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A novel methodology is presented for determining the velocity and location of multiple moving targets using a single strip-map synthetic aperture radar (SAR) sensor. The so-called azimuth position uncertainty problem is therefore solved. The method exploits the structure of the amplitude and phase modulations of the returned echo from a moving target in the Fourier domain. A crucial step in the whole processing scheme is a matched filtering, depending on the moving target parameters, that simultaneously accounts for range migration and compresses two-dimensional signatures into one-dimensional ones without losing moving target information. A generalized likelihood ratio test approach is adopted to detect moving targets and derive their trajectory parameters. The effectiveness of the method is illustrated with synthetic and real data covering a wide range of targets velocities and signal-to-clutter ratios (SCRs). Even in the case of parallel to platform moving target motion, the most unfavorable scenario, the proposed method yields good results for, roughly, SCR > 10 dB.