Many systems, including telecommunication systems, radar and imaging systems, biomedical systems, control and robotics systems, rely on powerful digital signal processing (DSP). DSP algorithms are hard pressed to provide accurate estimates of a signal from as few as possible noisy measurements. If the signal to be estimated is sparse and high dimensional, a novel DSP technique, called compressed sensing (CS), allows efficient recovery from (possibly noisy) low dimensional representation. Even

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... ough reconstruction guarantees of a number of CS recovery algorithms have been known for almost a decade, many nonlinear distortions introduced by a practical measurement system are often not considered in the analysis. Neglecting these distortions could in turn have a detrimental effect on the performance of a recovery algorithm in a practical application. In this thesis, the focus is on algorithms for recovering sparse vectors from measurements tampered with some of the most common nonlinear distortions that appear in practice, namely quantization and modulo distortions, which are not treated with classical CS recovery algorithms. To present date, many reconstruction algorithms have been proposed to solve noisy CS problems. Among them, the class of approximate message passing (AMP) algorithms stands out for its low computational complexity, low reconstruction error, and the ability to predict the states of the algorithm across iterations (at least in the large system limit). Furthermore, the Bayesian approximate message passing (BAMP) algorithm has the ability to incorporate a signal prior to additionally improve the estimate, while the generalized approximate message passing (GAMP) algorithm allows for reconstruction of sparse signals from nonlinear measurements. These facts make the AMP algorithms particularly interesting for our problems involving quantization and modulo distortions with known prior. BAMP follows the probabilistic estimation approach where a prior distribution is assumed for the unknown signal. A comm [...]