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）的角度出发，解决离散信号重构问题。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处理的。为了克服这一难题，我们引入一种新的离散化强制先验，以利用对信号离散性的了解。通过将离散化强制先验与SBL框架相结合，并采用变分贝叶斯推断（VBI）方法，我们设计了一种交替优化算法，以联合表征有限字母特征并重构未知信号。当测量矩阵是每个分量独立同分布的高斯时，我们进一步将广义近似消息传递（GAMP）嵌入到基于VBI的方法中，以直接采用理想先验并显著减少计算负担。仿真结果表明，所提出的两种方法相对现有方案实现了显著的性能提升。此外，基于GAMP的变异体在i.i.d.高斯测量矩阵上优于基于VBI的方法，但对于非i.i.d.高斯矩阵无法发挥作用。