This study addresses the problem of discrete signal reconstruction from the
perspective of sparse Bayesian learning (SBL). Generally, it is intractable to
perform the Bayesian inference with the ideal discretization prior under the
SBL framework. To overcome this challenge, we introduce a novel discretization
enforcing prior to exploit the knowledge of the discrete nature of the
signal-of-interest. By integrating the discretization enforcing prior into the
SBL framework and applying the variational Bayesian inference (VBI)
methodology, we devise an alternating update algorithm to jointly characterize
the finite alphabet feature and reconstruct the unknown signal. When the
measurement matrix is i.i.d. Gaussian per component, we further embed the
generalized approximate message passing (GAMP) into the VBI-based method, so as
to directly adopt the ideal prior and significantly reduce the computational
burden. Simulation results demonstrate substantial performance improvement of
the two proposed methods over existing schemes. Moreover, the GAMP-based
variant outperforms the VBI-based method with an i.i.d. Gaussian measurement
matrix but it fails to work for non i.i.d. Gaussian matrices.Comment: 13 pages, 7 figure