## 3 2-D Frequency-Domain Focusing Based on BiCoT-ESRM
### Modified Azimuth Preprocessing  
The echo data in (1) is transformed into range frequency domain by range FFT, and the phase term is given by  
\[\phi_{1}\big{(}f_{\tau},t\big{)}=-\pi\frac{f_{\tau}^{2}}{k_{\tau}}-2\pi\frac{ f_{0}+f_{\tau}}{c}R_{bi}(t) \tag{12}\]  
where \(f_{\tau}\) is the range frequency variable. Then, an azimuth phase multiplication is introduced to eliminate the spectrum skewing and align the azimuth spectrum to base band, which facilitates the subsequent processing. \[H_{1}\big{(}f_{\tau},t\big{)}=\exp\Big{\{}-j2\pi f_{dc}\big{(}f_{\tau}\big{)} \bigg{(}\frac{2R_{e}}{v_{e}}\sin\frac{\omega_{e}t}{2}\bigg{)}\Big{\}} \tag{13}\]  
where \(v_{e}=\omega_{e}R_{e}|_{ref}\) is the equivalent platform velocity with respect to the reference target. \(f_{dc}(f_{\tau})\) denotes the range frequency dependent Doppler centroid of the reference target. Based on (11), \(f_{dc}(f_{\tau})\) can be expressed as  
\[f_{dc}\big{(}f_{\tau}\big{)}=\frac{\big{(}f_{0}+f_{\tau}\big{)}\sin\theta_{e} ^{ref}\omega_{e}^{ref}R_{e}^{ref}}{c} \tag{14}\]  
where superscript '_ref_' represents the corresponding parameters for the reference target. According to (10) and the derivations in the Appendix A, the Doppler centroid in (14) is contributed by the first-order coefficients of the range models of GEO transmitter, LEO receiver and'stop-and-go' assumption error. In contrast to a monostatic LEO-SAR, the Doppler centroid in GEO-LEO BiSAR is dependent on the orbit position of the GEO transmitter [24], which should be removed. During the data acquisition of a sliding-spotlight mode, the steering of the antenna beam leads to the azimuth variance of Doppler centroid. In addition, the translational variant imaging configuration of GEO-LEO BiSAR further aggravates the azimuth variance of Doppler centroid [24]. This variance will increase the total Doppler bandwidth and causes azimuth aliasing. Therefore, the two-step preprocessing method [56] is modified to remove the azimuth variance of Doppler centroid and eliminate the azimuth aliasing. \[H_{2}(t)=\exp\left\{j\pi K_{rot}^{bi}\Big{(}\frac{2R_{e}}{v_{e}}\sin\frac{\omega_ {e}t}{2}\Big{)}^{2}\right\} \tag{15}\]  
where \(K_{rot}^{bi}\) is the bistatic deramping factor, which is calculated by  
\[K_{rot}^{bi}=\frac{v_{r}^{2}}{\lambda R_{rot}\cos\theta_{s}}+\frac{\Delta k_{T1 s}v_{b}}{\lambda} \tag{16}\]  
where \(v_{r}\) is the effective velocity of the LEO receiver and \(R_{rot}\) is the slant range of the sliding-spotlight rotation center [46]. \(\Delta k_{T1s}\) is the partial derivative of \(k_{T1s}\) with respect to azimuth position variable \(y\), which is derived in Appendix A. \(v_{b}\) is the beam footprint velocity of the LEO receiver with beam steering, which is given by  
\[v_{b}=\omega_{bs}(R_{rot}-R_{R0}) \tag{17}\]  
where \(\omega_{bs}\) is the constant beam rotation rate. In (16), the first term is caused by the beam steering of the LEO receiver and the second term is related to the azimuth variance of Doppler centroid introduced by the bistatic configuration. After the above bistatic deramping processing, the aliasing of Doppler spectrum is removed. Inspecting (11), the BiCoT-ESRM is the sum of a constant range and a square-root term with an additional fourth-order compensation term. However, the equivalent radar movement is still a circular orbit trajectory, which contains higher-order components of azimuth time \(t\). Direct solutions of 2-D spectrum using (11) is complicated and difficult. Moreover, the equivalent circular orbit trajectory will introduce azimuth variance of motion parameters, which should be eliminated.