# Delivering Extended Cellular Coverage and Capacity Using High-Altitude Platforms
## 2 System Model and Performance Metrics
### Performance Metrics  
In deciding the cell a user should associate with after the formation of cells, and whether service can be provided at a given performance level, the user's received power, expressed mathematically below, is used. Additionally, other metrics for performance evaluation are presented below.  
1. CINR \(\gamma_{i}\): For users already associated to cells, their performance with the proposed algorithm is evaluated using CINR \(\gamma_{i}\), which expresses the ratio of their carrier power to both interference and noise, which are further described below. For user \(i\), \(\gamma_{i}\) is defined as [14; 15] \[\gamma_{i}=\frac{P_{i}^{r}}{\sum\limits_{j=1}^{I}P_{i}^{j}+N_{i}},\] (8) where \(P_{i}^{r}\) is the user \(i\)'s useful received signal power, \(\sum\limits_{j=1}^{I}P_{i}^{j}\) is the sum of interference from all \(J=|\mathcal{C}|-1\) interfering beams, and \(N_{i}\) is user \(i\)'s thermal noise power, which is a zero mean Gaussian white noise and a function of system temperature and bandwidth (see Section 5 ). \(P_{i}^{r}\) is expressed as \[P_{i}^{r}=\frac{P_{i}^{t}G_{i}^{r}G_{i}^{t}}{L_{i,h}}.\] (9)where \(G_{i}^{r}\), is the receive antenna gain given in Section 5, \(G_{i}^{t}\) and \(L_{i,h}\) are as defined in (1) and (5), respectively. For interference, considering full reuse, all non-serving \(J\) beams can interfere with user \(i\) in the reference beam. The interference \(P_{i}^{j}\) on user \(i\) in the reference beam from the \(j\)th interfering beam is  
\[P_{i}^{j}=P_{j}^{r}G_{ij}^{t}, \tag{10}\]  
where \(P_{j}^{r}\) is the \(j\)th interfering beam receive power as per (9), and \(G_{ij}^{t}\) is the leakage antenna gain between the \(j\)th interfering beam and user \(i\) evaluated using (1)-(3) with \(\beta_{x}\) and \(\beta_{y}\) phase shifts based on \(\theta_{j}^{0}\) and \(\varphi_{j}^{0}\) denoting the elevation and azimuth angles at the centre of the \(j\)th interfering beam [28].
2. Throughput \(T_{i}\): This is evaluated per user in bits/s/Hz using the Truncated Shannon Bound expression as given below [29]. \[T_{i}=\begin{cases}0,&\gamma_{i}<\gamma_{min}\\ \alpha\log_{2}(1+\gamma_{i}),&\gamma_{min}\leq\gamma_{i}\leq\gamma_{max}\\ \alpha\log_{2}(1+\gamma_{max}),&\gamma_{i}>\gamma_{max},\end{cases},\] (11) where \(\alpha=0.65\) is the implementation loss, \(\gamma_{min}=1.8\) dB is the minimum allowed CINR and \(\gamma_{max}=22\) dB is the CINR resulting in the maximum achievable throughput [15]. \(\gamma_{min}\) is assumed for an acceptable long term evolution (LTE) signal quality [30], while \(\gamma_{max}\) relates to the LTE signal quality mapping with modulation and coding scheme, where maximum capacity is achieved at around 22 dB CINR [31].
3. Average Spectral Efficiency \(\eta_{i}^{f}\): Let the average capacity per user in cell \(i\) and the bandwidth allocated to a user in the cell pointed at any given distance be \(C_{pu}^{\vec{j}}\) (expression to be derived later) and \(B_{i}\), respectively. Thus, \(\eta_{i}^{f}~{}\triangleq~{}\frac{C_{pu}^{\vec{j}}}{B_{i}}\).
4. Average Area Spectral Efficiency \(S_{\eta_{j}}^{\vec{j}}\): This is the ratio of average spectral efficiency \(\eta_{f}^{i}\) to the area of the cell \(A_{i}\) in bit/s/Hz/km\({}^{2}\) for cell \(i\)[32; 33]. Thus, \(S_{\eta_{f}}^{\vec{j}}~{}\triangleq~{}\frac{\eta_{f}^{i}}{A_{i}}\).  
Several notations have been used throughout this paper. Notations part summarises the most prevalent notations used.