## 3 Resampling Filter-Based Wideband Flexible Monitoring
\begin{bmatrix}\cos\omega\mathrm{cos}\Omega-\mathrm{sin} \omega\mathrm{sin}\Omega\mathrm{cos}i\\ \cos\omega\mathrm{cos}\Omega+\mathrm{sin}\omega\mathrm{sin}\Omega\mathrm{cos}i \\ \mathrm{sin}\omega\mathrm{sin}i\end{bmatrix} . \tag{23}\]  
\[\mathbf{Q}= \begin{bmatrix}Q_{X}\\ Q_{y}\\ Q_{z}\end{bmatrix} = \begin{bmatrix}-\sin\omega\mathrm{cos}\Omega-\cos\omega \mathrm{sin}\Omega\mathrm{cos}i\\ -\mathrm{sin}\omega\mathrm{cos}\Omega+\mathrm{cos}\omega\mathrm{sin}\Omega \mathrm{cos}i\\ \mathrm{cos}\omega\mathrm{sin}i\end{bmatrix} . \tag{24}\]  
The transformation from the ECI coordinate system to the ECF coordinate system is necessary to facilitate the description of the satellite's relative position and velocity in terms of latitude and longitude. This conversion requires the utilization of the relationships between the ECI coordinate system and the instantaneous mean equatorial coordinate system, the instantaneous mean equatorial coordinate system and the instantaneous true equatorial coordinate system, the instantaneous true equatorial coordinate system and the quasi-inertial coordinate system, and the quasi-inertial coordinate system and the ECF coordinate system. To determine the position and velocity of the terminal in the ECF coordinate system, it is first necessary to establish the transformation matrices for position and velocity as \(\alpha\) and \(\beta\) (which are constant matrices). Consequently, the position and velocity of the satellite in the ECF coordinate system can be expressed as:  
\[ \begin{cases}\mathbf{R}_{\mathrm{s}}^{\prime}=\mathbf{\alpha}\cdot\mathbf{R}_{ \mathrm{s}},\\ \mathbf{V}_{\mathrm{s}}^{\prime}=\mathbf{\beta}\cdot\mathbf{V}_{\mathrm{s}}.\end{cases}  \tag{25}\]  
Typically, the geographical position of a receiver is represented using longitude (L), latitude (B), and elevation (H). This representation is converted to a vector \(\mathbf{R}_{0}\) in the ECF coordinate system through Equation (26). \[ \begin{split} R_{0}=\begin{bmatrix}X_{0}\\ Y_{0}\\ Z_{0}\end{bmatrix}=\begin{bmatrix}(N+H)\mathrm{cos}B\mathrm{cos}L\\ (N+H)\mathrm{cos}B\mathrm{sin}L\\ [N(1-e_{0}^{2})+H]\mathrm{sin}B\end{bmatrix}.\end{split}  \tag{26}\]  
\[N^{2}=\frac{b_{0}}{\sqrt{1-e_{0}^{2}\sin^{2}B}}. \tag{27}\]Within the context of the equation, \(a_{0}\) represents the semi-major axis of the ellipsoid, with a value of \(a_{0}=6378.137\) km; \(e_{0}\) denotes the first eccentricity, where \(e_{0}^{2}=0.0069438\). Consequently, the relative position and relative velocity between the satellite and the ground station are given by:  
\[ \begin{cases}\mathbf{R}={\mathbf{R_{s}}}^{\prime}-{\mathbf{R_{0}}},\\ \mathbf{V}={\mathbf{V_{s}}}^{\prime}-{\mathbf{V_{0}}}.\end{cases}  \tag{28}\]  
Furthermore, based on the original definition of the Doppler shift, the following can be determined:  
\[f_{\rm D}=f_{0}\cdot\frac{|V|\cos\alpha}{c}. \tag{29}\]  
In the equation, \(f_{0}\) denotes the nominal radio frequency of the received signal; \(c\) is the speed of light, with \(c=3.0\times 10^{8}\) m/s; and \(\alpha\) represents the angle between the relative velocity \(V\) of the satellite with respect to the ground station and the line connecting the satellite and the ground station, denoted by \(\mathbf{R}\). Within the OFDM parameter estimation module, the frequency offset compensation is applied to counteract the frequency offset generated by the actual motion, thereby ensuring that the processed signal is an OFDM signal from a Low Earth Orbit satellite with either no frequency offset or a very small frequency offset.