## 2 GTLR-BiSAR Imaging Geometry and Signal Properties
### GTLR-BiSAR Signal Model and Properties
\[S_{echo}(\tau,t_{a};P)=W(\frac{t_{a}}{T_{a}})W(\frac{\tau}{T_{\tau}})exp[-j2 \pi f_{c}\frac{R_{bi}(t_{a};P)}{c}]exp[j\pi K_{r}(\tau-\frac{R_{bi}(t_{a};P)}{ c})^{2}] \tag{2}\]0  
\[S_{echo}(\tau,t_{a};P)=W(\frac{t_{a}}{T_{a}})W(\frac{\tau}{T_{\tau}})exp[-j2 \pi f_{c}\frac{R_{bi}(t_{a};P)}{c}]exp[j\pi K_{r}(\tau-\frac{R_{bi}(t_{a};P)}{ c})^{2}] \tag{2}\]1  
Similarly, we bring \(t_{a}\) (Equation (11)) into \(\theta\) (Equation (9)), the 2D frequency domain expression is given as follows:  
\[S_{echo}(\tau,t_{a};P)=W(\frac{t_{a}}{T_{a}})W(\frac{\tau}{T_{\tau}})exp[-j2 \pi f_{c}\frac{R_{bi}(t_{a};P)}{c}]exp[j\pi K_{r}(\tau-\frac{R_{bi}(t_{a};P)}{ c})^{2}] \tag{2}\]2  
Set \(D(f_{a},V_{p})=\sqrt{1-\frac{c^{2}f_{a}^{2}}{f_{c}^{2}{V_{p}}^{2}}}\) and make a Taylor expansion for the root term in Equation (12), then the root term is approximately expressed in Equation (13). \[ \begin{split}\sqrt{D^{2}(f_{a},V_{p})+2\frac{f_{\tau}}{f_{c}}+ \frac{f_{\tau}^{2}}{f_{c}^{2}}}&\approx D(f_{a},V_{p})+\frac{f_ {\tau}}{D(f_{a},V_{p})f_{c}}+\frac{f_{\tau}^{2}}{2f_{c}}\frac{D^{2}(f_{a},V_{ p})-1}{D^{2}(f_{a},V_{p})}\\ &=D(f_{a},V_{p})+\frac{f_{\tau}}{D(f_{a},V_{p})f_{c}}-\frac{f_{ \tau}^{2}f_{a}^{2}c^{2}}{2f_{c}^{4}D^{3}(f_{a},V_{p})V_{p}^{2}}\end{split}  \tag{13}\]Resubmit Equation (13) into Equation (12), the 2D frequency domain expression is as below, after derivations. \[ \begin{split} S_{echo}(f_{\tau},f_{a};P)=exp[-j2\pi(f_{\tau}+f_{ \epsilon})\frac{R_{T_{p}}}{c}-j2\pi f_{\epsilon}\frac{R_{\tau_{0}p}D(f_{a},V_{p })}{c}-j2\pi f_{\tau}\frac{R_{\tau_{0}p}}{cD(f_{a},V_{p})}\\ +j\pi\frac{R_{\tau_{0}p}f_{a}^{2}c}{D^{3}(f_{a},V_{p})f_{c}^{3}V_ {p}^{2}}f_{\tau}^{2}-j\frac{\pi}{K_{\tau}}f_{\tau}^{2}]\end{split}  \tag{14}\]  
We set \(Z(f_{a},V_{p})=\frac{R_{\tau_{0}p}f_{a}^{2}c}{D^{3}(f_{a},V_{p})f_{c}^{3}V_{p}^ {2}}\), and rewrite Equation (14) in ascending power of \(f_{\tau}\), successively:  
\[ \begin{split} S_{echo}(f_{\tau},f_{a};P)=exp[-j2\pi f_{\epsilon} \frac{R_{T_{p}}+R_{\tau_{0}p}D(f_{a},V_{p})}{c}-j2\pi\frac{R_{T_{p}}+\frac{R_{ \tau_{0}p}}{D(f_{a},V_{p})}}{c}f_{\tau}+j\pi(Z(f_{a},V_{p})-\frac{1}{K_{\tau}} )f_{\tau}^{2}]\end{split}  \tag{15}\]  
Suppose that \(-\frac{1}{K_{\tau}}=Z(f_{a},V_{p})-\frac{1}{K_{\tau}}\), then Equation (15) becomes:  
\[ \begin{split} S_{echo}(f_{\tau},f_{a};P)=exp[-j2\pi f_{\epsilon} \frac{R_{T_{p}}+R_{\tau_{0}p}D(f_{a},V_{p})}{c}-j2\pi\frac{R_{T_{p}}+\frac{R_ {\tau_{0}p}}{D(f_{a},V_{p})}}{c}f_{\tau}-j\frac{\pi}{K_{m}}f_{\tau}^{2}]\end{split}  \tag{16}\]  
We obtain the concise 2D frequency spectrum expression. Finally, we perform inverse Fourier transform of Equation (16) in the range dimension to obtain the RD domain expression:  
\[ \begin{split} S_{echo}(\tau,f_{a};P)=\int_{-f_{s}/2}^{f_{s}/2}S_ {echo}(f_{\tau},f_{a};P)exp(j2\pi\tau f_{\tau})\,df_{\tau}\end{split}  \tag{17}\]  
where \(f_{s}\) is the sample rate of the receiver. The integral phase \(\theta\) of Equation (17) becomes:  
\[ \begin{split}\theta(f_{\tau},f_{a},\tau;P)=-2\pi f_{\epsilon} \frac{R_{T_{p}}+R_{\tau_{0}p}D(f_{a},V_{p})}{c}-2\pi\frac{R_{T_{p}}+\frac{R_{ \tau_{0}p}}{D(f_{a},V_{p})}}{c}f_{\tau}-\frac{\pi}{K_{m}}f_{\tau}^{2}+2\pi f_{ \tau}\tau\end{split}  \tag{18}\]  
Similar to the former operations, we obtain the relationship between \(f_{\tau}\) and \(\tau\). \[ \begin{split}\frac{\partial\theta(f_{\tau},f_{a},\tau;P)}{ \partial f_{\tau}}=-2\pi\frac{R_{T_{p}}+\frac{R_{\tau_{0}p}}{D(f_{a},V_{p})}} {c}-2\frac{\pi}{K_{m}}f_{\tau}+2\pi\tau=0\end{split}  \tag{19}\]  
\[ \begin{split} f_{\tau}=K_{m}[\tau-\frac{R_{T_{p}}+\frac{R_{\tau _{0}p}}{D(f_{a},V_{p})}}{c}]\end{split}  \tag{20}\]  
Submit \(f_{\tau}\) into Equation (18), the RD spectrum is given as follows:  
\[ \begin{split} S_{echo}(\tau,f_{a};P)=exp[-j2\pi f_{\epsilon} \frac{R_{T_{p}}+R_{\tau_{0}p}D(f_{a},V_{p})}{c}+j\pi K_{m}(\tau-\frac{R_{T_{p}} +\frac{R_{\tau_{0}p}}{D(f_{a},V_{p})}}{c})^{2}]\end{split}  \tag{21}\]  
So far, the signal expressions in time, 2D frequency, and RD domains have been derived in detail.