State of the art radar systems apply a large bandwidth and an increasing number of channels produce huge amount of data. The data easily exceeds that be stacked in the sensor or downlinked to the ground station. In order to solve this trouble, a novel synthetic aperture radar (SAR) raw data retrieval and a corresponding pulse compression method based on compressive sensing (CS) theory were presented. Under the assumption that the observed scene shows characteristic of a sparse reflectivity distribution, traditional matched filter can be replaced by CS for pulse compression. Benefits from this substitution include much lower data amount for scenario reconstruction than traditional SAR. In this method, the pulse compressed signal was reconstructed by solving an inverse problem through a greedy pursuit. The principle and process of the algorithm were given, and the effectiveness was validated by computer simulation. The new approach greatly simplifies the radar system, effectively reduces the huge amount of data, thus shifting emphasis from expensive receiver design to smart signal recovery algorithms.