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Abstract The orthogonal matching pursuit (OMP) algorithm is a classical greedy algorithm widely used in compressed sensing. The number of iterations required for the OMP algorithm to perform exact the recovery of sparse signals is a fundamental problem in signal processing. In this work, by investigating the relationship between the iteration number for OMP and the signal estimation error based on the restricted isometry property, the authors obtained tighter bounds on the number of iterations required to approximately recover a sparse signal with noise and exact support recovering for the noiseless cases of OMP.