## 2 Array Antenna Angle-of-Arrival Positioning Algorithm  
The schematic diagram of a circular antenna array is presented in Figure 1. In Figure 1, the middle antenna 9 is used as a single frequency estimation antenna corresponding to one receiving channel. Because this antenna is not switched, the number of sampling points collected is sufficient. FFT can better estimate the frequency offset. The surrounding six pairs of orthogonally polarized antennas, which include a total of 12 arrays, are used as circular antennas connected to another receiving channel. The schematic diagram of the positions of the array antenna and source during the observation is presented in Figure 2. The specific algorithm is given below. Figure 1: Circular array antenna illustration. Suppose that \(s(t)\) is an incident signal waveform, \(f_{0}\) is the signal carrier frequency, \(\lambda_{0}\) is the signal wavelength, \(\theta\) is the incidence pitch angle, and \(\varphi\) is the incidence azimuth. A two-channel circular array antenna is adopted to receive the incident signal, and the switching time interval of the acquisition antenna is \(T\). Then, the received echo signals of an alternating acquisition antenna and a single-channel antenna can be respectively expressed as:  
\[x_{n}(t)=b\times p_{n}(\theta,\varphi,\gamma,\eta)s[t+(n-1)T-\tau_{n}(\theta, \varphi)]e^{2\pi(f_{0}+f_{d})[t+(n-1)T-\tau_{n}(\theta,\varphi)]} \tag{1}\]  
\[x_{n0}(t)=b\times p_{0}(\theta,\varphi,\gamma,\eta)s[t+(n-1)T-\tau_{0}(\theta, \varphi)]e^{2\pi(f_{0}+f_{d})[t+(n-1)T-\tau_{n0}(\theta,\varphi)]} \tag{2}\]  
where \(b\) represents the component related to the transmitting power of the source and the distance from the array antenna; \(p_{n}(\theta,\varphi,\gamma,\eta)\) is the component related to the polarization; \(\gamma\) and \(\eta\) are two polarization parameters; \(\tau_{n}(\theta,\varphi)\) represents the time delay of the signal to the \(n\) array element; \(f_{d}\) is the Doppler frequency of the source. After the frequency-mixing signal processing, the echo signals can be expressed as:  
\[x_{n}(t)=b\times p_{n}(\theta,\varphi,\gamma,\eta)s[t]e^{-2\pi f_{0}\tau_{n}( \theta,\varphi)}e^{2\pi f_{d}[t+(n-1)T-\tau_{n}(\theta,\varphi)]} \tag{3}\]  
\[x_{n0}(t)=b\times p_{0}(\theta,\varphi,\gamma,\eta)s[t]e^{-2\pi f_{0}\tau_{0}( \theta,\varphi)}e^{2\pi f_{d}[t+(n-1)T-\tau_{n}(\theta,\varphi)]} \tag{4}\]  
The existence of the frequency synchronization error leads to the deviation between \(\overline{f_{0}}\) and \(f_{0}\), which can be expressed as \(f_{\Delta}=f_{0}-\overline{f_{0}}\). Since a base station is generally arranged at a high level and the incident signals are narrowband signals in the far-field, the delay along the array of the envelope of the received signal can be ignored. Then, Equations (3) and (4) can be respectively rewritten as:  
\[x_{n}(t)=b\times p_{n}(\theta,\varphi,\gamma,\eta)s[t]e^{-2\pi f_{0}\tau_{n}( \theta,\varphi)}e^{2\pi(f_{\Delta}+f_{d})[t+(n-1)T-\tau_{n}(\theta,\varphi)]} \tag{5}\]  
\[x_{n0}(t)=b\times p_{0}(\theta,\varphi,\gamma,\eta)s[t]e^{-2\pi f_{0}\tau_{0}( \theta,\varphi)}e^{2\pi(f_{\Delta}+f_{d})[t+(n-1)T-\tau_{n}(\theta,\varphi)]} \tag{6}\]  
For the \(N\)-point FFT transformation of \(x_{n0}(t)\) the spectrum is expressed as follows:  
\[X_{N}(k)=FFT[x_{n0}(t)]=\sum_{n=0}^{N-1}x_{n0}(t)e^{-j\frac{2\pi\mathrm{i}\pi }{N}} \tag{7}\]  
where the spectrum line corresponding to the amplitude is \(k\). Figure 2.0 Illustration of antenna. For the maximum value of the spectrum, the corresponding spectrum line is denoted as \(k_{1}\). According to the maximum likelihood estimation principle, the relationship between the frequency offset value and \(k_{1}\) can be obtained as follows:  
\[f_{\Delta}= \begin{cases}\frac{k1}{TN}&0\leq k1\leq\frac{N}{2}-1\\ -\frac{N-k1}{TN}&\frac{N}{2}\leq k1\leq N-1\end{cases}  \tag{8}\]  
According to Equation (8), the estimation accuracy of the frequency offset is directly related to the FFT length, and its maximum estimation error is obtained as \(\frac{k1}{2 TN}\).