  For wideband spectrum sensing, compressive sensing has been proposed as a
solution to speed up the high dimensional signals sensing and reduce the
computational complexity. Compressive sensing consists of acquiring the
essential information from a sparse signal and recovering it at the receiver
based on an efficient sampling matrix and a reconstruction technique. In order
to deal with the uncertainty, improve the signal acquisition performance, and
reduce the randomness during the sensing and reconstruction processes,
compressive sensing requires a robust sampling matrix and an efficient
reconstruction technique. In this paper, we propose an approach that combines
the advantages of a Circulant matrix with Bayesian models. This approach is
implemented, extensively tested, and its results have been compared to those of
l1 norm minimization with a Circulant or random matrix based on several
metrics. These metrics are Mean Square Error, reconstruction error,
correlation, recovery time, sampling time, and processing time. The results
show that our technique is faster and more efficient.
