# Robust Symbol and Frequency Synchronization Method for Burst OFDM Systems in UAV Communication
## 3 Improved Burst OFDM Synchronization Method
### Symbol Timing Synchronization Method Based on 2-Bit Non-Uniform Quantization
#### 3.2.2 Low-Complexity Timing Synchronization Method  
Based on the quantization method presented in the previous section, the complex multiplication in Equation (2) can be expressed as follows:  
\[c_{n,k} =\hat{r}_{n+k}s_{k}^{*} \tag{12}\] \[=\left(\hat{r}_{n+k}^{I}+\hat{r}_{n+k}^{Q}\right)\left(s_{k}^{I}-j _{k}^{Q}\right)\] \[=\hat{r}_{n+k}^{I}s_{k}^{I}+\hat{r}_{n+k}^{Q}s_{k}^{Q}+j\left(\hat {r}_{n+k}^{Q}s_{k}^{I}-\hat{r}_{n+k}^{I}s_{k}^{Q}\right)\]  
Let \(c_{1}=\hat{r}_{n+k}^{I}s_{k^{\prime}}^{I}\), \(c_{2}=\hat{r}_{n+k}^{Q}s_{k}^{Q}\), \(c_{3}=\hat{r}_{n+k}^{Q}s_{k^{\prime}}^{I}\), and \(c_{4}=\hat{r}_{n+k}^{I}s_{k}^{Q}\). We can determine the signs of these variables as  
\[\left\{\begin{array}{l}a_{c_{1}}=a_{\hat{r}_{n+k}}^{I}\oplus a_{s_{k}}^{I}\\ a_{c_{2}}=a_{\hat{r}_{n+k}}^{Q}\oplus a_{s_{k}}^{Q}\\ a_{c_{3}}=a_{\hat{r}_{n+k}}^{Q}\oplus a_{s_{k}}^{Q}\\ a_{c_{4}}=a_{\hat{r}_{n+k}}^{Q}\oplus a_{s_{k}}^{Q}\\ \end{array}\right., \tag{13}\]  
where \(\oplus\) represents a bitwise XOR operation. The magnitudes of these variables are expressed as  
\[\left\{\begin{array}{l}b_{c_{1}}=b_{\hat{r}_{n+k}}^{I}+b_{s_{k}}^{I}\\ b_{c_{2}}=b_{\hat{r}_{n+k}}^{Q}+b_{s_{k}}^{Q}\\ b_{c_{3}}=b_{\hat{r}_{n+k}}^{Q}+b_{s_{k}}^{I}\\ b_{c_{4}}=b_{\hat{r}_{n+k}}^{I}+b_{s_{k}}^{Q}\\ \end{array}\right.. \tag{14}\]  
The calculated result \(b_{c_{i}}(i=1,2,3,4)\) is represented by two bits 00,01,10, and the corresponding real values are as shown in Table 1. Since \(0<T_{0}<T_{1}\), we can conclude that \(T_{0}^{2}<T_{0}T_{1}<T_{1}^{2}\).  
As a result, the variables \(c_{i}(i=1,2,3,4)\) can be represented with 3 bits as \(\{a_{c_{i}},b_{c_{i}}\}\). However, the 3-bit result cannot be used directly when calculating \(c_{k}^{I}=c_{1}+c_{2}\) and \(c_{k}^{Q}=c_{3}-c_{4}\). They should be mapped to the true value before the addition operation. The specific calculation method can be expressed as  
\[c_{n,k}=f(c_{1})+f(c_{2})+j(f(c_{3})-f(c_{4})), \tag{15}\]  
where the reconstruction function is defined by  
\[f(c_{i})=\left\{\begin{array}{ll}(1-2a_{c_{i}})T_{0}^{2},&b_{c_{i}}=\text{ `00'}\\ (1-2a_{c_{i}})T_{0}T_{1},&b_{c_{i}}=\text{`01'}\\ (1-2a_{c_{i}})T_{1}^{2},&b_{c_{i}}=\text{`10'}\\ \end{array}\right.. \tag{16}\]  
Therefore, \(c_{n,k}\) can be computed using XOR and addition operations instead of complex multiplication operations. By substituting it into Equation (2), we can obtain the instantaneous correlation value \(V(n)\). A decision can be made to determine the starting position of the OFDM symbol by comparing \(V_{n}\) to the pre-established threshold \(V_{th}\). It should be noted that the performance of the proposed symbol timing synchronization method based on 2-bit non-uniform quantization depends largely on the selection of the  
\begin{table}
\begin{tabular}{c c} \hline \hline \(\boldsymbol{b_{c_{i}}(i=1,2,3,4)}\) & **Magnitude** \\ \hline
00 & \(T_{0}^{2}\) \\
01 & \(T_{0}T_{1}\) \\
10 & \(T_{1}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: A 2-bit magnitude representation and their real value.  
threshold \(V_{th}\) and mapping values \(T_{0}\) and \(T_{1}\). In this study, an optimal search method for these parameters based on the information rate distortion function is presented.