## 4 Navigation Signal Processing
### Signal Tracking  
In the signal tracking stage, the acquisition results are first used for initialization, followed by continuous processing of the navigation signals to obtain accurate measurements of delay and Doppler frequency. The proposed tracking module includes a frequency-locked loop-assisted phase-locked loop (PLL) and a carrier-aided delay-locked loop (DLL). Figure 6 shows the block diagram of DDBDM signal tracking. As can be observed from the Figure, most of the components of the tracking loop are the same as in a traditional navigation receiver, except for the generation of local replicas. Specifically, in contrast to GNSS signals that generate code replicas in the time domain, DDBDM is in the TF domain (DD to TF domain can be performed offline), and requires an additional modulation process, i.e., the IFFT and P/S part in the figure. This no doubt increases the computational overhead, which is one of the reasons why this paper considers the receival of part of the signal bandwidth.  
In the tracking process with the DLL, it is necessary to simultaneously generate local replicas for the early, prompt, and late branches, and then the complex correlation values are computed through the integration and dump operation. The general form of the correlation values parameterized by \(\Lambda\) is as follows:  
\[\left\{\begin{array}{l}I[\Lambda]=\sum\limits_{q=0}^{M_{\mathrm{t}}N-1}r[q] \cdot\cos(2\pi\hat{\upsilon}qT_{s}+\hat{\varphi})\cdot s_{\mathrm{loc}}[qT_{s} -\hat{\tau}+\Lambda]\\ Q[\Lambda]=\sum\limits_{q=0}^{M_{\mathrm{t}}N-1}r[q]\cdot\sin(2\pi\hat{ \upsilon}qT_{s}+\hat{\varphi})\cdot s_{\mathrm{loc}}[qT_{s}-\hat{\tau}+ \Lambda]\end{array}\right. \tag{24}\]  
Assuming \(d\) is the early-late spacing of the DLL, the early, prompt, and late correlation values can be produced by setting \(\Lambda\) to be \(-d/2\), \(0\), and \(d/2\), respectively.  
The DDBDM signal has in-phase and quadrature components, so the correlation values from both channels are combined with the same weighting to form the composite tracking error estimate, which continuously steers the shared NCO for both channels. For the PLL, the carrier phase error using an arctangent phase discriminator is as follows:  
\[\delta\phi=\frac{\delta\phi_{\mathrm{Re}}+\delta\phi_{\mathrm{Im}}}{2} \tag{25}\]  
where  
\[\delta\phi_{\mathrm{Re}}=\tan^{-1}\left(\frac{Q_{\mathrm{P}}g_{\mathrm{Re}}}{ l_{\mathrm{P}}g_{\mathrm{Re}}}\right) \tag{26}\]  
For the DLL, the code phase error using the noncoherent early-late processing (NELP) discriminator is as follows:  
\[\delta\hat{\tau}=\frac{\delta\tau_{\mathrm{Re}}+\delta\hat{\tau}_{\mathrm{Im}} }{2} \tag{27}\]  
where  
\[\delta\tau_{\mathrm{Re}}=\frac{\sqrt{l_{\mathrm{I,Re}}^{2}+Q_{\mathrm{Re}}^{2 }}-\sqrt{l_{\mathrm{I,Re}}^{2}+Q_{\mathrm{L,Re}}^{2}}}{\sqrt{l_{\mathrm{I,Re} }^{2}+Q_{\mathrm{Re}}^{2}+Q_{\mathrm{Re}}^{2}}+\sqrt{l_{\mathrm{I,Re}}^{2}+Q _{\mathrm{Re}}^{2}}} \tag{28}\]  
Figure 6: Block diagram of signal tracking.  
The subscripts \(E\), \(P\), and \(L\) represent the early, prompt, and late correlation values, respectively. The subscripts Re and Im represent the real and imaginary parts of the correlation values, respectively.  
Finally, the navigation receiver calculates the position, velocity, and clock bias of UE using the measurements and the navigation messages demodulated by the tracking loop. It is worth noting that the CP of the received signal has little impact on the main lobe of the ACF [52]; it could be considered part of the time-domain signal, aiming to utilize as much energy as possible for correlation.