We consider the problem of estimating a deterministic finite alphabet vector f from underdetermined measurements y = A f , where A is a given (random) n × N matrix. Two new convex optimization methods are introduced for the recovery of finite alphabet signals via ℓ1-norm minimization. The first method is based on regularization. In the second approach, the problem is formulated as the recovery of sparse signals after a suitable sparse transform. The regularization-based method is less complex than the transform-based one. When the alphabet size p equals 2 and (n, N) grows proportionally, the conditions under which the signal will be recovered with high probability are the same for the two methods. When p > 2, the behavior of the transform-based method is established. Experimental results support this theoretical result and show that the transform method outperforms the regularization-based one.

We address the problem of blind source separation in the underdetermined and instantaneous mixture case. The proposed method is based on an algorithm developed by Aissa-El-Bey and al.. This algorithm requires a good choice of the noise threshold and does not take into account the noise contribution in the inversion process. In order to overcome these drawbacks, this paper presents a robust underdetermined blind source separation approach. Robustness is achieved by estimating the noise standard deviation and using this estimate in the inversion process and the expression of the noise threshold. The good performance of the proposed method is shown by comparison with state-of-the-art methods.

This paper addresses an innovative approach to informed enhancement of damaged sound. It uses sparse approximations with a learned dictionary of atoms modeling the main components of the undamaged source spectra. The decomposition process aims at finding which of the atoms could constitute the decomposition of the undamaged source in order to recover it. The decomposition of the damaged signal is done with a Matching Pursuit algorithm and involves an adaptation of the dictionary learned on undamaged sources. Evaluation is performed on a bandwidth extension task for various classes of signals.