In contrary to the existing work related with compressed sensing based STAP technique, which adopts the original sensing matrix, the proposed noise driven compressed sensing method is to construct a new sensing matrix with weak coherence through incorporating the measurement noise. The proposed method tries to build an equivalent system of the classical model in compressed sensing, resulting in an equivalent sensing matrix. Inspired by the idea that low coherence guarantees the reconstruction of the sparse vector with large probability, the equivalent sensing matrix is updated iteratively in a Markov chain Monte Carlo (MCMC) based framework to reduce the large coherence between a set of specific columns in the original sensing matrix. At the same time, the proposed method tries to preserve most of the information of the original sensing matrix via adjusting a noise related matrix. The simulation results show that the proposed method obtains much less average reconstruction error compared with the existing compressed sensing based STAP methods, and it is also very efficient when coping with measurement noise with low SNR.