## 4 Signal Models  
### Uplink Signal Model for Different Satellite Systems  
This section presents the signal model for the uplink communication in different satellite systems, i.e., the GEO and LEO satellites in Figure 4. The signal model incorporates a receiver with a uniform linear array (ULA) geometry comprising \(N\) directional antenna elements, capable of receiving signals from a source in the far field. The computation of the time delay of arrival between the signals received at antenna element 0 and antenna element \(n\in[0,\cdots,N\)-1] in a multi-antenna system is based on a careful analysis of the system geometry and application of fundamental trigonometric principles. The time delay of arrival is mathematically expressed as \(\Delta t_{n}\) and given by the following expression:  
\[\Delta t_{n}=\frac{nD\sin\theta}{c};\theta\in[-90^{\circ},90^{\circ}], \tag{7}\]  
Equation (7) highlights the significance of the distance \(D\) between the antenna elements and the speed of light c for receiving signals accurately. To prevent aliasing in space, \(D\) is conventionally set to 0.5 \(\lambda_{c}\). The wavelength of the propagating wave is represented by \(\lambda_{c}=c/f_{c}\), where \(f_{c}\) signifies the frequency. Furthermore, the angle \(\theta\) reflects the direction of the transmitter, which provides crucial information about the received signal. Therefore, the received signal \(\mathbf{r}\) can be represented mathematically as below:  
\[\mathbf{r}=\mathbf{AHs}+\mathbf{v}, \tag{8}\]  
The signal vector \(\mathbf{s}\), with dimensions \(1\times\xi\), can be expressed as \(\mathbf{s}=[s_{0},s_{1},\cdots s_{(g-1)}]\), where \(\xi\) represents the number of samples. The noise vector \(\mathbf{v}\) is an \([N\times\xi]\) matrix given by:  
\[\mathbf{v}=[\mathbf{v}_{0},\mathbf{v}_{1},\cdots\mathbf{v}_{n},\cdots\mathbf{ v}_{(N-1)}]^{T}, \tag{9}\]  
Figure 3: Downlink satellite geometry for DoA estimation. In Equation (9), \(\mathbf{v}_{n}\) is a \(1\times\zeta\) row vector representing the noise at the \(n\)th element over the \(\zeta\) samples as below:  
\[\mathbf{v}_{n}=[v_{n0},v_{n1},\cdots,v_{ni},\cdots,v_{n(\zeta-1)}], \tag{10}\]  
The variable \(v_{ni}\) represents the complex zero-mean AWGN [13] at the receiver, with a noise power of \(\sigma_{v}^{2}\) for the \(i\)th sample of noise for the \(n\)th antenna element. The signal-to-noise ratio (SNR) is denoted as \(\gamma\) in this paper, with the signal power represented as \(E[\mathbf{s}]^{2}\), as given by the following mathematical expression:  
\[\gamma=E[\mathbf{s}]^{2}/\sigma_{v}^{2}. \tag{11}\]  
Finally, from Equation (8), the steer vector component \(\mathbf{A}\), with dimension \(N\times 1\), is given by:  
\[\mathbf{A}=[a_{0},a_{1},\cdots a_{n},\cdots a_{(N-1)}]^{T}, \tag{12}\]  
where \(a_{n}\) from Equation (12) is first described for a LEO satellite as follows:  
\[a_{n}=\exp\left(-\mathrm{j}2\pi n\frac{D}{\lambda_{\mathrm{d}}}\sin\theta+ \mathrm{j}\phi_{n}\right);\forall n, \tag{13}\]  
Thus, ignoring subscript \(n\) in the notation, \(\phi\) denotes the channel-phase of the receiver system. The phase is considered to be a random variable and has a uniform distribution within the range of (\(-\pi<\phi<\pi\)). Moreover, the variable \(\lambda_{\mathrm{d}}\) is defined as \(\lambda_{\mathrm{d}}=f_{\mathrm{c}}+f_{\mathrm{d}}\). Note that \(\mathrm{f_{d}}\) is the Doppler shift [22] considered in the LEO satellite system geometry. This effect is caused by the relative motion between the satellite and the Earth station on the ground given by:  
\[\mathrm{f_{d}}=\frac{V_{\mathrm{sat}}}{\lambda_{c}}\cos\varphi, \tag{14}\]  
In Equation (14), \(\varphi\) is the angle of the velocity vector, i.e., the driving direction concerning the satellite. It is a randomly distributed value between 0 and \(\pi\). However, this work assumes \(\varphi=0\), generating a maximum value of \(\mathrm{f_{d}}\) among values for different angles.