## 2 Digital Beamforming Theory Derivation
From Equation (9), it can be seen that the noise figure at the output decreases according to the rule \(10 LogX\), meaning that the corresponding receiver performance improves by \(10 LogX\). In practical engineering, the receiver performance can also be improved by increasing the antenna aperture area, performing coherent signal accumulation, and canceling out uncorrelated noise [15; 16; 17]. In the above derivation process, for the sake of simplicity, assumptions and approximations were made in Equations (7) and (9). However, in actual engineering problems, the inconsistencies in amplitude and phase between channels can lead to discrepancies between the actual and theoretical calculations. Nevertheless, this does not affect our ability to use the derived results for the analysis and design of the phased array antenna receiver. Figure 5 illustrates the signal processing flow of a digital array radar with two signal paths [17; 20], using a two-element linear array as an example [21; 22; 23]. DBF is implemented in baseband in digital form. By applying appropriate delay or phase shift, the desired signal incident from a certain direction can be added in phase to produce maximum output. From the previous derivation of beamforming gain and noise figure, it can be concluded that for a DBF network with two signals which is shown in Figure 5, the signal strength after beamforming is  
Figure 4: Schematic diagram of the beamforming array. \[x_{7}(t)=|x_{6,1}(t)|+|x_{6,2}(t)| \tag{10}\]  
When \(x_{6,1}(t)\) and \(x_{6,2}(t)\) are coherently added, the signal strength of DBF can reach its maximum value, which is twice the original expected direction of the input signal, that is,  
\[x_{7\max}(t)=|x_{6,1}(t)|+|x_{6,2}(t)|_{\varphi(x_{6,2})=\varphi(x_{6,1})}=2|x_ {6,1}(t)| \tag{11}\]  
According to Equation (9), the noise figure decreases by \(10 Log2\) times. Extending this to an N-element array, the gain becomes \(N\) times the original, and the noise figure decreases by \(10 LogN\). For the 32-element spherical array antenna in this design, where \(N=32\), the summed signal strength becomes 32 times that of the original single reception channel. For the beamforming network of two signals shown in Figure 5, the key to beamforming lies in determining the phase shift values \(\varphi(x_{6,1})\) and \(\varphi(x_{6,2})\) in Equation (11). In practical engineering applications, the problem of determining the phase shift values is transformed into the problem of determining the beamforming weights for each signal. Therefore, in the two-signal beamforming network, the determination of the phase shifts \(\varphi(x_{6,1})\) and \(\varphi(x_{6,2})\) can be simplified to the determination of the beamforming weights \(a\) and \(b\). The process of phase correction for a single channel to achieve DBF inside the FPGA is shown in Figure 6. The entire DBF module operates at a rate of 200 M, with the NCO using a 32-bit control word and a 6-bit bit width, essentially functioning as a complex multiplier. All operations are performed on the JFM7K325T chip produced by Fudan Microelectronics (Shanghai, China), which contains a large number of DSP units to handle multi-beam digital down-conversion and phase-shifting processing. The implementation method is that the signal is first multiplied by the local NCO to obtain the I/Q signals. Then, the I/Q data is multiplied by the beamforming weights \(a\) and \(b\) using complex multiplication to obtain the I/Q signals. After the 32 signals are summed, filtering is applied to obtain the final synthesized beam, completing the beamforming function. In the case of a beamforming network with \(N\) channels, the beamforming weight for each channel signal needs to be determined. For the LEO satellite broadcast signal receiving terminal in this design, a spherical cap array is used for the antenna array, and the specific derivation process for the beamforming weights is as follows. Figure 5: Signal processing flow diagram of a digital array radar with a two-path antenna array.