## 2 Digital Beamforming Theory Derivation
Assume there are \(N\) mutually independent far-field input signals, with one desired signal and \((N-1)\) interference signals. The working wavelength is denoted as \(\lambda\). In the three-dimensional coordinate system, let \((r_{x_{k}},r_{y_{k}},r_{z_{k}})\) represent the coordinates of the \(k\)-th antenna element, where \(x\), \(y\), and \(z\) correspond to the three directions in three-dimensional space. The beam pointing direction is denoted as \((\theta_{0},\varphi_{0},\delta_{0})\), as shown in Figure 7[18; 24; 25]. Define \((u_{0},v_{0},w_{0})\) as the projection components of the beam pointing direction \((\theta_{0},\varphi_{0},\delta_{0})\) on the \(x\), \(y\), and \(z\) axes; \((u_{n},v_{n},w_{n})\) are the projection components of the incident wave direction \((\theta_{n},\varphi_{n},\delta_{n})\) on the \(x\), \(y\), and \(z\) axes; then \((u,v,w)=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)\). Let \(\omega_{k}\) be the analog complex weighting coefficient of the phase shifter for the \(k\)-th antenna element, \(s_{n}(t)\) be the time-domain signal of the \(n\)-th signal source, and \(n(t)\) be the spatial Gaussian white noise signal of the antenna array. The output signal of the antenna array can be expressed as  
\[x(t)=\sum_{k=1}^{K}\omega_{k}^{*}(u_{0},v_{0},\omega_{0})\sum_{n=1}^{N}e^{-j \frac{2\pi}{\lambda}(r_{x_{k}}u_{n}+r_{y_{k}}v_{n}+r_{z_{k}}w_{n})}s_{n}(t)+n (t) \tag{12}\]  
Next, we derive the coordinates of the antenna array's radiation pattern. Figure 6: FPGA internal single-channel phase correction flowchart. \[NF_{0}=nf\cdot\frac{1}{X} \tag{8}\]0 Diagram of the direction of wave arrival. Let the relative coordinates of the \(k\)-th element relative to the first element be \((r^{\prime}_{x_{k}},r^{\prime}_{y_{k}},r^{\prime}_{z_{k}})\), then \(\omega_{k}\) can be expressed as  
\[NF_{0}=nf\cdot\frac{1}{X} \tag{8}\]1  
Then the antenna array radiation pattern function with the beam pointing at \((\theta_{0},\varphi_{0},\delta_{0})\) is  
\[NF_{0}=nf\cdot\frac{1}{X} \tag{8}\]2  
Let the absolute position coordinates (starting coordinates) of the first element be \((r_{x_{0}},r_{y_{0}},r_{z_{0}})\), and the coordinates of the \(k\)-th element be \((r_{x_{k}},r_{y_{k}},r_{z_{k}})=(r^{\prime}_{x_{k}}+r_{x_{0}},r^{\prime}_{y_{ k}}+r_{y_{0}},r^{\prime}_{z_{k}}+r_{z_{0}})\), then Equation (12) can be transformed as  
\[NF_{0}=nf\cdot\frac{1}{X} \tag{8}\]3  
The DBF complex weight vector for an antenna array with \(M\) elements is  
\[NF_{0}=nf\cdot\frac{1}{X} \tag{8}\]4  
The output signal after DBF of the antenna array is  
\[NF_{0}=nf\cdot\frac{1}{X} \tag{8}\]5  
Ignoring the configuration of the spherical cap array for now and treating the array as a regular cubic structure, let the number of elements along the \(x,y,\) and \(z\) axes be \(K_{x}\), \(K_{y}\), and \(K_{z}\), respectively. Let the element spacing along the \(x,y,\) and \(z\) axes be \(d_{x}\), \(d_{y}\), and \(d_{z}\), respectively. Then, the beam output in the direction of \((u,v,w)\) is  
\[NF_{0}=nf\cdot\frac{1}{X} \tag{8}\]6  
By performing the discrete Fourier transform (DFT) on the output signal of the antenna array, we can obtain  
\[NF_{0}=nf\cdot\frac{1}{X} \tag{8}\]7  
where \(l_{1}=0,1,2,\ldots,K_{x}-1\), \(From Equation (20), it can be seen that by detecting and making decisions on the DFT output and using the correspondence between the DFT output channels and the angles given by this equation, the results of interference detection or direction-of-arrival (DOA) can be obtained. Subsequently, the beamforming weights can be determined based on the above results [26; 27]. Now, consider the scenario of a spherical cap array. The previously discussed regular cubic array is transformed into a conformal spherical cap array coordinate system through coordinate transformation. A schematic diagram of spherical coordinates is shown in Figure 8[27; 28]. Assume that the initial coordinates of the \(k\)-th antenna element in Figure 7 are \((r_{x_{k}},r_{y_{k}},r_{z_{k}})\).