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

Abstract

Signals sparse in a transformation domain can be recovered from a reduced set of randomly positioned samples by using compressive sensing algorithms. Simple reconstruction algorithms are presented in the first part of the paper. The missing samples manifest themselves as a noise in this reconstruction. Once the reconstruction conditions for a sparse signal are met and the reconstruction is achieved, the noise due to missing samples does not influence the results in a direct way. It influences the possibility to recover a signal only. Additive input noise will remain in the resulting reconstructed signal. The accuracy of the recovery results is related to the additive input noise. Simple derivation of this relation is presented. If a reconstruction algorithm for a sparse signal is used in the reconstruction of a nonsparse signal then the noise due to missing samples will remain and behave as an additive input noise. An exact relation for the mean square error of this error is derived for the partial DFT matrix case in this paper and presented in form of a theorem. It takes into account very important fact that if all samples are available then the error will be zero, for both sparse and nonsparse recovered signals. Theory is illustrated and checked on statistical examples.