# Robust Symbol Timing Synchronization for Initial Access under LEO Satellite Channel
## 3 Robust Symbol Timing Synchronization  
### Previous Works  
As already mentioned in Section 2.2, the STR function can be divided into two parts: coarse timing recovery and fine-timing recovery processing. Typically, the coarse recovery part is to find the start of the burst during the HGT of a timeslot, and the fine part is to find the exact sampling time during the symbol interval of a burst.  
In most previous works and DVB-RCS2 demodulator designs, feedforward architecture-based NDA synchronization is prevalent for fine-timing recovery. In this regard, O&M introduced an NDA square law estimator using multiple samples per symbol, and others were suggested with a similar principle, i.e., different nonlinear functions such as Absolute Value Nonlinearity (AVN) arithmetic and fourth square, as in Figure 5[20]. They exploit the presence in the spectrum of the squared output of the Matched Filter in a demodulator of a frequency component at \(f=1/T\). Then, they use the delay property of the Fourier Transform to relate the phase of the spectrum to the time delay as follows:  
\[r(t-\tau)\leftrightarrow R(f)\cdot e^{-\gamma 2\pi\tau f}. \tag{2}\]  
After the coarse STR completes the burst detection process, the fine STR using the O&M algorithm can be applied to the entire burst duration for timing offset estimation. Coarse timing and CFO frequency recovery can be performed by pre-compensation function on the UT side through FLS signaling information that contains the satellite position, velocity, and common timing offset of UTs in a beam. Even though coarse synchronization can be achieved by an accurate GNSS receiver, fine synchronization should be necessarily performed in the receiver.  
The mathematical expression of the O&M algorithm can be expressed as follows:  
\[\hat{\tau}=-\frac{T}{2\pi}\text{arg}\left\{\sum_{k=0}^{NL-1}|r(kT_{s})|^{2}e^{- j\frac{2\pi k}{N}}\right\}, \tag{3}\]  
where \(\hat{\tau}\) is the estimated timing error, \(T_{\text{s}}\) is the sampling duration, \(T\) is the symbol duration, and \(N\) is the oversampling factor per symbol, which is equal to 4. The variable \(L\) is the number of symbols used for timing error estimation. The term \(r(kT_{\text{s}})\) is the input sample of the STR block. The computation is based on a large number of symbols to grow the reliability of estimation and to reduce the effects of noise. This estimator is very simple for implementation but not the best in the case of a short packet for log-on and control burst  
Figure 5: O&M algorithm architecture.  
transmission mode under low SNR and a large frequency offset channel. To overcome this drawback, a DA-based Maximum Likelihood (ML) estimator can be considered. The details will be described in Section 4.2.  
In our paper, we also derive the fine-timing processing step based on the ML estimator. As a measure of performance assessment, we can compute the standard deviation of the timing estimation error and compare it to the Modified Cramer-Rao Bound [21], given by  
\[\mathbb{E}[(\hat{\tau}-\tau)^{2}]\geqslant\text{MCRB}(\tau)=\frac{-1}{2g_{ \alpha}^{\prime\prime}(0)}\times\frac{1}{L}\times\frac{1}{\frac{E_{\alpha}}{N _{0}}}, \tag{4}\]  
where \(\hat{\tau}\) is the timing estimation, and \(g_{\alpha}(\cdot)\) is the raised cosine function with roll-off \(\alpha\). The right-hand side of Equation (4) will be derived in Appendix A.