# An Optimized Planning Tool for Microwave Terrestrial and Satellite Link Design
## 3 Tool Description
### Free-Space Attenuation  
To calculate free-space attenuation \(L_{bf}\), the ITU-R Recommendation P.525-4 [21] was used. In this recommendation, this attenuation is obtained as follows:  
\[L_{bf}=32.4+20\log(f_{\left[\text{MHz}\right]})+20\log(d_{\left[\text{km} \right]}) \tag{1}\]  
where \(f\) is the signal frequency in MHz, and \(d\) is the link distance in km. To calculate diffraction attenuation, the ITU-R Recommendation P.526-15 [22] was used. In this recommendation, this attenuation is obtained using three different methods:  
* Single knife-edge obstacle, when one obstacle is found;
* Double-isolated edges, when two obstacles are found;
* Bullington model, when three or more obstacles are found. For a single knife-edge obstacle, the diffraction attenuation \(J(v)\) is obtained using the following equation:  
\[J(v)=-20\log\left(\frac{\sqrt{[1-C(v)-S(v)]^{2}+[C(v)-S(v)]^{2}}}{2}\right) \tag{2}\]  
where \(v\) is defined as  
\[v=h\sqrt{\frac{2}{\lambda}\left(\frac{1}{d_{1}}+\frac{1}{d_{2}}\right)} \tag{3}\]  
where \(\lambda\) is the used wavelength, and \(h\), \(d_{1}\), and \(d_{2}\) can be obtained following the images provided in P.526-15 [22], and \(C(v)\), and \(S(v)\) represent the Fresnel integrals, also described in the recommendation. For double-isolated edges, the diffraction attenuation \(L\) is obtained using the following equation:  
\[L=L_{1}+L_{2}+L_{c} \tag{4}\]  
or  
\[L=L_{1}+L_{2}-T_{c} \tag{5}\]  
where \(L_{1}\) represents the signal diffraction caused by the first obstacle in an imaginary connection between the transmission antenna and the second obstacle, and \(L_{2}\) represents  
Figure 5: Example of created latitude and longitude tree search. the signal diffraction caused by the second obstacle in an imaginary connection between the first obstacle and the reception antenna. \(T_{c}\) and \(L_{c}\) represent the correction term used. Equation (4) is used if \(L_{1}\) and \(L_{2}\) have a value greater than 15 dB and (5), if otherwise. For **the Bullington model**, the diffraction attenuation \(L_{b}\) is obtained as follows:  
\[L_{b}=L_{uc}+[1-\exp(-L_{uc}/6)](10+0.02d) \tag{6}\]  
where \(L_{uc}\) represents the knife-edge loss obtained as described in [22]. The model to predict the atmospheric attenuation is described in ITU-R Recommendation P:676-13 [23] and is obtained as follows:  
\[A_{a}=\gamma\times r_{0}=0.1820\times f\times(N_{Oxygen}^{\prime\prime}(f)+N_{ WaterVapour}^{\prime\prime}(f))\times r_{0} \tag{7}\]  
where \(r_{0}\) represents the path length, \(N_{Oxygen}^{\prime\prime}(f)\) and \(N_{WaterVapour}^{\prime\prime}(f)\) are the imaginary parts of the frequency-dependent complex refractivities. As defined in ITU-R Recommendation P.838-3 [24], the attenuation due to rain in relation to the rain rate \(R\) is obtained as follows:  
\[A_{r}=d\times\gamma_{R}=d\times kR^{\alpha} \tag{8}\]  
where the coefficients \(k\) and \(\alpha\), in cases of frequencies between 1 and 1000 GHz, are obtained as follows:  
\[\log_{10}k=\sum_{j=1}^{4}\left(a_{j}\exp\left[-\left(\frac{\log_{10}f-b_{j}}{ c_{j}}\right)^{2}\right]\right)+m_{k}\log_{10}f+c_{k} \tag{9}\]  
\[J(v)=-20\log\left(\frac{\sqrt{[1-C(v)-S(v)]^{2}+[C(v)-S(v)]^{2}}}{2}\right) \tag{2}\]0  
The values of the constants \(a_{j}\), \(b_{j}\), \(c_{j}\), \(m_{k}\), \(m_{\alpha}\), \(c_{k}\), and \(c_{\alpha}\) follow Tables 1-4 in Recommendation P.838-3 [24].