Maximum entropy reconstruction of rotating-frame zeugmatography data

Abstract

For about fifteen years NMR spectroscopists have, broadly speaking, been content with the discrete Fourier transform as a means for converting time-domain data into spectra. Recently, lured perhaps by the siren call of simultaneous sensitivity and resolution enhancement, several NMR practitioners have been exploring alternative data processing techniques (I-11), among them the maximum entropy method (MEM) (I-8). Put simply, the MEM involves finding spectra that are consistent with the experimental data and rejecting all but the one with the minimum information content or equivalently the maximum entropy (12-14). By choosing the spectrum with the least structure one tries to ensure that it contains only features for which there is sufficient evidence in the data. This seductive philosophy may, in part, account for the recent rash of publications on the use of maximum entropy in NMR. While it has yet to be demonstrated convincingly that maximum entropy is any more successful than conventional methods for sensitivity and resolution enhancement (15, 16), the method does appear to be valuable in cases where the discrete FT is manifestly unsatisfactory. For example, to obtain a spectrum from a truncated free induction decay (FID) by discrete Fourier transformation, one is forced to assume the missing data points are nonexistent or equal to zero and tolerate the resulting “sine wiggles” or else to apodize the FID and accept the loss of resolution. Maximum entropy sidesteps these problems by using the FT in the opposite direction, namely, to convert “trial spectra” into “trial FIDs” that can be compared with the incomplete experimental FID (4, 6). The Fourier transform is also inappropriate when the FID contains phase shifts and intensity variations associated with insufficiently hard pulses. These distortions, which the FT simply carries over into the spectrum, are usually minor and easily eliminated by increasing the transmitter power or using offset-compensated composite pulses. Off-resonance effects are more serious, however, in experiments such as rotatingframe zeugmatography (I 7) where radiofrequency field gradients are used to achieve spatial localization. For example, consider the adaptation of Hoult’s original method suggested by Styles et al. (18, 2 9). This is a simple two-dimensional NMR experiment in which a one-dimensional radiofrequency field gradient pulse of duration tl is followed