# Fine Recognition of MEO SAR Ship Targets Based on a Multi-Level Focusing-Classification Strategy
## 2 Signal Modeling and Characteristic Analysis of MEO SAR Ships
### MEO SAR Image Defocusing Caused by Ship Rotation  
Ocean waves can induce rotational movements for ships, including rolling, pitching, and yawing, as shown in Figure 5. The oscillations of a ship exhibit distinct periodic characteristics. For LEO SAR systems, the synthetic aperture time is relatively short, and the impact of ship rotations on imaging quality is minimal. However, as the orbital altitude increases, to maintain the same resolution, the synthetic aperture time also becomes longer. For MEO SAR, during the synthetic aperture time, a target may experience one or even multiple oscillation cycles. These complex movements can cause spatial and temporal variations in the phase errors of SAR echo signals, leading to significant degradation of SAR image quality. To simplify the analysis, the ship's rotational motion is modeled as a sinusoidal function. The expression for the rotation angle is as follows:  
\[\beta_{i}=A_{i}\sin(\frac{2\pi}{T_{i}}\eta~{}+~{}\phi_{i}),~{}i~{}=~{}x,~{}y,~ {}z \tag{6}\]  
in which \(\beta_{i}\) is the instantaneous rotation angle, \(A_{i}\) is the amplitude, \(T_{i}\) is the period, \(\phi_{i}\) is the initial phase, \(i~{}=~{}x\) indicates the roll angle, \(i~{}=~{}y\) indicates the pitch angle, and \(i~{}=~{}z\) indicates the yaw angle. Unlike the translational motion, the ship's rotational motion is non-uniform, meaning that different scattering points on the ship exhibit different motion patterns. Taking roll motion as an example to analyze phase errors caused by target rotation, Figure 6 shows the motion error caused by ship rotation. For simplicity, the rotation axis is parallel to the azimuth direction. Let \(\gamma\) be the antenna beam incidence angle. The red rectangle denotes the ship target position in the stationary state. The black rectangle denotes the ship target position at azimuth time \(\eta\). The rotation angle at \(\eta\) is \(\beta_{x}\). The motion error of the point scatter at a distance \(\Delta L\) from the rotation axis is \(\Delta d\). According to the small angle approximation principle, projection of \(\Delta d\) in the SAR line of sight direction is  
\[\Delta R_{x}=\Delta L\cdot\beta_{x}\cdot\cos\gamma \tag{7}\]  
Figure 5: Three-dimensional rotations of a ship target. The above equation indicates that the slant distance error caused by ship rotation is linearly related to the distance from the point scatter to the rotation axis. The phase error caused by \(\Delta R_{x}\) in the azimuth time domain is  
\[4\pi\cdot\Delta L\cdot A_{x}\cdot\sin(2\pi/T_{x}\cdot\eta+\phi_{x})\cdot\cos \gamma/\lambda \tag{8}\]  
According to the principle of stationary phase (POSP), the azimuth signal in the frequency domain can be expressed as  
\[\begin{split}& S_{a}(f)\\ &=\int\limits_{-\infty}^{\infty}\omega_{a}(\eta)\exp\left\{j\left( 4\pi\frac{\Delta L\cdot A_{x}\cdot\cos\gamma}{\lambda}\sin\left(\frac{2\pi}{T_ {x}}\eta+\phi_{x}\right)-\pi K_{a}\eta^{2}\right)\right\}\exp\{-j2\pi f\eta\} d(\eta)\\ &\approx W_{a}(f)\exp\left\{j\pi\frac{f^{2}}{K_{a}}\right\}\exp \left\{-j4\pi\frac{\Delta L\cdot A_{x}\cdot\cos\gamma}{\lambda}\sin\left(\frac {2\pi}{T_{x}}\frac{f}{K_{a}}-\phi_{x}\right)\right\}\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quadFrom the analysis above, it is evident that the defocusing of imaging results caused by ship motions is more severe for MEO SAR, with the defocused ship targets being significantly distorted and unrecognizable in their direct form. It is necessary to perform refocusing processing of the images of ship targets to achieve further recognition. However, refocusing of ships with complex motions is computationally intensive. Therefore, this paper integrates the refocusing processing with a target recognition network.