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In cognitive radio networks, spectrum sensing is one of the most challenging technologies. However, traditional spectrum sensing methods need too much detection time in a very low SNR environment, and the detection probability is rather small. Compressive sensing provides a new and attractive perspective for spectrum sensing in cognitive radio network. Considering different influences of SNRs, a modified spectrum sensing algorithm based on compressive sensing is proposed in this paper. In order to achieve a good tradeoff between detection precision and detection time, a structure on the basis of adaptive compressive sampling for spectrum sensing is presented. Extensive simulation results are presented to justify the effectiveness of the proposed algorithms.