## 2 THz-SAR Echoes Signal Model with Vibration Error
### THz-SAR Echo Signal Model  
Figure 1 gives the THz-SAR imaging geometric model when there is high-frequency vibration error. The radar operates in right side-looking strip-map mode. Ideally, the radar platform moves horizontally in the X direction at a reference altitude \(H\) and a constant velocity \(v\).  
The coordinate of the ideal position of the APC is \((vt,\,0,H)\). Take a point scatterer, \(P\), with the coordinate \((vt_{0},\,y_{0},0)\) as an example, where \(t_{0}\) is the zero doppler time and \(y_{0}\) is the \(Y\)-axis position. The ideal instantaneous slant range \(r(t)\) from the APC to the point is  
\[r(t)=\sqrt{r_{0}^{2}+v^{2}(t-t_{0})^{2}}=r_{0}+\frac{v^{2}(t-t_{0})^{2}}{2r_{0}} \tag{4}\]  
where \(r_{0}=\sqrt{y_{0}^{2}+H^{2}}\) is the nearest slant range of point scatterer.  
However, in practical applications, the actual trajectory of the APC deviates from the ideal motion state because of the vibration error and is indicated by the curve line. Thus, the instantaneous slant range is affected by high-frequency vibration error. The actual instantaneous slant range \(R(t)\) is  
\[R(t)=r(t)+\Delta R(t)\]Assume that the linear frequency modulation (LFM) pulse signal emitted by THz-SAR is  
\[s(\tau)=\text{rect}\Big{(}\frac{\tau}{T}\Big{)}\exp(j2\pi f_{0}\tau+j\tau\gamma \tau^{2}) \tag{6}\]  
where \(\text{rect}(\cdot)\) denotes the rectangular window function, \(\tau\) represents the fast time, \(T\) denotes the signal pulse width, \(f_{0}\) represents the carrier frequency, and \(\gamma\) denotes the chirp rate.  
In THz-SAR imaging, the radar system usually uses the dechirp heterodyne receiver [28]. After compensating the residual video phase, the recorded baseband echo signal is  
\[s(\tau,t)=\sigma\cdot\text{rect}\bigg{[}\frac{\tau-2R(t)/c}{T}\bigg{]}\exp\{-j \frac{4\pi f_{0}}{c}R(t)+j\frac{4\pi\gamma R(t)}{c}\tau\} \tag{7}\]  
where \(c\) is the speed of light.  
Then, through performing the fast Fourier transform (FFT) upon \(\tau\) and a systematic range cell-migration correction (RCMC), the range-compressed signal is  
\[s(f,t)=\delta\bigg{(}f-\gamma\frac{2r_{0}}{c}\bigg{)}\exp\{-j\frac{4\pi}{ \lambda}r\}\exp\{-j\frac{4\pi}{\lambda}\Delta R\} \tag{8}\]  
where \(\delta(\cdot)\) denotes the impulse function and \(\lambda\) represents the wavelength.  
Substituting (3) and (4) into (8), we can obtain  
\[\begin{array}{l}s(f,t)=\delta\Big{(}f-\gamma\frac{2r_{0}}{c}\Big{)}\quad \exp\bigg{\{}-j\frac{4\pi}{\lambda}\bigg{(}r_{0}+\frac{\tau^{2}(t-t_{0})^{2}}{ 2r_{0}}\bigg{)}\bigg{\}}\\ \quad\exp\bigg{\{}-j\frac{4\pi}{\lambda}\sum\limits_{i=1}^{l}a_{i}\sin(2\pi f_ {i}t+\phi_{i})\bigg{\}}\end{array} \tag{9}\]  
Then, after performing the dechirp operation and the residual video phase compensation in azimuth, (9) can be rewritten as  
\[\begin{array}{l}s(f,t)=\delta\Big{(}f-\gamma\frac{2r_{0}}{c}\Big{)}\quad \exp\bigg{\{}-j\frac{4\pi}{\lambda}\bigg{(}r_{0}+\frac{\tau^{2}t_{0}}{r_{0}}t \bigg{)}\bigg{\}}\\ \quad\exp\bigg{\{}-j\frac{4\pi}{\lambda}\sum\limits_{i=1}^{l}a_{i}\sin(2\pi f _{i}t+\phi_{i})\bigg{\}}\end{array} \tag{10}\]  
Finally, the azimuth compression in the Doppler domain can be accomplished by performing the FFT upon \(t\) in (10). However, the high-frequency vibration error brings the additional period-modulated phase into the ideal echo, as shown in the last term of (10). The phase error results in the paired echo in the THz-SAR images.  
Figure 1: THz-SAR imaging geometric model.