# 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
#### 4.1.1 Correlator  
In Figure 4 the local BII signals generated on the left side and the local B1C signals generated on the right side are used to correlate with the incoming IF signal, respectively. The BII correlator is the conventional BPSK-type correlator. The locally generated signals are defined as the six possible combinations of in-phase (I) and quadra-phase (Q) local carrier replicas and early (E), prompt (P), and late (L) local code replicas [1]. The B1C correlator is the classical BOC(1,1)-type correlator. The locally generated signals are defined as the ten possible combinations of in-phase and quadra-phase local carrier replicas and very early (VE), early, prompt, late, and very late (VL) local code replicas. Figure 4: The tracking channel architecture of ASYM-DBT for the B1-ADS signal. This setting supports the Bump-Jump algorithm in detecting and recovering from a false lock in the single-band processing mode based on the comparison of the main and side peak amplitudes. According to Equation (12), it is not difficult to discover that B1I and B1C signals are mutually orthogonal frequency-wise and code-wise. This means one signal has little influence on the correlator outputs of the other signal on the other sideband. For simplicity, and to clarify the concepts underlying the algorithm design, the thermal noise term in (2) is dropped. Thus, the outputs of the two groups of correlators can be approximately expressed as follows:  
\[\left\{ \begin{array}{l}IE_{B1I}\simeq A_{B1I}D_{B1I}R_{B1I}(\Delta\tau-d_{B1 I}/2)\mathrm{T}\sin\mathrm{c}(\alpha)\cos(\alpha+\Delta\theta_{c}-\Delta\theta_{s}) \\ QE_{B1I}\simeq A_{B1I}D_{B1I}R_{B1I}(\Delta\tau-d_{B1I}/2)\mathrm{T}\sin \mathrm{c}(\alpha)\sin(\alpha+\Delta\theta_{c}-\Delta\theta_{s})\\ IP_{B1I}\simeq A_{B1I}D_{B1I}R_{B1I}(\Delta\tau)\mathrm{T}\sin\mathrm{c}( \alpha)\cos(\alpha+\Delta\theta_{c}-\Delta\theta_{s})\\ QP_{B1I}\simeq A_{B1I}D_{B1I}R_{B1I}(\Delta\tau)\mathrm{T}\sin\mathrm{c}( \alpha)\sin(\alpha+\Delta\theta_{c}-\Delta\theta_{s})\\ IL_{B1I}\simeq A_{B1I}D_{B1I}R_{B1I}(\Delta\tau+d_{B1I}/2)\mathrm{T}\sin \mathrm{c}(\alpha)\cos(\alpha+\Delta\theta_{c}-\Delta\theta_{s})\\ QL_{B1I}\simeq A_{B1I}D_{B1I}R_{B1I}(\Delta\tau+d_{B1I}/2)\mathrm{T}\sin \mathrm{c}(\alpha)\sin(\alpha+\Delta\theta_{c}-\Delta\theta_{s})\\ \end{array} \right. \tag{14a}\] \[\left\{ \begin{array}{l}IVE_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau-d_{B1C}) \mathrm{T}\sin\mathrm{c}(\beta)\cos(\beta+\Delta\theta_{c}+\Delta\theta_{s}) \\ QVE_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau-d_{B1C})\mathrm{T}\sin\mathrm{c}( \beta)\sin(\beta+\Delta\theta_{c}+\Delta\theta_{s})\\ IE_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau-d_{B1C}/2)\mathrm{T}\sin\mathrm{c}( \beta)\cos(\beta+\Delta\theta_{c}+\Delta\theta_{s})\\ QE_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau-d_{B1C}/2)\mathrm{T}\sin\mathrm{c} (\beta)\sin(\beta+\Delta\theta_{c}+\Delta\theta_{s})\\ IP_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau)\mathrm{T}\sin\mathrm{c}(\beta) \cos(\beta+\Delta\theta_{c}+\Delta\theta_{s})\\ QP_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau)\mathrm{T}\sin\mathrm{c}(\beta) \sin(\beta+\Delta\theta_{c}+\Delta\theta_{s})\\ IL_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau+d_{B1C}/2)\mathrm{T}\sin\mathrm{c}( \beta)\cos(\beta+\Delta\theta_{c}+\Delta\theta_{s})\\ QL_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau+d_{B1C})\mathrm{T}\sin\mathrm{c}( \beta)\sin(\beta+\Delta\theta_{c}+\Delta\theta_{s})\\ IVL_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau+d_{B1C})\mathrm{T}\sin\mathrm{c}( \beta)\cos(\beta+\Delta\theta_{c}+\Delta\theta_{s})\\ QVL_{B1C}\simeq A_{B1C}R_{B1C}(\Delta\tau+d_{B1C})\mathrm{T}\sin\mathrm{c}( \beta)\sin(\beta+\Delta\theta_{c}+\Delta\theta_{s})\end{array} \right.