## 2 Designing Nominal SAR Operational Parameters at Nominal Altitude
### PRF Selection Considering Ambiguity  
PRF should be determined considering its impact on RAR and AAR, whose values are presented in Table 1. The RAR and AAR are calculated according to (1) and (2), respectively [8; 9]. \[\text{RAR}=\frac{\sum_{m\
eq 0}G^{2}\big{[}\theta\big{(}\tau+\frac{n}{PRF} \big{)}\big{]}\frac{\sigma_{0}(\Psi_{\text{a}})}{R_{\text{s}}^{3}\sin\Psi_{ \text{s}}}}{G^{2}[\theta\big{(}\tau)]\frac{\sigma_{0}(\Psi_{\text{b}})}{R_{ \text{s}}^{3}\sin\Psi_{\text{t}}}} \tag{1}\]  
\[\text{AAR}=\frac{\sum_{m\
eq 0}\int_{-\frac{n}{2}}^{+\frac{n}{2}}G^{2}(f+m \cdot PRF)df}{\int_{-\frac{n}{2}}^{+\frac{n}{2}}G^{2}(f)df} \tag{2}\]  
Table 2 shows symbols in Equations (1)-(2) and description of each symbol. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline Satellite altitude & 509–540 km \\ Incidence angle range & 20–55\({}^{\circ}\) \\ Ground swath width & \(\geq\)5 km \\ NESZ & \(\leq\)\(-\)20 dB \\ RAR & \(\leq\)\(-\)20 dB \\ AAR & \(\leq\)\(-\)20 dB \\ \hline \end{tabular}
\end{table} 
Table 1: Mission requirement for sliding spotlight mode. Figure 2: General SAR design flow and iteration process. Figure 3 shows the ambiguity-to-signal ratio (ASR) for an exemplary swath #5. Figure 3a shows positions where the RAR and AAR performances are investigated while Figure 3b shows the RAR and AAR performances at each position. The RAR and AAR performances are better at the center than at the edges. RAR is degraded at higher PRF, but AAR has sinusoidal-like dependency for PRF. The AAR performance shows different trends along the scene in the azimuth direction. Usually, the performance at the edge of one scene is worse than one at the center of that scene. This is because the grating lobes are raised due to an azimuth beam steering in the sliding spotlight mode. PRF should be chosen to minimize the RAR and AAR performances in both azimuth and range direction. As shown in Figure 3, the best PRF range is about 9 kHz between the dotted lines in grey. Important factors in PRF selection are the echo timing constraints and ASR. To achieve the incidence angle from 20 to 55\({}^{\circ}\), a total of 115 different swaths are assigned for the sliding spotlight mode. Instead of showing all the assigned swaths, only eight swaths are selected to present in this paper. The selected swaths have angle space of 5\({}^{\circ}\) between the adjacent swaths. For simulation, the satellite altitude, swath width, and Tx duty ratio are set to 520 km, 5.3 km, and 30%, respectively. Figure 4 shows the simulated RAR performances for each of the selected swaths. The RAR values correspond to the worst ones shown in Figure 3b. For all the swaths, the RAR performance becomes better for the increasing PRF when PRF is larger than a specific value. In order to satisfy \(-20\)-dB RAR in Table 1, the PRF range should be decreased for the increasing incidence angle, i.e., the increasing swath number SW#. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Description** \\ \hline \(\tau\) & Echo delay time projected into the cross-track ground plane \\ \(\theta\) & Elevation angle \\ \(\varphi\) & Azimuth angle \\ \(G^{2}\) & Two-way far field antenna power pattern \\ \(\sigma_{0}\) & Backscattering coefficient \\ \(\psi_{\mathrm{t}}\) & Incidence angle from target signal \\ \(\psi_{\mathrm{a}}\) & Incidence angle from ambiguous signal \\ \(R_{\mathrm{t}}\) & Slant range of target signal \\ \(R_{\mathrm{a}}\) & Slant range of ambiguous signal \\ \(B_{\mathrm{D}}\) & Target Doppler bandwidth \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Symbols and description for (1) and (2). Figure 3: RAR and AAR performances for PRF: (a) position of RAR and AAR performances; (b) exemplary ASR performance in SW#5. Figure 5 shows the simulated AAR performances. The AAR performances are the same for all the swaths, so AAR can be easily estimated for all the swath numbers, respectively, as long as AAR is known for one swath.