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Automatic self-correcting in signal processing for magnetic resonance spectroscopy: noise reduction, resolution improvement and splitting overlapped peaks
Journal of Mathematical Chemistry ( IF 1.81 ) Pub Date : 2019-08-31 , DOI: 10.1007/s10910-019-01060-x
Dževad Belkić, Karen Belkić

Nuclear magnetic resonance spectroscopy originated in physics and quickly found versatile applications of paramount importance in other sciences, including chemistry. Signal processing in this methodology is a key to data analysis and interpretation. Herein, one of the most powerful tools from mathematical theory of approximations, known as rational polynomials, is the prime example of reliable handling of the two stumbling blocks that hamper further progress: noise suppression and resolution improvement. Within this realm resides the fast Padé transform (FPT), which simultaneously solves both these problems. It has a self-correcting procedure, which is automatically built in rational polynomials through noise suppression by pole-zero cancellations in spectra. Moreover, by solving the quantification problem (called spectral analysis in mathematics), the FPT can unequivocally separate overlapped peaks and thereby improve resolution. Further, lineshape estimations are provided by both non-parametric and parametric signal processing in the FPT. Since the FPT includes singularities (poles) of the expanded function, it achieves exponental convergence \(\exp {(-N)}\) (the so-named spectral resolution) with respect to the size N of the basis set. This is contrasted to merely the inverse-power-law convergence 1 / N in the fast Fourier transform because its basis functions do not describe the singularities of the expanded function. The present investigation reports on practical aspects of all these critical features and gives several representative illustrations for measured time signals heavily contaminated with noise.
更新日期:2019-11-04

 

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