# The Ionospheric Scintillation Effects on the BeiDou Signal Receiver
## 3 Solution of Signal Propagation Based on MPS Technique  
### Formulation  
Ionospheric scintillation is ultimately the fluctuation of phase and amplitude caused by irregularities. Currently, the BeiDou system has ten Medium Earth Orbit (MEO) satellites, five Geostationary Earth Orbit (GEO) satellites and five Inclined Geosynchronous Satellite Orbit (IGSO)  
Figure 1: (**top**) PSD and (**bottom**) correlation function of irregularities in the ionosphere. satellites. GEO and IGSO are both synchronous satellites. The orbit altitudes of MEO/GEO/IGSO are, 21528 km/35786 km/35786 km, respectively. Generally speaking, MEO and IGSO serve as positioning satellites and GEO communicate with ground stations to exchange telemetry information. This research would particularly focus on the impact of ionospheric scintillation on synchronous satellites. Figure 2a shows a schematic figure. In Figure 2a, O is the Earth center, S is a satellite, A and B are the tangency points. As shown, \(\angle\)ASB = 17\({}^{\circ}\) indicates the maximum coverage area of a synchronous satellite. Generally, a receiver's antenna elevation angle would be less than 45\({}^{\circ}\) to ensure a high ranging accuracy, so the practical maximum coverage angle \(\angle\)ESF would be less than 10\({}^{\circ}\). Thus the maximum angle of the incident wave from a synchronous satellite would be less than 5\({}^{\circ}\). In some cases, for synchronous satellite which is positioned above the equator, the effective coverage area is from equator to mid-latitude as shown in figure. Based on the reasons mentioned above, this paper make an assumption of vertical incidence situation. As shown in Figure 2b, ionospheric irregularities are assumed to be distributed in the space between layer \(\mathrm{z}=0\) and \(\mathrm{z}=\mathrm{L}\). A time-harmonic electromagnetic wave is incident on the irregular slab at \(\mathrm{z}=0\) and received at ground receiver. In MPS theory, the ionized region is divided into several thin screens with each perpendicular to the direction of propagation and each screen could cause a change on signal phase and amplitude during the propagation. MPS method obeys to Markov approximation [22]: the wave field in any altitude \(\mathrm{z^{\prime}}\) is only relevant to the irregularities located in \(\mathrm{z}<\mathrm{z^{\prime}}\); propagation distance is much higher  
Figure 2: (**a**) Schematic figure of Medium Earth Orbit (GEO); (**b**) Propagation of incident wave; (**c**) Coordinate system. than the radial correlation distance. Thus for the MPS method, it is assured to perfectly simulate the deviation of amplitude and phase as long as the thickness and number of phase screen are set small and large enough. Figure 2c shows the established coordinate system. Let us consider an incident wave with complex amplitude \(A_{0}\) and after the propagation through a phase screen the wave becomes:  
\[u(\stackrel{{\rightarrow}}{{\rho}},z)=A_{0}\,\exp[\psi(\stackrel{{ \rightarrow}}{{\rho}},z)]=A_{0}\,\exp[\chi(\stackrel{{ \rightarrow}}{{\rho}},z)-jS(\stackrel{{\rightarrow}}{{\rho}},z)] \tag{13}\]  
In Equation (13), \(\chi(\stackrel{{\rightarrow}}{{\rho}},z)\) is referred to as the log-amplitude and \(S(\stackrel{{\rightarrow}}{{\rho}},z)\) as the phase deviation of the wave. Every time the signal wave pass through a phase screen, \(S(\stackrel{{\rightarrow}}{{\rho}},z)\) will increase by \(\varphi(\stackrel{{\rightarrow}}{{\rho}})\):  
\[S=S+\varphi(\stackrel{{\rightarrow}}{{\rho}}) \tag{14}\]  
According to [14], when the thickness of the screen is set small enough, the irregularities in each phase screen layer are regarded to be distributed uniformly.