# A Robust Timing Synchronization Algorithm Based on PSSS for LTE-V2X
## 4 Improved PSSS-Based Time Synchronization Algorithm
### Frequency Offset Estimation  
The calculation of \(\hat{\theta}_{1}\) is based on the downsampled received signal and the local synchronization signal, so there is an error of \([-N,N]\) from the actual position. Therefore, fine synchronization is required to complete the synchronization process, but due to the high frequency offset characteristics of the LTE-V2X communication scene, performing fine synchronization directly without any processing on the received signal can easily lead to TO or even fail the fine synchronization.  
In this part, we use \(\hat{\theta}_{1}\) and frame struct to calculate the position of the Demodulation Reference Signal (DMRS) in the same frame of the broadcast channel and then extract them to estimate the frequency offset, and we preset the estimated frequency offset to the local synchronization signal to improve the accuracy and reliability of the fine synchronization.  
DMRS-based frequency offset estimation methods are mainly divided into the Adjacent DMRS-based algorithm [24] and the Half-symbol-based algorithm [25; 26]. The Adjacent DMRS-based algorithm estimates the frequency offset by calculating the phase difference between adjacent DMRSs. Its range is related to the minimum interval of the DMRS in the time domain. The Half-symbol-based algorithm estimates the frequency offset by comparing the phase changes in each half of a single DMRS in the time domain. As shown in Figure 8, the correction range is equal to the subcarrier spacing.  
In the LTE-V2X system, the high communication frequency point and the relative speed between nodes make the channel change rapidly over time. Assuming that the relative velocity between cars is 300 km/h, the communication frequency point is 5915 MHz. Therefore, the maximum Doppler frequency deviation is about 1643 Hz, and the channel coherence time is 0.1 ms. The period of the SC-FDM symbol is 0.07 ms, which is greater than the channel coherence time of V2X. Therefore, it can be considered that the channel is approximate within one SC-FDM symbol. However, the Adjacent DMRS-based algorithm uses an adjacent DMRS for frequency offset estimation and spans three SC-FDM symbols. Hence, its accuracy is lower than that of the Half-symbol-based algorithm, which uses one DMRS symbol frequency offset estimation. In summary, this paper chooses the Half-symbol-based algorithm to estimate the frequency offset after coarse synchronization.  
The DMRS in the PSBCH is located on the 5 th, 7 th and 10 th SC-FDMA symbols. The three received DMRSs are extracted according to the specific PSBCH frame struct and are cross-correlated successively with the local DMRS duplicate. The frequency offset estimation using the Half-symbol-based algorithm can be expressed as  
\[\hat{f}=-\frac{1}{2\pi}\frac{N}{3(N+N_{cp})}\text{arg}\left(\sum_{k=0}^{2}\left( \left(\sum_{n=0}^{N-1}r_{k}(n)P_{k}^{*}(n)\right)^{*}\left(\sum_{n=0}^{N-1}r_{ k+1}(n)P_{k+1}^{*}(n)\right)\right)\right) \tag{18}\]  
where \(r_{k}(n)\) is the received DMRS, \(P_{k}(n)\) is the local DMRS duplicate, \(k=0,1,2,3\) represents different SC-FDM symbols and \(N\) is the number of FFT points.  
The local synchronization signal that presets the estimated frequency offset \(\hat{P}(n)\) can be expressed as  
\[\hat{P}(n)=\exp(-(2\pi\hat{f}nj)/f_{s})p(n) \tag{19}\]