# Radio Frequency Compatibility Evaluation of S Band Navigation Signals for Future BeiDou
## 2 Radio Frequency Compatibility Evaluation Methods
### Compatibility Evaluation Method for Code Tracking  
Code tracking is used for the accurate estimation of code phase and requires two correlator channels namely early and late ones. A general method for measuring code tracking performance is to estimate the variance of smoothed time of arrival (TOA). When the noncoherent discriminator is employed using the delay between early and late of \(d\), the variance of smoothed TOA is expressed as [17]  
\[ \begin{array}{l}\sigma_{non}^{2}&=\left(\frac{B_{L}\left(1-0.5B_{L}T\right) \int_{-B/2}^{B/2}G_{s}\left(f\right)\sin^{2}\left(\pi fd\right)df}{(2\pi)^{2} \frac{C}{N_{0}}\left(\int_{-B/2}^{B/2}G_{s}\left(f\right)\sin\left(\pi fd \right)df\right)^{2}}+\frac{B_{L}\left(1-0.5B_{L}T\right)C_{J}\int_{-B/2}^{B/2 }G_{s}\left(f\right)G_{J}\left(f\right)\sin^{2}\left(\pi fd\right)df}{(2\pi)^ {2}C\left(\int_{-B/2}^{B/2}G_{s}\left(f\right)\sin\left(\pi fd\right)df\right) ^{2}}\right)\\ &\times\left(1+\frac{\int_{-B/2}^{B/2}G_{s}\left(f\right)\cos^{2}\left(\pi fd \right)df}{\frac{C}{N_{0}}\left(\int_{-B/2}^{B/2}G_{s}\left(f\right)\cos(\pi fd \right)df\right)^{2}}+\frac{C_{J}\int_{-B/2}^{B/2}G_{s}\left(f\right)G_{J} \left(f\right)\cos^{2}\left(\pi fd\right)df}{TC\left(\int_{-B/2}^{B/2}G_{s} \left(f\right)\cos\left(\pi fd\right)df\right)^{2}}\right)\\ &=\sigma_{coh}^{2}\times\vartheta\end{array} , \tag{10}\]  
where \(B_{L}\) is the loop bandwidth. Through Equation (10), we can observe that the carrier-to-noise ratio has very significant impacts on the error variance, so it can also be indirectly regarded as a measure of code tracking performance. Besides, the noncoherent error variance is the product of coherent error variance \(\sigma_{coh}^{2}\)[20] and squaring loss \(\vartheta\) greater than 1. The \(\vartheta\) is very close to 1 for the usual range of \(C/N_{0}\) greater than 30 dB-Hz [16], then the both of the \(\sigma_{non}^{2}\) and \(\sigma_{coh}^{2}\) are approximately equivalent. To attain the \(\left(C/N_{0}\right)_{\text{eff}}\) for code tracking, the error variance in white noise is obtained when the \(C_{J}\) equals 0, i.e.,  
\[\sigma_{coh}^{2}=\frac{B_{L}\left(1-0.5B_{L}T\right)\int_{-B/2}^{B/2}G_{s}\left( f\right)\sin^{2}\left(\pi fd\right)df}{\left(2\pi\right)^{2}\frac{C}{N_{0}} \left(\int_{-B/2}^{B/2}fG\left(f\right)\sin\left(\pi fd\right)df\right)^{2}}. \tag{11}\]  
Since the received \(C/N_{0}\) simply equals the \(\left(C/N_{0}\right)_{\text{eff}}\) in white noise, then the \(\left(C/N_{0}\right)_{\text{eff}}\) for code tracking is derived as  
\[\left(\frac{C}{N_{0}}\right)_{\text{eff,CTSSC}}=\frac{B_{L}\left(1-0.5B_{L}T \right)\int_{-B/2}^{B/2}G_{s}\left(f\right)\sin^{2}\left(\pi fd\right)df}{(2 \pi)^{2}\sigma_{coh}^{2}\left(\int_{-B/2}^{B/2}fG\left(f\right)\sin\left(\pi fd \right)df\right)^{2}}, \tag{12}\]  
When the white noise and non-white interference signal exist, we substitute Equation (10) into Equation (12) and attain the corresponding \(\left(C/N_{0}\right)_{\text{eff,CTSSC}}\) as  
\[\left(\frac{C}{N_{0}}\right)_{\text{eff,CTSSC}}=\frac{C}{N_{0}+\frac{C_{J}\int_{ -B/2}^{B/2}G_{s}\left(f\right)G_{J}\left(f\right)\sin^{2}\left(\pi fd\right)df} {\int_{-B/2}^{B/2}G_{s}\left(f\right)\sin^{2}\left(\pi fd\right)df}} \tag{13}\]  
with the CTSSC defined as \[\kappa_{I}^{s}=\frac{\int_{-B/2}^{B/2}G_{s}(f)G_{I}(f)\sin^{2}(\pi fd)df}{\int_{-B/2 }^{B/2}G_{s}(f)\sin^{2}(\pi fd)df}. \tag{14}\]  
When considering multiple space constellations, the \((C/N_{0})_{\text{eff},\text{CTSSC}}\) with the intersystem and intrasystem interference can be deduced as  
\[\begin{array}{ll}\left(\frac{C}{N_{0}}\right)_{\text{eff},\text{CTSSC}}&= \frac{C}{N_{0}+\frac{\text{[PERSON]}(t)}{\frac{\text{[PERSON]}(t)}{\frac{\text{[PERSON] }(t)}{\frac{\  
earth orbit (MEO) planes at an inclination of the orbital planes of 56\({}^{\circ}\) with reference to the equatorial plane. The BDS will accomplish a fully deployed constellation of 35 satellites by 2020, including five GSO satellites, twenty-seven MEO satellites and three IGSO satellites [25; 26; 27].