## 3 Quantitative Analyses of Disruptive Effects of Large Frequency Errors
### Effects on the C/N\({}_{0}\) Estimation  
In order to avoid the loss of signal energy induced by the code phase offset, the \(C/N_{0}\) is usually estimated using the coherent integration values from prompt correlators [20]. When a code loop is in a stable tracking state, the code phase tracking error is nominal. The autocorrelation function corresponding to the prompt code phase error \(R\left(\tau_{p}\right)\) is equal to one. When the coherent integration time does not cross data bits, the data bit is constant, and the influence on the coherent integration can be omitted. For these reasons, the coherent integration values of the prompt correlators can be simplified as  
\[ \begin{array}{l}I_{p}(n)=A\text{sinc}(\pi f_{e}T_{coh})\cos(\phi_{e})+w_{ lp,n}\\ Q_{p}(n)=A\text{sinc}(\pi f_{e}T_{coh})\sin(\phi_{e})+w_{Qp,n}\end{array} . \tag{13}\]  
In fact, in Equation (13), the attenuation of the signal amplitude caused by the frequency error \(f_{e}\) is very difficult to separate from variation in the signal amplitude \(A\)[26]. Accordingly, the \(\text{sinc}(\cdot)\) term will cause the estimated \(C/N_{0}\) to decrease significantly in the case of a large frequency error. It should be pointed out that the true \(C/N_{0}\) is not related to \(f_{e}\). The frequency error affects the estimated \(C/N_{0}\) from coherent integration values. In practice, GNSS receivers measure \(C/N_{0}\) by means of various \(C/N_{0}\) estimation algorithms [27; 28]. In the GNSS context, the narrow-to-wideband power ratio (NWPR) method is usually used as a standard estimator. Therefore, we used the NWPR method as an example to analyze the effects of large frequency tracking errors on the \(C/N_{0}\) estimation results. The narrow-to-wide power ratio is expressed as  
\[P_{n/w}=\frac{P_{n}}{P_{w}} \tag{14}\]  
where \(P_{n}\) is the narrowband power, and \(P_{w}\) is the wideband power. When the frequency error is negligible, the expectation of the narrow-to-wide power ratio is defined as [27]  
\[\mathrm{E}(P_{n/w})\approx\frac{M(c/n_{0}\tau+1)}{M+c/n_{0}\tau} \tag{15}\]  
where \(\tau=MT_{coh}\) and \(M\) is the length of the data bit in milliseconds. For the GPS L1 C/A code, \(M=20\) is commonly adopted. When the frequency error cannot be ignored, the expectation of the narrow-to-wide power ratio can be expressed as follows (see Appendix A):  
\[\mathrm{E}\Big{(}\overline{P}_{n/w}\Big{)}\approx\frac{M\Big{(}c/n_{0}\tau \mathrm{sinc}^{2}(\pi f_{e}\tau)+1\Big{)}}{M+c/n_{0}\tau\mathrm{sinc}^{2}(\pi f _{e}T_{coh})}. \tag{16}\]  
In the NWPR method, \(N\) power ratios are usually averaged to reduce the effects of noise on the statistical results. The average value is given by  
\[\mu_{P}=\frac{1}{N}\sum_{k=1}^{N}P_{n/w}(k). \tag{17}\]  
Suppose that the number of power ratios that can ignore the effect of frequency errors is \(N_{1}\), and the number of power ratios that are impaired by large frequency errors is \(N_{2}\). Then \(N=N_{1}+N_{2}\). The expectation of \(\mu_{P}\) is  
\[\mathrm{E}(\mu_{P})=\frac{N_{1}}{N}\mathrm{E}(P_{n/w})+\frac{N_{2}}{N}\mathrm{ E}\Big{(}\overline{P}_{n/w}\Big{)}. \tag{18}\]  
In the presence of large frequency errors, the estimated \(C/N_{0}\) is modeled as  
\[\overline{c}/\overline{n}_{0}=\frac{1}{T_{coh}}\frac{\mathrm{E}(\mu_{P})-1}{ M-\mathrm{E}(\mu_{P})}. \tag{19}\]  
If there is no large frequency error, the estimated \(C/N_{0}\) can be expressed as  
\[c/n_{0}=\frac{1}{T_{coh}}\frac{\mathrm{E}(P_{n/w})-1}{M-\mathrm{E}(P_{n/w})}. \tag{20}\]  
The theoretical \(C/N_{0}\) attenuation due to large frequency tracking errors is obtained as  
\[G_{muppr}=10\log_{10}\Bigg{(}\frac{\overline{c}/\overline{n}_{0}}{c/n_{0}} \Bigg{)}. \tag{21}\]  
Theoretical analysis and Monte Carlo (MC) simulation were used to quantify the influence of frequency errors on the \(C/N_{0}\) estimation. The theoretical \(C/N_{0}\) attenuation was obtained from Equations (19)-(21). \(M\) was set to 20. \(T_{coh}\) and \(\tau\) were 1 ms and 20 ms, respectively. In the Monte Carlo simulation, the true \(C/N_{0}\) was set to 40 dB-Hz.