# The Ionospheric Scintillation Effects on the BeiDou Signal Receiver
## 5 Effect on Tracking Process for BeiDou Receiver
### Determining PLL Variance for the BeiDou B11 Signal  
After the signal acquisition, the carrier frequency is evaluated, then the receiver performs a tracking process with carrier tracking loop (PLL and FLL) and PN code tracking loop (DLL). As the ionospheric scintillations won't have an effect on signal frequency, this paper would mainly discuss the influence on PLL. For the carrier tracking loop, once the receiver fails to track the carrier phase, the signal is lost. Loss of lock is related to the tracking error variance at the output of the PLL. This variance could be expressed as follows [7]:  
\[\sigma_{\Phi}^{2}=\sigma_{\Phi S}^{2}+\sigma_{\Phi T}^{2}+\sigma_{\Phi osc}^{2} \tag{42}\]  
In Equation (42), \(\sigma_{\Phi S}\) is the phase scintillation, \(\sigma_{\Phi T}\) is the thermal noise, \(\sigma_{\Phi osc}\) is the receiver oscillator noise (usually 0.122 rad). The phase variance at the output of the PLL is:  
\[\sigma_{\Phi S}^{2}=\int\limits_{-\infty}^{+\infty}\left|1-H(f)\right|^{2}\Phi _{S}(\overset{\rightarrow}{f})df \tag{43}\]  
In Equation (43), \(\Phi_{S}(\overset{\rightarrow}{f})\), given in Equation (12), is the PSD of phase scintillation provided by the MPS model. \(\left|1-H(f)\right|^{2}\) is the closed loop transfer function of the PLL. According to [28], the thermal noise tracking error could be characterized by \(S_{4}\), which is given by:  
\[\sigma_{\Phi T}^{2}=\frac{B_{n}(1+\frac{1}{2t_{0}(C/N_{0})(1-2S_{4}^{2})})}{(C /N_{0})(1-2S_{4}^{2})} \tag{44}\]  
In Equation (44), \(C/N_{0}\) is the SNR which contains the antenna gains about 40 dB, \(B_{n}\) is the third-order PLL one-sided bandwidth equal to 10 Hz for the BeiDou B11 signal, \(t_{0}\) is the pre-detection integration time, equal to 0.02 s for the BeiDou signal. Based on Equations (43) and (44), the PLL tracking error variance can be computed. Figure 9 shows the variance with \(S_{4}=0.70\) and \(S_{4}=0.41\). By the observation of 100-repetition simulation data, we find that the value of the phase scintillation is far smaller than that of the thermal noise, and the thermal noise appears to be the decisive contributor to the PLL tracking error. The figure  
shows that as the signal SNR increases, the phase error of the PLL output decreases and that the influence of intense scintillation on PLL is much higher than that of weak scintillation when the signal SNR is relative lower. For PLL, there always exits a tracking threshold: once the signal \(C/N_{0}\) is lower than this threshold, the receiver can't continue tracking such a weak signal stably, which is also called the loss of lock. According to [26], one method for establishing this threshold is that the triple of the PLL variance can't be more than a quarter of the scope of the phase discriminator:  
\[3\times\sigma_{PLL}\leq\frac{1}{4}\times 180^{\circ} \tag{45}\]  
Figure 9 also shows this threshold. From the figure, loss of lock is highly probable for values above the \(15^{\circ}\) threshold, therefore a receiver can't tolerate scintillation if the \(C/N_{0}\) (in dB) is below a maximum, which is 24 dB for \(S_{4}\) = 0.41 and 29.5 dB for \(S_{4}\) = 0.70. On the other hand, one of the outputs of PLL is the measurement of carrier phase, which is an essential parameter for carrier phase ranging. Carrier phase ranging uses the phase difference between the received signal and the initial signal to calculate the distance from the satellite to the receiver. Figure 9 shows that if the signal \(C/N_{0}\) is 26 dB, the phase error under \(S_{4}\) = 0.70 and \(S_{4}\) = 0 could be \(27^{\circ}\) and \(8^{\circ}\). Then for the BeiDou B1I signal (\(\lambda\) = 19.2 cm), the error of carrier phase ranging caused by scintillations \(S_{4}\) = 0.70 could be \(\frac{2\times(27^{\circ}-8^{\circ})}{360^{\circ}}\cdot\lambda\) = 2.02 cm, which would have a relatively severe effect on the precise point position in the carrier phase ranging method.