In this paper, a new algorithm is proposed for the design of sparse FIR filters. Traditional l₁-optimization-based methods take all the coefficients into l₁-norm minimization. However, it is unnecessary since some of them can only take nonzero values to satisfy design specifications. Furthermore, minimizing l₁ norm of all the coefficients could drive the design results to deviate from the optimal ones. The proposed algorithm aims to identify nonzero coefficients at some crucial positions in each iteration to minimize the number of nonzero coefficients. Simulation results demonstrate that the proposed algorithm can achieve better design results than traditional l₁-optimization methods.