# A Linear Strong Constraint Joint Solution Method Based on Angle Information Enhancement
## 3 A Synchronization and Fine Ranging Method Based on the 5G Signal
### Fine Robust Tracking Based on the PRS  
In the previous text, a coarse timing estimation method based on sliding windows and a fine timing synchronization method based on the PSS sliding cross-correlation were used for preliminary timing, synchronizing the received signal to the specific sampling points of each time-domain symbol. This is sufficient for communication performance in 5G systems, but for positioning systems, it still has a relatively rough time resolution and cannot obtain accurate ranging information. Since the positioning terminal processes all digital signals, the value between two samples of the transmitted signal can be regarded as the continuous level of the corresponding value of the previous sampling point. For the convenience of expression and not to be confused with the following text, this article refers to it as a chip. To improve time resolution, it is necessary to oversample the received signal by non-integer multiples, so that the sampling points are evenly distributed at different phase positions of the chip, ensuring accurate capture of the chip's transition time and improving the measurement resolution of the code phase. To obtain the ranging accuracy within the chip, it is necessary to track the code phase. In this section, a delay phase-locked loop based on lead lag is used for tracking to improve measurement accuracy, while suppressing high-frequency noise in the signal and improving signal-to-noise ratio. The delay phase-locked loop consists of a correlator, a code phase discriminator, a loop filter, and a voltage-controlled oscillator. After time-frequency synchronization, the received signal has been synchronized to chip accuracy. As the time delay of the 5G signal in the time domain can be equivalent to phase shift in the frequency domain, the locally generated leading and lagging signals in the frequency domain can be represented as:  
\[S_{\text{early}}(k)=e^{-\beta x\hat{\delta}/K}S(k) \tag{17}\]\[S_{\text{late}}(k)=e^{\beta 2\pi\xi k/K}S(k) \tag{18}\]  
Among them, \(k=0,1\ldots,K-1\), \(k\) representing the number of subcarrier indices for the PRS; K represents the number of the PRS subcarriers used for positioning; \(\xi\in[0.5,1]\) represents the normalized offset in the time domain. By generating local lead and lag sequences, the lead and lag correlations in the frequency domain can be calculated separately [22]  
\[\mathcal{R}_{\text{early}}=\sum_{k=0}^{K-1}R^{\prime}(k)S_{\text{early}}^{*}(k) \tag{19}\]  
\[\mathcal{R}_{\text{late}}=\sum_{k=0}^{K-1}R^{\prime}(k)S_{\text{late}}^{*}(k) \tag{20}\]  
Among them, \(R^{\prime}(k)\) represents the sequence received by the kth subcarrier at the current time. The output of the code phase discriminator can be expressed as:  
\[D_{\text{DLL}}\triangleq\left|\mathcal{R}_{\text{early}}\right.{}^{1}{}^{2}- \left|\mathcal{R}_{\text{late}}\right.{}^{1}{}^{2}\triangleq K^{2}C\Lambda_ {\text{DLL}}\left(\widetilde{e}_{\tau},\xi\right)+N_{\text{DLL}} \tag{21}\]  
Among them, \(C\) represents the energy of the received signal, and the normalized s-curve \(\Lambda_{\text{DLL}}\left(\widetilde{e}_{\tau},\xi\right)\) can be further expressed as:  
\[\Lambda_{\text{DLL}}\left(\widetilde{e}_{\tau},\xi\right)\triangleq\left[ \frac{\sin\left(\pi\left(\widetilde{e}_{\tau}-\xi\right)\right)}{K\sin\left( \pi\left(\widetilde{e}_{\tau}-\xi\right)/K\right)}\right]^{2}-\left[\frac{ \sin\left(\pi\left(\widetilde{e}_{\tau}+\xi\right)\right)}{K\sin\left(\pi \left(\widetilde{e}_{\tau}+\xi\right)/K\right)}\right]^{2} \tag{22}\]  
Among them, \(\xi\) represents the normalized time offset, and the relative relationship between the normalized s-curve and \(\xi\) variation is shown in the following Figure 3:  
To ensure high tracking sensitivity, the slope at the zero point of the normalized s-curve should be increased, so \(\xi=0.5\) was selected in subsequent experiments.