Digital signals Nyquist frequency

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 (if digital signal filters are not employed to eliminate this). 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]. 

A digital signal has an upper bound on the frequencies it can represent. This is called the Nyquist frequency and is exactly half the sample rate. 

The ADC needs at least two samples per waveform cycle in order to represent the frequency of the sound if not, then the frequency information is lost. Digital recording systems place a low-pass filter before the ADC in order to ensure that only signals below the Nyquist frequency enter the converter. Otherwise, the conversion process creates foldback frequencies, thus causing a phenomenon known as aliasing distortion (Figure 1.3). 

This concept is one of the most difficult to quantitate. There are some relatively explicit definitions of information content for electronic communications. (For example, the Nyquist theorem defines the minimum sampling rate required in order to preserve the maximum frequency information in a periodic signal. And, the relationships between digital encoding formats and information content of a data base can be quantitated.) However, for the general problem of evaluating the results of instrumental measurements of chemical systems, the definitions for information content of data are very clear. 

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. 

The above illustrates what happens when the signal frequency lies outside the range of frequencies used in the Fourier analysis, in which case the digital Fourier transform will misread that frequency as one within its range. As already indicated, another problem occurs when the frequency lies within the analysis range, and also satisfies the Nyquist criterion (i.e., is sampled more than twice during the repeat cycle of that signal), but has a frequency that does not quite fit those of the analysis, as illustrated below. 

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