Super-resolution Harmonic Inversion of Time Signals using Filter Diagonalization Method with Applications to NMR Diffusion-Ordered Spectroscopy and Fourier Transform Mass Spectrometry

This thesis demonstrates current development in theory and applications of the Filter Diagonalization Method (FDM). While it has been used primarily for spectral inversion of NMR data, FDM can be applied to time domain data collected from many types of spectroscopic experiments in chemistry.;The beginning of this document will outline the improvements made to the core FDM algorithm as applied to 1D signals. These improvements include the development of a new spectral mode: Hybrid FT-FDM, which maintains the enhanced resolution of FDM while providing a more robust tolerance for noise which is more typical of the Fourier Transform (FT). We have also developed a high-performance implementation of FDM, which utilizes the Fast Fourier Transform (FFT) for computation of the Fourier-type Krylov basis. This modification provides a significant runtime improvement over the previous implementation, making FDM a feasible option for experiments which produce large datasets.;Extending FDM into the realm of 2D data processing, our new code, written in FOR-TRAN90 and ported to MATLAB, allows for the simultaneous determination of NMR spectra for individual compounds in a mixture. Using 2D NMR Pulsed Field Gradient (PFG) experiments, we can use FDM to isolate NMR signals according to the diffusion properties of the corresponding molecules. The results are favorably compared to existing Diffusion-Ordered Spectroscopy (DOSY) algorithms.;Finally we present our findings in adapting FDM to work with a new Ion Cyclotron Resonance Mass Spectrometer (ICR MS) called "Orbitrap". In response to a demand by the instrument's developer for a high resolution processing method, we analyzed real data collected from an Orbitrap, and performed an extensive analysis of the computed FDM spectra. We were able to obtain results superior to the existing FT-based method. Our results are the first reported quantitative analysis of FDM resolution limits with respect to signal characteristics such as signal-to-noise ratio, peak separation, and dynamic range.