## 2 System Model  
The model of the embedded pilot signal is shown in Figure 1. The pilot symbol is placed on \((m_{p},n_{p})\). Let \(T\) and \(\Delta f\) denote the sampling interval and frequency interval, respectively. The discrete DD domain grid is represented as \(\{(k/NT,l/M\Delta f),k=0,1,\ldots,N,l=0,1,\ldots,M\}\), where \(1/NT\) and \(1/M\Delta f\) represent the Doppler resolution and delay resolution of the DD domain, respectively. Accordingly, \(k\) and \(l\) represent the indices of these two dimensions, respectively. The transmitted signal is given as follows:  
\[X_{dd}(l,k)=X_{dd}^{pilot}(l_{p},k_{p})+X_{dd}^{data}(l_{d},k_{d}) \tag{1}\]  
where the pilot symbols can be expressed as follows:  
\[X_{dd}^{pilot}(l_{p},k_{p})= \begin{cases}0,&else\\ a,&l_{p}=m_{p},k_{p}=n_{p}\end{cases}  \tag{2}\]  
and the data symbols can be expressed as follows:  
\[X_{dd}^{data}(l_{d},k_{d})= \begin{cases}0,&k_{d}\in[n_{p}-\tau_{max},n_{p}+\tau _{max}]\\ &l_{d}\in[m_{p}-2k_{max}-k_{g},m_{p}+2k_{max}+k_{g}]\\ data,&else\end{cases}  \tag{3}\]  
where \(\tau_{max}\) and \(k_{max}\) represent the maximum delay and Doppler indices in the channel, respectively, while \(k_{g}\) represents the additional guard interval. Through the inverse symplectic finite fourier transform (ISFFT), the DD domain signal \(X_{dd}(l,k)\) is transformed into the time-frequency (TF) domain signal \(X_{tf}(n,m)\), which can be expressed as follows:  
\[X_{tf}(n,m)=\frac{1}{N}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1}X_{dd}(l,k)e^{j2\pi( \frac{nk}{N}-\frac{ml}{M})} \tag{4}\]  
where, \(n\in[0,N-1]\) and \(m\in[0,M-1]\) are the time-frequency grid indices. Then, the TF-domain signal is transformed into the time-domain transmission signal \(s(t)\) through the Heisenberg transform, which is expressed as follows:\[s(t)=\sum_{k=0}^{N-1}\sum_{l=0}^{M-1}X_{tf}(n,m)g_{tx}(t-nT)e^{j2\pi m\Delta f(t-nT)} \tag{5}\]  
where \(g_{tx}(t)\) denotes the transmit pulse shaping filter at the transmitter. The sparsity channel impulse response with \(P\) paths in the DD domain can be expressed as follows:  
\[h(\tau,v)=\sum_{i=1}^{p}h_{i}\delta(\tau-\tau_{i})\delta(v-v_{i}) \tag{6}\]  
where \(h_{i}\), \(\tau_{i}=l_{i}/M\Delta f\), and \(v_{i}=\left(k_{i}+\kappa_{i}\right)/NT\) are the channel gain, delay, and Doppler shift of the \(i^{th}\) path, respectively. \(k_{i}\) and \(\kappa_{i}\) donate the integer and fractional Doppler of the path, respectively. At the receiver, the time-domain signal \(r(t)\) can be expressed as follows:  
\[r(t)=\int\int h(\tau,v)e^{j2\pi v(t-\tau)}s(t-\tau)d\tau dv+\omega(t) \tag{7}\]  
where \(\omega(t)\sim CN(0,\sigma^{2})\) is the Gaussian white noise. The received signal is transformed into the TF-domain signal through the Wigner transform, and then the DD domain signal \(Y_{dd}(l,k)\) is obtained through the short-time Fourier transform (STFT), which can be expressed as follows:  
\[Y_{dd}(l,k)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}r(t)g_{rx}^{*}(t- nT)e^{-j2\pi m\Delta f(t-nT)}e^{-j2\pi(\frac{\omega}{N}-\frac{\omega}{M})}dndm \tag{8}\]  
where \(g_{rx}(t)\) is the receiver pulse shaping filter. Similar to the transmitted DD-domain signal \(X_{dd}(l,k)\), the received DD domain signal can be represented as: \(Y_{dd}(l,k)=Y_{dd}^{pilot}(l_{p},k_{p})+Y_{dd}^{data}(l_{d},k_{d})+\omega(l,k)\), where the received pilot symbols \(Y_{dd}^{pilot}(l_{p},k_{p})\) are given by:  
\[ \begin{split} Y_{dd}^{pilot}(l_{p},k_{p})=&\sum_{i=1 }^{p}h_{i}\kappa_{i}e^{j2\pi\frac{(l_{i}+\kappa_{i})\kappa_{p}}{M\Delta t}}X_{ dd}^{pilot}(m_{p},n_{p})\\ & l_{p}=m_{p}+l_{i},k_{p}=n_{p}+k_{i}-j\end{split}  \tag{9}\]  
Figure 1: Placement of the pilot and data symbol.