# Interpolation Methods with Phase Control for Backprojection of Complex-Valued SAR Data +
## 4 Simulations
### Results  
In Figure 2 is shown SAR images \(h(\xi_{\mathrm{n}},\rho_{\mathrm{n}})\) of \(251\times 251\) pixels reconstructed with the GBP algorithm (2) for sampling rate \(f_{\mathrm{s}}=f_{\mathrm{max}}=0.33\,\mathrm{THz}\). Here, the intensity is normalized with the peak intensity value, the range \(\rho\) and the azimuth \(\xi\) are normalized with \(-3\,\mathrm{dB}\) beamwidths. The reconstruction procedure has been performed by corresponding incorporation of the nearest neighbor (7), linear (13), cubic (23), and sinc (25) interpolation algorithms to GBP. To evaluate the efficiency of the proposed interpolation algorithms, SAR scenes reconstructed at the rate \(f_{\mathrm{s}}=f_{\mathrm{max}}\) based on conventional linear, cubic, and sinc interpolators (i.e., those, in which the phase-control procedure is _not included_), are depicted in Figure 2b-d, respectively. The results demonstrate that linear, cubic, and sinc interpolators provide more accurate results at the Nyquist rate \(f_{\mathrm{s}}=f_{\mathrm{max}}\) in comparison with the nearest neighbor approach and corresponding conventional interpolators, where distortions over the normalized range interval \(\rho_{\mathrm{n}}\in[-2,2]\) are observed; see Figure 2a-g. Furthermore, sinc interpolation without the phase-control procedure provides a defocused reconstruction of the point target along the azimuth axis, as depicted in Figure 2d. It has been investigated that the involvement of the phase control procedure (6) increases the computational costs of linear, cubic, and sinc interpolators additionally by 14, 21, and \(7(2L+1)\) operations per each iteration of the GBP algorithm, respectively, where \(L\) is the summation limit in the truncated sinc kernel of the extended sinc interpolator. However, the order of computational complexity of the GBP algorithm remains the same, i.e., \(\mathcal{O}\{N_{\xi}M_{\rho}M_{\xi}\}\), where \(N_{\xi}=345\) denotes the number of aperture positions, \(M_{\rho}=M_{\xi}=251\) the number of SAR-image pixels in range and azimuth directions, respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline The highest frequency processed, \(f_{\mathrm{max}}\) & \(0.33\,\mathrm{THz}\) \\ The lowest frequency processed, \(f_{\mathrm{min}}\) & \(0.22\,\mathrm{THz}\) \\ Number of aperture positions, \(N_{\xi}\) & \(345\) \\ Aperture step, \(\Delta_{\xi}\) & \(0.997\,\mathrm{mm}\) \\ Integration angle, \(\phi_{0}\) & \(\approx\)\(9.8^{\circ}\) \\ Reference range, \(R_{0}\) & \(2\,\mathrm{m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation Setup Parameters. When the sampling rate is twice higher, i.e., \(f_{\rm s}=2f_{\rm max}\), the point target can be reconstructed accurately with all the interpolation methods discussed in this paper, as shown in Figure 3a-d. However, the azimuthal and some minor range distortions are still observed over the normalized range interval \(\rho_{\rm n}\in[-2,2]\), when the nearest neighbor approach is used; see Figure 3a. Note that the upsampling procedure for the nearest neighbor interpolator can be provided through the FFT interpolation, the computational complexity which is \(\mathcal{O}\{2N_{g}uN_{\rho}\log_{2}\{uN_{\rho}\}\}\), where \(u\geq 2\), \(u\in\mathbb{Z}\), is the upsampling factor, \(N_{\rho}=66017\) the number of range samples. It has been investigated that for upsampling factors \(u\geq 2\), the computational complexity of the FFT interpolation becomes dominant over the computational complexity of the global backprojection algorithm. This fact demonstrates additional advantages of the extended interpolators, which provide the opportunity to avoid the FFT-based upsampling procedure, in terms of computational costs and memory resources. It should also be noted that the kernel of the sinc interpolator used in this numerical example is truncated to \(2L+1=25\) normalized sinc functions, where \(L=12\) and based on which the reconstruction results are accurate; see Figures 2g and 3d.