# OQPSK Synchronization Parameter Estimation Based on Burst Signal Detection
## 2 Algorithm Description
### Synchronization Parameter Estimation
#### 2.2.3 Timing Error Estimation  
After the burst signal detection is completed, in addition to estimating the above carrier synchronization parameters, it is also necessary to estimate the timing error. The signal model with timing error is:  
\[r(t)=e^{i\theta_{0}}\sum_{l}a(l)g(t-lT-\tilde{\tau})+n(t) \tag{17}\]  
where \(\{a(l)\}\) represents the independent and identically distributed information sequence and g(t) and \(\tau\) denote the baseband shaping function and timing error, respectively. When the received signal is sampled at \(t=kT+\tilde{\tau}\), the sampled data \(r(kT+\tilde{\tau})\) can be obtained. The ML-based data-aided timing error estimation formula given in [20] is as follows:  
\[\Lambda(\tilde{\tau})=\left|\sum_{k=0}^{N-1}\tilde{c}^{*}(k)r(kT+\tilde{\tau}) \right|^{2} \tag{18}\]  
If removing the square, Equation (18) is accurate for the burst signal detection based on cross-correlation. The process of capturing the frame header is the process of searching for correlation peaks. When Equation (18) achieves the maximum value, the timing error can be calculated as:  
\[\tilde{\tau}=arg\underset{\tilde{\tau}}{\max}\left\{\left|\sum_{k=0}^{N-1} \tilde{c}^{*}(k)r(kT+\tilde{\tau})\right|^{2}\right\} \tag{19}\]  
As mentioned above, the cross-correlation algorithm has the fatal disadvantage of no anti-frequency offset. Fortunately, we found that the segment correlation used in this article is also the process of capturing the frame header, and its maximum value position is the same as Equation (18), that is when the local \(UW\) is completely aligned with the \(UW\) in the received signal. Therefore, the cross-correlation peak in Equation (19) can be replaced by the segment correlation peak in Equation (6), and the timing error estimation algorithm at this time obtains the ability of anti-frequency offset. Thus, burst signal detection and timing error estimation are linked via:  
\[\tilde{\tau}=argmax\underset{\tilde{\tau}}{\max}\left\{\left|\sum_{i=1}^{N/L- 1}x_{i}x_{i-1}^{*}\right|^{2}\right\} \tag{20}\]  
The interpolation method is usually adopted to calculate the maximum value of Equation (20); here, we chose a simple and highly accurate trigonometric interpolation method [21]. It should be noted that the number of points used must be an even number. In this article, there were four points employed for interpolation, and each symbol took two points; thus, a total of two symbols were required. The four-point interpolation formula given in [21] is as follows:  
\[z(\mu)=\frac{1}{4}Re(C_{0}+2C_{1}e^{i\pi\mu/2}+C_{2}e^{j\pi\mu}) \tag{21}\]  
Among them:  
\[C_{0} =z(-1)+z(0)+z(1)+z(2) \tag{22}\] \[C_{1} =[z(0)-z(2)]+j[-z(1)+z(-1)]\] \[C_{2} =z(0)-z(1)+z(2)-z(-1)\]  
To find the timing error, we only need to find out what the value of \(\mu\) is to maximize \(z(\mu)\):  
\[\hat{\mu}=arg\left\{\underset{\mu}{\max}\left|Re(C_{0}+2C_{1}e^{i\pi\mu/2}+C_ {2}e^{i\pi\mu})\right|^{2}\right\} \tag{23}\]Therefore, finding the timing error is transformed into the problem of finding the maximum value of \(Re(C_{0}+2C_{1}e^{i\pi\mu/2}+C_{2}e^{i\pi\mu})\). We define:  
\[Z(\mu)\triangleq Re(C_{0}+2C_{1}e^{i\pi\mu/2}+C_{2}e^{i\pi\mu}) \tag{24}\]  
For the convenience of calculation, we define \(C_{1}\triangleq A+Bj\), where \(A=z(0)-z(2)\), \(B=-z(1)+z(-1)\).