Noises in randomly sampled sparse signals

Abstract

Sparse signals can be recovered from a reduced set of randomly positioned samples by using compressive sensing algorithms. Two main reconstruction directions are in the sparse transformation domain analysis of signals and the gradient based algorithms. In the transformation domain analysis, that will be considered here, the estimation of nonzero signal coefficients is based on the signal transform calculated using available samples only. The missing samples manifest themselves as a noise. This kind of noise is analyzed in the case of random sampling, when the sampling instants do not coincide with the sampling theorem instants. Analysis of the external noise influence to the results, with randomly sampled sparse signals, is done as well. Theory is illustrated and checked on statistical examples.

Reconstruction of Sparse and Nonsparse Signals from a Reduced Set of Samples

“…where X KS and X KN are the reconstructed signal and noise components respectively. Assume that the reconstruction conditions are met and the positions of nonzero coefficients K ={k 1 , k 2 , ..., k K } can be determined through a single step or iterative procedure [13,20,22]. The equations to find the unknown coefficients are written for M > K time instants…”