# A Linear Strong Constraint Joint Solution Method Based on Angle Information Enhancement
## 3 A Synchronization and Fine Ranging Method Based on the 5G Signal
### A Three-Stage Time-Frequency Synchronization Method Based on Group Peak Time Sequence Tracing
In the traditional timing coarse synchronization method based on a single sliding window, each sliding window has two sliding windows with a length of \(N_{CP}\) sampling points, and the length between the two windows is \((N_{FFT}-N_{CP})\) sampling points. The starting position of a subframe can be estimated by the correlation between the cyclic prefix and the tail of the symbol [21],  
\[\delta=\arg\!\!\max_{\delta}\left\{\sum_{i=\delta}^{N_{CP}-1+\delta}|y[i]y^{*} [i+N_{FFT}]|\right\} \tag{4}\]  
Among them, \(\delta\) is the time-domain starting position of the subframe obtained through estimation. When the long CP falls exactly into the window, the similarity between the two sliding windows is the highest. When the short CP falls within the sliding window, partial similarity also occurs, but its similarity is smaller than that of the long CP, which can distinguish different symbols in the same subframe. In simple open spaces, the above method can accurately perform coarse timing synchronization. However, in complex scenarios, signal attenuation is significant and the SNR of the received signal is low, resulting in poor coarse synchronization performance using the above method. The following Figure 2 compares the timing coarse synchronization performance of a single sliding window under different signal-to-noise ratios, assuming a subcarrier spacing of 30 kHz. When SNR = 30 dB, prominent peaks can be obtained through sliding windows, and the relative position relationship between adjacent peaks is obvious. The first and fifteenth peaks correspond to two long CPs in a subframe, with high correlation. When SNR = 10 dB, the performance of this method deteriorates significantly, and clear peaks cannot be obtained, resulting in a serious decrease in coarse synchronization accuracy. To enhance the robustness of coarse synchronization, this chapter improves the timing synchronization algorithm of a single sliding window and proposes a group peak accumulation coarse timing synchronization method using a multi-window joint estimation mechanism. In a subframe, the index information \(peak_{index}(i)\) of the group peaks is as follows:  
\[peak_{index}(i)=\left\{ \begin{array}{ccc}1&,&i=1\\ 1+N^{\prime}&,&i=2\\ 1+N^{\prime}+(i-2)\big{(}N_{FFT}+N_{cp}\big{)}&,2<i\leq\frac{n_{s}}{2}\\ 1+\frac{N_{orb}}{2}&,&i=\frac{n_{s}}{2}+1\\ 1+\frac{N_{orb}}{2}+N^{\prime}&,&i=\frac{n_{s}}{2}+2\\ 1+\frac{N_{orb}}{2}+N^{\prime}+\big{(}i-\frac{n_{s}}{2}-2\big{)}\big{(}N_{FFT}+ N_{cp}\big{)}&,\frac{n_{s}}{2}<i\leq n_{s}\end{array} \right. \tag{5}\]  
Among them, \(peak_{index}(i)\) represents the index of the sampling point corresponding to the \(i^{th}\) peak in a subframe, \(N^{\prime}=N_{FFT}+\frac{N_{CP}+N_{CP}}{2}\), \(n_{s}\) is the number of symbols contained in each subframe, and \(N_{sub}\) is the number of sampling points in each subframe. The detection function of the long CP can be expressed as:  
\[cp_{detect}(i)=\sum_{i=1}^{N_{CP}}|y[i]y^{*}[i+N_{FFT}]| \tag{6}\]  
The rough timing estimation result obtained by multi-window group peak accumulation based on the CP can be expressed as:  
\[\hat{\delta}=\underset{\delta}{\text{argmax}}\left\{\sum_{i=1}^{n_{s}}cp_{detect }[\hat{\delta}+peak_{index}(i)-1\ ]\right\} \tag{7}\]  
The coarse timing result represents the first subframe of the signal received at the time corresponding to the \(\hat{\delta}^{th}\) sample. The first SSB is usually located in the first subframe of the received signal header, and rough timing information of the SSB can be obtained through the relative position relationship of the group peaks. Figure 2: Comparison of timing coarse synchronization performance of a single sliding window under different signal-to-noise ratios.