Compressive sensing offers a new wideband spectrum sensing scheme in cognitive radio. In this paper, a sparse signal recovery algorithm based on sparse Bayesian learning (SBL) framework is proposed. By exploiting intrablock correlation in a block sparse model and using Expectation-Maximization (EM) method, this algorithm achieves superior performance. The results of experiments show that this algorithm is robust to noise and has better performance than other algorithms in signal recovery. Then we apply it to wideband spectrum sensing, we find that proposed algorithm not only guarantees accurate signal estimation, but also obtains higher correct detection probability.