# Analysis and Design of an X-Band Reflectarray Antenna for Remote Sensing Satellite System
## 5 Flat-Top Pattern Synthesized Using a Genetic Algorithm  
In order to start the optimization process, the requirements of the flat-top pattern that can be applied in the optimization procedure should be first introduced. To obtain a nearly constant communication link along the coverage angle, an antenna with a flat-top pattern is required. The maximum coverage angle \(\theta_{max}\) is defined from the Earth-satellite geometry shown in Figure 5, as:  
\[\theta_{max}=\cos^{-1}\biggl{(}\frac{d_{max}}{h_{o}+R_{e}}\biggr{)} \tag{5}\]  
where \(R_{e}\) is the radius of the Earth and \(h_{o}\) is the vertical distance from the satellite to the Earth's surface. \(d_{max}\) is maximum distance from the satellite to the Earth given by the trigonometric equation as:  
\[d_{max}=\sqrt{\left(R_{e}+h_{o}\right)^{2}-R_{e}{}^{2}} \tag{6}\]  
Figure 4: Phase distribution of the pencil beam reflectarray: (**a**) broadside beam, (**b**) tilted beam with a tilting angle of \(15^{\circ}\). where \(\Delta\theta\) is the allowable angle between the upper and the lower mask. To achieve the flat-top radiation pattern, the phase distribution on the reflectarray elements should be determined through an optimization of a properly defined cost function. The cost function is defined as the error between the obtained normalized array factor \(AF\) and the required upper and lower masks. The cost function is normalized for \(N_{total}\) angles for both the upper and lower limits of the mask as follows:  
\[cost=\frac{\varepsilon_{u}(\theta_{i})+\varepsilon_{l}(\theta_{i})}{N_{total}} \tag{10}\]  
where \(\varepsilon_{u}(\theta_{i})\) and \(\varepsilon_{l}(\theta_{i})\) are given by Equations (11) and (12), respectively:  
\[\varepsilon_{u}(\theta_{i})=\sum_{i}^{N_{total}}[AF(\theta_{i})_{\text{dB}}- Mask_{u}(\theta_{i})_{\text{dB}}]\frac{[1+\text{sgn}(AF(\theta_{i})_{\text{dB}}- Mask_{u}(\theta_{i})_{\text{dB}})]}{2} \tag{11}\]  
\[\varepsilon_{l}(\theta_{i})=\sum_{i}^{N_{total}}[Mask_{l}(\theta_{i})_{\text{ dB}}-AF(\theta_{i})_{\text{dB}}]\frac{[1+\text{sgn}(Mask_{l}(\theta_{i})_{\text{dB}}- AF(\theta_{i})_{\text{dB}})]}{2} \tag{12}\]  
where \(AF(\theta_{i})_{\text{dB}}=20\log(|AF(\theta_{i})|)\) and \(\text{sgn}(x)=1\) for \(x>0\) and \(\text{sgn}(x)=-1\) for \(x<0\). Thus, a better match between the obtained pattern and the required pattern is obtained for the minimum value of this cost function. After determining the required radiation pattern for the proposed MEO satellite communication system and the amplitude distribution of the fields due to the feeding element, the corresponding phase distribution on the aperture of the reflectarray must be determined. This phase distribution is obtained using GA. In GA, the optimization starts with an initial population comprising a number of candidate solutions (designated as chromosomes). These parents are controlled using different factors (combination, crossover, or mutation) to make a new set of chromosomes for the next generation. During the advancement of the arrangement, chromosomes are reviewed with respect to the enhancement of the fitness between the obtained radiation pattern and the required mask. The higher-positioned chromosomes are chosen to proceed to the next generation. Once the new generation is formed, the fitness of its chromosomes is estimated and the process continues until the convergence condition is satisfied. The algorithm stops when the value of the fitness function for the best point in the current population is less than or equal to the fitness limit. The important basic genetic algorithm steps are presented in Figure 7.  
The main problem when applying GA is the large number of optimization variables, which correspond to all reflecting elements on the reflectarray. This large number of variables requires a large computational time, which affects the overall convergence of the optimization process.