Sparse signal recovery in the presence of colored noise and rank-deficient noise covariance matrix: An SBL approach

In this work, we address the recovery of sparse and compressible vectors in the presence of colored noise possibly with a rank-deficient noise covariance matrix, from overcomplete noisy linear measurements. We exploit the structure of the noise covariance matrix in a Bayesian framework. In particular, we propose the CoNo-SBL algorithm based on the popular and efficient Sparse Bayesian Learning (SBL) technique. We also derive Bayesian and Marginalized Cramér Rao lower Bounds (CRB) for the problem of estimating compressible vectors. We consider an unknown compressible vector drawn from a Student-t prior distribution, and derive CRBs that encompass the random nature of the unknown compressible vector and the parameters of the prior distribution, in the presence of colored noise and rank-deficient noise covariance matrix. Using Monte Carlo simulations, we demonstrate the efficacy of the proposed CoNo-SBL algorithm as compared to compressed sensing and greedy techniques. Further, we demonstrate the mean squared error performance of the proposed estimator compared to the CRBs, for different ranks of the noise covariance matrix.