# The Ionospheric Scintillation Effects on the BeiDou Signal Receiver
## 3 Solution of Signal Propagation Based on MPS Technique
### Formulation
 \tag{21}\]  
In Equations (21) and (22), \(z\) is the diffraction distance equal to the distance between two adjacent phase screens. The PSD form of the recursion formula is then achieved by Fourier transformation:  
\[ \begin{array}{l}\Phi_{\chi}(\stackrel{{\rightharpoonup}}{{ \kappa}})=\frac{1}{2}[\Phi_{S}(\stackrel{{\rightharpoonup}}{{ \kappa}})+\Phi_{\varphi}(\stackrel{{\rightharpoonup}}{{\kappa}})+ \[PERSON]_{\chi}(\stackrel{{\rightharpoonup}}{{\kappa}})]\\ +\frac{1}{2}[\Phi_{\chi}(\stackrel{{\rightharpoonup}}{{\kappa}})- \[PERSON]_{S}(\stackrel{{\rightharpoonup}}{{\kappa}})+\[PERSON]_{\varphi}( \stackrel{{\rightharpoonup}}{{\kappa}})]\text{cos}(\kappa^{2}z/k+ \[PERSON]_{\chi S}(\stackrel{{\rightharpoonup}}{{\kappa}})\text{sin}( \kappa^{2}z/k)\\ \\ \[PERSON]_{S}(\stackrel{{\rightharpoonup}}{{\kappa}})=\frac{1}{2}[\[PERSON]_{S }(\stackrel{{\rightharpoonup}}{{\kappa}})+\[PERSON]_{\varphi}( \stackrel{{\rightharpoonup}}{{\kappa}})+\[PERSON]_{\chi}(\stackrel{{ \rightharpoonup}}{{\kappa}})]\\ +\frac{1}{2}[\[PERSON]_{S}(\stackrel{{\rightharpoonup}}{{\kappa}})+ \[PERSON]_{\varphi}(\stackrel{{\rightharpoonup}}{{\kappa}})-\[PERSON]_{ \chi}(\stackrel{{\rightharpoonup}}{{\kappa}})]\text{cos}(\kappa^{2}z/ k)-\[PERSON]_{\chi S}(\stackrel{{\rightharpoonup}}{{\kappa}})\text{sin}( \kappa^{2}z/k)\\ \\ \[PERSON]_{\chi S}(\stackrel{{\rightharpoonup}}{{\kappa}})=\frac{1}{2}[ \[PERSON]_{S}(\stackrel{{\rightharpoonup}}{{\kappa}})+\[PERSON]_{\varphi}( \stackrel{{\rightharpoonup}}{{\kappa}})-\Phi_{\chi}(\stackrel{{ \rightharpoonup}}{{\kappa}})]\text{sin}(\kappa^{2}z/k)\\ \end{array}  \tag{25}\]  
According to MPS theory, for each single phase screen, the calculation becomes a one-dimensional process due to the infinite distribution of irregularities in \(y\)-direction, so all the \(\stackrel{{\rightharpoonup}}{{\kappa}}\) used above should be one-dimensional variables:  
\[\left.\stackrel{{\rightharpoonup}}{{\kappa}}=\sqrt{\kappa_{x}^{2} +\kappa_{y}^{2}+\kappa_{z}^{2}}\right|_{\kappa y=0,\kappa_{z}=0}=\kappa_{x}= \frac{2\pi f}{V_{\mathit{eff}}} \tag{26}\]  
In Equation (26), \(V_{\mathit{eff}}\) is the speed at which signals scan the ionospheric phase screen, and \(V_{\mathit{eff}}=\frac{V_{x}}{\gamma}\), \(\gamma\) is the ratio of satellite altitude and ionosphere altitude, \(V_{s}\) is the linear velocity of the satellite. For the MPS method, when it comes to the calculation of \(V_{\mathit{eff}}\), this paper considers the ionosphere as a whole with an altitude of 350 km, which means that for a given satellite, the parameter \(V_{\mathit{eff}}\) in the calculation of each phase screen is regarded as a constant. By recursion formula, the PSD of log-amplitude and phase deviation in each phase screen can be obtained. Especially, when signal passes through the last screen to ground receiver, \(z\) should be assigned with the value of the distance between the last screen and ground. After the final step of recursion, the PSD on the ground screen is achieved. To generate the fluctuations of signal phase and amplitude on space field, a sequence of uniformly distributed random numbers is used. The calculation is related to the 1-D discrete Fourier inverse transform of the PSD as described in the following equation:  
\[f(z)=\frac{1}{L_{h}}\sum\limits_{k=0}^{N-1}F(k)\exp(j\frac{2\pi k}{N}n)\quad n =0,1,2\Lambda,N \tag{27}\]  
In Equation (27), \(L_{h}=\frac{N}{f_{s}}\cdot V_{eff}\) is the length of the phase screen, \(N\) is the number of sample points, which is equal to grid points on phase screen, \(f_{s}\) is the sample frequency, \(\frac{N}{f_{s}}\) is determined by the navigation message. \(F(k)\) is calculated as follows:  
\[F(k)=\sqrt{2\pi L_{h}\cdot PSD(\frac{2\pi k}{L_{h}})}\cdot\exp(j\varphi_{k}) \tag{28}\]  
In Equation (28), \(\varphi_{k}\) is a series of random phase angle with the uniform distribution in (0, 2\(\pi\)) and meet the requirement \(\varphi_{k}=-\varphi_{-k}\).