## 5 Results of Testing the Omega-k Algorithm
### Simulated SAR Raw Data  
Figure 10 shows the simulation of the synthetic aperture radar geometry used to simulate SAR raw data with the main parameters of Table 1 and Equation (17). Additionally, Figure 10 shows a magenta circle located at a certain height that represents the SAR platform. On the other hand, the green triangle represents the antenna footprint that once contained the black asterisk, which is an isolated and isotropic white dot, and it generatesa chirp signal. Otherwise, if the footprint does not contain the black asterisk, it does not generate a chirp signal.  
Table 1 shows the initial parameters used to simulate a synthetic aperture radar that transmits a chirp signal or an electromagnetic pulse signal with a linear frequency modulation, as seen in Figure 10. This SAR system operates in the C band, for which the center frequency of the chirp signal is 5 \(\mathrm{GH_{z}}\), and the bandwidth of 150 \(\mathrm{MH_{z}}\) determines the distance resolution.  
Figure 11 shows the magnitude plot of the SAR raw data, because this data have elements that are complex values; therefore, Figure 12 shows the SAR raw data phase of an isolated and isotropic point target that serves to test the proposed Omega-k algorithm and test the approximate quality of the SAR images.  
In Figure 12a, it is observed that the SAR raw data phase has the shape of a hyperbola and it indicates that the phase of the transmitted signal has a positive slope or is also known as Up-Chirp. On the other hand, in Figure 12b, the the SAR raw data phase have the shape of an ellipse, which indicates that the derivative of the phase of the transmitted signal has a negative slope and is known as Down-Chirp.  
Figure 13 shows a SAR image of a point target located in the center of the image with the azimuth and range directions generated from the simulated SAR raw data shown in Figure 11.  
The SAR image in Figure 13 has pixels that are complex values; therefore, the image shown in said figure is the magnitude of the SAR image demonstrating the correct operation of the modified Omega-k algorithm for the SAR system (pulsed), and simulating the SAR raw data using the proposed Omega-k algorithm is validated.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Main Parameters** & **Symbol** & **Values** \\ \hline Carrier frequency & \(f_{0}\) & 5.3 \(\mathrm{GH_{z}}\) \\ Bandwidth & \(B\) & 150 \(\mathrm{MH_{z}}\) \\ Pulse repetition rate & \(PRF\) & 100 \(\mathrm{H_{z}}\) \\ Pulse width & \(T_{p}\) & 5 \(\mathrm{\mu s}\) \\ Horizontal beam width & \(\theta_{H}\) & 7\(\mathrm{\SIUnitSymbolDegree}\) \\ SAR platform speed & \(V_{s}\) & 15 \(\mathrm{m/s}\) \\ Speed of light & \(c\) & \(3\times 10^{8}\) \(\mathrm{m/s}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: SAR system parameters (pulsed).  
Figure 10: SAR system (pulsed).  
On the other hand, Table 2 shows the approximate values of the parameters of the SAR images of a point target by making a cut in the azimuth and range at the point or pixel of the maximum intensity of the SAR image shown in Figure 13. The quality parameters of the Omega-k algorithm are shown in Table 2, for which the values of the quality parameters are acceptable. These approximate parameters were calculated by simulating the antenna  
Figure 11: SAR raw data magnitude.  
Figure 12: This Figure shows: (**a**) SAR raw data phase (Up-Chirp). (**b**) SAR raw data phase (Down-Chirp).  
Figure 13: SAR image of a point target.  
footprint with the parameters of Table 1 along the azimuth and a point target. These quality parameters vary depending on the dimensions of the antenna footprint and observed scene.