## 1 Introduction
The more types of prior information errors there are, the more complex the correction process becomes, and the more demanding the correction environment. To address this, quantitative analysis of the range of prior information errors and their impact on receiver performance is necessary to simplify the calibration process for the antenna array and ensure the performance of DBF algorithms. Since prior information-based methods are limited by errors, this proposal also incorporates a method combining spatial domain search for beam control [18]. After completing the beam control step, the beamforming process begins. Beamforming involves determining the required amplitude and phase weighting coefficients for each array element in different directions. In engineering implementations, there are two main methods for calculating the weighting coefficients: real-time calculation and lookup table calculation. Real-time calculation uses certain decision criteria (e.g., detecting the signal power after beamforming) and adaptive algorithms to adjust the weighting coefficients. The advantage of this method is that it adjusts the coefficients in real time based on the synthesized signal characteristics, suppressing side lobes and optimizing the received signal. For the angle-of-arrival (AoA) estimation of low Earth orbit (LEO) satellite receiving terminals, [PERSON] and [PERSON] et al. proposed a method for coarse positioning based on AoA estimation of signals of opportunity from LEO satellites, using a dedicated hardware architecture and real-time phase interferometry, achieving angular position tracking of fast-moving spacecraft with a precision of tenths of a degree [19]. It can adapt to different application environments, but it requires high computational power and optimized hardware platforms. The lookup table method calculates the weighting coefficients for different directions through simulation analysis, stores them in memory, and retrieves the appropriate coefficients for beamforming based on specific pointing parameters. This method has the advantage of low computational load, fast execution speed, and simple algorithm design, but it cannot adjust the coefficients in real time and is less adaptable to interference signals. Recently, in the literature, there have been proposed techniques which are able to estimate parameters like the angle-of-arrival in real time using fully digital reconfigurable architectures synthesized on FPGA. Some of those have been employed to build positioning systems based on the AoA estimation of signals of opportunity coming from the LEO orbit, on which basis was the estimation of the LEO satellite position itself. However, con sidering that LEO satellites have short communication times and fast beam switching, the lookup table method is adopted in this proposal for LEO satellites [11]. This paper addresses the issue of suboptimal beamforming performance in digital beamforming modules of LEO satellite signal receiving terminals, where beamforming precision and gain are limited. Based on a hemispherical array for LEO satellite broadcast signal reception and processing, a beamforming method is proposed that combines prior information and spatial search for beam control, along with the use of a lookup table for beam synthesis. The method is theoretically demonstrated and experimentally validated. The research content of this paper includes:  
(1) Deriving the formulas for the SNR and noise figure in the DBF network and analyzing the case of some array elements failure. Using a two-element linear array as an example, the signal processing process of a digital radar array with two channels is demonstrated, and the final processing results are extrapolated to an N-element array scenario. The process of phase adjustment and weighted summation of multiple array signals in DBF is shown. Finally, the method for determining the beamforming weights in a spherical array antenna system is derived.