# High-Temporal-Resolution High-Spatial-Resolution Spaceborne SAR Based on Continuously Varying PRF
## 2 CVPRF Concept  
For spaceborne SAR systems, the antenna length imposes a lower bound on the selected PRF [18]. In return, the PRF limits its ability to continuously acquire echo data in the slant range. As indicated in Figure 3, the span of the slant range \(\Delta R_{slant}\), which is related to the length of the receive window, can be expressed as  
\[\Delta R_{slant}\leq\left(\frac{1}{PRF}-2\tau-2t_{prot}\right)\cdot\frac{c}{2} \tag{1}\]  
where \(c\) is the speed of light, \(\tau\) is the transmitted pulse duration, and \(t_{prot}\) is the guard interval, which is usually equal to \(\tau\). The span of the slant range consists of two parts: \(\Delta R_{GS}\) (equivalent to \(\frac{t_{GS}\cdot c}{2}\)), caused by the ground swath in the range direction, as illustrated on the left of Figure 6, and \(\Delta R_{squi}\) (equivalent to \(\frac{t_{SGM}\cdot c}{2}\)), caused by the squint illumination, as illustrated on the right of Figure 6. The span of the slant range caused by the ground swath can be estimated by  
\[\Delta R_{GS}\approx SW\cdot\frac{R_{e}+H}{R_{e}}\cdot\sin\theta \tag{2}\]  
where \(SW\) is the ground swath, \(R_{e}\) is the Earth's radius, \(H\) is the orbit height, and \(\theta\) is the look angle. For convenience of presentation, the path of the sensor is represented by a linear model. Of course, we need to note that the actual path is substantially more complicated for spaceborne SAR systems. Thus, the span of the slant range caused by the squint illumination can also be estimated by  
\[\Delta R_{squi}\approx\frac{R_{0}}{\cos\varphi}-\frac{R_{0}}{\cos\beta} \tag{3}\]  
where \(R_{0}\) is the slant range at the Doppler center time, \(\varphi\) is the start illumination angle, and \(\beta\) is the end illumination angle. Figure 5: Timing diagram of Staggered-SAR, also called sweep SAR. Both red and gray color represents for beams of Staggered-SAR. The red beam means pulses of the current PRFs is in conflict with transmitted pulse blockage and cannot be received. However, pulses of this slant range still can be received at other PRFs. With Equations ( 2) and ( 3), Equation ( 1) can be expressed as  
\[\left(SW\cdot\frac{R_{e}+H}{R_{e}}\cdot\sin\theta\right)+\left(\frac{R_{0}}{\cos \varphi}-\frac{R_{0}}{\cos\beta}\right)\leq\left(\frac{1}{PRF}-2\tau-2t_{prot} \right)\cdot\frac{c}{2} \tag{4}\]  
With the given orbit height, look angle, PRF and transmitted pulse duration, a compromise between ground swath and azimuth squint angle should be made. Since the antenna beam is always orthogonal to the flight direction for conventional stripmap SAR systems, the span of the slant range caused by squint illumination is negligible. However, for the sliding spotlight mode, apart from the factor of the ground range, the squint illumination angle must be considered. To obtain a more intuitive description of the factor \(\Delta R_{squit}\), simulations are performed. The slant range history of the sliding spotlight mode is illustrated in Figure 7. The dark line represents the slant range history of the center target of the scene, similar to the conventional stripmap mode. With increasing squint angle, the spans of the slant range increase almost exponentially. Moreover, the distance between Point 1 and Point 3 is 5 km in azimuth, and the \(\Delta R_{squit}\) for a 5 km azimuth swath is 35.0 km. Figure 6: Spaceborne SAR geometry: side-looking geometry (**left**) and along-track geometry (**right**). Figure 7: Slant range history of the sliding spotlight SAR. Lines of different color indicate different targets, with a distance of 2.5 km in azimuth, and each line records the slant range history of a target with a 0.25 m resolution; the dark line is the center target of the scene. The distance between Point 1 and Point 3 is 5 km in azimuth; hence, the span of the squint angle for a 5-km azimuth swath is 17.3\({}^{\circ}\)\(\sim\)23.5\({}^{\circ}\). The comparison of the acquisition capability for a Uniform PRF system and a CVPRF system in the timing diagram would be necessary for a straightforward explanation.