# Research on Synthetic Aperture Radar Processing for the Spaceborne Sliding Spotlight Mode
## 3 Ambiguity Elimination in Sliding Spotlight SAR  
In spaceborne SAR, the selection of the PRF is usually only about 1.2 times the Doppler bandwidth produced by the antenna illumination range [17]. The relationship between the Doppler bandwidth required in the SAR and the resolution is as follows:  
\[B_{\min}=\frac{V_{r}}{\rho_{a}} \tag{12}\]  
In the spaceborne sliding spotlight SAR, since the PRF selection is limited by many conditions, the PRF is usually only slightly larger than the instantaneous Doppler bandwidth, and is much smaller than the entire signal Doppler bandwidth. The Doppler ambiguity phenomenon exists in the echos, so it is necessary to study the solution of eliminating ambiguity in sliding spotlight SAR. This section uses the dechirp operation to resolve the Doppler ambiguity [18]. The reference function of the SPECAN operation is selected to be related to the frequency of the distance. The difference in the SPENCAN operation results in different operations of phase compensation and interpolation. The specific algorithm flow is shown in Figure 4.  
The biggest difference between sliding spotlight SAR and spotlight SAR is that the imaging area of sliding spotlight SAR is larger than the radiation area of the antenna. Therefore, if we do the dechirp operation at the center of scene, the Doppler bandwidth is still greater than the pulse repetition frequency, so the azimuth data need to be divided into several sub-apertures, each sub-aperture can have a certain overlap [19]. In the analysis of algorithm, we only consider the oblique plane imaging. In sliding spotlight mode, the echo signal can be written as follows:  
\[S(\omega,u_{mohu})=\sum\sigma(x_{n},r_{n})\exp\biggl{(}-j\frac{2(\omega+\omega _{c})}{c}\sqrt{\left(x_{n}-u_{mo}\right)^{2}+\left(R_{c}+r_{n}\right)^{2}} \biggr{)}. \tag{13}\]  
In the above formula, \(R_{c}\) is the satellite's distance to the center of the scene, \(u_{mo}\) is the position of the satellite at each moment. Use the following signal forms to dechirp data:  
\[S_{de}(\omega,u)=\exp\biggl{(}j\frac{2(\omega+\omega_{c})}{c}\sqrt{\left(u_{ a}\right)^{2}+\left(R_{c}\right)^{2}}\biggr{)}. \tag{14}\]  
The echo after dechirp can be expressed as:  
\[S_{1}(\omega,n)=\sum_{i=1}^{N}\sigma(x_{n},r_{n})\exp\left(-j\frac{2(\omega+ \omega_{c})}{c}\left(\frac{R_{c}r_{n}-n\cdot\Delta u_{a}x_{n}}{\sqrt{n^{2} \cdot\left(\Delta u_{a}\right)^{2}+R_{c}^{2}}}+\xi(n\cdot\Delta u_{a},x_{n},r_{n})\right)\right). \tag{15}\]  
\(\Delta u_{mohu}=V_{r}/PRF\), \(\xi(n\cdot\Delta u_{mohu},x_{n},r_{n})\) is residual error term based on the plane wave assumption. The Doppler bandwidth of the signal is only determined by the scene-induced Doppler bandwidth. That is to say, if the scene-induced Doppler bandwidth in the sub-aperture is more than the pulse repetition rate, then we can use the interpolation method to increase PRF. In the interpolation process, we can perform the Fourier transform on the above equation, and then perform the inverse Fourier  
Figure 4: Flow chart of ambiguity elimination. transform after the zero-padding, which is equivalent to interpolating the above equation. The number of zeroes is determined by the number of ambiguity. After interpolation, the signal can be expressed as:  
\[S_{1}(\omega,i)=\sum_{i=1}^{N}\sigma(x_{n},r_{n})\exp\left(-j\frac{2(\omega+ \omega_{c})}{c}\left(\frac{R_{c}r_{n}-i\cdot\Delta ux_{n}}{\sqrt{i^{2}\cdot\left( \Delta u\right)^{2}+R_{c}^{2}}}+\xi(i\cdot\Delta u,x_{n},r_{n})\right)\right). \tag{16}\]  
In order to recover the echo signal without ambiguity, inverse dechirp processing also needs to be performed.