## 2 System Model and Signal Processing Methods
### Observation Model for Real Signals  
After data acquisition via the integrated navigation model discussed in Section 2.1, all the real and spoofed GNSS signals received by each antenna are efficiently processed, and MIMU measurements are recorded. This ensures accurate acquisition, tracking, and carrier-phase measurements of all signal sources. The carrier-phase observation equation is given as follows:  
\[\widetilde{\phi}_{j}^{i}=\frac{1}{\lambda}(\left|\vec{r}^{i}-\vec{r}\right|- \delta\rho_{kn}^{i}-\delta\rho_{kp}^{i})+f\cdot\delta t^{i}-N^{i} \tag{1}\]  
where \(\lambda\) is the wavelength of the carrier phase, \(\Delta\widetilde{\phi}_{j}^{i}\) is the real value of the carrier phase, and \(N\) is the integer ambiguity of the carrier phase; \(\vec{r}\) is the real position of Satellite \(i\), \(\vec{r}\) is the real position of Antenna 1, and \(\delta t^{i}\) is the clock difference between Antenna 1 and the receiver; \(\delta\rho_{kn}^{i}\) and \(\delta\rho_{kp}^{i}\) are ionospheric and tropospheric errors, respectively. When the distance between the two stations is short, the ionospheric and tropospheric errors experienced by the two stations are approximately identical. Therefore, errors can be eliminated in the differences. The single-difference carrier-phase ambiguity  
Figure 2: Integrated navigation model. observation equations are constructed for the two frequency points of the two stations as follows:  
\[\left\{ \begin{array}{l}\Delta\widehat{\varphi}_{12,B_{1}}^{i}=\frac{1}{\lambda_ {B_{1}}}\left(\left|\vec{r}_{1}^{i}-\vec{r}_{1}\right|-\left|\vec{r}_{2}^{i}- \vec{r}_{2}\right|\right)+f_{B_{1}}\cdot\delta{t_{12,B_{1}}}^{i}-N_{12,B_{1}} ^{i}\\ \Delta\widehat{\varphi}_{12,B_{3}}^{i}=\frac{1}{\lambda_{B_{3}}}\left(\left|\vec {r}_{1}^{i}-\vec{r}_{1}\right|-\left|\vec{r}_{2}^{i}-\vec{r}_{2}\right|\right) +f_{B_{3}}\cdot\delta{t_{12,B_{3}}}^{i}-N_{12,B_{3}}^{i}\end{array} \right. \tag{2}\]  
where \(\Delta\widehat{\varphi}_{12,B_{1}}^{i}\) and \(\Delta\widehat{\varphi}_{12,B_{3}}^{i}\) denote the real values of the single-difference carrier phases between the two stations at two frequency points, \(B_{1}\) and \(B_{3}\), respectively; \(\vec{r}_{1}^{i}\) and \(\vec{r}_{2}^{i}\) denote the positions of the satellite \(i\) corresponding to the signals received by Antennas 1 and 2, respectively; \(\vec{\tau}_{1}\) and \(\vec{\tau}_{2}\) represent the real positions of Antennas 1 and 2, respectively; \(f_{B_{1}}\) and \(f_{B_{3}}\) are the frequencies of the two frequency points, \(B_{1}\) and \(B_{3}\), respectively; \(\delta{t_{12,B_{1}}}^{i}\) and \(\delta{t_{12,B_{3}}}^{i}\) are the inter-station clock difference between Antennas 1 and 2 at the two frequency points, \(B_{1}\) and \(B_{3}\), respectively; \(N_{12,B_{1}}^{i}\) and \(N_{12,B_{1}}^{i}\) indicate the single-difference integer ambiguities of Antennas 1 and 2 relative to satellite \(i\) at the two frequency points, \(B_{1}\) and \(B_{3}\), respectively. As depicted in Figure 3, \(\theta\) (i.e., the angle between the DOA and the baseline vector formed by Antennas 1 and 2) can be expressed as follows:  
\[\lambda\cdot(\Delta\phi_{12}+N_{12})=L\cos\theta \tag{3}\]  
As illustrated in Figure 4, \(\theta\) in Equation (3) can be decomposed into the pitch angle (\(\beta\)) and yaw angle (\(\alpha\)) from the carrier to the signal source. Figure 4: Signal source position in the body coordinate system. Figure 3: Single-differenced carrier phase between two antennas on the baseline vector-source plane. The yaw angle \(\alpha\) and pitch angle \(\beta\) in the figure are the objectives to be solved in this study. When the signal source is a real satellite signal, the geometric configuration of the cosine function is given by  
\[\cos\theta=\frac{ \begin{bmatrix}L&0&0\end{bmatrix}