Noise Folding in Completely Perturbed Compressed Sensing

Abstract

This paper first presents a new generally perturbed compressed sensing (CS) model $y=(A+E)(x+u)+e$, which incorporated a general nonzero perturbation $E$ into sensing matrix $A$ and a noise $u$ into signal $x$ simultaneously based on the standard CS model $y=Ax+e$ and is called noise folding in completely perturbed CS model. Our construction mainly will whiten the new proposed CS model and explore in restricted isometry property ($\mathrm{R}\mathrm{I}\mathrm{P}$) and coherence of the new CS model under some conditions. Finally, we use OMP to give a numerical simulation which shows that our model is feasible although the recovered value of signal is not exact compared with original signal because of measurement noise $e$, signal noise $u$, and perturbation $E$ involved.