# Radio Frequency Compatibility Evaluation of S Band Navigation Signals for Future BeiDou
## 2 Radio Frequency Compatibility Evaluation Methods  
### Compatibility Evaluation Method for Acquisition  
Acquisition refers to the coarse estimation of code phase and Doppler by the prompt correlation of received signal with hypotheses reference signal. The output signal-to-noise-plus-interference ratio (SNIR) of prompt correlator can describe signal acquisition performance well. For noncoherent processing, the output SNIR at the prompt correlator is expressed as [17; 18]  
\[\rho_{n}=\frac{T\frac{C}{N_{0}}\left(\int_{-B/2}^{B/2}G_{s}(f)df\right)^{2}}{ \int_{-B/2}^{B/2}G_{s}(f)df+\frac{C_{f}}{N_{0}}\int_{-B/2}^{B/2}G_{s}(f)G_{I} (f)df}, \tag{1}\]  
where the \(G_{s}(f)\) and \(G_{J}(f)\) are respectively the power spectrum density (PSD) of desired signal and interfering signal, normalized to unit power over infinite transmission bandwidth, the \(C\) and \(C_{J}\) are receiver antenna output power of desired signal and interfering signal, \(B\) is the receiver pre-filtering bandwidth, \(T\) is the integration time and \(N_{0}\) is the PSD of white noise taking value of \(-204\) dBW/Hz. The SSC is the inner product of PSDs between desired and interfering signals, defined as \[\chi_{J}^{s}=\int_{-B/2}^{B/2}G_{s}(f)G_{J}(f)df. \tag{2}\]  
As reported in Reference [19], when the white noise exists alone, the received \(C/N_{0}\) is simply equivalent to the effective carrier-to-noise ratio, resulting in  
\[\left(\frac{C}{N_{0}}\right)_{\text{eff,SSC}}=\frac{\rho_{n}}{T\int_{-B/2}^{B/ 2}G_{s}(f)df}. \tag{3}\]  
When the both of white noise and non-white interference are present, the general expression of the \(\left(C/N_{0}\right)_{\text{eff,SSC}}\) can be derived by substituting Equation (1) into Equation (3), i.e.,  
\[\left(\frac{C}{N_{0}}\right)_{\text{eff,SSC}}=\frac{C}{N_{0}+\frac{C_{J}\int_ {-B/2}^{B/2}G_{s}(f)G_{J}(f)df}{\int_{-B/2}^{B/2}G_{s}(f)df}}. \tag{4}\]  
Obviously, the \(\left(C/N_{0}\right)_{\text{eff,SSC}}\) can be treated as the other interpretation of output SNIR to clarify how the combination of noise and interference affects signal acquisition. In real environment, several space constellations may coexist in same frequency band, then the intersystem and intrasystem interference must be considered for more accurate compatibility evaluation. Thus, the \(\left(C/N_{0}\right)_{\text{eff,SSC}}\) can further be deduced as  
\[\left(\frac{C}{N_{0}}\right)_{\text{eff,SSC}} =\frac{C}{N_{\text{N}\text{N}\text{N}\text{N}\text{N}\text{N} \text{N}\text{N  
To reflect the relative weakening of a desired signal due to the intrasystem or intersystem interference, an indicator called effective carrier-to-noise ratio degradation \(\Delta\left(C/N_{0}\right)_{\text{eff}}\) is introduced by ITU-R [15]. The \(\Delta\left(C/N_{0}\right)_{\text{eff}}\) of desired signal induced by the intrasystem interference in acquisition is derived by  
\[\Delta\left(C/N_{0}\right)_{\text{eff,SSC}}^{\text{intra}}=\frac{C}{N_{0}}/ \frac{C}{N_{0}+I_{\text{intra}}^{\text{SSC}}\left(t\right)}=1+\frac{I_{\text {intra}}^{\text{SSC}}\left(t\right)}{N_{0}}. \tag{8}\]  
Similarly, the \(\Delta\left(C/N_{0}\right)_{\text{eff}}\) of desired signal induced by the intersystem interference in acquisition is given by  
\[\Delta\left(C/N_{0}\right)_{\text{eff,SSC}}^{\text{inter}}=\frac{C}{N_{0}+I_{ \text{intra}}^{\text{SSC}}\left(t\right)}/\frac{C}{N_{0}+I_{\text{intra}}^{ \text{SSC}}\left(t\right)+I_{\text{inter}}^{\text{SSC}}\left(t\right)}=1+ \frac{I_{\text{inter}}^{\text{SSC}}\left(t\right)}{N_{0}+I_{\text{intra}}^{ \text{SSC}}\left(t\right)}. \tag{9}\]