## 2 Digital Beamforming Theory Derivation  
Before signal summation in DBF, the phase of the data from each channel needs to be adjusted to ensure that the phase errors are within an acceptable range. This allows for the summation of the channel data, thereby improving the received signal power and SNR. The software flow structure of DBF is divided into two steps: beam control and beamform  
Figure 2: The organizational structure of the entire paper. ing synthesis. The function of beam control is to align the main beam of the antenna array with the direction of the useful signal, so that the subsequent beamforming synthesis can maximize the gain of the combined useful signal. The purpose of beamforming synthesis is to combine the useful signals. In this scheme, the control method is a combination of prior information and spatial domain search, where logical control determines the beam direction. Then, the phase difference between the signals to be summed is extracted, and after compensating for the phase difference, the beamforming is completed. Figure 3 shows a typical dual-port single-channel receiver RF front-end [16; 20], where the noise power \(N\) at the output end is  
\[N=kT_{S}B=k\cdot(T_{A}+T_{e})\cdot B \tag{1}\]  
In the above equation, \(k\) is the Boltzmann constant, with a value of \(k\approx 1.38\times 10^{-23}\) J/K; \(T_{S}\) is the noise temperature at the output end of the receiver; \(T_{A}\) is the noise temperature at the input end of the receiver; and \(T_{e}\) is the equivalent noise temperature of the receiver's internal noise, which is equivalent to the input end. Its value is  
\[T_{e}=(NF_{S}-1)\cdot T_{A} \tag{2}\]  
After equivalence, the ideal receiver does not generate internal noise, that is  
\[T_{e}=0 \tag{3}\]  
\(B\) is the monitoring bandwidth of the receiver. One of the most commonly used detection metrics for detecting signals in noise is the signal-to-noise ratio \(S/N\). To ensure normal reception, the signal-to-noise ratio at the receiver input \(SNR_{i}\) must satisfy  
\[SNR_{i}=S_{i}/N_{i}\geq M \tag{4}\]  
\(S_{i}\) and \(N_{i}\) represent the input signal and input noise of the receiver, respectively, and \(M\) represents the recognition factor of the receiver. The minimum resolvable signal power of the receiver \(S_{\text{min}}\) is  
\[S_{\text{min}}=MN_{i} \tag{5}\]  
For the purpose of illustrating the impact of different receivers on sensitivity, let \(M=1\) and \(T_{a}=T_{0}\). According to Equations (1) and (3), we can obtain  
\[S_{\text{min}}=kT_{S}B=k\cdot T_{0}\cdot NF_{S}\cdot B \tag{6}\]  
Taking \(T_{0}=290K\), Equation (6) is transformed into dB form, expressed as  
\[ \begin{array}{l}\left|S_{\text{min}}\right|_{\text{dB}}=-228.6+24.6+10\text {lg}B(\text{MHz})+60+\left|NF_{S}\right|_{\text{dB}}\\ =-144+10\text{lg}B+\left|NF_{S}\right|_{\text{dB}}\end{array}  \tag{7}\]  
Figure 3: Schematic diagram of the dual-port single-channel receiver RF front-end. In the equation, the unit of \(B\) is MHz; the unit of \(NF_{5}\) is dB. Now we extend the case of a single-channel receiver to the situation where the receiver performs beamforming on multiple received signals. The receiving array with \(X\) elements is shown in Figure 4 [15]. Assuming that the noise figure of each branch in the beamforming network is \(nf\), the noise figure \(NF_{0}\) of the beamforming end's SNR relative to the SNR of a single branch in the beamforming network can be calculated as  
\[NF_{0}=nf\cdot\frac{1}{X} \tag{8}\]  
Converting Equation (8) into the dB form, we get  
\[\left|NF_{0}\right|_{dB}=10 Lognf-10 LogX \tag{9}\]  
From the above derivation, it can be seen that the SNR at the output of the beamforming network is increased by a factor of \(X\) compared to the SNR of the single-channel input. In other words, the SNR at the output of the phased array antenna is \(X\) times higher than the SNR of the single-channel input.