# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 1 Introduction  
By using azimuth antenna beam-steering technology, spaceborne synthetic aperture radar (SAR) systems can operate in beam-steering (BS) modes, such as sliding spotlight mode [1; 2] and Terrain observation with progressive scan (TOPS) mode [3; 4], which can achieve higher resolution or wider swaths than traditional strip-map mode. However, spaceborne SAR suffers from a conflict between azimuth resolution and range swath. By increasing the number of azimuth-receiving channels, multi-channel technology [5; 6; 7] can overcome this dichotomy and significantly increase the swath without resolution loss. As a result, multi-channel technology has been adopted in advanced spaceborne SAR systems [8; 9; 10; 11] and processing algorithms for multi-channel strip-map imaging have been proposed, including spectrum reconstruction algorithm [12] and digital beamforming algorithm [13]. Increasing requirements for high-resolution wide-swath data are currently driving the development of multi-channel BS-SAR [14]. With the combination of beam-steering technology and multi-channel technology, multi-channel sliding spotlight SAR can provide very high resolution over awide swath and multi-channel TOPS can produce imaging at high resolution with ultra-wide swath. However, two main challenges that must be faced by multi-channel BS-SAR are spectrum aliasing caused by the azimuth-varying Doppler centroid and non-uniform sampling [15]. Due to the azimuth beam steering, the Doppler centroid (DC) varies linearly along targets at different azimuths [3], and can be divided into two parts: the azimuth-varying Doppler centroid and the constant Doppler centroid (CDC). The azimuth-varying Doppler centroid, whose slope is denoted as the derotation rate (DRR), is related to the illumination geometry and the beam-steering rate, while the CDC is determined by the antenna pointing direction at the central time of observation. An azimuth-varying Doppler centroid results in the azimuth bandwidth spanning several pulse repetition frequencies (PRF) and Doppler spectrum aliasing. In addition, Doppler spectrum aliasing will be much more severe in multi-channel systems because the PRF is several times smaller than in single-channel BS-SAR and the data are usually non-uniform in time. To resolve these challenges, several imaging algorithms have been proposed [14; 15; 16; 17], including use of sub-aperture blocking and filter reconstruction [14], a full-aperture algorithm which combines derotation and a space-time adaptive processor [16] and an approach combining derotation and filter reconstruction [15; 17]. In most algorithms, however, the deration requires accurate values of the DRR and CDC to remove the spectrum aliasing. If the DRR or CDC have errors, the azimuth-varying Doppler centroid cannot be fully eliminated, resulting in inaccurate correction of the azimuth antenna pattern. In addition, for a multi-channel SAR system, phase inconsistency errors between channels are inherent. These can strongly influence the performance of multi-channel signal reconstruction and even produce ghost targets in images. In addition, amplitude inconsistency errors, timing delay errors, and channel position errors should be taken into account for multi-channel systems. Therefore, it is important to develop methods to estimate parameters and correct errors that are properly suited to multi-channel BS-SAR. In [18], the sensitivity of the DRR to the beam-steering rate was analyzed, but its effect on imaging and its estimation were not considered. The effect of CDC error on azimuth ambiguity was analyzed in [19]. The spatial cross-correlation coefficient (SCCC) method to estimate CDC error was presented in [20] in multi-channel strip-map and involved the phase inconsistency errors, but it is sensitive to non-uniformity in the spatial sampling.