Using the Nyquist Sampling Theorem to Obtain Higher Resolution Solid State NMR Spectra of Spin I = n/2 Quadrupolar Nuclei.

In favorable circumstances, the solid state MAS NMR spectra of spin I = n/2 quadrupolar nuclei can show a strong central transition (free of spinning sidebands) and the satellite transitions in an extended spinning sideband manifold. Although the central transition is very strong, it is often broadened out significantly by the second order quadrupolar interaction. This broadening often presents resolution problems when more than one site is present. The problem can of course be reduced by going to higher magnetic field strengths where the second order interaction is reduced. If this option is not available, it is sometimes advantageous to look at the satellite transitions which can be affected to a lesser extent by the second order quadrupolar interaction and therefore exhibit narrower lines. Such is the case for the first satellite transition for ²⁷Al. The problem is that since the satellite transitions are spread over a very large sideband manifold, any one sideband is of very low intensity. It is desirable to add the intensities of all of the sidebands into the 0^th order sideband. This is conveniently accomplished by taking advantage of the Nyquist sampling theorem and collecting the data in simultaneous mode without digital filtering. If the dwell time is made equal to a single rotor period (i.e. the spectral width is set to one half of the spinning speed) and the analog filter bandwidth is maximized, the sidebands of the satellite transitions will all fold into the 0^th. order sideband. If the magic angle is set very precisely, in the case of the first satellite transition of ²⁷Al, the intensity of this sharper line is greater than that of the central transition. Further, the central transition can be suppressed with a double quantum filter (Ashbrook and Wimperis, Journal of Magnetic Resonance, 177, 44 (2006)) to produce a clean spectrum with a much sharper line than the central transition. This is illustrated for the ²⁷Al MAS NMR spectrum of Al(acac)₃ in the figure below.

Nyquist Fold-backs and the Mode of Data Acquisition

Some older Bruker NMR spectrometers allow the collection of the two quadrature channels of the FID either simultaneously (where complex pairs of points are collected at the same time) or sequentially (where a real data point is collected then an imaginary point etc...). In the case of simultaneous acquisition, a Nyquist fold-back will fold in at the far side of the spectrum, whereas for sequential data acquisition, it will fold in at the near side of the spectrum. This is illustrated in the figure below. The data were collected without digital filtering.

Thank you to Dr. Michael Lumsden, the NMR Facility Manager at Dalhousie University for suggesting this entry.

Digital Filtering, Nyquist Fold-Backs and Signal-to-Noise Ratio

Modern NMR spectrometers use digital filtering to improve the quality of the data. Digital filtering is achieved in three steps:

1. Despite the spectral width requested by the user, the spectrometer oversamples the FID as if a very large spectral width was requested (i.e. short dwell time).

2. Depending on the requested spectral width, a digital filter is calculated and applied to the oversampled FID by the spectrometer.

3. The digitally filtered oversampled data is decimated according to the originally requested spectral width and then Fourier transformed to produce an NMR spectrum.

These steps are schematically illustrated in the figure below.Note that digital filtering applied as in the figure, will suppress Nyquist fold-back signals but more importantly, it will suppress noise from outside the requested spectral width from folding into the spectrum. This suppression of folded in noise represents a significant improvement in the signal-to-noise ratio compared to a spectrum acquired without the use of digital filters..

Nyquist Fold-Back Signals

The Nyquist sampling theorem states that an FID must be sampled at a rate at least twice the highest frequency in the FID in order to faithfully reproduce the correct frequencies in an NMR spectrum. In the FID, the highest frequency is plus or minus 1/2 the spectral width. If a resonance falls within plus or minus 1/2 the spectral width, it will be correctly represented in the spectrum. In the absence of digital filters, if a resonance is outside of the spectral width but within the analog filter band width of the spectrometer, it will still appear in the spectrum but at the wrong frequency (and often with a different phase than the correctly represented resonances). The figure below shows an example of this.The bottom trace is a properly recorded NMR spectrum. The top trace shows a spectrum of the same sample with the spectral width set smaller than necessary to capture all of the peaks. One can see that the resonance outside of the spectral width by delta f is folded into the other side of the spectrum by delta f. This phenomenon is also observed in the indirect dimension of a 2D data set as well as in magnetic resonance images.

Nyquist Fold-backs in Magnetic Resonance Images

Nyquist fold-backs are not just limited to 1D and 2D NMR data. They can also be observed in magnetic resonance images. The figures below are magnetic resonance images of seedless grapes acquired on our AVANCE 500. The Nyquist fold-backs are circled in blue.

Nyquist Fold-Backs in HMBC Spectra

The digital filters of modern NMR spectrometers have not only improved the signal-to-noise specification quoted by instrument companies but they have also eliminated Nyquist fold-backs for signals outside of the spectral window. This is not the case however for the indirect dimension of 2D experiments. In the left panel of the figure below is a properly recorded HMBC spectrum of 3-heptanone. In the right panel is a spectrum acquired with the 13C spectral window set too small. The carbonyl correlations are folded into the low frequency end of the 13C axis.
If you inadvertently collect a spectrum with a Nyquist fold-back you can still calculate the correct chemical shift, as the signal will be the same number of ppm away from the wrong end of the axis as it is outside of the correct end of the axis.