## 1 Introduction
[12], [PERSON], [PERSON] and [PERSON] et al. [13] have explored hybrid beamforming designs for intelligent transportation systems, employing alternating optimization methods to design DBF vectors and phase shifts, thereby improving communication efficiency, as described in their study from 2024. DBF uses digital phase shifting to compensate for phase differences, caused by path differences resulting from inconsistent RF channel delays, and spatial distance differences in signals incident from a particular direction. This allows for coherent signal addition, thereby achieving the maximum energy reception in that direction and forming the beam to receive the desired useful signal[PERSON] Beamforming ensures phase consistency across multiple channels by correcting the signal phases, then directly summing the multi-channel data to increase signal power. The signal-to-noise ratio (SNR) after summing the data from \(N\) channels improves by a factor of \(N\) compared to the original single-channel network. During signal transmission, noise is approximated as white noise (with strong directional interference not considered, as it is dealt with through anti-jamming techniques), and the noise figure at the output decreases following a \(10 LogN\) rule. The carrier-to-noise ratio (CNR) improves by \(10 LogN\) times, which represents the ideal DBF gain, and a detailed derivation will be provided next. Furthermore, antennas using DBF have good self-calibration and low sidelobe performance, advantages that are not present in ABF, significantly impacting the improvement of receiver performance [14; 15]. DBF processes the received signals from the antenna array in the digital domain, using DBF algorithms to compute the optimal adaptive weighting vector. This directs the main lobe of the antenna pattern towards the desired direction, while steering the nulls towards interference directions, achieving spatial filtering. Array signal processing in the digital domain can use advanced digital techniques, making the processing process more flexible and controllable. When the array and target parameters are known accurately, DBF can provide an approximate optimal output SNR, enabling the antenna array to achieve good signal resolution and parameter estimation performance [6]. However, in practical applications, errors and non-ideal factors can cause deviations in the direction vector of the desired signal, leading to reduced gain or even cancellation of the desired signal, thus degrading output performance. Additionally, high computational complexity is a challenge for DBF in engineering applications. Therefore, efficient and robust beamforming algorithms have significant research and practical value [16]. In terms of DBF algorithm implementation, the FPGA implementation of DBF involves two steps: beam control and beamforming. In the beam control step, commonly used methods in the LEO satellite industry are based on prior information, with two technical approaches: one uses directional prior information, and the other uses waveform prior information. Array weighting criteria based on directional information include the BS criterion, mean signal-to-interference-plus-noise ratio (MSINR) criterion, and linearly constrained minimum variance (LCMV) criterion. In satellite navigation systems, the satellite signal direction can be computed using ephemeris data, receiver position, and array platform attitude, and combining the antenna array element positions to calculate the steering vector, thus guiding the antenna array's main beam towards the signal direction. However, these prior information sources may have errors, such as different accuracies in ephemeris data (e.g., almanac data, broadcast ephemeris, and precise ephemeris), and receiver position estimation errors may be several meters or even tens of meters [17]. Prior information errors are a key factor limiting the performance of DBF technology.