## 6 SAR Payload Performance Analysis
### Range Ambiguity Ratio  
Ambiguity is measured by the ambiguity-to-signal ratio of the SAR, i.e., the ratio of the ambiguity power to the power of the useful signals [2; 3; 4]. Range ambiguity refers to echoes occurring outside the swath, and it enters the radar receiver along with the useful  
Figure 26: Impulse response function analysis in the (**a**) range and (**b**) azimuth of the S-STEP SAR payload in the wide-swath mode. The red circles indicate the peak side lobes.  
Figure 25: Impulse response function analysis in the (**a**) range and (**b**) azimuth of the S-STEP SAR payload in the high-resolution mode. The red circles indicate the peak side lobes.  
echoes inside the swath, which deteriorates the radar image quality after signal processing. The range ambiguity-to-signal ratio (RASR or RAR) can be expressed as [45]  
\[\text{RASR}=\frac{\sum_{n=1}^{N_{\text{s}}}S_{a,n}}{\sum_{n=1}^{N_{\text{s}}}S_{ n}}, \tag{10}\]  
where \(N_{\text{s}}\) denotes the total number of samples within the Rx echo window, \(S_{a,n}\) represents the ambiguity power of the \(n\)-th sampling point within the Rx echo window.  
\[S_{a,n}=\sum_{\begin{subarray}{c}k\to N_{p}\\ k\
eq 0\end{subarray}}^{N_{p}}\frac{\sigma^{0}(\theta_{i}(n,k))G_{\text{r}}( \theta_{i}(n,k))G_{\text{r}}(\theta_{i}(n,k))}{R_{\text{s}}^{3}(\theta_{i}(n,k ))\sin(\theta_{i}(n,k))}, \tag{11}\]  
where \(k\) indicates the range ambiguity number index (\(k\
eq 0\)), \(N_{p}\) denotes the total number of pulses within the radar horizon, \(S_{n}\) represents the signal power of the \(n\)-th sampling point inside the echo receiving window, \(\theta_{i}(n,k)\) denotes the angle of the \(n\)-th sampling point and \(k\)-th range ambiguity, and \(R_{s}\) reflects the slant range from the antenna to the range ambiguity.  
\[S_{n}=\frac{\sigma^{0}(\theta_{i}(n,k))G_{\text{r}}(\theta_{i}(n,k))G_{\text{r }}(\theta_{i}(n,k))}{R_{\text{s}}^{3}(\theta_{i}(n,k))\sin(\theta_{i}(n,k))}, \tag{12}\]  
where \(\sigma^{0}\) denotes the backscattering coefficient of the surface clutter, \(G_{\text{r}}\) reflects the transmit antenna pattern at elevation, \(G_{\text{r}}\) indicates the receive antenna pattern at elevation, and \(k=0\).  
The RAR was obtained using the average backscattering model for the land clutter by [PERSON] [46]. The geometry and antenna gain of the range ambiguities at an incidence angle of \(25^{\circ}\) are portrayed in Figure 27a,b, respectively. The forward and backward range ambiguity points can be obtained by evaluating the slant range corresponding to the multiples of PRI selected for the specific antenna beam [47].  
Figure 27: (**a**) Forward range ambiguities (blue), backward range ambiguities (red), and antenna beam boresight (green), and (**b**) range ambiguities in the elevation angle.  
The RARs at incidence angles of 15 and 25\({}^{\circ}\) are illustrated in Figure 28a,b, respectively. As observed in Figure 28, the RAR deteriorates with the increasing PRF. The RAR according to the incidence angle in high-resolution and wide-swath modes is presented in Figure 29a,b, respectively. As observed from Figure 29, the RAR is satisfied for all operating incidence angle intervals of the selected PRFs in the high-resolution and wide-swath modes.  
Figure 28: Range ambiguity ratio vs. PRF at an incidence angle of (**a**) 15 and (**b**) 25\({}^{\circ}\).  
Figure 29: Range ambiguity ratio vs. incidence angle in (**a**) high-resolution and (**b**) wide-swath modes. RAR was lower than the required level of \(-17\) dB at the incidence angle of 25\({}^{\circ}\). Each color shows the corresponding swath.