Sparse Signal Recovery (SSR) has an essential role in a number of modern engineering applications. This thesis focuses on Bayesian algorithms for sparse signal recovery, where it addresses some of the shortcomings associated with such algorithms. The high complexity of the sparse Bayesian learning algorithm is addressed first, where we present an algorithm that incorporates damped Gaussian generalized approximate message passing (GGAMP) into Expectation-Maximization (EM)-based sparse Bayesian learning (SBL). In particular, GGAMP is used to implement the E-step in SBL in place of matrix inversion. We propose an algorithm that is much more robust to arbitrary measurement matrices than the standard damped GAMP algorithms while being much lower complexity than the standard SBL algorithm. We then extend the approach from the single measurement vector (SMV) case to the temporally correlated multiple measurement vector (MMV) case.
The approach developed for the standard SBL is extended to address the sparse non-negative least squares (NNLS) problem using a rectified Gaussian scale mixture approach combined with the generalized approximate message passing algorithm. This approach enhances convergence compared to existing GAMP based sparse NNLS algorithms. Other advantages include significant reduction in derivation complexity of the algorithm, and the ability to impose different priors on the signal, simply by changing the less computationally demanding M-step. Moreover, extending the algorithm to the multiple measurement vector case is straightforward, and is achieved by a simple modification to the M-step as well.
Next, we provide a new perspective of the SBL algorithm. A novel interpretation of the SBL algorithm's iterations is developed, where the iterations are divided into a minimum power distortionless receiver (MPDR) step and a denoising step. This interpretation provides significant intuitive insights into the SBL, which can potentially enable enhancing the algorithm and extending it beyond some of its current limitations. To demonstrate this potential, we propose a low complexity algorithm that can handle a wide range of sparsity promoting priors. We also show how the new perspective on SBL extends to other variants of the algorithm, such as the sequential fast-SBL algorithm and the multiple measurement vector (MMV) SBL variants. We demonstrate potential benefits of such interpretation by extending the fast-SBL to incorporate more general priors, and by developing a low complexity MMV algorithm.
Finally, we address the MIMO semi-blind channel estimation problem, benefiting from the insights gained from previous results. We propose an eigenvalue decomposition based technique to significantly reduce the dimensionality of the Gaussian EM based estimation algorithm, greatly lowering the computational complexity. In addition to that, we apply the MPDR based decoupling principle to derive a tractable EM algorithm that uses the actual discrete prior of the data symbols.
