## 6 Conclusions  
The SAR and LFM-CW SAR raw signals were transformed by adding the Doppler factor using the two-dimensional Fourier transform from the time domain to the two-dimensional Fourier domain, then the Omega-k algorithm was modified by adding the Doppler factor for continuous and pulsed wave synthetic aperture radars.  
The modified Omega-k algorithm for SAR (pulsed) systems was validated using simulated SAR raw data from a point target; additionally, this modified Omega-k algorithm was validated with real SAR raw data that were obtained by the ERS-2 satellite through its SAR radar (pulsed). On the other hand, in the modified Omega-k algorithm for LFM-CW SAR systems, in addition to adding the Doppler factor, the SAR system aerial platform movement compensation was added. This Omega-k algorithm was validated with simu  
Figure 23: Noise reduction in LFM-CW SAR image obtained with the proposed Omega-k algorithm.  
Figure 22: (a) LFM-CW SAR image obtained with the proposed Omega-k algorithm. (b) Georectified LFM-CW SAR image obtained with the proposed Omega-k algorithm.  
lated SAR raw data and real SAR raw data obtained by the MicroASAR system that was installed on board an unmanned aerial vehicle.  
SAR images generated by the modified Omega-k algorithm for continuous wave and pulsed SAR systems have pixels with complex values and have remote observation information of the surface of interest. These SAR images had range and azimuth directions, and these SAR images obtained with the range-Doppler algorithm and the modified Omega-k algorithm were compared, obtaining similar results and thus validating these proposed Omega-k algorithms. On the other hand, the modified Omega-k algorithm for the LFM-CW SAR system showed better results because of residual video phase elimination, and Doppler factor and motion compensation were added.  
As a contribution to the scientific community, the Doppler factor was successfully added to the Omega-k algorithm for the pulsed SAR system, generating good quality SAR images. For future work, the application of the Omega-k algorithm is recommended with the addition of the Doppler factor for bistatic and circular synthetic aperture radars. It is also recommended that the effects of adding the Doppler factor in the range-Doppler algorithm and the chirp scaling algorithm in the approach of SAR raw data obtained on SAR platforms with high velocities are examined.  
conceptualization, methodology, and algorithm, J.J.-C. and F.P.-Q.; validation and formal analysis, J.J.-C., F.P.-Q., R.J.C.-C. and R.F.; resources, F.P.-Q., R.J.C.-C. and J.C.H.-L.; data curation, J.J.-C.; writing--original draft preparation, J.J.-C. and F.P.-Q.; writing--review and editing, J.J.-C. and R.F.; supervision, R.F. and F.P.-Q.; project administration, F.P.-Q.; funding acquisition, F.P.-Q., R.J.C.-C. and J.C.H.-L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The SAR raw data (Level Zero) used in this paper can be found in [[https://search.nsf.alaska.edu](https://search.nsf.alaska.edu)]([https://search.nsf.alaska.edu](https://search.nsf.alaska.edu)) or [[https://doi.org/10.5067/4](https://doi.org/10.5067/4) KVOYM7 NQ19L]([https://doi.org/10.5067/4](https://doi.org/10.5067/4) KVOYM7 NQ19L). More information about the LFM-CW SAR raw data can be found in [[https://api.semanticscholar.org/CorpusID:201822364](https://api.semanticscholar.org/CorpusID:201822364)]([https://api.semanticscholar.org/CorpusID:201822364](https://api.semanticscholar.org/CorpusID:201822364))  
The research was supported by the Institutional laboratory for research, entrepreneurship, and innovation in automatic control systems, automation, and robotics (LIECAR) at the University of San Antonio Abad del Cusco UNSAAC.  
The authors declare no conflict of interest.