# Generalized Nonlinear Chirp Scaling Algorithm for High-Resolution Highly Squint SAR Imaging
## 2 Gnlcsa
### Azimuth Filter and Coefficients
The last term, \(\phi_{res}\), is the residual phase, which needs to be eliminated in the applications, e.g., SAR interferometry and differential SAR interferometry. The coefficients are shown in Table 1. \begin{table}
\begin{tabular}{l c} \hline \multicolumn{1}{c}{**Coefficients**} & \multicolumn{1}{c}{**Value**} \\ \hline \(B(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\) & \(-\frac{2\pi\epsilon_{LRW\text{C}}}{K_{a}+\epsilon_{3}}\) \\ \hline \(\text{C}(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\) & \(\frac{\pi\epsilon_{LRW\text{C}}(K_{a},K_{a}+\epsilon_{3}+3q_{3}-3q_{3}(x_{3}))}{(K_{ \text{LRW}}+q_{2})^{3}}\) \\ \hline \(D(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\) & \(\pi K_{LRW\text{C}}\frac{-2\epsilon_{L}\text{C}(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5 })\eta_{p}^{2}f_{\eta}^{3}}{(K_{\text{LRW}}+q_{2})^{3}}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Coefficients for different order. The coefficients of \(q_{i}(i=2,3,4,5)\) and \(X_{i}(i=3,4,5)\) can be solved by the following equations:  
\[\left\{ \begin{array}{l}B(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})=-\frac{ \pi}{\alpha}\\ C(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})=0\\ D(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})=0\\ E(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})=0\\ F(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})=0\\ G(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})=0\\ H(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})=0\\ \end{array} \right. \tag{31}\]  
The parameters are solved by Equation (31):  
\[\left\{ \begin{array}{l}q_{2}=(-1+2\alpha)K_{LRW\mathrm{C}}\\ q_{3}=\frac{1}{3}(-1+2\alpha)K_{LRW\mathrm{C}}K_{s}\\ q_{4}=\frac{K_{LRW\mathrm{C}}}{21}\left(K_{s}^{2}(-3+7\alpha)-3K_{LRW\mathrm{C }}^{2}\phi_{3,s}^{\prime}(1-2\alpha)\right)\\ q_{5}=\frac{K_{LRW\mathrm{C}}}{60(-1+2\alpha)}\left((9-50\alpha+82\alpha^{2})K _{s}^{3}+6K_{LRW\mathrm{C}}^{2}K_{s}\phi_{3,s}^{\prime}(5-24\alpha+25\alpha^{2 })+12K_{LRW\mathrm{C}}^{3}\phi_{4,s}^{\prime}(1-2\alpha)^{2}\right)\\ X_{3}=\frac{(-1+4\alpha)K_{s}}{3(-1+2\alpha)K_{LRW\mathrm{C}}^{2}}\\ X_{4}=\frac{-6\alpha K_{s}^{2}-3K_{LRW\mathrm{C}}^{2}\phi_{3,s}^{\prime}(1-4 \alpha)}{12(-1+2\alpha)K_{LRW\mathrm{C}}^{2}}\\ X_{5}=\frac{-3\alpha K_{s}\phi_{3,s}^{\prime}-K_{LRW\mathrm{C}}^{2}\phi_{4,s} ^{\prime}(1-4\alpha)}{5(-1+2\alpha)K_{LRW\mathrm{C}}^{2}}\end{array} \right. \tag{32}\]  
The coefficient \(A\Big{(}q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5},f_{\eta},f_{\eta}^{2},f_{ \eta}^{3},f_{\eta}^{4},f_{\eta}^{5}\Big{)}\) is rewritten as:  
\[A\Big{(}q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5},f_{\eta},f_{\eta}^{2},f_{ \eta}^{3},f_{\eta}^{4},f_{\eta}^{5}\Big{)}=\sum_{i=1}^{5}A_{i}(q_{2},q_{3},q_ {4},q_{5},X_{3},X_{4},X_{5})f_{\eta}^{i} \tag{33}\]  
Table 2 shows the expressions of the coefficients \(A_{i}(i=1\sim 5)\). \begin{table}
\begin{tabular}{l l} \hline \multicolumn{1}{c}{**Coefficients**} & \multicolumn{1}{c}{**Value**} \\ \hline \(E(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\) & \(\frac{\pi K_{LRW\mathrm{C}}}{2(K_{s,\mathrm{inc}}+6)^{2}}\Bigg{(} \begin{array}{ c}2K_{\mathrm{d}}q_{2}^{2}(K_{\mathrm{d}}q_{2}+3q_{3})+K_{LRW\mathrm{C}}\big{(}4K_{ \mathrm{d}}^{2}q_{2}^{2}-6K_{\mathrm{d}}q_{2}q_{3}-27q_{3}^{2}+12q_{2}q_{4} \big{)}\\ -3K_{LRW\mathrm{C}}^{2}\big{(}3q_{3}X_{3}+q_{2}(-9q_{2}^{2}X_{3}^{2}-4q_{2}X_{4 }+2\psi_{3,s}^{\prime})\big{)}\end{array} \) \\ \hline \(F(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\) & \(\frac{-\pi K_{LRW\mathrm{C}}}{(K_{\mathrm{LRW\mathrm{C}}}+6)^{2}}\Bigg{(} \begin{array}{ c}3K_{\mathrm{d}}q_{2}q_{3}+9q_{3}^{2}+K_{\mathrm{LRW\mathrm{C}}}\big{(}3K_{ \mathrm{d}}q_{3}-4q_{4}\big{)}+K_{LRW\mathrm{C}}^{2}(9q_{2}^{2}X_{3}^{2}+4q_{2}X_{ 4}-\phi_{3,s}^{\prime})\end{array} \Bigg{)}\) \\ \hline \(G(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\) & \(\frac{-\pi K_{LRW\mathrm{C}}}{(K_{\mathrm{LRW\mathrm{C}}}+6)^{2}}\Bigg{(} \begin{array}{c}3K_{\mathrm{d}}q_{2}q_{3}+9q_{3}^{2}+K_{\mathrm{LRW\mathrm{C} }}\big{(}3K_{\mathrm{d}}q_{3}-4q_{4}\big{)}+K_{LRW\mathrm{C}}^{2}(9q_{2}^{2}X_{ 3}^{2}+4q_{2}X_{4}-q_{3}^{2})\end{array}