# Forward Link Optimization for the Design of VHTS Satellite Networks
## 3 Fundamentals and Analytical Models
### Spot Beam Model and Link Budget Calculation  
In a multibeam system, the coverage footprint of each beam can be described as the curve formed by the intersection of the conical model of the antenna beam with a spherical model of the Earth [22]. The variation in the number of beams depends on the beamwidth of the antenna beams. If the beamwidth is reduced, the area covered by each beam will be smaller, so the number of beams will be increased to match the total coverage area. This affects other parameters such as directivity, the diameter of the satellite antenna, as well as the definition of the feed cluster. If \(\theta_{x}\) represents the position of the user as a function of the angle between the center of the beam and the position of the receiver with respect to the satellite, the beam gain can be modelled as [23]:  
\[G(\theta_{x},b)=G_{max}(b)\bigg{(}\frac{J_{1}(u)}{2u}+\frac{36\,J_{3}(u)}{u^{3 }}\bigg{)}^{2} \tag{6}\]  
\[G_{max}(b)=\bigg{(}\frac{\lambda}{4\pi}\bigg{)}^{2}\frac{1}{{d_{0}}^{2}+{d(b)} ^{2}} \tag{7}\]  
where \(G_{max}(b)\) represents the gain at the \(b\)-\(th\) beam centre (\(1\leq b\leq B\)) when free space loss is taken into account, and the \(d(b)\) denotes the distance (in thousands of km) of the \(b\)-\(th\) beam centre from centre of the central beam and d\({}_{0}\simeq\) 35,786 km, \(u=2.07123 sin\theta_{x}/sin\theta_{3 MB}\), \(\theta_{3 MB}\) is the one-sided half-power beamwidth, and \(J_{x}\) is the first-kind Bessel function of order \(x\)[23]. The EIRP (Effective Isotropic Radiated Power) footprints are obtained by multiplying \(G(\theta_{x},b)\) by the power assigned to each beam. The beam gain must be calculated for the latitude and longitude (_lat_, _lon_) grid in the service area. The sidelobes of the satellite antenna pattern are taken into account only for the calculation of the co-channel interference induced by spatially separated co-channel beams. In [16], it is shown that the carrier-to-interference ratio (_CIR_) in each (_lat_, _lon_) decreases alongside the number of beams due to the presence of more cochannel beams. Co-channel interference power \(I\) depends on frequency reuse scheme and can be calculated considering the co-channel beams set in the system as [16]:  
\[I(\textit{lat},\textit{lon},\textit{b})=\sum\
olimits_{q=1}^{Q(b)}P_{\alpha}( \textit{q},\textit{lat},\textit{lon},\textit{b}) \tag{8}\]  
where  
* \(q\) represents the _q-th_ interferer spot,
* \(Q(b)\) is the total number of interfering beams of the beam \(b\), and
* \(P_{\alpha}\) is the power level of _q-th_ interference inside the _b-th_ beam in the geographical position (_lat_, _lon_). Rain attenuation for each point in the grid depends on the required link availability, elevation, frequency, and polarization. In this sense, the service area percentage will be the percentage of those geographic coordinates that meet the minimum _CINR_ requirement for the specified annual availability. A bent-pipe transponder architecture is considered in the satellite. The uplink is not considered in the forward link because there are different technologies to guarantee a value not affecting the total link budget, such as ULPC (Uplink Power Control), or gateway diversity [24; 25]. The total _CINR_ (in dB) in the downlink of the user link is calculated for each geographical position using (9):  
\[\textit{CINR}\ (\textit{lat},\textit{lon})=-10 log_{10}\Big{(}\textit{cnr}^{-1}( \textit{lat},\textit{lon})+\textit{cir}^{-1}(\textit{lat},\textit{lon})\Big{)} \tag{9}\]  
where _cnr_ and _cir_ represent the carrier-to-noise and carrier-to-interference ratios in natural units, respectively. The _CNR_ (dB) calculation is based on the well-known link-budget models.