## 3 Proposed Scheme  
### Synchronization Scheme  
PSS and SSS, which have good cross-correlation and autocorrelation properties, are adopted for synchronization. The PSS and SSS are located on SS/PBCH Block, which is defined in [31]. PSS is adopted to estimate delay and coarse doppler offset estimation. PSS and SSS are used to fine doppler offset estimation. As the satellite adopts the staring mode and the terminal adopts the reflector antenna, the interference of another satellite can be ignored. The received signal of antenna 1 can be simplified as  
\[y_{1}(n)=G_{1}x_{1}(n)\exp(j2\pi n\frac{\varepsilon_{1}}{N})\exp(-j2\pi f_{c} \tau_{1})+z_{1}(n) \tag{16}\]  
where \(G_{1}\) represents the complex gain. \(\tau_{1}\) denotes the delay from satellite 1 to antenna 1. In order to estimate the large frequency shift of satellite, the two-dimensional search method is adopted for time synchronization and coarse frequency shift estimation. Multiply the received signal by the conjugate of the local PSS and perform Fourier transform to obtain  
\[ \begin{split} R_{1}(n,k)&=\sum_{m=0}^{N_{1}-1}y_{1} (n+m)s_{pss}^{*}(m)\exp\left(-j2\pi\frac{m}{N_{1}}k\right)\\ &=\sum_{m=0}^{N_{1}-1}\Big{[}G_{1}x_{1}(n+m)s_{pss}^{*}(m)\exp \left(j2\pi m\frac{\varepsilon_{1}}{N}\right)\\ &\quad\exp(-j2\pi f_{c}\tau_{1})\exp\left(-j2\pi\frac{m}{N_{1}}k \right)+\psi_{1}\Big{]}\\ &=\sum_{m=0}^{N_{1}-1}\Big{[}G_{1}x_{1}(m)s_{pss}^{*}(m)exp\left( -j2\pi\frac{m}{N}\left(k\frac{N}{N_{1}}-\varepsilon_{1}\right)\right)\\ &\quad\exp(-j2\pi f_{c}\tau_{1})+\psi_{1}(n)\Big{]}\end{split}  \tag{17}\]\(R_{1}(n,k)\) denotes the \(k\)-th Fourier transform value obtained from FFT and cross-correlation sequences, which are calculated from the \(n\)-th point of the received sequence, where \(k=0,1,\ldots,N_{1}-1\) represents the number of Fourier transform points. \(s_{pss}(n)\) represents the local PSS. \(N\) denotes the number of FFT points modulated by OFDM. \(N_{1}\) represents the Fourier transform points, \(N_{1}>N\). \(\psi_{1}(n)=\sum\limits_{m=0}^{N_{1}-1}z_{1}(n+m)s_{pss}^{*}(m)\exp\left(-j2\pi \frac{m}{N_{1}}k\right)\) represents the influence of noise on synchronization algorithm. Then, the estimated delay is given by  
\[\hat{n}_{1}=\underset{n}{\arg\max}\{\arg\max}\{\underset{k}{R_{1}(n,k)}\} \tag{18}\]  
where \(\hat{n}_{1}\) denotes the estimated delay. In (18), slide in the frequency domain to search for the peak and then slide in the time domain to search for the peak. It can effectively resist the influence of the large frequency shift. When the estimated delay \(\hat{n}_{1}\) is accurate, bring \(\hat{n}_{1}\) into (17), (17) can be rewritten as  
\[R_{1}(\hat{n}_{1},k)=\sum\limits_{m=0}^{N_{1}-1}\left[G_{1}\big{|}s_{pss}(m) \big{|}^{2}\exp(-j2\pi\frac{m}{N}(k\frac{N}{N_{1}}-\varepsilon_{1}))+\psi_{1} (n)\right] \tag{19}\]  
When the value of \((kN/N_{1}-\varepsilon_{1})\) reaches its minimum value, the \(R_{1}(\hat{n}_{1},k)\) can obtain the maximum value. Thus, the estimated normalized frequency can be expressed as  
\[\hat{k}_{1}=\underset{n}{\arg\max}\,R_{1}(\hat{n}_{1},k) \tag{20}\]  
\[\varepsilon_{c}=\hat{k}_{1}\frac{N}{N_{1}} \tag{21}\]  
The accuracy of \(\varepsilon_{c}\) is affected by the Fourier transform points \(N_{1}\) in equation (17). The estimated frequency offset \(\varepsilon_{c}\) is used to compensate for the signal. Then, the residual frequency offset is estimated by the fractional frequency offset estimation. Take out the signal corresponding to the local PSS mapping position in the received signal, which can be expressed as  
\[y^{\prime}_{1}(n)=G_{1}s_{pss}(n)e^{j2\pi n\frac{\varphi}{N}}+z_{1}(n) \tag{22}\]  
where \(G_{1}\) denotes the complex gain. \(s_{pss}(n)\) represents the local PSS. \(\varepsilon_{r}\) denotes the residual frequency shift. \(N\) represents the number of Fourier transform points in OFDM modulation. \(z_{1}(n)\) denotes the additive Gaussian white noise. The conjugate of the local PSS and \(y^{\prime}_{1}(n)\) are multiplied to construct \(r_{1}(n)\). \(r_{1}(n)\) can be expressed as  
\[