## 3 DNN-Based Synchronization and the Proposed Receiver
### Full Receiver in Baseband  
The proposed synchronization was based on the model-driven DNN. The core algorithm in the proposed synchronization, which was built by the DNN, was used to suppress the correlation peak ambiguity between the delay and the Doppler; the other parts in the receiver used classical methods to reduce the dimension of the learning parameters. As a result of the accurate compensation of the time-varying relative motion, the subsequent equalizer did not need to be time-invariant. The receiver structure is shown in Figure 2 and described in detail as follows. The first step was to convert the passband waveform \(r_{\text{PB}}(t)\) into the baseband one \(r[n]\) with a low sampling rate, \(T_{\text{BB}}\). Conversion in accordance with the digital down converter (DDC) is given as:  
\[r[n]=\left.\text{LPF}\!\left[r_{\text{PB}}(mT_{\text{PB}})e^{-2\pi mT_{\text{ PB}}/\text{c}}\right]\right|_{n=\frac{mT_{\text{PB}}}{T_{\text{BB}}}}, \tag{15}\]  
where \(\text{LPF}\!\left\{\cdot\right\}\) is the low pass filter, \(f_{\text{C}}\) is the carrier frequency, and \(T_{\text{PB}}\) is the passband sampling interval of the DDC input. For the timing offset estimation, we first obtained the discrete timing offsets at the start of the synchronization signals using traditional correlation. Then, the bias vector was learned from the DNN and subtracted. Finally, the continuous timing offsets were interpolated by a cubic spline function. Figure 2: Receiver structure. LFM, linear frequency modulation; DDC, digital down converter; LS, least-squared; ZF, zero forcing equalization. For the \(k\)-th synchronization signal in the received waveform, correlation to the baseband waveform is carried out as:  
\[z_{k}[n]=\left|\sum_{m}r[m+n+kT_{\text{Frm}}]s^{*}_{\text{Sync}}(mT_{\text{BB}})e ^{\hat{\jmath}2\pi f_{c}(mT_{\text{BB}}-kT_{\text{Frm}})}\right|. \tag{16}\]  
The integer timing offset is written as:  
\[\tau_{\text{I, }k}=\underset{n}{\text{argmax}}\big{\{}z_{k}[n]\big{\}}, \tag{17}\]  
and then the fractional offset is acquired by maximizing the parabolic fitting curve around it as follows:  
\[\tau_{\text{F, }k}=\frac{z_{k}\big{[}\tau_{\text{I, }k}+1\big{]}-z_{k}\big{[}\tau_{\text{I, }k}-1\big{]}}{4z_{k}\big{[}\tau_{\text{I, }k}\big{]}-2z_{k}\big{[}\tau_{\text{I, }k}-1\big{]}-2z_{k}\big{[}\tau_{\text{I, }k}+1\big{]}}. \tag{18}\]  
The timing offset estimate with the bias is as follows:  
\[\hat{\tau}_{\text{B}}(kT_{\text{Frm}})=\big{(}\tau_{\text{I, }k}+\tau_{\text{F, }k}\big{)}T_{\text{BB}}. \tag{19}\]  
The biased timing offset vector in the whole packet is fed into the DNN, and the bias vector \(\big{\{}\hat{\Delta}(kT_{\text{Frm}})\big{\}}\) is then obtained, which is eliminated to get the unbiased estimation, as follows:  
\[\hat{\tau}(kT_{\text{Frm}})=\hat{\tau}_{\text{B}}(kT_{\text{Frm}})-\hat{\Delta }(kT_{\text{Frm}}),\qquad 0\leq k\leq N_{\text{Frm}}. \tag{20}\]  
Finally, the timing offset for each sample in the packet is obtained by cubic spline interpolation from the timing offsets at the synchronization signals:  
\[\hat{\tau}(nT_{\text{BB}})=\text{Spline}\big{\{}\big{\{}(kT_{\text{Frm}},\hat {\tau}(kT_{\text{Frm}}))\big{\}}_{k=0}^{N_{\text{Frm}}};nT_{\text{BB}}\big{\}}, \tag{21}\]  
where \(\text{Spline}\{\cdot\}\) is the cubic spline interpolation function in the \"not-a-knot\" condition. The timing offset compensation was carried out to recover the transmitted signal without the motion effects. As shown in Figure 3, the compensation in the low sample-rate baseband form included three steps: the inversion of the timing offset, the Farrow filtering, and the carrier offset ration compensation. The timing offset curve was resampled to meet the requirement of the Farrow filter's input being equispaced.