Abstract: Radar process is one of the most important applications of compressive sensing (CS) theory. Compressive sensing theory can reduce the sampling rate of echo signals and improve the processing performance in some radar applications. However, the huge computational complexity in signal reconstruction puts strict constraint on some practical radar applications, especially on large scale problems. This paper proposes a novel fast reconstruction algorithm for compressive sensing radar signal. The proposed algorithm realizes the direct implementation of the costly matrix-vector multiplications in conventional reconstruction algorithms with fast Fourier transform (FFT) and nonuniform fast Fourier transform, i.e., NUFFT, which greatly reduces the computational complexity of the reconstruction algorithm and therefore significantly speeds up the recovery of compressive sensing radar signal. In addition, the algorithm eliminates the huge storage requirement of sensing matrix in most common compressive sensing recovery algorithms, since the matrix-vector multiplication is realized with fast Fourier transform algorithm here. Numerical simulation results validate the feasibility and efficiency of the proposed algorithm.