# Delivering Extended Cellular Coverage and Capacity Using High-Altitude Platforms
## 2 System Model and Performance Metrics
### Beam Deployment  
Consider a quasi-stationary HAP located at an altitude \(h_{p}\) above sea level and at the centre of a service area of radius \(R\). Conventionally, \(h_{p}\) ranges between 17-22 km due to the considerably lower wind speeds at these altitudes, which enables HAPs to use lower energy for station keeping compared with other altitudes. Generally, most HAP studies and projects consider an altitude of 20 km [4]. The HAP supports a radio unit (RU) with a multi-element uniform planar phased array antenna of \(M\times N\) antenna elements. Multiple beams are formed from the antenna such that the footprints of the main lobes provide coverage to a set of users \(\mathcal{I}\) within the service area, as shown in Figure 1. These beams are pointed at a set of coordinates \(\mathcal{C}\), such that the resulting footprints produce a regular tessellated structure of contiguous cells \(c\in\mathcal{C}\) over an extended HAP service area. Each user associates to a cell that maximises its received signal strength. The HAP transmit antenna gain profile \(G_{i}^{t}\) for signal quality evaluation, as observed by user \(i\in\mathcal{I}\), is given as follows [24],  
\[G_{i}^{t}=g_{e}AF_{i}^{x}AF_{i}^{y}, \tag{1}\]  
where \(g_{e}\) is the gain of an isotropic antenna element. Array factors \(AF_{i}^{x}\) and \(AF_{i}^{y}\) are given as  
\[AF_{i}^{x}=\sum_{n=1}^{N}I_{n1}e^{i(n-1)(kd^{x}\sin(\theta_{i})\cos(\varphi_{i })+g_{i}^{y})}, \tag{2}\]  
\[AF_{i}^{y}=\sum_{m=1}^{M}I_{1m}e^{i(m-1)(kd^{y}\sin(\theta_{i})\sin(\varphi_{i })+g_{i}^{y})}, \tag{3}\]  
where angular wave number \(k=\frac{2\pi}{A_{s}}\), \(A_{s}\) is the wavelength, \(d^{x}\) and \(d^{y}\) are the inter-element spacings in the \(x\)- and \(y\)-axes of the antenna array with array factors \(AF_{i}^{x}\) and \(AF_{i}^{y}\) at user \(i\), respectively. \(I_{nj}\) and \(I_{jm}\) represent the excitation amplitudes of the antenna elements as depicted by Figure 2, \(\theta_{i}\) and \(\varphi_{i}\) define user elevation and azimuth angles as seen by the HAP. \(\beta_{x}=-kd^{x}\sin(\theta_{i}^{0})\cos(\varphi_{i}^{0})\) and \(\beta_{y}=-kd^{y}\sin(\theta_{i}^{0})\sin(\varphi_{i}^{0})\) are phase shifts with \(\theta_{i}^{0}\) and \(\varphi_{i}^{0}\) being the boresight elevation and azimuth angles, respectively. Figure 1: HAP phased array antenna beamforming for cellular coverage. Coordinates of the multiple beam boresight points on the ground, which are obtained as a set of cell coordinates \(\mathcal{C}\) from the proposed algorithm and then converted into their corresponding elevation and azimuth angles of the HAP from the ground, are supplied to the beamformer implementing (1) to obtain the distribution of HAP antenna transmit gain on the ground. The beams formed are then used to create HAP cells on the ground. In defining a cell in this paper, let \(\tilde{\mathcal{A}}_{b}\) be the footprint of the \(b\)th beam on the ground, \(p\) be any interior point in \(\tilde{\mathcal{A}}_{b}\) and \(\Gamma_{p}\) be the carrier-to-noise ratio (CNR) at point \(p\), the cell \(c\) is a bounded region around the beam boresight with boundary \(\partial c:=p\in\tilde{\mathcal{A}}_{b}\;:\;\forall p,\Gamma_{p}\geq 9\) dB. The 9 dB value is a global system for mobile communications (GSM) standard for cell delineation [25], which ensures that every user in a cell can receive signal with good quality. \(\Gamma_{i}\) for user \(i\) is evaluated as  
\[\Gamma_{i}=\frac{P_{i}^{t}G_{i}^{t}G_{i}^{t}}{NL_{i,h}}, \tag{4}\]  
where \(P_{i}^{t}\) is the HAP transmit power, and \(G_{i}^{t}\) and \(N\) are the antenna gain and noise power of the receiver, respectively. The path loss \(L_{i,h}\) between user \(i\) and the HAP \(h\) is modelled as free-space path loss with log-normally distributed fading due to shadowing [14], which follows the 3 rd generation partnership project (3 GPP) non-terrestrial network (NTN) channel model [26] and allows for a realistic large-scale representation of the HAP propagation channel.