# Interpolation Methods with Phase Control for Backprojection of Complex-Valued SAR Data +
## 1 Introduction
According to the evaluation [21], the four-point cubic convolution interpolator has been considered as the optimal interpolator among the described interpolation algorithms. However, for high-resolution applications, the six-point cubic convolution interpolator has been recommended. To improve coherency in SAR interferometry, it has been proposed to combine the truncated sinc interpolator with the Hanning window [22]. Nevertheless, the results obtained in [22] have demonstrated that there is no interpolator for SAR image resampling that is optimal for all SAR data types and quality. [PERSON] et al. have introduced in [23] Knab and approximate prolate windows for sinc interpolator that is used for SAR backprojection and compared their efficiency in terms of the root mean square error (RMSE). The results demonstrate that windowed sinc interpolators provide better accuracy (the RMSE is about 0.01%) than the other interpolators. To the knowledge of the authors, none of the interpolators described above, with the exception of the nearest neighbor, the performance of which can be improved through the support of the FFT interpolation [20], has been clearly formulated to interpolate SAR data with complex-valued representation. The FFT interpolation in combination with the nearest neighbor interpolator can lead to a huge amount of unused data, which might not be stored or processed by the real-time THz SAR system due to constructive limitations (e.g., installation of such a system on compact UAV platforms to monitor the environment, which brings limitations on energy and computational power). Furthermore, none of these interpolators accounts for the relationship between the range distance from the SARplatform antenna to the point in the defined image plane and the phase of interpolated SAR data value, which affects the accuracy of reconstructed SAR scenes. In the rest of the paper, we define SAR data with complex-valued representation as complex SAR data. In this paper, we introduce the extended versions of linear, cubic, and sinc interpolation algorithms that, in comparison with the interpolators described in [21; 22; 23], include the phase control procedure. The initial results on the phase control of complex SAR data under linear and cubic interpolation have been partially reported in [24]. The idea of the proposed control procedure is naturally related to the complex SAR data processing. To control the argument of the interpolated SAR data value, a priori known information about the range distance between the SAR platform antenna and the given point in the defined image plane has to be assigned to the phase of surrounding nearest neighbor sample points. The procedure is implemented through the multiplication of surrounding nearest neighbor data samples with corresponding phase compensation terms. The developed interpolators can be incorporated into the backprojection algorithms to process complex SAR data, and GBP is selected as an example in this paper. The results achieved with the developed interpolators are compared with the results obtained with the nearest neighbor and corresponding conventional (existing) interpolation algorithms. Furthermore, the efficiency of the developed interpolators is tested on simulated and real data at THz frequencies, where the accuracy of interpolation methods plays a crucial role, and is verified by comparison with the results provided by the analytical approach introduced in [4]. The effects of the sampling rate on the accuracy of SAR image formation are also considered in this paper. The novelty of the proposed scheme is highly beneficial for SAR sensing at THz frequencies. The computational complexity drastically increases at the THz spectrum due to the large number of pixels to be processed. Especially for UAV-based THz SARs, the payload and energy resources are limited and hence the limited computational power for on-board processing. The rest of the paper is organized as follows.