# Interpolation Methods with Phase Control for Backprojection of Complex-Valued SAR Data +
## 6 Conclusions  
In this paper, the existing linear, cubic, and sinc interpolation algorithms have been extended and adapted to process complex SAR data. The extended algorithms include the phase control procedure that relates the phase value of interpolated complex SAR data to the given range-time sample. The proposed phase control procedure (6) has increased computational costs of linear, cubic, and sinc interpolators additionally by 14, 21, and \(7(2L+1)\) operations per each iteration of the global backprojection algorithm, respectively, where \(L\) denotes the summation limit in the truncated sinc kernel of the extended sinc interpolator. The developed interpolation approaches are incorporated into the GBP algorithm, and their efficiency is tested at THz frequencies. Similarly, the interpolation methods can be incorporated into other backprojection algorithms, such as fast and fast factorized backprojections. It can be concluded that the use of the phase control procedure (6) provides the opportunity to achieve accurate image reconstruction results both in azimuth and range directions, even at the Nyquist sampling rate. At the same time, to reach an approximately similar level of reconstruction accuracy with the nearest neighbor approach, sixteen times higher sampling  
Figure 8: Reconstructed SAR scenes \(h\) of \(251\times 251\) pixels with nearest neighbor, linear, cubic, and sinc interpolations: (**a–d**) without the phase-control procedure; (**e–g**) with the phase control procedure. Here, the sampling rate \(f_{\rm s}=f_{\rm max}=0.33\,\rm THz\).  
Figure 9: SAR scenes \(h\) of \(251\times 251\) pixels reconstructed with: (**a**) nearest neighbor interpolation for the sampling rate \(f_{\rm s}=16f_{\rm max}=5.28\,\rm THz\); (**b**) sinc interpolation for the sampling rate \(f_{\rm s}=f_{\rm max}=0.33\,\rm THz\).  
rate, \(f_{\rm s}=16f_{\rm max}\), is required. It has been investigated that for such a sampling rate and at THz frequencies, the computational complexity of the nearest neighbor interpolator becomes much higher than the computational complexity of the extended interpolation algorithms at the Nyquist rate. Correspondingly, the amount of raw data needed in the image reconstruction procedure can be significantly reduced, which is of great importance in the SAR hardware realization, especially at THz frequencies.  
Conceptualization, Y.I. and V.T.V.; methodology, Y.I.; software, Y.I.; validation, Y.I.; formal analysis, Y.I.; investigation, Y.I.; resources, A.B.; data curation, Y.I.; writing--original draft preparation, Y.I.; writing--review and editing, V.T.V., A.B., M.I.P., T.K.; visualization, Y.I.; supervision, M.I.P., V.T.V.; project administration, M.I.P.; funding acquisition, M.I.P., T.K. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the ELILITY research environment under the project Multistatic High-resolution Sensing at THz, Project-ID A17, and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project-ID 287022738 TRR 196 for associated project S05.  
Not applicable.  
The data presented in this study are available on request from [PERSON].  
The authors declare no conflict of interest.