## 3 Derivation of 2-D Point Target Frequency Spectrum  
Generally, we can obtain the 2-D frequency spectrum easily after the azimuth FFT. However, due to the existence of velocities and accelerations in both along-track and along-height directions, it is difficult to calculate the stationary point directly. Here, we expand the instantaneous slant range \(R(t_{\rm a})\) into a Taylor series at \(t_{\rm a}=0\) as:  
\[R(t_{\rm a})=R_{\rm t}+R_{\rm r0}+(k_{\rm 1c}+k_{\rm 1})t_{\rm a}+k_{\rm 2}t_{ \rm a}^{2}+k_{\rm 3}t_{\rm a}^{3}+k_{\rm 4}t_{\rm a}^{4}+o\left(t_{\rm a}^{4}\right) \tag{2}\]  
where:  
\[R_{\rm r0}=\sqrt{x_{\rm P}^{2}+y_{\rm P}^{2}+H_{\rm r}^{2}} \tag{3}\]  
\[k_{\rm 1c}=\frac{-V_{\rm ry0}y_{\rm pc}+V_{\rm rz0}H_{\rm r}}{R_{\rm r0}} \tag{4}\]  
\[k_{\rm 1}=\frac{V_{\rm ry0}\left(y_{\rm pc}-y_{\rm p}\right)}{R_{\rm r0}} \tag{5}\]  
\[k_{\rm 2}=\frac{-a_{\rm ry}y_{\rm p}+a_{\rm rz}H_{\rm r}+V_{\rm ry0}{}^{2}+V_{ \rm rz0}{}^{2}-\left(k+k_{\rm 1c}\right)^{2}}{2R_{\rm r0}} \tag{6}\]  
\[k_{\rm 3}=\frac{V_{\rm ry0}a_{\rm ry}+V_{\rm rz0}a_{\rm rz}-2(k_{\rm 1}+k_{ \rm 1c})k_{\rm 2}}{2R_{\rm r0}} \tag{7}\]  
\[k_{\rm 4}=\frac{a_{\rm ry}{}^{2}+a_{\rm rz}{}^{2}}{8R_{\rm r0}}-\frac{2(k_{\rm 1 }+k_{\rm 1c})k_{\rm 3}+k_{\rm 2}^{2}}{2R_{\rm r0}} \tag{8}\]  
where \(R_{\rm r0}\) is the instantaneous range at \(t_{\rm a}=0\); \(y_{\rm pc}\) is the y-axis of scene center point \(P_{\rm c}\); \(k_{\rm 1c}\) is the linear coefficient at \(P_{\rm c}\) which represents the coefficient of the linear range walk (LRW); \(k_{\rm 1}\) is the residual LRW coefficient of the target, which is far from the scene center; and \(k_{\rm 2}\), \(k_{\rm 3}\), and \(k_{\rm 4}\) are the quadratic, cubic, and quartic coefficients, respectively. Here, the Taylor expansion series is kept up to fourth-order term. In this bistatic configuration, we compensate the LRW at the scene center point \(P_{\rm c}\), so the LRWC function can be written as:  
\[H_{\rm 1}(f_{\rm r},t_{\rm a})=\exp\left[j2\pi\frac{\left(f_{\rm c}+f_{\rm r} \right)}{c}k_{\rm 1c}t_{\rm a}\right] \tag{9}\]  
After LRWC, the signal can be written as:  
\[ \begin{array}{l}s_{\rm r3}(f_{\rm r},t_{\rm a})=W_{\rm r}(f_{\rm r})a_{\rm a }(t_{\rm a})\cdot\exp\left(-j\pi\frac{f_{\rm r}^{2}}{\gamma}\right)\\ \times\exp\left[-j2\pi\frac{\left(f_{\rm c}+f_{\rm r}\right)}{c}\left(R_{\rm t }+R_{\rm r0}+k_{\rm 1}t_{\rm a}+k_{\rm 2}t_{\rm a}^{2}+k_{\rm 3}t_{\rm a}^{3}+k_{\rm 4 }t_{\rm a}^{4}\right)\right]\end{array}  \tag{10}\]  
Then, we can derive the 2-D frequency spectrum by the method of POSP and series reversion as:  
\[s_{\rm r4}(f_{\rm r},f_{\rm a})=W_{\rm r}(f_{\rm r})W_{\rm a}(f_{\rm a})\cdot \exp[j\psi(f_{\rm r},f_{\rm a})] \tag{11}\]where:  
\[ \begin{split}\psi(f_{\rm r},f_{\rm a})&=-\pi\frac{f_{ \rm r}^{2}}{\gamma}-2\pi\frac{R_{\rm r}+R_{\rm r0}}{A}+\frac{2\pi AA_{1}}{2} \Big{(}f_{\rm a}+\frac{k_{1}}{A}\Big{)}^{2}\\ &\quad-2\pi\frac{A^{2}A_{2}}{3}\Big{(}f_{\rm a}+\frac{k_{1}}{A} \Big{)}^{3}+2\pi\frac{A^{3}A_{3}}{4}\Big{(}f_{\rm a}+\frac{k_{1}}{A}\Big{)}^{4 }\end{split}  \tag{12}\]  
and:  
\[ \begin{cases}A=\frac{c}{f_{\rm c}+f_{\rm r}}&A_{1}=\frac{1}{2k_{2}}\\ A_{2}=-\frac{3k_{3}}{8k_{2}^{3}}&A_{3}=\frac{9k_{3}^{2}-4k_{2}k_{4}}{16 k_{2}^{3}}\end{cases}  \tag{13}\]  
For further analysis, the phase term should be decomposed into its Taylor series of \(f_{\rm r}\) at \(f_{\rm r}=0\), then reorganized as:  
\[ \begin{split}\psi_{\rm e}&(f_{\rm r},f_{\rm a})=\psi_ {0}(f_{\rm a})+\psi_{1}(f_{\rm a})f_{\rm r}+\psi_{2}(f_{\rm a})f_{\rm r}^{2}\\ &+\psi_{3}(f_{\rm a})f_{\rm r}^{3}+\psi_{4}(f_{\rm a})f_{\rm r}^{4}+ \psi_{\rm a}(f_{\rm a})+\psi_{\rm r}(f_{\rm r})\end{split}  \tag{14}\]  
where:  
\[