## 2 Parabolic Antenna Analysis Method
### Induced Amplitude Calculation  
In a parabolic reflector surface replaced by charge point sources, each charge point source is influenced by the radiated electromagnetic waves from the feeder, resulting in induced signals. In this paper, we derived the amplitudes of the induced signals for each charge point source based on such influences. Here, the impact from the feeder is considered using only scalar functions, and the electromagnetic field theory for induced signal analysis is simplified. The amplitude of induced signals is divided into two categories. The first is the influence of the electric field intensity due to wave propagation from the feeder, and the second is the influence of the radiation pattern of the feeder. In the electric field integral equation (EFIE) [31] used to determine the electric field intensity at observation positions (the positions of each charge point source) resulting from wave propagation from the feeder, the electric field intensity based on the far-field approximation is inversely proportional to the distance between the feeder and the observation point. In this paper, relying on this phenomenon, we took the decrease in electric field intensity at each charge point source location as the first induced signal amplitude. Therefore, the induced signal amplitude due to wave propagation from the feeder is defined as in Equation (11). \[I_{n}^{R}=\frac{1}{\left|\overrightarrow{r_{f}}-\overrightarrow{r_{n}} \right|} \tag{11}\]  
where \(\overrightarrow{r_{f}}\) is a position vector according to the position \(\left(f_{x},\ f_{y},\ f_{z}\right)\) of the feeder, and \(\overrightarrow{r_{n}}\) is a position vector according to the position \(\left(P_{n}^{x},\ f_{y}^{y},\ f_{z}\right)\) of the \(n\)-th charge point source. Generally, the feeder of a parabolic reflector antenna is composed of a horn antenna, which exhibits a directional radiation pattern. Therefore, the electric field intensity at each charge point source is influenced by the radiation pattern of the feeder. The amplitude of the induced signal at each charge point source can be defined as in Equation (12), where the radiation pattern of the feeder is assumed by the cosine-q model [32] and the angle between the beam axis of the feeder and each point source is denoted by \(\Theta_{n}\). \[I_{n}^{P}=\cos^{q}(\Theta_{n}) \tag{12}\]  
For a defocus-fed parabolic antenna, not only is the position of the feed axis displaced, but the beam axis may also not steer toward the center of the reflector. Therefore, in calculating \(\Theta_{n}\) in Equation (12), the position \(\left(f_{x},\ f_{y},\ f_{z}\right)\) of the feeder and the beam steering angle \(\left(\theta_{s},\ \ \phi_{s}\right)\) need to be considered. In this paper, considering the influence of the feeder configuration, each charge point source is moved and rotated according to Equation (13), and \(\Theta_{n}\) is calculated based on Equation (14). \[ \begin{bmatrix}R_{n}^{Px}\\ R_{n}^{Py}\\ R_{n}^{Pz}\\ \end{bmatrix} = \begin{bmatrix}\cos(\pi-\theta_{s})\cos(-\phi_{s})&-\cos(\pi- \theta_{s})\sin(-\phi_{s})&\sin(\pi-\theta_{s})\\ \sin(-\phi_{s})&\cos(-\phi_{s})&0\\ -\sin(\pi-\theta_{s})\cos(-\phi_{s})&\sin(\pi-\theta_{s})\sin(-\phi_{s})&\cos (\pi-\theta_{s})\\ \end{bmatrix}\begin{bmatrix}P_{n}^{x}-f^{x}\\ P_{n}^{y}-f^{y}\\ P_{n}^{z}-f^{z}\\ \end{bmatrix}  \tag{13}\]  
\[\Theta_{n}=\frac{\sqrt{\left(R_{n}^{Px}\right)^{2}+\left(R_{n}^{Py}\right)^{ 2}}}{RP_{n}^{2}} \tag{14}\]  
Figure 3 shows the geometry of the divided parabolic reflector surface for \(\left(P_{n}^{x},\ \ P_{n}^{y},\ \ P_{n}^{z}\right)\) based on Equations (8)-(10) and \(\left(R_{n}^{Px},\ R_{n}^{Py},\ R_{n}^{z}\right)\) based on Equation (13).