In this paper, we propose a new method for wideband spectrum sensing using compressed measurements of the received wideband signal; we can directly separate information of the sub-channels and perform detection in each. Wideband spectrum sensing empowers us to rapidly access the vacant sub-channels in high utilization regime. Regarding the fact that at each time instant some sub-channels are vacant, the received signal is sparse in some bases. Then we could apply the Compressive Sensing (CS) algorithms and take the compressed measurements. On the other hand, the primary user signals in different sub-channels could have different modulation types; therefore, the signal in each sub-channel is chosen among a signal space. Knowing these signal spaces, the secondary user could separate information of different sub-channels employing the compressed measurements. We perform filtering and detection based on these compressed measurements; this decreases the computational complexity of the wideband spectrum sensing. In addition, we model the received wideband signal as a vector which has a block-sparse representation on a basis consisting of all sub-channel bases whose elements occur in clusters. Based on this feature of the received signal, we propose another wideband spectrum sensing method with lower computational complexity. In order to evaluate the performance of the proposed method, we employ the Monte-Carlo simulation. According to simulations if the compression rate is selected appropriately according to the CS theorems and the problem model, the detection performance of our method leads to the performance of the ideal filter bank-based method, which uses the ideal and impractical narrow band filters.