## 3 Digital Beam Control and Beamforming Algorithm Design
The beam direction with the highest power is selected as the tracking result and assigned to the corresponding satellite's beamforming channel [30; 32]. The overall flowchart of the beam control process is shown in Figure 12.  
After completing the beam control process, the system transitions to the beam synthesis process. As described in Section 1, the primary task of beam synthesis is determining  
Figure 12: Overall flowchart of beam control. the amplitude and phase weighting coefficients for array element signals in different directions. In this scheme, beam synthesis is implemented using a lookup table method. The lookup table-based beam synthesis approach stores the DBF parameters in RAM. In this design, the DBF parameters are updated every \(5^{\circ}\) change in elevation or azimuth angles [33]. The spherical-cap antenna array in this design has an elevation angle range of \(15^{\circ}\)\(\sim\)\(85^{\circ}\) and an azimuth angle range of \(5^{\circ}\)\(\sim\)\(360^{\circ}\). Thus, the parameters are divided into \([(85-15)/5+1]\times[(360-5)/5+1]=1080\) groups. Actual verification ensures that within a \(5^{\circ}\) range, the DBF gain variation does not exceed 1.5 dB. The DBF parameter bit width is 10 bits, corresponding to an elevation angle precision of \(90^{\circ}/1024=0.09^{\circ}\) per bit and an azimuth angle precision of \(360^{\circ}/1024=0.35^{\circ}\) per bit. During data reception, signals undergo processing through the antenna, mixing, cable transmission, amplification, filtering, and ADC sampling, which introduces phase shifts. Thus, when determining the beamforming weights, the hardware-induced phase offsets must be considered. These phase offsets are determined through phase detection. For this issue, [PERSON] et al. proposed a technique that simultaneously implements mutual coupling compensation and beamspace transform in a digital antenna array with low complexity [34]. [PERSON] and [PERSON] et al. proposed a method that uses precomputed lookup tables and satellite orbital positions for prediction, reducing the fixed C/N margin in adaptive coding and modulation systems by eliminating the time delays for signal transmission and modulation and coding rate switching [35]. By inputting signals with the same frequency, phase, and source into all channels, the phase differences between channels can be calculated using a phase reference method, yielding the phase values that require compensation [19]. Simulations show that with a 25 MHz signal, the final phase measurement error is less than 0.0002 radians. The corresponding time error is at the picosecond level, under 10 ps, meeting precision requirements. The method for extracting phase differences is illustrated in Figure 13[36; 37]. After the FPGA chip samples the raw data via the ADC, it performs down-conversion processing. The 31 ADC signals are uniformly down-converted to baseband signals and then passed through an FIR (low-pass filter) to filter out high-frequency components, retaining only the desired baseband signals. Each signal is then multiplied with the reference signal in a complex operation to produce two IQ signals containing phase differences. Figure 13: Schematic diagram of the phase difference extraction principle. Finally, integration and ARCTAN2(Q/I) operations are performed to obtain the desired phase difference angles for the 31 channels [38]. The phase difference calculation formulas are derived based on Figure 13 as follows:  
Let the signal frequency be \(f\), the initial phase be \(\varphi\), and the signal amplitude be \(A\).