PadéFroissart exact signal-noise separation in nuclear magnetic resonance spectroscopy

Dževad Belkić (1) , Karen Belkić (1)

1 Karolinska Institute

Nuclear magnetic resonance spectroscopy is one of the key methods for studying the structure of matter on vastly different levels (sub-nuclear, nuclear, atomic, molecular, cellular , etc). Its overall success critically depends upon reliable mathematical analysis and interpretation of the studied data. This is especially aided by parametric signal processing with the ensuing data quantification, which can yield the abundance or concentrations of the constituents in the examined matter. Such reconstructed information for the most relevant constituents is one of the prerequisites for an accurate assessment of the overall function of the investigated substance. The sought reliability of signal processing rests upon the possibility of the solution of the quantification problem as well as on the separation of true from false information in the spectrally analyzed data. We presently demonstrate that the diagonal fast Padé transform, as a ratio of two polynomials PK /QK of the common degree K, can yield exact quantification and exact signal-noise separation for noise-free and noise corrupted time signals built from 25 molecules. Spurious (noise or noise-like) resonances appear in every parametric estimator. The fast Padé transform makes use of pole-zero cancellations via Froissart doublets in the response function PK /QK to unequivocally distinguish genuine (physical) from spurious (unphysical) resonances. Once identified, unphysical resonances are discarded from the final output of the data analysis. Invariably, the number of spurious resonances is by orders of magnitude larger than that of the true ones. The computation is carried out by gradually and systematically increasing the degree K of the Padé polynomials PK and QK. As this degree K changes, the reconstructed parameters and spectra fluctuate until stabilization occurs. The polynomial degree K at which the full stabilization is achieved represents the sought exact number of resonances. An illustrative set of results is reported in this work to show an unequivocal separation of genuine from spurious information by using the denoising Froissart filter. The fast Padé transform for optimal quantification of the physical constituents of the studied matter and the accompanying Froissart filter for unambiguous signal-noise separation is expected to significantly aid nuclear magnetic resonance spectroscopy in achieving the most reliable data analysis and interpretation.

Mots clés: Physical Sciences

Dževad Belkić, Karen Belkić. PadéFroissart exact signal-noise separation in nuclear magnetic resonance spectroscopy. Journal of Physics B: Atomic, Molecular and Optical Physics, 2011, 44 (12), pp.125003. ⟨10.1088/0953-4075/44/12/125003⟩. ⟨hal-00627152⟩