# OQPSK Synchronization Parameter Estimation Based on Burst Signal Detection
## 2 Algorithm Description
### Burst Signal Detection  
Without considering the timing error, the received signal model can be represented as:  
\[r(k)=exp(j(\theta+2\pi f_{d}kT+\theta_{0}))+n(k) \tag{1}\]  
where \(e^{i\theta}\) is the \(M\)-ary phase-modulated symbol, \(n(k)\) denotes Additive White Gaussian Noise (AWGN), \(T\) stands for the symbol period, and \(f_{d}\) and \(\theta_{0}\) represent the frequency offset and phase offset, respectively. Applying double correlation between \(r(k)\) in Equation (1) and the known local Unique Word (UW) \(c(k)\) yields:  
\[L_{0}(v)=\sum_{i=1}^{N-1}\left\{\left|\sum_{k=i}^{N-1}r(v+k)c(k)^{*}r^{*}(v+k-i )c(k-i)\right|-\sum_{k=v+i}^{v+N-1}|r(k)||r(k-i)|\right\} \tag{2}\]  
where \(v\) is the moment of judgment. In Equation (2), \(\sum_{k=i}^{N-1}r(v+k)c(k)^{*}r^{*}(v+k-i)c(k-i)\) is the double correlation term, and \(\sum_{k=v+i}^{v+N-1}|r(k)||r(k-i)|\) is the random data correction term. Usually, the random data correction term can be omitted [6], such that Equation (2) can be approximated as:  
\[L_{1}(v)=\sum_{i=1}^{N-1}\left\{\left|\sum_{k=i}^{N-1}r(v+k)c(k)^{*}r^{*}(v+k -i)c(k-i)\right|\right\} \tag{3}\]  
The reason we chose double correlation in Equations (2) and (3) is that cross-correlation is dysfunctional when frequency offset exists. Unfortunately, this brings unacceptable complexity and makes its implementation in hardware impossible. In order to reduce the computational complexity, the double correlation method can be improved by the concept of segments [7]; that is:  
\[x_{i}=\sum_{k=0}^{L-1}r(v+k+iL)c(k+iL)^{*} \tag{4}\]  
\[L_{2}(v)=\sum_{j=1}^{N/L-1}\left|\sum_{i=j}^{N/L-1}x_{i}x_{i-j}^{*}\right| \tag{5}\]  
where \(N\) and \(L\) are the lengths of the local \(UW\) and each segment of \(UW\), respectively. The segment correlation takes the conjugate multiplication between different segments to eliminate the influence of frequency offset. It can be seen from Equations (4) and (5) that the complexity of \(L_{2}(\mu)\) is greatly reduced compared to that of \(L_{1}(\mu)\).  
Figure 1: Algorithm flowchart.  
Normally, the received signal should be up-sampled, but when the local \(UW\) is correlated with the received signal, we can choose whether the local \(UW\) is up-sampled or not. In the simulation, we find that using local \(UW\) without up-sampling in QPSK modulation yields acceptable performance, but this does not work for OQPSK modulation. This is because OQPSK staggers the in-phase component and quadrature component by half a symbol period, and the local \(UW\) will lose the information of the half symbol if the local \(UW\) is not up-sampled. Obviously, the solution to that is to up-sample the local \(UW\) and stagger its in-phase component and quadrature component by half a symbol period.  
We find it a convenient approximation of Equation (5) to only consider the case of \(j=1\). This is because the correlation of the segment correlation algorithm is mainly dominated by the adjacent segments (\(j=1\) case). Therefore, Equation (5) can be written as follows:  
\[L_{3}(v)=\left|\sum_{i=1}^{N/L-1}x_{i}x_{i-1}^{*}\right| \tag{6}\]  
Undoubtedly, \(L_{3}(v)\) has lower complexity, but results in extra performance loss. Fortunately, our simulation shows that this loss is acceptable. More in detail, the complexity comparison between \(L_{1}(v)\), \(L_{2}(v)\), and \(L_{3}(v)\) is explained in Table 1.