In this paper, a novel wideband spectrum sensing algorithm based on Compressive Sensing (CS) and reconstruction of second order signal statistics from covariance matrix of the acquired samples for Cognitive Radio (CR) systems is presented. This allows cognitive users to sense the spectrum without apriori knowledge of signal characteristics in the radio environment by minimizing the amount of samples to be processed. Simulation results show that the proposed algorithm allows to sense the spectrum efficiently, improving the performance in terms of detection probability, false alarm probability and miss detection probability regarding previously proposed algorithms.