# Interpolation Methods with Phase Control for Backprojection of Complex-Valued SAR Data +
## 1 Introduction  
Synthetic aperture radar (SAR) is a remote sensing technique that has been developed since the 1950s as an alternative to the optical imaging system and with the aim to acquire images with a high cross-range resolution. Nowadays, it is used in a wide range of applications, including stationary and moving target detection [1, 2]. The range of SAR applications is directly correlated to the frequency band of the SAR systems. For example, SAR systems, such as spaceborne and airborne, commonly operate in the microwave frequency range below 30 GHz. The spaceborne SAR systems, including TerraSAR-X, RADARSAT-2, and COSMO-SkyMed, provide an opportunity to study geoscience and hydrology at a distance of hundred kilometres from the Earth's surface [3]. The airborne SARs that operate in this frequency range include ultrawideband-ultrawidebeam SAR imaging systems (UWB SAR) that have been developed to perform high-resolution ground imaging, even under adverse weather conditions [4]. The UWB SARs refer to the systems that utilize radar signals with large fractional bandwidth and synthesize a wide integration angle [4, 5]. The typical examples of UWB SAR systems areCARABAS operating in the frequency range [20; 90]  MHz [6] and LORA that operates in the frequency range [200; 800]  MHz [7]. The THz frequency spectrum is also suitable to achieve high-resolution imaging. The SAR systems that operate at these frequencies are suitable for short-range applications, including the indoor environment purpose. The development of such type of systems is quite an active research topic, which has potential applications in the areas of security, logistics, and medicine. Most of the state-of-the-art THz SARs have been designed to process bandpass signals and are realized as ground- and rail-based systems. For example, in [8; 9; 10], SAR imaging is performed at 0.3 THz, and, in [11; 12], the imaging has been implemented at frequencies 0.6 THz and 0.75 THz, respectively. Recent results include imaging at 1.1 THz that have been published in [13]. Nevertheless, modern SAR systems face different technical issues caused by the deviation of SAR platforms from the path. For example, unmanned aerial vehicle (UAV) SAR imaging systems deal with platform vibrations. This problem becomes crucial for UAV-based SARs that operate in the THz frequency range, which are extremely sensitive to the platform path deviation [14; 15]. To deal with this issue, the use of signal processing algorithms that are capable to handle motion error compensation is required. Among the SAR image formation algorithms, the backprojection algorithms, such as Global Backprojection (GBP) [16], Fast Backprojection [17], and Fast Factorized Backprojection [18], have their own capability to manage the motion error compensation [19] based on a priori knowledge on SAR platform deviation from the expected path [20]. Due to the fact that in these algorithms the backprojection procedure always requires range calculation for each aperture position, information about the platform deviations can be included in the backprojection algorithm as a motion compensation procedure at this stage. In addition, the backprojection algorithms are capable to handle radar signals with a large fractional bandwidth [5; 19]. For these reasons, the backprojection algorithms play an important role in SAR data processing despite their high computational cost. One of the contributions to the high computational cost is the interpolation procedure, at which a complex data value is estimated based on given complex SAR data values for a given range-time delay. Then, the interpolated complex value is backprojected into the defined image plane. There exist various interpolation algorithms that are used for SAR applications. [PERSON] and [PERSON] introduced an evaluation of nearest neighbor, piecewise linear, four- and six-point cubic convolution, and truncated sinc interpolators in [21] and presented their application to SAR interferometry.