An efficient algorithm for automatic phase correction of NMR spectra based on entropy minimization

Abstract

A new algorithm for automatic phase correction of NMR spectra based on entropy minimization is proposed. The optimal zero-order and first-order phase corrections for a NMR spectrum are determined by minimizing entropy. The objective function is constructed using a Shannon-type information entropy measure. Entropy is defined as the normalized derivative of the NMR spectral data. The algorithm has been successfully applied to experimental ${}^1\text{H}\text{NMR}$ spectra. The results of automatic phase correction are found to be comparable to, or perhaps better than, manual phase correction. The advantages of this automatic phase correction algorithm include its simple mathematical basis and the straightforward, reproducible, and efficient optimization procedure. The algorithm is implemented in the Matlab program ACME—Automated phase Correction based on Minimization of Entropy.

Introduction

Normally, zero-order and first-order phase corrections are required for Fourier transform NMR spectra in order to obtain the desired appearance of the real part of the spectra. The zero-order phase misadjustment arises from the phase difference between the reference phase and the receiver detector phase. The first-order phase misadjustment arises from the time delay between excitation and detection, flip-angle variation across the spectrum, and phase shifts from the filter employed to reduce noise outside the spectral bandwidth [1], [2], [3]. Zero-order phase correction is frequency-independent, while first-order phase correction is frequency-dependent.

The misphased Fourier transform NMR spectrum consisting of the complex points can be phase corrected by applying the equations

$$\text{R}{}_{\mathrm{i}}\text{=R}{}_{\mathrm{i}}{}^0\text{cos}\text{(φ}{}_{\mathrm{i}}\text{)−I}{}_{\mathrm{i}}{}^0\text{sin}\text{(φ}{}_{\mathrm{i}}\text{),}$$

$$\text{I}{}_{\mathrm{i}}\text{=I}{}_{\mathrm{i}}{}^0\text{cos}\text{(φ}{}_{\mathrm{i}}\text{)+R}{}_{\mathrm{i}}{}^0\text{sin}\text{(φ}{}_{\mathrm{i}}\text{),}$$

where R_i⁰ and I_i⁰ represent the misphased ith data points for the real and imaginary parts of the NMR spectral data, respectively, R_i and I_i denote the phase-corrected ith data points, and φ_i is the total phase correction in radians applied to the ith data points as shown in

$$\text{φ}{}_{\mathrm{i}}\text{=phc0+phc1×}\text{i}\text{n}\text{,}$$

where phc0 is the zero-order term, phc1 is the constant part of the first-order term, and n is the total number of real data points.

There already exist algorithms automating the procedure of phase correction. As early as 1969, Ernst [4] introduced an automatic phase correction method using the Hilbert transform to find a dispersion spectrum with total integral zero and absorption spectrum with maximum integral. This approach is limited by the spectral signal-to-noise ratio and baseline distortion. Craig and Marshall [1] proposed a method based on dispersion versus absorption plots (DISPA). However, when there are no isolated spectral lines, the phases cannot be calculated correctly because overlap distorts the DISPA circle. Van Vaals and Gerwen [5] developed a method employing the measured phases at the peak positions of spectral lines to calculate the phase distortion. Brown et al. [6] introduced an automatic method by baseline optimization. Heuer [7] reported a noniterative method, APSL (automatic phasing by symmetrizing lines), for automatic phase correction. This approach exploits the fact that symmetric lines are maximally symmetric in correctly phased absorption spectra.

Although many algorithms have been proposed for automatic phase correction, some of them involve rather complicated mathematical models and some are limited by signal-to-noise ratio or overlapping bands. In this paper, an efficient algorithm for automatic phase correction is proposed based on entropy minimization.