# Asymmetric Dual-Band Tracking Technique for Optimal Joint Processing of B1I and B1C Signals
## 4 The ASYM-DBT Method and Thermal Noise Performance Analysis
### Theoretical Noise Analysis  
After detailing the ASYM-DBT method, we analyze its theoretical ranging performance for the B1-ADS signals. For simplicity, thermal noise is generally treated as the dominant source of error for signal tracking. Generally, the thermal noise manifests in the code performance, the subcarrier performance, and the carrier performance.  
According to Equation (12), we can safely ignore the interaction between adjacent frequency signals when analyzing the thermal noise performance. By first calculating the signal part and noise part of each single sideband, the combination correlations of the B1-ADS signal and theoretical errors induced by the thermal noise can then be obtained. For simplicity, this paper will not cover the derivation process. With the coherent DLL discriminator, the dual-band thermal noise can be obtained as follows [27, 28]:  
\[\sigma_{IDLL}^{2}\simeq\frac{B_{nDLL}(\int_{-\beta_{r}/2}^{\beta_{r}/2}\left(w_{ B1I}G_{B1I}(f)\sin^{2}(\pi fd_{B1I}T)+w_{B1C}G_{B1C}(f)\sin^{2}(\pi fd_{B1C}T) \right)df)}{(2\pi)^{2}C/N_{0}\Big{(}\int_{-\beta_{r}/2}^{\beta_{r}/2}\left(w_{ B1I}fG_{B1I}(f)\sin(\pi fd_{B1I}T)+w_{B1C}fG_{B1C}(f)\sin(\pi fd_{B1C}T) \right)df)^{2}}, \tag{19}\]  
where \(B_{nDLL}\) represents the equivalent noise bandwidth of the code tracking loop; \(w_{B1C}\) and \(w_{B1I}\) are the power proportion of the B1C and B1I signals; \(G(f)\) represents the PSD of the single-sideband signal, which is normalized to unity over infinite bandwidth; \(\beta_{r}\) represents the equivalent dual-band front-end bandwidth; \(C/N_{0}\) represents the carrier-to-noise ratio of the total dual band; \(d\) represents the E-L correlator spacing. In case of a non-coherent discriminator, the joint dual-band thermal noise is shown below:  
\[\sigma_{IDLL}^{2}\simeq\frac{B_{nDLL}(\int_{-\beta_{r}/2}^{\beta_{r}/2}\left( w_{B1I}G_{B1I}(f)\sin^{2}(\pi fd_{B1I}T)+w_{B1C}G_{B1C}(f)\sin^{2}(\pi fd_{B1C}T) \right)df)}{(2\pi)^{2}C/N_{0}\Big{(}\int_{-\beta_{r}/2}^{\beta_{r}/2}\left(w _{B1I}G_{B1I}(f)\sin(\pi fd_{B1I}T)+w_{B1C}fG_{B1C}(f)\sin(\pi fd_{B1C}T)df \right)^{2}}\times \tag{20}\] \[\left(1+\frac{B_{nDLL}(\int_{-\beta_{r}/2}^{\beta_{r}/2}\left(w _{B1I}G_{B1I}(f)\cos^{2}(\pi fd_{B1I}T)+w_{B1C}G_{B1C}(f)\cos^{2}(\pi fd_{B1C} T)df)}{TC/N_{0}\Big{(}\int_{-\beta_{r}/2}^{\beta_{r}/2}\left(w_{B1I}G_{B1I}(f) \cos(\pi fd_{B1I}T)df+w_{B1C}G_{B1C}(f)\cos(\pi fd_{B1C}T)df)\Big{)}^{2}}\right)\]  
The term in the second line represents the squaring loss.  
Considering the asymmetric property of the B1-ADS, the associated subcarrier phase error is given by (21a) [27, 28]:  
\[\sigma_{tPLLS}^{2}\simeq\frac{B_{nPLLS}}{C/N_{0}\int_{-\beta_{r}/2}^{\beta_{r }/2}\left(w_{B1I}G_{B1I}(f)+w_{B1C}G_{B1C}(f)\right)df}.\] (21a) Similarly, the carrier phase error is given by ( 21b ): \[\sigma_{tPLLC}^{2}\simeq\frac{B_{nPLLC}}{C/N_{0}\int_{-\beta_{r}/2}^{\beta_{r }/2}\left(w_{B1I}G_{B1I}(f)+w_{B1C}G_{B1C}(f)\right)df}, \tag{21b}\]  
where \(B_{nPLLS}\) and \(B_{nPLLC}\) represent the equivalent noise bandwidth of the subcarrier and carrier tracking loops, respectively. These theoretical results of the thermal noise performance are compared with the experiment results in the next section.