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In this paper, we propose a new method to recover band-limited signals corrupted by impulsive noise. The proposed method successively uses adaptive thresholding and soft decisioning to find the locations and amplitudes of the impulses. In our proposed method, after estimating the positions and amplitudes of the additive impulsive noise, an adaptive algorithm, followed by soft decision, is employed to detect and attenuate the impulses. In the next step, by using an iterative method, an approximation of the signal is obtained. This signal approximation is successively used to improve the noise estimate. The algorithm is analyzed and verified by computer simulations. Simulation results confirm the robustness of the proposed algorithm, even if the impulsive noise exceeds the theoretical reconstruction capacity.