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VA Mandelshtam

Publications and source records attributed to VA Mandelshtam.

6 recordsLinked to original sources
PubMed2000

Regularization of the two-dimensional filter diagonalization method: FDM2K

We outline an important advance in the problem of obtaining a two-dimensional (2D) line list of the most prominent features in a 2D high-resolution NMR spectrum in the presence of noise, when using the Filter Diagonalization Method (FDM) to sidestep limitations of conventional FFT processing. Although respectable absorption-mode spectra have been obtained previously by the artifice of "averaging" several FDM calculations, no 2D line list could be directly obtained from the averaged spectrum, and each calculation produced numerical artifacts that were demonstrably inconsistent with the measured data, but which could not be removed a posteriori. By regularizing the intrinsically ill-defined generalized eigenvalue problem that FDM poses, in a particular quite plausible way, features that are weak or stem from numerical problems are attenuated, allowing better characterization of the dominant spectral features. We call the new algorithm FDM2K. Copyright 2000 Academic Press.

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PubMed2000

RRT: the regularized resolvent transform for high-resolution spectral estimation

A new numerical expression, called the regularized resolvent transform (RRT), is presented. RRT is a direct transformation of the truncated time-domain data into a frequency-domain spectrum and is suitable for high-resolution spectral estimation of multidimensional time signals. One of its forms, under the condition that the signal consists only of a finite sum of damped sinusoids, turns out to be equivalent to the exact infinite time discrete Fourier transformation. RRT naturally emerges from the filter diagonalization method, although no diagonalization is required. In RRT the spectrum at each frequency s is expressed in terms of the resolvent R(s)(-1) of a small data matrix R(s), that is constructed from the time signal. Generally, R is singular, which requires certain regularization. In particular, the Tikhonov regularization, R(-1) approximately [R(dagger)R + q(2)](-1)R(dagger) with regularization parameter q, appears to be computationally both efficient and very stable. Numerical implementation of RRT is very inexpensive because even for extremely large data sets the matrices involved are small. RRT is demonstrated using model 1D and experimental 2D NMR signals. Copyright 2000 Academic Press.

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PubMed1998

Application of the filter diagonalization method to one- and two-dimensional NMR spectra

A new non-Fourier data processing algorithm, the filter diagonalization method (FDM), is presented and applied to phase-sensitive 1D and 2D NMR spectra. FDM extracts parameters (peak positions, linewidths, amplitudes, and phases) directly from the time-domain data by fitting the data to a sum of damped complex sinusoids. Grounded in a quantum-mechanical formalism, FDM shares some of the features of linear prediction and other linear algebraic approaches, but is numerically more efficient, scaling like the fast Fourier transform algorithm with respect to data size, and has the ability to correctly handle spectra with thousands or even millions of lines where the competing methods break down. Results obtained on complex spectra are promising. Copyright 1998 Academic Press.

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