# A CSAR 3D Imaging Method Suitable for Edge Computation
## 3 A CSAR 3D Imaging Method Suitable for Edge Computation  
### Principle of the Method  
Let \(R=R_{\text{p}}(\varphi)-R_{\text{c}}\). When the radar and the target meet the far-field conditions, i.e., \(\left(\frac{r_{\text{p}}}{R_{\text{xy}}}\right)^{2}\to 0\), \(\left(\frac{z_{\text{p}}}{H}\right)^{2}\to 0\), then Formula (1) can be expressed as follows:  
\[ \begin{array}{ll}R_{\text{p}}(\varphi)-R_{\text{c}}&=\sqrt{{R_{\text{xy}}}^ {2}+{r_{\text{p}}}^{2}+\left(H-z_{\text{p}}\right)^{2}-2R_{\text{xy}}r_{\text {p}}\cos(\varphi-\theta_{\text{p}})}-R_{\text{c}}\\ &\approx-r_{\text{p}}\sin(\psi)\cos(\varphi-\theta_{\text{p}})-z_{\text{p}} \cos(\psi)\end{array}  \tag{6}\]  
During the CSAR movement, the incident angle \(\psi\) of the beam remains unchanged. Substituting Formula (6) into Formula (2) yields the following:  
\[s_{\text{r}}(\tau,\varphi)\approx\sigma_{\text{p}}\cdot\sin\text{c}\left\{B \left[\tau-\frac{2\left(-r_{\text{p}}\sin(\psi)\cos(\varphi-\theta_{\text{p}} )-z_{\text{p}}\cos(\psi)\right)}{c}\right]\right\}\cdot\exp\left[-j\frac{4\pi f }{c}\cdot\left(-r_{\text{p}}\sin(\psi)\cos(\varphi-\theta_{\text{p}})-z_{ \text{p}}\cos(\psi)\right)\right] \tag{7}\]  
where \(B\left[\tau-\frac{2\left(-r_{\text{p}}\sin(\psi)\cos(\varphi-\theta_{\text{p }})-z_{\text{p}}\cos(\psi)\right)}{c}\right]\) represents the position tracking of target point P in the echo after range compression. When point P is located at the center of the scene, Formula (7) can be expressed as follows:  
\[s_{\text{r}}(\tau,\varphi)\approx\sigma_{\text{p}}\cdot\sin\text{c}\left\{B \left[\tau-\frac{2\left(-z_{\text{p}}\cos(\psi)\right)}{c}\right]\right\} \cdot\exp\left[-j\frac{4\pi f}{c}\cdot\left(-z_{\text{p}}\cos(\psi)\right)\right] \tag{8}\]  
Here, the position trajectory of the target \(B\left[\tau-\frac{2\left(-z_{\text{p}}\cos(\psi)\right)}{c}\right]\) is constant; that is, the trajectory of the center point of the scene will not change with a position change in the SAR platform after the target range is compressed. Next, ignore the change in the scattering coefficient for target \(\sigma_{\text{p}}\). Except for the central point of the scene, the trajectories of all other target points have the characteristics of a sinusoidal curve change, and the change frequency is \(\frac{\varphi}{2\pi}\), which is consistent with the CSAR motion platform. The amplitude of oscillation is \(2 Br_{\text{p}}\sin(\psi)\), which is related to the bandwidth of the radar signal \(B\), the distance between the target point and the center of the scene \(r_{\text{p}}\), and the incident angle of the radar beam \(\psi\). When the CSAR observation geometry is determined, the amplitude of the sinusoidal change in the trajectory of a fixed-point target is only related to \(r_{\text{p}}\). The larger \(r_{\text{p}}\) is, the greater the amplitude of sinusoidal oscillation becomes. The initial phase is \(-\theta_{\text{p}}\), which is the inversion of the azimuth angle of the target point. When there are multiple targets located at different positions and azimuth angles in the scene, the range-compressed echoes form multiple sinusoidal curves that have the same frequency, different amplitudes, and different initial phases. These sinusoids overlap each other with the locus of the central point of the scene as the symmetrical center \(B\bigg{[}\tau-\frac{2\left(-z_{\mathrm{p}}\cos(\psi)\right)}{c}\bigg{]}\). By detecting the sinusoidal curve of range-compressed CSAR echo data along the azimuth direction, the position of the target in the scene can be extracted, and 2D imaging of the scene can be realized. IRT can accumulate and focus the sinusoidal curve in the image, transforming it into a point on the imaging plane [12].