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The electronic counter-countermeasures capabilities of staggered pulse repetition interval (PRI) signal is limited by its repetitive character, and a random PRI radar is an alternative to improve the range and velocity coverage. However, the high sidelobe pedestal of the target Doppler spectrum caused by randomness prevents its development. In this paper, based on the compressed sensing theory, we present a novel framework in random PRI radar to generate the Doppler spectrum with low sidelobe for moving target detection. As a precondition, the equivalent sensing matrix is proved to satisfy the restricted isometry property by testifying its independent sub-Gaussian elements and comparing its mutual coherence as well as eigenvalues of its Gram matrices to a typical random compressed sensing matrix in a statistical sense. To cover the concerned velocity, multichannel processing is used and the iterative grid optimization algorithm is employed to eliminate the grid mismatch effect. The simulation results demonstrate that this scheme has high performance of detection and free aliasing characteristic, which can also shorten the coherent processing interval compared with traditional staggered PRI mode.