Abstract: This paper tackles the well-known problem of basis mismatch in compressed channel estimation for wireless channels. The basis mismatch problem is caused by the fact that multipath delays exist in the continuous space, not exactly on the discretized grid. Any deviation from the grid points leads to performance degradation in channel estimation. This problem is known as the basis mismatch problem in the compressed channel estimation. In the paper, we propose a new convex optimization based alternating iteration recovery (AIR) algorithm to alternately solve the sparse channel coefficient vector and the basis mismatch vector. The optimization problems in each stage of the iterations are convex. The results show that the proposed AIR algorithm achieves better estimation accuracy compared with the comparative algorithms.