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In this paper, a novel algorithm is proposed to design sparse FIR filters. It is known that this design problem is highly nonconvex due to the existence of -norm of a filter coefficient vector in its objective function. To tackle this difficulty, an iterative procedure is developed to search a potential sparsity pattern, which is then used to compute the final solution by solving a convex optimization problem. In each iterative step, the original sparse filter design problem is successively transformed to a simpler subproblem. It can be proved that under a weak condition, globally optimal solutions of these subproblems can be attained by solving their dual problems. In this case, the overall iterative procedure converges to a locally optimal solution of the original design problem. The design procedure described above can be repeated for several times to further improve the sparsity of design results. The output of the previous stage can be used as the initial point of the subsequent design. The performance of the proposed algorithm is evaluated by two sets of design examples, and compared to other sparse FIR filter design algorithms.