  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 optimization 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 i.i.d. Gaussian
measurement matrices but it fails to work for non i.i.d. Gaussian matrices.
