# OQPSK Synchronization Parameter Estimation Based on Burst Signal Detection
## 2 Algorithm Description
### Synchronization Parameter Estimation
#### 2.2.1 Frequency Offset Estimation  
We start our derivation of the estimation of frequency offset in Equation (4). Specifically, the term:  
\[r(k)c^{*}(k) =exp(j(2\pi f_{d}kT+\theta_{0}))+n(k)c^{*}(k) \tag{7}\] \[=exp(j(2\pi f_{d}kT+\theta_{0}))+n^{\prime}(k)\]  
In order to facilitate the derivation, we temporarily ignore the influence of the noise term \(n^{\prime}(k)\) and define \(f^{\prime}\triangleq 2\pi f_{d}T\). Equation (7) can be written as follows:  
\[r(k)c^{*}(k)=exp(j(f^{\prime}k+\theta_{0})) \tag{8}\]  
As we mentioned earlier, the value of \(v\) indicates the judgment time, which only represents the starting position of the burst signal and does not affect the parameter estimation. Thus, we can set take the value of \(v\) to zero. Then, incorporating Equation (8) into Equation (4):  
\[x_{i}=exp(j(f^{\prime}iL+\theta_{0}))\big{\{}1+exp(jf^{\prime})+...+exp(j(L-1) f^{\prime})\big{\}} \tag{9}\]  
\[x_{i-1}=exp(j(f^{\prime}(i-1)L+\theta_{0}))\big{\{}1+exp(jf^{\prime})+...+exp( j(L-1)f^{\prime})\big{\}} \tag{10}\]  
In order to obtain \(L_{3}(v)\) in Equation (6), we first simplify \(x_{i}x_{i-1}^{*}\):  
\[x_{i}x_{i-1}^{*}=Aexp(jf^{\prime}L) \tag{11}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Multiplication** & **Addition** \\ \hline \(L_{1}(v)\) & \(6N^{2}-6N\) & \(N^{2}-2N\) \\ \(L_{2}(v)\) & \(2(N/L)^{2}+4N-2N/L\) & \((N/L)^{2}+2N-4N/L\) \\ \(L_{3}(v)\) & \(4N+4N/L-4\) & \(2N-4\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Algorithmic complexity comparison of burst signal detection.  
where:  
\[A=L+2(L-1)cosf^{\prime}+2(L-2)cos2f^{\prime}+...+2 cos(L-1)f^{\prime} \tag{12}\]  
Equation (12) implies that the value of \(A\) is a real number, and accordingly, it will not affect the result of the complex angle of Equation (11). Obviously, the following results can be obtained:  
\[f^{\prime}=\frac{1}{L}arg\{x_{i}x_{i-1}^{*}\} \tag{13}\]  
The result of Equation (13) is in an ideal case, that is no noise is considered. Unfortunately, noise has a non-negligible effect on the frequency offset estimation. In order to improve the performance of the estimation, we accumulate the value of \(x_{i}x_{i-1}^{*}\) to smooth the effect of noise. We should also note that in order to facilitate the derivation, we defined \(f^{\prime}\triangleq 2\pi f_{d}T\) above. Consequently, the final frequency offset can be calculated as:  
\[\hat{f}_{d}=\frac{1}{2\pi L}arg\Bigg{\{}\sum_{i=1}^{N/L-1}x_{i}x_{i-1}^{*} \Bigg{\}} \tag{14}\]  
Equation (14) indicates that the frequency offset estimation range is inversely proportional to \(L\), and its estimation accuracy is directly proportional to it. It is particularly important to note that \(\sum_{i=1}^{N/L-1}x_{i}x_{i-1}^{*}\) in Equation (14) equals Equation (6) after removing the modulus. Therefore, the burst signal detection and parameter estimation algorithm are connected here. In this case, we can directly use the results of burst signal detection to estimate the frequency offset, which is why the complexity of the algorithm in this article is greatly reduced. Table 2 is a comparison of the complexity of the frequency offset estimation algorithms. \(N_{0}\) in L&R and Fitz is usually \(N/2\).