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Due to the independence of azimuth invariance and high implementing efficiency, a fast time-domain algorithm has significant advantages for airborne bistatic synthetic aperture radar (BiSAR) data process with general geometric configuration. In this article, the practical problem of unexpected motion errors of the airborne platform is carefully analyzed under a fast factorized back-projection (FFBP) framework for a general BiSAR process and a coherent data-driven motion compensation (MOCO) algorithm integrated with FFBP is proposed. By utilizing wavenumber decomposition, the analytical spectrum of a polar grid image is obtained where the motion error can be conveniently investigated in image spectrum domain and the coherence between azimuthal phase error (APE) and motion-induced nonsystematic range cell migration (NsRCM) can be perfectly revealed. Then, a new data-driven MOCO method for both APE and NsRCM correction is developed with the FFBP process. Different from the data-driven MOCO in most frequency-domain algorithms, the residual NsRCM introduced by the FFBP process is particularly analyzed and addressed in the MOCO, which significantly improves the image quality in focusing. Promising results from both simulation and raw data experiments are presented and analyzed to validate the advantages of the proposed algorithm for the airborne BiSAR process.