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Current Cognitive Radios are limited in their operational bandwidth by existing hardware devices, much of the extensive theoretical work on spectrum sensing is impossible to realize in practice over a wide frequency band. To alleviate the sampling bottleneck, some researchers have begun to use a technique called Compressive Sensing (CS), which allows for the acquisition of sparse signals at sub-Nyquist rates, in conjunction with CRs. These researchers have sequentially combined the two techniques: first acquiring compressed samples, then reconstructing the Nyquist rate signal, and lastly performing CR spectrum sensing on the reconstructed signal. While CS alleviates the bandwidth constraints imposed by front-end ADCs, the resulting increase in computation/complexity is non-trivial, especially in a power-constrained mobile CR. In addition, the computation time of the signal reconstruction introduces significant delay into the spectrum sensing operation. This motivates us to look at different ways to reduce computational complexity while achieving the same goals. In this paper, we will demonstrate how utilizing a Bayesian Compressive Sensing (BCS) framework can achieve the sampling reduction advantage of Compressive Sensing with significantly less computational complexity. Our key observation is that the CR does not have to reconstruct the entire signal because it is only interested in detecting the presence of Primary Users. Our BCS PU detection algorithm takes advantage of this observation by estimating signal parameters directly from the compressed signal, thereby eliminating the reconstruction stage and reducing the computational complexity. In addition, the BCS framework provides a measure of the quality of estimation allowing the system to optimize its data acquisition process to always acquire the minimum number of compressed measurements, even in a dynamic spectral environment.