# Maximum Transmit Power for UE in an LTE Small Cell Uplink
## 3 Network and Propagation Model
### HetNet Deployment with Uniformly Distributed Small Cells
#### 3.2.2 Interference  
At first, let us consider the small cell interference offered to the central small cell. The interference \(I_{n}\) offered by one interfering ring to the central small cell eNodeB \(x\) can be given using Equation (9) as:  
\[ \begin{array}{l}I_{n}=2\pi\int_{2n-1R}^{2 nR}\rho_{ue}p_{o\_{\textrm{small}}} \textrm{A}^{-\alpha_{\textrm{small}}}(2 nR-r)^{\alpha_{\textrm{small}}\eta} \textrm{A}r^{-\eta}rdr\\ +2\pi\int_{2 nR}^{(2n+1)R}\rho_{ue}p_{o\_{\textrm{small}}}\textrm{A}^{-\alpha_ {\textrm{small}}}(-2 nR+r)^{\alpha_{\textrm{small}}\eta}\textrm{A}r^{-\eta}rdr \end{array}  \tag{14}\]  
As there are two interfering rings of small cells at a distance of \(2R\) and \(4R\), therefore the total interference offered by the interfering rings of small cells to the central small cell eNodeB \(x\) can be given using Equations (10) and (14) as:  
\[I_{\textrm{small}}=\sum\
olimits_{n=1}^{2}I_{n} \tag{15}\]  
To compute the macro cell interference, the macro cell containing small cells as well as the neighboring macro cells were also considered. At first, the interference from the macro cell containing the small cells was computed, followed by the interference from the two rings of macro cells around the central macro cell. From Figure 2, it is clear that the central small cell was in the middle of the macro cell. Consider user \(k\) scheduled by macro cell eNodeB \(m\), co-located with the central small cell eNodeB \(x\). The distance range of user \(k\) from eNodeB \(x\) is \(r\in(0;R_{\textrm{macro}})\) as depicted in Figure 2. The interference offered by the area \(2\pi rdr\) around UE \(k\) is \(\rho_{ue\_macro}p_{o\_macro}\textrm{A}^{-\alpha_{\textrm{macro}}}(R_{\textrm{ macro}}-r)^{\alpha_{\textrm{macro}}\eta}\textrm{A}r^{-\eta}2\pi rdr\). Therefore,  
Figure 2: HetNet scenario for uniformly distributed small cells within a macro cell. the expression for the interference offered by the central macro cell to the central small cell eNodeB \(x\) can be given using Equation (7) as:  
\[I_{c\_macro}=2\pi\int_{0}^{R_{\rm macro}}\rho_{ue\_macro}P_{\rm o\_macro}\rm A^{ -\alpha_{macro}}r^{\alpha_{macro}\eta}\rm Ar^{-\eta}rdr \tag{16}\]  
To calculate the interference offered by the interfering ring of macro cells, consider user \(k\) scheduled by eNodeB \(n\), located on the first interfering ring of cells around the central cell. The distance range of user \(k\) from eNodeB \(x\) is \(r\in(R_{\rm macro};2~{}R_{\rm macro})\) as depicted in Figure 2. The user \(k\) is at a distance of \(2R_{\rm macro}\)-\(r\) from eNodeB \(n\). The interference offered by the area \(2\pi rdr\) around UE \(k\) is \(\rho_{ue\_macro}\rho_{\rm o\_macro}\rm A^{-\alpha_{macro}}(2R_{\rm macro}-r)^ {\alpha_{macro}\eta}\rm Ar^{-\eta}2\pi rdr\). Similarly, if UE \(k\) is located in the distance range \(r\in(2R_{\rm macro};3R_{\rm macro})\) from eNodeB \(x\), it is at a distance \(r-2R_{\rm macro}\) from eNodeB \(n\). Hence, the interference offered now by the area \(2\pi rdr\) around UE \(k\) can be estimated to be \(\rho_{ue\_macro}\rho_{\rm o\_macro}\rm A^{-\alpha_{macro}}(-2R_{\rm macro}+r )^{\alpha_{macro}\eta}\rm Ar^{-\eta}2\pi rdr\). The interference offered by the \(nth\) interfering ring of macro cells around the central small cell can be formulated from Equation (9) as:  
\[ \begin{array}{l}I_{n}=2\pi\int_{(2n-1)R_{\rm macro}}^{2 nR_{\rm macro}}\rho_{ue \_macro}\rm{\rm{\it p}\_{o\_macro}\rm{\it A}^{-\alpha_{macro}}(2 nR_{\rm macro }-r)^{\alpha_{macro}\eta}\rm Ar^{-\eta}rdr\\ +2\pi\int_{2 nR_{\rm macro}}^{(2n+1)R_{\rm macro}}\rho_{ue\_macro}\rho_{\rm o \_macro}\rm{\it A}^{-\alpha_{macro}}(-2 nR_{\rm macro}+r)^{\alpha_{macro}\eta }\rm Ar^{-\eta}rdr\end{array}  \tag{17}\]  
Hence, the total macro cell interference offered to the central small cell can be approximated as:  
\[I_{\rm macro}=I_{c\_macro}+\sum\
olimits_{n=1}^{N_{i}}I_{n} \tag{18}\]  
where \(N_{i}\) represents the number of macro cell interfering rings around the central small cell. \(N_{i}=2\) was considered for this section as well.