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To achieve high-resolution images, synthetic aperture radar (SAR) faces considerable technical challenges such as huge amount of data samples and high hardware complexity. Compressed sensing (CS) theory shows that the super-resolved images can be reconstructed from an extremely smaller set of measurements than what is generally considered necessary by Nyquist/Shannon theorem. In this paper, a new algorithm of SAR imaging based on the concept of CS is presented, in which a random fractional Fourier transform (FRFT) matrix is used as the sensing matrix. By utilizing the FRFT matrix the demodulator for de-ramping the linear frequency modulation signal can be eliminated. Simulation results with both simulated and real data exhibit the validity of the proposed algorithm.