# Generalized Nonlinear Chirp Scaling Algorithm for High-Resolution Highly Squint SAR Imaging
## 2 Gnlcsa
### Azimuth Filter and Coefficients
The effect of high order phase filter (HOPF) is consistent with the one in [4], which is given as follows:  
\[H_{2}\big{(}f_{\eta}\big{)}=\exp\left[-j\sum_{i=3}^{n}\varphi_{i,LRWC}(R_{0}) f_{\eta}^{i}\right] \tag{15}\]  
Filtered by \(H_{2}\big{(}f_{\eta}\big{)}\), the signal of Equation (8) can be rewritten as:  
\[s_{3}\big{(}\tau,f_{\eta}\big{)}\approx\omega_{r}(\tau)W_{a}\big{(}f_{\eta} \big{)}\cdot\text{sinc}(\tau-2R_{LRWC}/c)\cdot\exp\left(j\left(-2\pi\eta_{p}f _{\eta}+\varphi_{2}R_{0}f_{\eta}^{2}+\sum_{i=3}^{n}\varphi_{i,s}\eta_{p}f_{ \eta}^{i}\right)\right) \tag{16}\]  
\(\varphi_{2}R_{0}\) can be rewritten as \(\varphi_{2}R_{0}=-\frac{\pi}{-2V^{2}_{c}\cos^{2}\theta_{s}/\lambda/R_{0}}=- \frac{\pi}{K_{a}}\), where \(K_{a}\) is the FM rates along azimuth direction. In Section 2, we have discussed that the equivalent nearest slant range is \(R_{LRWC}\) after the LRWC. Substitute the Equation (6) to the expression of \(K_{a}\) and expand \(K_{a}\) near \(\eta_{p}=0\):  
\[ \begin{array}{rl}K_{a}&=-\frac{2V^{2}_{c}\cos^{2}\theta_{s}}{\lambda R_{LRWC }}\frac{-2V^{2}_{c}\cos^{2}\theta_{s}}{\lambda R_{LRWC}}\frac{V_{s}\sin\theta _{s}}{R_{LRWC}}\eta_{p}-\frac{2V^{2}_{c}\cos^{2}\theta_{s}}{\lambda R_{LRWC}} \frac{V^{2}_{c}\sin^{2}\theta_{s}}{R_{LRWC}^{2}}\eta_{p}^{2}\\ &-\frac{2V^{2}_{c}\cos^{2}\theta_{s}}{\lambda R_{LRWC}}\frac{V^{3}_{c}\sin^{ 3}\theta_{s}}{R_{LRWC}^{2}}\eta_{p}^{3}+O\big{[}\eta_{p}\big{]}^{4}\\ &=K_{LRWC}\Big{(}1+K_{s}\eta_{p}+K_{s}^{2}\eta_{p}^{2}+K_{s}^{3}\eta_{p}^{3} \Big{)}+O\big{[}\eta_{p}\big{]}^{4}\\ &\hskip 14.226378 ptK_{LRWC}=-\frac{2V^{2}_{c}\cos^{2}\theta_{s}}{\lambda R_{LRWC }}\\ &\hskip 14.226378 ptK_{s}=\frac{V_{s}\sin\theta_{s}}{R_{LRWC}}\end{array}  \tag{17}\]It turns that the FM rates of different slant range along the azimuth direction are azimuth variant according to the Equations (16)-(18). This paper adds a degree of freedom to achieve a coincident FM rate form of different slant range along the azimuth direction. The turbulent function is expressed as:  
\[H_{3}\big{(}f_{\eta}\big{)}=\exp\Biggl{(}j\pi\sum_{i=3}^{n}X_{i}f_{\eta}^{i} \Biggr{)} \tag{19}\]  
Assume the coefficients \(\varphi_{i,s}^{\prime}(i=3,4,5)\) are proportional to \(\varphi_{i,s}\) as follows:  
\[\varphi_{i,s}^{\prime}=\varphi_{i,s}/\,\pi \tag{20}\]  
Multiplying (19) with (16) results in:  
\[s_{4}(\tau,f_{\eta})\approx\omega_{r}(\tau)W_{a}\big{(}f_{\eta}\big{)}\cdot \mathrm{sinc}(\tau-2R_{LRWC}/c)\cdot\exp\Biggl{(}j\biggl{(}-2\pi\eta_{p}f_{\eta }-\frac{\pi}{K_{a}}f_{\eta}^{2}+\sum\limits_{i=3}^{n}\left(\varphi_{i,s}^{ \prime}\eta_{p}+X_{i}\right)f_{\eta}^{i}\biggr{)}\Biggr{)} \tag{21}\]  
The MSR is applied to calculate the solution of stationary phase (SP) of the Equation (21). To ensure the precision of the solution, this paper reserves a second order form for the solution, which is given as:  
\[f_{\eta,SP}=K_{a}\big{(}\eta-\eta_{p}\big{)}+\frac{3}{2}k_{3}K_{a}^{3}\big{(} \eta-\eta_{p}\big{)}^{2} \tag{22}\]  
We can get \(s_{5}(\tau,\eta)\) by substituting the solution of SP into the IFFT form of \(s_{4}(\tau,\eta)\):  
\[ \begin{array}{rl}s_{5}(\tau,\eta)&=\int\limits_{f_{\eta}=-\infty}^{+\infty}s _{4}\big{(}\tau,f_{\eta}\big{)}\exp\bigl{(}j2\pi f_{\eta}\eta\big{)}df_{\eta} \\ &\approx s_{4}\big{(}\tau,f_{\eta}\big{)}\exp\bigl{(}j2\pi f_{\eta}\eta\big{)} \big{|}_{f_{\eta}=K_{a}(\eta-\eta_{p})+\frac{3}{2}k_{3}K_{a}^{3}(\eta-\eta_{p })^{2}}\\ &=\omega_{a}(\eta-\eta_{c})\mathrm{sinc}(\tau-2R_{LRWC}/c)\exp\bigl{(}j\pi \theta_{5}\big{(}f_{\eta}\big{)}\bigr{)}\end{array}  \tag{23}\]  
\(\theta_{5}(\eta)\) is rewritten as:  
\[ \begin{array}{rl}\theta_{5}(\eta)&=\pi K_{a}\big{(}\eta-\eta_{p}\big{)}^{2} +\pi k_{3}K_{a}^{3}\big{(}\eta-\eta_{p}\big{)}^{3}+\frac{1}{4}\pi K_{a}^{4} \big{(}4k_{4}+9k_{3}^{2}K_{a}\big{)}\big{(}\eta-\eta_{p}\big{)}^{4}\\ &+\frac{1}{4}\pi K_{a}^{5}\big{(}4k_{5}+3k_{3}K_{a}\big{(}8k_{4}+9k_{3}^{2}K_{ a}\big{)}\big{)}\big{(}\eta-\eta_{p}\big{)}^{5}\\ &=\pi K_{a}\big{(}\eta-\eta_{p}\big{)}^{2}+\pi\sum\limits_{i=3}^{5}p_{i}(\eta -\eta_{p})^{i}\end{array}