## 3 2-D Frequency-Domain Focusing Based on BiCoT-ESRM
### 2-D Frequency-Domain Focusing Method Based on BiCoT-Esrm  
Based on the above analysis, the two-dimensional spectrum of the signal after preprocessing can be derived by principle of stationary phase based on (12) and (20)  
\[\phi_{3}\big{(}f_{\tau},f_{t}\big{)} =-\frac{f_{\tau}^{2}}{k_{r}}-2\pi\frac{f_{0}+f_{\tau}}{c}R_{T0s}\] \[-2\pi R_{R0}\cos\theta_{e}\sqrt{\left(\frac{f_{0}+f_{\tau}}{c} \right)^{2}-\left(\frac{f_{t}}{v_{e}}\right)^{2}}\] \[-2\pi f_{t}{t^{\prime}}_{0}-2\pi f_{t}\frac{R_{R0}\sin\theta_{e}} {v_{e}}+\phi_{err4}\big{(}f_{\tau},f_{t}\big{)} \tag{23}\]  
where \({t^{\prime}}_{0}\) is the beam center crossing time of an arbitrary target \(P\). \(\phi_{err4}\) is the fourth-order compensation term given by  
\[\phi_{err4}\big{(}f_{\tau},f_{t}\big{)} =-2\pi\frac{f_{0}+f_{\tau}}{c}\Big{(}k_{T4s}+\Delta k_{err}^{4} \Big{)}\bigg{(}\frac{2}{w_{e}}\arcsin\frac{{v_{e}{t^{\prime}}_{posp}}}{2R_{e} }\bigg{)}^{4} \tag{24}\]  
where \(t^{\prime}_{posp}\) is the position of stationary phase given by  
\[t^{\prime}_{posp} =\frac{R_{R0}\sin\theta_{e}+R_{R0}\cos\theta_{e}\frac{1}{\sqrt{ \frac{v_{e}^{2}\big{(}f_{0}+f_{\tau}\big{)}^{2}}{c^{2}f_{t}^{2}}-1}}}{v_{e}}+ t^{\prime}_{0}. \tag{25}\]  
During the derivation of (23), the fourth-order term is ignored for solving \(t^{\prime}_{posp}\). In (23), the first term is the range modulation phase, and the second term represents a range offset. The third term contains azimuth modulation, range cell migration (RCM) and secondary range compression (SRC). The fourth and fifth terms are linear phases of azimuth frequency, respectively, which are determined by the azimuth position of the targets. Then, the reference function is multiplied with (23) to achieve the range compression, bulk azimuth demodulation, bulk RCM correction, bulk SRC and fourth-order phase compensation. The reference function is given by  
\[H_{5}\big{(}f_{\tau},f_{t}\big{)} =\exp\left\{\pi\frac{f_{\tau}^{2}}{k_{r}}+2\pi\frac{f_{0}+f_{ \tau}}{c}R_{T0s}^{ref}\right\}\] \[\times\exp\left\{2\pi f_{t}\frac{R_{R0}^{ref}\sin\theta_{e}^{ref} }{v_{e}}+\phi_{err4}^{ref}\right\}\] \[\times\exp\left\{2\pi R_{R0}^{ref}\cos\theta_{e}^{ref}\sqrt{\left( \frac{f_{0}+f_{\tau}}{c}\right)^{2}-\left(\frac{f_{t}}{v_{e}}\right)^{2}} \right\}. \tag{26}\]After reference function multiplication (RFM), the phase of the reference target is completely compensated. However, residual phase errors exist for the targets away from the reference target, which is given by  
\[\phi_{4}\big{(}f_{\tau},f_{t}\big{)}= -2\pi\frac{f_{0}+f_{\tau}}{c}\Delta R_{T0s}-2\pi f_{t}\big{(}\Delta T _{1}+t^{\prime}{}_{0}\big{)}\] \[-2\pi\Delta R_{1}\sqrt{\frac{\big{(}f_{0}+f_{\tau}\big{)}^{2}}{c^ {2}}-\frac{f_{t}^{2}}{v_{e}^{2}}} \tag{27}\]  
where  
\[-2\pi R_{R0}\cos\theta_{e}\sqrt{\left(\frac{f_{0}+f_{\tau}}{c} \right)^{2}-\left(\frac{f_{t}}{v_{e}}\right)^{2}}\]0 \[-2\pi R_{R0}\cos\theta_{e}\sqrt{\left(\frac{f_{0}+f_{\tau}}{c} \right)^{2}-\left(\frac{f_{t}}{v_{e}}\right)^{2}}\]1 \[-2\pi R_{R0}\cos\theta_{e}\sqrt{\left(\frac{f_{0}+f_{\tau}}{c} \right)^{2}-\left(\frac{f_{t}}{v_{e}}\right)^{2}}\]2  
The spatial variance of \(\phi_{err4}\) is extremely small and can be arguably ignored. Then, according to the 2-D Stolt mapping [24], \(\Delta R_{T0s}\) and \(\Delta T_{1}\) are expanded to the first-order terms with respect to spatial variable \(y\) and \(R_{R0}\cos\theta_{e}\), given by  
\[-2\pi R_{R0}\cos\theta_{e}\sqrt{\left(\frac{f_{0}+f_{\tau}}{c} \right)^{2}-\left(\frac{f_{t}}{v_{e}}\right)^{2}}\]3 \[-2\pi R_{R0}\cos\theta_{e}\sqrt{\left(\frac{f_{0}+f_{\tau}}{c} \right)^{2}-\left(\frac{f_{t}}{v_{e}}\right)^{2}}\]4  
where \(p_{1}\), \(q_{1}\), \(p_{2}\) and \(q_{2}\) are the first-order expanding coefficients of \(\Delta R_{T0s}\) and \(\Delta T_{1}\). The detailed derivations and expressions of these coefficients are given in the Appendix A.