Nyquist sampling criterion

A single-mode He Ne laser has an emission at 632.8 nm (visible, red) which corresponds to a frequency of 15,800 wavenumbers. The reference He Ne laser beam is coupled to the moving mirror of the interferometer by either its own beamsplitter in parallel with the infrared beamsplitter, or directly through the infrared optics. The laser radiation is detected separately from the infrared radiation and is recorded as a cosine wave, as indicated by Equation 6. According to the Nyquist sampling criterion, to correctly sample the He Ne signal data would have to be collected at twice its frequency. 

If the Doppler frequency characteristics of a target are to be determined, then the sampling rate (PRF) must be sufficient to satisfy the Nyquist sampling criterion, two samples per Doppler cycle. In this case, the minimum PRF is found from... 

To minimize data acquisition time, the overall magnification of this system was made as small as possible, consistent with the spatial resolution desired. Since the image is sampled by the discrete pixels of the CCD camera, the Nyquist sampling criterion must be obeyed to avoid the generation of artifacts. There is an explicit relationship describing how the number of counts per pixel depends on the optical design. We know from Equation (1.2) that the radius of the Airy disk r is equal to 0.61A/NA (since 2r). If the size of the CCD pixel is p, then the Nyquist sam-... 

It should also be noted that the time constant for the LIA must be quite short (less than 1 ms) to ensure that the Nyquist sampling criterion is fulfilled. 

Nyquist sampling criterion Criterion of the allowable maximum sampling interval that can be given to regu-... 

The other factor determining the resolution of a reconstructed super-resolution image is the labeling density. According to the Nyquist sampling criterion, the resolution will not exceed twice... 

Sampled-data control systems can be designed in the frequency domain by using the same techniques that we employed for continuous systems. The Nyquist stability criterion is applied to the appropriate closedloop characteristic equation to find the number of zeros outside the unit circle. 

Thus the Nyquist stability criterion can be applied directly to sampled-data... 

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). 

If the. system is openloop stable, there will be no poles of HGm z) (z) outside the unit circle, and P = 0. Thus, the Nyquist stability criterion can be applied directly sampled-data systems.. ... 

For a finite sampling interval A<5, more than one superposition of cosine/sine waves can give rise to the recorded interferogram. For the transformed spectrum to be unique, the sampling interval A<5 must be sufficiently small to detect modulations in the interferogram due to the shortest wavelength present in the spectrum, the so-called Nyquist criterion [66] ... 

It can be shown that there is a Nyquist criterion for sampled data systems which is equivalent to that for continuous systems (see Section 7.10.5) and equation 7.131 can be applied in its comparable r-transformed form(42). In practice it is generally sufficient to ascertain whether the polar plot of G(z) in the complex z-plane encircles the (-1,0) point (as with continuous systems in the j-plane) where 1 + G(r) = 0 is the system z-transformed characteristic equation. The polar plot is constructed from... 

Figure 3.32. Enhanced sensitivity can be realised in favourable cases by oversampling the data and hence reducing digitisation noise. Spectrum (a) shows part of a conventional proton spectrum sampled according to the Nyquist criterion. Oversampling the data by a factor of 24 as in (b) provides a sensitivity gain (all other conditions as for (a)).

Figure 7.4-4 illustrates the Nyquist criterion the signal is sampled at exactly two points per cycle, which is just not good enough. In this borderline case the frequency is recovered, but the amplitudes of the possible sine and cosine components of the signal are not. 

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. 

Sampling Frequency. The sampling frequency selected for data acquisition obviously should be related to the bandwidth of the sampled waveform. From information theory, the criterion for adequate sampling is that the minimum sampling frequency Nyquist frequency) must be twice the bandwidth of the sampled waveform. Thus, for a 100-Hz signal, the sampling frequency must be at least 200 Hz to retain the information inherent in the waveform. 

Figure 3.11. The Nyquist condition. To correctly characterise the ftequency of a NMR signal, it must be sampled at least twice per wavelength, it is then said to fall within the spectral width. The sampUng of signals (a) and (b) meet this criterion. Signals with frequencies too high to meet this condition are abased 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.

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