## 2 Analysis of Modulation Process
### The Models of Antenna and Spinning Vehicle  
The microstrip antenna has been extensively applied to GNSS (Global Navigation Satellite System) receivers for its low cost and low profile [19]. In this paper, a microstrip single-patch antenna is taken as an example to analyze its radiation pattern and the effect of rotation on the amplitude of received signals. A local spherical coordinate system is set up at the center of the microstrip antenna, as shown in Figure 1a,b. The figure shows a typical normalized radiation pattern for the microstrip antenna. Figure 1: Schematic diagram (**a**) and radiation pattern (**b**) of the rectangular microstrip antenna. According to transmission line theory [20], the normalized far-field radiation pattern of a microstrip antenna can be expressed as:  
\[E_{\theta}=\cos\phi\frac{\sin\left(\frac{k_{0}h}{2}\sin\theta\cos\phi\right)}{ \frac{k_{0}h}{2}\sin\theta\cos\phi}\cdot\frac{\sin\left(\frac{k_{0}W}{2}\sin \theta\sin\phi\right)}{\frac{k_{0}W}{2}\sin\theta\sin\phi}\cos\left(\frac{k_{0 }L_{e}}{2}\sin\theta\cos\phi\right) \tag{1}\]  
\[E_{\phi}=\cos\theta\sin\phi\frac{\sin\left(\frac{k_{0}h}{2}\sin\theta\cos\phi \right)}{\frac{k_{0}h}{2}\sin\theta\cos\phi}\cdot\frac{\sin\left(\frac{k_{0}W} {2}\sin\theta\sin\phi\right)}{\frac{k_{0}W}{2}\sin\theta\sin\phi}\cos\left( \frac{k_{0}L_{e}}{2}\sin\theta\sin\phi\right) \tag{2}\]  
where \(k_{0}\) is the wave number, \(k_{0}=2\pi/\lambda\), \(\lambda\) is the length of the wave, \(L_{e}=L+2\Delta L\) is the effective length of the antenna, \(L\) is the length of the antenna, \(\Delta L\) is the length of radiating slots, \(W\) is the width, \(h\) is the thickness of the antenna. Therefore, the radiation pattern of E surface (\(\varphi=0\)) is given by:  
\[E_{\theta}=\sin\left(\frac{k_{0}h}{2}\sin\theta\right)\cos\left(\frac{k_{0}L_ {e}}{2}\sin\theta\right)/\frac{k_{0}h}{2}\sin\theta \tag{3}\]  
The radiation pattern of H surface (\(\varphi=\pi/2\)) is given by:  
\[E_{\phi}(\theta)=\cos\theta\sin\left(\frac{k_{0}W}{2}\sin\theta\right)/\frac{ k_{0}W}{2}\sin\theta \tag{4}\]  
The relationship between the radiation pattern and the power of the receiver signals \(P_{R}(\theta,\varphi)\) can be expressed as follows:  
\[P_{R}(\theta,\varphi)=\left(\frac{\lambda}{4\pi d}\right)^{2}P_{T}G_{R}G_{T}= \left(\frac{\lambda}{4\pi d}\right)^{2}P_{T}\eta_{A}D(\theta,\varphi)G_{T} \tag{5}\]  
where \(\lambda\) is the wavelength of signal, \(d\) is the transmission distance, \(P_{T}\) is the power of transmitting signals, \(G_{R}\) and \(G_{T}\) are the gain of receiving and transmitting antenna respectively, \(\eta_{A}\) is the efficiency of antenna, \(D(\theta,\varphi)\) is the directivity of the receiving antenna, which can be represented as:  
\[D(\theta,\varphi)=P_{0}\left[\left|E_{\theta}(\theta,\varphi)\right|^{2}+ \left|E_{\phi}(\theta,\varphi)\right|^{2}\right] \tag{6}\]  
where \(P_{0}=\pi/\int_{0}^{2\pi}\int_{0}^{\pi}\left[\left|E_{\theta}(\theta,\varphi) \right|^{2}+\left|E_{\phi}(\theta,\varphi)\right|^{2}\right]\sin\theta d\theta d\varphi\). Therefore, we can conclude from (4)-(6) that power of the receiver signals received by a single-patch microstrip antenna changes in the form of a sine wave in H surface, and the radiation pattern in Figure 1b also indicates the same results. To thoroughly analyze the relationship between the amplitude of receiver signals and the radiation pattern of the microstrip antenna during rotation, a schematic diagram of the spinning vehicle is presented in Figure 2.  
As shown in Figure 2, the antenna, denoted by a black rectangle, is mounted on the surface of a cylindrical vehicle, and the body frame of the vehicle is defined as \(ox_{1}y_{1}z_{1}\), \(ox_{1}\) along the spinning axis, \(oy_{1}\) pointing to the phase center of the antenna, and \(oz_{1}\) completing the right-handed reference frame. The spherical radiation coordinate system of the antenna is defined as \(oxyz\), which shares the same origin and axes with body frame \(ox_{1}y_{1}z_{1}\).