Sparse Estimation using Bayesian Hierarchical Prior Modeling for Real and Complex Models

Sparse modeling and estimation of complex signals is not uncommon in practice. However, historically, much attention has been drawn to real-valued system models, lacking the research of sparse signal modeling and estimation for complex-valued models. This paper introduces a unifying sparse Bayesian formalism that generalizes to complex- as well as real-valued systems. The methodology relies on hierarchical Bayesian sparsity-inducing prior modeling of the parameter of interest. This approach allows for the Bayesian modeling of l1-norm constraint for complex-valued as well as real-valued models. In addition, the proposed two-layer hierarchical model allows for the design of novel priors for sparse estimation that outperform the Bayesian formulation of the l1-norm constraint and lead to estimators approximating a soft-thresholding rule. An extension of the two-layer model to a three-layer model is also presented. Varying the free parameters of the three-layer model leads to estimators that approximate a hard-thresholding rule. Finally, a variational message-passing (VMP) implementation of the proposed Bayesian method that effectively exploits the hierarchical structure of the inference problem is presented. The simulation results show that the VMP algorithm outperforms existing sparse methods both in terms of the sparsity of the estimation results and achieved mean squared error in low and moderate SNR regimes.