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In cognitive radio system, the spectrum hole of primary users can be used by secondary users while avoiding interference with primary users. We propose a multi-user cooperative wideband spectrum sensing framework based on compressed sensing to reduce the heavy burdens of high sampling rate of front end in cognitive radio. The proposed method can achieve a good performance of the reconstructed wideband spectrum with fewer samples, owing to making use of the correlation of secondary users' compressed sensing measurements by utilizing bayesian compressed sensing. The bigger the correlation of secondary users' measurements, the fewer measurements of each secondary user are required to sense the wideband spectrum accurately in cognitive radio. Our simulation results confirm the validity of the proposed method.