## 5 Results of Testing the Omega-k Algorithm
### Real SAR Raw Data and Discussion  
Figure 14a shows the magnitude of the actual SAR raw data acquired by the Active Microwave Instrument that is installed on the ERS-2 satellite. Figure 14b shows the SAR image obtained with the unmodified Omega-k algorithm, also called precise version algorithm; meanwhile, in Figure 14c, the SAR image obtained using the range-Doppler algorithm is observed. The focusing algorithm used by the ERS-2 system is the range-Doppler algorithm (SAR Training Processor v1.1). On the other hand, comparing Figure 14b,c it is observed that they are similar; however, some parts of the surface are better observed in the SAR image using the Omega-k algorithm.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Values** \\ \hline Range Resolution & \(\delta_{R}\) & 0.81 m \\ Azimuth Resolution & \(\delta_{A}\) & 0.204 m \\ \hline Peak Side Lobe Ratio & \(PSLR_{A}\) & \(-\)7.05 dB \\ \(PSLR_{R}\) & \(-\)10.57 dB \\ \hline Integrated Side Lobe Ratio & \(ISLR_{A}\) & \(-\)4.31 dB \\ \(ISLR_{R}\) & \(-\)2.74 dB \\ \hline \hline \end{tabular}
\end{table}
Table 2: Omega-k algorithm quality parameters (SAR system).  
Figure 14: (**a**) Real SAR raw data obtained by the European Remote Sensing satellite (ERS-2) through the Active Microwave Instrument (AMI), known as the SAR sensor [24]. (**b**) Real SAR image obtained with the Omega-k algorithm [19; 24]. (**c**) Real SAR image obtained with the range-Doppler algorithm (SAR Training Processor v1.1) [24].  
The SAR raw data come from the surface of Lake Tticaca in Bolivia, a photograph of which is shown in Figure 15a [26] and the magnitude of SAR raw data is shown in Figure 15b [26]. These raw data were acquired by the European Remote Sensing satellite (ERS-2) through its active sensor, known as the synthetic aperture radar with pulsed linear frequency modulation (LFM), whose SAR system's main parameters are shown in Table 3. then, the SAR image generated with the proposed Omega-k algorithm is shown in Figure 16, and it is observed that the SAR images of Figures 14c and 16 are similar; furthermore, Figure 17 shows three samples obtained from Figures 14b,c and 16, that is, images obtained with the Omega-k algorithm, range-Doppler algorithm, and the proposed Omega-k algorithm. A selected area with a blue outline is seen clearer and sharper in the SAR image obtained with the proposed Omega-k algorithm compared with the range-Doppler algorithm and similar to the Omega-k algorithm; therefore, the proposed Omega-k algorithm works correctly. On the other hand, it was also tested using other real SAR raw data captured by the ERS-2 satellite, and the result of the modified Omega-k algorithm focusing is shown in Figure 15c, where a SAR image of a part of Lake Tticaca located in South America is observed. One limitation is the lack of SAR raw data obtained in air or space vehicles with high speeds or hypersonic speeds, so it would be advisable to test this proposed Omega-k algorithm on such SAR raw data to more effectively evaluate its performance.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Main Parameters** & **Symbol** & **Values** \\ \hline Carrier frequency & \(f_{0}\) & \(5.3\,\mathrm{GH}_{z}\) \\ Bandwidth & \(B\) & \(15.5\,\mathrm{MH}_{z}\) \\ Pulse repetition rate & \(PRF\) & \(1679.90233438\,\mathrm{H}_{z}\) \\ Sample rate & \(f_{s}\) & \(18.86\,\mathrm{MH}_{z}\) \\ chirp modulation rate & \(k_{r}\) & \(4.17788\times 10^{11}\,\mathrm{H}_{z}/\mathrm{s}\) \\ Angle of incidence & \(\theta_{H}\) & \(23^{\circ}\) \\ SAR platform speed & \(V_{s}\) & \(7543.41\,\mathrm{m}/\mathrm{s}\) \\ Speed of light & \(c\) & \(3\times 10^{8}\,\mathrm{m}/\mathrm{s}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: SAR ERS-2 parameters.  
Figure 15: (a) Geographic location in ASF Data Search Vertex where the SAR raw data [27] is located. (b) Magnitude of SAR raw data. (c) SAR image obtained with the modified Omega-k algorithm.