{-\ (\alpha_{t}(d_{\text{ linear}}^{*})/10)}\ (y_{\text{max,linear}}/\alpha_{\text{thr,linear}})\right)^{1/3} \tag{13}\]

where \(\alpha_{\text{thr,planer}}\) and \(\alpha_{\text{thr,linear}}\) denote co-channel interference thresholds set by the operator at a serving UE in intra-floor and inter-floor levels, respectively. \(d_{\text{min}}=5\) m for the \(10\times 10\) m\({}^{2}\) apartment. \(y_{\text{max,planer}}=8\) is the maximum number of intra-floor co-channel nodes and \(y_{\text{max, linear}}=1\) for the single-sided and \(y_{\text{max, linear}}=2\) for the double-sided inter-floor co-channel nodes. Hence, the number of nodes in the intra-floor level can be expressed as follows:

\[\Delta_{\text{planer}}=\varepsilon_{\text{planer}}\text{{}^{2}}\]

where \(\varepsilon_{\text{planer}}\) denotes the maximum number of tiers of nodes corresponding to \(d_{\text{planer}}\)\({}^{*}\) in Figure 11 and is given by

\[\varepsilon_{\text{planer}}=\text{ceil}\left(d_{\text{planer}}^{*}+(a/2)/a\right)\]

where \(a=10\) m. Likewise, if \(\varepsilon_{\text{linear}}\) represents the maximum number of tiers of nodes at the inter-floor level and is given by

\[\varepsilon_{\text{linear}}=\text{ceil}\left(d_{\text{linear}}^{*}/d_{\text{ ver}}\right)\]

where \(d_{\text{ver}}\) denotes the vertical distance between adjacent floors.