# SATCOM Earth Station Arrays Anti-Jamming Based on MVDR Algorithm
## 2 Satellite Communication Earth Station Array Model  
Reflector antennas have the advantages of simple structure and good directionality. Portable small satellite communication earth stations usually use this type of antenna, with a gain of [14]  
\[[G(0)]=10\lg\left(\frac{4\pi A\eta}{\lambda^{2}}\right)=10\lg\left(\left(\frac{ \pi D}{\lambda}\right)^{2}\eta\right) \tag{1}\]  
In the formula, \(A\) is the antenna area, \(D\) is the antenna diameter, \(\lambda\) is the signal wavelength, and \(\eta\) is the antenna efficiency. Variables in this article with square brackets represent their dB values. The half-power beamwidth of the antenna can be expressed as  
\[\theta_{1/2}=70^{\circ}\frac{\lambda}{D} \tag{2}\]  
When the error of the antenna main beam deviation from the satellite direction (antenna pointing error) is less than half the power beamwidth, the antenna gain can be estimated using the following formula [15]  
\[[G(\theta)]=[G(0)]-12\left(\frac{\theta}{\theta_{1/2}}\right)^{2} \tag{3}\]  
In the formula, \(\theta\) is the antenna pointing error. When the ratio of the diameter of the receiving antenna to the wavelength is less than 50, the gain of the antenna sidelobes is not greater than the following formula [16; 17]  
\[[G(\theta)]=\left\{ \begin{matrix}32-25\lg\theta&100^{\circ}\frac{\lambda}{D} \leq\theta\leq 48^{\circ}\\ -10&\theta>48^{\circ}\end{matrix} \right. \tag{4}\]  
In this article, Equation (4) is used to represent the antenna gain for the corresponding angle. The gains of point and point are connected by a straight line to represent the antenna gain between two angles (the method of representing negative angle gains is similar). Therefore, the gain of the satellite communication earth station reflector antenna can be expressed as  
\[[G(\theta)]=\left\{ \begin{matrix}10\lg\left(\left(\frac{\pi D}{\lambda}\right) ^{2}\eta\right)-12\left(\frac{\theta}{\theta_{1/2}}\right)^{2}&|\theta|\leq 35 ^{\circ}\frac{\lambda}{D}\\ \frac{\beta}{\gamma}+10\lg\left(\left(\frac{\pi D}{\lambda}\right)^{2}\eta \right)-3&35^{\circ}\frac{\lambda}{D}<|\theta|\leq 100^{\circ}\frac{\lambda}{D}\\ 32-25\lg\theta&100^{\circ}\frac{\lambda}{D}<|\theta|\leq 48^{\circ}\\ -10&|\theta|>48^{\circ}\end{matrix} \right. \tag{5}\]  
In the formula, \(\beta\) and \(\gamma\) can be expressed as  
\[\beta =35-10\lg\left((100)^{\frac{5}{2}}\pi^{2}\eta\left(\frac{\lambda}{D} \right)^{\frac{1}{2}}\right) \tag{6}\]  
\[\gamma =65\frac{\lambda}{D\cdot\left(\theta-\frac{1}{2}\theta_{1/2}\right)} \tag{7}\]  
Consider N narrowband signals/jammings entering the uniform linear array composed of reflector antennas of the earth station (as shown in Figure 1) in the far field. Assuming that the angle between the earth station reflector antenna and the satellite is \(0^{\circ}\), the jamming is in the plane composed of the array and the incoming wave from the satellite. The distance between the reflector antennas is \(d\), and the first reflector antenna onthe left (point \(O\)) is the reference antenna. Then, the position of the lth reflector antenna is (\((l-1)\)\(d\),0). Assuming that the signal incident angle is \(\theta_{0}\), the i-th jamming incident angle is \(\theta_{i}\), and the gain of the earth station reflector antenna for it satisfies Equation (5), let the gain of the lth reflector antenna for the signal be [\(G_{l}(\theta_{0})\)], and the gain for the i-th jamming be [\(G_{l}(\theta_{i})\)].