# Three-Dimensional Drone Cell Placement: Drone Placement for Optimal Coverage
## 3 System Model  
Expanding more on the A2G communication links, [PERSON] has proposed the A2G communication model taking into consideration the Line of Sight (LoS) and Non-Line of Sight (NLoS) probability links in different urban scenarios [27]. The link is calculated as:  
\[P_{LoS}(r,h)=\frac{1}{1+a\exp{(-b(\arctan(\frac{h}{r})-a))}} \tag{1}\]  
where h is the height of the DC (altitude), r is the horizontal distance between the user and DC, i.e., \(r=\sqrt{(x_{i}-x_{D})^{2}+(y_{i}-y_{D})^{2}}\)), where \((x_{D},y_{D})\) is the location of the DC in the horizontal dimension, \((x_{i},y_{i})\) denotes the location of the user, a and b are constant values which are determined by the terrestrial environment of the DC operation. Moreover, the probability of Non-Line of Sight is:  
\[P_{NLoS}=1-P_{LoS} \tag{2}\]  
In addition to the path loss suffered in free space propagation, there is additional path loss experienced due to the random behavior and structure of the specific environment. In this report, only the mean path loss values are taken into consideration since DC or BS placement deals with long-term variation rather than short-term behavioral changes in the signal model [34]. From the previous model, the path loss models for NLoS and LoS are:  
\[LoS=20\log(\frac{4\pi f_{c}d}{c})+\eta_{LoS} \tag{3}\]  
\[NLoS=20\log(\frac{4\pi f_{c}d}{c})+\eta_{NLoS} \tag{4}\]  
where \(d=\sqrt{h^{2}+r^{2}}\), d is the distance between DC and user, and \(\eta_{LoS}\) and \(\eta_{NLoS}\) are the average additional losses for LoS and NLoS links; these were calculated depending on the environment in [42]. The mean path loss covered by the LoS and NLoS probability link is [42]:  
\[PL(r,h)=P_{LoS}(r,h)\eta_{LoS}+P_{LoS}(r,h)\eta_{NLoS} \tag{5}\]  
where \(f_{c}\) is the frequency of the carrier and c is the speed of light. Let us consider \(A=\eta LoS-\eta NLoS\), \(B=20\log(\frac{4\pi f_{c}}{c})+\eta_{NLoS}\), \(\theta=\tan^{-1}(\frac{h}{r})\) and \(h^{2}+r^{2}=(\frac{r}{\cos\theta})^{2}\); substituting Equations (1), (3) and (4) into (5), we obtain:  
\[P_{LoS}(r,h)=\frac{A}{1+a\exp{(-b(\frac{180}{\pi}\theta-a))}}+20\log(\frac{r}{ cos\theta})+B \tag{6}\]  
For a UAV flying at height h, the coverage can be classified as a circular disk of radius R. Any users within the radius R can be considered as covered since the path loss experienced is less than or equal to the threshold (\(PL_{th}\)):  
\[R=r|_{\Lambda=PL_{max}} \tag{7}\]  
The optimal height of the DC was calculated in [42]. It can be said that for a given path loss, and coverage region R, there can be only one altitude h that maximizes the coverage region. Since R is an implicit function of h, the optimal height or the stationary point can be found by [29]:  
\[\frac{\partial R}{\partial h}=\frac{\partial R}{\partial\theta}\frac{\partial \theta}{\partial h}=0 \tag{8}\]  
where \(\frac{\partial\theta}{\partial h}=\frac{\partial}{\partial h}\tan^{-1}(\frac{ h}{r})=\frac{r}{R^{2}+r^{2}}>0\). Finding h that maximizes R can be performed by searching for optimal \(\theta\) or \(\theta_{opt}\). Solving \(\frac{\partial R}{\partial\theta}=0\) yields the equation [42]:  
\[\frac{\pi}{9\ln(10)}\tan\theta_{opt}+\frac{abA\exp(-b(\frac{180}{\pi}\theta_{ opt}-a))}{(a\exp(-b(\frac{180}{\pi}\theta_{opt}-a))+1)^{2}}=0 \tag{9}\]  
Solving Equation (9) numerically yields the following optimal elevation angles for different environments, which were calculated in [29] and tabulated in Table 1:With \(PL_{thr}\), \(\theta_{opt}\) known, the maximum coverage radius was calculated using:  
\[P_{NLoS}=1-P_{LoS} \tag{2}\]0  
the optimal height can be found by:  
\[P_{NLoS}=1-P_{LoS} \tag{2}\]1  
and finally, the coverage region can be calculated by:  
\[P_{NLoS}=1-P_{LoS} \tag{2}\]2  
Based on these experiments, it was discovered that the optimal DC height can exceed beyond the Earth's atmosphere, and hence, it was advised to add a height constraint on the DC placement problem [42]. This iteration of the DC problem is devoid of any power consumption or transmission power criteria.