## 3 Effect of Error Sources on Antenna Gain
### The Influence of the Scanning Error on Antenna Gain  
It can be seen from Equation (8) that \(\Delta u\) and \(\Delta v\) are related to the far-field coordinates (\(u,v\)), so the two must have one-to-one correspondence when using FFT, which is difficult to deal with. Therefore, this section uses the discrete Fourier transform (DFT) to solve the far-field pattern caused by the scanning error. In the process of antenna microwave holographic measurement, the angle encoder measures and records the azimuth angle and elevation angle of the antenna in real time and compares them with the predetermined direction to obtain the scanning error \(\Delta AZ\) and \(\Delta EL\) of each scanning point. The offsets \(\Delta u\) and \(\Delta v\) of the coordinates (\(u,v\)) of each scanning point are calculated and brought into Equation (8) to solve the far-field pattern caused by the scanning error. Due to the randomness of the scanning error, it cannot be directly quantitatively analyzed. The scanning error of each scanning point in the whole scanning area can be statistically analyzed by the method of probability statistics, and the influence of the scanning error on the far-field pattern of the antenna can be studied. The mean represents the pointing error of the system, and the pointing is offset by a fixed value as a whole, which leads to the phase error of the linear distribution aperture surface, which can be deducted by two-dimensional linear fitting of the aperture surface phase. The standard deviation represents the scanning error, which describes the degree of deviation between the random scanning error and the mean. In order to clearly demonstrate the effect of scanning error on the antenna far-field pattern, the antenna simulation parameters are changed so that the antenna diameter is 60 m and the rest of the parameters remain unchanged. The scanning errors \(\Delta AZ\) and \(\Delta EL\) of each sampling point are randomly generated by using the Gaussian distribution with the mean \(\mu\) being zero and the standard deviation being 25 arcsec, and the far-field pattern is calculated, as shown in Figure 8. The absolute values of all the generated \(\Delta AZ\) and \(\Delta EL\) are averaged to get the average value of the elevation error and azimuth error for the whole scanning area, which is about 20 arcsec. The random scanning error causes the main lobe of the far-field pattern to become uneven, and the side lobes show left-right asymmetry. Due to the randomness of the scanning error, it may cause distortion in the far-field directional map, producing a large gain loss. With the increase in the antenna aperture surface, the effect of the scanning error will become more and more obvious. The phase error of the antenna aperture field is solved by Equation (10) and then brought into Equation (12) to solve the relationship between the effect of the scanning error on the antenna gain loss by using the solved gain to make a ratio with the ideal gain. In order to understand more deeply the effect of antenna diameter on the gain loss caused by scanning error, we conducted a series of simulation experiments, as shown in Figure 9. When the standard deviation of the scanning error is controlled at 30 arc seconds, the gain loss caused by the 30 m radio telescope antenna is about \(-\)0.014 dB, which is negligible. For a 110 m large-aperture, high-precision radio telescope antenna, the gain loss is about \(-\)0.36 dB, which is intolerable for the gain loss caused by a single error. This result indicates that the increase in antenna diameter can improve the signal reception ability of the antenna, and at the same time, it also makes the antenna more sensitive to the scanning error, which negatively affects the gain.