## 2 Analysis of Modulation Process
### Coordinate Transformations in Roll Angle Measurement  
However, \(\alpha\) is not the real roll angle that we want. It is related to the true roll angle. To obtain the true roll angle, several coordinate systems are set up on the body of a simplified cylinder vehicle, as shown in Figure 4, in which the single-patch antenna denoted by a black rectangle is mounted on the surface of a cylinder vehicle. \(OX_{b}Y_{b}Z_{b}\) is a right-handed body coordinate system with origin \(O\). The \(X\)-axis (denoted by \(X_{b}\)) pointing forward, is the spinning axis of the vehicle. \(Z\)-axis (denoted by \(Z_{b}\)) points to the phase center of the single-patch antenna, and \(Y\)-axis (denoted by \(Y_{b}\)) is orthogonal to  
Figure 3: The variation map of signal amplitude received by a rotating antenna. Figure 2: Schematic diagram of the single-patch antenna mounted on a cylindrical vehicle. \(Z\)- and \(X\)-axes with the usual right-hand rule. \(OX_{r}Y_{r}Z_{r}\) is a reference coordinate system, and it shares the same origin with the body coordinate system. The \(X\)-axis (denoted by \(X_{r}\)) points forward and it overlaps with \(X_{b}\). The \(Y\)-axis (denoted by \(Y_{r}\)) is parallel to the horizon, and the \(Z\)-axis (denoted by \(Z_{r}\)) is pointing downward to comply with the right-hand rule. The body frame spins with the vehicle, while the reference coordinate system is rigidly attached to the spin axis, but not the rest of the vehicle body. The roll angle \(\gamma\) is between the \(OZ_{b}\) and \(OZ_{r}\) axis. \(\alpha_{i}\) is the angle between \(OZ_{b}\) axis and the projection of LOS on the \(OY_{i}Z_{i}\) surface, which is the phase we are going to obtain from the roll modulation signals. Therefore, the roll angle \(\gamma\) can be described as:  
\[\gamma=\alpha_{i}-\psi_{i} \tag{7}\]  
where \(\Psi_{i}\) can be obtained by the LOS vector in a reference coordinate system, and it differs in circumstances of different satellites and attitudes of the vehicle. The true roll angles detected from different satellites signals should be approximately equal. The roll angles obtained from various satellite signals could be averaged to get a more accurate estimation. To obtain \(\Psi_{i}\), the positions of the satellite and vehicle in ECEF (Earth-Centered Earth-Fixed) and geodetic coordinate systems, as well as the attitude of the vehicle, such as yaw and pitch angles, must be known. The definition of ECEF and vehicle-carried NED (North-East-Down) coordinate systems are identical to [PERSON] et al. [21]. \(\Psi_{i}\) is obtained by the following procedures. 1. Get the LOS vector from ECEF positions of the \(i\)-th satellite and the vehicle, which are obtained by GPS receiver. \[P_{\text{LOS}}^{i}=P_{s}^{i}-P_{v}=\left[ \begin{array}{cc}x_{s}^{i}-x_{v}&y_ {s}^{i}-y_{v}&y_{s}^{i}-y_{v}\\ \end{array} \right]^{T}=\left[ \begin{array}{cc}x_{e}^{i}&y_{e}^{i}&z_{e}^{i} \\ \end{array} \right]^{T}\] (8) where \(P_{s}^{i}\) is the \(i\)-th satellite position, \(P_{v}\) is the vehicle position in the ECEF coordinate system. 2. Transform the LOS vector from the ECEF coordinate system to the vehicle-carried NED coordinate system. \[P_{n/e}^{i}=R_{n/e}P_{\text{LOS}}^{i}=\left[ \begin{array}{cc}x_{n}^{i}&y_{ n}^{i}&z_{n}^{i}\\ \end{array} \right]^{T}\] (9) where \(R_{n/e}\) is the rotation matrix from the ECEF frame to the vehicle-carried NED frame, which is given by  
Figure 4: Schematic diagram of roll angle detection. \[R_{n/e}=\left[ \begin{array}{ccc}-\sin\varphi_{geo}\cos\lambda_{geo}&-\sin\varphi_ {geo}\sin\lambda_{geo}&\cos\varphi_{geo}\\ -\sin\lambda_{geo}&\cos\lambda_{geo}&0\\ -\cos\varphi_{geo}\cos\lambda_{geo}&-\cos\varphi_{geo}\sin\lambda_{geo}&- \sin\varphi_{geo}\end{array} \right] \tag{10}\]  
where \(\phi_{geo}\) and \(\lambda_{geo}\) are the geodetic longitude and latitude position of the vehicle respectively. 3. Then transform LOS vector from vehicle-carried NED frame to the reference coordinate system \(OX_{r}Y_{r}Z_{r}\) which is set up in Figure 4. \[P_{r/n}^{i}=R_{r/n}P_{n/e}^{i}=\left[