Whatever the field of application, optimizing the results and sometimes even solving problems requires taking advantage of the whole prior information. In this context, sparsity has emerged as a fundamental prior in recent years. A signal is said to be sparse in a certain basis if it can be described by a few small number of non-zero coefficients in that basis. The purpose of this thesis is to study new contributions of sparsity to signal processing. Two fields of application are considered. In addition to using sparsity, these two fields have in common the resolution of underdetermined inverse problems. The first one concerns the source separation problem. In this field, the sparsity leads to the development of several source separation methods. The performance of these sparseness-based methods relies on parameters that are usually chosen empirically. In this thesis, we propose a statistical formalism which reduces the number of parameters, while maintaining the source separation performance. The second field of application is the compressed sensing of finite alphabet signals. Such sensing seeks to reduce the number of measurements to be taken from finite alphabet signals while still keeping enough information to reconstruct them. We formulate this problem as a recovery of sparse signals from highly incomplete measurements. Therefore, this thesis is an exploration of emerging issues where integration of sparsity leads to good performance.
