# Delivering Extended Cellular Coverage and Capacity Using High-Altitude Platforms
## 3 Capacity and Spectral Efficiency Analysis
The radius varies with respect to the azimuth angle \(\varphi\). With origin at the centre of the cell, therefore, \(r(\varphi)\) is defined as [35]  
\[ \begin{split} r(\varphi)&=\frac{x_{i}y_{i}}{\sqrt{(y_ {i}\cos(\varphi))^{2}+(x_{i}\sin(\varphi))^{2}}}\\ &=\frac{y_{i}}{\sqrt{1-(e\cos(\varphi))^{2}}},\end{split}  \tag{32}\]  
where \(e\) is the cell eccentricity, \(x_{i}\) and \(y_{i}\) are the semi-major and semi-minor axes defined in (19) and (23), respectively. \(e\) is expressed as  
\[e=\sqrt{1-\left(\frac{y_{i}}{x_{i}}\right)^{2}}. \tag{33}\]  
By applying the cosine rule on \(\Delta POC\) in the above figure, \(D_{i}\) can be expressed as a function of \(\varphi\). Firstly, the distance \(q_{i}\) of an arbitrary user from the SPP is obtained as follows. \[q_{i}^{2}=d_{i}^{2}+r(\varphi)^{2}+2r(\varphi)d_{i}\cos(\varphi). \tag{34}\]  
Then, considering \(\Delta HOC\) and substituting for \(q_{i}\),  
\[D_{i}^{2}=h_{p}^{2}+q_{i}^{2}=h_{p}^{2}+d_{i}^{2}+r(\varphi)^{2}+2r(\varphi)d _{i}\cos(\varphi). \tag{35}\]  
Finally, substituting for \(D_{i}\) in (31), the average per-user capacity \(\tilde{C_{pu}}\) is given as  
\[\tilde{C_{pu}^{2}}=\frac{B_{i}}{A_{i}}\int\limits_{0}^{2\pi}\int\limits_{0}^{ \pi_{max}}\log_{2}\left(1+\frac{P_{i}^{t}G_{i}^{t}G_{i}^{t}\lambda^{2}}{\xi} \right)r(\varphi)dr(\varphi)d\varphi, \tag{36}\]  
where  
\[\xi=N_{o}(4\pi)^{2}(h_{p}^{2}+d_{i}^{2}+r(\varphi)^{2}+2r(\varphi)d_{i}\cos( \varphi)). \tag{37}\]  
Unfortunately, there is no closed-form expression for (36). This was the conclusion after trying Chebyshev-Gauss quadrature approximation and using Mathematica, a software tool for mathematical analysis. However, it can be evaluated using numerical methods. Consequently, the average ASE \(\tilde{S_{\eta_{f}}^{i}}\) in cell \(i\) is then expressed as  
\[\tilde{S_{\eta_{f}}^{i}}=\frac{\tilde{C_{pu}^{i}}}{B_{i}A_{i}}=\frac{\eta_{f} ^{i}}{A_{i}}. \tag{38}\]  
Figure 5: HAP elliptical cell geometry (polar coordinates). where \(\eta_{f}^{i}=\frac{C_{lm}^{i}}{B_{i}}\) is the SE. The average ASE \(\vec{S}_{\eta_{f}}^{i}\) highlights the effect of cell broadening on the capacity of a cell, which also skews the maximum power point of a beam away from the centre of the beam, by also considering the cell area. This effect becomes increasingly severe as the HAP service area increases. The degree of service area extension may be based on a given minimum average ASE required, which can be a design parameter. The widely used capacity density metric can be obtained by multiplying the average ASE by bandwidth. The expressions derived above can be used to model a HAP extended coverage communication system and understand the analytical limit of coverage and capacity extension achievable from a HAP while guaranteeing a minimum QoS. Practically, apart from the shape, size, and capacity of HAP cells, the appropriate placement of the cells plays a key role in minimising interference while enhancing coverage. Section 4 discusses how the disproportionately sized beams due to varied elevation angles can be formed and pointed in practice for better system performance.