# Delivering Extended Cellular Coverage and Capacity Using High-Altitude Platforms
## 4 HAP Beam Pointing for Extended Coverage
### Beam-Pointing Algorithm  
The proposed algorithm involves five steps, which are described as follows:  
Step 1:  
Initially, the boundary of the SPPC \(c_{1}\) is defined by the angle \(\rho\) it subtends at the HAP with its centre as highlighted in Figure 6. In addition, the look angle of the cell boresight from the HAP horizontal plane \(\beta\) is set to \(90^{\circ}\). The angle between the cell centre and boundary is assumed to be equal for all cells. Similarly, the initial path distance \(d_{1}\) between the HAP and the SPPC centre \(c_{1}\) is set to \(h_{p}\). A second cell \(c_{2}\) points along the \(x\)-axis of the SPPC by making the angular distance between the centres of \(c_{1}\) and \(c_{2}\) equal to \(2\rho\) and \(d_{1}\) updated to \(d_{2}\). A set of the tessellated cell centre coordinates \(\mathcal{C}_{T}\) along the \(x\)-axis is maintained and updated for each new coordinate. The update of \(\mathcal{C}_{T}\) for subsequent cell centres \(c_{i+1},\cdots,c_{n}\) along the \(x\)-axis of \(c_{i}\) is carried out using an update function \(U^{d}(\rho,\beta_{i})\), which is derived as follows. In order to derive \(U^{d}(\rho,\beta_{i})\), the geometry shown in Figure 7 is obtained from Figure 6. The solid lines in Figure 7 represent the cell boresight and the dashed lines represent the cell edges. Let \(d_{1},\ d_{2},\ d_{3},\ \cdots\) be the distances of cell boresights from the SPP along an axis. The distance of cell boresight \(\overline{Oa}\), i.e., \(Oa=d_{2}-d_{1}=d_{2}\) since \(d_{1}=0\). Using sine rule on \(\Delta OHa\),  
\[\frac{d_{2}}{\sin(2\rho)}=\frac{h_{1}}{\sin(\beta_{1}-2\rho)}. \tag{39}\]  
Therefore,  
\[d_{2}=h_{1}\bigg{(}\frac{\sin(2\rho)}{\sin(\beta_{1}-2\rho)}\bigg{)}+d_{1}. \tag{40}\]  
Similarly, the distance of cell boresight \(d_{3}=d_{2}+\overline{ab}\). From \(\Delta aHb\),  
\[\frac{\overline{ab}}{\sin(2\rho)}=\frac{h_{2}}{\sin(\beta_{2}-2\rho)}. \tag{41}\]  
Thus,  
\[\overline{ab}=h_{2}\bigg{(}\frac{\sin(2\rho)}{\sin(\beta_{2}-2\rho)}\bigg{)}. \tag{42}\]  
Figure 7: The HAP cell boresight geometry. This implies that,  
\[d_{3}=\overline{ab}+d_{2}=h_{2}\bigg{(}\frac{\sin(2\rho)}{\sin(\beta_{2}-2\rho)} \bigg{)}+d_{2}. \tag{43}\]  
Therefore, a generic equation with a common update function \(U^{d}(\rho,\beta_{i})\) can be obtained from (40) and (43) as  
\[d_{i+1}=h_{i}U^{d}(\rho,\beta_{i})+d_{i}, \tag{44}\]  
where,  
\[U^{d}(\rho,\beta_{i})=\frac{\sin(2\rho)}{\sin(\beta_{i}-2\rho)}. \tag{45}\]  
Note that \(d_{1}=0\) implies that the distance of the SPPC boresight from the SPP is zero and SPPC is the cell at the centre of the HAP service area. In order to include some level of overlap between \(c_{i}\) and \(c_{i+1}\), as depicted in Figure 6, to minimise outage for mobile users moving between cells, the distance between \(c_{i}\) and \(c_{i+1}\) is reduced by a distance corresponding to an overlap angle \(\Phi\), which is a function of the overlap rate \(\varepsilon\) and inter-cell distance. The value of \(\Phi\) between cells \(c_{i}\) and \(c_{i+1}\) is derived as follows. Consider \(c_{1}\) and \(c_{2}\) in Figure 6 with the geometry partially reproduced in Figure 6.0  
Let \(\Phi_{i,i+1}\) be the angle of overlap between neighbouring cells \(c_{i}\) and \(c_{i+1}\). Since the coordinate of \(a\) is determined based on the size of the SPPC centred at \(O\), \(h_{2}\) can be evaluated while \(h_{1}\) is the platform height. The overlap distance \(\overline{Aa}\) is determined by a given overlap rate \(\varepsilon\) and the distance between the neighbouring cell centre boresight from the SPPC boresight. If \(d_{2}\) is the distance of the neighbour to the SPPC and \(d_{1}\) is the distance of the SPPC boresight from the SPP, then  
Figure 6.1  
Hence, using the sine rule on \(\Delta OHA\) and \(\Delta OHa\),  
Figure 6.2  
Therefore,  
Figure 6.3  
Extending (48) to subsequent cells results in  
Figure 6.4  
Figure 6.5 The HAP cell overlap geometry. This process, as described above and depicted in Figure 6, continues for subsequent cells along the \(x\)-axis.