# Delivering Extended Cellular Coverage and Capacity Using High-Altitude Platforms
## 3 Capacity and Spectral Efficiency Analysis
\begin{split} A_{i}&=\pi x_{i}y_{i}=\pi\left(\frac {\sqrt{h_{p}^{2}+d_{i}^{2}}}{\cos(\beta_{i})+\sin(\beta_{i})\cot(\rho)}\right) \left(\left(\sqrt{h_{p}^{2}+d_{i}^{2}}\right)\tan(\rho)\right)\\ &=\frac{\pi\left(h_{p}^{2}+d_{i}^{2}\right)\tan(\rho)}{\cos( \beta_{i})+\sin(\beta_{i})\cot(\rho)}.\end{split}  \tag{24}\]  
Multiplying (24) by \(\frac{\cot(\rho)}{\cot(\rho)}\), and using the trigonometric identity \(\tan(\rho)\cot(\rho)=1\), the area \(A_{i}\) of cell \(i\) at a given distance \(d_{i}\) from the SPP is derived as  
Figure 4: HAP elliptical cell geometry (semi-minor axis). \[A_{i}=\frac{\pi(h_{p}^{2}+d_{i}^{2})}{\cos(\beta_{i})\cot(\rho)+\sin(\beta_{i}) \cot^{2}(\rho)}. \tag{25}\]  
With the expression for the area \(A_{i}\) of a HAP cell \(i\) derived, the average capacity per-user \(C_{pu}^{\vec{l}}\) and the average ASE \(\tilde{S}_{\eta_{j}}^{\vec{l}}\) in the cell can now be derived. Let \(B_{i}\) be the user-allocated bandwidth. If \(P_{i}^{t}\), \(G_{i}^{t}\), and \(G_{i}^{r}\) are the HAP antenna transmit power, transmit antenna gain and receive antenna gain, respectively, \(\lambda_{s}\) is the signal wavelength, \(r(\varphi)\) is the distance of any arbitrary point from the cell centre at an angle \(\varphi\) from its \(x\)-axis, \(N_{0}\) is the receiver noise density, and \(r_{max}\) is the distance from the boundary to the centre of cell \(i\); \(C_{pu}^{\vec{l}}\) is then derived. Considering a uniform distribution of users over cell \(i\), a normalised bandwidth density \(\tilde{b}_{i}\) (in Hz/unit area) detected at position \((x,y)\) is expressed as  
\[\tilde{b}_{i}(x,y)=\frac{B_{i}}{A_{i}}. \tag{26}\]  
Hence, the expected capacity density \(\tilde{c_{A_{i}}}\) (in bit/s/unit of surface) for a given realisation is defined using the Shannon capacity equation as [34],  
\[\tilde{c_{A_{i}}}(x,y)=\tilde{b}_{i}(x,y)\log_{2}(1+\gamma_{i}(x,y)), \tag{27}\]  
where \(\gamma_{i}(x,y)\) is the CINR at point \((x,y)\) in cell \(i\). Considering a user distribution with density \(u_{i}(x,y)\) in cell \(i\), the average per-user capacity (in bit/s/user) in cell \(i\) is expressed as [34]  
\[\tilde{C_{pu}^{\vec{l}}}=\frac{1}{A_{i}}\iint\limits_{\mathcal{A}}\frac{c_{A_ {i}}(x,y)}{u_{i}(x,y)}dxdy, \tag{28}\]  
where \(\mathcal{A}\) denotes the \(i\)th cell region. Assuming constant user and bandwidth densities \(u_{i}\) and \(B_{i}\), respectively, throughout the cell and substituting (27) in (28),  
\[ \begin{split}\tilde{C_{pu}^{\vec{l}}}&=\frac{1}{A_ {i}}\iint\limits_{\mathcal{A}}\frac{B_{i}}{u_{i}}\log_{2}(1+\gamma_{i}(x,y))dxdy \\ &=\frac{B}{A_{i}}\iint\limits_{\mathcal{A}}\log_{2}(1+\gamma_{i} (x,y))dxdy.\end{split}  \tag{29}\]  
With the assumption of constant user density throughout the cell, the term \(u_{i}\) is eliminated in (29). Thus, the concept of per-user capacity becomes the same as capacity density [34]. Due to the elliptical geometry of the HAP cell, it is more logical to express (29) in polar coordinate form. This involves specifying \(\gamma_{i}(x,y)\) and \(dxdy\) in their polar coordinate form. Considering free-space path loss, \(\gamma_{i}(x,y)\) can be expressed as follows:  
\[\gamma_{i}(x,y)=\gamma_{i}(r(\varphi),\varphi)=\frac{P_{i}^{t}G_{i}^{t}G_{i}^{ r}\lambda^{2}}{N_{0}(4\pi D_{i})^{2}}. \tag{30}\]  
Therefore,  
\[\tilde{C_{pu}^{\vec{l}}}=\frac{B_{i}}{A_{i}}\iint\limits_{\mathcal{A}}\log_{2 }\left(1+\frac{P_{i}^{t}G_{i}^{t}G_{i}^{r}\lambda^{2}}{N_{0}(4\pi D_{i})^{2}} \right)r(\varphi)dr(\varphi)d\varphi, \tag{31}\]  
where \(D_{i}\) is the slant distance of an arbitrary user in cell \(i\) to the HAP. This is based on a single cell scenario, which is noise limited and gives the maximum achievable capacity. \(D_{i}\) needs to be resolved as a function of \(d_{i}\) and \(h_{i}\), and it is represented in polar coordinates as derived below. This can be achieved by considering the geometry presented in Figure 5.  
Imagine that the elliptical cell is flat on the ground and is bounded within a 2D space of radius \(r(\varphi)\).