Nyquist condition

Aliased signals Signals that fall outside the spectral window (i.e., those that fail to meet the Nyquist condition). Such signals still appear in the spectrum but at the wrong frequency because they become folded back into the spectrum and are characterised by being out of phase with respect to the other signals. 

Nyquist condition Sampling of all signals within an FID such that each is sampled at least twice per wavelength. 

Figure 3.11. The Nyquist condition. To correctly characterise the frequency of an NMR signal it must be sampled at least twice per wavelength, it is then said to fall within the spectral width. The sampling of signals (a) and (b) meet this criteria. Signals with frequencies too high to meet this condition are alia.sed back within the spectral width and so appear with the wrong frequency. The sampling pattern of signal (c) matches that of (a) and hence it is incorrectly recorded as having the lower frequency.

For either scheme, the total number of data points digitised, acquisition times and spectral widths are identical, so the resulting spectra are largely equivalent. The most obvious difference is in the appearance of aliased signals, that is, those that violate the Nyquist condition, as described below. Experimentally, flatter baselines are observed for the simultaneous method as a result of the symmetrical sampling of the initial data points in the FID and this is the recommended protocol. 

In Section 3.2.3 it was shown that a resonance falling outside the spectral window (because it violates the Nyquist condition) will still be detected but will appear at an incorrect frequency and is said to be aliased or folded back into the spectrum. This can be confusing if one is unable to tell whether the resonance exhibits the correct chemical shift or not. The precise location of the aliased signal in the spectrum depends on the quadrature detection scheme in use and on how far outside the window it truly resonates. With the simultaneous (complex FT) scheme, signals appear to be wrapped around the spectral window and appear at the opposite end of the spectrum (Fig. 3.24b), whereas with the sequential (real FT) scheme signals are folded back at the same end of the spectrum (Fig. 3.24c). If you are interested to know why this difference occurs see reference [7]. 

This Nyquist condition quickly brings us to a problem. A typical NMR frequency is of the order of hundreds of MHz but there simply are no ADCs available which work fast enough to digitize such a waveform with the kind of accuracy (i.e. number of bits) we need for NMR. The solution to this problem is to mix down the signal to a lower frequency, as is described in the next section. 

Fig. 5.8 Illustration of the concept of folding. In spectrum (a) the peak (shown in grey) is at a higher frequency than the maximum set by the Nyquist condition. In practice, such a peak would appear in the position shown in (b).

Because the number of time-domain slices (and hence the number of recorded projections) is relatively small, the density of sampling points is far lower than the density used in the conventional experiment, which must examine every point on the complete Cartesian grid while satisfying the Nyquist condition and the requirement for adequate resolution. This is where the critical time saving occurs. With this limited radial sampling [13], the speed advantage increases by an order of magnitude for each new evolution dimension beyond the first. This opens up the... 

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