## 2 Designing Nominal SAR Operational Parameters at Nominal Altitude
### PRF Selection and Timing  
A swath should be allocated in such a way that each pulse repetition interval (PRI) is wide enough to receive all echoes returning from the swath on the ground. For the Rx sampling, the total window length should be wider than the sum of all the differences of the near and far slant ranges with the Rx chirp duration, which is equal to the Tx chirp duration. As shown in Figure 7, this criterion is used to determine the allocable area for the Rx sampling after setting PRI and Tx pulse duration, \(\tau_{\text{p}}\), respectively. Rank in Figure 7 means the number of pulses within the delay time that is experienced by echo signal from the near slant range.  
Like in a general radar system, the SAR receiver is in a blind stage during the transmitting event due to weak isolation between the transmitter and receiver. When the returning echo coincides with the transmitting event, the received signal is eclipsed. PRF should be set to avoid the eclipsing. For this, PRF must satisfy the following inequality.  
\[\frac{N-1}{\tau_{\text{near}}-\tau_{\text{p}}-\tau_{\text{TxRxGuard}}}<PRF< \frac{N}{\tau_{\text{far}}+\tau_{\text{p}}+\tau_{\text{RxTxGuard}}}\ (N=1,\ 2,\ 3,\ \ldots) \tag{3}\]  
As shown in Figure 7, \(\tau_{\text{near}}\) and \(\tau_{\text{far}}\) represent the respective return time for the slant range, which is defined as a time interval between the near and far edges in one swath. Also, \(\tau_{\text{p}}\) is the transmitter pulse length, and \(\tau_{\text{TxRxGuard}}\) and \(\tau_{\text{RxTxGuard}}\) are the guard times, respectively, between Tx and Rx and vice versa. The available sampling time is limited by hardware constraints. Examples of such constraints are the commanding transfer time between the SAR controller unit and the designated control module in the RF unit, activation time of the RF pulse gate for the Standardized Modular Transmit/Receive-Module (SMRR-Module) in the RF unit, and delay of the timing signal from the controller to the SMRR-Module. The hardware constraints are considered by setting the guard times.  
An echo resulting from nadir exists for every transmitted pulse, so every echo occurs at time \(\tau_{\text{nadir}}\), which is defined as follows.  
\[\tau_{\text{nadir}}=\frac{2h}{c} \tag{4}\]  
Here, \(h\) is the altitude over the nadir point. Like the transmitter pulse eclipsing, PRFs should be determined to avoid coincidence between nadir echoes and the scene echo, so the following inequality should be satisfied.  
\[\frac{M-1}{\tau_{\text{near}}-\tau_{\text{p}}-\tau_{\text{nadir}}-\tau_{ \text{TxRxGuard}}}<PRF<\frac{M}{\tau_{\text{far}}+\tau_{\text{p}}+\tau_{\text {nadir}}+\tau_{\text{RxTxGuard}}}\ (M=1,\ 2,\ 3,\ \ldots) \tag{5}\]  
PRF depends on the return times, \(\tau_{\text{near}}\) and \(\tau_{\text{far}}\), which correspond to time at each slant, respectively. Also, it depends on a length of the transmitting pulse, \(\tau_{\text{p}}\), and the altitude of the radar over the nadir point. The swath design and the PRF determination  
Figure 7: Allocable swath area.  
should be validated using a PRF diamond diagram. As an example, Figure 8 shows the PRF determination for SW#5. The optimum PRF is determined by comparing Figure 8 with Figure 3: the PRF zone is determined to satisfy the requirement on the RAR and AAR performances. As shown in Figure 8b, this optimization procedure results in PRF at 8956.5 Hz.