Fast NMR Data Acquisition: Beyond the Fourier Transform
Chapter 9: Maximum Entropy Reconstruction
Published:18 May 2017
M. Mobli, A. S. Stern, and J. C. Hoch, in Fast NMR Data Acquisition: Beyond the Fourier Transform, ed. M. Mobli and J. C. Hoch, The Royal Society of Chemistry, 2017, ch. 9, pp. 252-266.
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The introduction of pulsed NMR revolutionised the field, enabling amongst other things construction of multidimensional experiments. The time response in a pulsed NMR experiment, however, requires further processing prior to analysis of a frequency spectrum. The discrete Fourier Transform (DFT) is a general method for achieving the transformation of the time response to a frequency spectrum. The DFT, however, introduces a number of implicit assumptions about the NMR signal, and when applied to a finite, decaying and noisy NMR signal can lead to significant distortions. The DFT further imposes a number of constraints on the signal that in multidimensional NMR experiments can lead to lengthy data acquisition times. To address these shortcomings, an alternative, statistical, processing method was introduced, where the entropy of the signal is used to iteratively reconstruct a maximum entropy spectrum from the measured NMR signal. The maximum entropy method (MaxEnt) does not make any implicit assumptions about the nature of the signals and does not require the data to be complete or uniformly sampled. These attributes make MaxEnt a robust and general method for reconstruction of NMR data that has been demonstrated to outperform the DFT in many cases. In this chapter, we discuss the history and theory of MaxEnt in NMR and provide examples of its applications to nonuniformly sampled data. We demonstrate how the method can be used to perform linewidth and J-coupling deconvolution as well as how reconstruction parameters affect the MaxEnt spectrum.