# High-Temporal-Resolution High-Spatial-Resolution Spaceborne SAR Based on Continuously Varying PRF
## 3 Design of the Sequence of PRF  
As mentioned, the sequence of the continuously varying PRFs is designed according to the history of the slant range to keep the location of the echo data unchanged in the receive window. Consider a case in which the reflected pulse is received at the \(M\)-th pulse interval (PI) after being transmitted. To facilitate the calculation of \(PI_{k+M-2}\), it will be necessary to assume that the pulse intervals from 1 to \(M\)-1 are equal to \(PI_{1}\) at the beginning of the calculation.  
As shown in Figure 10, the \(k\)-th pulse is transmitted and received at \(t_{k\_trans}\) and \(t_{k\_reci}\), respectively. \(PI_{k+M-2}\) is the unknown variable that we need to solve based on the preceding pulse intervals \(PI_{k}\sim P_{k+M-3}\). Then, the transmit time of the \(k\)-th pulse and the receive time of the \(k\)-th pulse can be expressed as  
\[t_{k\_trans}=\sum_{i=1}^{k-1}PI_{i} \tag{5}\]  
\[t_{k\_reci}=t_{k\_trans}+\sum_{i=k}^{k+M-3}PI_{i}+PI_{k+M-2}+\tau+t_{prot} \tag{6}\]  
Figure 9: Comparison of the uniform PRF and the CVPRF system with a diagram of the transmitted and received pulses. Different color represents different pulse. The purple box indicates the receive window, and the colored feature in purple box is the echo for its corresponding transmitted pulse. (**a**–**c**) represent the varying position of the echo data for the uniform PRF system. For the sliding spotlight mode, the slant range decreases with decreasing squint illumination angle, and the position of the echo data shifts closer to the beginning of the receive window, eventually running out of the receive window; For the CVPRF system in (**d**), the pulse interval decreases for each new pulse, which is equivalent to a corresponding movement of the receive window, and ensures a proper position for the echo data.  
where \(k\_trans\) denotes the \(k\)-th transmitted pulse and \(k\_recei\) denotes the receiving of the \(k\)-th transmitted pulse. \(Pl_{k+M-2}\) is the variable that we need to determine.  
Using the equation as follows, we can obtain the optimal value of \(Pl_{k+M-2}\) by an iterative calculation. Both \(R(t_{k\_trans})\) and \(R(t_{k\_reci})\) can be accurately acquired via a high-precision simulation.  
\[(t_{k\_reci}-t_{k\_trans})\cdot c=R(t_{k\_trans})+R(t_{k\_reci}) \tag{7}\]  
Figure 11 shows the result from a simulation performed with the orbit and radar parameters listed in Table 2. The pulse frequency varies from 3535 Hz to 3714 Hz with decreasing squint angle from 23.5\({}^{\circ}\) to 17.3\({}^{\circ}\).  
To verify the ability to acquire high-resolution and wide-swath SAR images, we study a simulation of the uniform PRF system and the CVPRF system. The simulation scene is shown in Figure 12a, where the distances of the different targets are 5 km in the azimuth and range direction. The echo data of the uniform PRF system and those of the CVPRF system are shown in Figure 12b,c. It can be observed that the echo data of the uniform PRF system cannot be acquired, as the range-walk proliferates at high squint angles. In contrast, the CVPRF system not only can record the echo data of a high squint angle but also has the potential to image larger swaths. The simulation results validate the analysis in Section 2.  
Under ideal conditions, the PRFs change continuously. For a compromise and practical strategy, PRFs would change discretely according to the precision of hardware. According to Figure 11, if the step size of PRF is limited to 1 Hz, the PRFs would vary about every 250 pulses and the echo data is shown in Figure 12d. In addition, the echo data with the step size of PRF up to 2 Hz is also shown in Figure 12e.  
Figure 11: A sequence of a pulse frequency for the CVPRF system. The frequency of the first pulse is 3535 Hz and there are 44,584 pulses during the squint angle changes from 23.5\({}^{\circ}\) to 17.3\({}^{\circ}\).  
Figure 10: Illustration of the transmit and receive of the \(k\)-th pulse.