# A Novel Spaceborne Sliding Spotlight Range Sweep Synthetic Aperture Radar: System and Imaging
## 2 Continuous Varying Pulse Interval Technique  
Similar to the discussion for the SS-RSSAR in [17], the CBP of the SSS-RSSAR is assumed to point vertically to the ROI strip as shown in Figure 2, where \(m\) denotes the azimuth sampling index and \(m=0\) denotes the central data acquisition time. The antenna should be continuously manipulated in two dimensions to make sure the ground beam center always moves along the X-axis. A Cartesian coordinate is built to simplify the discussions. The origin \(O\) overlaps with the swath center. The X-axis is along the along-BIS direction. The Y-axis is along the cross-BIS direction in the ground plane, corresponding to the projection of CBSR at \(m=0\). The Z-axis is vertical to the ground plane and points inversely to the Earth center. The directions along the X- and Y-axis are defined as the azimuth and range directions, respectively. Other parameters are defined as follows. \(H\) denotes the constant height of the SSS-RSSAR. \(\theta\) denotes the tile angle between the X-axis and the nadir of the satellite tract. \(\beta\) denotes the instant look angle with respect to the swath center \(O\). \(\alpha\) denotes the instant azimuth angle between the Y-axis and the projection of the instant slant range from the SAR antenna center to the swath center \(O\). \(\beta_{\rm c}\) and \(R_{\rm c0}\) denote the central look angle and the central slant range with respect to the swath center at \(m=0\). \(p\) and \(k\) denote two azimuth sampling indexes before and after the central data acquisition time. The CBSR at \(m=p\) and \(m=k\) are \(R_{\rm cp}\) and \(R_{\rm ck}\), respectively. \(W_{\rm r}\) and \(W_{\rm a}\) denote the expected range and azimuth BIS swaths. The real curved satellite orbit is assumed to be rectilinear here for simplicity. \(V_{\rm s}\) and \(V_{\rm g}\) denote the satellite velocity and the beam velocity along the X-axis, respectively. They have the following relationship  
\[V_{\rm g}=\eta V_{\rm s}\cos\theta \tag{1}\]  
where \(\eta\) denotes the sliding spotlight factor. The SS-RSSAR mode discussed in [17] corresponds to the very case with \(\eta=1\). If a traditional constant PI was applied for the SSS-RSSAR, the continuous range-azimuth beam sweeping would lead to slant range variation and inevitably cause the transmission blockage. To solve this problem, a new CVPI technique is developed for the full echo receiving by non-uniformly adjusting the PIs based on the instant data acquisition geometry. Unlike the Multi-Interval Sampling (MIS) technique used by the SS-RSSAR [17], the CVPI technique continuously varies the PI sequence by employing a polynomial fitting method. Furthermore, unlike the MIS technique that changes the number of PI between the transmitting and receiving, the CVPI technique keeps this number unchanged during the entire DAP. Thus, the CVPI-based non-uniform azimuth sampling array will be easier to design and generate. The key idea of the CVPI method is to locate the echo of the targets along X-axis to the center of every PI. In this case, all the echoes from the ROI can be set around the central part of every PI. In other words, the expected PI variation should counteract the CBSR variation within the DAP. For the three sampling positions with the CBSRs \(R_{\rm cp}\), \(R_{\rm c0}\) and \(R_{\rm ck}\) in Figure 2, the expected CVPI-based echo  
Figure 2: SSS-RSSAR data acquisition geometry. distribution is shown in Figure 3, where, despite the differences of the \(R_{\text{cp}}\), \(R_{\text{c0}}\) and \(R_{\text{ck}}\), all the echoes are all located at the central part of their individual PIs. A fifth-order polynomial of the azimuth time is used for the PI variation description. In this case, the \(m\)th PI can be expressed by  
\[PI(m)=\sum_{n=0}^{5}\mu_{\text{n}}n^{n}(m) \tag{2}\]  
where \(\mu_{\text{n}}\) denotes the polynomial coefficient and \(t\) denotes the azimuth time. A referential PI, denoted as \(PI_{0}\), should be set firstly to initiate the non-uniform PI array generation.