Nyquist sampling

For image slicers, contiguous slices of the sky are re-arranged end-to-end to form the pseudo-slit. In that case it is obvious that the sky can be correctly sampled (according to the Nyquist sampling theorem) by the detector pixels on which the slices are projected in the same way as required for direct imaging. 

Figure 6 Liquid-state hi spectra in chloroform acquired in 11.7 T (A) on a Varian INOVA 500 MHz spectrometer, and (B) OPENCORE NMR spectrometer. The intermediate frequency was 180 MHz. In the transmitter, the 180-MHz DDS image was used, while super-Nyquist sampling was employed in the receiver.

ADC sampling rate, and is producing the fundamental wave at a quarter (20 MHz) of the ADC sampling rate. In this very configuration can one extend the intermediate frequency in the simple way that was described above, by utilizing the DDS-image frequency and the super-Nyquist sampling scheme. 

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). 

Harris, 1990] Harris, S. (1990). The Effects of Sampling Clock Jitter on Nyquist Sampling Analog-to-Digital Converters, and on Oversampling Delta-Sigma ADCs. J. Audio Eng. Soc., 38(7/8) 537-542. 

In order to compare with the experimentally measured spectrum, one would ideally like to have the spectra simulated with equal or better energy resolution and with a good statistical reliability for the predicted intensity. Therefore, a suitable estimate of these quantities is important, in order to minimise the output to a manageable level (i.e. to output the trajectories and velocities as less as possible). According to the sampling theorem [23], the smallest time step for the Z(co) calculations is a factor of n times larger than MD simulation step which is determined by the maximum frequency, (Omox, of the system to be simulated. Hence the appropriate time step n At is given by the Nyquist sample rate, as ... 

Equation (2.3.13) states the Nyquist sampling theorem. If At is set too high, the spectral width is too low, and signals at higher frequencies will appear at false positions (Fig. 2.3.9). This phenomenon is called signal aliasing [Deri]. 

When running the spectral arm, data is acquired every 16 p,m of travel, or 32 p,m of optical path difference (OPD), which corresponds to a Nyquist sampling of 4.7 THz. The data presented corresponds to a scanning of the spectral arm from the nominal zero path difference along 3.2cm, at 0.005cm/s. 

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. 

Figure 2 Nyquist sampling in the spectral and time domains.

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... 

Nyquist sampling theorem A law of sampling theory stating that if a continuous bandwidth-limited signal contains no frequency components higher than half the frequency at which it is sampled, then the original signal can be recovered without distortion. 

One critical mathematical preHminary for the processing of discrete-time waveforms, which originate from sampling continous-time waveforms, is the issue of sampUngrate and aliasing. The Nyquist sampling theorem can be summarized as follows ... 

Failure to obey the Nyquist sampling theorem results in aliasing, that is, the effect of frequencies greater than the Nyquist frequency are reflected into the band of frequencies between DC and one-half the sampling rate. This situation is demonstrated in Fig. 20.70, where the samples of the high-frequency sinusoid are indistinguishable from a uniform sampling of the low-frequency sinusoid. 

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