INTRODUCTION:   Time-series analysis has become an integral part of signal interpretation in magnetic resonance and other experiments. As an example, linear prediction with singular value decomposition (LPSVD) and the related method, Hankel SVD (HSVD), have been recently applied to problems in NMR, 2D NMR, 2D electron spin-echo spectroscopy (2D ESE), Fourier transform ESR correlation spectroscopy, time-resolved femtosecond spectroscopy, and nonstationary neurological currents. The minimal goal of time-series analysis is to separate signal from noise. Spectral transform methods, such as FFT filtering, maximum entropy (MEM), and LPZ, replace the original data with a new time series (or its spectrum) that has an enhanced signal-to-noise ratio. In contrast spectral decomposition methods, such as LPSVD or HSVD, which assume a certain functional form for the time series, provide both a recipe for separating signal from noise and a listing of the harmonic components contained in the original data. This harmonic list simplifies spectral analysis and can be used to (1) extend the noise-reduced time series, (2) selectively reconstruct certain regions of the harmonic spectrum, and (3) data-compress the signal. The substantial benefits of spectral decomposition are often offset by the requirements of considerable computational time and memory storage. For a time series of N data points spectral transform techniques usually require computational time of O(N log₂N) to O(N²) and storage of O(N) whereas spectral decomposition, such as HSVD, requires computational time of O(N³) and storage of O(N²).
We have investigated a new approach for solving the spectral decomposition problem which is based on the Lanczos algorithm (LA). In the past the LA has provided an effective means for solving slow-motional magnetic resonance spectra as well as other complicated problems in chemical physics. The LA is an extremely fast method for solving large eigenelement problems when either (1) all the eigenelements of a sparse matrix are required or (2) a small set of relevant eigenelements of a dense (or sparse) matrix of special structure are required. (A Lanczos-Prony method for time-series problems has been previously proposed, but it does not benefit from SVD as an intermediate step nor is it based on the LA that we employ.) The goal of this report is to discuss our recent findings on the applicability of the LA for spectral decomposition. We base our approach on the HSVD. The time-consuming singular value decomposition is performed with the LA and we find that the computational time for Lanczos-HSVD (LA-HSVD) is of O(N²) and the storage is of O(N).