In this paper we introduce a new method for wideband spectrum sensing which uses the received signal compressed measurements to determine the occupancy status of each subchannel. In the cognitive radio systems, wideband spectrum sensing in high utilization regimes increases the speed of the access to the spectrum holes. In order to send data, the primary users employ different modulation types in each sub-channel; therefore, the transmitted signal is chosen among a signal space. Since just a few numbers of sub-channels are occupied by the primary user signals in each time instant, the received wideband signal has a sparse representation on the sub-channel signal space bases; then, the secondary user can compress the received signal samples. Knowing about the modulation types and therefore, the signal spaces, the information in each subchannel is separated using a projection matrix applied to the compressed measurements, directly. The separated compressed measurements are employed to derive a Generalized Likelihood Ratio detector and determine the presence or absence of the primary user signal in each sub-channel. We try to determine the sensing matrix so that the filtering and detection steps and, therefore, wideband spectrum sensing have low error.