The application of the basis pursuit method to SAR super-resolution processing was extended. Firstly, based on the SAR attributed scattering model, a Fourier dictionary with fast implicit algorithm was designed in phase history domain. Secondly, in light of the separable feature of Fourier atom, The dictionaries in column-wise and row-wise of the image were construct, the dimension of the dictionary and the computation burden were greatly reduced. A new and fast iterative algo- rithm was proposed to solve the optimization problem of the basis pursuit. Finally, higher resolution image was obtained by IFFT to the coefficient. Simulation experiments and computational results of measured MSTAR data demonstrate that basis pursuit can be implemented speedy and stably, and it can provide super-resolution at the same time.