## 5 Modified Chirp Scaling and Azimuth Compression for GTLR-BiSAR System
### Two-Dimensional Spectrum Expression after Chirp Scaling Operation  
According to the range variance analysis in Section 4.3, we should multiply the RD signal (referring to Equation (21) in Section 2) with the chirp scaling function (referring to Equations (30) and (32) in Section 4.3) to remove the range variance:  
\[ \begin{split} S_{echo}(\tau,f_{a};P)\times S_{CS}(\tau,f_{a})& =exp[-j2\pi f_{c}\frac{R_{T_{p}}+R_{\tau_{0}p}D(f_{a},V_{p})}{c}+j \pi K_{m}[\tau-\frac{R_{T_{p}}+\frac{R_{\tau_{0}p}}{D(f_{a},V_{p})}}{c}]^{2}]\\ & exp\{j\pi K_{m}[\frac{D(f_{a,ref},V_{r})}{D(f_{a},V_{r})}-1][\tau- \frac{1}{c}[R_{T\_ref}+\frac{R_{\tau_{0},ref}}{D(f_{a},V_{r\_ref})}]]^{2}\}\\ & exp\{-j2\pi K_{m}\frac{R_{T_{p}}-R_{T\_ref}}{c}[\frac{D(f_{a,ref },V_{r})}{D(f_{a},V_{r})}-1][\tau-\frac{1}{c}[R_{T\_ref}+\frac{R_{\tau_{0},ref} }{D(f_{a},V_{r\_ref})}]]\}\end{split}  \tag{33}\]  
We adopt POSP again to deduce the 2D spectrum expression, and the integral phase \(\theta(\tau,f_{a},f_{\tau};P)\) is shown as follows:  
\[ \begin{split}\theta(\tau,f_{a},f_{\tau};P)&=-2\pi f _{c}\frac{R_{T_{p}}+R_{\tau_{0}p}D(f_{a},V_{p})}{c}+\pi K_{m}[\tau-\frac{R_{T_{ p}}+\frac{R_{\tau_{0}p}}{D(f_{a},V_{p})}}{c}]^{2}\\ &\quad+\pi K_{m}[\frac{D(f_{a\_ref},V_{r})}{D(f_{a},V_{r})}-1]\{ \tau-\frac{1}{c}[R_{T\_ref}+\frac{R_{\tau_{0},ref}}{D(f_{a},V_{r\_ref})}]\}^{2 }\\ &\quad-2\pi K_{m}\frac{R_{T_{p}}-R_{\_ref}}{c}[\frac{D(f_{a\_ref {a\_ref}},V_{r})}{D(f_{a},V_{r})}-1]\{\tau-\frac{1}{c}[R_{T\_ref}+\frac{R_{ \tau_{0},ref}}{D(f_{a},V_{r\_ref})}]\}-2\pi f_{\tau}\tau\end{split}  \tag{34}\]  
Calculate the partial derivative of \(\theta\) to \(\tau\) and find the zero point, we obtain the relationship between \(\tau\) and \(f_{\tau}\):  
\[ \begin{split}\frac{\partial\theta(\tau,f_{a},f_{\tau};P)}{ \partial\tau}&=2\pi K_{m}[\tau-\frac{R_{T_{p}}+\frac{R_{\tau_{0}p }}{D(f_{a},V_{p})}}{c}]+2\pi K_{m}[\frac{D(f_{a\_ref},V_{r})}{D(f_{a},V_{r})}-1] \{\tau-\frac{1}{c}[R_{T\_ref}+\frac{R_{\tau_{0},ref}}{D(f_{a},V_{r\_ref})}]\} \\ &\quad-2\pi K_{m}\frac{R_{T_{p}}-R_{T\_ref}}{c}[\frac{D(f_{a\_ref {a\_ref}},V_{r})}{D(f_{a},V_{r})}-1]-2\pi f_{\tau}=0\end{split}  \tag{35}\]  
\[ \begin{split}\tau&=\frac{R_{T_{p}}+\frac{R_{\tau_{0}p }}{D(f_{a\_ref},V_{r})}}{c}+\frac{R_{\tau_{0},ref}}{cD(f_{a},V_{r\_ref})}+\frac{ D(f_{a},V_{r})f_{\tau}}{K_{m}D(f_{a\_ref},V_{r})}\\ &\quad\quad\quad\quad\quad\quad-\frac{R_{\tau_{0},ref}D(f_{a},V_ {r})}{cD(f_{a\_ref},V_{r})D(f_{a},V_{r\_ref})}\end{split}  \tag{36}\]  
Submit \(\tau\) into Equation (34), we obtain the 2D frequency spectrum in ascending power of \(c\) successively:  
\[\begin{split} S_{2D}(f_{\tau},f_{a};P)&=exp\{j\pi \frac{K_{m}}{c^{2}}[R_{T_{p}}^{2}+R_{T\_ref}^{2}-2R_{T_{p}}R_{T\_ref}+\frac{R_{ \tau_{0}p}^{2}}{D^{2}(f_{a},V_{r})}+\frac{R_{\tau_{0},ref}^{2}}{D^{2}(f_{a},V_{ r\_ref})}-\frac{D(f_{a\_ref},V_{r})R_{T_{p}}^{2}}{D(f_{a},V_{r})}\\ &\quad\quad\quad\quad\quad\quad\quad-\frac{D(f_{a\_ref},V_{r})R_{T \_ref}^{2}}{D(f_{a},V_{r})}-\frac{R_{\tau_{0}p}^{2}}{D(f_{a},V_{r})D(f_{a\_ref },V_{r})}+\frac{2D(f_{a\_ref},V_{r})R_{T_{p}}R_{T\_ref}}{D(f_{a},V_{r})}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{2R_{\tau_{0}p }R_{\tau_{0},ref}}{D(f_{a},V_{r})D(f_{a},V_{r\_ref})}+\frac{2R_{\tau_{0}p}R_{ \tau_{0},ref}}{D(f_{a},V_{r\_ref})D(f_{a\_ref},V_{r})}-\frac{D(f_{a},V_{r})R_{ \tau_{0},ref}^{2}}{D(f_{a\_ref},V_{r})}]\}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quadAfter proper derivations, Equation (37) is simply expressed as follows:  
\[