Based on the compressed sensing theory, if a signal is sparse in a suitable space, by using the optimization methods, signal could be accurately reconstructed from measurements that are significantly less than the theoretical Shannon requirements. The sparse representation may exist for the signal and it is not available for the noise; this could be used to distinguish these two. On the other hand, in compressed sensing, finding the answer hinges on finding the most sparse solution; thus this technique can separate clean signal from the noise. In FMCW radar, the distance of a target could be obtained from the frequency of the receiver output signal. Since this signal has a sparse representation in the frequency domain, based on compressed sensing theory, it could be reconstructed from a few number of data. In this paper, a new method for signal processing of FMCW radar is presented based on compressed sensing. Moreover, by considering noise removal feature that is in the nature of this technique, it is shown that the effect of noise on the receiver output signal can be reduced and the system performance of the radar can be improved.