## 5 Errors Compensation
### Compensation Examples  
The feed offset error compensation method is verified by simulation experiments. The input parameters are as follows: the antenna aperture is \(d\) = 12 m, the wavelength is \(\lambda\) = 5 cm, the focal length is \(f\) = 4.8 m, and the edge illumination level is \(-\)12 dB. The feed offset error in the aperture field phase is fitted using two methods: (1) polynomial fitting [11] and (2) the proposed compensation method. In the polynomial fitting method, the fitting coefficients are obtained by two-dimensional nonlinear fitting of the aperture field phase, and the aperture field phase error due to the feed offset error is solved by the coefficient values, and the offset position of the feed is calculated by the coefficients. In the proposed method, using Equation (18), the most suitable predicted phase \(\varphi\) is calculated. The inverse solution of \(\delta a\) and \(\delta l\) gives the feeder offset position directly without the need to solve complex geometrical relations. Figure 14 displays the distribution of aperture field phase errors during the error compensation process for both methods. Table 5 lists the RMS errors of the antenna surface after compensation with the two methods. Under the influence of the feed offset error, the RMS error before compensation is 2.604 mm, and the RMS errors of method (1) and method (2) are reduced to 0.115 mm and 0.117 mm, respectively. Although the surface RMS error compensated with the polynomial method is smaller than that with the proposed method, it is evident from the compensation results in sub-figure (e) for the polynomial method and sub-figure (f) for the proposed method in Figure 14, that the surface error distribution in sub-figure (f) is more closely aligned with the original reflector surface error distribution (a), indicating a better compensation effect. This situation is because there is a big difference between the polynomial fitting aperture field phase and the actual result, resulting in too much deduction of the error phase after compensation, making the RMS error smaller. In the following, the simulation experiment is carried out to verify the scanning error compensation of the 12 m antenna. The input parameters remain unchanged, and two circles of protrusions are set on the ideal parabolic surface. The amount and position distribution of the protrusions are shown in Table 6, and the RMS error is 0.27 mm. Figure 15a shows the distribution of surface shape measurement error with scanning error, and the RMS error is 0.35 mm. Figure 15b shows the distribution of surface shape measurement error after compensation, and the RMS error is reduced to 0.27 mm. As shown in Table 7, the RMS error after compensation is equal to the RMS error of the actual setting, and the influence of the scanning error is basically eliminated. \begin{table}
\begin{tabular}{c c} \hline \hline
**Position (m)** & **Convexity (mm)** \\ \hline
0 \(<\) r \(\leq\) 1 & 0 \\
1 \(<\) r \(\leq\) 2 & 1 \\
2 \(<\) r \(\leq\) 3 & 0 \\
3 \(<\) r \(\leq\) 4 & 1 \\
4 \(<\) r \(\leq\) 5 & 0 \\
5 \(<\) r \(\leq\) 6 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Simulation setting of convex position and convex amount. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Terms** & \begin{tabular}{c} **The Antenna** \\ **Surface Error** \\ \end{tabular} & \begin{tabular}{c} **Before** \\ **Compensation** \\ \end{tabular} & \
\begin{tabular}{c} **The Polynomial** \\ **Compensation** \\ \end{tabular} &
\begin{tabular}{c} **The Proposed** \\ **Compensation** \\ \end{tabular} \\ \hline RMS error/mm & 0.120 & 2.604 & 0.115 & 0.117 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The antenna surface shape measurement error RMS.