## 2 System Model
### MUSIC Algorithm  
When considering a typical LEO Ka-band signal with a bandwidth of 3.5 GHz and a center frequency \(f\) exceeding 26.5 GHz, the incident signal source can be modeled as a far-field narrowband signal. Its complex envelope can be expressed as follows:  
\[s_{i}(t)=u_{i}(t)e^{i(\omega t+\phi(t))} \tag{12}\]  
In Equation (12), \(u_{i}(t)\) represents the amplitude of the far-field source, while \(\omega=2\pi f\) denotes the angular frequency of the received source, and \(\phi(t)\) is its phase. In the study of antenna array-based AOA estimation algorithms, the primary focus is on the effect of the time delay \(\Delta t\) on the incident signal source. When the incident signal experiences a time delay \(\Delta t\), its expression is given as follows:  
\[s_{i}(t-\Delta t)=u_{i}(t-\Delta t)e^{i(\omega(t-\Delta t)+\phi(t-\Delta t))} \tag{13}\]  
Given that the incident signal source is a narrowband far-field source with an effective bandwidth much smaller than the center frequency \(f\), both \(u_{i}(t)\) and \(\phi(t)\) vary at a relatively  
Figure 1: Visual representation for GDOP [29]. slow rate over time. Under this condition, after a small time delay \(\Delta t\), the equivalent expression can be written as follows:  
\[ \begin{split} u_{i}(t-\Delta t)&\approx u_{i}(t)\\ \phi(t-\Delta t)&\approx\phi(t)\end{split}  \tag{14}\]  
From Equations (12)-(14), we have the following:  
\[ \begin{split} s_{i}(t-\Delta t)&=u_{i}(t-\Delta t)e^ {j(\omega(t-\Delta t)+\phi(t-\Delta t))}\\ &\approx u_{i}(t)e^{j(\omega(t-\Delta t)+\phi(t))}\\ &=u_{i}(t)e^{j(\omega t-\omega\Delta t+\phi(t))}\\ &=u_{i}(t)e^{j(\omega t+\phi(t))}e^{-j\omega\Delta t}\\ &=s_{i}(t)e^{-j\omega\Delta t}\end{split}  \tag{15}\]  
Based on the expression in Equation (15), it can be concluded that the propagation of a far-field narrowband signal source with a small time delay between different elements of an antenna array can be equated to a phase shift in the target signal. In contrast, the variation in the signal source amplitude across the array elements is negligible and can be treated as a constant. This key conclusion simplifies the signal reception model for far-field narrowband sources. Specifically, when the sensor array elements are arranged in a particular configuration, target signals from different incidence angles induce phase differences between the array elements. By applying suitable techniques to extract these phase differences, the performance parameters of the target signal source can be accurately obtained, thus enabling effective estimation of the target signal source parameters. Consider a uniform surface array antenna commonly used for LEO satellites. The receiving system consists of a single-channel receiver and an \(M\times N\) array of strip dipole antennas, each operating at half the wavelength. Only one antenna is connected to the receiver at any given time. These \(M\times N\) antennas are uniformly distributed with a spacing of \(d=\lambda/2\) to form a planar array, where \(\lambda\) represents the wavelength of the incident signal. Assume that \(K\) uncorrelated narrowband signals are incident on the array plane from space at two-dimensional angles \((\theta_{k},\varphi_{k})\) for \(k=1,2,\cdots,K\). Here, \(\theta_{k}\) and \(\varphi_{k}\) represent the elevation and azimuth angles of the \(k\)th incident signal, respectively, as shown in Figure 2.  
Figure 2: Diagram of signal incidence on an antenna array. Based on the described array structure, the output of an individual array element can be expressed as  
\[y_{k,mn}(t)=\sum_{k=1}^{K}\delta_{k,mn}s_{k}(t-\tau_{k,mn})+n_{mn}(t) \tag{16}\]  
In this equation, the parameters are defined as follows:  
* \(y_{k,mn}(t)\) denotes the signal received by the \(k\)th signal wave at the array element with sequence number \(m\) in the \(x\)-axis and \(n\) in the \(y\)-axis (Equation \(m=0,1,\ldots,M-1\), \(n=0,1,\ldots,N-1\)).