# Interpolation Methods with Phase Control for Backprojection of Complex-Valued SAR Data +
## 4 Simulations
### Results
Nevertheless, the optimal length of the kernel has to be investigated. Figure 4a,b depict SAR-scene cuts reconstructed by the GBP algorithm that involves sinc interpolation, as well as SAR-scene cuts obtained based on the impulse response function (A1), which is defined in Appendix A, for \(\varphi=0\) and \(\varphi=\pi/2\), respectively. The scene cuts are plotted as functions of the normalized range \(\rho_{\rm n}\) (for \(\zeta_{\rm n}=0\)) and the normalized azimuth \(\zeta_{\rm n}\) (for \(\rho_{\rm n}=0\)), respectively, where the intensity is normalized with the peak intensity value. Here, the sampling rate \(f_{\rm s}=f_{\rm max}=0.33\,\rm THz\), the range and the azimuth are normalized similarly as in the example described above, and the kernel is truncated to the length \(2L+1\), where \(L\in\{4,6,8,10,12,14\}\). It has been observed that for \(L\geq 12\), the reconstruction results converge both for range and azimuth SAR-scene cuts. Furthermore, the results have a good agreement with the solution based on the approach (A1) in range and azimuth directions. It should be noted that intensity deviation in the third sidelobe, which is observed in Figure 4b, can be reduced by increasing the sampling  
Figure 3: Reconstructed SAR scenes \(h\) of \(251\times 251\) pixels with nearest neighbor, linear, cubic, and sinc (for \(L=12\)) interpolations. Here, the sampling rate \(f_{\rm s}=2f_{\rm max}=0.66\,\rm THz\). Figure 2: Reconstructed SAR scenes \(h\) of \(251\times 251\) pixels with nearest neighbor, linear, cubic, and sinc (for \(L=12\)) interpolations. Here, the sampling rate \(f_{\rm s}=f_{\rm max}=0.33\,\rm THz\). rate \(f_{\rm s}\). Hence, the sinc interpolator with the kernel truncated up to \(2L+1=25\) normalized sinc functions, where \(L=12\), is sufficient to obtain accurate interpolation results. Therefore, let the truncated sinc interpolator (25) for \(L=12\) be defined as the _sinc interpolator_ and used in the rest of the paper. In Figure 5a,b are shown SAR-scene cuts \(h(\xi_{\rm n},\rho_{\rm n})\) that are reconstructed for the Nyquist rate based on the GBP algorithm and the interpolation methods described in Section 3 and plotted as functions of normalized range (for \(\xi_{\rm n}=0\)) and normalized azimuth (for \(\rho_{\rm n}=0\)). Here, the reconstructed SAR-scene cuts are compared the results provided by the impulse response function (A1) for \(\varphi=0\) and \(\varphi=\pi/2\). Here, the intensity of cuts is normalized with the intensity value of the SAR-scene center, i.e., with \(h(0,0)\). Furthermore, range and azimuth are normalized similarly to the examples described above. It has been observed that reconstructions obtained with the cubic and sinc interpolation techniques are more accurate both in range and azimuth directions than the results obtained based on the nearest neighbor and linear interpolation algorithms. The results based on nearest neighbor interpolation contain strong distortions in range and indistinguishable sidelobes in azimuth. The reconstructions based on cubic and sinc interpolations have the same azimuth resolution as the analytically-based result (A1); see Figure 5b. However, in comparison with the SAR-scene cuts based on sinc interpolation and (A1), \(h(0,\rho_{\rm n})\) based on the cubic interpolation algorithm has lower range resolution; see Figure 5a. It has also been investigated that sinc interpolator provides the most accurate reconstruction results in terms of the peak-sidelobe ratio (PSLR) (A4), which is introduced in Appendix B, and the root mean square error (RMSE). The PSLR-deviation of SAR scene \(h\), which is reconstructed with sinc interpolation, from the analytical solution at the Nyquist rate is around 0.5%; see the calculated PSLRs in Table 2. It should be noted that PSLR of \(h\) obtained with the extended linear interpolation procedure is higher (18.9%-deviation from the analytical PSLR) in comparison with the ratios of scenes obtained with the extended cubic and sinc interpolation approaches.