Abstract: This paper proposes a novel wideband compressive spectrum sensing scheme based on the Generalized Likelihood Ratio Test (GLRT), in which the GLRT statistic and the decision threshold are derived according to Random Matrix Theory (RMT). The proposed scheme exploits only compressive measurements to detect the occupancy status of each sub-band in a wide spectral range without requiring signal reconstruction or priori information. In addition, to alleviate the communication and data acquisition overhead of Secondary Users (SUs), a Sensor Node (SN)-assisted cooperative sensing framework is also addressed. In this sensing framework, the sensor nodes perform compressive sampling instead of the SUs at the sub-Nyquist rate. Both theoretical analysis and simulation results show that compared with the traditional GLRT algorithm with signal reconstruction and the Roys Largest Root Test (RLRT) algorithm, the proposed scheme not only has lower computational complexity and cost and more robust sensing performance, but also can achieve better detection performance with a fewer number of SNs.