# Generalized Nonlinear Chirp Scaling Algorithm for High-Resolution Highly Squint SAR Imaging
## 2 Gnlcsa
### Azimuth Filter and Coefficients  
A 2D fast Fourier transform (FFT) yields the signal of Equation (7) in the 2D frequency domain:  
\[s_{1}\big{(}f_{\tau},f_{\eta}\big{)}=W_{r}(f_{\tau})W_{a}\big{(}f_{\eta}\big{)} \cdot\exp\big{(}-j2\pi f_{\eta}\eta_{p}\big{)}\cdot\exp\bigg{(}-j4\pi\frac{ \big{(}f_{\tau}+f_{0}\big{)}}{c}x_{p}\sin\theta_{s}\bigg{)}\cdot\exp\big{(}j \Phi\big{(}f_{\tau},f_{\eta}\big{)}\big{)} \tag{8}\]  
The first exponential term implies the position of target, and the second one indicates the offset of the target's position, which can be corrected after the accomplishment of azimuth compression. \(\Phi\big{(}f_{\tau},f_{\eta}\big{)}\) represents the phase coupling between the range and azimuth directions. \[\Phi\big{(}f_{\tau},f_{\eta}\big{)}\approx\phi_{0}\big{(}f_{\eta}\big{)}R_{0 }+\sum_{i=1}^{n}\phi_{i}\big{(}f_{\eta},R_{0}\big{)}f_{\tau}^{i} \tag{9}\]  
In Section 2, it has been discussed that the processing of range dimension is accessible in the References [18; 21]. It is considered that \(\sum\limits_{i=1}^{n}\phi_{i}\big{(}f_{\eta},R_{0}\big{)}f_{\tau}^{i}\) is eliminated after the processing of range dimension, so the coefficients \(\phi_{i}(i=1,2,\cdots,n)\) are not discussed again in this paper. The \(\phi_{0}\big{(}f_{\eta}\big{)}\) is shown as the following equation:  
\[\phi_{0}\big{(}f_{\eta}\big{)} \approx-\frac{4\pi}{\lambda}-\frac{2\pi\eta_{p}}{R_{0}}f_{\eta}+ \frac{\pi}{\lambda}\frac{\sec^{2}\theta_{s}}{2V^{2}}f_{\eta}^{2}+\pi\frac{ \sin\theta_{s}}{\cos^{4}\theta_{s}}\frac{\lambda^{2}}{4V^{2}}f_{\eta}^{3}- \pi\lambda^{3}\frac{(-3+2\cos(2\theta_{s}))}{32V^{2}\cos^{6}\theta_{s}}f_{\eta}^ {4} \tag{10}\] \[-\pi\lambda^{4}\frac{(-6\sin\theta_{s}+\sin(3\theta_{s}))}{64V^{ 2}\cos^{6}\theta_{s}}f_{\eta}^{5}+O\big{[}f_{\eta}\big{]}^{5}\] \[=\sum\limits_{i=0}^{5}\varphi_{i}f_{\eta}^{i}+O\big{[}f_{\eta}\big{]} ^{5}\]  
Figure 2: Geometry of squint SAR after the linear range walk correction (LRWC). The red dot line represents the equivalent trajectory of the platform after the LRWC. After the RCMC/SRC/RC processing of the signal in Equation (8), the signal in RD domain is expressed as:  
\[s_{2}\big{(}\tau,f_{\eta}\big{)}=\omega_{r}(\tau)W_{a}\big{(}f_{\eta}\big{)}\cdot \text{sinc}(\tau-2R_{LRWC}/c)\cdot\exp\big{(}j\theta_{2}\big{(}f_{\eta}\big{)} \big{)} \tag{11}\]  
where \(\theta_{2}\big{(}f_{\eta}\big{)}\) is \(\phi_{0}\big{(}f_{\eta}\big{)}R_{0}\). The coefficients \(\varphi_{0}\sim\varphi_{5}\) are as follows:  
\[\left\{ \begin{array}{ccc}\varphi_{0}=-\frac{4\pi}{\lambda};&\varphi_{1}=- \frac{2\pi\eta_{p}}{R_{0}}\\ \varphi_{2}=\pi\lambda\frac{\sec^{2}\theta_{s}}{2V^{2}};&\varphi_{3}=\pi \lambda^{2}\frac{\sin\theta_{s}}{\cos^{4}\theta_{s}}\frac{1}{4V^{3}}\\ \varphi_{4}=-\pi\lambda^{3}\frac{(-3+2\cos(2\theta_{s}))}{32V^{4}_{c}\cos^{ 6}\theta_{s}};&\varphi_{5}=-\pi\lambda^{4}\frac{(-6\sin\theta_{s}+\sin(3\theta _{s}))}{64V^{3}_{c}\cos^{6}\theta_{s}}\end{array} \right. \tag{12}\]  
Also, we define the coefficients \(\varphi_{i,LRWC}\) and \(\varphi_{i,s}(i=3,4,5\cdots)\), respectively. \[\left\{ \begin{array}{ccc}\varphi_{i,LRWC}(R_{LRWC})=\varphi_{i}R_{LRWC}\\ \varphi_{i,s}=-\varphi_{i}V_{r}\sin\theta_{s}\end{array} \right. \tag{13}\]  
Substituting the Equation (6) to \(\phi_{0}\big{(}f_{\eta}\big{)}R_{0}\), we can rewrite \(\theta_{2}\big{(}f_{\eta}\big{)}\):  
\[\theta_{2}\big{(}f_{\eta}\big{)}=-\frac{4\pi\big{(}R_{LRWC}-V_{r}\eta_{p}\sin \theta_{s}\big{)}}{\lambda}-2\pi\eta_{p}f_{\eta}+\varphi_{2}R_{0}f_{\eta}^{2} +\sum_{i=3}^{n}\big{(}\varphi_{i,LRWC}+\varphi_{i,s}\eta_{p}\big{)}f_{\eta}^{i} \tag{14}\]  
In this paper, the method of series reversion (MSR) is used to enhance the focusing precision of the algorithm and the ADOF for the high-resolution highly squint SAR. The first step of azimuth processing is the compensation of high order phase.