# Research on Synthetic Aperture Radar Processing for the Spaceborne Sliding Spotlight Mode
## 4 Convolution Backprojection Algorithm for Sliding Spotlight SAR
### Fast CBP Algorithm Based on Image Segmentation  
The backprojection is a point to point image reconstruction process, which requires large amounts of interpolation operations resulting in huge computation [21]. Therefore, in this paper a fast implementation method based on Quadtree sub-image segmentation is discussed below. In the pre-processing phase of the CBP algorithm, match filtering and motion compensation are needed according to the center of the scene, which are equivalent to the two-dimension dechirp processing of the original echo signal, eliminating the second order phase of the signal. The echo signal of single target after the two-dimensional dechirp is as follows [21]:  
\[S_{PB}(t,f_{r})=\text{rect}\bigg{(}\frac{t}{T_{a}}\bigg{)}\cdot\text{rect} \bigg{(}\frac{f}{\gamma T_{r}}\bigg{)}\exp\bigg{[}\frac{4\pi}{c}f_{r}\big{(}R _{a}-R_{p}\big{)}\bigg{]}\cdot\exp\bigg{\{}i\frac{4\pi}{c}f_{c}\big{[}R_{a}(t )-R_{p}(t)\big{]}\bigg{\}}. \tag{21}\]  
The first exponential term is a nearly single frequency signal related to the distance from radar to the target, and the range profile can be obtained through Fourier transform with \(f_{r}\). Suppose \(w_{r}\) is the range scope of the scene, then the bandwidth in range domain after dechirp processing is:  
\[B_{r}=\frac{2w_{r}}{c}. \tag{22}\]  
The second exponential term is also a nearly single frequency signal related to the azimuth position of the target, and the azimuth profile can be obtained through Fourier transform with \(t\). Suppose \(w_{a}\) is the azimuth scope of the scene, then the bandwidth in azimuth domain after dechirp processing is [22]:  
\[B_{a}=\frac{2w_{a}v}{\lambda_{c}R_{ac}}. \tag{23}\]where \(\lambda_{c}\) is the wavelength, \(R_{ac}\) is the distance between the center point of the aperture and the center of the scene. In the CBP algorithm, if the scene decreases, the bandwidth in azimuth and range domain corresponding to the phase history will be reduced. Accordingly, the sampling rate can be reduced while increasing the sampling interval in azimuth and range domain to reduce the computation. Therefore, a CBP algorithm based on sub-image processing is used below. The schematic diagram is shown in Figure 6, and the specific process includes the following steps [23]:  
Step 1: Sub-image segmentation  
The segmentation of sub-image is based on the quadrant, and the whole image is divided into four sub-images according to the quadrant. The pixels are evenly distributed in the range and azimuth domain. For an image that includes \(N\times N\) pixels, corresponding to Figure 6, each sub-image should contain \(N/2\times N/2\) pixels. The scene is reduced to half of the original in both range and azimuth domain, so the sampling rate of data in range and azimuth domain can also be reduced to half in processing. Step 2: Filtering the original phase history, then down-sample in the spatial frequency domain. The filtering should be based on the range and the center point of each sub-image. The original phase history in the full scene is considered:  
\[S_{W}(t,f_{\tau})=\text{rect}\bigg{(}\frac{t}{T_{a}}\bigg{)}\cdot\iint\limits _{(x,y)\in W}g(x,y)\cdot\text{rect}\bigg{(}\frac{f_{\tau}}{B_{r}}\bigg{)} \cdot\exp\bigg{[}\text{j}\frac{4\pi}{c}(f_{c}+f_{\tau})(R_{a}-R_{t})\bigg{]} \text{dxdy}. \tag{24}\]  
where \(W\) represents the ground illuminated area, and \(g(x,y)\) represents the reflectivity of point target with the coordinate of \((x,y)\). The original phase history array size is \(N\times N\), and the sampling intervals in \((t,f_{\tau})\) domain are \(T_{0}\) and \(F_{s}\)/\(N\). So the discrete values of \(t\) and \(f_{\tau}\) are:  
\[t=-\frac{N}{2}\cdot T_{0}+m\cdot Tm=0,1,2,\ldots\ldots,N-1 \tag{25}\]  
\[f_{\tau}=-\frac{F_{s}}{2}+\frac{n\cdot F_{s}}{N}n=0,1,2,\ldots\ldots,N-1 \tag{26}\]  
A basic image can be obtained from this data array through the basic linear RD algorithm [23]. Except the center of the scene, the other points could have some defocus.