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Signals sparse in a transformation domain can be recovered from a reduced setof randomly positioned samples by using compressive sensing algorithms. Simplere- construction algorithms are presented in the first part of the paper. Themissing samples manifest themselves as a noise in this reconstruction. Once thereconstruction conditions for a sparse signal are met and the reconstruction isachieved, the noise due to missing samples does not influence the results in adirect way. It influences the possibility to recover a signal only. Additiveinput noise will remain in the resulting reconstructed signal. The accuracy ofthe recovery results is related to the additive input noise. Simple derivationof this relation is presented. If a reconstruction algorithm for a sparsesignal is used in the reconstruction of a nonsparse signal then the noise dueto missing samples will remain and behave as an additive input noise. An exactrelation for the mean square error of this error is derived for the partial DFTmatrix case in this paper and presented in form of a theorem. It takes intoaccount very important fact that if all samples are available then the errorwill be zero, for both sparse and nonsparse recovered signals. Theory isillustrated and checked on statistical examples.

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