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Review
. 2014 Nov:83:21-41.
doi: 10.1016/j.pnmrs.2014.09.002. Epub 2014 Oct 13.

Nonuniform sampling and non-Fourier signal processing methods in multidimensional NMR

Mehdi Mobli  1 , Jeffrey C Hoch  2
Affiliations
Review

Nonuniform sampling and non-Fourier signal processing methods in multidimensional NMR

Mehdi Mobli et al. Prog Nucl Magn Reson Spectrosc. 2014 Nov.

Erratum in

  • Prog Nucl Magn Reson Spectrosc. 2015 Apr;86-87;80

Abstract

Beginning with the introduction of Fourier Transform NMR by Ernst and Anderson in 1966, time domain measurement of the impulse response (the free induction decay, FID) consisted of sampling the signal at a series of discrete intervals. For compatibility with the discrete Fourier transform (DFT), the intervals are kept uniform, and the Nyquist theorem dictates the largest value of the interval sufficient to avoid aliasing. With the proposal by Jeener of parametric sampling along an indirect time dimension, extension to multidimensional experiments employed the same sampling techniques used in one dimension, similarly subject to the Nyquist condition and suitable for processing via the discrete Fourier transform. The challenges of obtaining high-resolution spectral estimates from short data records using the DFT were already well understood, however. Despite techniques such as linear prediction extrapolation, the achievable resolution in the indirect dimensions is limited by practical constraints on measuring time. The advent of non-Fourier methods of spectrum analysis capable of processing nonuniformly sampled data has led to an explosion in the development of novel sampling strategies that avoid the limits on resolution and measurement time imposed by uniform sampling. The first part of this review discusses the many approaches to data sampling in multidimensional NMR, the second part highlights commonly used methods for signal processing of such data, and the review concludes with a discussion of other approaches to speeding up data acquisition in NMR.

Keywords: Fast acquisition; Multidimensional NMR; Non-uniform sampling; Reduced dimensionality; Signal processing.

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Figures

Figure 1
Figure 1
Radial, Poisson gap, random, and exponentially biased sampling (envelope-matched sampling, EMS) schemes (left column) for 30% sampling coverage of the underlying Nyquist grid, and corresponding point spread functions (PSFs) for 30%, 10%, and 5% sampling coverage (left to right). The intensity scale for the PSFs is shown on the right. The central (zero frequency) component for random sampling is so sharp as to be barely visible. Adapted from Mobli et al..
Figure 2
Figure 2
Two-dimensional cross-sections from a four-dimensional N,C-NOESY spectrum, obtained from (A) uniformly sampled hypercomplex (States-Haberkorn-Ruben) data, (B) real values only in the indirect dimensions, and (C) using random phase detection. One-dimensional cross-sections at the frequencies depicted by colored lines crossing the contour plots are depicted at the top. Reprinted with permission from Maciejewski et al. (2011).
Figure 3
Figure 3
A timeline showing the introduction of various NMR methods aimed at speeding up NMR data acquisition (below the horizontal line). Entries above the timeline refer to fundamental advances in NMR spectroscopy that indirectly impacted or enabled these methods.
Figure 4
Figure 4
Relationships among different methods used for speeding up NMR data acquisition. Processing methods are categorised based on the type of data they are applicable to. The dotted line indicates that the methods appropriate for non-deterministic sampling are also applicable to the other types of sampling, whereas the converse is not true. Parametric methods are italicised and post-processing methods are underlined.
Figure 5
Figure 5
Schematic depiction of RD, GFT and BPR. (A) The RD method gives rise to chemical shift multiplets, which are separated by the chemical shift of one nucleus (ω2) and centered at the chemical shift of the other nucleus (ω1). In GFT processing of RD data, the phases are manipulated (B) so that their combination results in “basic spectra” of reduced complexity (C–D). In projection spectroscopy the delays are scaled by a projection angle. The resulting spectra can be directly projected onto the higher dimensional plane, where an intersection in this higher dimensional spectrum reveals the true chemical shifts (X marks the spot).
Figure 6
Figure 6
Sampling grid showing on- and off-grid sampling of the same number of points (circles). The off-grid samples are according to radial sampling and the on-grid samples are distributed randomly. The distances between the radial points are sufficient to reconstruct the same spectral window as the underlying grid (black dots) without aliasing.
Figure 7
Figure 7
Comparison of spectra obtained with uniform sampling (A and B) and nonuniform sampling (C and D). In C the spectrum was computed using maximum entropy reconstruction, using the same number of samples employed in B. In D the spectrum was computed using nuDFT (FT in which samples not collected are set to zero). Reproduced from Hoch et al. with permission from the PCCP Owner Societies.
Figure 8
Figure 8
Comparison of DFT, CLEAN, and CLEAN with SCRUB post-processing applied to NUS data. Reprinted with permission from Coggins et al. (2012).
Figure 9
Figure 9
Comparison of a spectrum obtained using uniform sampling and conventional DFT processing (left) with a spectrum obtained using the same number of samples in the indirect dimension (the same experiment time) using NUS and FM processing (right). Adapted from Fig. 1 of Hyberts et al. (2014). Reprinted with permission.
Figure 10
Figure 10
Spectra computed from uniformly sampled data using IST (A) and MaxEnt (B). A single exponentially decaying sinusoid was added to the experimental time-domain data, with a frequency indicated by the red oval.
Figure 11
Figure 11
IST spectral reconstruction from NUS data illustrating the non-Gaussian (spiky) noise distribution. Reprinted with permission from Fig. 3D in Stern et al..
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