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The combination of frequency-modulated continuous-wave (FMCW) technology and synthetic aperture radar (SAR) techniques leads to lightweight cost-effective imaging sensors of high resolution. One limiting factor to the use of FMCW sensors is the well-known presence of nonlinearities in the transmitted signal. This results in contrast- and range-resolution degradation, particularly when the system is intended for high-resolution long-range applications, as it is the case for SAR. This paper presents a novel processing solution, which solves the nonlinearity problem for the whole range profile. Additionally, the conventional stop-and-go approximation used in pulse-radar algorithms is not valid in FMCW SAR applications under certain circumstances. Therefore, the motion within the sweep needs to be taken into account. Analytical development of the FMCW SAR signal model, starting from the deramped signal and without using the stop-and-go approximation, is presented in this paper. The model is then applied to stripmap, spotlight, and single-transmitter/multiple-receiver digital-beamforming SAR operational mode. The proposed algorithms are verified by processing real FMCW SAR data collected with the demonstrator system built at the Delft University of Technology.