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Bayesian compressive sensing framework for spectrum reconstruction in Rayleigh fading channels

Compressive sensing (CS) is a novel digital signal processing technique that has found great interest in 
many applications including communication theory and wireless communications. In wireless communications, CS 
is particularly suitable for its application in the area of spectrum sensing for cognitive radios, where the complete 
spectrum under observation, with many spectral holes, can be modeled as a sparse wide-band signal in the frequency 
domain. Considering the initial works performed to exploit the benefits of Bayesian CS in spectrum sensing, the fading 
characteristic of wireless communications has not been considered yet to a great extent, although it is an inherent feature 
for all sorts of wireless communications and it must be considered for the design of any practically viable wireless system. 
In this paper, we extend the Bayesian CS framework for the recovery of a sparse signal, whose nonzero coefficients follow 
a Rayleigh distribution. It is then demonstrated via simulations that mean square error significantly improves when 
appropriate prior distribution is used for the faded signal coefficients and thus, in turns, the spectrum reconstruction 
improves. Different parameters of the system model, e.g., sparsity level and number of measurements, are then varied 
to show the consistency of the results for different cases.