# Delivering Extended Cellular Coverage and Capacity Using High-Altitude Platforms
## 3 Capacity and Spectral Efficiency Analysis  
In order to objectively assess the limits of extension of the proposed HAP extended coverage, it is important to highlight the theoretical performance bounds of the system. In this section, an expression for the average per-user capacity \(C_{pu}^{\vec{j}}\) achievable in cell \(i\) pointing at any given distance away from the sub-platform point (SPP) within the HAP service area is derived. Furthermore, an expression for the average ASE \(S_{\eta_{i}}^{\vec{j}}\) in cell \(i\) is obtained as a function of \(C_{pu}^{\vec{j}}\). Calculating the average ASE of a cell pointed at a given distance away from the SPP requires the area of the cell, taking the cell geometry and broadening into consideration. In deriving an expression for HAP cell area, it is assumed that the maximum power point is at the centre of the cell. Ideally, this is not exactly the case, especially for cells formed at low elevation angles. The maximum power point, which might not be at the centre of the beam depending on the boresight point distance from SPP, is skewed towards the HAP antenna. This is also exacerbated by the broadening effect. In this paper, we assume that the beam footprint is elliptical in shape to simplify the mathematical analysis. It is important to highlight that this assumption is plausible. Imagine that a beam from the antenna is approximately conical in shape; then, cutting through the cone at an angle using a straight surface gives an elliptical footprint on the ground. The effect of the curvature of the earth is neglected. For an elliptical cell \(i\) pointing at a distance \(d_{i}\) from the SPP, let \(x_{i}\) and \(y_{i}\) represent its semi-major and semi-minor axes, respectively. If \(h_{p}\) is the height of the platform with the other variables retaining their definition from the previous section, the area \(A_{i}\) of cell \(i\) is derived as follows. Considering the geometry in Figure 3, \(\alpha_{i}\) and \(h_{i}\) are derived from \(\Delta OHP\) as given below. \[\alpha_{i}=\tan^{-1}\left(\frac{d_{i}}{h_{p}}\right), \tag{12}\]  
\[h_{i}=\sqrt{h_{p}^{2}+d_{i}^{2}}. \tag{13}\]  
Using the sum of angles in a triangle rule in \(\Delta OHP\), the elevation angle of the HAP \(\beta_{i}\) from the centre of cell \(i\) is,  
\[\beta_{i}=\frac{\pi}{2}-\alpha_{i}. \tag{14}\]  
Hence, using sine rule in \(\Delta HEP\), \(x_{i}\) is expressed as follows. \[x_{i}=\frac{h_{i}\sin(\rho)}{\sin(\rho+\beta_{i})}. \tag{15}\]  
Substituting for \(h_{i}\) in (15),  
\[x_{i}=\left(\frac{\sin(\rho)}{\sin(\rho+\beta_{i})}\right)\sqrt{h_{p}^{2}+d_{ i}^{2}}. \tag{16}\]  
Using trigonometric identity, (16) can be re-expressed as  
\[\frac{1}{x_{i}}=\frac{\sin(\rho)\cos(\beta_{i})+\sin(\beta_{i})\cos(\rho)}{ \sin(\rho)\sqrt{h_{p}^{2}+d_{i}^{2}}}. \tag{17}\]  
Expanding (17),  
\[ \begin{split}\frac{1}{x_{i}}&=\left(\frac{\sin(\rho )\cos(\beta_{i})}{\sin(\rho)}+\frac{\sin(\beta_{i})\cos(\rho)}{\sin(\rho)} \right)\frac{1}{\sqrt{h_{p}^{2}+d_{i}^{2}}}\\ &=\left(\cos(\beta_{i})+\frac{\sin(\beta_{i})}{\tan(\rho)} \right)\frac{1}{\sqrt{h_{p}^{2}+d_{i}^{2}}}.\end{split}  \tag{18}\]  
Figure 3: HAP elliptical cell geometry (semi-major axis). Therefore, the semi-major axis \(x_{i}\) is expressed as follows. \[x_{i}=\frac{\sqrt{h_{p}^{2}+d_{i}^{2}}}{\cos(\beta_{i})+\sin(\beta_{i})\cot( \rho)}. \tag{19}\]  
Furthermore, the semi-minor axis \(y_{i}\) can be derived from Figure 4 redrawn from Figure 3 for convenience. Applying the sum of angle in a triangle and sine rules in \(\Delta HSP\), \(\angle HSP\) yields  
\[\chi_{i}=\frac{\pi}{2}-\rho, \tag{20}\]  
and  
\[\frac{h_{i}}{\sin(\frac{\pi}{2}-\rho)}=\frac{y_{i}}{\sin(\rho)}. \tag{21}\]  
Thus,  
\[y_{i}=h_{i}\tan(\rho). \tag{22}\]  
Therefore, substituting (13) into (22) gives,  
\[y_{i}=\left(\sqrt{h_{p}^{2}+d_{i}^{2}}\right)\tan(\rho). \tag{23}\]  
Using (19) and (23), the expression for surface the area of cell \(i\) can be obtained. Recall that the area of an ellipse is given as \(\pi x_{i}y_{i}\). Thus,  
\[