## 2 ISAR Imaging Model of Maneuvering Targets
\tag{3}\]  
where \(v_{0}\) and \(a\) respectively represent the initial velocity and acceleration of the target, \(\big{(}\omega_{x},\omega_{y},\omega_{z}\big{)}\) and \(\big{(}\Omega_{x},\Omega_{y},\Omega_{z}\big{)}\) respectively represent the angular velocity and angular acceleration of the decomposed \(\big{(}\omega_{ex},\omega_{ey},\omega_{ez}\big{)}\) along three coordinate axes, and \(t_{m}\) is the azimuth time. Combining Equations (2) and (3), the range from the scattering point \(P\) to the radar is:  
\[R_{p}(t_{m})=R_{p}(t_{0})+\int_{0}^{t_{m}}\frac{\lambda}{2}f_{d}dt_{m}=R_{p}(t _{0})+(v_{0}+\mathbf{\omega}_{l}\cdot\mathbf{\gamma})t_{m}+\frac{1}{2}(a+\mathbf{\Omega}_ {l}\cdot\mathbf{\gamma})t_{m}^{2} \tag{4}\]  
Figure 1: Inverse synthetic aperture radar (ISAR) imaging geometry. (**a**) Three-dimensional (3D) imaging geometry and (**b**) two-dimensional (2D) turntable imaging geometry. where \(R_{p}(t_{0})\) represents the range from the radar to the target origin O at the initial moment \(t_{0}\). \(\mathbf{\omega}_{l}\)\(=(\omega_{x},\omega_{y},\omega_{z})\), \(\mathbf{\Omega}_{l}\)\(=(\Omega_{x},\Omega_{y},\Omega_{z})\), and \(\mathbf{\gamma}\)\(=(r_{py}R_{z}-r_{py}R_{z},r_{pz}R_{x}-r_{px}R_{z},r_{px}R_{y}-r_{py}R_{x})\). From Equation (4), for the maneuvering target, the instantaneous range exhibits a quadratic characteristic of the azimuth time, corresponding to the frequency modulation characteristic in the azimuth phase, which seriously affects the azimuth Doppler imaging. In order to obtain a high-range resolution, a radar usually uses an ultra-bandwidth linear frequency modulation (LFM) signal. The expression of the transmitted signal is:  
\[s_{t}(t_{k},t_{m})=\text{rect}(\frac{t_{k}}{T_{p}})\exp\left(j2\pi f_{c}t+j\pi k _{r}{t_{k}}^{2}\right) \tag{5}\]  
where \(\text{rect}(\frac{t_{k}}{T_{p}})=\left\{ \begin{array}{cc}1,&|t_{k}|\leq T_{ p}/2\\ 0,&|t_{k}|>T_{p}/2\end{array} \right.\),\(T_{p}\) is the pulse width, \(f_{c}\) is the carrier frequency, \(k\) is the chirp-rate, \(t_{k}\) is the range fast time, and \(t_{m}\) is the azimuth slow time. After the time delay \(\tau_{p}=2R_{p}(t_{m})/c\), the radar receives the echo signal of the point \(P\):  
\[s_{r}(t_{k},t_{m})=\text{rect}\Big{(}\frac{t_{k}-\tau_{p}}{T_{p}}\Big{)}\ \cdot\exp\Big{(}j2\pi f_{c}(t_{k}-\tau_{p})+j\pi k_{r}(t_{k}-\tau_{p})^{2}\Big{)} \tag{6}\]  
ISAR uses the \"dechirp\" reception mode to reduce the sampling rate of the echo signal. The reference signal is:  
\[s_{r}(t_{k},t_{m})=\text{rect}\Big{(}\frac{t_{k}-\tau_{ref}}{T_{p}}\Big{)}\ \cdot\exp\Big{(}j2\pi f_{c}(t_{k}-\tau_{ref})+j\pi k_{r}(t_{k}-\tau_{ref})^{2} \Big{)} \tag{7}\]  
where \(\tau_{ref}=2R_{ref}(t_{m})/c\), \(R_{ref}(t_{m})\) is the reference range. For ease of analysis, it is usually assumed that \(R_{ref}(t_{m})=R_{p}(t_{0})\) is satisfied. Therefore, the range compression signal after the dechirp processes compensates for the residual video phase (RVP) and the envelope skew term is [15]:  
\[S(f,t_{m})=AT_{p}\text{sin}\,\text{c}\Big{[}T_{p}\Big{(}f+2\frac{k_{r}}{c}R_ {\Delta}\Big{)}\Big{]}\cdot\exp\bigg{(}-j\frac{4\pi R_{\Delta}}{\lambda}\bigg{)} \tag{8}\]  
where \(R_{\Delta}=R_{p}(t_{m})-R_{ref}(t_{m})=v_{0}t_{m}+\frac{1}{2}at_{m}^{2}+(\mathbf{ \omega}_{l}\cdot\mathbf{\gamma})\,t_{m}+\frac{1}{2}(\mathbf{\Omega}_{l}\cdot\mathbf{\gamma })\,t_{m}^{2}\). It can be seen from Equation (8) that \(R_{\Delta}\) in the phase of the range compressed signal determines the azimuth focusing effect. The first and second terms cause envelope migration and need to be compensated for. The third term is necessary to achieve high-resolution azimuth imaging; the fourth term is due to azimuth defocusing caused by azimuth Doppler time-varying and needs to be compensated for. Therefore, envelope migration correction and azimuth focusing processing are needed to achieve high-resolution imaging. Envelope migration has been studied in many papers and achieved good results [24; 25; 26], and we will not go into detail in this paper. This paper focuses on azimuth focusing imaging.