Compressed sensing for finite-valued signals

Kurzfassung/Abstract

The need of reconstructing discrete-valued sparse signals from few measurements, that is solving an underdetermined system of linear equations, appears frequently in science and engineering. Whereas classical compressed sensing algorithms do not incorporate the additional knowledge of the discrete nature of the signal, classical lattice decoding approaches such as the sphere decoder do not utilize sparsity constraints.
In this work, we present an approach that incorporates discrete values prior into basis pursuit. We consider bipolar finite-valued and unipolar finite-valued sparse signals, i.e., sparse signals with entries in , respectively in , with . For those signals, we will show that the phase transition for our approach takes place earlier than in the case of basis pursuit. We will in particular derive highly improved performance guarantees for the special type of unipolar binary and bipolar ternary sparse signals, i.e., sparse signals having entries in , respectively in . More precisely, we will show that independently of the sparsity of the signal, at most , respectively , measurements are necessary to recover a unipolar binary, and a bipolar ternary signal uniquely, where N is the dimension of the ambient space. We will further discuss robustness of the algorithm and phase transition under noisy measurements.