# Generalized Nonlinear Chirp Scaling Algorithm for High-Resolution Highly Squint SAR Imaging
## 2 Gnlcsa
### Azimuth Filter and Coefficients
 \tag{24}\]  
The coefficients are as follows:  
\[ \begin{array}{rl}k_{i}=Y_{i}+\varphi_{i,s}^{\prime}\eta_{p}(i=3,4,5)\\ p_{3}=k_{3}K_{a}^{3}\\ p_{4}=\frac{1}{4}K_{a}^{4}\big{(}4k_{4}+9k_{3}^{2}K_{a}\big{)}\\ p_{5}=\frac{1}{4}K_{a}^{5}\big{(}4k_{5}+3k_{3}K_{a}\big{(}8k_{4}+9k_{3}^{2}K_{ a}\big{)}\big{)}\end{array}  \tag{25}\]  
To compensate the azimuth variance and the difference of Doppler rates along the azimuth, a high-order nonlinear azimuth chirp scaling filter is applied:  
\[H_{4}(\eta)=\exp\Biggl{(}j\pi\frac{5}{i=2}\,a_{i}\eta^{i}\Biggr{)} \tag{26}\]Multiplying (26) with (23) results in \(s_{6}(\tau,\eta)\):  
\[ \begin{array}{ll}s_{6}(\tau,\eta)&=s_{5}(\tau,\eta)H_{4}(\eta)\\ &=\omega_{a}(\eta-\eta_{c})\text{sinc}(\tau-2R_{LRW\text{C}}/c)\exp\left(j\pi \theta_{5}\big{(}f_{\eta}\big{)}+j\pi\frac{5}{i=2}\,q_{i}\eta^{i}\right)\end{array}  \tag{27}\]  
In order to achieve a fast calculation for azimuth compression, an FFT for the Equation (27) yields the \(s_{7}\big{(}\tau,f_{\eta}\big{)}\) in RD domain. Also, the solution of SP is calculated by MSR:  
\[ \begin{array}{ll}\eta_{SP}&=\frac{f_{\eta}+\eta_{p}K_{a}+\frac{5}{i-3}\,\frac {i}{2}\big{(}-\eta_{p}\big{)}^{i-1}p_{i}}{K_{a}+\frac{5}{i-3}\,\frac{i}{i-1}(i -1)\big{(}-\eta_{p}\big{)}^{i-1}p_{i}+q_{2}}\\ &-\frac{\Big{(}\sum\limits_{i=3}^{5}p_{i}^{3}p_{i}\big{(}-\eta_{p} \big{)}^{i-3}+6q_{3}\Big{)}\Big{(}f_{\eta}+\eta_{p}K_{a}+\sum\limits_{i=3}^{5} \frac{i}{2}\big{(}-\eta_{p}\big{)}^{i-1}p_{i}\Big{)}^{2}}{4\Big{(}K_{a}+\sum \limits_{i=3}^{5}i(i-1)\big{(}-\eta_{p}\big{)}^{i-1}p_{i}+q_{2}\Big{)}^{3}} \end{array}  \tag{28}\]  
Expanding (28) near \(\eta_{p}=0\) by the principle of [PERSON] expansion yields the results:  
\[ \begin{array}{ll}\frac{1}{K_{a}+\frac{5}{i-3}\,\frac{i}{i-1}\big{(}-\eta_{p }\big{)}^{i-1}p_{i}+q_{2}}\approx\frac{1}{K_{LRW\text{C}}+q_{2}}+\frac{\big{(} -K_{LRW\text{C}}K_{a}+3K_{LRW\text{C}}^{3}X_{3}\big{)}\eta_{p}}{(K_{LRW\text{C }}+q_{2})^{2}}\\ +\frac{\big{(}-2K_{LRW\text{C}}K_{a}^{2}q_{2}+6K_{LRW\text{C}}^{ \prime}K_{a}X_{3}+18K_{LRW\text{C}}^{\prime}K_{a}q_{2}X_{3}-9K_{LRW\text{C}} ^{\prime}X_{3}\big{)}\eta_{p}^{2}}{2\big{(}K_{LRW\text{C}}+q_{2}\big{)}^{3}} \\ \frac{1}{\Big{(}K_{a}+\sum\limits_{i=3}^{5}i(i-1)\big{(}-\eta_{p} \big{)}^{i-1}p_{i}+q_{2}\Big{)}^{3}}\approx\frac{1}{(K_{LRW\text{C}}+q_{2})^{ 3}}-\frac{3\big{(}K_{LRW\text{C}}K_{a}-3K_{LRW\text{C}}^{\prime}X_{3}\big{)} \eta_{p}}{(K_{LRW\text{C}}+q_{2})^{4}}\\ +\frac{\frac{12\big{(}K_{LRW\text{C}}K_{a}-3K_{LRW\text{C}}^{ \prime}X_{3}\big{)}^{2}}{(K_{LRW\text{C}}+q_{2})^{2}}-\frac{\big{(}\epsilon _{LRW\text{C}}K_{a}^{2}-9K_{LRW\text{C}}^{3}K_{LRW\text{C}}K_{3}+\frac{9}{2 }K_{LRW\text{C}}^{\prime}Y_{a}^{2}+6K_{LRW\text{C}}^{\prime}X_{4}\big{)}}{2(K _{LRW\text{C}}+q_{2})^{3}}\eta_{p}^{2}}{2\big{(}K_{LRW\text{C}}+q_{2}\big{)}^{3 }}\end{array}  \tag{29}\]  
Then we can get the phase \(\theta_{7}\big{(}f_{\eta}\big{)}\) of the \(s_{7}\big{(}\tau,f_{\eta}\big{)}\) after the Taylor expansion:  
\[ \begin{array}{ll}\theta_{7}\big{(}f_{\eta}\big{)}&=A\Big{(}q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5},f_{\eta},f_{\eta}^{2},f_{\eta}^{3},f_{\eta}^{4},f_{ \eta}^{5}\Big{)}\\ &+B(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\eta_{p}f_{\eta}+C(q_{2},q_{3},q_ {4},q_{5},X_{3},X_{4},X_{5})\eta_{p}^{2}f_{\eta}\\ &+D(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\eta_{p}f_{\eta}^{2}+E(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\eta_{p}^{2}f_{\eta}^{2}\\ &+F(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\eta_{p}f_{\eta}^{3}+G(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\eta_{p}^{2}f_{\eta}^{3}\\ &+H(q_{2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\eta_{p}f_{\eta}^{4}+\phi_{res}(q_ {2},q_{3},q_{4},q_{5},X_{3},X_{4},X_{5})\end{array}  \tag{30}\]  
The first term on the right of (30) is the azimuth frequency modulation. The second term is the real target azimuth position. The third-the eighth term represents the nonlinear geometric deviation along the azimuth direction.