# Asymmetric Dual-Band Tracking Technique for Optimal Joint Processing of B1I and B1C Signals
## 3 Reception Model of B1-ADS Signal  
While the theoretical analysis of the frequency domain and ACF in the previous section shows that the B1-ADS signal has an excellent potential for ranging accuracy, we will further discuss the time domain reception model of the B1-ADS signal in this section for designing the optimal reception algorithm. As mentioned earlier, the B1C signal adopts the data-plus-pilot signal structure, and the pilot signal also includes the components of BOC(1,1) and BOC(6,1). To facilitate subsequent analysis and expression, without losing generality, a pilot tracking and data demodulation strategy that neglects the BOC(6,1) component will be adopted for the B1C signal. In other words, the B1-ADS signal discussed below will be simplified to consisting of the B1I signal on the lower band and the BOC(1,1) component of the B1C pilot signal on the upper band. In this paper, the B1-ADS signal is assumed to be received in the presence of thermal noise at the receiver front end that can be modeled as stationary additive white Gaussian noise. As a result, the received signal can be expressed as follows:  
\[r(t)=s_{B1-ADS}(t-\tau)+n(t) \tag{5}\]  
where \(\tau\) is the signal propagation delay, and \(n(t)\) is the thermal noise. In Equation (6), the delayed B1-ADS signal is represented as a composite of B1I and B1C signals:  
\[ \begin{array}{ll}s_{B1-ADS}(t-\tau)=&A_{B1I}s_{B1I}(t-\tau)\cos(2\pi f_{B1I}( t-\tau)+\theta_{B1I})\\ &+A_{B1C}s_{B1C}(t-\tau)\cos(2\pi f_{B1C}(t-\tau)+\theta_{B1C})\end{array}  \tag{6}\]  
where \(f\), \(\theta\) and \(A\) represent received frequency, initial carrier phase, and amplitude of the corresponding components, respectively. The values \(s_{B1I}(t-\tau)\) and \(s_{B1C}(t-\tau)\) are the baseband waveforms of the B1I and B1C signals, which are the combination of subcarrier, ranging code, secondary code,  
Figure 3: The auto-correlation functions (ACFs) of B1-ADS and other signals. and navigation data bit. The received frequency may be further expressed as a sum of the transmitted nominal frequency \(f_{\cdot,n}\) and Doppler frequency \(f_{\cdot,D}\) as follows,  
\[\left\{ \begin{array}{l}f_{\mathrm{B1I}}=f_{\mathrm{B1I,n}}+f_{\mathrm{B1I,D}}\\ f_{\mathrm{B1C}}=f_{\mathrm{B1C,n}}+f_{\mathrm{B1C,D}}\end{array} \right. \tag{7}\]  
Next, we want to rewrite Equation (6) with the parameters of the B1-ADS signal. The nominal carrier frequency and subcarrier frequency of the B1-ADS signal are introduced and defined in Equation (8), and the initial phases of carrier and subcarrier are given in Equation (9):  
\[\left\{ \begin{array}{l}f_{\mathrm{c,n}}=\frac{f_{\mathrm{B1C,n}}+f_{\mathrm{ B1I,2}}}{2}=1568.259\;\mathrm{MHz}\\ f_{\mathrm{s,n}}=\frac{f_{\mathrm{B1C,n}}-f_{\mathrm{B1I,n}}}{2}=7.161\; \mathrm{MHz}\end{array} \right. \tag{8}\]  
\[\left\{ \begin{array}{l}\theta_{c}=\frac{\theta_{\mathrm{B1C}}+\theta_{ \mathrm{B1I}}}{2}\\ \theta_{s}=\frac{\theta_{\mathrm{B1C}}-\theta_{\mathrm{B1I}}}{2}\end{array} \right. \tag{9}\]  
where the subscripts \(c\) and \(s\) are used to indicate carrier and subcarrier, respectively. Since the Doppler frequency scales linearly with the signal center frequency, the Doppler frequencies of both the carrier and subcarrier of B1-ADS signal can be represented in terms of the Doppler frequencies of the B1I and B1C signals as follows:  
\[\left\{ \begin{array}{l}f_{\mathrm{c,D}}=\frac{f_{\mathrm{B1C,D}}+f_{\mathrm{ B1 LD}}}{2}\\ f_{\mathrm{s,D}}=\frac{f_{\mathrm{B1C,D}}-f_{\mathrm{B1D}}}{2}\end{array} \right. \tag{10}\]  
Then, the received frequencies of the carrier and subcarrier of the B1-ADS signal are given in Equation (11),  
\[\left\{ \begin{array}{l}f_{\mathrm{c}}=f_{\mathrm{c,n}}+f_{\mathrm{c,D}}\\ f_{\mathrm{s}}=f_{\mathrm{s,n}}+f_{\mathrm{s,D}}\end{array} \right. \tag{11}\]  
By Equations (7)-(11), Equation (6) is rewritten based on the parameters of the B1-ADS signal as:  
\[