Wideband spectrum sensing is an important aspect of cognitive radio systems. In current models of wide spectrum sensing, a discrete frequency denotes a continuous frequency band. This type of model is divorced from practice and cannot reflect the reality of spectrum occupation. In this paper, we propose a novel wideband spectrum sensing scheme in terms of a modulated wideband converter (MWC) and sparse Bayesian learning (SBL). By exploiting the MWC, a practical wideband signal denoted by a block-sparse model is sampled to acquire the compressed measurements. We then employ SBL to directly extract the relevant information from the compressed measurements for estimating the support set. The estimated support set is chosen as the test statistic to facilitate spectrum sensing and analyze the detection performance. For applications in actual situations of support sets, different matching criteria are presented according to the requirements imposed by wideband spectrum sensing. Finally, we analyze the detection performance when a frequency band's location is given as well as when it is randomly generated by conducting simulations. These simulations show that the proposed method outperforms the MWC-based orthogonal matching pursuit method, and the best performance can be achieved under matching criterion 3.