Fourier Nyquist frequency

FIGURE 19-22 Folding of a spectral line brought about by sampling al a frequency that is less than the Nyquist frequency of 1600 Hz and that is sampled al a frequency of 2000 samples per second as shown by dots solid line is a cosine wave having a frequency of 400 Hz. (b) Frequency-domain spectrum of dashed signal in (a) showing Ihe folded line at 400 Hz. (Adapted from 0. Shaw. Fourier Transform NMR Spectroscopy. 2nd ed.. p, 159, New York Elsevier. 1987. With permission. Elsevier Science Publishers.)... 

The highest frequency at which non-redundant information is available from a discrete cosine (or sine) Fourier transform is called the Nyquist frequency, Nyquist> given by... 

Figure 17. Demonstrate of foldover aliasing, (a) Hypothetical spectrum, with peaks located at their true frequencies, (b) Discrete cosine Fourier transform of the time-domain signal corresponding to (a), with sampling and Nyquist frequencies as shown. The peaks in (b) have correct relative intensities, but are folded-back to lower apparent displayed frequencies.

Because computerized Fourier Transforms are performed on discrete, digital data arrays of finite length, two well-known problems arise. The discreteness of the data array leads to a phenomenon referred to as "aliasing" in which frequencies which are higher than one-half the data point acquisition frequency (the Nyquist frequency) appear at values which are lower than the true frequency. This effect is illustrated in Figure 6 for the case of a sine wave. 

The combination of NUS schemes appeared very promising, as it accounted for a significant gain in experimental time. Covariance processing proved robust for use with limited, sparsely sampled data sets in contrast to Fourier transformation. Nevertheless, care must be taken with respect to sampling in the range of the Nyquist frequency. No loss in resolution was observed, either, due to the projection or mapping of the acquisition dimension onto the sparsely sampled dimension. Application examples were reported, in particular, for solid-state NMR. 

The advantage of the EFT is that this analysis allows one to determine the response of each periodic function when their sum is applied. It should be stressed, however, that the frequency information is for/between/mm = 1/T and the Nyquist frequency/max = l/2At. For example, the FT of the curve displayed in Fig. 2.15 shows that it is composed of four cosine (only real values in the Fourier domain) and three sine functions (only imaginary values in the Fourier domain) (Fig. 2.16). 

Fig. 2.19 Example of wave of 750 Hz continuous line) being sampled with frequency of 1,000 Hz. The Nyquist frequency is 500 Hz. The Fourier transform finds a phantom frequency of 250 Hz (dashed line) that does not exist in the system...

The Fourier frequency bandpass of the spectrometer is determined by the diffraction limit. In view of this fact and the Nyquist criterion, the data in the aforementioned application were oversampled. Although the Nyquist sampling rate is sufficient to represent all information in the data, it is not sufficient to represent the estimates o(k) because of the bandwidth extension that results from information implicit in the physical-realizability constraints. Although it was not shown in the original publication, it is clear from the quality of the restoration, and by analogy with other similarly bounded methods, that Fourier bandwidth extrapolation does indeed occur. This is sometimes called superresolution. The source of the extrapolation should be apparent from the Fourier transform of Eq. (13) with r(x) specified by Eq. (14). 

Answer The spectral width is too narrow to allow the Nyquist limit ( 1.3.1) to be satisfied for all the frequencies in the spectrum and the methyl signals are folded into the window. On spectrometers that use a different version of the Fourier transformation, the aliased data may appear at the other end of the frequency spectrum but will still be out of phase with the rest of the signals. Clearly, the spectral width needs to be increased. 

Figure 8.4 Impedance results obtained for the time-domain results presented in Figure 8.3 by use of the Fourier analysis presented in Section 7.3.3 with potential perturbation amplitude AV as a parameter a) Nyquist representation b) real part of the impedance as a function of frequency and c) imaginary part of the impedance as a function of frequency.

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. 

Keep in mind that, because of the Nyquist criterion, the number of available frequencies is only half the number of data points. Or, to put it differently, all frequencies / in the Fourier transform have their negative counterparts at - /The largest number you can therefore select for filtering is half the number of data points in the set, and at that point you would filter out everything ... 

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