# Asymmetric Dual-Band Tracking Technique for Optimal Joint Processing of B1I and B1C Signals
## 4 The ASYM-DBT Method and Thermal Noise Performance Analysis
### The ASYM-DBT Method
#### 4.1.2 Processing Unit
\begin{array}{l}\gamma QP_{B1C}-\widetilde{D}_{B1I}QP_{B1I}\simeq 2A_{B1I} \text{Tcos}(\pi\Delta\text{f}_{\text{c}}\text{T}+\Delta\theta_{\text{c}}) \text{sin}(\pi\Delta\text{f}_{\text{s}}\text{T}+\Delta\theta_{\text{s}})\\ \gamma QP_{B1C}+\widetilde{D}_{B1I}QP_{B1I}\simeq 2A_{B1I}\text{Tsin}(\pi \Delta\text{f}_{\text{c}}\text{T}+\Delta\theta_{\text{c}})\text{cos}(\pi \Delta\text{f}_{\text{s}}\text{T}+\Delta\theta_{\text{s}})\\ \gamma IP_{B1C}+\widetilde{D}_{B1I}IP_{B1I}\simeq 2A_{B1I}\text{Tcos}(\pi \Delta\text{f}_{\text{c}}\text{T}+\Delta\theta_{\text{c}})\text{cos}(\pi \Delta\text{f}_{\text{s}}\text{T}+\Delta\theta_{\text{s}})\end{array}  \tag{15}\]  
Figure 5: Two working modes of the processing unit. (a) Single-band mode; (b) Dual-band mode. As can be observed from Equation (15), subcarrier and carrier phases are completely separated. Then, phase discriminations of subcarrier and carrier can be achieved by employing the pure PLL discriminator by the optimal four-quadrant arctangent discriminator algorithm as follows:  
\[ \begin{array}{rl}disc(\theta_{s})&=\text{atan2}(\gamma QP_{B1C}-\widetilde{D}_{ B1I}QP_{B1I},\gamma IP_{B1C}+\widetilde{D}_{B1I}IP_{B1I})\\ &\simeq\pi\Delta\mathbb{f}_{s}T+\Delta\theta_{s}\\ disc(\theta_{c})&=\text{atan2}(\gamma QP_{B1C}+\widetilde{D}_{B1I}QP_{B1I}, \gamma IP_{B1C}+\widetilde{D}_{B1I}IP_{B1I})\\ &\simeq\pi\Delta\mathbb{f}_{c}T+\Delta\theta_{c}\end{array}  \tag{16}\]  
Note that the pure PLL discriminator is superior to the Costas PLL discriminator with a 6 dB gain in signal tracking threshold [1]. * Discriminators for code phase  
The coherent and non-coherent discrimination results of the code phase are calculated based on the early and late branches of two groups of correlators. In the design of the code phase discriminator for the B1-ADS signal, the difference in ACFs between the B1I and B1C signals and the difference of the correlator spacing between the two groups of correlators must be considered. The coherent code phase discriminator of the B1-ADS signal is the weighted sum of the coherent discriminators of the B1I and B1C signals as follows [1; 11]:  
\[disc_{Coh}(\tau)=\frac{1}{2}\frac{2-k_{B1I}d_{B1I}}{4k_{B1I}}\frac{IE_{B1I}- IL_{B1I}}{IP_{B1I}}+\frac{1}{2}\frac{2-k_{B1C}d_{B1C}}{4k_{B1C}}\frac{IE_{B1C}- IL_{B1C}}{IP_{B1C}} \tag{17}\]  
where \(k_{B1I}\) and \(k_{B1C}\) represent the slopes of the main peaks of B1I ACF and B1C ACF, i.e., one and three chip\({}^{-1}\), respectively. Similarly, the non-coherent code discriminator of the B1-ADS signal is given in the form of the weighted sum of the non-coherent discriminators of the B1I and B1C signals, as written below:  
\[disc_{Non}(\tau)=\frac{1}{2}\frac{2-k_{B1I}d_{B1I}}{4k_{B1I}}\frac{IE_{B1I}^{2} +QE_{B1I}^{2}-IL_{B1I}^{2}-QL_{B1I}^{2}}{IE_{B1I}^{2}+QE_{B1I}^{2}+IL_{B1I}^{2 }+QL_{B1I}^{2}}+\frac{1}{2}\frac{2-k_{B1C}d_{B1C}}{4k_{B1C}}\frac{IE_{B1C}^{2} +QE_{B1C}^{2}-IL_{B1C}^{2}-QL_{B1C}^{2}}{IE_{B1C}^{2}+QE_{B1C}^{2}+IL_{B1C}^{ 2}+QL_{B1C}^{2}} \tag{18}\]  
* Loop filters  
Since the outputs of the discriminators mentioned above are noisy, the discriminators of code, subcarrier, and carrier are followed by the corresponding loop filters to reduce the noise. In our design, two separate second-order filters are selected for DLL and SLL, while a third-order filter is selected for PLL. Note that carrier-aiding of the code phase and subcarrier phase might be employed in order to achieve more accurate measurements. The three filtered discriminator outputs are then used to drive code and carrier NCOs of the B1I and B1C signals, respectively. Since there is no essential difference between the DBT loop filters and ASYM-DBT loop filters, for simplicity this paper will not discuss different filter configurations; refer to [1; 17] for details.