## 2 System Model
### Parameter Estimation Based on CNR  
The receiving antenna has different radiation and reception capabilities in different directions and has different signal gains for signals with varying angles of the incident. First, the changes in antenna gain during the rotation process of the antenna are analyzed. There is a requirement to establish a station center coordinate system with the antenna center as the origin, with the Z-axis pointing towards the zenith, the Y-axis pointing north, and the X-axis pointing east. The unit vectors corresponding to the antenna axis direction and signal direction are:  
\[ \begin{split}\mathbf{A}&=[\cos(\beta)\cos(\varphi _{0}-\omega t),\cos(\beta)\sin(\varphi_{0}-\omega t),\sin(\beta)]^{T}\\ \mathbf{B}&=[\cos(\theta)\cos(\varphi),\cos(\theta) \sin(\varphi),\sin(\theta)]^{T}\end{split}  \tag{1}\]  
where \(\beta\) is the elevation angle of the antenna axis, \(\varphi_{0}\) is the initial azimuth angle of the antenna axis, \(t\) is the rotation time, \(\theta\) is the signal incident elevation angle, and \(\varphi\) is the signal incident azimuth angle. Assuming that the gain direction of the antenna is consistent, the antenna's gain to the incident signal is mainly affected by the angle between the two unit vectors. The antenna gain can be expressed as [18]:  
\[G_{R}=f(\hat{\theta})=f\Big{[}\frac{\pi}{2}-\arccos(\mathbf{A}\cdot\mathbf{B})\Big{]}= f\Big{[}\frac{\pi}{2}-\arccos(\cos(\theta)\cos(\beta)\cos(\varphi-\varphi_{0}+ \omega t)+\sin(\beta)\sin(\theta))\Big{]} \tag{2}\]  
where \(\hat{\theta}\) represents the elevation angle of the signal direction relative to the antenna's normal plane, indicating that the antenna gain is affected by the direction of the incident signal, the tilt angle of the rotating antenna, the speed of rotation, and time. Therefore, the antenna gain can be expressed as the function \(G_{R}=f(\varphi,\theta,\beta,\omega,t)\). The most widely used GNSS antenna is fixed reception pattern antennas (FRPs). Figure 3 shows the typical FRPAs directional gain in commercial receivers. The relationship between the antenna gain and elevation angle is roughly a trigonometric one, so the antenna gain in a rotating antenna can be further simplified as [18]:  
\[G_{R}=f(\varphi,\theta,\beta,\omega,t)\approx u+g\cos(\omega t+\varphi_{0}- \varphi)+w \tag{3}\]  
where \(u\) represents the constant component of the gain, \(g\) represents the varying component of the gain, and \(w\) represents Gaussian white noise. Figure 2: Rotating antenna coordinate system. The CNR of the navigation signal received by the receiver can be expressed as [17]:  
\[CNR=\frac{P_{R}}{N_{0}}=\frac{P_{T}G_{T}G_{R}(\theta,\varphi)\lambda^{2}}{(4\pi D )^{2}N_{0}} \tag{4}\]  
where \(P_{R}\) represents the received power of the signal, \(N_{0}\) represents the spectral power density of the noise, \(P_{T}\) represents the transmission power of the signal, \(G_{T}\) represents the gain of the transmitting antenna, \(G_{R}(\theta,\varphi)\) represents the gain of the receiving antenna when the incident elevation angle of the signal is \(\varphi\), and the azimuth angle is \(\theta\), \(\lambda\) is the wavelength of the signal, and \(D\) is the distance between the transmitting antenna and the receiving antenna. If \(p=P_{T}G_{T}\lambda^{2}/[{(4\pi D)}^{2}N_{0}]\), the received airborne noise ratio can be expressed as:  
\[CNR=G_{R}(\theta,\varphi)p \tag{5}\]  
Due to the distance between the navigation satellite and the ground receiver, \(\lambda\), \(N_{0}\), and \(D\) can be considered constant values in a short period, and the transmission power and gain of the navigation satellite remain unchanged, so \(p\) can be regarded as a constant value. Therefore, the change in antenna gain caused by rotating the antenna can be reflected through the CNR of the signal [17].