NMR data processing using the maximum entropy method

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NMR spectroscopists have become accustomed to the Fourier transform for converting free induction decays into spectra. Indeed the FT is now so familiar that there is a tendency to ignore its deficiencies and forget that alternative data processing techniques do exist and may sometimes be superior. Perhaps the most important failing of the discrete F’T is its inability to handle satisfactorily a free induction decay the duration of which is less than about 3TT. Fourier transformation of such a truncated decay produces “sine wiggle” artifacts in the spectrum while apodization prior to transformation, for example by exponential weighting, gives an unwelcome loss in spectral resolution. Sine wiggles are a direct consequence of the unwarranted assumption, implicit in the Fourier analysis of truncated signals, that the unrecorded data points are either nonexistent or equal to zero. While these problems can usually be avoided simply by extending the acquisition time, truncated responses are often unavoidable or even desirable in two-dimensional NMR and NMR imaging. An undesirable feature of some two-dimensional NMR spectra is the “phase twist” lineshape produced by Fourier transformation of a 2D data matrix in which the FiDs in t2 are phase modulated as a function of the other time variable, t,. Unless the two quadrature amplitude modulated signals are available separately, it is impossible by Fourier transformation alone to separate the two-dimensional absorption and dispersion lineshapes. So there are situations in which the FT is less than ideal and an alternative procedure would seem desirable. The properties required of such a technique are (a) absence of prejudice concerning missing data points, (b) no requirement to perform a IFourier transform on the experimental data with its inherent imperfections, and (c) that no feature should appear in the final spectrum for which there is not su:t&ient evidence in the data. As will be shown below a data handling technique known as the maximum entropy method (I-3) (MEM) meets these conditions and so circumvents some of the pitfalls of Fourier transformation. Maximum entropy has enjoyed widespread use in recent years in such fields as radioastronomy, crystallography, and medical tomography, but has barely been exjplored in the context of magnetic resonance. Van Ormondt et al. (4, 5) were the first, using the method for processing electron spin-echo spectra. More recently, Sibisi et al. (6, 7) have applied maximum entropy to NMR, claiming simultaneous improvements in both resolution and signal-to-noise, as well as seemingly spectacular