# Delivering Extended Cellular Coverage and Capacity Using High-Altitude Platforms
## 4 HAP Beam Pointing for Extended Coverage
### Algorithm Validation Using Simulated Annealing  
Given the resulting boresight coordinates from Algorithm 1, it is relevant to understand if these locations are near optimum for contiguous and full coverage as part of radio network planning and optimisation. We used SA to validate that the coordinates are near optimum for beam deployment to achieve full coverage. Figure 10: The full cellular structure for the HAP extended coverage. ```
1:Declare \(h\), \(\rho\), \(\varepsilon\), \(\beta\): HAP height, broadside cell subtended angle, overlap ratio, and SPPC boresight angle. 2:Set \(i=1\) and \(d_{1}=0\). 3:Initialise \(\beta_{1}=90^{\circ}\), \(h_{1}=h_{p}\), \(\mathcal{C}_{T}=\{(0,d_{1},0)\}\), \(\mathcal{C}_{P}=\emptyset\) and \(\Phi\). 4:while\(\beta_{i}>0\)do
5: Compute \(d_{i+1}:=h_{i}U^{d}(\rho,\beta_{i})+d_{i}\). 6: Update \(\mathcal{C}_{T}\mid\mathcal{C}_{T}=\mathcal{C}_{T}\cup(0,d_{i+1},0)\). 7: Update \(\beta\mid\beta_{i+1}:=\beta_{i}-2\rho\). 8: Update \(h\mid h_{i+1}:=\frac{h_{i}}{\sin(\beta_{i+1})}\). 9:\(i\gets i+1\). 10:endwhile
11:Adjust \(\mathcal{C}_{T}\) for overlap, i.e., \(\forall d_{i}\
eq 0\mid i\in\{1,\ 2,\ 3,\ \cdots\}\wedge d_{i}\in x_{i}\subseteq \mathcal{C}_{T}\), \(d_{i}:=d_{i}-h_{i}\sin(\Phi_{i})\). 12:Obtain new set \(\mathcal{C}_{R}:=60^{\circ}\) rotation of \(\mathcal{C}_{T}\). 13:Set ring \(j=0\). 14:for each \(m_{i}\subseteq\mathcal{C}_{T}\) and corresponding \(n_{i}\subseteq\mathcal{C}_{R}\mid i\in\{1,\ 2,\ 3,\ \cdots\},m_{i}\
eq\emptyset,n_{i}\
eq\emptyset\)do
15:\(V_{I}(j+1):=\) Interpolation between \(m_{i}(j+1)\) and \(n_{i}(j+1)\) using (51). 16: Set \(V_{I}^{\circ}(j+1):=\) mirror image of \(V_{I}(j+1)\). 17: Obtain \(\mathcal{C}_{P}:=\mathcal{C}_{P}\cup\frac{V_{I}^{\circ}(j+1)}{j+1}\) as new cells equidistant from other cells in each ring \(j\). 18:\(j\gets j+1\). 19:endfor
20:Evaluate \(\mathcal{C}_{F}:=5\) sets of coordinates where each is a \(60^{\circ}\) step rotation of \(\mathcal{C}_{T}\cup\mathcal{C}_{R}\cup\mathcal{C}_{P}\). 21:Obtain a set of cell coordinates \(\mathcal{C}:=\mathcal{C}_{T}\cup\mathcal{C}_{R}\cup\mathcal{C}_{P}\cup \mathcal{C}_{F}\). 22:Collect \(\mathcal{C}\): Coordinates for beamforming. ```  
**Algorithm 1**Cell-Pointing Algorithm  
The implementation of SA for validating Algorithm 1 is similar to the approach proposed in [10]. Firstly, the initial boresight coordinates are set as the resulting coordinates from Algorithm 1. Then, a coordinate \((x,y)\) is randomly selected and modified as \((x+\delta x,y+\delta y)\), where \(\delta x\) and \(\delta y\) are random scalars drawn from standard normal distribution. Cells are formed using the previous and updated coordinates with users associated to the cells. The sum user CINRs with the previous and updated coordinates are evaluated as \(\gamma_{SA}^{i-1}\) and \(\gamma_{SA}^{i}\), respectively. If \(\gamma_{SA}^{i}>\gamma_{SA}^{i-1}\), the coordinate adjustment is accepted; otherwise, it is only accepted with a probability of \(e^{-\frac{V_{SA}^{i}-\gamma_{SA}^{i-1}}{T}}\), where \(T^{\circ}\) is the temperature of the annealing process. The entire process is repeated with \(T^{\circ}\) decreased after each run as \(T_{i+1}^{\circ}=pT_{i}^{\circ}|p\in[0,1]\), where \(p\) is the decay factor. The percentage of coordinates adjusted to improve user CINR compared to the initial coordinates from the proposed beam-pointing algorithm is evaluated for varying \(T\) and \(p\) and presented in Table 1. Note that only about 5% of the boresight coordinates obtained from the beam-pointing algorithm are adjusted to improve user CINR. This adjustment can be ignored considering the added complexity of running simulated annealing on the overall HAP system with its characteristic energy and weight limitations, and other wider environmental factors that will affect performance, which highlights the practical benefits of the beam-pointing algorithm.