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A novel algorithm is proposed for detecting and estimating vibrating targets in synthetic aperture radar (SAR) data based on a pulse-repetition-interval (PRI) transform. Azimuthal signals of vibrating targets can be modelled as sinusoidal frequency-modulated (SFM) ones. The algorithm utilises the resemblance between the Doppler spectrum of vibrating-target SFM signals (or ghost image) and a pulse train, and applies to the spectrum the PRI transform originally used for estimating PRIs of pulse trains. The algorithm can detect SAR vibrating targets under moderate signal-to-noise/clutter ratios, and is also capable of accurately estimating the vibration frequencies even if there are multiple targets in a single range cell. The algorithm proposed has been successfully applied to both simulated and quasi-real data, and compared with that of the autocorrelation method, showing its superiority.