# Asymmetric Dual-Band Tracking Technique for Optimal Joint Processing of B1I and B1C Signals
## 2 Summary of the BDS B1 Signal
### Characteristic Analysis of the Virtual Broadband Signal B1-ADS  
Admittedly, B1I and B1C signals with unequal power and different modulation schemes have been expected to be processed as two independent signals in receivers. However, when considering the fact that both signals are generated based on the same onboard reference clock and their propagation paths to the receiver antenna are nearly the same, we can regard these two signals as a whole and define it as a new signal, that is, B1-ADS. This virtual wideband signal has a center frequency of 1568.259 MHz ((1561.098 + 1575.42)/2) and has two asymmetrical sidebands. The characteristics of the B1-ADS signal are given in terms of PSD, Gabor bandwidth, and ACF, as follows.  
First, the PSD of the B1-ADS signal is approximated by the sum of the PSDs of the B1I and B1C signals, as shown below:  
\[G_{B1-ADS}(f)=G_{B1I}(f+f_{s})+G_{B1C}(f-f_{s}) \tag{3}\]  
where \(f_{s}\) represents half the center frequency spacing between the B1I and B1C signals, i.e., 7.161 MHz. The rationality of this approximation is that the center frequencies of the B1I and B1C signals are spaced by 14.332 MHz, and the vast majority of the spectrum of the B1I and B1C signals does not overlap, which leads to negligible interaction between the two components of the spectrum. The PSD of the B1-ADS signal is plotted in Figure 1 where it can be seen that the two components, B1I and B1C, are modulated on the lower and upper sidebands, respectively.  
Then, the Gabor bandwidth, as an important index used to characterize a signal's theoretical ranging capability, can be calculated based on its PSD. It is well known that the broader the Gabor bandwidth is, the better the ranging performance that can be achieved. Given the PSD and front-end bandwidth \(\beta_{r}\), the formula to calculate Gabor bandwidth is as follows [24; 25]:  
\[\overline{F}^{2}=\frac{\int_{-\beta_{r}/2}^{+\beta_{r}/2}f^{2}G(f)df}{\int_{- \beta_{r}/2}^{+\beta_{r}/2}G(f)df}. \tag{4}\]  
Figure 2 gives the results of the Gabor bandwidth of the B1-ADS signal with different front-end bandwidths. At the same time, the Gabor bandwidth results of B1I, B1C, and BPSK(10) modulation signal are also given in the same figure for comparison. Among them, the Gabor bandwidths of two wideband signals, B1-ADS and BPSK(10), are calculated for the front-end bandwidths in the [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60] MHz range. Figure 2 shows that the Gabor bandwidth of B1-ADS is not only significantly greater than those of the B1I and B1C signals as expected, but also greater than the Gabor bandwidth of the conventional wideband signal with BPSK(10) modulation. This indicates the excellent potential for ranging performance of the B1-ADS signal.  
Figure 1: The power spectral density (PSD) of the B1-ADS signal.  
Figure 2: The Gabor bandwidths of B1-ADS and other signals.  
Finally, the ACF of the B1-ADS signal is depicted in Figure 3. The same figure also gives the ACFs of B1I, B1C, and a signal with BOC(7,2) modulation. Comparing the ACFs of these signals, it is not difficult to see that the ACF of the B1-ADS has multiple peaks similar to that of the BOC(7,2) signal. The reason will be explained in the next section. While the steep main peak in the ACF of B1-ADS signal is another indicator of its excellent ranging performance, the feature of multiple peaks can also lead to the problem of tracking ambiguity similar to BOC signals.