Joint-sparse recovery in compressed sensing with dictionary mismatch

In traditional compressed sensing theory, the dictionary matrix is given a priori, while in real applications this matrix suffers from random noise and fluctuations. This paper considers a signal model where each column in the dictionary matrix is affected by a structured noise. This formulation is common in radar related applications and direction-of-arrival estimation. We propose to use joint-sparse signal recovery in this compressed sensing problem with dictionary mismatch and also give a theoretical result on the performance bound for this joint-sparse method. We show that under mild conditions the reconstruction error of the original sparse signal is bounded by both the sparsity and the noise level in the measurement model. Moreover, a fast first-order method is implemented to speed up the computing process. Numerical examples demonstrate the good performance of the proposed algorithm, and also show that the joint-sparse recovery method converges faster and gives a better reconstruction result than previous methods.