# A Robust Timing Synchronization Algorithm Based on PSSS for LTE-V2X
## 4 Improved PSSS-Based Time Synchronization Algorithm
### Coarse Synchronization  
The bandwidth is assumed to be 20 MHz, and the baseband sampling rate is set to 30.72 MHz. Considering the large synchronization signal period in LTE-V2X, the improved algorithm only takes 4,915,200 points for the baseband signal \(R(\theta)\) (one synchronization period) to perform time synchronization.  
To reduce the computational complexity, two kinds of local synchronization signals are downsampled N times. To ensure that the amount of information is not lost, the number of the local synchronization signal points after downsampling should be greater than the  
Figure 7: Flow chart of improved time synchronization algorithm.  
length of the PSSS sequence (62 points) in the frequency domain, so the sampling multiple \(N=32\) is taken here.  
The local PSSS \(p_{i}(n)\) and downsampled received signal \(r(\theta)\) (taking into account the time domain symmetry of the primary synchronization sequence, here the number of blocks is M = 2) are divided into several blocks, and each block of \(p_{i}(n)\) and \(r^{*}(\theta)\) is respectively correlated and superimposed to obtain two correlation values: \(C_{1i}(\theta),i=0,1\). After that, the maximum likelihood decision is used to obtain the type of receiving synchronization signal \(i\). \(C_{1i}(\theta)\) and the maximum likelihood decision are defined as  
\[C_{1i}(\theta)=\sum_{j=0}^{M-1}\left|\sum_{n=0}^{L-1}r^{*}(\theta+n+jL)p_{i}( n+jL)\right|^{2} \tag{9}\]  
\[\hat{i}=\underset{i}{\text{argmax}}\{C_{1i}(\theta)\} \tag{10}\]  
where \(r(\cdot)\) represents the downsampled received signal, \(p_{i}(\cdot),i=0,1\) represents the downsampled local PSSS of a different type, \(\hat{i}\) represents the type of PSSS in the received signal obtained by the maximum likelihood decision, \(M\) represents the number of blocks and \(L\) is the length of each block sequence.  
After comparing the correlated samples \(C_{1i}(\theta)\) with the detection threshold \(T_{11}(\theta)\), the coarse synchronization peak \(S_{11}(\theta)\) and \(T_{11}(\theta)\) can be expressed by the following equations:  
\[T_{11}(\theta)=\beta_{1}\text{max}(C_{1i}(\theta)) \tag{11}\]  
\[S_{11}(\theta)=\underset{\theta}{\text{arg}}\{C_{1i}(\theta)-T_{11}(\theta)>0\} \tag{12}\]  
where \(\beta_{1}\) is the false alarm rate. The false alarm rate of detection is higher when \(\beta_{1}\) increases. Since \(S_{11}(\theta)\) requires a second decision, to avoid missing the PSSS and improve the effectiveness of the second decision, \(\beta_{1}\) is 0.2 here, so Equation (12) can be rewritten as  
\[S_{11}(\theta)=\underset{\theta}{\text{arg}}\{C_{1i}(\theta)-0.2\text{max}(C_ {1i}(\theta))>0\} \tag{13}\]  
The values in \(S_{11}(\theta)\) equal to the relative distance between the two PSSSs are selected and added to obtain the summing peak value \(S_{11}^{sum}(\theta)\). Then, the coarse peak \(S_{12}(\theta)\) is obtained by comparing the correlated samples \(S_{11}^{sum}(\theta)\) with the detection threshold \(T_{12}(\theta)\). The above can be expressed as  
\[S_{11}^{sum}(\theta)=sum_{11}(\theta)+sum_{11}(\theta\pm d) \tag{14}\]  
\[T_{12}(\theta)=\beta_{2}\text{max}(S_{11}^{sum}(\theta)) \tag{15}\]  
\[S_{12}(\theta)=\underset{\theta}{\text{arg}}\{S_{11}^{sum}(\theta)-T_{12}( \theta)>0\} \tag{16}\]  
To prevent \"secondary peaks\" in \(S_{11}^{sum}(\theta)\) from being misjudged as the correct peak, the false alarm coefficient of the secondary decision threshold \(\beta_{2}\) is set to 0.7, which effectively reduces the false alarm rate.  
The coarse estimate point \(\hat{\theta}_{1}\) can be expressed as  
\[\left\{\begin{array}{l}\theta_{11}=\text{argmax}(S_{12}(\theta))\\ \theta_{12}=\text{argmax}(S_{12}(\theta)),\theta\
otin(\theta-N/2,\theta+N/2) \\ \hat{\theta}_{1}=\text{min}(\theta_{11},\theta_{12})\end{array}\right. \tag{17}\]