Abstract

Phase correction of high-resolution NMR spectra is a routine data processing task. It used to be done by a manual adjustment of the two parameters it involves and it was left to the operator to subjectively judge whether the final result was satisfactory. More recently, due to a growing need for automatic evaluation procedures, phase-correction has become progressively more operator-independent, due to the appearance of a growing list of automatic phasing algorithms [1-3].

We feel, however, that most of these algorithms are not yet satisfactory because, rather than relying on clearly defined global parameters of the spectrum, they entail loosely defined steps such as picking two sufficiently isolated and sufficiently distant peaks. Another problem is that most phase correction algorithms have not been subjected to statistical (Monte-Carlo) tests extensive enough for a quantitative analysis of their bias and error distributions, especially in the presence of severe baseline artefacts. In many practical cases, a manual fine-tuning of the phase after an automatic correction provides a better final result, thus indicating that the phase correction estimators coded into the automatic procedures are not optimal.

Here we present a new algorithm, inspired by the old idea [4] of DISPA (Dispersion-Absorption) plots in which the in-phase component (u) of a spectrum is plotted against its out-of-phase component (v), disregarding the frequency coordinate. The original DISPA was introduced in ESR spectroscopy to help experimentalists phase the ESR signals. It was later used also in NMR to estimate the phases of selected lines with the goal of phasing the whole spectrum [5-7].

We show that, first of all, DISPA plots can be conveniently applied to a complete NMR spectrum, avoiding any a-priori selection or discrimination of particular intervals. In a similar way one can also exploit DISPA-like plots of the (complex) powers of the complex spectral data and/or DISPA-like plots weighed by a power of the magnitude of each data point.

By themselves, such extended DISPA plots can be used as is for interactive, manual phasing of the NMR spectra. More important, however, is the fact that the insights gained from the eDISPA plots make it possible to define real-valued functionals Q of the spectra which, when evaluated as functions Q(φ0,φ1) of the two phase correction angles, exhibit a maximum at the 'correct' phase correction values. Standard numerical maximization of Q with respect to (φ0,φ1) therefore amounts to a new phasing algorithm which is completely objective and void of any a-priori discrimination between various experimental data points. It also has the advantage of being eminently insensitive to baseline imperfections, even those which are severe enough to preclude reliable manual phasing. The only problem which we have had to solve by means other than plain maximization is that of multiple maxima of Q for large φ1 values.

The new, fully automatic (φ0,φ1) estimators have been extensively tested using simulated experimental data with various numbers of spectral lines and various amounts of baseline artifacts and applying Monte Carlo methods. The results show that the estimators are quite robust, their statistical bias is small enough to be negligible, and their mean errors are smaller than those one can reasonably expect from manual phasing.