Abstract: Compressive sensing offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to reconstruct the original signal from the observed sample. The time-frequency and time-scale communities have developed a large number of overcomplete waveform dictionaries --- stationary wavelets, wavelet packets, cosine packets, chirplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB).This paper describes a improve proposal of the iterative recovery algorithm called sparsity adaptive matching pursuit (SAMP) that delivers the same guarantees as the best optimization-based approaches. Sparsity adaptive matching pursuit adopts fixed-step in the iteration which causes over-estimated, therefore, according to the variation of the energy residual error between adjacent signals, “variable steps” during the iteration is used in the improved algorithm, namely, a large step size is used at the initial stage, and then becomes smaller. Compared with the original algorithmic, this proposal avoids the over estimation of the sparseness of the original signal. Moreover, the proposal solves the contradiction between the recovery precision and the recovery speed that exists in the original algorithm.