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.

翻译：本研究从稀疏贝叶斯学习（SBL）角度出发，解决离散信号重建问题。通常情况下，在SBL框架下使用理想离散化先验进行贝叶斯推断是难以实现的。为克服这一挑战，我们引入一种新型离散化强制先验，以充分利用目标信号的离散特性。通过将该离散化强制先验融入SBL框架，并应用变分贝叶斯推断（VBI）方法，我们设计了一种交替优化算法，以联合刻画有限字母表特征并重建未知信号。当测量矩阵为独立同分布（i.i.d.）高斯矩阵时，我们进一步将广义近似消息传递（GAMP）嵌入基于VBI的方法中，从而直接采用理想先验并显著降低计算负担。仿真结果表明，所提出的两种方法相较于现有方案均有显著性能提升。此外，基于GAMP的变体在使用i.i.d.高斯测量矩阵时优于基于VBI的方法，但在非i.i.d.高斯矩阵场景下失效。