# Robust Symbol Timing Synchronization for Initial Access under LEO Satellite Channel
## 4 Numerical Results
### PER Performance Assessment  
Firstly, we compare the PER performances of the O&M algorithm of the conventional scheme and the proposed one for different waveform IDs [8,22]. The decoding algorithm to be used for the turbo decoder is the log-MAP (Maximum A Posteriori) algorithm. The iteration number of the turbo decoder is set to 8. In the legend of the chart in Figure 8, the symbolic marks represent the applied algorithm scheme, HGT, and CFO in order, respectively. For example, O&M-H100-F2 means that the O&M algorithm is used, HGT has a maximum 100 symbol length, and the normalized CFO value is 2% vs. symbol rate. Here, a blue curve with a diamond mark stands for ideal PER performance in AWGN and perfect synchronization conditions which can be found in chapter 10.2 of the reference [22]. Note that in Figure 8, we are mainly interested in the two specific PER points of \(10^{-3}\) or \(10^{-5}\), where they are related to the achievable Quasi Error Free (QEF) in the physical layer.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Waveform ID** & **LB (ID\#1)** & **CB (ID\#2)** & **TB (ID\#3, \#13)** \\ \hline Maximum HGT & \(\pm\)200 symbols & \(\pm\)100 symbols & \(\pm\)20 symbols \\ (ToA in symbols) & (\(\pm\)20 μs & (\(\pm\)10 μs & (\(\pm\)2 μs \\ Uniform distribution & in 10 Mbaud) & in 10 Mbaud) & in 10 Mbaud) \\ Normalized maximum & \(\pm\)0.05 & \(\pm\)0.02 & \(\pm\)0.004 \\ CFO vs. symbol rate & (\(\pm\)500 kHz & (\(\pm\)200 kHz & (\(\pm\)40 kHz \\ Uniform distribution & in 10 Mbaud) & in 10 Mbaud) & in 10 Mbaud) \\ \hline \end{tabular}
\end{table}
Table 1: Channel condition for simulation in symbol rate 10 Mbaud.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Waveform ID** & **Burst Length** & **Modulation** & **Code Rate** & **Known** \\  & **(Symbol)** & & **Length** & **The Ratio of Known** \\ \hline \#1(LB) & 664 & QPSK & 1/3 & 208 & 31\% \\ \#2(CB) & 262 & QPSK & 1/3 & 94 & 36\% \\ \#3(STB3) & 536 & QPSK & 1/3 & 80 & 15\% \\ \#13(LTB3) & 1616 & QPSK & 1/3 & 140 & 9\% \\ \hline \end{tabular}
\end{table}
Table 2: Waveform ID and transmission parameters for simulation.  
Figure 8: _Cont_.  
As we can see in Figure 8, the _FFT_ algorithm we proposed always shows better performance than the O&M algorithm in terms of PER performance for LB, CB burst, and Short Traffic Burst (STB) 3 (i.e., waveform ID#3) burst regardless of channel condition, as in Figure 8a-c. Meanwhile, the O&M algorithm is superior to the _FFT_ scheme in the Long Traffic Burst (LTB)3 (i.e., waveform ID #13) waveform, as in Figure 8d. Intuitively, we analyze that the performance can be determined by the known symbol length or the ratio of known symbols in burst for the DA estimator (_FFT_) and by the estimation length for the NDA estimator (i.e., O&M scheme). The SNR performance gap in Figure 8b between the two schemes is remarkably large, and this can be interpreted as the CB burst has the shortest length among waveform IDs and has relatively many known symbols compared to other waveforms at the same time as in Table 2. As a result, we can draw the analytic findings from the simulation results. In the process of uplink initial access, the packet for control data transmission is typically characterized as short length but lengthy known symbols for reliable packet detection. In such a case, the proposed scheme can reduce the performance gap around SNR 1.5 dB at PER \(10^{-5}\) from the ideal AWGN condition compared to the existing schemes.