  Multidimensional NMR spectroscopy is one of the basic tools for determining
the structure of biomolecules. Unfortunately, the resolution of the spectra is
often limited by inter-nuclear couplings. This limitation cannot be overcome by
common ways of increasing resolution, i.e. non-uniform sampling (NUS) followed
by compressed sensing (CS) reconstruction. In this paper, we show how to enrich
CS processing with virtual decoupling leading to an increase in resolution,
sensitivity, and overall quality of NUS reconstruction. A mathematical
description of the decoupling by deconvolution approach explains the effects of
noise, modulation of the sampling schedule, and reveals relation with the
underlying assumption of the CS. The gain in resolution and sensitivity is
demonstrated for the basic experiment used for protein backbone assignment 3D
HNCA applied to two large protein systems: intrinsically disordered 441-residue
Tau and a 509-residue globular bacteriophytochrome fragment.
