# The Ionospheric Scintillation Effects on the BeiDou Signal Receiver
## 3 Solution of Signal Propagation Based on MPS Technique
### Formulation
For a thin phase screen with thickness \(\Delta z\), the value of \(\varphi(\stackrel{{\rightarrow}}{{\rho}})\) depends on irregularities density \(N_{f}(\stackrel{{\rightarrow}}{{\rho}})\):  
\[\varphi(\stackrel{{\rightarrow}}{{\rho}})=-\lambda r_{e}\int_{0} ^{\Delta z}N_{f}(\stackrel{{\rightarrow}}{{\rho}})<N_{e}>dz \tag{15}\]  
In Equation (15), \(\stackrel{{\rightarrow}}{{\rho}}\) is a two-dimensional coordinate vector on the phase screen plane, \(\lambda\) is the signal wavelength, \(<\)\(N_{e}\)\(>\) is the mean value of the electron density in the ionosphere. For ionospheric irregularities, the autocorrelation function of the phase change is related to the autocorrelation function of ionization fluctuations by the following equation [14]:  
\[B_{\varphi}(\xi)=N_{e}{}^{2}r_{e}^{2}\lambda^{2}\Delta z\int B_{Nf}(\xi,z)dz \tag{16}\]  
In Equation (16), \(B_{N}\)(\(\xi\),\(z\)) is the autocorrelation function of the electron density fluctuation. Based on Equations (15) and (16), the PSD of \(\varphi(\stackrel{{\rightarrow}}{{\rho}})\) can be obtained by Fourier transform:  
\[\Phi_{\varphi}(\stackrel{{\rightarrow}}{{\kappa}})=N_{e}^{2}( \lambda r_{e})^{2}2\pi\Delta z\Phi_{\xi}(\stackrel{{\rightarrow}} {{\kappa}}) \tag{17}\]  
\(\Phi_{\xi}(\stackrel{{\rightarrow}}{{\kappa}})\) is shown in Equation (12). Figure 3 shows the PSD of \(\varphi(\stackrel{{\rightarrow}}{{\rho}})\). Figure 3: Phase PSD of single phase screen (\(\sigma_{\xi}^{2}=0.1\), \(p=4\)). The parabolic wave equation is given by [19; 23]:  
\[-2 ik\frac{\partial u}{\partial z}+\
abla^{2}u=-k^{2}\Delta\varepsilon_{1}u(0<z \leq L) \tag{18}\]  
In Equation (18), \(\
abla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\) is the transverse Laplace operator. \(k\) is the wavenumber and \(\Delta\varepsilon_{1}\) is the fluctuating part characterizing the random variations caused by irregularities. According to [14], in this case of a vertical incident wave, the irregularities could be assumed infinitely elongated in the \(y\)-direction, so the equation is limited in two-dimensions with no variable in the \(y\)-direction. Then the wave propagation can be characterized by the two following equations:  
\[\frac{\partial^{2}u}{\partial x^{2}}-2 ik\frac{\partial u}{\partial z}+k^{2} \Delta\varepsilon_{1}u=0(z_{n}^{-}<z\leq z_{n}^{+}) \tag{19}\]  
Figure 2:0  
Equation (19) characterizes the propagation through the \(n\)-th phase screen and Equation (20) characterizes the propagation in the vacuum space between adjacent screens. Then by using the [PERSON]'s diffraction formula [24] and Fresnel diffraction theory [24], the recursion formula of log-amplitude and phase deviation from one screen to the next can be obtained:  
\[ \begin{array}{l}\chi(\stackrel{{\rightharpoonup}}{{\rho}})= \frac{k}{2\pi z}\iint S(\stackrel{{\rightharpoonup}}{{\rho}}) \text{cos}(k\left|\stackrel{{\rightharpoonup}}{{\rho}}-\stackrel{{ \rightharpoonup}}{{\rho^{\prime}}}\right|^{2}/2z)\\ +\chi(\stackrel{{\rightharpoonup}}{{\rho^{\prime}}})\text{sin}(k \left|\stackrel{{\rightharpoonup}}{{\rho}}-\stackrel{{ \rightharpoonup}}{{\rho^{\prime}}}\right|^{2}/2z)+\varphi( \stackrel{{\rightharpoonup}}{{\rho^{\prime}}})\text{cos}(k \left|\stackrel{{\rightharpoonup}}{{\rho}}-\stackrel{{ \rightharpoonup}}{{\rho^{\prime}}}\right|^{2}/2z)d^{2}\rho^{\prime}\\ \\ S(\stackrel{{\rightharpoonup}}{{\rho}})=\frac{k}{2\pi z}\iint S( \stackrel{{\rightharpoonup}}{{\rho^{\prime}}})\text{sin}(k \left|\stackrel{{\rightharpoonup}}{{\rho}}-\stackrel{{ \rightharpoonup}}{{\rho^{\prime}}}\right|^{2}/2z)\\ -\chi(\stackrel{{\rightharpoonup}}{{\rho^{\prime}}})\text{cos}(k \left|\stackrel{{\rightharpoonup}}{{\rho}}-\stackrel{{ \rightharpoonup}}{{\rho^{\prime}}}\right|^{2}/2z)+\varphi( \stackrel{{\rightharpoonup}}{{\rho^{\prime}}})\text{sin}(k \left|\stackrel{{\rightharpoonup}}{{\rho}}-\stackrel{{ \rightharpoonup}}{{\rho^{\prime}}}\right|^{2}/2z)d^{2}\rho^{\prime}\\ \end{array}