# Generalized Nonlinear Chirp Scaling Algorithm for High-Resolution Highly Squint SAR Imaging
## 2 Gnlcsa  
### Preprocess in Range Direction by Generalized Chirp Scaling  
Assume that target P is an object of the observation. In Figure 1, \(R(\eta;R_{0})\) is the range between target P and the sensor at the azimuth time of \(\eta\). \(x_{p}\) is defined as the separation distance bias between the footprint of the beam center and target P.  
Figure 1: Geometry of squint synthetic aperture radar (SAR). The geometry of a general squint SAR provides the model of the signal acquisition. The radar flies along the X-axis with the velocity of \(V_{r}\). \(\overset{\rightarrow}{OO}^{\prime}\) is the vector of the beam illumination. The angle of which away from the X-axis is \((\frac{\pi}{2}-\theta_{s})\), where \(\theta_{s}\) is the squint angle. \(R_{0}\) is the slant range between the sensor and the boresight intersection. In \(\triangle SQP\), based on the cosine theorem, the instantaneous slant range \(R(\eta;R_{0})\) can be given by the following equation:  
\[R(\eta;R_{0})=\sqrt{R_{0}^{2}+\big{(}V_{r}\eta-x_{p}\big{)}^{2}+2R_{0}\big{(}V_{ r}\eta-x_{p}\big{)}\cos\Bigl{(}\frac{\pi}{2}-\theta_{s}\Bigr{)}} \tag{1}\]  
Assume that a linear FM (LFM) pulse is transmitted by the sensor, and the signal reflected from target P can be expressed as:  
\[ \begin{array}{ll}s_{r}(\tau,\eta)&=\omega_{r}(\tau-2R(\eta;R_{0})/c)\omega_{ a}\bigl{(}\eta-\eta_{p}\bigr{)}\cdot\exp\Bigl{[}-j\frac{4\pi}{\lambda}R(\eta;R_{0}) \Bigr{]}\\ &\cdot\exp\Bigl{[}-j\pi K_{r}(\tau-2R(\eta;R_{0})/c)^{2}\Bigr{]}\end{array}  \tag{2}\]  
where \(\tau\) is the fast time; \(\eta_{p}\) is the center Doppler moment of target P. \(c\) is the speed of light; \(\omega_{r}(\bullet)\) and \(\omega_{a}(\bullet)\) are the antenna patterns in range and azimuth directions, respectively [18]. \(f_{0}\) is the carrier frequency of the radar. \(\lambda=f_{0}/c\) represents the wavelength of the carrier frequency. \(K_{r}\) is the chirp rate of the LFM signal. By Taylor expansion, \(R(\eta;R_{0})\) can be further approximated as:  
\[ \begin{array}{ll}R(\eta;R_{0})&\approx R_{0}+\frac{\cos^{2}\theta_{s}\big{(} V_{r}\eta-x_{p}\big{)}^{2}}{2R_{0}}+\frac{\sin\theta_{s}\cos^{2}\theta_{s}}{2R_{0} ^{2}}\bigl{(}V_{r}\eta-x_{p}\big{)}^{3}-\frac{\cos^{2}\theta_{s}\left(-3+5 \cos(2\theta_{s})\right)\big{(}V_{r}\eta-x_{p}\big{)}^{4}}{16R_{0}^{3}}\\ &-\frac{\cos^{2}\theta_{s}\left(-1+7\cos(2\theta_{s})\sin\theta_{s}\right) \big{(}V_{r}\eta-x_{p}\big{)}^{3}}{16R_{0}^{3}}-\bigl{(}V_{r}\eta-x_{p}\big{)} \sin\theta_{s}+O[\eta]^{5}\end{array}  \tag{3}\]  
In Equation (3), the term \(V_{r}\eta\sin\theta_{s}\) represents linear range walk, which can be compensated by the function of \(H_{1}\):  
\[H_{1}(f_{\tau},\eta)=\exp\biggl{(}-j4\pi\frac{f_{\tau}+f_{0}}{c}V_{r}\eta\sin \theta_{s}\biggr{)} \tag{4}\]  
where \(f_{\tau}\) is the range frequency. After the linear range walk correction (LRWC) in range frequency domain, the signal can be given as:  
\[ \begin{array}{ll}s_{1}(f_{\tau},\eta)&=W_{r}(f_{\tau})\omega_{a}\bigl{(}\eta -\eta_{p}\bigr{)}\exp\Bigl{(}-j4\pi\frac{f_{\tau}+f_{0}}{c}V_{r}\eta_{p}\sin \theta_{s}\Bigr{)}\exp\Bigl{(}-j\pi\frac{f_{\tau}^{2}}{K_{r}}\Bigr{)}\\ &\cdot\exp\biggl{(}-j4\pi\frac{f_{\tau}+f_{0}}{c}\sqrt{R_{0}^{2}+\cos^{2} \theta_{s}\big{(}V_{r}\eta-x_{p}\big{)}^{2}}\biggr{)}\\ &\cdot\exp\biggl{(}-j4\pi\frac{f_{\tau}+f_{0}}{c}\frac{\sin\theta_{s}\cos^{2} \theta_{s}}{2R_{0}^{2}}\bigl{(}V_{r}\eta-x_{p}\big{)}^{3}\biggr{)}\end{array}  \tag{5}\]  
Another way to understand LRWC is shown in the Figure 2. \(T_{a}\) is the synthetic aperture time of radar in the original sampling geometry. The flight path during the synthetic aperture time of target P is \(\stackrel{{\rightarrow}}{{S_{1}S_{2}}}\). After the LRWC, its equivalent trajectory turns to \(S_{1}{}^{\prime}S_{2}{}^{\prime}\). Accordingly, the equivalent velocity becomes \(V_{r}\cos\theta_{s}\). After the LRWC, the signal model can be treated as the \"broadside SAR\".