# Robust Symbol Timing Synchronization for Initial Access under LEO Satellite Channel
## 3 Robust Symbol Timing Synchronization
### The Proposed Structure  
From Equation (1) in Section 2.2, the objective of timing offset synchronization is to determine the desired timing offset, i.e., sampling timing estimate \(\hat{\tau}\) from the observation. If the sampling on time may be aligned with the symbol on time by the decimator, the log-likelihood function for the timing offset \(\tau\) and carrier offset \(\theta_{n}=2\pi\Delta fn+\phi(n)+\phi_{o}\) given the observed signal \(r(t)\), which incorporates the known data symbols \(\{a_{k}\}\) for \(k\) such that \(n-L<k\leq n\), denoted by vector \(a_{k}\) during the timing offset estimation  
\[\Lambda(\tau,\theta_{n})=\frac{2}{N_{o}}\text{Re}\left[e^{-\beta_{n}}\sum_{k= n-L+1}^{n}a_{k}^{*}r(kT-\tau)\right], \tag{5}\]  
where \(\theta_{n}\) is carrier offset at \(nT\), which may be known or unknown during the detection. From the log-likelihood function \(\Lambda(\cdot)\) of \(\tau\) and \(\theta_{n}\) of Equation (5) when carrier offset information is unknown and very low SNR condition. The maximization of the likelihood function typically entails a two-step operation. The first step is called a coarse search to compute \(\Lambda(\tau,\theta_{n})\) over timeslot duration, and then the symbol time index that maximizes \(\Lambda(\tau,\theta_{n})\) is chosen during burst known symbol length \(L\), where \(n=0,1,\ldots,L-1\). In practice, the peak search from the cross-correlation scheme between the received symbol and the known symbol is used as the maximization method. The second step is a fine search to find the maximum within a symbol duration using interpolation or oversampling in time domain computation. However, coherent, non-coherent, and differential correlation scheme in the time domain that is typically used as cross-correlation scheme is not robust to high Doppler and low SNR condition because it is difficult to collect signal energy due to rapid phase rotation and noisy component. Therefore, this paper proposes to find peak as a coarse search through the consecutive _FFT_, which is widely used to analyze the characteristic in the frequency domain from a sequence of discrete time signals and magnitude computation during a timeslot. The proposed scheme can be regarded as an approximate ML approach and is to design a finite discrete time and frequency bin in the coarse estimate step and try to fine search within a high hypothesis area. We assume that the \(P\) (<\(K\)) known symbols composed of preamble, postamble, and distributed pilot symbols are given, and \(K\)_FFT_ size is given. This approach requires PK computation for each sample time position. However, to perform the coherent correlation during the constant phase, the partial coherent correlation approach is introduced in the initial step. It is indicated as the subcorrelation window in Figure 6. The output complex signals, which are the received samples multiplied with the complex conjugate vector values of known symbols, are accumulated and indicated as the partial complex vector sum. According to partial coherent correlation length and _FFT_ size, the computation complexity can be reduced drastically. The position n with the maximum magnitude of _FFT_ output in each sample time can have the most probable CFO estimate value among discrete K bin CFO hypothesis values. Based on the grid area to acquire a CFO estimate, the fine-timing search process can be performed among local maximum values at the same time. The \(FFT\) magnitude output \(FFT\{\Lambda(\tau_{n})\}\) is shown in Figure 7, and the blue stick is represented as the maximum one in every sampling time index. The interpolation scheme can provide the fractional sampling time offset estimate \(\hat{\tau}\) between n and \((n+1)\). Specifically, \(FFT\{\Lambda(\tau_{n})\}\) is revealed as the maximum among all \(\Lambda(\tau)\) in the first step.